Preprints
- R. K. Bansal, E. Mallada, and P. Hidalgo-Gonzalez, A Market Mechanism for a Two-stage Settlement Electricity Market with Energy Storage, 2024, submitted.
[BibTeX] [Abstract] [Download PDF]
Electricity markets typically clear in two stages: a day-ahead market and a real-time market. In this paper, we propose market mechanisms for a two-stage multi-interval electricity market with energy storage, generators, and demand uncertainties. We consider two possible mixed bidding strategies: storage first bids cycle depths in the day ahead and then charge-discharge power bids in the real-time market for any last-minute adjustments. While the first strategy only considers day-ahead decisions from an individual participant’s perspective as part of their individual optimization formulation, the second strategy accounts for both the market operator’s and participants’ perspectives. We demonstrate that the competitive equilibrium exists uniquely for both mechanisms. However, accounting for the day-ahead decisions in the bidding function has several advantages. Numerical experiments using New York ISO data provide bounds on the proposed market mechanism.
@unpublished{bmh2024a-preprint, abstract = {Electricity markets typically clear in two stages: a day-ahead market and a real-time market. In this paper, we propose market mechanisms for a two-stage multi-interval electricity market with energy storage, generators, and demand uncertainties. We consider two possible mixed bidding strategies: storage first bids cycle depths in the day ahead and then charge-discharge power bids in the real-time market for any last-minute adjustments. While the first strategy only considers day-ahead decisions from an individual participant's perspective as part of their individual optimization formulation, the second strategy accounts for both the market operator's and participants' perspectives. We demonstrate that the competitive equilibrium exists uniquely for both mechanisms. However, accounting for the day-ahead decisions in the bidding function has several advantages. Numerical experiments using New York ISO data provide bounds on the proposed market mechanism.}, author = {Bansal, Rajni Kant and Mallada, Enrique and Hidalgo-Gonzalez, Patricia}, grants = {CPS-2136324, Global-Centers-2330450}, month = {03}, title = {A Market Mechanism for a Two-stage Settlement Electricity Market with Energy Storage}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-BMH.pdf}, year = {2024, submitted} } - R. K. Bansal, P. You, Y. Chen, and E. Mallada, Intercept Function and Quantity Bidding in Two-stage Electricity Market with Market Power Mitigation, 2023, under revision, submitted Aug 2023.
[BibTeX] [Abstract] [Download PDF]
Electricity markets typically operate in two stages, day-ahead and real-time. Despite best efforts striving efficiency, evidence of price manipulation has called for system-level market power mitigation (MPM) initiatives that substitute noncompetitive bids with default bids. Implementing these policies with a limited understanding of participant behavior may lead to unintended economic losses. In this paper, we model the competition between generators and inelastic loads in a two-stage market with stage-wise MPM policies. The loss of Nash equilibrium and lack of guarantee of stable market outcome in the case of conventional supply function bidding motivates the use of an alternative market mechanism where generators bid an intercept function. A Nash equilibrium analysis for a day-ahead MPM policy leads to a Stackelberg-Nash game with loads exercising market power at the expense of generators. A comparison of the resulting equilibrium with the standard market (not implementing any MPM policy) shows that a day-ahead policy completely mitigates the market power of generators. On the other hand, the real-time MPM policy increases demand allocation to real-time, contrary to current market practice with most electricity trades in the day-ahead market. Numerical studies illustrate the impact of the slope of the intercept function on the standard market.
@unpublished{bcym2023a-preprint, abstract = {Electricity markets typically operate in two stages, day-ahead and real-time. Despite best efforts striving efficiency, evidence of price manipulation has called for system-level market power mitigation (MPM) initiatives that substitute noncompetitive bids with default bids. Implementing these policies with a limited understanding of participant behavior may lead to unintended economic losses. In this paper, we model the competition between generators and inelastic loads in a two-stage market with stage-wise MPM policies. The loss of Nash equilibrium and lack of guarantee of stable market outcome in the case of conventional supply function bidding motivates the use of an alternative market mechanism where generators bid an intercept function. A Nash equilibrium analysis for a day-ahead MPM policy leads to a Stackelberg-Nash game with loads exercising market power at the expense of generators. A comparison of the resulting equilibrium with the standard market (not implementing any MPM policy) shows that a day-ahead policy completely mitigates the market power of generators. On the other hand, the real-time MPM policy increases demand allocation to real-time, contrary to current market practice with most electricity trades in the day-ahead market. Numerical studies illustrate the impact of the slope of the intercept function on the standard market.}, author = {Bansal, Rajni Kant and You, Pengcheng and Chen, Yue and Mallada, Enrique}, month = {11}, pages = {1-14}, title = {Intercept Function and Quantity Bidding in Two-stage Electricity Market with Market Power Mitigation}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-BCYMb.pdf}, year = {2023, under revision, submitted Aug 2023} } - H. Min, R. Pates, and E. Mallada, A Frequency Domain Analysis of Slow Coherency in Networked Systems, 2024, revised, submitted Feb 2022.
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Network coherence generally refers to the emergence of simple aggregated dynamical behaviors, despite heterogeneity in the dynamics of the network’s subsystems. In this paper, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system’s input-output response. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network’s effective algebraic connectivity grows, the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further show that many networked systems can exhibit similar coherent behavior by establishing a concentration result in a setting with randomly chosen individual nodal dynamics.
@unpublished{mpm2023a-preprint, abstract = {Network coherence generally refers to the emergence of simple aggregated dynamical behaviors, despite heterogeneity in the dynamics of the network's subsystems. In this paper, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system's input-output response. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network's effective algebraic connectivity grows, the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further show that many networked systems can exhibit similar coherent behavior by establishing a concentration result in a setting with randomly chosen individual nodal dynamics.}, author = {Min, Hancheng and Pates, Richard and Mallada, Enrique}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {2}, pages = {1-15}, title = {A Frequency Domain Analysis of Slow Coherency in Networked Systems}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-MPM.pdf}, year = {2024, revised, submitted Feb 2022} } - H. Min, S. Tarmoun, R. Vidal, and E. Mallada, Convergence and Implicit Bias of Gradient Flow on Overparametrized Linear Networks, 2023, submitted.
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Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.
@unpublished{mtvm2023a-preprint, abstract = {Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution. }, author = {Min, Hancheng and Tarmoun, Salma and Vidal, Rene and Mallada, Enrique}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {02}, title = {Convergence and Implicit Bias of Gradient Flow on Overparametrized Linear Networks}, url = {https://mallada.ece.jhu.edu/pubs/2022-Preprint-MTVM.pdf}, year = {2023, submitted} } - P. You, M. Fernandez, D. F. Gayme, and E. Mallada, Mixed Supply Function and Quantity Bidding in Two-Stage Settlement Markets, 2023, under revision, submitted Mar 2023.
[BibTeX] [Abstract] [Download PDF]
Motivated by electricity markets, we study the incentives of heterogeneous participants (firms and consumers) in a two-stage settlement market with a mixed bidding mechanism, in which firms participate using supply function bids and consumers use quantity bids. We carry out an equilibrium analysis of the market outcome and obtain closed-form solutions. The characterization of the equilibria allows us to gain insights into the market-power implications of mixed bidding and uncover the importance of accounting for consumers’ strategic behavior in a two-stage market, even when their demand is completely inelastic with respect to price. We show that strategic consumers are able to exploit firms’ strategic behavior to maintain a systematic difference between the forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payment and undermines the supply-side market power. However, it is only effective when firms are behaving strategically. We also observe situations where firms lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Our results further suggest that market competition has a heterogeneous impact across consumer sizes, particularly benefiting small consumers. Our analysis can accommodate other market policies, and we demonstrate this versatility by examining the impact of some example policies, including virtual bidding, on the market outcome.
@unpublished{yfgm2023a-preprint, abstract = {Motivated by electricity markets, we study the incentives of heterogeneous participants (firms and consumers) in a two-stage settlement market with a mixed bidding mechanism, in which firms participate using supply function bids and consumers use quantity bids. We carry out an equilibrium analysis of the market outcome and obtain closed-form solutions. The characterization of the equilibria allows us to gain insights into the market-power implications of mixed bidding and uncover the importance of accounting for consumers' strategic behavior in a two-stage market, even when their demand is completely inelastic with respect to price. We show that strategic consumers are able to exploit firms' strategic behavior to maintain a systematic difference between the forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payment and undermines the supply-side market power. However, it is only effective when firms are behaving strategically. We also observe situations where firms lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Our results further suggest that market competition has a heterogeneous impact across consumer sizes, particularly benefiting small consumers. Our analysis can accommodate other market policies, and we demonstrate this versatility by examining the impact of some example policies, including virtual bidding, on the market outcome.}, author = {You, Pengcheng and Fernandez, Marcelo and Gayme, Dennice F. and Mallada, Enrique}, grants = {CAREER-1752362;TRIPODS-1934979;CPS-2136324}, month = {8}, pages = {1-45}, title = {Mixed Supply Function and Quantity Bidding in Two-Stage Settlement Markets}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-YFGM.pdf}, year = {2023, under revision, submitted Mar 2023} } - P. You, Y. Jiang, E. Yeung, D. Gayme, and E. Mallada, On the Stability, Economic Efficiency and Incentive Compatibility of Electricity Market Dynamics, 2023, under revision, submitted Dec 2021.
[BibTeX] [Abstract] [Download PDF]
This paper focuses on the operation of an electricity market that accounts for participants that bid at a sub-minute timescale. To that end, we model the market-clearing process as a dynamical system, called market dynamics, which is temporally coupled with the grid frequency dynamics and is thus required to guarantee system-wide stability while meeting the system operational constraints. We characterize participants as price-takers who rationally update their bids to maximize their utility in response to real-time schedules of prices and dispatch. For two common bidding mechanisms, based on quantity and price, we identify a notion of alignment between participants’ behavior and planners’ goals that leads to a saddle-based design of the market that guarantees convergence to a point meeting all operational constraints. We further explore cases where this alignment property does not hold and observe that misaligned participants’ bidding can destabilize the closed-loop system. We thus design a regularized version of the market dynamics that recovers all the desirable stability and steady-state performance guarantees. Numerical tests validate our results on the IEEE 39-bus system.
@unpublished{yjygm2023a-preprint, abstract = {This paper focuses on the operation of an electricity market that accounts for participants that bid at a sub-minute timescale. To that end, we model the market-clearing process as a dynamical system, called market dynamics, which is temporally coupled with the grid frequency dynamics and is thus required to guarantee system-wide stability while meeting the system operational constraints. We characterize participants as price-takers who rationally update their bids to maximize their utility in response to real-time schedules of prices and dispatch. For two common bidding mechanisms, based on quantity and price, we identify a notion of alignment between participants' behavior and planners' goals that leads to a saddle-based design of the market that guarantees convergence to a point meeting all operational constraints. We further explore cases where this alignment property does not hold and observe that misaligned participants' bidding can destabilize the closed-loop system. We thus design a regularized version of the market dynamics that recovers all the desirable stability and steady-state performance guarantees. Numerical tests validate our results on the IEEE 39-bus system.}, author = {You, Pengcheng and Jiang, Yan and Yeung, Enoch and Gayme, Dennice and Mallada, Enrique}, grants = {CAREER-1752362, CPS-2136324}, month = {10}, pages = {1-16}, title = {On the Stability, Economic Efficiency and Incentive Compatibility of Electricity Market Dynamics}, url = {https://mallada.ece.jhu.edu/pubs/2021-Preprint-YJYGM.pdf}, year = {2023, under revision, submitted Dec 2021} } - P. You, Y. Liu, and E. Mallada, A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design, 2024, submitted.
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This work examines the conditions for asymptotic and exponential convergence of saddle flow dynamics of convex-concave functions. First, we propose an observability-based certificate for asymptotic convergence, directly bridging the gap between the invariant set in a LaSalle argument and the equilibrium set of saddle flows. This certificate generalizes conventional conditions for convergence, e.g., strict convexity-concavity, and leads to a novel state-augmentation method that requires minimal assumptions for asymptotic convergence. We also show that global exponential stability follows from strong convexity-strong concavity, providing a lower-bound estimate of the convergence rate. This insight also explains the convergence of proximal saddle flows for strongly convex-concave objective functions. Our results generalize to dynamics with projections on the vector field and have applications in solving constrained convex optimization via primal-dual methods. Based on these insights, we study four algorithms built upon different Lagrangian function transformations. We validate our work by applying these methods to solve a network flow optimization and a Lasso regression problem.
@unpublished{ylm2024a-preprint, abstract = {This work examines the conditions for asymptotic and exponential convergence of saddle flow dynamics of convex-concave functions. First, we propose an observability-based certificate for asymptotic convergence, directly bridging the gap between the invariant set in a LaSalle argument and the equilibrium set of saddle flows. This certificate generalizes conventional conditions for convergence, e.g., strict convexity-concavity, and leads to a novel state-augmentation method that requires minimal assumptions for asymptotic convergence. We also show that global exponential stability follows from strong convexity-strong concavity, providing a lower-bound estimate of the convergence rate. This insight also explains the convergence of proximal saddle flows for strongly convex-concave objective functions. Our results generalize to dynamics with projections on the vector field and have applications in solving constrained convex optimization via primal-dual methods. Based on these insights, we study four algorithms built upon different Lagrangian function transformations. We validate our work by applying these methods to solve a network flow optimization and a Lasso regression problem.}, author = {You, Pengcheng and Liu, Yingzhu and Mallada, Enrique}, grants = {CAREER-1752362, CPS-2136324, Global-Centers-2330450}, month = {9}, pages = {1-16}, title = {A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-YLM.pdf}, year = {2024, submitted} } - T. Zheng, J. W. Simpson-Porco, and E. Mallada, Closed-Loop Motion Planning for Differentially Flat Systems: A Time-Varying Optimization Framework, 2023, submitted.
[BibTeX] [Abstract] [Download PDF]
Motion planning and control are two core components of the robotic systems autonomy stack. The standard approach to combine these methodologies comprises an offline/open-loop stage, \textitplanning, that designs a feasible and safe trajectory to follow, and an online/closed-loop stage, \textittracking, that corrects for unmodeled dynamics and disturbances. Such an approach generally introduces conservativeness into the planning stage, which becomes difficult to overcome as the model complexity increases and real-time decisions need to be made in a changing environment. This work addresses these challenges for the class of differentially flat nonlinear systems by integrating planning and control into a cohesive closed-loop task. Precisely, we develop an optimization-based framework that aims to steer a differentially flat system to a trajectory implicitly defined via a constrained \textittime-varying optimization problem. To that end, we generalize the notion of feedback linearization, which makes non-linear systems behave as linear systems, and develop controllers that effectively transform a differentially flat system into an optimization algorithm that seeks to find the optimal solution of a (possibly time-varying) optimization problem. Under sufficient regularity assumptions, we prove global asymptotic convergence for the optimization dynamics to the minimizer of the time-varying optimization problem. We illustrate the effectiveness of our method with two numerical examples: a multi-robot tracking problem and a robot obstacle avoidance problem.
@unpublished{zsm2023a-preprint, abstract = {Motion planning and control are two core components of the robotic systems autonomy stack. The standard approach to combine these methodologies comprises an offline/open-loop stage, \textitplanning, that designs a feasible and safe trajectory to follow, and an online/closed-loop stage, \textittracking, that corrects for unmodeled dynamics and disturbances. Such an approach generally introduces conservativeness into the planning stage, which becomes difficult to overcome as the model complexity increases and real-time decisions need to be made in a changing environment. This work addresses these challenges for the class of differentially flat nonlinear systems by integrating planning and control into a cohesive closed-loop task. Precisely, we develop an optimization-based framework that aims to steer a differentially flat system to a trajectory implicitly defined via a constrained \textittime-varying optimization problem. To that end, we generalize the notion of feedback linearization, which makes non-linear systems behave as linear systems, and develop controllers that effectively transform a differentially flat system into an optimization algorithm that seeks to find the optimal solution of a (possibly time-varying) optimization problem. Under sufficient regularity assumptions, we prove global asymptotic convergence for the optimization dynamics to the minimizer of the time-varying optimization problem. We illustrate the effectiveness of our method with two numerical examples: a multi-robot tracking problem and a robot obstacle avoidance problem.}, author = {Zheng, Tianqi and Simpson-Porco, John W. and Mallada, Enrique}, grants = {CAREER-1752362,CPS-2136324,EPICS-2330450}, month = {10}, pages = {1-14}, title = {Closed-Loop Motion Planning for Differentially Flat Systems: A Time-Varying Optimization Framework}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-ZSM.pdf}, year = {2023, submitted} }
Journals
- T. Zheng, N. Loizou, P. You, and E. Mallada, “Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization,” IEEE Control Systems Letters (L-CSS), vol. 8, pp. 2009-2014, 2024. doi:10.1109/LCSYS.2024.3413004
[BibTeX] [Abstract] [Download PDF]
Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA’s effectiveness in solving saddle point problems.
@article{zlym2024lcss, abstract = {Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA's effectiveness in solving saddle point problems.}, author = {Zheng, Tianqi and Loizou, Nicolas and You, Pengcheng and Mallada, Enrique}, doi = {10.1109/LCSYS.2024.3413004}, grants = {CPS-2136324, Global-Centers-2330450}, journal = {IEEE Control Systems Letters (L-CSS)}, month = {06}, pages = {2009-2014}, record = {published, accepted May 2024, submitted Mar 2024}, title = {Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization}, url = {https://mallada.ece.jhu.edu/pubs/2024-LCSS-ZLYM.pdf}, volume = {8}, year = {2024} } - R. K. Bansal, Y. Chen, P. You, and E. Mallada, “Market Power Mitigation in Two-stage Electricity Market with Supply Function and Quantity Bidding,” IEEE Transactions on Energy Markets, Policy and Regulation, vol. 1, iss. 4, pp. 512-522, 2023. doi:10.1109/TEMPR.2023.3318149
[BibTeX] [Abstract] [Download PDF]
The main goal of a sequential two-stage electricity market—e.g., day-ahead and real-time markets—is to operate efficiently. However, the price difference across stages due to inadequate competition and unforeseen circumstances leads to undesirable price manipulation. To mitigate this, some Inde- pendent System Operators (ISOs) proposed system-level market power mitigation (MPM) policies in addition to existing local policies. These policies aim to substitute noncompetitive bids with a default bid based on estimated generator costs. However, these policies may lead to unintended consequences when implemented without accounting for the conflicting interest of participants. In this paper, we model the competition between generators (bidding supply functions) and loads (bidding quantity) in a two-stage market with a stage-wise MPM policy. An equilibrium analysis shows that a real-time MPM policy leads to equilibrium loss, meaning no stable market outcome (Nash equilibrium) exists. A day-ahead MPM policy, besides, leads to a Stackelberg-Nash game with loads acting as leaders and generators as followers. In this setting, loads become winners, i.e., their aggregate payment is always less than competitive payments. Moreover, comparison with standard market equilibrium highlights that markets are better off without such policies. Finally, numerical studies highlight the impact of heterogeneity and load size on market equilibrium.
@article{bcym2023tempr, abstract = {The main goal of a sequential two-stage electricity market---e.g., day-ahead and real-time markets---is to operate efficiently. However, the price difference across stages due to inadequate competition and unforeseen circumstances leads to undesirable price manipulation. To mitigate this, some Inde- pendent System Operators (ISOs) proposed system-level market power mitigation (MPM) policies in addition to existing local policies. These policies aim to substitute noncompetitive bids with a default bid based on estimated generator costs. However, these policies may lead to unintended consequences when implemented without accounting for the conflicting interest of participants. In this paper, we model the competition between generators (bidding supply functions) and loads (bidding quantity) in a two-stage market with a stage-wise MPM policy. An equilibrium analysis shows that a real-time MPM policy leads to equilibrium loss, meaning no stable market outcome (Nash equilibrium) exists. A day-ahead MPM policy, besides, leads to a Stackelberg-Nash game with loads acting as leaders and generators as followers. In this setting, loads become winners, i.e., their aggregate payment is always less than competitive payments. Moreover, comparison with standard market equilibrium highlights that markets are better off without such policies. Finally, numerical studies highlight the impact of heterogeneity and load size on market equilibrium.}, author = {Bansal, Rajni Kant and Chen, Yue and You, Pengcheng and Mallada, Enrique}, doi = {10.1109/TEMPR.2023.3318149}, grants = {CAREER-1752362, CPS-2136324, EPICS-2330450}, journal = {IEEE Transactions on Energy Markets, Policy and Regulation}, month = {12}, number = {4}, pages = {512-522}, record = {published, online Sep 2023, revised July 2023, under revision May 2023, submitted Jan 2023}, title = {Market Power Mitigation in Two-stage Electricity Market with Supply Function and Quantity Bidding}, url = {https://mallada.ece.jhu.edu/pubs/2023-TEMPR-BCYM.pdf}, volume = {1}, year = {2023} } - A. Castellano, H. Min, J. Bazerque, and E. Mallada, “Learning to Act Safely with Limited Exposure and Almost Sure Certainty,” IEEE Transactions on Automatic Control, vol. 68, iss. 5, pp. 2979-2994, 2023. doi:10.1109/TAC.2023.3240925
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This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to navigate trade-offs between optimality, level of exposure to unsafe events, and the maximum detection time of unsafe actions. We illustrate this concept in two complementary settings. We first focus on the canonical multi-armed bandit problem and seek to study the intrinsic trade-offs of learning safety in the presence of uncertainty. Under mild assumptions on sufficient exploration, we provide an algorithm that provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. We then consider the problem of finding optimal policies for a Markov Decision Process (MDP) with almost sure constraints. We show that the (action) value function satisfies a barrier-based decomposition which allows for the identification of feasible policies independently of the reward process. Using this decomposition, we develop a Barrier-learning algorithm, that identifies such unsafe state-action pairs in a finite expected number of steps. Our analysis further highlights a trade-off between the time lag for the underlying MDP necessary to detect unsafe actions, and the level of exposure to unsafe events. Simulations corroborate our theoretical findings, further illustrating the aforementioned trade-offs, and suggesting that safety constraints can further speed up the learning process.
@article{cmbm2023tac, abstract = {This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to navigate trade-offs between optimality, level of exposure to unsafe events, and the maximum detection time of unsafe actions. We illustrate this concept in two complementary settings. We first focus on the canonical multi-armed bandit problem and seek to study the intrinsic trade-offs of learning safety in the presence of uncertainty. Under mild assumptions on sufficient exploration, we provide an algorithm that provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. We then consider the problem of finding optimal policies for a Markov Decision Process (MDP) with almost sure constraints. We show that the (action) value function satisfies a barrier-based decomposition which allows for the identification of feasible policies independently of the reward process. Using this decomposition, we develop a Barrier-learning algorithm, that identifies such unsafe state-action pairs in a finite expected number of steps. Our analysis further highlights a trade-off between the time lag for the underlying MDP necessary to detect unsafe actions, and the level of exposure to unsafe events. Simulations corroborate our theoretical findings, further illustrating the aforementioned trade-offs, and suggesting that safety constraints can further speed up the learning process.}, author = {Castellano, Agustin and Min, Hancheng and Bazerque, Juan and Mallada, Enrique}, doi = {10.1109/TAC.2023.3240925}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, journal = {IEEE Transactions on Automatic Control}, month = {5}, number = {5}, pages = {2979-2994}, record = {published, online May 2023, accepted Jan 2023, revised Oct 2022, submitted May 2021}, title = {Learning to Act Safely with Limited Exposure and Almost Sure Certainty}, url = {https://mallada.ece.jhu.edu/pubs/2023-TAC-CMBM.pdf}, volume = {68}, year = {2023} } - B. K. Poolla, Y. Lin, A. Bernstein, E. Mallada, and D. Groß, “Frequency shaping control for weakly-coupled grid-forming IBRs,” IEEE Control Systems Letters (L-CSS), pp. 937-942, 2022. doi:10.1109/LCSYS.2022.3228855
[BibTeX] [Abstract] [Download PDF]
We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.
@article{plbmg2023lcss, abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.}, author = {Poolla, Bala Kameshwar and Lin, Yashen and Bernstein, Andrey and Mallada, Enrique and Groß, Dominic}, doi = {10.1109/LCSYS.2022.3228855}, grants = {CAREER-1752362, CPS-2136324}, journal = {IEEE Control Systems Letters (L-CSS)}, month = {12}, pages = {937-942}, record = {published, online Dec 2022, accepted Nov 2022, submitted Sep 2022.}, title = {Frequency shaping control for weakly-coupled grid-forming IBRs}, url = {https://mallada.ece.jhu.edu/pubs/2022-LCSS-PLBMG.pdf}, year = {2022} } - G. H. Oral, E. Mallada, and D. Gayme, “On the Role of Interconnection Directionality in the Quadratic Performance of Double-Integrator Networks,” IEEE Transactions on Automatic Control, vol. 67, iss. 11, pp. 6211-6218, 2022. doi:10.1109/TAC.2021.3135358
[BibTeX] [Abstract] [Download PDF]
This paper provides a framework to evaluate the performance of single and double integrator networks over arbitrary directed graphs. Adopting vehicular network terminology, we consider quadratic performance metrics defined by the L2-norm of position and velocity based response functions given impulsive inputs to each vehicle. We exploit the spectral properties of weighted graph Laplacians and output performance matrices to derive a novel method of computing the closed-form solutions for this general class of performance metrics, which include H2-norm based quantities as special cases. We then explore the effect of the interplay between network properties (such as edge directionality and connectivity) and the control strategy on the overall network performance. More precisely, for systems whose interconnection is described by graphs with normal Laplacian L, we characterize the role of directionality by comparing their performance with that of their undirected counterparts, represented by the Hermitian part of L. We show that, for single-integrator networks, directed and undirected graphs perform identically. However, for double-integrator networks, graph directionality -expressed by the eigenvalues of L with nonzero imaginary part- can significantly degrade performance. Interestingly, in many cases, well-designed feedback can also exploit directionality to mitigate degradation or even improve the performance to exceed that of the undirected case. Finally we focus on a system coherence metric -aggregate deviation from the state average- to investigate the relationship between performance and degree of connectivity, leading to somewhat surprising findings. For example, increasing the number of neighbors on a ω-nearest neighbor directed graph does not necessarily improve performance. Similarly, we demonstrate equivalence in performance between all-to-one and all-to-all communication graphs.
@article{omg2022tac, abstract = {This paper provides a framework to evaluate the performance of single and double integrator networks over arbitrary directed graphs. Adopting vehicular network terminology, we consider quadratic performance metrics defined by the L2-norm of position and velocity based response functions given impulsive inputs to each vehicle. We exploit the spectral properties of weighted graph Laplacians and output performance matrices to derive a novel method of computing the closed-form solutions for this general class of performance metrics, which include H2-norm based quantities as special cases. We then explore the effect of the interplay between network properties (such as edge directionality and connectivity) and the control strategy on the overall network performance. More precisely, for systems whose interconnection is described by graphs with normal Laplacian L, we characterize the role of directionality by comparing their performance with that of their undirected counterparts, represented by the Hermitian part of L. We show that, for single-integrator networks, directed and undirected graphs perform identically. However, for double-integrator networks, graph directionality -expressed by the eigenvalues of L with nonzero imaginary part- can significantly degrade performance. Interestingly, in many cases, well-designed feedback can also exploit directionality to mitigate degradation or even improve the performance to exceed that of the undirected case. Finally we focus on a system coherence metric -aggregate deviation from the state average- to investigate the relationship between performance and degree of connectivity, leading to somewhat surprising findings. For example, increasing the number of neighbors on a ω-nearest neighbor directed graph does not necessarily improve performance. Similarly, we demonstrate equivalence in performance between all-to-one and all-to-all communication graphs.}, author = {Oral, H. Giray and Mallada, Enrique and Gayme, Dennice}, doi = {10.1109/TAC.2021.3135358}, grants = {ENERGISE-DE-EE0008006, EPCN-1711188,AMPS-1736448, CPS-1544771, CAREER-1752362, AMPS-1736448, ARO-W911NF-17-1-0092}, journal = {IEEE Transactions on Automatic Control}, month = {11}, number = {11}, pages = {6211-6218}, record = {published, online Dec. 2021, accepted Nov. 2021, conditionally accepted Apr 2021, revised Nov. 2020, submitted Nov. 2019}, title = {On the Role of Interconnection Directionality in the Quadratic Performance of Double-Integrator Networks}, url = {https://mallada.ece.jhu.edu/pubs/2019-Preprint-OMG.pdf}, volume = {67}, year = {2022} } - R. K. Bansal, P. You, D. F. Gayme, and E. Mallada, “A Market Mechanism for Truthful Bidding with Energy Storage,” Electric Power Systems Research, vol. 211, iss. 108284, pp. 1-7, 2022. doi:https://doi.org/10.1016/j.epsr.2022.108284
[BibTeX] [Abstract] [Download PDF]
This paper proposes a market mechanism for multiinterval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energycycling functions that map prices to cycle depths. The resulting market-clearing process–implemented via convex programming–yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. We illustrate the benefits of our solution by comparing the competitive equilibrium of the resulting mechanism to that of an alternative solution that uses prosumer-based bids. Our solution shows several advantages over the prosumerbased approach. It does not require a priori price estimation. It also incentivizes participants to reveal their truthful cost, thus leading to an efficient, competitive equilibrium. Numerical experiments using New York Independent System Operator (NYISO) data validate our findings.
@article{bygm2022epsr, abstract = {This paper proposes a market mechanism for multiinterval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energycycling functions that map prices to cycle depths. The resulting market-clearing process--implemented via convex programming--yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. We illustrate the benefits of our solution by comparing the competitive equilibrium of the resulting mechanism to that of an alternative solution that uses prosumer-based bids. Our solution shows several advantages over the prosumerbased approach. It does not require a priori price estimation. It also incentivizes participants to reveal their truthful cost, thus leading to an efficient, competitive equilibrium. Numerical experiments using New York Independent System Operator (NYISO) data validate our findings.}, author = {Bansal, Rajni Kant and You, Pengcheng and Gayme, Dennice F. and Mallada, Enrique}, doi = {https://doi.org/10.1016/j.epsr.2022.108284}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, issn = {0378-7796}, journal = {Electric Power Systems Research}, month = {10}, note = {also in PSCC 2022}, number = {108284}, pages = {1-7}, record = {published, accepted Feb. 2022, submitted Oct. 2021}, title = {A Market Mechanism for Truthful Bidding with Energy Storage}, url = {https://mallada.ece.jhu.edu/pubs/2022-EPSR-BYGM.pdf}, volume = {211}, year = {2022} } - R. Pates, A. Ferragut, E. Pivo, P. You, F. Paganini, and E. Mallada, “Respect the Unstable: Delays and Saturation in Contact Tracing for Disease Control,” SIAM Journal on Control and Optimization, vol. 60, iss. 2, p. S196-S220, 2022. doi:https://doi.org/10.1137/20M1377825
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Motivated by the novel coronavirus disease (COVID-19) pandemic, this paper aims to apply Gunter Stein’s cautionary message of respecting the unstable to the problem of controlling the spread of an infectious disease. With this goal, we study the effect that delays and capacity constraints in the TeTrIs process have on preventing exponential disease spread. Our analysis highlights the critical importance of speed and scale in the TeTrIs process. Precisely, having a delay in the TeTrIs process smaller than the doubling time of the disease spread is necessary for achieving acceptable performance. Similarly, limited TeTrIs capacity introduces a threshold on the size of an outbreak beyond which the disease spreads almost like the uncontrolled case. Along the way, we provide numerical illustrations to highlight these points.
@article{pfpypm2022sicon, abstract = {Motivated by the novel coronavirus disease (COVID-19) pandemic, this paper aims to apply Gunter Stein's cautionary message of respecting the unstable to the problem of controlling the spread of an infectious disease. With this goal, we study the effect that delays and capacity constraints in the TeTrIs process have on preventing exponential disease spread. Our analysis highlights the critical importance of speed and scale in the TeTrIs process. Precisely, having a delay in the TeTrIs process smaller than the doubling time of the disease spread is necessary for achieving acceptable performance. Similarly, limited TeTrIs capacity introduces a threshold on the size of an outbreak beyond which the disease spreads almost like the uncontrolled case. Along the way, we provide numerical illustrations to highlight these points.}, author = {Pates, Richard and Ferragut, Andres and Pivo, Elijah and You, Pengcheng and Paganini, Fernando and Mallada, Enrique}, doi = {https://doi.org/10.1137/20M1377825}, grants = {CAREER-1752362, EPCN-1711188, AMPS-1736448, TRIPODS-1934979}, journal = {SIAM Journal on Control and Optimization}, month = {4}, number = {2}, pages = {S196-S220}, record = {accepted Dec 2021, revised Nov 2021, submitted Nov 2020}, title = {Respect the Unstable: Delays and Saturation in Contact Tracing for Disease Control}, url = {https://mallada.ece.jhu.edu/pubs/2022-SICON-PFPYPM.pdf}, volume = {60}, year = {2022} } - Y. Jiang, A. Bernstein, P. Vorobev, and E. Mallada, “Grid-forming frequency shaping control in low inertia power systems,” IEEE Control Systems Letters (L-CSS), vol. 5, iss. 6, pp. 1988-1993, 2021. doi:10.1109/LCSYS.2020.3044551
[BibTeX] [Abstract] [Download PDF]
We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.
@article{jbvm2021lcss, abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.}, author = {Jiang, Yan and Bernstein, Andrey and Vorobev, Petr and Mallada, Enrique}, doi = {10.1109/LCSYS.2020.3044551}, grants = {CAREER-1752362, AMPS-1736448, TRIPODS-1934979, EPCN-1711188, CPS-2136324}, journal = {IEEE Control Systems Letters (L-CSS)}, month = {12}, note = {also in ACC 2021}, number = {6}, pages = {1988-1993}, record = {early access Dec 2020, accepted Nov 2020, revised Nov 2020, submitted Sep 2020}, title = {Grid-forming frequency shaping control in low inertia power systems}, url = {https://mallada.ece.jhu.edu/pubs/2021-LCSS-JBVM.pdf}, volume = {5}, year = {2021} } - Y. Jiang, E. Cohn, P. Vorobev, and E. Mallada, “Storage-Based Frequency Shaping Control,” IEEE Transactions on Power Systems, vol. 36, iss. 6, pp. 5006-5019, 2021. doi:10.1109/TPWRS.2021.3072833
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With the decrease in system inertia, frequency security becomes an issue for power systems around the world. Energy storage systems (ESS), due to their excellent ramping capabilities, are considered as a natural choice for the improvement of frequency response following major contingencies. In this manuscript, we propose a new strategy for energy storage — frequency shaping control — that allows to completely eliminate the frequency Nadir, one of the main issue in frequency security, and at the same time tune the rate of change of frequency (RoCoF) to a desired value. With Nadir eliminated, the frequency security assessment can be performed via simple algebraic calculations, as opposed to dynamic simulations for conventional control strategies. Moreover, our proposed control is also very efficient in terms of the requirements on storage peak power, requiring up to 40% less power than conventional virtual inertia approach for the same performance.
@article{jcvm2021tps, abstract = {With the decrease in system inertia, frequency security becomes an issue for power systems around the world. Energy storage systems (ESS), due to their excellent ramping capabilities, are considered as a natural choice for the improvement of frequency response following major contingencies. In this manuscript, we propose a new strategy for energy storage -- frequency shaping control -- that allows to completely eliminate the frequency Nadir, one of the main issue in frequency security, and at the same time tune the rate of change of frequency (RoCoF) to a desired value. With Nadir eliminated, the frequency security assessment can be performed via simple algebraic calculations, as opposed to dynamic simulations for conventional control strategies. Moreover, our proposed control is also very efficient in terms of the requirements on storage peak power, requiring up to 40% less power than conventional virtual inertia approach for the same performance.}, author = {Jiang, Yan and Cohn, Eliza and Vorobev, Petr and Mallada, Enrique}, doi = {10.1109/TPWRS.2021.3072833}, grants = {CAREER-1752362;CPS-2136324}, journal = {IEEE Transactions on Power Systems}, month = {11}, number = {6}, pages = {5006-5019}, record = {early access Apr 2021, accepted Mar 2021, revised Oct 2020, submitted May 2020}, title = {Storage-Based Frequency Shaping Control}, url = {https://mallada.ece.jhu.edu/pubs/2021-TPS-JCVM.pdf}, volume = {36}, year = {2021} } - L. S. P. Lawrence, J. W. Simpson-Porco, and E. Mallada, “Linear-Convex Optimal Steady-State Control,” IEEE Transactions on Automatic Control, pp. 5377-5385, 2021. doi:10.1109/TAC.2020.3044275
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We consider the problem of designing a feedback controller for a multivariable nonlinear system that regulates an arbitrary subset of the system states and inputs to the solution of a constrained optimization problem, despite parametric modelling uncertainty and time-varying exogenous disturbances; we term this the optimal steady-state (OSS) control problem. We derive necessary and sufficient conditions for the existence of an OSS controller by formulating the OSS control problem as an output regulation problem wherein the regulation error is unmeasurable. We introduce the notion of an optimality model, and show that the existence of an optimality model is sufficient to reduce the OSS control problem to an output regulation problem with measurable error. This yields a design framework for OSS control that unifies and extends many existing designs in the literature. We present a complete and constructive solution of the OSS control problem for the case where the plant is linear time-invariant with structured parametric uncertainty, and disturbances are constant in time. We illustrate these results via an application to optimal frequency control of power networks, and show that our design procedure recovers several frequency controllers from the recent literature.
@article{lsm2020tac, abstract = {We consider the problem of designing a feedback controller for a multivariable nonlinear system that regulates an arbitrary subset of the system states and inputs to the solution of a constrained optimization problem, despite parametric modelling uncertainty and time-varying exogenous disturbances; we term this the optimal steady-state (OSS) control problem. We derive necessary and sufficient conditions for the existence of an OSS controller by formulating the OSS control problem as an output regulation problem wherein the regulation error is unmeasurable. We introduce the notion of an optimality model, and show that the existence of an optimality model is sufficient to reduce the OSS control problem to an output regulation problem with measurable error. This yields a design framework for OSS control that unifies and extends many existing designs in the literature. We present a complete and constructive solution of the OSS control problem for the case where the plant is linear time-invariant with structured parametric uncertainty, and disturbances are constant in time. We illustrate these results via an application to optimal frequency control of power networks, and show that our design procedure recovers several frequency controllers from the recent literature.}, author = {Lawrence, Liam S. P. and Simpson-Porco, John W. and Mallada, Enrique}, doi = {10.1109/TAC.2020.3044275}, grants = {CAREER-1752362;TRIPODS-1934979;CPS-2136324}, journal = {IEEE Transactions on Automatic Control}, month = {11}, pages = {5377-5385}, record = {early access Dec 2020, accepted Nov 2020, conditionally accepted Aug. 2020, 2nd revision May 2020, revised Sept 2019, submitted Oct. 2018}, title = {Linear-Convex Optimal Steady-State Control}, url = {https://mallada.ece.jhu.edu/pubs/2020-TAC-LSM.pdf}, year = {2021} } - H. Min, F. Paganini, and E. Mallada, “Accurate Reduced Order Models for Coherent Heterogeneous Generators,” IEEE Control Systems Letters (L-CSS), vol. 5, iss. 5, pp. 1741-1746, 2021. doi:10.1109/LCSYS.2020.3043733
[BibTeX] [Abstract] [Download PDF]
We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.
@article{mpm2021lcss, abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.}, author = {Min, Hancheng and Paganini, Fernando and Mallada, Enrique}, doi = {10.1109/LCSYS.2020.3043733}, grants = {CAREER-1752362, CPS-1544771, ENERGISE-DE-EE0008006, AMPS-1736448, TRIPODS-1934979, EPCN-1711188, ARO-W911NF-17-1-0092}, journal = {IEEE Control Systems Letters (L-CSS)}, month = {11}, note = {also in ACC 2021}, number = {5}, pages = {1741-1746}, record = {early accesss Nov 2020, accepted Nov 2020, revised Nov 2020, submitted Sep 2020}, title = {Accurate Reduced Order Models for Coherent Heterogeneous Generators}, url = {https://mallada.ece.jhu.edu/pubs/2021-LCSS-MPM.pdf}, volume = {5}, year = {2021} } - Y. Jiang, R. Pates, and E. Mallada, “Dynamic Droop Control in Low Inertia Power Systems,” IEEE Transactions on Automatic Control, vol. 66, iss. 8, pp. 3518-3533, 2021. doi:10.1109/TAC.2020.3034198
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A widely embraced approach to mitigate the dynamic degradation in low-inertia power systems is to mimic generation response using grid-connected inverters to restore the grid’s stiffness. In this paper, we seek to challenge this approach and advocate for a principled design based on a systematic analysis of the performance trade-offs of inverterbased frequency control. With this aim, we perform a qualitative and quantitative study comparing the effect of conventional control strategies –droop control (DC) and virtual inertia (VI)– on several performance metrics induced by L2 and L∞ signal norms. By extending a recently proposed modal decomposition method, we capture the effect of step and stochastic power disturbances, and frequency measurement noise, on the overall transient and steady-state behavior of the system. Our analysis unveils several limitations of these solutions, such as the inability of DC to improve dynamic frequency response without increasing steady-state control effort, or the large frequency variance that VI introduces in the presence of measurement noise. We further propose a novel dynam-i-c Droop controller (iDroop) that overcomes the limitations of DC and VI. More precisely, we show that iDroop can be tuned to achieve high noise rejection, fast system-wide synchronization, or frequency overshoot (Nadir) elimination without affecting the steady-state control effort share, and propose a tuning recommendation that strikes a balance among these objectives. Extensive numerical experimentation shows that the proposed tuning is effective even when our proportionality assumptions are not valid, and that the particular tuning used for Nadir elimination strikes a good trade-off among various performance metrics.
@article{jpm2021tac, abstract = {A widely embraced approach to mitigate the dynamic degradation in low-inertia power systems is to mimic generation response using grid-connected inverters to restore the grid's stiffness. In this paper, we seek to challenge this approach and advocate for a principled design based on a systematic analysis of the performance trade-offs of inverterbased frequency control. With this aim, we perform a qualitative and quantitative study comparing the effect of conventional control strategies --droop control (DC) and virtual inertia (VI)-- on several performance metrics induced by L2 and L∞ signal norms. By extending a recently proposed modal decomposition method, we capture the effect of step and stochastic power disturbances, and frequency measurement noise, on the overall transient and steady-state behavior of the system. Our analysis unveils several limitations of these solutions, such as the inability of DC to improve dynamic frequency response without increasing steady-state control effort, or the large frequency variance that VI introduces in the presence of measurement noise. We further propose a novel dynam-i-c Droop controller (iDroop) that overcomes the limitations of DC and VI. More precisely, we show that iDroop can be tuned to achieve high noise rejection, fast system-wide synchronization, or frequency overshoot (Nadir) elimination without affecting the steady-state control effort share, and propose a tuning recommendation that strikes a balance among these objectives. Extensive numerical experimentation shows that the proposed tuning is effective even when our proportionality assumptions are not valid, and that the particular tuning used for Nadir elimination strikes a good trade-off among various performance metrics.}, author = {Jiang, Yan and Pates, Richard and Mallada, Enrique}, doi = {10.1109/TAC.2020.3034198}, grants = {ENERGISE-DE-EE0008006, EPCN-1711188,AMPS-1736448, CPS-1544771, CAREER-1752362, AMPS-1736448, ARO-W911NF-17-1-0092}, journal = {IEEE Transactions on Automatic Control}, month = {8}, number = {8}, pages = {3518-3533}, record = {available online Nov. 2020, accepted Aug. 2020, revised Mar. 2020, submitted Aug. 2019}, title = {Dynamic Droop Control in Low Inertia Power Systems}, url = {https://mallada.ece.jhu.edu/pubs/2021-TAC-JPM.pdf}, volume = {66}, year = {2021} } - M. D. Kaba, M. Zhao, R. Vidal, D. R. Robinson, and E. Mallada, “What is the Largest Sparsity Pattern that Can Be Recovered by 1-Norm Minimization?,” IEEE Transactions on Information Theory, vol. 67, iss. 5, pp. 3060-3074, 2021. doi:10.1109/TIT.2021.3067280
[BibTeX] [Abstract] [Download PDF]
We propose a new framework for studying the recovery of signals with structured sparsity patterns via $l_1$-minimization. We achieve this by generalizing the well-known nullspace property for sparse recovery. We show that for each dictionary there is a maximum sparsity pattern—described by a mathematical object called an abstract simplicial complex—that can be recovered via $l_1$-minimization. We provide two different characterizations of this maximum sparsity pattern. In addition, we show how this new framework is useful in the study of sparse recovery problems when the dictionary takes the form of a graph incidence matrix or a partial Discrete Fourier Transform. In both cases we successfully characterize the collection of all support sets for which exact recovery via $l_1$-minimization is possible. When the dictionary is an incidence matrix, we show that the success of exact recovery can be certified in polynomial time, although this is known to be NP-hard for general matrices.
@article{kzvrm2021tit, abstract = {We propose a new framework for studying the recovery of signals with structured sparsity patterns via $l_1$-minimization. We achieve this by generalizing the well-known nullspace property for sparse recovery. We show that for each dictionary there is a maximum sparsity pattern---described by a mathematical object called an abstract simplicial complex---that can be recovered via $l_1$-minimization. We provide two different characterizations of this maximum sparsity pattern. In addition, we show how this new framework is useful in the study of sparse recovery problems when the dictionary takes the form of a graph incidence matrix or a partial Discrete Fourier Transform. In both cases we successfully characterize the collection of all support sets for which exact recovery via $l_1$-minimization is possible. When the dictionary is an incidence matrix, we show that the success of exact recovery can be certified in polynomial time, although this is known to be NP-hard for general matrices.}, author = {Kaba, Mustafa Devrim and Zhao, Mengnan and Vidal, Rene and Robinson, Daniel R. and Mallada, Enrique}, doi = {10.1109/TIT.2021.3067280}, grants = {AMPS:1736448;CAREER-1752362}, journal = {IEEE Transactions on Information Theory}, month = {5}, number = {5}, pages = {3060-3074}, record = {early access Mar. 2021, accepted Jan. 2021, revised Apr. 2020, submitted Oct. 2019}, title = {What is the Largest Sparsity Pattern that Can Be Recovered by 1-Norm Minimization?}, url = {https://mallada.ece.jhu.edu/pubs/2021-TIT-KZVRM.pdf}, volume = {67}, year = {2021} } - M. Almassalkhi, S. Brahma, N. Nazir, H. Ossareh, P. Racherla, S. Kundu, S. P. Nandanoori, T. Ramachandran, A. Singhal, D. Gayme, C. Ji, E. Mallada, Y. Shen, P. You, and D. Anand, “Hierarchical, Grid-Aware, and Economically Optimal Coordination of Distributed Energy Resources in Realistic Distribution Systems,” Energies, vol. 13, iss. 23, pp. 1-35, 2020. doi:10.3390/en13236399
[BibTeX] [Abstract] [Download PDF]
Forward-looking renewable portfolio standards will lead to extreme levels of variable solar PV in electric distribution systems, which makes reliability more challenging to maintain for distribution system operators (DSOs). Distributed energy resources (DERs), including smart, connected appliances and PV inverters, represent responsive grid resources that can provide flexibility to support the DSO in actively managing their networks to facilitate reliability under extreme levels of solar PV. This flexibility can also be used to optimize system operations with respect to economic signals from wholesale energy and ancillary service markets. Here, we present a novel hierarchical scheme that actively controls behind-the-meter DERs to reliablymanage each unbalanced distribution feeder and exploits the available flexibility to ensure reliable operation and economically optimize the entire distribution network. Each layer of the scheme employs advanced optimization methods at different timescales to ensure that the system operates within both grid and device limits. The hierarchy is validated in a large-scale realistic simulation based on data from the industry. Simulation results show that coordination of flexibility improves both system reliability and economics, and enables greater penetration of solar PV. Discussion is also provided on the practical viability of the required communications and controls to implement the presented scheme within a large DSO.
@article{abetal2020energies, abstract = {Forward-looking renewable portfolio standards will lead to extreme levels of variable solar PV in electric distribution systems, which makes reliability more challenging to maintain for distribution system operators (DSOs). Distributed energy resources (DERs), including smart, connected appliances and PV inverters, represent responsive grid resources that can provide flexibility to support the DSO in actively managing their networks to facilitate reliability under extreme levels of solar PV. This flexibility can also be used to optimize system operations with respect to economic signals from wholesale energy and ancillary service markets. Here, we present a novel hierarchical scheme that actively controls behind-the-meter DERs to reliablymanage each unbalanced distribution feeder and exploits the available flexibility to ensure reliable operation and economically optimize the entire distribution network. Each layer of the scheme employs advanced optimization methods at different timescales to ensure that the system operates within both grid and device limits. The hierarchy is validated in a large-scale realistic simulation based on data from the industry. Simulation results show that coordination of flexibility improves both system reliability and economics, and enables greater penetration of solar PV. Discussion is also provided on the practical viability of the required communications and controls to implement the presented scheme within a large DSO.}, author = {Almassalkhi, Mads and Brahma, Sarnaduti and Nazir, Nawaf and Ossareh, Hamid and Racherla, Pavan and Kundu, Soumya and Nandanoori, Sai P. and Ramachandran, Thiagarajan and Singhal, Ankit and Gayme, Dennice and Ji, Chengda and Mallada, Enrique and Shen, Yue and You, Pengcheng and Anand, Dhananjay}, doi = {10.3390/en13236399}, grants = {CAREER-1752362,EPCN-1711188,AMPS-1736448,TRIPODS-1934979}, journal = {Energies}, month = {12}, number = {23}, pages = {1-35}, title = {Hierarchical, Grid-Aware, and Economically Optimal Coordination of Distributed Energy Resources in Realistic Distribution Systems}, url = {https://mallada.ece.jhu.edu/pubs/2020-ENERGIES-ABetal.pdf}, volume = {13}, year = {2020} } - A. Bahram, M. H. Hajiesmaili, Z. Lee, N. Crespi, and E. Mallada, “Online EV Scheduling Algorithms for Adaptive Charging Networks with Global Peak Constraints,” IEEE Transactions on Sustainable Computing, 2020. doi:10.1109/TSUSC.2020.2979854
[BibTeX] [Abstract] [Download PDF]
Electricity bill constitutes a significant portion of operational costs for large scale data centers. Empowering data centers with on-site storages can reduce the electricity bill by shaping the energy procurement from deregulated electricity markets with real-time price fluctuations. This paper focuses on designing energy procurement and storage management strategies to minimize the electricity bill of storage-assisted data centers. Designing such strategies is challenging since the net energy demand of the data center and electricity market prices are not known in advance, and the underlying problem is coupled over time due to evolution of the storage level. Using competitive ratio as the performance measure, we propose an online algorithm that determines the energy procurement and storage management strategies using a threshold based policy. Our algorithm achieves the optimal competitive ratio of as a function of the price fluctuation ratio. We validate the algorithm using data traces from electricity markets and data-center energy demands. The results show that our algorithm achieves close to the offline optimal performance and outperforms existing alternatives.
@article{bhlcm2019tsusc, abstract = {Electricity bill constitutes a significant portion of operational costs for large scale data centers. Empowering data centers with on-site storages can reduce the electricity bill by shaping the energy procurement from deregulated electricity markets with real-time price fluctuations. This paper focuses on designing energy procurement and storage management strategies to minimize the electricity bill of storage-assisted data centers. Designing such strategies is challenging since the net energy demand of the data center and electricity market prices are not known in advance, and the underlying problem is coupled over time due to evolution of the storage level. Using competitive ratio as the performance measure, we propose an online algorithm that determines the energy procurement and storage management strategies using a threshold based policy. Our algorithm achieves the optimal competitive ratio of as a function of the price fluctuation ratio. We validate the algorithm using data traces from electricity markets and data-center energy demands. The results show that our algorithm achieves close to the offline optimal performance and outperforms existing alternatives.}, author = {Bahram, Alina and Hajiesmaili, Mohammad H. and Lee, Zachary and Crespi, Noel and Mallada, Enrique}, doi = {10.1109/TSUSC.2020.2979854}, grants = {CAREER-1752362, CPS-1544771, ENERGISE-DE-EE0008006, AMPS-1736448, TRIPODS-1934979,EPCN-1711188,}, journal = {IEEE Transactions on Sustainable Computing}, month = {1}, title = {Online EV Scheduling Algorithms for Adaptive Charging Networks with Global Peak Constraints}, url = {https://mallada.ece.jhu.edu/pubs/2020-TSUSC-BHLCM.pdf}, year = {2020} } - F. Paganini and E. Mallada, “Global analysis of synchronization performance for power systems: bridging the theory-practice gap,” IEEE Transactions on Automatic Control, vol. 67, iss. 7, pp. 3007-3022, 2020. doi:10.1109/TAC.2019.2942536
[BibTeX] [Abstract] [Download PDF]
The issue of synchronization in the power grid is receiving renewed attention, as new energy sources with different dynamics enter the picture. Global metrics have been proposed to evaluate performance, and analyzed under highly simplified assumptions. In this paper we extend this approach to more realistic network scenarios, and more closely connect it with metrics used in power engineering practice. In particular, our analysis covers networks with generators of heterogeneous ratings and richer dynamic models of machines. Under a suitable proportionality assumption in the parameters, we show that the step response of bus frequencies can be decomposed in two components. The first component is a system-wide frequency that captures the aggregate grid behavior, and the residual component represents the individual bus frequency deviations from the aggregate. Using this decomposition, we define –and compute in closed form– several metrics that capture dynamic behaviors that are of relevance for power engineers. In particular, using the system frequency, we define industry-style metrics (Nadir, RoCoF) that are evaluated through a representative machine. We further use the norm of the residual component to define a synchronization cost that can appropriately quantify inter-area oscillations. Finally, we employ robustness analysis tools to evaluate deviations from our proportionality assumption. We show that the system frequency still captures the grid steady-state deviation, and becomes an accurate reduced-order model of the grid as the network connectivity grows. Simulation studies with practically relevant data are included to validate the theory and further illustrate the impact of network structure and parameters on synchronization. Our analysis gives conclusions of practical interest, sometimes challenging the conventional wisdom in the field.
@article{pm2020tac, abstract = {The issue of synchronization in the power grid is receiving renewed attention, as new energy sources with different dynamics enter the picture. Global metrics have been proposed to evaluate performance, and analyzed under highly simplified assumptions. In this paper we extend this approach to more realistic network scenarios, and more closely connect it with metrics used in power engineering practice. In particular, our analysis covers networks with generators of heterogeneous ratings and richer dynamic models of machines. Under a suitable proportionality assumption in the parameters, we show that the step response of bus frequencies can be decomposed in two components. The first component is a system-wide frequency that captures the aggregate grid behavior, and the residual component represents the individual bus frequency deviations from the aggregate. Using this decomposition, we define --and compute in closed form-- several metrics that capture dynamic behaviors that are of relevance for power engineers. In particular, using the system frequency, we define industry-style metrics (Nadir, RoCoF) that are evaluated through a representative machine. We further use the norm of the residual component to define a synchronization cost that can appropriately quantify inter-area oscillations. Finally, we employ robustness analysis tools to evaluate deviations from our proportionality assumption. We show that the system frequency still captures the grid steady-state deviation, and becomes an accurate reduced-order model of the grid as the network connectivity grows. Simulation studies with practically relevant data are included to validate the theory and further illustrate the impact of network structure and parameters on synchronization. Our analysis gives conclusions of practical interest, sometimes challenging the conventional wisdom in the field.}, author = {Paganini, Fernando and Mallada, Enrique}, doi = {10.1109/TAC.2019.2942536}, grants = {CPS-1544771, AMPS-1736448, EPCN-1711188, CAREER-1752362, ENERGISE-DE-EE0008006}, journal = {IEEE Transactions on Automatic Control}, month = {7}, number = {7}, pages = {3007-3022}, title = {Global analysis of synchronization performance for power systems: bridging the theory-practice gap}, url = {https://mallada.ece.jhu.edu/pubs/2020-TAC-PM.pdf}, volume = {67}, year = {2020} } - L. Yang, M. H. Hajiesmaili, R. Sitaraman, A. Wierman, E. Mallada, and W. S. Wong, “Online Linear Optimization with Inventory Management Constraints,” Proceedings of ACM on Measurement and Analysis of Computing Systems (POMACS), vol. 3, iss. 2, 2020. doi:10.1145/3379482
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This paper considers the problem of online linear optimization with inventory management constraints. Specifically, we consider an online scenario where a decision-maker needs to satisfy her time-varying demand for some units of an asset, either from a market with a time-varying price or from her own inventory. In each time slot, the decision-maker is presented a (linear) price and must immediately decide the amount to purchase to cover the demand and/or to store in inventory for future use. The inventory has a limited capacity and can be used to buy and store assets at a low price and in order to cover the demand when the price is high. The ultimate goal of the decision-maker is to cover the demand at each time slot while minimizing the cost of buying assets from the market. We propose ARP, an online algorithm for linear programming with inventory constraints, and ARPRate, an extended version that handles rate constraints to/from the inventory. Both ARP and ARPRate achieve optimal competitive ratios, meaning that no other online algorithm can achieve a better theoretical guarantee. To illustrate the results, we use the proposed algorithms in a case study focused on energy procurement and storage management strategies for data centers.
@article{yhswmw2020pomacs, abstract = {This paper considers the problem of online linear optimization with inventory management constraints. Specifically, we consider an online scenario where a decision-maker needs to satisfy her time-varying demand for some units of an asset, either from a market with a time-varying price or from her own inventory. In each time slot, the decision-maker is presented a (linear) price and must immediately decide the amount to purchase to cover the demand and/or to store in inventory for future use. The inventory has a limited capacity and can be used to buy and store assets at a low price and in order to cover the demand when the price is high. The ultimate goal of the decision-maker is to cover the demand at each time slot while minimizing the cost of buying assets from the market. We propose ARP, an online algorithm for linear programming with inventory constraints, and ARPRate, an extended version that handles rate constraints to/from the inventory. Both ARP and ARPRate achieve optimal competitive ratios, meaning that no other online algorithm can achieve a better theoretical guarantee. To illustrate the results, we use the proposed algorithms in a case study focused on energy procurement and storage management strategies for data centers.}, address = {New York, NY, USA}, author = {Yang, Lin and Hajiesmaili, Mohammad H. and Sitaraman, Ramesh and Wierman, Adam and Mallada, Enrique and Wong, Wing S.}, date = {2020}, doi = {10.1145/3379482}, grants = {CAREER-1752362, CPS-1544771, ENERGISE-DE-EE0008006, AMPS-1736448,EPCN-1711188}, journal = {Proceedings of ACM on Measurement and Analysis of Computing Systems (POMACS)}, month = {6}, note = {also in ACM Sigmetrics 2020}, number = {2}, publisher = {ACM}, title = {Online Linear Optimization with Inventory Management Constraints}, url = {https://mallada.ece.jhu.edu/pubs/2020-POMACS-YHSWMW.pdf}, volume = {3}, year = {2020} } - E. Weitenberg, Y. Jiang, C. Zhao, E. Mallada, C. De Persis, and F. Dorfler, “Robust Decentralized Secondary Frequency Control in Power Systems: Merits and Trade-Offs,” IEEE Transactions on Automatic Control, vol. 64, iss. 10, pp. 3967-3982, 2019. doi:10.1109/TAC.2018.2884650
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Frequency restoration in power systems is conventionally performed by broadcasting a centralized signal to local controllers. As a result of the energy transition, technological advances, and the scientific interest in distributed control and optimization methods, a plethora of distributed frequency control strategies have been proposed recently that rely on communication amongst local controllers. In this paper we propose a fully decentralized leaky integral controller for frequency restoration that is derived from a classic lag element. We study steady-state, asymptotic optimality, nominal stability, input-to-state stability, noise rejection, transient performance, and robustness properties of this controller in closed loop with a nonlinear and multivariable power system model. We demonstrate that the leaky integral controller can strike an acceptable trade-off between performance and robustness as well as between asymptotic disturbance rejection and transient convergence rate by tuning its DC gain and time constant. We compare our findings to conventional decentralized integral control and distributed-averaging-based integral control in theory and simulations.
@article{wjzmdd2019tac, abstract = {Frequency restoration in power systems is conventionally performed by broadcasting a centralized signal to local controllers. As a result of the energy transition, technological advances, and the scientific interest in distributed control and optimization methods, a plethora of distributed frequency control strategies have been proposed recently that rely on communication amongst local controllers. In this paper we propose a fully decentralized leaky integral controller for frequency restoration that is derived from a classic lag element. We study steady-state, asymptotic optimality, nominal stability, input-to-state stability, noise rejection, transient performance, and robustness properties of this controller in closed loop with a nonlinear and multivariable power system model. We demonstrate that the leaky integral controller can strike an acceptable trade-off between performance and robustness as well as between asymptotic disturbance rejection and transient convergence rate by tuning its DC gain and time constant. We compare our findings to conventional decentralized integral control and distributed-averaging-based integral control in theory and simulations.}, author = {Weitenberg, Erik and Jiang, Yan and Zhao, Changhong and Mallada, Enrique and De Persis, Claudio and Dorfler, Florian}, doi = {10.1109/TAC.2018.2884650}, grants = {CPS-1544771, EPCN-1711188, AMPS-1736448, CAREER-1752362, ENERGISE-DE-EE0008006}, issn = {0018-9286}, journal = {IEEE Transactions on Automatic Control}, keywords = {Power Networks}, month = {10}, number = {10}, pages = {3967-3982}, title = {Robust Decentralized Secondary Frequency Control in Power Systems: Merits and Trade-Offs}, url = {https://mallada.ece.jhu.edu/pubs/2019-TAC-WJZMDD.pdf}, volume = {64}, year = {2019} } - R. Pates and E. Mallada, “Robust Scale Free Synthesis for Frequency Regulation in Power Systems,” IEEE Transactions on Control of Network Systems, vol. 6, iss. 3, pp. 1174-1184, 2019. doi:10.1109/TCNS.2019.2922503
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This paper develops a framework for power system stability analysis, that allows for the decentralised design of frequency controllers. The method builds on a novel decentralised stability criterion, expressed as a positive real requirement, that depends only on the dynamics of the components at each individual bus, and the aggregate susceptance of the transmission lines connected to it. The criterion is both robust to network uncertainties as well as heterogeneous network components, and it can be verified using several standard frequency response, state space, and circuit theory analysis tools. Moreover, it allows to formulate a scale free synthesis problem, that depends on individual bus dynamics and leverages tools from Hinf optimal control. Notably, unlike similar passivity methods, our framework certifies the stability of several existing (non-passive) power system control schemes and allows to study robustness with respect to delays.
@article{pm2019tcns, abstract = {This paper develops a framework for power system stability analysis, that allows for the decentralised design of frequency controllers. The method builds on a novel decentralised stability criterion, expressed as a positive real requirement, that depends only on the dynamics of the components at each individual bus, and the aggregate susceptance of the transmission lines connected to it. The criterion is both robust to network uncertainties as well as heterogeneous network components, and it can be verified using several standard frequency response, state space, and circuit theory analysis tools. Moreover, it allows to formulate a scale free synthesis problem, that depends on individual bus dynamics and leverages tools from Hinf optimal control. Notably, unlike similar passivity methods, our framework certifies the stability of several existing (non-passive) power system control schemes and allows to study robustness with respect to delays.}, author = {Pates, Richard and Mallada, Enrique}, doi = {10.1109/TCNS.2019.2922503}, grants = {CPS:1544771, EPCN-1711188, AMPS-1736448, CAREER-1752362}, journal = {IEEE Transactions on Control of Network Systems}, keywords = {Network Control; Power Networks}, month = {9}, number = {3}, pages = {1174-1184}, title = {Robust Scale Free Synthesis for Frequency Regulation in Power Systems}, url = {https://mallada.ece.jhu.edu/pubs/2019-TCNS-PM.pdf}, volume = {6}, year = {2019} } - C. Zhao, E. Mallada, S. H. Low, and J. W. Bialek, “Distributed plug-and-play optimal generator and load control for power system frequency regulation,” International Journal of Electric Power and Energy Systems, vol. 101, pp. 1-12, 2018. doi:https://doi.org/10.1016/j.ijepes.2018.03.014
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A distributed control scheme, which can be implemented on generators and controllable loads in a plug-and-play manner, is proposed for power system frequency regulation. The proposed scheme is based on local measurements, local computation, and neighborhood information exchanges over a communication network with an arbitrary (but connected) topology. In the event of a sudden change in generation or load, the proposed scheme can restore the nominal frequency and the reference inter-area power flows, while minimizing the total cost of control for participating generators and loads. Power network stability under the proposed control is proved with a relatively realistic model which includes nonlinear power flow and a generic (potentially nonlinear or high-order) turbine-governor model, and further with first- and second-order turbine-governor models as special cases. In simulations, the proposed control scheme shows a comparable performance to the existing automatic generation control (AGC) when implemented only on the generator side, and demonstrates better dynamic characteristics that AGC when each scheme is implemented on both generators and controllable loads.
@article{zmlb2018ijepes, abstract = {A distributed control scheme, which can be implemented on generators and controllable loads in a plug-and-play manner, is proposed for power system frequency regulation. The proposed scheme is based on local measurements, local computation, and neighborhood information exchanges over a communication network with an arbitrary (but connected) topology. In the event of a sudden change in generation or load, the proposed scheme can restore the nominal frequency and the reference inter-area power flows, while minimizing the total cost of control for participating generators and loads. Power network stability under the proposed control is proved with a relatively realistic model which includes nonlinear power flow and a generic (potentially nonlinear or high-order) turbine-governor model, and further with first- and second-order turbine-governor models as special cases. In simulations, the proposed control scheme shows a comparable performance to the existing automatic generation control (AGC) when implemented only on the generator side, and demonstrates better dynamic characteristics that AGC when each scheme is implemented on both generators and controllable loads.}, author = {Zhao, Changhong and Mallada, Enrique and Low, Steven H and Bialek, Janusz W}, doi = {https://doi.org/10.1016/j.ijepes.2018.03.014}, grants = {W911NF-17-1-0092, 1544771, 1711188, 1736448, 1752362}, issn = {0142-0615}, journal = {International Journal of Electric Power and Energy Systems}, keywords = {Power Networks; Frequency Control}, month = {10}, pages = {1 -12}, title = {Distributed plug-and-play optimal generator and load control for power system frequency regulation}, url = {https://mallada.ece.jhu.edu/pubs/2018-IJEPES-ZMLB.pdf}, volume = {101}, year = {2018} } - A. Cherukuri, E. Mallada, S. H. Low, and J. Cortes, “The role of convexity on saddle-point dynamics: Lyapunov function and robustness,” IEEE Transactions on Automatic Control, vol. 63, iss. 8, pp. 2449-2464, 2018. doi:10.1109/TAC.2017.2778689
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This paper studies the projected saddle-point dynamics associated to a convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity-concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity-concavity of the saddle function. When strong convexity-concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results.
@article{cmlc2018tac, abstract = {This paper studies the projected saddle-point dynamics associated to a convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity-concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity-concavity of the saddle function. When strong convexity-concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results.}, author = {Cherukuri, Ashish and Mallada, Enrique and Steven H. Low and Jorge Cortes}, doi = {10.1109/TAC.2017.2778689}, grants = {W911NF-17-1-0092}, journal = {IEEE Transactions on Automatic Control}, keywords = {Saddle-Point Dynamics; Caratheodory solutions}, month = {08}, number = {8}, pages = {2449-2464}, title = {The role of convexity on saddle-point dynamics: Lyapunov function and robustness}, url = {https://mallada.ece.jhu.edu/pubs/2018-TAC-CMLC.pdf}, volume = {63}, year = {2018} } - E. Mallada, C. Zhao, and S. H. Low, “Optimal load-side control for frequency regulation in smart grids,” IEEE Transactions on Automatic Control, vol. 62, iss. 12, pp. 6294-6309, 2017. doi:10.1109/TAC.2017.2713529
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Frequency control rebalances supply and demand while maintaining the network state within operational margins. It is implemented using fast ramping reserves that are expensive and wasteful, and which are expected to grow with the increasing penetration of renewables. The most promising solution to this problem is the use of demand response, i.e. load participation in frequency control. Yet it is still unclear how to efficiently integrate load participation without introducing instabilities and violating operational constraints. In this paper we present a comprehensive load-side frequency control mechanism that can maintain the grid within operational constraints. Our controllers can rebalance supply and demand after disturbances, restore the frequency to its nominal value and preserve inter-area power flows. Furthermore, our controllers are distributed (unlike generation-side), can allocate load updates optimally, and can maintain line flows within thermal limits. We prove that such a distributed load-side control is globally asymptotically stable and robust to unknown load parameters. Simulations are used to illustrate the properties of our solution.
@article{mzl2017tac, abstract = {Frequency control rebalances supply and demand while maintaining the network state within operational margins. It is implemented using fast ramping reserves that are expensive and wasteful, and which are expected to grow with the increasing penetration of renewables. The most promising solution to this problem is the use of demand response, i.e. load participation in frequency control. Yet it is still unclear how to efficiently integrate load participation without introducing instabilities and violating operational constraints. In this paper we present a comprehensive load-side frequency control mechanism that can maintain the grid within operational constraints. Our controllers can rebalance supply and demand after disturbances, restore the frequency to its nominal value and preserve inter-area power flows. Furthermore, our controllers are distributed (unlike generation-side), can allocate load updates optimally, and can maintain line flows within thermal limits. We prove that such a distributed load-side control is globally asymptotically stable and robust to unknown load parameters. Simulations are used to illustrate the properties of our solution.}, author = {Mallada, Enrique and Zhao, Changhong and Low, Steven H}, doi = {10.1109/TAC.2017.2713529}, grants = {1544771}, journal = {IEEE Transactions on Automatic Control}, keywords = {Power Networks}, month = {12}, number = {12}, pages = {6294-6309}, title = {Optimal load-side control for frequency regulation in smart grids}, url = {https://mallada.ece.jhu.edu/pubs/2017-TAC-MZL.pdf}, volume = {62}, year = {2017} } - D. Cai, E. Mallada, and A. Wierman, “Distributed optimization decomposition for joint economic dispatch and frequency regulation,” IEEE Transactions on Power Systems, vol. 32, iss. 6, pp. 4370-4385, 2017. doi:10.1109/TPWRS.2017.2682235
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Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co-optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.
@article{cmw2017tps, abstract = {Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co-optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.}, author = {Cai, Desmond and Mallada, Enrique and Wierman, Adam}, doi = {10.1109/TPWRS.2017.2682235}, grants = {1544771}, journal = {IEEE Transactions on Power Systems}, keywords = {Power Networks; Markets}, month = {11}, number = {6}, pages = {4370-4385}, title = {Distributed optimization decomposition for joint economic dispatch and frequency regulation}, url = {https://mallada.ece.jhu.edu/pubs/2017-TPS-CMW.pdf}, volume = {32}, year = {2017} } - K. Dvijotham, E. Mallada, and J. W. Simpson-Porco, “High-Voltage Solution in Radial Power Networks: Existence, Properties, and Equivalent Algorithms,” IEEE Control Systems Letters, vol. 1, iss. 2, pp. 322-327, 2017. doi:10.1109/LCSYS.2017.2717578
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The AC power flow equations describe the steady-state behavior of the power grid. While many algorithms have been developed to compute solutions to the power flow equations, few theoretical results are available characterizing when such solutions exist, or when these algorithms can be guaranteed to converge. In this paper, we derive necessary and sufficient conditions for the existence and uniqueness of a power flow solution in balanced radial distribution networks with homogeneous (uniform R/X ratio) transmission lines. We study three distinct solution methods: fixed point iterations, convex relaxations, and energy functions – we show that the three algorithms successfully find a solution if and only if a solution exists. Moreover, all three algorithms always find the unique high-voltage solution to the power flow equations, the existence of which we formally establish. At this solution, we prove that (i) voltage magnitudes are increasing functions of the reactive power injections, (ii) the solution is a continuous function of the injections, and (iii) the solution is the last one to vanish as the system is loaded past the feasibility boundary.
@article{dms2017ieee-csl, abstract = {The AC power flow equations describe the steady-state behavior of the power grid. While many algorithms have been developed to compute solutions to the power flow equations, few theoretical results are available characterizing when such solutions exist, or when these algorithms can be guaranteed to converge. In this paper, we derive necessary and sufficient conditions for the existence and uniqueness of a power flow solution in balanced radial distribution networks with homogeneous (uniform R/X ratio) transmission lines. We study three distinct solution methods: fixed point iterations, convex relaxations, and energy functions - we show that the three algorithms successfully find a solution if and only if a solution exists. Moreover, all three algorithms always find the unique high-voltage solution to the power flow equations, the existence of which we formally establish. At this solution, we prove that (i) voltage magnitudes are increasing functions of the reactive power injections, (ii) the solution is a continuous function of the injections, and (iii) the solution is the last one to vanish as the system is loaded past the feasibility boundary.}, author = {Dvijotham, Krishnamurthy and Mallada, Enrique and Simpson-Porco, John W.}, doi = {10.1109/LCSYS.2017.2717578}, grants = {1544771}, journal = {IEEE Control Systems Letters}, keywords = {Power Networks; Power Flow Solutions}, month = {10}, number = {2}, pages = {322-327}, title = {High-Voltage Solution in Radial Power Networks: Existence, Properties, and Equivalent Algorithms}, url = {https://mallada.ece.jhu.edu/pubs/2017-IEEE-CSL-DMS.pdf}, volume = {1}, year = {2017} } - A. Gushchin, E. Mallada, and A. Tang, “Phase-coupled oscillators with plastic coupling: Synchronization and stability,” IEEE Transactions on Network Science and Engineering, vol. 3, iss. 4, pp. 240-256, 2016. doi:10.1109/TNSE.2016.2605096
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In this article we study synchronization of systems of homogeneous phase-coupled oscillators with plastic coupling strengths and arbitrary underlying topology. The dynamics of the coupling strength between two oscillators is governed by the phase difference between these oscillators. We show that, under mild assumptions, such systems are gradient systems, and always achieve frequency synchronization. Furthermore, we provide sufficient stability and instability conditions that are based on results from algebraic graph theory. For a special case when underlying topology is a tree, we formulate a criterion (necessary and sufficient condition) of stability of equilibria. For both, tree and arbitrary topologies, we provide sufficient conditions for phase-locking, i.e. convergence to a stable equilibrium almost surely. We additionally find conditions when the system possesses a unique stable equilibrium, and thus, almost global stability follows. Several examples are used to demonstrate variety of equilibria the system has, their dependence on system’s parameters, and to illustrate differences in behavior of systems with constant and plastic coupling strengths.
@article{gmt2016tnse, abstract = {In this article we study synchronization of systems of homogeneous phase-coupled oscillators with plastic coupling strengths and arbitrary underlying topology. The dynamics of the coupling strength between two oscillators is governed by the phase difference between these oscillators. We show that, under mild assumptions, such systems are gradient systems, and always achieve frequency synchronization. Furthermore, we provide sufficient stability and instability conditions that are based on results from algebraic graph theory. For a special case when underlying topology is a tree, we formulate a criterion (necessary and sufficient condition) of stability of equilibria. For both, tree and arbitrary topologies, we provide sufficient conditions for phase-locking, i.e. convergence to a stable equilibrium almost surely. We additionally find conditions when the system possesses a unique stable equilibrium, and thus, almost global stability follows. Several examples are used to demonstrate variety of equilibria the system has, their dependence on system's parameters, and to illustrate differences in behavior of systems with constant and plastic coupling strengths.}, author = {Gushchin, Andrey and Mallada, Enrique and Tang, Ao}, doi = {10.1109/TNSE.2016.2605096}, journal = {IEEE Transactions on Network Science and Engineering}, keywords = {Synchronization}, month = {09}, number = {4}, pages = {240-256}, title = {Phase-coupled oscillators with plastic coupling: Synchronization and stability}, url = {https://mallada.ece.jhu.edu/pubs/2016-TNSE-GMT.pdf}, volume = {3}, year = {2016} } - A. Cherukuri, E. Mallada, and J. Cortes, “Asymptotic convergence of constrained primal–dual dynamics,” Systems & Control Letters, vol. 87, pp. 10-15, 2016. doi:10.1016/j.sysconle.2015.10.006
[BibTeX] [Abstract] [Download PDF]
This paper studies the asymptotic convergence properties of the primal–dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the solutions of the primal–dual dynamics in the Caratheodory sense and characterize their existence, uniqueness, and continuity with respect to the initial condition. We use the invariance principle for discontinuous Caratheodory systems to establish that the primal–dual optimizers are globally asymptotically stable under the primal–dual dynamics and that each solution of the dynamics converges to an optimizer.
@article{cmc2016scl, abstract = {This paper studies the asymptotic convergence properties of the primal--dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the solutions of the primal--dual dynamics in the Caratheodory sense and characterize their existence, uniqueness, and continuity with respect to the initial condition. We use the invariance principle for discontinuous Caratheodory systems to establish that the primal--dual optimizers are globally asymptotically stable under the primal--dual dynamics and that each solution of the dynamics converges to an optimizer. }, author = {Ashish Cherukuri and Mallada, Enrique and Jorge Cortes}, doi = {10.1016/j.sysconle.2015.10.006}, issn = {0167-6911}, journal = {Systems & Control Letters}, keywords = {Caratheodory solutions; Optimization; Saddle-Point Dynamics}, month = {01}, pages = {10 - 15}, title = {Asymptotic convergence of constrained primal--dual dynamics}, url = {https://mallada.ece.jhu.edu/pubs/2016-SCL-CMC.pdf}, volume = {87}, web = {http://www.sciencedirect.com/science/article/pii/S0167691115002078}, year = {2016} } - E. Mallada, R. A. Freeman, and A. Tang, “Distributed synchronization of heterogeneous oscillators on networks with arbitrary topology,” IEEE Transactions on Control of Network Systems, vol. 3, iss. 1, pp. 12-23, 2016. doi:10.1109/TCNS.2015.2428371
[BibTeX] [Abstract] [Download PDF]
Many network applications rely on the synchronization of coupled oscillators. For example, such synchronization can provide networked devices with a common temporal reference necessary for coordinating actions or decoding transmitted messages. In this paper, we study the problem of using distributed control to achieve both phase and frequency synchronization of a network of coupled heterogeneous nonlinear oscillators. Not only do our controllers guarantee zero phase error in steady state under arbitrary frequency heterogeneity, but they also require little knowledge of the oscillator nonlinearities and network topology. Furthermore, we provide a global convergence analysis, in the absence of noise and propagation delay, for the resulting nonlinear system whose phase vector evolves on the n-torus.
@article{mft2016tcns, abstract = {Many network applications rely on the synchronization of coupled oscillators. For example, such synchronization can provide networked devices with a common temporal reference necessary for coordinating actions or decoding transmitted messages. In this paper, we study the problem of using distributed control to achieve both phase and frequency synchronization of a network of coupled heterogeneous nonlinear oscillators. Not only do our controllers guarantee zero phase error in steady state under arbitrary frequency heterogeneity, but they also require little knowledge of the oscillator nonlinearities and network topology. Furthermore, we provide a global convergence analysis, in the absence of noise and propagation delay, for the resulting nonlinear system whose phase vector evolves on the n-torus.}, author = {Mallada, Enrique and Freeman, Randy A and Tang, Ao}, doi = {10.1109/TCNS.2015.2428371}, journal = {IEEE Transactions on Control of Network Systems}, keywords = {Coupled Oscillators; Synchronization}, month = {3}, number = {1}, pages = {12-23}, title = {Distributed synchronization of heterogeneous oscillators on networks with arbitrary topology}, url = {https://mallada.ece.jhu.edu/pubs/2016-TCNS-MFT.pdf}, volume = {3}, year = {2016} } - E. Mallada, X. Meng, M. Hack, L. Zhang, and A. Tang, “Skewless network clock synchronization without discontinuity: Convergence and performance,” IEEE/ACM Transactions on Networking (TON), vol. 23, iss. 5, pp. 1619-1633, 2015. doi:10.1109/TNET.2014.2345692
[BibTeX] [Abstract] [Download PDF]
This paper examines synchronization of computer clocks connected via a data network and proposes a skewless algorithm to synchronize them. Unlike existing solutions, which either estimate and compensate the frequency difference (skew) among clocks or introduce offset corrections that can generate jitter and possibly even backward jumps, our solution achieves synchronization without these problems. We first analyze the convergence property of the algorithm and provide explicit necessary and sufficient conditions on the parameters to guarantee synchronization. We then study the effect of noisy measurements (jitter) and frequency drift (wander) on the offsets and synchronization frequency, and further optimize the parameter values to minimize their variance. Our study reveals a few insights, for example, we show that our algorithm can converge even in the presence of timing loops and noise, provided that there is a well-defined leader. This marks a clear contrast with current standards such as NTP and PTP, where timing loops are specifically avoided. Furthermore, timing loops can even be beneficial in our scheme as it is demonstrated that highly connected subnetworks can collectively outperform individual clients when the time source has large jitter. The results are supported by experiments running on a cluster of IBM BladeCenter servers with Linux.
@article{mmhzt2015ton, abstract = {This paper examines synchronization of computer clocks connected via a data network and proposes a skewless algorithm to synchronize them. Unlike existing solutions, which either estimate and compensate the frequency difference (skew) among clocks or introduce offset corrections that can generate jitter and possibly even backward jumps, our solution achieves synchronization without these problems. We first analyze the convergence property of the algorithm and provide explicit necessary and sufficient conditions on the parameters to guarantee synchronization. We then study the effect of noisy measurements (jitter) and frequency drift (wander) on the offsets and synchronization frequency, and further optimize the parameter values to minimize their variance. Our study reveals a few insights, for example, we show that our algorithm can converge even in the presence of timing loops and noise, provided that there is a well-defined leader. This marks a clear contrast with current standards such as NTP and PTP, where timing loops are specifically avoided. Furthermore, timing loops can even be beneficial in our scheme as it is demonstrated that highly connected subnetworks can collectively outperform individual clients when the time source has large jitter. The results are supported by experiments running on a cluster of IBM BladeCenter servers with Linux.}, author = {Mallada, Enrique and Meng, Xiaoqiao and Hack, Michel and Zhang, Li and Tang, Ao}, doi = {10.1109/TNET.2014.2345692}, issn = {1063-6692}, journal = {IEEE/ACM Transactions on Networking (TON)}, keywords = {Synchronization; Networking}, month = {10}, number = {5}, pages = {1619--1633}, rating = {1}, title = {Skewless network clock synchronization without discontinuity: Convergence and performance}, url = {https://mallada.ece.jhu.edu/pubs/2015-ToN-MMHZT.pdf}, volume = {23}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6883234}, year = {2015} } - M. Wang, W. Xu, E. Mallada, and A. Tang, “Sparse recovery with graph constraints,” IEEE Transactions on Information Theory, vol. 61, iss. 2, pp. 1028-1044, 2015. doi:10.1109/TIT.2014.2376955
[BibTeX] [Abstract] [Download PDF]
Sparse recovery can recover sparse signals from a set of underdetermined linear measurements. Motivated by the need to monitor the key characteristics of large-scale networks from a limited number of measurements, this paper addresses the problem of recovering sparse signals in the presence of network topological constraints. Unlike conventional sparse recovery where a measurement can contain any subset of the unknown variables, we use a graph to characterize the topological constraints and allow an additive measurement over nodes (unknown variables) only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs, and the number of measurements by our construction is less than that needed by existing random constructions. Moreover, our construction for a line network is provably optimal in the sense that it requires the minimum number of measurements. A measurement construction algorithm for general graphs is also proposed and evaluated. For any given graph G with n nodes, we derive bounds of the minimum number of measurements needed to recover any k-sparse vector over G (Mk,nG). Using the Erdös-Rényi random graph as an example, we characterize the dependence of Mk,nG on the graph structure. This paper suggests that Mk,nG may serve as a graph connectivity metric.
@article{wxmt2015tit, abstract = {Sparse recovery can recover sparse signals from a set of underdetermined linear measurements. Motivated by the need to monitor the key characteristics of large-scale networks from a limited number of measurements, this paper addresses the problem of recovering sparse signals in the presence of network topological constraints. Unlike conventional sparse recovery where a measurement can contain any subset of the unknown variables, we use a graph to characterize the topological constraints and allow an additive measurement over nodes (unknown variables) only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs, and the number of measurements by our construction is less than that needed by existing random constructions. Moreover, our construction for a line network is provably optimal in the sense that it requires the minimum number of measurements. A measurement construction algorithm for general graphs is also proposed and evaluated. For any given graph G with n nodes, we derive bounds of the minimum number of measurements needed to recover any k-sparse vector over G (Mk,nG). Using the Erdös-Rényi random graph as an example, we characterize the dependence of Mk,nG on the graph structure. This paper suggests that Mk,nG may serve as a graph connectivity metric.}, author = {Wang, M. and Xu, W. and Mallada, Enrique and Tang, A.}, doi = {10.1109/TIT.2014.2376955}, journal = {IEEE Transactions on Information Theory}, keywords = {Sparse Recovery}, month = {02}, number = {2}, pages = {1028-1044}, title = {Sparse recovery with graph constraints}, url = {https://mallada.ece.jhu.edu/pubs/2015-TIT-WXMT.pdf}, volume = {61}, year = {2015} } - E. Mallada and A. Tang, “Synchronization of weakly coupled oscillators: coupling, delay and topology,” Journal of Physics A: Mathematical and Theoretical, vol. 46, iss. 50, p. 505101, 2013. doi:10.1088/1751-8113/46/50/505101
[BibTeX] [Abstract] [Download PDF]
There are three key factors in a system of coupled oscillators that characterize the interaction between them: coupling (how to affect), delay (when to affect) and topology (whom to affect). The existing work on each of these factors has mainly focused on special cases. With new angles and tools, this paper makes progress in relaxing some assumptions on these factors. There are three main results in this paper. Firstly, by using results from algebraic graph theory, a sufficient condition is obtained that can be used to check equilibrium stability. This condition works for arbitrary topology, generalizing existing results and also leading to a sufficient condition on the coupling function which guarantees that the system will reach synchronization. Secondly, it is known that identical oscillators with sin () coupling functions are guaranteed to synchronize in phase on a complete graph. Our results prove that in many cases certain structures such as symmetry and concavity, rather than the exact shape of the coupling function, are the keys for global synchronization. Finally, the effect of heterogenous delays is investigated. Using mean field theory, a system of delayed coupled oscillators is approximated by a non-delayed one whose coupling depends on the delay distribution. This shows how the stability properties of the system depend on the delay distribution and allows us to predict its behavior. In particular, we show that for sin () coupling, heterogeneous delays are equivalent to homogeneous delays. Furthermore, we can use our novel sufficient instability condition to show that heterogeneity, i.e. wider delay distribution, can help reach in-phase synchronization.
@article{mt2013jopa, abstract = {There are three key factors in a system of coupled oscillators that characterize the interaction between them: coupling (how to affect), delay (when to affect) and topology (whom to affect). The existing work on each of these factors has mainly focused on special cases. With new angles and tools, this paper makes progress in relaxing some assumptions on these factors. There are three main results in this paper. Firstly, by using results from algebraic graph theory, a sufficient condition is obtained that can be used to check equilibrium stability. This condition works for arbitrary topology, generalizing existing results and also leading to a sufficient condition on the coupling function which guarantees that the system will reach synchronization. Secondly, it is known that identical oscillators with sin () coupling functions are guaranteed to synchronize in phase on a complete graph. Our results prove that in many cases certain structures such as symmetry and concavity, rather than the exact shape of the coupling function, are the keys for global synchronization. Finally, the effect of heterogenous delays is investigated. Using mean field theory, a system of delayed coupled oscillators is approximated by a non-delayed one whose coupling depends on the delay distribution. This shows how the stability properties of the system depend on the delay distribution and allows us to predict its behavior. In particular, we show that for sin () coupling, heterogeneous delays are equivalent to homogeneous delays. Furthermore, we can use our novel sufficient instability condition to show that heterogeneity, i.e. wider delay distribution, can help reach in-phase synchronization.}, author = {Mallada, Enrique and Tang, Ao}, doi = {10.1088/1751-8113/46/50/505101}, journal = {Journal of Physics A: Mathematical and Theoretical}, keywords = {Coupled Oscillators; Synchronization}, month = {12}, number = {50}, pages = {505101}, title = {Synchronization of weakly coupled oscillators: coupling, delay and topology}, url = {https://mallada.ece.jhu.edu/pubs/2013-JOPA-MT.pdf}, volume = {46}, year = {2013} } - F. Paganini and E. Mallada, “A unified approach to congestion control and node-based multipath routing,” IEEE/ACM Transactions on Networking (TON), vol. 17, iss. 5, pp. 1413-1426, 2009. doi:10.1109/TNET.2008.2011902
[BibTeX] [Abstract] [Download PDF]
The paper considers a TCP/IP-style network with flow control at end-systems based on congestion feedback and routing decisions at network nodes on a per-destination basis. The main generalization with respect to standard IP is to allow routers to split their traffic in a controlled way between the outgoing links. We formulate global optimization criteria, combining those used in the congestion control and traffic engineering, and propose decentralized controllers at sources and routers to reach these optimal points, based on congestion price feedback. We first consider adapting the traffic splits at routers to follow the negative price gradient; we prove this is globally stabilizing when combined with primal congestion control, but can exhibit oscillations in the case of dual congestion control. We then propose an alternative anticipatory control of routing, proving its stability for the case of dual congestion control. We present a concrete implementation of such algorithms, based on queueing delay as congestion price. We use TCP-FAST for congestion control and develop a multipath variant of the distance vector routing protocol RIP. We demonstrate through ns2-simulations the collective behavior of the system, in particular that it reaches the desired equilibrium points.
@article{pm2009ton, abstract = {The paper considers a TCP/IP-style network with flow control at end-systems based on congestion feedback and routing decisions at network nodes on a per-destination basis. The main generalization with respect to standard IP is to allow routers to split their traffic in a controlled way between the outgoing links. We formulate global optimization criteria, combining those used in the congestion control and traffic engineering, and propose decentralized controllers at sources and routers to reach these optimal points, based on congestion price feedback. We first consider adapting the traffic splits at routers to follow the negative price gradient; we prove this is globally stabilizing when combined with primal congestion control, but can exhibit oscillations in the case of dual congestion control. We then propose an alternative anticipatory control of routing, proving its stability for the case of dual congestion control. We present a concrete implementation of such algorithms, based on queueing delay as congestion price. We use TCP-FAST for congestion control and develop a multipath variant of the distance vector routing protocol RIP. We demonstrate through ns2-simulations the collective behavior of the system, in particular that it reaches the desired equilibrium points.}, author = {Paganini, Fernando and Mallada, Enrique}, doi = {10.1109/TNET.2008.2011902}, journal = {IEEE/ACM Transactions on Networking (TON)}, keywords = {Networking}, month = {10}, number = {5}, pages = {1413--1426}, publisher = {IEEE}, title = {A unified approach to congestion control and node-based multipath routing}, url = {https://mallada.ece.jhu.edu/pubs/2009-ToN-MP.pdf}, volume = {17}, web = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5200323}, year = {2009} }
Conference Proceedings
- Y. Liu, E. Mallada, Z. Li, and P. You, “Accelerated Saddle Flow Dynamics for Bilinearly Coupled Minimax Problems,” in 63rd IEEE Conference on Decision and Control (CDC), 2024.
[BibTeX] [Abstract] [Download PDF]
Minimax problems have attracted much attention due to various applications in constrained optimization problems and zero-sum games. Identifying saddle points within these problems is crucial, and saddle flow dynamics offer a straightforward yet useful approach. This study focuses on a class of bilinearly coupled minimax problems and designs an accelerated algorithm based on saddle flow dynamics that achieves a convergence rate beyond stereotype limits. The algorithm is derived based on a sequential two-step transformation of the objective function. First, a change of variable is aimed at a better-conditioned saddle function. Second, a proximal regularization, when staggered with the first step, guarantees strong convexity-strong concavity of the objective function that can be tuned for accelerated exponential convergence. Besides, such an approach can be extended to a class of weakly convex-weakly concave functions and still achieve exponential convergence to one stationary point. The theory is verified by a numerical test on an affine equality-constrained convex optimization problem.
@inproceedings{lmly2024cdc, abstract = {Minimax problems have attracted much attention due to various applications in constrained optimization problems and zero-sum games. Identifying saddle points within these problems is crucial, and saddle flow dynamics offer a straightforward yet useful approach. This study focuses on a class of bilinearly coupled minimax problems and designs an accelerated algorithm based on saddle flow dynamics that achieves a convergence rate beyond stereotype limits. The algorithm is derived based on a sequential two-step transformation of the objective function. First, a change of variable is aimed at a better-conditioned saddle function. Second, a proximal regularization, when staggered with the first step, guarantees strong convexity-strong concavity of the objective function that can be tuned for accelerated exponential convergence. Besides, such an approach can be extended to a class of weakly convex-weakly concave functions and still achieve exponential convergence to one stationary point. The theory is verified by a numerical test on an affine equality-constrained convex optimization problem.}, author = {Liu, Yingzhu and Mallada, Enrique and Li, Zhongkui and You, Pengcheng}, booktitle = {63rd IEEE Conference on Decision and Control (CDC)}, grants = {CPS-2136324, Global-Centers-2330450}, month = {07}, pubstate = {accepted, submitted Mar 2024}, title = {Accelerated Saddle Flow Dynamics for Bilinearly Coupled Minimax Problems}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-LMLY.pdf}, year = {2024} } - K. Poe, E. Mallada, and R. Vidal, “Invertibility of Discrete-Time Linear Systems with Sparse Inputs,” in 63rd IEEE Conference on Decision and Control (CDC), 2024.
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One of the fundamental problems of interest for discrete-time linear systems is whether its input sequence may be recovered given its output sequence, a.k.a. the left inversion problem. Many conditions on the state space geometry, dynamics, and spectral structure of a system have been used to characterize the well-posedness of this problem, without assumptions on the inputs. However, certain structural assumptions, such as input sparsity, have been shown to translate to practical gains in the performance of inversion algorithms, surpassing classical guarantees. Establishing necessary and sufficient conditions for left invertibility of systems with sparse inputs is therefore a crucial step toward understanding the performance limits of system inversion under structured input assumptions. In this work, we provide the first necessary and sufficient characterizations of left invertibility for linear systems with sparse inputs, echoing classic characterizations for standard linear systems. The key insight in deriving these results is in establishing the existence of two novel geometric invariants unique to the sparse-input setting, the weakly unobservable and strongly reachable subspace arrangements. By means of a concrete example, we demonstrate the utility of these characterizations. We conclude by discussing extensions and applications of this framework to several related problems in sparse control.
@inproceedings{pmv2024cdc, abstract = {One of the fundamental problems of interest for discrete-time linear systems is whether its input sequence may be recovered given its output sequence, a.k.a. the left inversion problem. Many conditions on the state space geometry, dynamics, and spectral structure of a system have been used to characterize the well-posedness of this problem, without assumptions on the inputs. However, certain structural assumptions, such as input sparsity, have been shown to translate to practical gains in the performance of inversion algorithms, surpassing classical guarantees. Establishing necessary and sufficient conditions for left invertibility of systems with sparse inputs is therefore a crucial step toward understanding the performance limits of system inversion under structured input assumptions. In this work, we provide the first necessary and sufficient characterizations of left invertibility for linear systems with sparse inputs, echoing classic characterizations for standard linear systems. The key insight in deriving these results is in establishing the existence of two novel geometric invariants unique to the sparse-input setting, the weakly unobservable and strongly reachable subspace arrangements. By means of a concrete example, we demonstrate the utility of these characterizations. We conclude by discussing extensions and applications of this framework to several related problems in sparse control.}, author = {Poe, Kyle and Mallada, Enrique and Vidal, Rene}, booktitle = {63rd IEEE Conference on Decision and Control (CDC)}, grants = {CPS-2136324, Global-Centers-2330450}, month = {07}, pubstate = {accepted, submitted Mar 2024}, title = {Invertibility of Discrete-Time Linear Systems with Sparse Inputs}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-PMV.pdf}, year = {2024} } - Y. Shen, H. Sibai, and E. Mallada, “Generalized Barrier Functions: Integral Conditions & Recurrent Relaxations,” in 60th Allerton Conference on Communication, Control, and Computing, 2024, pp. 1-8.
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Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function $h$ that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set $h_\ge 0$. This methodology, however, does not prescribe a general method for finding the function $h$ that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for $h$ easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of $h$. Subsequently, we further replace the stringent invariance requirement on $h≥0$ with a more flexible concept known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the ($τ$-)recurrence of the super-level set $h_≥ 0$ is sufficient to guarantee the system’s safety.
@inproceedings{ssm2024allerton, abstract = {Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function $h$ that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set $h_\ge 0$. This methodology, however, does not prescribe a general method for finding the function $h$ that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for $h$ easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of $h$. Subsequently, we further replace the stringent invariance requirement on $h≥0$ with a more flexible concept known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the ($τ$-)recurrence of the super-level set $h_≥ 0$ is sufficient to guarantee the system's safety.}, author = {Shen, Yue and Sibai, Hussein and Mallada, Enrique}, booktitle = {60th Allerton Conference on Communication, Control, and Computing}, grants = {CPS-2136324, Global-Centers-2330450}, keywords = {Barrier Functions}, month = {07}, pages = {1-8}, pubstate = {accepted, submitted Jul 2024}, record = {submitted Jul 2024}, title = {Generalized Barrier Functions: Integral Conditions & Recurrent Relaxations}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-SSM.pdf}, year = {2024} } - T. Zheng, N. Loizou, P. You, and E. Mallada, “Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization,” in 63rd IEEE Conference on Decision and Control (CDC), 2024.
[BibTeX] [Abstract] [Download PDF]
Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA’s effectiveness in solving saddle point problems.
@inproceedings{zlym2024cdc, abstract = {Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA's effectiveness in solving saddle point problems.}, author = {Zheng, Tianqi and Loizou, Nicolas and You, Pengcheng and Mallada, Enrique}, booktitle = {63rd IEEE Conference on Decision and Control (CDC)}, grants = {CPS-2136324, Global-Centers-2330450}, month = {07}, note = {also in L-CSS}, pubstate = {accepted, submitted Mar 2024}, title = {Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization}, url = {https://mallada.ece.jhu.edu/pubs/2024-LCSS-ZLYM.pdf}, year = {2024} } - B. K. Poolla, Y. Lin, A. Bernstein, E. Mallada, and D. Groß, “Dynamic Shaping of Grid Response of Multi-Machine Multi-Inverter Systems Through Grid-Forming IBRs,” in PES General Meeting, 2024, pp. 1-5.
[BibTeX] [Abstract] [Download PDF]
We consider the problem of controlling the frequency response of weakly-coupled multi-machine multi-inverter low-inertia power systems via grid-forming inverter-based resources (IBRs). In contrast to existing methods, our approach relies on dividing the larger system into multiple strongly-coupled subsystems, without ignoring either the underlying network or approximating the subsystem response as an aggregate harmonic mean model. Rather, through a structured clustering and recursive dynamic shaping approach, the frequency response of the overall system to load perturbations is shaped appropriately. We demonstrate the proposed approach for a three-node triangular configuration and a small-scale radial network. Furthermore, previous synchronization analysis for heterogeneous systems requires the machines to satisfy certain proportionality property. In our approach, the effective transfer functions for each cluster can be tuned by the IBRs to satisfy such property, enabling us to apply the shaping control to systems with a wider range of heterogeneous machines.
@inproceedings{plbmg2024pesgm, abstract = {We consider the problem of controlling the frequency response of weakly-coupled multi-machine multi-inverter low-inertia power systems via grid-forming inverter-based resources (IBRs). In contrast to existing methods, our approach relies on dividing the larger system into multiple strongly-coupled subsystems, without ignoring either the underlying network or approximating the subsystem response as an aggregate harmonic mean model. Rather, through a structured clustering and recursive dynamic shaping approach, the frequency response of the overall system to load perturbations is shaped appropriately. We demonstrate the proposed approach for a three-node triangular configuration and a small-scale radial network. Furthermore, previous synchronization analysis for heterogeneous systems requires the machines to satisfy certain proportionality property. In our approach, the effective transfer functions for each cluster can be tuned by the IBRs to satisfy such property, enabling us to apply the shaping control to systems with a wider range of heterogeneous machines.}, author = {Poolla, Bala Kameshwar and Lin, Yashen and Bernstein, Andrey and Mallada, Enrique and Groß, Dominic}, booktitle = {PES General Meeting}, month = {06}, pages = {1-5}, pubstate = {presented}, record = {accepted Mar. 2024, submitted Nov. 2023}, title = {Dynamic Shaping of Grid Response of Multi-Machine Multi-Inverter Systems Through Grid-Forming IBRs}, url = {https://mallada.ece.jhu.edu/pubs/2024-PESGM-PLBMG.pdf}, year = {2024} } - Z. Siahaan, E. Mallada, and S. Geng, “Decentralized Stability Criteria for Grid-Forming Control in Inverter-Based Power Systems,” in PES General Meeting, 2024, pp. 1-5.
[BibTeX] [Abstract] [Download PDF]
This paper presents a decentralized stability analysis of power systems comprising grid-forming (GFM) inverters. We leverage a decentralized stability framework capable of ensuring the stability of the entire interconnection through individual assessments at each bus. The key novelty lies in incorporating voltage dynamics and their coupling with reactive power, in addition to the angle dynamics and their coupling with active power. We perform loop transformation to address the challenge posed by the non-Laplacian nature of the network Jacobian matrix in this case. This methodology is applied to characterize conditions on the droop gains of GFM controllers that can preserve system-wide stability. Our proposed stability criteria exhibit scalability and robustness, and can be extended to accommodate delays, variations in network conditions, and plug-and-play of new components in the network.
@inproceedings{smg2024pesgm, abstract = {This paper presents a decentralized stability analysis of power systems comprising grid-forming (GFM) inverters. We leverage a decentralized stability framework capable of ensuring the stability of the entire interconnection through individual assessments at each bus. The key novelty lies in incorporating voltage dynamics and their coupling with reactive power, in addition to the angle dynamics and their coupling with active power. We perform loop transformation to address the challenge posed by the non-Laplacian nature of the network Jacobian matrix in this case. This methodology is applied to characterize conditions on the droop gains of GFM controllers that can preserve system-wide stability. Our proposed stability criteria exhibit scalability and robustness, and can be extended to accommodate delays, variations in network conditions, and plug-and-play of new components in the network.}, author = {Siahaan, Zudika and Mallada, Enrique and Geng, Sijia}, booktitle = {PES General Meeting}, month = {06}, pages = {1-5}, pubstate = {presented}, record = {accepted Mar. 2024, submitted Nov. 2023}, title = {Decentralized Stability Criteria for Grid-Forming Control in Inverter-Based Power Systems}, url = {https://mallada.ece.jhu.edu/pubs/2024-PESGM-SMG.pdf}, year = {2024} } - H. Min, R. Vidal, and E. Mallada, “Early Neuron Alignment in Two-layer ReLU Networks with Small Initialization,” in International Conference on Representation Learning (ICLR), 2024.
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This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons’ directional dynamics allows us to provide an $\mathcalO(\fracłog n\sqrtμ)$ upper bound on the time it takes for all neurons to achieve good alignment with the input data, where $n$ is the number of data points and $μ$ measures how well the data are separated. After the early alignment phase, the loss converges to zero at a $\mathcalO(\frac1t)$ rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.
@inproceedings{mvm2024iclr, abstract = {This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons' directional dynamics allows us to provide an $\mathcalO(\fracłog n\sqrtμ)$ upper bound on the time it takes for all neurons to achieve good alignment with the input data, where $n$ is the number of data points and $μ$ measures how well the data are separated. After the early alignment phase, the loss converges to zero at a $\mathcalO(\frac1t)$ rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.}, author = {Min, Hancheng and Vidal, Rene and Mallada, Enrique}, booktitle = {International Conference on Representation Learning (ICLR)}, grants = {CAREER-1752362}, month = {05}, record = {published, accepted Jan 2024, submitted Sep 2023}, title = {Early Neuron Alignment in Two-layer ReLU Networks with Small Initialization}, url = {https://mallada.ece.jhu.edu/pubs/2024-ICLR-MVM.pdf}, year = {2024} } - H. Sibai and E. Mallada, “Recurrence of Nonlinear Control Systems: Entropy and Bit Rates,” in Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control (HSCC), New York, NY, USA, 2024, pp. 1-9. doi:https://doi.org/10.1145/3641513.3650121
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In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also present an algorithm for achieving recurrence asymptotically.
@inproceedings{sm2024hscc, abstract = {In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also present an algorithm for achieving recurrence asymptotically.}, address = {New York, NY, USA}, author = {Sibai, Hussein and Mallada, Enrique}, booktitle = {Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control (HSCC)}, doi = {https://doi.org/10.1145/3641513.3650121}, grants = {CPS-2136324, Global-Centers-2330450}, month = {05}, number = {23}, pages = {1--9}, publisher = {Association for Computing Machinery}, record = {accepted Jan 2024, submitted Nov 2023}, series = {HSCC '24}, title = {Recurrence of Nonlinear Control Systems: Entropy and Bit Rates}, url = {https://mallada.ece.jhu.edu/pubs/2024-HSCC-SM.pdf}, year = {2024} } - K. Poe, E. Mallada, and R. Vidal, “Necessary and Sufficient Conditions for Simultaneous State and Input Recovery of Linear Systems with Sparse Inputs by $\ell_1$-Minimization,” in 62nd IEEE Conference on Decision and Control (CDC), 2023, pp. 6499-6506. doi:10.1109/CDC49753.2023.10383682
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The study of theoretical conditions for recovering sparse signals from compressive measurements has received a lot of attention in the research community. In parallel, there has been a great amount of work characterizing conditions for the recovery both the state and the input to a linear dynamical system (LDS), including a handful of results on recovering sparse inputs. However, existing sufficient conditions for recovering sparse inputs to an LDS are conservative and hard to interpret, while necessary and sufficient conditions have not yet appeared in the literature. In this work, we provide (1) the first characterization of necessary and sufficient conditions for the existence and uniqueness of sparse inputs to an LDS, (2) the first necessary and sufficient conditions for a linear program to recover both an unknown initial state and a sparse input, and (3) simple, interpretable recovery conditions in terms of the LDS parameters. We conclude with a numerical validation of these claims and discuss implications and future directions.
@inproceedings{pmv2023cdc, abstract = {The study of theoretical conditions for recovering sparse signals from compressive measurements has received a lot of attention in the research community. In parallel, there has been a great amount of work characterizing conditions for the recovery both the state and the input to a linear dynamical system (LDS), including a handful of results on recovering sparse inputs. However, existing sufficient conditions for recovering sparse inputs to an LDS are conservative and hard to interpret, while necessary and sufficient conditions have not yet appeared in the literature. In this work, we provide (1) the first characterization of necessary and sufficient conditions for the existence and uniqueness of sparse inputs to an LDS, (2) the first necessary and sufficient conditions for a linear program to recover both an unknown initial state and a sparse input, and (3) simple, interpretable recovery conditions in terms of the LDS parameters. We conclude with a numerical validation of these claims and discuss implications and future directions. }, author = {Poe, Kyle and Mallada, Enrique and Vidal, Rene}, booktitle = {62nd IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC49753.2023.10383682}, grants = {CPS-2136324,CAREER-1752362}, month = {12}, pages = {6499--6506}, record = {presented, accepted Jul 2023, submitted Mar 2023}, title = {Necessary and Sufficient Conditions for Simultaneous State and Input Recovery of Linear Systems with Sparse Inputs by $\ell_1$-Minimization}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-PMV.pdf}, year = {2023} } - R. Siegelmann, Y. Shen, F. Paganini, and E. Mallada, “A Recurrence-based Direct Method for Stability Analysis and GPU-based Verification of Non-monotonic Lyapunov Functions,” in 62nd IEEE Conference on Decision and Control (CDC), 2023, pp. 6665-6672. doi:10.1109/CDC49753.2023.10383373
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Lyapunov direct method is a powerful tool that provides a rigorous framework for stability analysis and control design for dynamical systems. A critical step that enables the application of the method is the existence of a Lyapunov function $V$—a function whose value monotonically decreases along the trajectories of the dynamical system. Unfortunately, finding a Lyapunov function is often tricky and requires ingenuity, domain knowledge, or significant computational power. At the core of this challenge is the fact that the method requires every sub-level set of $V$ ($V_łeq c$) to be forward invariant, thus implicitly coupling the geometry of $V_łeq c$ and the trajectories of the system. In this paper, we seek to disentangle this dependence by developing a direct method that substitutes the concept of invariance with a more flexible notion known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, the recurrence of level sub-level sets is sufficient to guarantee stability, asymptotic stability, and exponential stability. We further provide a GPU-based algorithm that can to verify whether $V$ satisfies such conditions up to an arbitrarily small subset of the equilibrium.
@inproceedings{sspm2023cdc, abstract = {Lyapunov direct method is a powerful tool that provides a rigorous framework for stability analysis and control design for dynamical systems. A critical step that enables the application of the method is the existence of a Lyapunov function $V$---a function whose value monotonically decreases along the trajectories of the dynamical system. Unfortunately, finding a Lyapunov function is often tricky and requires ingenuity, domain knowledge, or significant computational power. At the core of this challenge is the fact that the method requires every sub-level set of $V$ ($V_łeq c$) to be forward invariant, thus implicitly coupling the geometry of $V_łeq c$ and the trajectories of the system. In this paper, we seek to disentangle this dependence by developing a direct method that substitutes the concept of invariance with a more flexible notion known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, the recurrence of level sub-level sets is sufficient to guarantee stability, asymptotic stability, and exponential stability. We further provide a GPU-based algorithm that can to verify whether $V$ satisfies such conditions up to an arbitrarily small subset of the equilibrium.}, author = {Siegelmann, Roy and Shen, Yue and Paganini, Fernando and Mallada, Enrique}, booktitle = {62nd IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC49753.2023.10383373}, grants = {CPS-2136324, CAREER-1752362, EPICS-2330450}, month = {12}, organization = {IEEE}, pages = {6665--6672}, record = {presented, accepted Jul 2023, submitted Mar 2023}, title = {A Recurrence-based Direct Method for Stability Analysis and GPU-based Verification of Non-monotonic Lyapunov Functions}, url = {https://mallada.ece.jhu.edu/pubs/2023-CDC-SSPM.pdf}, year = {2023} } - A. Castellano, H. Min, J. Bazerque, and E. Mallada, “Learning safety critics via a non-contractive binary Bellman operator,” in 57th Asilomar Conference on Signals, Systems, and Computers, 2023, pp. 814-821. doi:10.1109/IEEECONF59524.2023.10476995
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The inability to naturally enforce safety in Reinforcement Learning (RL), with limited failures, is a core challenge impeding its use in real-world applications. One notion of safety of vast practical relevance is the ability to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing–except for a spurious solution–maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.
@inproceedings{cmbm2023asilomar, abstract = {The inability to naturally enforce safety in Reinforcement Learning (RL), with limited failures, is a core challenge impeding its use in real-world applications. One notion of safety of vast practical relevance is the ability to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing--except for a spurious solution--maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.}, author = {Castellano, Agustin and Min, Hancheng and Bazerque, Juan and Mallada, Enrique}, booktitle = {57th Asilomar Conference on Signals, Systems, and Computers}, doi = {10.1109/IEEECONF59524.2023.10476995}, grants = {CAREER-1752362, CPS-2136324, Global-Centers-2330450}, month = {11}, organization = {IEEE}, pages = {814--821}, record = {presented Nov 2023, accepted Sep 2023, submitted Apr 2023}, title = {Learning safety critics via a non-contractive binary Bellman operator}, url = {https://mallada.ece.jhu.edu/pubs/2023-Asilomar-CMBM.pdf}, year = {2023} } - H. Min and E. Mallada, “Spectral clustering and model reduction for weakly-connected coherent network systems,” in American Control Conference (ACC), 2023, pp. 2957-2962. doi:10.23919/ACC55779.2023.10156212
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We propose a novel model-reduction methodology for large-scale dynamic networks with tightly-connected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the network feedback. Then, a reduced network is built, where each node represents the aggregate dynamics of each coherent group, and the reduced network captures the dynamic coupling between the groups. Our approach is theoretically justified under a random graph setting. Finally, numerical experiments align with and validate our theoretical findings.
@inproceedings{mm2023acc, abstract = {We propose a novel model-reduction methodology for large-scale dynamic networks with tightly-connected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the network feedback. Then, a reduced network is built, where each node represents the aggregate dynamics of each coherent group, and the reduced network captures the dynamic coupling between the groups. Our approach is theoretically justified under a random graph setting. Finally, numerical experiments align with and validate our theoretical findings.}, author = {Min, Hancheng and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC55779.2023.10156212}, grants = {CAREER-1752362, CPS-2136324}, month = {6}, pages = {2957-2962}, record = {presented Jun 2023, accepted Jan 2023, submitted Sep 2022}, title = {Spectral clustering and model reduction for weakly-connected coherent network systems}, url = {https://mallada.ece.jhu.edu/pubs/2023-ACC-MM.pdf}, year = {2023} } - H. Min and E. Mallada, “Learning Coherent Clusters in Weakly-Connected Network Systems,” in Proceedings of The 5th Annual Learning for Dynamics and Control Conference, 2023, pp. 1167-1179.
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We propose a structure-preserving model-reduction methodology for large-scale dynamic networks with tightly-connected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the network feedback. Then, a reduced network is built, where each node represents the aggregate dynamics of each coherent group, and the reduced network captures the dynamic coupling between the groups. We provide an upper bound on the approximation error when the network graph is randomly generated from a weight stochastic block model. Finally, numerical experiments align with and validate our theoretical findings.
@inproceedings{mm2023l4dc, abstract = {We propose a structure-preserving model-reduction methodology for large-scale dynamic networks with tightly-connected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the network feedback. Then, a reduced network is built, where each node represents the aggregate dynamics of each coherent group, and the reduced network captures the dynamic coupling between the groups. We provide an upper bound on the approximation error when the network graph is randomly generated from a weight stochastic block model. Finally, numerical experiments align with and validate our theoretical findings.}, author = {Min, Hancheng and Mallada, Enrique}, booktitle = {Proceedings of The 5th Annual Learning for Dynamics and Control Conference}, editor = {Matni, Nikolai and Morari, Manfred and Pappas, George J.}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {6}, pages = {1167--1179}, publisher = {PMLR}, record = {presented, accepted Mar 2022, submitted Nov 2022}, series = {Proceedings of Machine Learning Research}, title = {Learning Coherent Clusters in Weakly-Connected Network Systems}, url = {https://mallada.ece.jhu.edu/pubs/2023-L4DC-MM.pdf}, volume = {211}, year = {2023} } - H. Min, R. Vidal, and E. Mallada, “On the Convergence of Gradient Flow on Multi-layer Linear Models,” in International Conference on Machine Learning (ICML), 2023, pp. 1-8.
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Much of the theory for classical sparse recovery is based on conditions on the dictionary that are both necessary and sufficient (e.g., nullspace property) or only sufficient (e.g., incoherence and restricted isometry). In contrast, much of the theory for subspace-preserving recovery, the theoretical underpinnings for sparse subspace classification and clustering methods, is based on conditions on the subspaces and the data that are only sufficient (e.g., subspace incoherence and data inner-radius). This paper derives a necessary and sufficient condition for subspace-preserving recovery that is inspired by the classical nullspace property. Based on this novel condition, called here subspace nullspace property, we derive equivalent characterizations that either admit a clear geometric interpretation, relating data distribution and subspace separation to the recovery success, or can be verified using a finite set of extreme points of a properly defined set. We further exploit these characterizations to derive new sufficient conditions, based on inner-radius and outer-radius measures and dual bounds, that generalize existing conditions, while preserving the geometric interpretations. These results fill an important gap in the subspace-preserving recovery literature
@inproceedings{mvm2023icml, abstract = {Much of the theory for classical sparse recovery is based on conditions on the dictionary that are both necessary and sufficient (e.g., nullspace property) or only sufficient (e.g., incoherence and restricted isometry). In contrast, much of the theory for subspace-preserving recovery, the theoretical underpinnings for sparse subspace classification and clustering methods, is based on conditions on the subspaces and the data that are only sufficient (e.g., subspace incoherence and data inner-radius). This paper derives a necessary and sufficient condition for subspace-preserving recovery that is inspired by the classical nullspace property. Based on this novel condition, called here subspace nullspace property, we derive equivalent characterizations that either admit a clear geometric interpretation, relating data distribution and subspace separation to the recovery success, or can be verified using a finite set of extreme points of a properly defined set. We further exploit these characterizations to derive new sufficient conditions, based on inner-radius and outer-radius measures and dual bounds, that generalize existing conditions, while preserving the geometric interpretations. These results fill an important gap in the subspace-preserving recovery literature}, author = {Min, Hancheng and Vidal, Rene and Mallada, Enrique}, booktitle = {International Conference on Machine Learning (ICML)}, grants = {TRIPODS-1934979, CAREER-1752362}, month = {4}, pages = {1-8}, pubstate = {accepted, submitted Jan 2023}, title = {On the Convergence of Gradient Flow on Multi-layer Linear Models}, url = {https://mallada.ece.jhu.edu/pubs/2023-ICML-MVM.pdf}, year = {2023} } - B. K. Poolla, Y. Lin, A. Bernstein, E. Mallada, and D. Groß, “Frequency shaping control for weakly-coupled grid-forming IBRs,” in American Control Conference (ACC), 2023, pp. 1-6.
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We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.
@inproceedings{plbmg2023acc, abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.}, author = {Poolla, Bala Kameshwar and Lin, Yashen and Bernstein, Andrey and Mallada, Enrique and Groß, Dominic}, booktitle = {American Control Conference (ACC)}, grants = {CAREER-1752362, CPS-2136324}, month = {1}, pages = {1-6}, record = {presented Jun 2023, accepted Jan 2023, submitted Sep 2022}, title = {Frequency shaping control for weakly-coupled grid-forming IBRs}, url = {https://mallada.ece.jhu.edu/pubs/2023-ACC-PLBMG.pdf}, year = {2023} } - Z. Xu, H. Min, S. Tarmoun, E. Mallada, and R. Vidal, “Linear Convergence of Gradient Descent For Overparametrized Finite Width Two-Layer Linear Networks with General Initialization,” in International Conference on Artificial Intelligence and Statistics (AISTATS), 2023, pp. 2262-2284.
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Recent theoretical analyses of the convergence of gradient descent (GD) to a global minimum for over-parametrized neural networks make strong assumptions on the step size (infinitesimal), the hidden-layer width (infinite), or the initialization (spectral, balanced). In this work, we relax these assumptions and derive a linear convergence rate for two-layer linear networks trained using GD on the squared loss in the case of finite step size, finite width and general initialization. Despite the generality of our analysis, our rate estimates are significantly tighter than those of prior work. Moreover, we provide a time-varying step size rule that monotonically improves the convergence rate as the loss function decreases to zero. Numerical experiments validate our findings.
@inproceedings{xmtmv2023aistats, abstract = {Recent theoretical analyses of the convergence of gradient descent (GD) to a global minimum for over-parametrized neural networks make strong assumptions on the step size (infinitesimal), the hidden-layer width (infinite), or the initialization (spectral, balanced). In this work, we relax these assumptions and derive a linear convergence rate for two-layer linear networks trained using GD on the squared loss in the case of finite step size, finite width and general initialization. Despite the generality of our analysis, our rate estimates are significantly tighter than those of prior work. Moreover, we provide a time-varying step size rule that monotonically improves the convergence rate as the loss function decreases to zero. Numerical experiments validate our findings.}, author = {Xu, Ziqing and Min, Hancheng and Tarmoun, Salma and Mallada, Enrique and Vidal, Rene}, booktitle = {International Conference on Artificial Intelligence and Statistics (AISTATS)}, editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem}, grants = {No Grant}, month = {4}, pages = {2262--2284}, publisher = {PMLR}, record = {published, accepted Jan 2023, submitted Oct 2022}, series = {Proceedings of Machine Learning Research}, title = {Linear Convergence of Gradient Descent For Overparametrized Finite Width Two-Layer Linear Networks with General Initialization}, url = {https://mallada.ece.jhu.edu/pubs/2023-AISTATS-XMTMV.pdf}, volume = {206}, year = {2023} } - Y. Shen, M. Bichuch, and E. Mallada, “Model-free Learning of Regions of Attraction via Recurrent Sets,” in 61st IEEE Conference on Decision and Control (CDC), 2022, pp. 4714-4719. doi:10.1109/CDC51059.2022.9993280
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We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be $τ$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $τ$ seconds (resp. $k$ steps). We show that under mild assumptions a $τ$-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allow them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.
@inproceedings{sbm2022cdc, abstract = {We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be $τ$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $τ$ seconds (resp. $k$ steps). We show that under mild assumptions a $τ$-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allow them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.}, author = {Shen, Yue and Bichuch, Maxim and Mallada, Enrique}, booktitle = {61st IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC51059.2022.9993280}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {12}, pages = {4714-4719}, record = {presented Dec. 2022, accepted Sep 2022, submitted Mar 2022}, title = {Model-free Learning of Regions of Attraction via Recurrent Sets}, url = {https://mallada.ece.jhu.edu/pubs/2022-CDC-SBM.pdf}, year = {2022} } - T. Zheng, P. You, and E. Mallada, “Constrained Reinforcement Learning via Dissipative Saddle Flow Dynamics,” in 56th Asilomar Conference on Signals, Systems, and Computers, 2022, pp. 1362-1366. doi:10.1109/IEEECONF56349.2022.10052060
[BibTeX] [Abstract] [Download PDF]
We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be $τ$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $τ$ seconds (resp. $k$ steps). We show that under mild assumptions a $τ$-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allow them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.
@inproceedings{zym2022asilomar, abstract = {We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be $τ$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $τ$ seconds (resp. $k$ steps). We show that under mild assumptions a $τ$-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allow them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.}, author = {Zheng, Tianqi and You, Pengcheng and Mallada, Enrique}, booktitle = {56th Asilomar Conference on Signals, Systems, and Computers}, doi = {10.1109/IEEECONF56349.2022.10052060}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {12}, pages = {1362-1366}, pubstate = {presented, accepted Sep 2022, submitted Apr 2022}, title = {Constrained Reinforcement Learning via Dissipative Saddle Flow Dynamics}, url = {https://mallada.ece.jhu.edu/pubs/2022-Asilomar-ZYM.pdf}, year = {2022} } - R. K. Bansal, Y. Chen, P. You, and E. Mallada, “Equilibrium Analysis of Electricity Markets with Day-Ahead Market Power Mitigation and Real-Time Intercept Bidding,” in Proceedings of the Thirteenth ACM International Conference on Future Energy Systems (e-Energy), 2022, pp. 47-62. doi:https://doi.org/10.1145/3538637.3538839
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Electricity markets are cleared by a two-stage, sequential process consisting of a forward (day-ahead) market and a spot (real-time) market. While their design goal is to achieve efficiency, the lack of sufficient competition introduces many opportunities for price manipulation. To discourage this phenomenon, some Independent System Operators (ISOs) mandate generators to submit (approximately) truthful bids in the day-ahead market. However, without fully accounting for all participants’ incentives (generators and loads), the application of such a mandate may lead to unintended consequences. In this paper, we model and study the interactions of generators and inelastic loads in a two-stage settlement where generators are required to bid truthfully in the day-ahead market. We show that such mandate, when accounting for generator and load incentives, leads to a generalized Stackelberg-Nash game where load decisions (leaders) are performed in day-ahead market and generator decisions (followers) are relegated to the real-time market. Furthermore, the use of conventional supply function bidding for generators in real-time, does not guarantee the existence of a Nash equilibrium. This motivates the use of intercept bidding, as an alternative bidding mechanism for generators in the real-time market. An equilibrium analysis in this setting, leads to a closed-form solution that unveils several insights. Particularly, it shows that, unlike standard two-stage markets, loads are the winners of the competition in the sense that their aggregate payments are less than that of the competitive equilibrium. Moreover, heterogeneity in generators cost has the unintended effect of mitigating loads market power. Numerical studies validate and further illustrate these insights.
@inproceedings{bcym2022e-energy, abstract = {Electricity markets are cleared by a two-stage, sequential process consisting of a forward (day-ahead) market and a spot (real-time) market. While their design goal is to achieve efficiency, the lack of sufficient competition introduces many opportunities for price manipulation. To discourage this phenomenon, some Independent System Operators (ISOs) mandate generators to submit (approximately) truthful bids in the day-ahead market. However, without fully accounting for all participants' incentives (generators and loads), the application of such a mandate may lead to unintended consequences. In this paper, we model and study the interactions of generators and inelastic loads in a two-stage settlement where generators are required to bid truthfully in the day-ahead market. We show that such mandate, when accounting for generator and load incentives, leads to a generalized Stackelberg-Nash game where load decisions (leaders) are performed in day-ahead market and generator decisions (followers) are relegated to the real-time market. Furthermore, the use of conventional supply function bidding for generators in real-time, does not guarantee the existence of a Nash equilibrium. This motivates the use of intercept bidding, as an alternative bidding mechanism for generators in the real-time market. An equilibrium analysis in this setting, leads to a closed-form solution that unveils several insights. Particularly, it shows that, unlike standard two-stage markets, loads are the winners of the competition in the sense that their aggregate payments are less than that of the competitive equilibrium. Moreover, heterogeneity in generators cost has the unintended effect of mitigating loads market power. Numerical studies validate and further illustrate these insights.}, author = {Bansal, Rajni Kant and Chen, Yue and You, Pengcheng and Mallada, Enrique}, booktitle = {Proceedings of the Thirteenth ACM International Conference on Future Energy Systems (e-Energy)}, doi = {https://doi.org/10.1145/3538637.3538839}, grants = {CAREER-1752362,EPCN-1711188,CPS-2136324}, month = {6}, pages = {47--62}, record = {published, accepted Jun 2022, submitted Feb 2022.}, title = {Equilibrium Analysis of Electricity Markets with Day-Ahead Market Power Mitigation and Real-Time Intercept Bidding}, url = {https://mallada.ece.jhu.edu/pubs/2022-e-Energy-BCYM.pdf}, year = {2022} } - R. K. Bansal, P. You, D. F. Gayme, and E. Mallada, “A Market Mechanism for Truthful Bidding with Energy Storage,” in Power Systems Computation Conference (PSCC), 2022, pp. 1-9.
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This paper proposes a market mechanism for multi-interval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energy cycling functions that map prices to cycle depths. The resulting market-clearing process–implemented via convex programming–yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. We illustrate the benefits of our solution by comparing the competitive equilibrium of the resulting mechanism to that of an alternative solution that uses prosumer-based bids. Our solution shows several advantages over the prosumer-based approach. It does not require a priori price estimation. It also incentivizes participants to reveal their truthful costs, thus leading to an efficient, competitive equilibrium. Numerical experiments using New York Independent System Operator (NYISO) data validate our findings.
@inproceedings{bygm2022pscc, abstract = {This paper proposes a market mechanism for multi-interval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energy cycling functions that map prices to cycle depths. The resulting market-clearing process--implemented via convex programming--yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. We illustrate the benefits of our solution by comparing the competitive equilibrium of the resulting mechanism to that of an alternative solution that uses prosumer-based bids. Our solution shows several advantages over the prosumer-based approach. It does not require a priori price estimation. It also incentivizes participants to reveal their truthful costs, thus leading to an efficient, competitive equilibrium. Numerical experiments using New York Independent System Operator (NYISO) data validate our findings.}, author = {Bansal, Rajni Kant and You, Pengcheng and Gayme, Dennice F. and Mallada, Enrique}, booktitle = {Power Systems Computation Conference (PSCC)}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {7}, pages = {1-9}, record = {presented, accepted Feb. 2022, submitted Oct. 2021}, title = {A Market Mechanism for Truthful Bidding with Energy Storage}, url = {https://mallada.ece.jhu.edu/pubs/2022-PSCC-BYGM.pdf}, year = {2022} } - A. Castellano, H. Min, J. Bazerque, and E. Mallada, “Reinforcement Learning with Almost Sure Constraints,” in Proceedings of The 4th Annual Learning for Dynamics and Control Conference, 2022, pp. 559-570.
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In this work we address the problem of finding feasible policies for Constrained Markov Decision Processes under probability one constraints. We argue that stationary policies are not sufficient for solving this problem, and that a rich class of policies can be found by endowing the controller with a scalar quantity, so called budget, that tracks how close the agent is to violating the constraint. We show that the minimal budget required to act safely can be obtained as the smallest fixed point of a Bellman-like operator, for which we analyze its convergence properties. We also show how to learn this quantity when the true kernel of the Markov decision process is not known, while providing sample-complexity bounds. The utility of knowing this minimal budget relies in that it can aid in the search of optimal or near-optimal policies by shrinking down the region of the state space the agent must navigate. Simulations illustrate the different nature of probability one constraints against the typically used constraints in expectation.
@inproceedings{cmbm2022l4dc, abstract = {In this work we address the problem of finding feasible policies for Constrained Markov Decision Processes under probability one constraints. We argue that stationary policies are not sufficient for solving this problem, and that a rich class of policies can be found by endowing the controller with a scalar quantity, so called budget, that tracks how close the agent is to violating the constraint. We show that the minimal budget required to act safely can be obtained as the smallest fixed point of a Bellman-like operator, for which we analyze its convergence properties. We also show how to learn this quantity when the true kernel of the Markov decision process is not known, while providing sample-complexity bounds. The utility of knowing this minimal budget relies in that it can aid in the search of optimal or near-optimal policies by shrinking down the region of the state space the agent must navigate. Simulations illustrate the different nature of probability one constraints against the typically used constraints in expectation.}, author = {Castellano, Agustin and Min, Hancheng and Bazerque, Juan and Mallada, Enrique}, booktitle = {Proceedings of The 4th Annual Learning for Dynamics and Control Conference}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {6}, organization = {PMLR}, pages = {559--570}, pubstate = {presented, Feb 2022 accepted, submitted Dec 2021}, series = {Proceedings of Machine Learning Research}, title = {Reinforcement Learning with Almost Sure Constraints}, url = {https://mallada.ece.jhu.edu/pubs/2022-L4DC-CMBM.pdf}, volume = {168}, year = {2022} } - J. Guthrie, M. Kobilarov, and E. Mallada, “Closed-Form Minkowski Sum Approximations for Efficient Optimization-Based Collision Avoidance,” in American Control Conference (ACC), 2022, pp. 3857-3864. doi:10.23919/ACC53348.2022.9867524
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Motion planning methods for autonomous systems based on nonlinear programming offer great flexibility in incorporating various dynamics, objectives, and constraints. One limitation of such tools is the difficulty of efficiently representing obstacle avoidance conditions for non-trivial shapes. For example, it is possible to define collision avoidance constraints suitable for nonlinear programming solvers in the canonical setting of a circular robot navigating around $M$ convex polytopes over $N$ time steps. However, it requires introducing $(2+L)MN$ additional constraints and $LMN$ additional variables, with $L$ being the number of halfplanes per polytope, leading to larger nonlinear programs with slower and less reliable solving time. In this paper, we overcome this issue by building closed-form representations of the collision avoidance conditions by outer-approximating the Minkowski sum conditions for collision. Our solution requires only $MN$ constraints (and no additional variables), leading to a smaller nonlinear program. On motion planning problems for an autonomous car and quadcopter in cluttered environments, we achieve speedups of 4.0x and 10x respectively with significantly less variance in solve times and negligible impact on performance arising from the use of outer approximations.
@inproceedings{gkm2022acc, abstract = {Motion planning methods for autonomous systems based on nonlinear programming offer great flexibility in incorporating various dynamics, objectives, and constraints. One limitation of such tools is the difficulty of efficiently representing obstacle avoidance conditions for non-trivial shapes. For example, it is possible to define collision avoidance constraints suitable for nonlinear programming solvers in the canonical setting of a circular robot navigating around $M$ convex polytopes over $N$ time steps. However, it requires introducing $(2+L)MN$ additional constraints and $LMN$ additional variables, with $L$ being the number of halfplanes per polytope, leading to larger nonlinear programs with slower and less reliable solving time. In this paper, we overcome this issue by building closed-form representations of the collision avoidance conditions by outer-approximating the Minkowski sum conditions for collision. Our solution requires only $MN$ constraints (and no additional variables), leading to a smaller nonlinear program. On motion planning problems for an autonomous car and quadcopter in cluttered environments, we achieve speedups of 4.0x and 10x respectively with significantly less variance in solve times and negligible impact on performance arising from the use of outer approximations.}, author = {Guthrie, James and Kobilarov, Marin and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC53348.2022.9867524}, grants = {CAREER-1752362, CPS-2136324, TRIPODS-1934979}, month = {6}, pages = {3857-3864}, pubstate = {presented}, record = {accepted Feb. 2022, submitted Oct. 2021}, title = {Closed-Form Minkowski Sum Approximations for Efficient Optimization-Based Collision Avoidance}, url = {https://mallada.ece.jhu.edu/pubs/2022-ACC-GKM.pdf}, year = {2022} } - T. Zheng, J. Guthrie, and E. Mallada, “Inner Approximations of the Positive-Semidefinite Cone via Grassmannian Packings,” in 60th IEEE Conference on Decision and Control (CDC), 2021, pp. 981-986. doi:10.1109/CDC45484.2021.9682923
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We investigate the problem of finding tight inner approximations of large dimensional positive semidefinite (PSD) cones. To solve this problem, we develop a novel decomposition framework of the PSD cone by means of conical combinations of smaller dimensional sub-cones. We show that many inner approximation techniques could be summarized within this framework, including the set of (scaled) diagonally dominant matrices, Factor-width k matrices, and Chordal Sparse matrices. Furthermore, we provide a more flexible family of inner approximations of the PSD cone, where we aim to arrange the sub-cones so that they are maximally separated from each other. In doing so, these approximations tend to occupy large fractions of the volume of the PSD cone. The proposed approach is connected to a classical packing problem in Riemannian Geometry. Precisely, we show that the problem of finding maximally distant sub-cones in an ambient PSD cone is equivalent to the problem of packing sub-spaces in a Grassmannian Manifold. We further leverage the existing computational methods for constructing packings in Grassmannian manifolds to build tighter approximations of the PSD cone. Numerical experiments show how the proposed framework can balance accuracy and computational complexity, to efficiently solve positive-semidefinite programs.
@inproceedings{zgm2021cdc, abstract = {We investigate the problem of finding tight inner approximations of large dimensional positive semidefinite (PSD) cones. To solve this problem, we develop a novel decomposition framework of the PSD cone by means of conical combinations of smaller dimensional sub-cones. We show that many inner approximation techniques could be summarized within this framework, including the set of (scaled) diagonally dominant matrices, Factor-width k matrices, and Chordal Sparse matrices. Furthermore, we provide a more flexible family of inner approximations of the PSD cone, where we aim to arrange the sub-cones so that they are maximally separated from each other. In doing so, these approximations tend to occupy large fractions of the volume of the PSD cone. The proposed approach is connected to a classical packing problem in Riemannian Geometry. Precisely, we show that the problem of finding maximally distant sub-cones in an ambient PSD cone is equivalent to the problem of packing sub-spaces in a Grassmannian Manifold. We further leverage the existing computational methods for constructing packings in Grassmannian manifolds to build tighter approximations of the PSD cone. Numerical experiments show how the proposed framework can balance accuracy and computational complexity, to efficiently solve positive-semidefinite programs.}, author = {Zheng, Tianqi and Guthrie, James and Mallada, Enrique}, booktitle = {60th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC45484.2021.9682923}, grants = {EPCN-1711188, CAREER-1752362, AMPS-1736448, TRIPODS-1934979}, month = {12}, pages = {981-986}, record = {presented Dec. 2022, accepted Jul. 2021, submitted Mar. 2021}, title = {Inner Approximations of the Positive-Semidefinite Cone via Grassmannian Packings}, url = {https://mallada.ece.jhu.edu/pubs/2021-CDC-ZGM.pdf}, year = {2021} } - M. D. Kaba, C. You, D. R. Robinson, E. Mallada, and R. Vidal, “Characterization of Subspace-Preserving Recovery by a Nullspace Property,” in International Conference on Machine Learning (ICML), 2021, pp. 5180-5188.
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Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co-optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.
@inproceedings{kcrmv2021icml, abstract = {Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co-optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.}, author = {Kaba, Mustafa Devrim and You, Chong and Robinson, Daniel R. and Mallada, Enrique and Vidal, Rene}, booktitle = {International Conference on Machine Learning (ICML)}, grants = {CAREER-1752362;TRIPODS-1934979;CPS-2136324}, month = {11}, note = {(21.5$%$ acceptance)}, pages = {5180--5188}, publisher = {PMLR}, record = {accepted May 2021}, series = {Proceedings of Machine Learning Research}, title = {Characterization of Subspace-Preserving Recovery by a Nullspace Property}, url = {https://mallada.ece.jhu.edu/pubs/2021-ICML-KCRMV.pdf}, volume = {139}, year = {2021} } - R. K. Bansal, P. You, D. F. Gayme, and E. Mallada, “Storage Degradation Aware Economic Dispatch,” in American Control Conference (ACC), 2021, pp. 589-595. doi:10.23919/ACC50511.2021.9482838
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In this paper, we formulate a cycling cost aware economic dispatch problem that co-optimizes generation and storage dispatch while taking into account cycle based storage degradation cost. Our approach exploits the Rainflow cycle counting algorithm to quantify storage degradation for each charging and discharging half-cycle based on its depth. We show that the dispatch is optimal for individual participants in the sense that it maximizes the profit of generators and storage units, under price taking assumptions. We further provide a condition under which the optimal storage response is unique for given market clearing prices. Simulations using data from the New York Independent System Operator (NYISO) illustrate the optimization framework. In particular, they show that the generation-centric dispatch that does not account for storage degradation is insufficient to guarantee storage profitability.
@inproceedings{bygm2021acc, abstract = {In this paper, we formulate a cycling cost aware economic dispatch problem that co-optimizes generation and storage dispatch while taking into account cycle based storage degradation cost. Our approach exploits the Rainflow cycle counting algorithm to quantify storage degradation for each charging and discharging half-cycle based on its depth. We show that the dispatch is optimal for individual participants in the sense that it maximizes the profit of generators and storage units, under price taking assumptions. We further provide a condition under which the optimal storage response is unique for given market clearing prices. Simulations using data from the New York Independent System Operator (NYISO) illustrate the optimization framework. In particular, they show that the generation-centric dispatch that does not account for storage degradation is insufficient to guarantee storage profitability. }, author = {Bansal, Rajni Kant and You, Pengcheng and Gayme, Dennice F. and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC50511.2021.9482838}, grants = {CPS-1544771, EPCN-1711188, CAREER-1752362, TRIPODS-1934979}, month = {5}, pages = {589-595}, record = {submitted Sep. 2020, accepted Jan. 2021}, title = {Storage Degradation Aware Economic Dispatch}, url = {https://mallada.ece.jhu.edu/pubs/2021-ACC-BYGM.pdf}, year = {2021} } - A. Castellano, J. Bazerque, and E. Mallada, “Learning to be safe, in finite time,” in American Control Conference (ACC), 2021, pp. 909-916. doi:10.23919/ACC50511.2021.9482829
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This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to relax its optimality requirements mildly. We focus on the canonical multi-armed bandit problem and seek to study the exploration-preservation trade-off intrinsic within safe learning. More precisely, by defining a handicap metric that counts the number of unsafe actions, we provide an algorithm for discarding unsafe machines (or actions), with probability one, that achieves constant handicap. Our algorithm is rooted in the classical sequential probability ratio test, redefined here for continuing tasks. Under standard assumptions on sufficient exploration, our rule provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. Our decision rule can wrap around any other algorithm to optimize a specific auxiliary goal since it provides a safe environment to search for (approximately) optimal policies. Simulations corroborate our theoretical findings and further illustrate the aforementioned trade-offs.
@inproceedings{cbm2021acc, abstract = {This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to relax its optimality requirements mildly. We focus on the canonical multi-armed bandit problem and seek to study the exploration-preservation trade-off intrinsic within safe learning. More precisely, by defining a handicap metric that counts the number of unsafe actions, we provide an algorithm for discarding unsafe machines (or actions), with probability one, that achieves constant handicap. Our algorithm is rooted in the classical sequential probability ratio test, redefined here for continuing tasks. Under standard assumptions on sufficient exploration, our rule provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. Our decision rule can wrap around any other algorithm to optimize a specific auxiliary goal since it provides a safe environment to search for (approximately) optimal policies. Simulations corroborate our theoretical findings and further illustrate the aforementioned trade-offs.}, author = {Castellano, Agustin and Bazerque, Juan and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC50511.2021.9482829}, grants = {CPS-1544771, CAREER-1752362, TRIPODS-1934979}, month = {5}, pages = {909-916}, record = {submitted Sep. 2020, accepted Jan. 2021}, title = {Learning to be safe, in finite time}, url = {https://mallada.ece.jhu.edu/pubs/2021-ACC-CBM.pdf}, year = {2021} } - J. Guthrie and E. Mallada, “Outer Approximations of Minkowski Operations on Complex Sets via Sum-of-Squares Optimization,” in American Control Conference (ACC), 2021, pp. 2367-2373. doi:10.23919/ACC50511.2021.9482940
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We study the problem of finding closed-form outerapproximations of Minkowski sums and products of sets inthe complex plane. Using polar coordinates, we pose this asan optimization problem in which we find a pair of contoursthat give lower and upper bounds on the radial distance ata given angle. Through a series of variable transformationswe rewrite this as a sum-of-squares optimization problem.Numerical examples are given to demonstrate the performance.
@inproceedings{gm2021acc, abstract = {We study the problem of finding closed-form outerapproximations of Minkowski sums and products of sets inthe complex plane. Using polar coordinates, we pose this asan optimization problem in which we find a pair of contoursthat give lower and upper bounds on the radial distance ata given angle. Through a series of variable transformationswe rewrite this as a sum-of-squares optimization problem.Numerical examples are given to demonstrate the performance.}, author = {Guthrie, James and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC50511.2021.9482940}, grants = {CPS-1544771, EPCN-1711188, CAREER-1752362, TRIPODS-1934979}, month = {5}, pages = {2367-2373}, record = {submitted Sep. 2020, accepted Jan. 2021}, title = {Outer Approximations of Minkowski Operations on Complex Sets via Sum-of-Squares Optimization}, url = {https://mallada.ece.jhu.edu/pubs/2021-ACC-GM.pdf}, year = {2021} } - Y. Jiang, A. Bernstein, P. Vorobev, and E. Mallada, “Grid-forming frequency shaping control in low inertia power systems,” in American Control Conference (ACC), 2021, pp. 4184-4189. doi:10.23919/ACC50511.2021.9482678
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We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.
@inproceedings{jbvm2021acc, abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.}, author = {Jiang, Yan and Bernstein, Andrey and Vorobev, Petr and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC50511.2021.9482678}, grants = {CAREER-1752362, AMPS-1736448, TRIPODS-1934979, EPCN-1711188}, month = {5}, pages = {4184-4189}, record = {submitted Sep. 2020, accepted Jan. 2021}, title = {Grid-forming frequency shaping control in low inertia power systems}, url = {https://mallada.ece.jhu.edu/pubs/2021-ACC-JBVM.pdf}, year = {2021} } - H. Min, F. Paganini, and E. Mallada, “Accurate Reduced Order Models for Coherent Heterogeneous Generators,” in American Control Conference (ACC), 2021, pp. 570-575. doi:10.23919/ACC50511.2021.9483031
[BibTeX] [Abstract] [Download PDF]
We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.
@inproceedings{mpm2021acc, abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.}, author = {Min, Hancheng and Paganini, Fernando and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC50511.2021.9483031}, grants = {CAREER-1752362, CPS-1544771, ENERGISE-DE-EE0008006, AMPS-1736448, TRIPODS-1934979, EPCN-1711188, ARO-W911NF-17-1-0092}, month = {5}, pages = {570-575}, record = {submitted Sep. 2020, accepted Jan. 2021}, title = {Accurate Reduced Order Models for Coherent Heterogeneous Generators}, url = {https://mallada.ece.jhu.edu/pubs/2021-ACC-MPM.pdf}, year = {2021} } - H. Min, S. Tarmoun, R. Vidal, and E. Mallada, “On the Explicit Role of Initialization on the Convergence and Implicit Bias of Overparametrized Linear Networks,” in International Conference on Machine Learning (ICML), 2021, pp. 7760-7768.
[BibTeX] [Abstract] [Download PDF]
Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.
@inproceedings{mtvm2021icml, abstract = {Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution. }, author = {Min, Hancheng and Tarmoun, Salma and Vidal, Rene and Mallada, Enrique}, booktitle = {International Conference on Machine Learning (ICML)}, grants = {TRIPODS-1934979, CAREER-1752362, AMPS-1736448}, month = {7}, note = {(21.5$%$ acceptance)}, pages = {7760--7768}, publisher = {PMLR}, record = {accepted May 2021}, series = {Proceedings of Machine Learning Research}, title = {On the Explicit Role of Initialization on the Convergence and Implicit Bias of Overparametrized Linear Networks}, url = {https://mallada.ece.jhu.edu/pubs/2021-ICML-MTVM.pdf}, volume = {139}, year = {2021} } - P. You and E. Mallada, “Saddle Flow Dynamics: Observable Certificates and Separable Regularization,” in American Control Conference (ACC), 2021, pp. 4817-4823. doi:10.23919/ACC50511.2021.9483346
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This paper proposes a certificate, rooted in observability, for asymptotic convergence of saddle flow dynamics of convex-concave functions to a saddle point. This observable certificate directly bridges the gap between the invariant set and the equilibrium set in a LaSalle argument, and generalizes conventional conditions such as strict convexity-concavity and proximal regularization. We further build upon this certificate to propose a separable regularization method for saddle flow dynamics that makes minimal requirements on convexityconcavity and yet still guarantees asymptotic convergence to a saddle point. Our results generalize to saddle flow dynamics with projections on the vector field and have an immediate application as a distributed solution to linear programs.
@inproceedings{ym2021acc, abstract = {This paper proposes a certificate, rooted in observability, for asymptotic convergence of saddle flow dynamics of convex-concave functions to a saddle point. This observable certificate directly bridges the gap between the invariant set and the equilibrium set in a LaSalle argument, and generalizes conventional conditions such as strict convexity-concavity and proximal regularization. We further build upon this certificate to propose a separable regularization method for saddle flow dynamics that makes minimal requirements on convexityconcavity and yet still guarantees asymptotic convergence to a saddle point. Our results generalize to saddle flow dynamics with projections on the vector field and have an immediate application as a distributed solution to linear programs.}, author = {You, Pengcheng and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC50511.2021.9483346}, grants = {CPS-1544771, EPCN-1711188, CAREER-1752362, TRIPODS-1934979}, month = {5}, pages = {4817-4823}, record = {submitted Sep. 2020, accepted Jan. 2021}, title = {Saddle Flow Dynamics: Observable Certificates and Separable Regularization}, url = {https://mallada.ece.jhu.edu/pubs/2021-ACC-YM.pdf}, year = {2021} } - J. Guthrie and E. Mallada, “Minimum-Time Charging of Energy Storage in Microgrids via Approximate Conic Relaxation,” in 19th IEEE European Control Conference (ECC), 2020, pp. 1713-1718. doi:10.23919/ECC51009.2020.9143992
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We study the problem of maximizing energy transfer to a load in a DC microgrid while respecting constraints on bus voltages and currents, and accounting for the impact of neighboring constant power loads. Both the objective and dynamics give rise to indefinite quadratic terms, resulting in a non-convex optimization problem. Through change of variables and relaxations we develop a closely related second-order cone program. The problem retains the same feasible set as the original problem but utilizes a linear approximation of the non-convex objective. We demonstrate how this can be used to design approximately optimal charging profiles for periodic pulsed loads in real time.
@inproceedings{gm2020ecc, abstract = {We study the problem of maximizing energy transfer to a load in a DC microgrid while respecting constraints on bus voltages and currents, and accounting for the impact of neighboring constant power loads. Both the objective and dynamics give rise to indefinite quadratic terms, resulting in a non-convex optimization problem. Through change of variables and relaxations we develop a closely related second-order cone program. The problem retains the same feasible set as the original problem but utilizes a linear approximation of the non-convex objective. We demonstrate how this can be used to design approximately optimal charging profiles for periodic pulsed loads in real time.}, author = {Guthrie, James and Mallada, Enrique}, booktitle = {19th IEEE European Control Conference (ECC)}, doi = {10.23919/ECC51009.2020.9143992}, grants = {CAREER-1752362, CPS-1544771, ENERGISE-DE-EE0008006, AMPS-1736448, TRIPODS-1934979, EPCN-1711188, ARO-W911NF-17-1-0092}, keywords = {Power Networks}, month = {5}, pages = {1713-1718}, title = {Minimum-Time Charging of Energy Storage in Microgrids via Approximate Conic Relaxation}, url = {https://mallada.ece.jhu.edu/pubs/2020-ECC-GM-b.pdf}, year = {2020} } - Y. Shen, M. Bichuch, and E. Mallada, “On the Value of Energy Storage in Generation Cost Reduction,” in 19th IEEE European Control Conference (ECC), 2020, pp. 1526-1532. doi:10.23919/ECC51009.2020.9143772
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This work seeks to quantify the benefits of using energy storage toward the reduction of the energy generation cost of a power system. A two-fold optimization framework is provided where the first optimization problem seeks to find the optimal storage schedule that minimizes operational costs. Since the operational cost depends on the storage capacity, a second optimization problem is then formulated with the aim of finding the optimal storage capacity to be deployed. Although, in general, these problems are difficult to solve, we provide a lower bound on the cost savings for a parametrized family of demand profiles. The optimization framework is numerically illustrated using real-world demand data from ISO New England. Numerical results show that energy storage can reduce energy generation costs by at least 2.5 percent.
@inproceedings{sbm2020ecc, abstract = {This work seeks to quantify the benefits of using energy storage toward the reduction of the energy generation cost of a power system. A two-fold optimization framework is provided where the first optimization problem seeks to find the optimal storage schedule that minimizes operational costs. Since the operational cost depends on the storage capacity, a second optimization problem is then formulated with the aim of finding the optimal storage capacity to be deployed. Although, in general, these problems are difficult to solve, we provide a lower bound on the cost savings for a parametrized family of demand profiles. The optimization framework is numerically illustrated using real-world demand data from ISO New England. Numerical results show that energy storage can reduce energy generation costs by at least 2.5 percent.}, author = {Shen, Yue and Bichuch, Maxim and Mallada, Enrique}, booktitle = {19th IEEE European Control Conference (ECC)}, doi = {10.23919/ECC51009.2020.9143772}, grants = {CAREER-1752362, CPS-1544771, ENERGISE-DE-EE0008006, AMPS-1736448, TRIPODS-1934979, EPCN-1711188, ARO-W911NF-17-1-0092}, keywords = {Power Networks}, month = {5}, pages = {1526-1532}, title = {On the Value of Energy Storage in Generation Cost Reduction}, url = {https://mallada.ece.jhu.edu/pubs/2020-ECC-SBM.pdf}, year = {2020} } - L. Yang, M. H. Hajiesmaili, R. Sitaraman, A. Wierman, E. Mallada, and W. S. Wong, “Online Linear Optimization with Inventory Management Constraints,” in ACM Sigmetrics, 2020, pp. 1-19. doi:10.1145/3393691.3394207
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This paper considers the problem of online linear optimization with inventory management constraints. Specifically, we consider an online scenario where a decision maker needs to satisfy her time-varying demand for some units of an asset, either from a market with a time-varying price or from her own inventory. In each time slot, the decision maker is presented a (linear) price and must immediately decide the amount to purchase for covering the demand and/or for storing in the inventory for future use. The inventory has a limited capacity and can be used to buy and store assets at low price and cover the demand when the price is high. The ultimate goal of the decision maker is to cover the demand at each time slot while minimizing the cost of buying assets from the market. We propose ARP, an online algorithm for linear programming with inventory constraints, and ARPRate, an extended version that handles rate constraints to/from the inventory. Both ARP and ARPRate achieve optimal competitive ratios, meaning that no other online algorithm can achieve a better theoretical guarantee. To illustrate the results, we use the proposed algorithms in a case study focused on energy procurement and storage management strategies for data centers.
@inproceedings{yhswmw2020sigmetrics, abstract = {This paper considers the problem of online linear optimization with inventory management constraints. Specifically, we consider an online scenario where a decision maker needs to satisfy her time-varying demand for some units of an asset, either from a market with a time-varying price or from her own inventory. In each time slot, the decision maker is presented a (linear) price and must immediately decide the amount to purchase for covering the demand and/or for storing in the inventory for future use. The inventory has a limited capacity and can be used to buy and store assets at low price and cover the demand when the price is high. The ultimate goal of the decision maker is to cover the demand at each time slot while minimizing the cost of buying assets from the market. We propose ARP, an online algorithm for linear programming with inventory constraints, and ARPRate, an extended version that handles rate constraints to/from the inventory. Both ARP and ARPRate achieve optimal competitive ratios, meaning that no other online algorithm can achieve a better theoretical guarantee. To illustrate the results, we use the proposed algorithms in a case study focused on energy procurement and storage management strategies for data centers.}, author = {Yang, Lin and Hajiesmaili, Mohammad H. and Sitaraman, Ramesh and Wierman, Adam and Mallada, Enrique and Wong, Wing S.}, booktitle = {ACM Sigmetrics}, doi = {10.1145/3393691.3394207}, grants = {CAREER-1752362, CPS-1544771, ENERGISE-DE-EE0008006, AMPS-1736448,EPCN-1711188,}, month = {6}, pages = {1-19}, title = {Online Linear Optimization with Inventory Management Constraints}, url = {https://mallada.ece.jhu.edu/pubs/2020-Sigmetrics-YHSWMW.pdf}, year = {2020} } - T. Zheng, J. W. Simpson-Porco, and E. Mallada, “Implicit Trajectory Planning for Feedback Linearizable Systems: A Time-varying Optimization Approach,” in American Control Conference (ACC), 2020, pp. 4677-4682. doi:10.23919/ACC45564.2020.9147997
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We develop an optimization-based framework for joint real-time trajectory planning and feedback control of feedback-linearizable systems. To achieve this goal, we define a target trajectory as the optimal solution of a time-varying optimization problem. In general, however, such trajectory may not be feasible due to , e.g., nonholonomic constraints. To solve this problem, we design a control law that generates feasible trajectories that asymptotically converge to the target trajectory. More precisely, for systems that are (dynamic) full-state linearizable, the proposed control law implicitly transforms the nonlinear system into an optimization algorithm of sufficiently high order. We prove global exponential convergence to the target trajectory for both the optimization algorithm and the original system. We illustrate the effectiveness of our proposed method on multi-target or multi-agent tracking problems with constraints.
@inproceedings{zsm2020acc, abstract = { We develop an optimization-based framework for joint real-time trajectory planning and feedback control of feedback-linearizable systems. To achieve this goal, we define a target trajectory as the optimal solution of a time-varying optimization problem. In general, however, such trajectory may not be feasible due to , e.g., nonholonomic constraints. To solve this problem, we design a control law that generates feasible trajectories that asymptotically converge to the target trajectory. More precisely, for systems that are (dynamic) full-state linearizable, the proposed control law implicitly transforms the nonlinear system into an optimization algorithm of sufficiently high order. We prove global exponential convergence to the target trajectory for both the optimization algorithm and the original system. We illustrate the effectiveness of our proposed method on multi-target or multi-agent tracking problems with constraints.}, author = {Zheng, Tianqi and Simpson-Porco, John W. and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC45564.2020.9147997}, grants = {CPS-1544771, CAREER-1752362, ARO-W911NF-17-1-0092}, month = {7}, pages = {4677-4682}, title = {Implicit Trajectory Planning for Feedback Linearizable Systems: A Time-varying Optimization Approach}, url = {https://mallada.ece.jhu.edu/pubs/2020-ACC-ZSM.pdf}, year = {2020} } - J. Guthrie and E. Mallada, “Adversarial Model Predictive Control via Second Order Cone Programming,” in 58th IEEE Conference on Decision and Control (CDC), 2019, pp. 1403-1409. doi:10.1109/CDC40024.2019.9029244
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We study the problem of designing attacks to safety critical systems in which the adversary seeks to maximize the overall system cost within a model predictive control framework. Although in general this problem is NP-hard, we characterize a family of problems that can be solved in polynomial time via a second-order cone programming relaxation. In particular, we show that positive systems fall under this family. We provide examples demonstrating the design of optimal attacks on an autonomous vehicle and a microgrid.
@inproceedings{gm2019cdc, abstract = {We study the problem of designing attacks to safety critical systems in which the adversary seeks to maximize the overall system cost within a model predictive control framework. Although in general this problem is NP-hard, we characterize a family of problems that can be solved in polynomial time via a second-order cone programming relaxation. In particular, we show that positive systems fall under this family. We provide examples demonstrating the design of optimal attacks on an autonomous vehicle and a microgrid.}, author = {Guthrie, James and Mallada, Enrique}, booktitle = {58th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC40024.2019.9029244}, grants = {ARO-W911NF-17-1-0092, CPS-1544771, EPCN-1711188, CAREER-1752362, AMPS-1736448}, month = {12}, pages = {1403-1409}, title = {Adversarial Model Predictive Control via Second Order Cone Programming}, url = {https://mallada.ece.jhu.edu/pubs/2019-CDC-GM.pdf}, year = {2019} } - H. Min and E. Mallada, “Dynamics Concentration of Tightly-Connected Large-Scale Networks,” in 58th IEEE Conference on Decision and Control (CDC), 2019, pp. 758-763. doi:10.1109/CDC40024.2019.9029796
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The ability to achieve coordinated behavior –engineered or emergent– on networked systems has attracted widespread interest over several fields. This has led to remarkable advances on the development of a theoretical understanding of the conditions under which agents within a network can reach agreement (consensus) or develop coordinated behaviors such as synchronization. However, fewer advances have been made toward explaining another commonly observed phenomena in tightly-connected networks systems: output responses of nodes in the networks are almost identical to each other despite heterogeneity in their individual dynamics. In this paper, we leverage tools from high-dimensional probability to provide an initial answer to this phenomena. More precisely, we show that for linear networks of nodal random transfer functions, as the networks size and connectivity grows, every node in the network follows the same response to an input or disturbance — irrespectively of the source of this input. We term this behavior as dynamics concentration as it stems from the fact that the network transfer matrix uniformly converges in probability to a unique dynamic response –i.e., it concentrates– determined by the distribution of the random transfer function of each node. We further discuss the implications of our analysis in the context of model reduction and robustness, and provide numerical evidence that similar phenomena occur in small deterministic networks over a properly defined frequency band.
@inproceedings{mm2019cdc, abstract = {The ability to achieve coordinated behavior --engineered or emergent-- on networked systems has attracted widespread interest over several fields. This has led to remarkable advances on the development of a theoretical understanding of the conditions under which agents within a network can reach agreement (consensus) or develop coordinated behaviors such as synchronization. However, fewer advances have been made toward explaining another commonly observed phenomena in tightly-connected networks systems: output responses of nodes in the networks are almost identical to each other despite heterogeneity in their individual dynamics. In this paper, we leverage tools from high-dimensional probability to provide an initial answer to this phenomena. More precisely, we show that for linear networks of nodal random transfer functions, as the networks size and connectivity grows, every node in the network follows the same response to an input or disturbance -- irrespectively of the source of this input. We term this behavior as dynamics concentration as it stems from the fact that the network transfer matrix uniformly converges in probability to a unique dynamic response --i.e., it concentrates-- determined by the distribution of the random transfer function of each node. We further discuss the implications of our analysis in the context of model reduction and robustness, and provide numerical evidence that similar phenomena occur in small deterministic networks over a properly defined frequency band.}, author = {Min, Hancheng and Mallada, Enrique}, booktitle = {58th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC40024.2019.9029796}, grants = {ARO-W911NF-17-1-0092, CPS-1544771, EPCN-1711188, CAREER-1752362, AMPS-1736448, ENERGISE-DE-EE0008006}, month = {12}, pages = {758-763}, title = {Dynamics Concentration of Tightly-Connected Large-Scale Networks}, url = {https://mallada.ece.jhu.edu/pubs/2019-CDC-MM.pdf}, year = {2019} } - P. You, D. F. Gayme, and E. Mallada, “The Role of Strategic Load Participants in Two-Stage Settlement Electricity Markets,” in 58th IEEE Conference on Decision and Control (CDC), 2019, pp. 8416-8422. doi:10.1109/CDC40024.2019.9029514
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We consider the problem of designing a feedback controller that guides the input and output of a linear timeinvariant system to a minimizer of a convex optimization problem. The system is subject to an unknown disturbance, piecewise constant in time, which shifts the feasible set defined by the system equilibrium constraints. Our proposed design combines proportional-integral control with gradient feedback, and enforces the Karush-Kuhn-Tucker optimality conditions in steady-state without incorporating dual variables into the controller. We prove that the input and output variables achieve optimality in steady-state, and provide a stability criterion based on absolute stability theory. The effectiveness of our approach is illustrated on a simple example system.
@inproceedings{ygm2019cdc, abstract = {We consider the problem of designing a feedback controller that guides the input and output of a linear timeinvariant system to a minimizer of a convex optimization problem. The system is subject to an unknown disturbance, piecewise constant in time, which shifts the feasible set defined by the system equilibrium constraints. Our proposed design combines proportional-integral control with gradient feedback, and enforces the Karush-Kuhn-Tucker optimality conditions in steady-state without incorporating dual variables into the controller. We prove that the input and output variables achieve optimality in steady-state, and provide a stability criterion based on absolute stability theory. The effectiveness of our approach is illustrated on a simple example system.}, author = {You, Pengcheng and Gayme, Dennice F. and Mallada, Enrique}, booktitle = {58th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC40024.2019.9029514}, grants = {ARO-W911NF-17-1-0092, CPS-1544771, EPCN-1711188, CAREER-1752362, AMPS-1736448, ENERGISE-DE-EE0008006}, month = {12}, pages = {8416-8422}, title = {The Role of Strategic Load Participants in Two-Stage Settlement Electricity Markets}, url = {https://mallada.ece.jhu.edu/pubs/2019-CDC-YGM.pdf}, year = {2019} } - C. Avraam, J. Rines, A. Sarker, F. Paganini, and E. Mallada, “Voltage Collapse Stabilization in Star DC Networks,” in American Control Conference (ACC), 2019, pp. 1957-1964. doi:10.23919/ACC.2019.8814708
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Voltage collapse is a type of blackout-inducing dynamic instability that occurs when the power demand exceeds the maximum power that can be transferred through the network. The traditional (preventive) approach to avoid voltage collapse is based on ensuring that the network never reaches its maximum capacity. However, such an approach leads to inefficiencies as it prevents operators to fully utilize the network resources and does not account for unprescribed events. To overcome this limitation, this paper seeks to initiate the study of voltage collapse stabilization. More precisely, for a DC network, we formulate the problem of voltage stability as a dynamic problem where each load seeks to achieve a constant power consumption by updating its conductance as the voltage changes. We show that such a system can be interpreted as a dynamic game, where each player (load) seeks to myopically maximize their utility, and where every stable power flow solution amounts to a Local Nash Equilibrium. Using this framework, we show that voltage collapse is equivalent to the non-existence of a Local Nash Equilibrium in the game and, as a result, it is caused by the lack of cooperation between loads. Finally, we propose a Voltage Collapse Stabilizer (VCS) controller that uses (flexible) loads that are willing to cooperate and provides a fair allocation of the curtailed demand. Our solution stabilizes voltage collapse even in the presence of non-cooperative loads. Numerical simulations validate several features of our controllers.
@inproceedings{arspm2019acc, abstract = {Voltage collapse is a type of blackout-inducing dynamic instability that occurs when the power demand exceeds the maximum power that can be transferred through the network. The traditional (preventive) approach to avoid voltage collapse is based on ensuring that the network never reaches its maximum capacity. However, such an approach leads to inefficiencies as it prevents operators to fully utilize the network resources and does not account for unprescribed events. To overcome this limitation, this paper seeks to initiate the study of voltage collapse stabilization. More precisely, for a DC network, we formulate the problem of voltage stability as a dynamic problem where each load seeks to achieve a constant power consumption by updating its conductance as the voltage changes. We show that such a system can be interpreted as a dynamic game, where each player (load) seeks to myopically maximize their utility, and where every stable power flow solution amounts to a Local Nash Equilibrium. Using this framework, we show that voltage collapse is equivalent to the non-existence of a Local Nash Equilibrium in the game and, as a result, it is caused by the lack of cooperation between loads. Finally, we propose a Voltage Collapse Stabilizer (VCS) controller that uses (flexible) loads that are willing to cooperate and provides a fair allocation of the curtailed demand. Our solution stabilizes voltage collapse even in the presence of non-cooperative loads. Numerical simulations validate several features of our controllers.}, author = {Avraam, Charalampos and Rines, Jesse and Sarker, Aurik and Paganini, Fernando and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC.2019.8814708}, grants = {CAREER-1752362,EPCN-1711188,ENERGISE-DE-EE0008006,ARO-W911NF-17-1-0092,EPCN-1711188,CPS-1544771}, keywords = {Power Networks}, month = {06}, pages = {1957-1964}, title = {Voltage Collapse Stabilization in Star DC Networks}, url = {https://mallada.ece.jhu.edu/pubs/2019-ACC-ARSPM.pdf}, year = {2019} } - C. Ji, M. H. Hajiesmaili, D. F. Gayme, and E. Mallada, “Coordinating Distribution System Resources for Co-optimized Participation in Energy and Ancillary Service Transmission System Markets,” in American Control Conference (ACC), 2019, pp. 1315-1322. doi:10.23919/ACC.2019.8815242
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This paper investigates the potential of using aggregate controllable loads and energy storage systems from multiple heterogeneous feeders to jointly optimize a utility’s energy procurement cost from the real-time market and their revenue from ancillary service markets. Toward this, we formulate an optimization problem that co-optimizes real-time and energy reserve markets based on real-time and ancillary service market prices, along with available solar power, storage and demand data from each of the feeders within a single distribution network. The optimization, which includes all network system constraints, provides real/reactive power and energy storage set-points for each feeder as well as a schedule for the aggregate system’s participation in the two types of markets. We evaluate the performance of our algorithm using several trace-driven simulations based on a real-world circuit of a New Jersey utility. The results demonstrate that active participation through controllable loads and storage significantly reduces the utility’s net costs, i.e., real-time energy procurement costs minus ancillary market revenues.
@inproceedings{jhgm2019acc, abstract = {This paper investigates the potential of using aggregate controllable loads and energy storage systems from multiple heterogeneous feeders to jointly optimize a utility's energy procurement cost from the real-time market and their revenue from ancillary service markets. Toward this, we formulate an optimization problem that co-optimizes real-time and energy reserve markets based on real-time and ancillary service market prices, along with available solar power, storage and demand data from each of the feeders within a single distribution network. The optimization, which includes all network system constraints, provides real/reactive power and energy storage set-points for each feeder as well as a schedule for the aggregate system's participation in the two types of markets. We evaluate the performance of our algorithm using several trace-driven simulations based on a real-world circuit of a New Jersey utility. The results demonstrate that active participation through controllable loads and storage significantly reduces the utility's net costs, i.e., real-time energy procurement costs minus ancillary market revenues.}, author = {Ji, Chengda and Hajiesmaili, Mohammad H. and Gayme, Dennice F. and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC.2019.8815242}, grants = {CAREER-1752362, ENERGISE-DE-EE0008006, EPCN-1711188}, month = {06}, pages = {1315-1322}, title = {Coordinating Distribution System Resources for Co-optimized Participation in Energy and Ancillary Service Transmission System Markets}, url = {https://mallada.ece.jhu.edu/pubs/2019-ACC-JHGM.pdf}, year = {2019} } - C. Ji, E. Mallada, and D. Gayme, “Evaluating Robustness of Consensus Algorithms Under Measurement Error over Digraph,” in 57th IEEE Conference on Decision and Control (CDC), 2018, pp. 1238-1244. doi:10.1109/CDC.2018.8619283
[BibTeX] [Abstract] [Download PDF]
Consensus algorithms constitute a powerful tool for computing average values or coordinating agents in many distributed applications. Unfortunately, the same property that allows this computation (i.e., the nontrivial nullspace of the state matrix) leads to unbounded state variance in the presence of measurement errors. In this work, we explore the trade-off between relative and absolute communication (feedback) in the presence of measurement errors. We evaluate the robustness of first and second order integrator systems under a parameterized family of controllers (homotopy) that continuously trade between relative and absolute feedback interconnections in terms of the H2 norm an appropriately defined inputoutput system. Our approach extends the previous H2 norm based analysis to systems with directed feedback interconnections whose underlying weighted graph Laplacians are diagonalizable. Our results indicate that any level of absolute communication is sufficient to achieve a finite H2 norm but that purely relative feedback can only achieve finite norms when the measurement error is not exciting subspace associated with the consensus state. Numerical examples demonstrate that smoothly reducing the proportion of relative feedback in double integrator systems smoothly decreases the system performance and that this performance degradation is more rapid systems with relative feedback in only the first state (position).
@inproceedings{jmg2018cdc, abstract = {Consensus algorithms constitute a powerful tool for computing average values or coordinating agents in many distributed applications. Unfortunately, the same property that allows this computation (i.e., the nontrivial nullspace of the state matrix) leads to unbounded state variance in the presence of measurement errors. In this work, we explore the trade-off between relative and absolute communication (feedback) in the presence of measurement errors. We evaluate the robustness of first and second order integrator systems under a parameterized family of controllers (homotopy) that continuously trade between relative and absolute feedback interconnections in terms of the H2 norm an appropriately defined inputoutput system. Our approach extends the previous H2 norm based analysis to systems with directed feedback interconnections whose underlying weighted graph Laplacians are diagonalizable. Our results indicate that any level of absolute communication is sufficient to achieve a finite H2 norm but that purely relative feedback can only achieve finite norms when the measurement error is not exciting subspace associated with the consensus state. Numerical examples demonstrate that smoothly reducing the proportion of relative feedback in double integrator systems smoothly decreases the system performance and that this performance degradation is more rapid systems with relative feedback in only the first state (position).}, author = {Ji, Chengda and Mallada, Enrique and Gayme, Dennice}, booktitle = {57th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2018.8619283}, grants = {CPS:1544771, ARO:W911NF-17-1-0092, CAREER-1752362}, issn = {2576-2370}, month = {12}, pages = {1238-1244}, title = {Evaluating Robustness of Consensus Algorithms Under Measurement Error over Digraph}, url = {https://mallada.ece.jhu.edu/pubs/2018-CDC-JMG.pdf}, year = {2018} } - L. S. P. Lawrence, Z. Nelson, E. Mallada, and J. W. Simpson-Porco, “Optimal Steady-State Control for Linear Time-Invariant Systems,” in 57th IEEE Conference on Decision and Control (CDC), 2018, pp. 3251-3257. doi:10.1109/CDC.2018.8619812
[BibTeX] [Abstract] [Download PDF]
We consider the problem of designing a feedback controller that guides the input and output of a linear timeinvariant system to a minimizer of a convex optimization problem. The system is subject to an unknown disturbance, piecewise constant in time, which shifts the feasible set defined by the system equilibrium constraints. Our proposed design combines proportional-integral control with gradient feedback, and enforces the Karush-Kuhn-Tucker optimality conditions in steady-state without incorporating dual variables into the controller. We prove that the input and output variables achieve optimality in steady-state, and provide a stability criterion based on absolute stability theory. The effectiveness of our approach is illustrated on a simple example system.
@inproceedings{lnms2018cdc, abstract = {We consider the problem of designing a feedback controller that guides the input and output of a linear timeinvariant system to a minimizer of a convex optimization problem. The system is subject to an unknown disturbance, piecewise constant in time, which shifts the feasible set defined by the system equilibrium constraints. Our proposed design combines proportional-integral control with gradient feedback, and enforces the Karush-Kuhn-Tucker optimality conditions in steady-state without incorporating dual variables into the controller. We prove that the input and output variables achieve optimality in steady-state, and provide a stability criterion based on absolute stability theory. The effectiveness of our approach is illustrated on a simple example system.}, author = {Lawrence, Liam S. P. and Nelson, Zachary and Mallada, Enrique and Simpson-Porco, John W.}, booktitle = {57th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2018.8619812}, grants = {CPS:1544771, ARO:W911NF-17-1-0092, CAREER-1752362}, issn = {2576-2370}, month = {12}, pages = {3251-3257}, title = {Optimal Steady-State Control for Linear Time-Invariant Systems}, url = {https://mallada.ece.jhu.edu/pubs/2018-CDC-LNMS.pdf}, year = {2018} } - M. Zhao, M. D. Kaba, R. Vidal, D. R. Robinson, and E. Mallada, “Sparse Recovery over Graph Incidence Matrices,” in 57th IEEE Conference on Decision and Control (CDC), 2018, pp. 364-371. doi:10.1109/CDC.2018.8619666
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Classical results in sparse representation guarantee the exact recovery of sparse signals under assumptions on the dictionary that are either too strong or NP hard to check. Moreover, such results may be too pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the form of an incidence matrix. We show that in this case necessary and sufficient conditions can be derived in terms of properties of the cycles of the graph, which can be checked in polynomial time. Our analysis further allows us to derive location dependent conditions for recovery that only depend on the cycles of the graph that intersect this support. Finally, we exploit sparsity properties on the measurements to a specialized sub-graph-based recovery algorithm that outperforms the standard $l_1$-minimization.
@inproceedings{zkvrm2018cdc, abstract = {Classical results in sparse representation guarantee the exact recovery of sparse signals under assumptions on the dictionary that are either too strong or NP hard to check. Moreover, such results may be too pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the form of an incidence matrix. We show that in this case necessary and sufficient conditions can be derived in terms of properties of the cycles of the graph, which can be checked in polynomial time. Our analysis further allows us to derive location dependent conditions for recovery that only depend on the cycles of the graph that intersect this support. Finally, we exploit sparsity properties on the measurements to a specialized sub-graph-based recovery algorithm that outperforms the standard $l_1$-minimization.}, author = {Zhao, Mengnan and Kaba, Mustafa Devrim and Vidal, Rene and Robinson, Daniel R. and Mallada, Enrique}, booktitle = {57th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2018.8619666}, grants = {AMPS:1736448}, issn = {2576-2370}, month = {12}, pages = {364-371}, title = {Sparse Recovery over Graph Incidence Matrices}, url = {https://mallada.ece.jhu.edu/pubs/2018-CDC-ZKVRM.pdf}, year = {2018} } - Z. Nelson and E. Mallada, “An integral quadratic constraint framework for steady state optimization of linear time invariant systems,” in American Control Conference (ACC), 2018. doi:10.23919/ACC.2018.8431231
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Achieving optimal steady-state performance in real-time is an increasingly necessary requirement of many critical infrastructure systems. In pursuit of this goal, this paper builds a systematic design framework of feedback controllers for Linear Time-Invariant (LTI) systems that continuously track the optimal solution of some predefined optimization problem. The proposed solution can be logically divided into three components. The first component estimates the system state from the output measurements. The second component uses the estimated state and computes a drift direction based on an optimization algorithm. The third component computes an input to the LTI system that aims to drive the system toward the optimal steady-state. We analyze the equilibrium characteristics of the closed-loop system and provide conditions for optimality and stability. Our analysis shows that the proposed solution guarantees optimal steady-state performance, even in the presence of constant disturbances. Furthermore, by leveraging recent results on the analysis of optimization algorithms using integral quadratic constraints (IQCs), the proposed framework is able to translate input-output properties of our optimization component into sufficient conditions, based on linear matrix inequalities (LMIs), for global exponential asymptotic stability of the closed loop system. We illustrate the versatility of our framework using several examples.
@inproceedings{nm2018acc, abstract = {Achieving optimal steady-state performance in real-time is an increasingly necessary requirement of many critical infrastructure systems. In pursuit of this goal, this paper builds a systematic design framework of feedback controllers for Linear Time-Invariant (LTI) systems that continuously track the optimal solution of some predefined optimization problem. The proposed solution can be logically divided into three components. The first component estimates the system state from the output measurements. The second component uses the estimated state and computes a drift direction based on an optimization algorithm. The third component computes an input to the LTI system that aims to drive the system toward the optimal steady-state. We analyze the equilibrium characteristics of the closed-loop system and provide conditions for optimality and stability. Our analysis shows that the proposed solution guarantees optimal steady-state performance, even in the presence of constant disturbances. Furthermore, by leveraging recent results on the analysis of optimization algorithms using integral quadratic constraints (IQCs), the proposed framework is able to translate input-output properties of our optimization component into sufficient conditions, based on linear matrix inequalities (LMIs), for global exponential asymptotic stability of the closed loop system. We illustrate the versatility of our framework using several examples.}, author = {Nelson, Zachary and Mallada, Enrique}, booktitle = {American Control Conference (ACC)}, doi = {10.23919/ACC.2018.8431231}, grants = {1544771, W911NF-17-1-0092, 1711188}, issn = {2378-5861}, keywords = {Optimization, IQCs}, month = {06}, title = {An integral quadratic constraint framework for steady state optimization of linear time invariant systems}, url = {https://mallada.ece.jhu.edu/pubs/2018-ACC-NM.pdf}, year = {2018} } - R. Pates and E. Mallada, “Damping, Inertia, and Delay Robustness Trade-offs in Power Systems,” in 23rd International Symposium on Mathematical Theory of Networks and Systems, 2018.
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Electro-mechanical oscillations in power systems are typically controlled by simple decentralised controllers. We derive a formula for computing the delay margin of such controllers when the power system is represented by a simple mechanical network. This formula reveals a clear trade-off between system damping, inertia, and robustness to delays. In particular, it shows that reducing system inertia, which is a common consequence of increased renewable generation, can reduce robustness to unmodelled dynamics.
@inproceedings{pm2018mtns, abstract = {Electro-mechanical oscillations in power systems are typically controlled by simple decentralised controllers. We derive a formula for computing the delay margin of such controllers when the power system is represented by a simple mechanical network. This formula reveals a clear trade-off between system damping, inertia, and robustness to delays. In particular, it shows that reducing system inertia, which is a common consequence of increased renewable generation, can reduce robustness to unmodelled dynamics.}, author = {Pates, Richard and Mallada, Enrique}, booktitle = {23rd International Symposium on Mathematical Theory of Networks and Systems}, grants = {CPS:1544771, ARO:W911NF-17-1-0092, 1711188, CAREER-1752362}, month = {7}, title = {Damping, Inertia, and Delay Robustness Trade-offs in Power Systems}, url = {https://mallada.ece.jhu.edu/pubs/2018-MTNS-PM.pdf}, year = {2018} } - E. Weitenberg, Y. Jiang, C. Zhao, E. Mallada, F. Dorfler, and C. De Persis, “Robust decentralized frequency control: A leaky integrator approach,” in 17th IEEE European Control Conference (ECC), 2018. doi:10.23919/ECC.2018.8550060
[BibTeX] [Download PDF]@inproceedings{wjzmdd2018ecc, author = {Weitenberg, Erik and Jiang, Yan and Zhao, Changhong and Mallada, Enrique and Dorfler, Florian and De Persis, Claudio}, booktitle = {17th IEEE European Control Conference (ECC)}, doi = {10.23919/ECC.2018.8550060}, grants = {1544771, 1711188, 1736448}, keywords = {Power Networks}, month = {6}, title = {Robust decentralized frequency control: A leaky integrator approach}, url = {https://mallada.ece.jhu.edu/pubs/2018-ECC-WJZMDD.pdf}, year = {2018} } - M. H. Hajiesmaili, D. Cai, and E. Mallada, “Understanding the Inefficiency of Security-Constrained Economic Dispatch,” in 56th IEEE Conference on Decision and Control (CDC), 2017, pp. 2035-2040. doi:10.1109/CDC.2017.8263947
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The security-constrained economic dispatch (SCED) problem tries to maintain the reliability of a power network by ensuring that a single failure does not lead to a global outage. The previous research has mainly investigated SCED by formulating the problem in different modalities, e.g. preventive or corrective, and devising efficient solutions for SCED. In this paper, we tackle a novel and important direction, and analyze the economic cost of incorporating security constraints in economic dispatch. Inspired by existing inefficiency metrics in game theory and computer science, we introduce notion of price of security as a metric that formally characterizes the economic inefficiency of security-constrained economic dispatch as compared to the original problem without security constraints. Then, we focus on the preventive approach in a simple topology comprising two buses and two lines, and investigate the impact of generation availability and demand distribution on the price of security. Moreover, we explicitly derive the worst-case input instance that leads to the maximum price of security. By extensive experimental study on two test-cases, we verify the analytical results and provide insights for characterizing the price of security in general networks.
@inproceedings{hcm2017cdc, abstract = {The security-constrained economic dispatch (SCED) problem tries to maintain the reliability of a power network by ensuring that a single failure does not lead to a global outage. The previous research has mainly investigated SCED by formulating the problem in different modalities, e.g. preventive or corrective, and devising efficient solutions for SCED. In this paper, we tackle a novel and important direction, and analyze the economic cost of incorporating security constraints in economic dispatch. Inspired by existing inefficiency metrics in game theory and computer science, we introduce notion of price of security as a metric that formally characterizes the economic inefficiency of security-constrained economic dispatch as compared to the original problem without security constraints. Then, we focus on the preventive approach in a simple topology comprising two buses and two lines, and investigate the impact of generation availability and demand distribution on the price of security. Moreover, we explicitly derive the worst-case input instance that leads to the maximum price of security. By extensive experimental study on two test-cases, we verify the analytical results and provide insights for characterizing the price of security in general networks.}, author = {Hajiesmaili, Mohammad H. and Cai, Desmond and Mallada, Enrique}, booktitle = {56th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2017.8263947}, grants = {1544771, 1711188, 1736448}, keywords = {Power Networks}, month = {12}, pages = {2035-2040}, title = {Understanding the Inefficiency of Security-Constrained Economic Dispatch}, url = {https://mallada.ece.jhu.edu/pubs/2017-CDC-HCM.pdf}, year = {2017} } - Y. Jiang, R. Pates, and E. Mallada, “Performance tradeoffs of dynamically controlled grid-connected inverters in low inertia power systems,” in 56th IEEE Conference on Decision and Control (CDC), 2017, pp. 5098-5105. doi:10.1109/CDC.2017.8264414
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Implementing frequency response using grid-connected inverters is one of the popular proposed alternatives to mitigate the dynamic degradation experienced in low inertia power systems. However, such solution faces several challenges as inverters do not intrinsically possess the natural response to power fluctuations that synchronous generators have. Thus, to synthetically generate this response, inverters need to take frequency measurements, which are usually noisy, and subsequently make changes in the output power, which are therefore delayed. This paper explores the system-wide performance tradeoffs that arise when measurement noise, delayed actions, and power disturbances are considered in the design of dynamic controllers for grid-connected inverters. Using a recently proposed dynamic droop (iDroop) control for grid-connected inverters that is inspired by classical first order lead-lag compensation, we show that the sets of parameters that result in highest noise attenuation, power disturbance mitigation, and delay robustness do not necessarily have a common intersection. In particular, lead compensation is desired in systems where power disturbances are the predominant source of degradation, while lag compensation is a better alternative when the system is dominated by delays or frequency noise. Our analysis further shows that iDroop can outperform the standard droop alternative in both joint noise and disturbance mitigation, and delay robustness.
@inproceedings{jpm2017cdc, abstract = {Implementing frequency response using grid-connected inverters is one of the popular proposed alternatives to mitigate the dynamic degradation experienced in low inertia power systems. However, such solution faces several challenges as inverters do not intrinsically possess the natural response to power fluctuations that synchronous generators have. Thus, to synthetically generate this response, inverters need to take frequency measurements, which are usually noisy, and subsequently make changes in the output power, which are therefore delayed. This paper explores the system-wide performance tradeoffs that arise when measurement noise, delayed actions, and power disturbances are considered in the design of dynamic controllers for grid-connected inverters. Using a recently proposed dynamic droop (iDroop) control for grid-connected inverters that is inspired by classical first order lead-lag compensation, we show that the sets of parameters that result in highest noise attenuation, power disturbance mitigation, and delay robustness do not necessarily have a common intersection. In particular, lead compensation is desired in systems where power disturbances are the predominant source of degradation, while lag compensation is a better alternative when the system is dominated by delays or frequency noise. Our analysis further shows that iDroop can outperform the standard droop alternative in both joint noise and disturbance mitigation, and delay robustness.}, author = {Jiang, Yan and Pates, Richard and Mallada, Enrique}, booktitle = {56th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2017.8264414}, grants = {1544771, 1711188, W911NF-17-1-0092}, keywords = {Power Networks}, month = {12}, pages = {5098-5105}, title = {Performance tradeoffs of dynamically controlled grid-connected inverters in low inertia power systems}, url = {https://mallada.ece.jhu.edu/pubs/2017-CDC-JPM.pdf}, year = {2017} } - G. H. Oral, E. Mallada, and D. Gayme, “Performance of first and second order linear networked systems over digraphs,” in 56th IEEE Conference on Decision and Control (CDC), 2017, pp. 1688-1694. doi:10.1109/CDC.2017.8263893
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In this paper we investigate the performance of linear networked dynamical systems over strongly connected digraphs. We consider first and second order systems subject to distributed disturbance inputs, and define an appropriate system output so that the performance measure is quantified through the input-output $\mathcal H_2$ norm of the system. We first develop a generalized framework for the computation of the $\mathcal H_2$ norm. We apply this framework to systems whose underlying network graphs result in normal weighted graph Laplacian matrices. We consider two performance metrics and find closed form solutions for the first and bounds for the other; which both depend on the eigenvalues of these graph Laplacians. Numerical examples indicate that: (i) the tightness of the bounds are highly dependent on the graph structure, (ii) the $\mathcal H_2$ norm of a symmetric system is less than or equal to that of the corresponding perturbed non-symmetric system for either line or complete graphs when the network size is sufficiently large.
@inproceedings{omg2017cdc, abstract = {In this paper we investigate the performance of linear networked dynamical systems over strongly connected digraphs. We consider first and second order systems subject to distributed disturbance inputs, and define an appropriate system output so that the performance measure is quantified through the input-output $\mathcal H_2$ norm of the system. We first develop a generalized framework for the computation of the $\mathcal H_2$ norm. We apply this framework to systems whose underlying network graphs result in normal weighted graph Laplacian matrices. We consider two performance metrics and find closed form solutions for the first and bounds for the other; which both depend on the eigenvalues of these graph Laplacians. Numerical examples indicate that: (i) the tightness of the bounds are highly dependent on the graph structure, (ii) the $\mathcal H_2$ norm of a symmetric system is less than or equal to that of the corresponding perturbed non-symmetric system for either line or complete graphs when the network size is sufficiently large.}, author = {Oral, H. Giray and Mallada, Enrique and Gayme, Dennice}, booktitle = {56th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2017.8263893}, grants = {1544771, W911NF-17-1-0092}, keywords = {Power Networks}, month = {12}, pages = {1688-1694}, title = {Performance of first and second order linear networked systems over digraphs}, url = {https://mallada.ece.jhu.edu/pubs/2017-CDC-OMG.pdf}, year = {2017} } - F. Paganini and E. Mallada, “Global performance metrics for synchronization of heterogeneously rated power systems: The role of machine models and inertia,” in 55th Allerton Conference on Communication, Control, and Computing, 2017, pp. 324-331. doi:10.1109/ALLERTON.2017.8262755
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A recent trend in control of power systems has sought to quantify the synchronization dynamics in terms of a global performance metric, compute it under very simplified assumptions, and use it to gain insight on the role of system parameters, in particular, inertia. In this paper, we wish to extend this approach to more realistic scenarios, by incorporating the heterogeneity of machine ratings, more complete machine models, and also to more closely map it to classical power engineering notions such as Nadir, Rate of Change of Frequency (RoCoF), and inter-area oscillations. We consider the system response to a step change in power excitation, and define the system frequency as a weighted average of generator frequencies (with weights proportional to each machine’s rating); we characterize Nadir and RoCoF by the Linf norm of the system frequency and its derivative, respectively, and inter-areas oscillations by the L2 norm of the error of the vector of bus frequencies w.r.t. the system frequency. For machine models where the dynamic parameters (inertia, damping, etc.) are proportional to rating, we analytically compute these norms and use them to show that the role of inertia is more nuanced than in the conventional wisdom. With the classical swing dynamics, inertia constant plays a secondary role in performance. It is only when the turbine dynamics are introduced that the benefits of inertia become more prominent.
@inproceedings{pm2017allerton, abstract = {A recent trend in control of power systems has sought to quantify the synchronization dynamics in terms of a global performance metric, compute it under very simplified assumptions, and use it to gain insight on the role of system parameters, in particular, inertia. In this paper, we wish to extend this approach to more realistic scenarios, by incorporating the heterogeneity of machine ratings, more complete machine models, and also to more closely map it to classical power engineering notions such as Nadir, Rate of Change of Frequency (RoCoF), and inter-area oscillations. We consider the system response to a step change in power excitation, and define the system frequency as a weighted average of generator frequencies (with weights proportional to each machine's rating); we characterize Nadir and RoCoF by the Linf norm of the system frequency and its derivative, respectively, and inter-areas oscillations by the L2 norm of the error of the vector of bus frequencies w.r.t. the system frequency. For machine models where the dynamic parameters (inertia, damping, etc.) are proportional to rating, we analytically compute these norms and use them to show that the role of inertia is more nuanced than in the conventional wisdom. With the classical swing dynamics, inertia constant plays a secondary role in performance. It is only when the turbine dynamics are introduced that the benefits of inertia become more prominent.}, author = {Paganini, Fernando and Mallada, Enrique}, booktitle = {55th Allerton Conference on Communication, Control, and Computing}, doi = {10.1109/ALLERTON.2017.8262755}, grants = {1544771, 1711188, 1736448}, keywords = {Power Networks; Synchronization}, month = {10}, pages = {324-331}, title = {Global performance metrics for synchronization of heterogeneously rated power systems: The role of machine models and inertia}, url = {https://mallada.ece.jhu.edu/pubs/2017-Allerton-PM.pdf}, year = {2017} } - M. H. Hajiesmaili, M. Chen, E. Mallada, and C. Chau, “Crowd-Sourced Storage-Assisted Demand Response in Microgrids,” in Proceedings of the Eighth International Conference on Future Energy Systems, New York, NY, USA, 2017, pp. 91-100. doi:10.1145/3077839.3077841
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This paper studies the problem of utilizing heterogeneous energy storage systems, including electric vehicles and residential batteries, to perform demand-response in microgrids. The objective is to minimize the operational cost while fulfilling the demand-response requirement. The design space is to select and schedule a subset of available storage devices that are heterogeneous in operating cost, capacity, and availability in time. Designing a performance-optimized solution, however, is challenging due to the combinatorial nature of the problem with mixed packing and covering constraints, and the essential need for online solution design in practical scenarios where both demand-response requirement and the profile of user-owned storage systems arrive online. We tackle these challenges and design several online algorithms by leveraging a recent theoretical computer science technique which uses a problem-specific exponential potential function to solve online mixed packing and covering problems. We show that the fractional version of the algorithm achieves a logarithmic bi-criteria competitive ratio. Empirical trace-driven experiments demonstrate that our algorithms perform much better than the theoretical bounds and achieve close-to-optimal performance.
@inproceedings{hcmc2017e-energy, abstract = {This paper studies the problem of utilizing heterogeneous energy storage systems, including electric vehicles and residential batteries, to perform demand-response in microgrids. The objective is to minimize the operational cost while fulfilling the demand-response requirement. The design space is to select and schedule a subset of available storage devices that are heterogeneous in operating cost, capacity, and availability in time. Designing a performance-optimized solution, however, is challenging due to the combinatorial nature of the problem with mixed packing and covering constraints, and the essential need for online solution design in practical scenarios where both demand-response requirement and the profile of user-owned storage systems arrive online. We tackle these challenges and design several online algorithms by leveraging a recent theoretical computer science technique which uses a problem-specific exponential potential function to solve online mixed packing and covering problems. We show that the fractional version of the algorithm achieves a logarithmic bi-criteria competitive ratio. Empirical trace-driven experiments demonstrate that our algorithms perform much better than the theoretical bounds and achieve close-to-optimal performance.}, acmid = {3077841}, address = {New York, NY, USA}, author = {Hajiesmaili, Mohammad H. and Chen, Minghua and Mallada, Enrique and Chau, Chi-Kin}, booktitle = {Proceedings of the Eighth International Conference on Future Energy Systems}, doi = {10.1145/3077839.3077841}, grants = {1544771, W911NF-17-1-0092}, isbn = {978-1-4503-5036-5}, keywords = {Microgrid, competitive online algorithm design, crowd-sourced storage-assisted demand response, scheduling}, location = {Shatin, Hong Kong}, month = {5}, numpages = {10}, pages = {91--100}, publisher = {ACM}, series = {e-Energy '17}, title = {Crowd-Sourced Storage-Assisted Demand Response in Microgrids}, url = {https://mallada.ece.jhu.edu/pubs/2017-eEnery-HCMC.pdf}, year = {2017} } - R. Pates and E. Mallada, “Decentralised Robust Inverter-based Control in Power Systems,” in IFAC World Congress, 2017, pp. 5548-5553. doi:https://doi.org/10.1016/j.ifacol.2017.08.1097
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This paper develops a novel framework for power system stability analysis, that allows for the decentralised design of inverter based controllers. The method requires that each individual inverter satisfies a standard H1 design requirement. Critically each requirement depends only on the dynamics of the components and inverters at each individual bus, and the aggregate susceptance of the transmission lines connected to it. The method is both robust to network and delay uncertainties, as well as heterogeneous network components, and when no network information is available it reduces to the standard decentralised passivity sufficient condition for stability. We illustrate the novelty and strength of our approach by studying the design of inverter-based control laws in the presence of delays.
@inproceedings{pm2017ifac-wc, abstract = {This paper develops a novel framework for power system stability analysis, that allows for the decentralised design of inverter based controllers. The method requires that each individual inverter satisfies a standard H1 design requirement. Critically each requirement depends only on the dynamics of the components and inverters at each individual bus, and the aggregate susceptance of the transmission lines connected to it. The method is both robust to network and delay uncertainties, as well as heterogeneous network components, and when no network information is available it reduces to the standard decentralised passivity sufficient condition for stability. We illustrate the novelty and strength of our approach by studying the design of inverter-based control laws in the presence of delays.}, author = {Pates, Richard and Mallada, Enrique}, booktitle = {IFAC World Congress}, doi = {https://doi.org/10.1016/j.ifacol.2017.08.1097}, grants = {1544771}, keywords = {Power Networks}, month = {7}, number = {1}, pages = {5548 - 5553}, title = {Decentralised Robust Inverter-based Control in Power Systems}, url = {https://mallada.ece.jhu.edu/pubs/2017-IFAC-WC-PM.pdf}, volume = {50}, year = {2017} } - E. Mallada, “iDroop: A dynamic droop controller to decouple power grid’s steady-state and dynamic performance,” in 55th IEEE Conference on Decision and Control (CDC), 2016, pp. 4957-4964. doi:10.1109/CDC.2016.7799027
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This paper presents a novel Dynam-i-c Droop (iDroop) control mechanism to perform primary frequency control with gird-connected inverters that improves the network dynamic performance while maintaining the same steady-state characteristics of droop control. The work is motivated by the increasing dynamic degradation experienced by the power grid due to the increment on asynchronous inverted-based generation. We show that the widely suggested virtual inertia solution suffers from unbounded noise amplification (infinite H2 norm) and therefore could potentially degrade further the grid performance once widely deployed. This motivates the proposed solution on this paper that over- comes the limitations of virtual inertia controllers while sharing the same advantages of traditional droop control. In particular, our iDroop controllers are decentralized, rebalance supply and demand, and provide power sharing. Furthermore, our solution improves the dynamic performance without affecting the steady state solution. Our algorithm can be incrementally deployed and can be guaranteed to be stable using a decentralized sufficient stability condition on the parameter values. We illustrate several features of our solution using numerical simulations.
@inproceedings{m2016cdc, abstract = {This paper presents a novel Dynam-i-c Droop (iDroop) control mechanism to perform primary frequency control with gird-connected inverters that improves the network dynamic performance while maintaining the same steady-state characteristics of droop control. The work is motivated by the increasing dynamic degradation experienced by the power grid due to the increment on asynchronous inverted-based generation. We show that the widely suggested virtual inertia solution suffers from unbounded noise amplification (infinite H2 norm) and therefore could potentially degrade further the grid performance once widely deployed. This motivates the proposed solution on this paper that over- comes the limitations of virtual inertia controllers while sharing the same advantages of traditional droop control. In particular, our iDroop controllers are decentralized, rebalance supply and demand, and provide power sharing. Furthermore, our solution improves the dynamic performance without affecting the steady state solution. Our algorithm can be incrementally deployed and can be guaranteed to be stable using a decentralized sufficient stability condition on the parameter values. We illustrate several features of our solution using numerical simulations.}, author = {Mallada, Enrique}, booktitle = {55th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2016.7799027}, grants = {1544771}, keywords = {Power Networks}, month = {12}, pages = {4957-4964}, title = {iDroop: A dynamic droop controller to decouple power grid's steady-state and dynamic performance}, url = {https://mallada.ece.jhu.edu/pubs/2016-CDC-M.pdf}, year = {2016} } - A. Cherukuri, E. Mallada, S. H. Low, and J. Cortes, “The role of strong convexity-concavity in the convergence and robustness of the saddle-point dynamics,” in 54th Allerton Conference on Communication, Control, and Computing, 2016, pp. 504-510. doi:10.1109/ALLERTON.2016.7852273
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This paper studies the projected saddle-point dynamics for a twice differentiable convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We provide a novel characterization of the omega-limit set of the trajectories of these dynamics in terms of the diagonal Hessian blocks of the saddle function. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexityconcavity of the saddle function. If this property is global, and for the case when the saddle function takes the form of the Lagrangian of an equality constrained optimization problem, we establish the input-to-state stability of the saddlepoint dynamics by providing an ISS Lyapunov function. Various examples illustrate our results.
@inproceedings{cmlc2016allerton, abstract = {This paper studies the projected saddle-point dynamics for a twice differentiable convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We provide a novel characterization of the omega-limit set of the trajectories of these dynamics in terms of the diagonal Hessian blocks of the saddle function. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexityconcavity of the saddle function. If this property is global, and for the case when the saddle function takes the form of the Lagrangian of an equality constrained optimization problem, we establish the input-to-state stability of the saddlepoint dynamics by providing an ISS Lyapunov function. Various examples illustrate our results.}, author = {Ashish Cherukuri and Mallada, Enrique and Steven H. Low and Jorge Cortes}, booktitle = {54th Allerton Conference on Communication, Control, and Computing}, doi = {10.1109/ALLERTON.2016.7852273}, grants = {1544771}, keywords = {Saddle-Point Dynamics; Caratheodory solutions}, month = {09}, pages = {504-510}, title = {The role of strong convexity-concavity in the convergence and robustness of the saddle-point dynamics}, url = {https://mallada.ece.jhu.edu/pubs/2016-Allerton-CMLC.pdf}, year = {2016} } - C. Zhao, E. Mallada, S. H. Low, and J. W. Bialek, “A Unified Framework for Frequency Control and Congestion Management,” in Power Systems Computation Conference, 2016, pp. 1-7. doi:10.1109/PSCC.2016.7541028
[BibTeX] [Abstract] [Download PDF]
The existing frequency control framework in power systems is challenged by lower inertia and more volatile power injections. We propose a new framework for frequency control and congestion management. We formulate an optimization problem that rebalances power, restores the nominal frequency, restores inter-area flows and maintains line flows below their limits in a way that minimizes the control cost. The cost can be squared deviations from the reference generations, minimizing the disruption from the last optimal dispatch. Our control thus maintains system security without interfering with the market operation. By deriving a primal-dual algorithm to solve this optimization, we design a completely decentralized primary frequency control without the need for explicit communication among the participating agents, and a distributed unified control which integrates primary and secondary frequency control and congestion management. Simulations show that the unified control not only achieves all the desired control goals in system equilibrium, but also improves the transient compared to traditional control schemes.
@inproceedings{zmlb2016pscc, abstract = {The existing frequency control framework in power systems is challenged by lower inertia and more volatile power injections. We propose a new framework for frequency control and congestion management. We formulate an optimization problem that rebalances power, restores the nominal frequency, restores inter-area flows and maintains line flows below their limits in a way that minimizes the control cost. The cost can be squared deviations from the reference generations, minimizing the disruption from the last optimal dispatch. Our control thus maintains system security without interfering with the market operation. By deriving a primal-dual algorithm to solve this optimization, we design a completely decentralized primary frequency control without the need for explicit communication among the participating agents, and a distributed unified control which integrates primary and secondary frequency control and congestion management. Simulations show that the unified control not only achieves all the desired control goals in system equilibrium, but also improves the transient compared to traditional control schemes.}, author = {Zhao, Changhong and Mallada, Enrique and Low, Steven H and Bialek, Janusz W}, booktitle = {Power Systems Computation Conference}, doi = {10.1109/PSCC.2016.7541028}, keywords = {Power Networks; Frequency Control; Congestion Management}, month = {06}, pages = {1--7}, title = {A Unified Framework for Frequency Control and Congestion Management}, url = {https://mallada.ece.jhu.edu/pubs/2016-PSCC-ZMLB.pdf}, year = {2016} } - D. Cai, E. Mallada, and A. Wierman, “Distributed optimization decomposition for joint economic dispatch and frequency regulation,” in 54th IEEE Conference on Decision and Control (CDC), 2015, pp. 15-22. doi:10.1109/CDC.2015.7402081
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Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co- optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.
@inproceedings{cmw2015cdc, abstract = {Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co- optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.}, author = {Cai, Desmond and Mallada, Enrique and Wierman, Adam}, booktitle = {54th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2015.7402081}, keywords = {Power Networks;Markets; Optimization}, month = {12}, pages = {15-22}, title = {Distributed optimization decomposition for joint economic dispatch and frequency regulation}, url = {https://mallada.ece.jhu.edu/pubs/2015-CDC-CMW.pdf}, web = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7402081}, year = {2015} } - A. Cherukuri, E. Mallada, and J. Cortes, “Convergence of Caratheodory solutions for primal-dual dynamics in constrained concave optimization,” in SIAM Conference on Control and its Applications, 2015. doi:10.1137/1.9781611974072.40
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This paper characterizes the asymptotic convergence properties of the primal-dual dynamics to the solutions of a constrained concave optimization problem using classical notions from stability analysis. We motivate our study by providing an example which rules out the possibility of employing the invariance principle for hybrid automata to analyze the asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and establish their existence, uniqueness, and continuity with respect to the initial conditions. We employ the invariance principle for Caratheodory solutions of a discontinuous dynamical system to show that the primal-dual optimizers are globally asymptotically stable under the primal-dual dynamics and that each solution of the dynamics converges to an optimizer.
@inproceedings{cmc2015siam-ct, abstract = {This paper characterizes the asymptotic convergence properties of the primal-dual dynamics to the solutions of a constrained concave optimization problem using classical notions from stability analysis. We motivate our study by providing an example which rules out the possibility of employing the invariance principle for hybrid automata to analyze the asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and establish their existence, uniqueness, and continuity with respect to the initial conditions. We employ the invariance principle for Caratheodory solutions of a discontinuous dynamical system to show that the primal-dual optimizers are globally asymptotically stable under the primal-dual dynamics and that each solution of the dynamics converges to an optimizer.}, author = {Cherukuri, Ashish and Mallada, Enrique and Cortes, Jorge}, booktitle = {SIAM Conference on Control and its Applications}, doi = {10.1137/1.9781611974072.40}, keywords = {Saddle-Point Dynamics; Caratheodory solutions}, month = {07}, title = {Convergence of Caratheodory solutions for primal-dual dynamics in constrained concave optimization}, url = {https://mallada.ece.jhu.edu/pubs/2015-SIAM-CT-CMC.pdf}, web = {http://epubs.siam.org/doi/abs/10.1137/1.9781611974072.40}, year = {2015} } - A. Gushchin, E. Mallada, and A. Tang, “Synchronization of heterogeneous Kuramoto oscillators with arbitrary topology,” in American Control Conference (ACC), 2015. doi:10.1109/ACC.2015.7170807
[BibTeX] [Abstract] [Download PDF]
We study synchronization of coupled Kuramoto oscillators with heterogeneous inherent frequencies and general underlying connectivity. We provide conditions on the coupling strength and the initial phases which guarantee the existence of a Positively Invariant Set (PIS) and lead to synchronization. Unlike previous works that focus only on analytical bounds, here we introduce an optimization approach to provide a computational-analytical bound that can further exploit the particular features of each individual system such as topology and frequency distribution. Examples are provided to illustrate our results as well as the improvement over previous existing bounds.
@inproceedings{gmt2015acc, abstract = {We study synchronization of coupled Kuramoto oscillators with heterogeneous inherent frequencies and general underlying connectivity. We provide conditions on the coupling strength and the initial phases which guarantee the existence of a Positively Invariant Set (PIS) and lead to synchronization. Unlike previous works that focus only on analytical bounds, here we introduce an optimization approach to provide a computational-analytical bound that can further exploit the particular features of each individual system such as topology and frequency distribution. Examples are provided to illustrate our results as well as the improvement over previous existing bounds.}, author = {Gushchin, Andrey and Mallada, Enrique and Tang, Ao}, booktitle = {American Control Conference (ACC)}, doi = {10.1109/ACC.2015.7170807}, keywords = {Coupled Oscillators; Synchronization}, month = {07}, title = {Synchronization of heterogeneous Kuramoto oscillators with arbitrary topology}, url = {https://mallada.ece.jhu.edu/pubs/2015-ACC-GMT.pdf}, web = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7170807}, year = {2015} } - C. Zhao, E. Mallada, and F. Dorfler, “Distributed frequency control for stability and economic dispatch in power networks,” in American Control Conference (ACC), 2015, pp. 2359-2364. doi:10.1109/ACC.2015.7171085
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We explore two different frequency control strategies to ensure stability of power networks and achieve economic dispatch between generators and controllable loads. We first show the global asymptotic stability of a completely decentralized frequency integral control. Then we design a distributed averaging-based integral (DAI) control which operates by local frequency sensing and neighborhood communication. Equilibrium analysis shows that DAI recovers nominal frequency with minimum total generation cost and user disutility for load control after a change in generation or load. Local asymptotic stability of DAI is established with a Lyapunov method. Simulations demonstrate improvement in both transient and steady state performance achieved by the proposed control strategies, compared to droop control.
@inproceedings{zmd2015acc, abstract = {We explore two different frequency control strategies to ensure stability of power networks and achieve economic dispatch between generators and controllable loads. We first show the global asymptotic stability of a completely decentralized frequency integral control. Then we design a distributed averaging-based integral (DAI) control which operates by local frequency sensing and neighborhood communication. Equilibrium analysis shows that DAI recovers nominal frequency with minimum total generation cost and user disutility for load control after a change in generation or load. Local asymptotic stability of DAI is established with a Lyapunov method. Simulations demonstrate improvement in both transient and steady state performance achieved by the proposed control strategies, compared to droop control.}, author = {Zhao, Changhong and Mallada, Enrique and Dorfler, Florian}, booktitle = {American Control Conference (ACC)}, doi = {10.1109/ACC.2015.7171085}, keywords = {Power Networks; Synchronization}, month = {07}, pages = {2359--2364}, title = {Distributed frequency control for stability and economic dispatch in power networks}, url = {https://mallada.ece.jhu.edu/pubs/2015-ACC-ZMD.pdf}, web = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7171085}, year = {2015} } - A. Gushchin, E. Mallada, and A. Tang, “Synchronization of heterogeneous Kuramoto oscillators with graphs of diameter two,” in Conference on Information Sciences and Systems, 2015. doi:10.1109/CISS.2015.7086426
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In this article we study synchronization of Kuramoto oscillators with heterogeneous frequencies, and where underlying topology is a graph of diameter two. When the coupling strengths between every two connected oscillators are the same, we find an analytic condition that guarantees an existence of a Positively Invariant Set (PIS) and demonstrate that existence of a PIS suffices for frequency synchronization. For graphs of diameter two, this synchronization condition is significantly better than existing general conditions for an arbitrary topology. If the coupling strengths can be different for different pairs of connected oscillators, we formulate an optimization problem that finds sufficient for synchronization coupling strengths such that their sum is minimal.
@inproceedings{gmt2015ciss, abstract = {In this article we study synchronization of Kuramoto oscillators with heterogeneous frequencies, and where underlying topology is a graph of diameter two. When the coupling strengths between every two connected oscillators are the same, we find an analytic condition that guarantees an existence of a Positively Invariant Set (PIS) and demonstrate that existence of a PIS suffices for frequency synchronization. For graphs of diameter two, this synchronization condition is significantly better than existing general conditions for an arbitrary topology. If the coupling strengths can be different for different pairs of connected oscillators, we formulate an optimization problem that finds sufficient for synchronization coupling strengths such that their sum is minimal.}, author = {Gushchin, Andrey and Mallada, Enrique and Tang, Ao}, booktitle = {Conference on Information Sciences and Systems}, doi = {10.1109/CISS.2015.7086426}, keywords = {Coupled Oscillators; Synchronization}, month = {03}, title = {Synchronization of heterogeneous Kuramoto oscillators with graphs of diameter two}, url = {https://mallada.ece.jhu.edu/pubs/2015-CISS-GMT.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7086426}, year = {2015} } - C. Zhao, E. Mallada, and S. H. Low, “Distributed generator and load-side secondary frequency control in power networks,” in Conference on Information Sciences and Systems, 2015. doi:10.1109/CISS.2015.7086825
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We design distributed secondary frequency control scheme for both generators and controllable loads. The proposed scheme operates via local sensing and computation, and communication between neighbors. Equilibrium and stability analysis of the closed-loop system is performed with a power network model that includes turbines and governors of generators and nonlinear AC power flows. After an unexpected change in power supply or demand, the proposed control is able to stabilize the system, restore bus frequencies and net inter-area power exchanges, and minimize total generation cost minus user utility at equilibrium.
@inproceedings{zml2015ciss, abstract = {We design distributed secondary frequency control scheme for both generators and controllable loads. The proposed scheme operates via local sensing and computation, and communication between neighbors. Equilibrium and stability analysis of the closed-loop system is performed with a power network model that includes turbines and governors of generators and nonlinear AC power flows. After an unexpected change in power supply or demand, the proposed control is able to stabilize the system, restore bus frequencies and net inter-area power exchanges, and minimize total generation cost minus user utility at equilibrium.}, author = {Zhao, Changhong and Mallada, Enrique and Low, Steven H}, booktitle = {Conference on Information Sciences and Systems}, doi = {10.1109/CISS.2015.7086825}, keywords = {Power Networks; Synchronization}, month = {03}, title = {Distributed generator and load-side secondary frequency control in power networks}, url = {https://mallada.ece.jhu.edu/pubs/2015-CISS-ZML.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7086825}, year = {2015} } - A. Gushchin, E. Mallada, and A. Tang, “Synchronization of phase-coupled oscillators with plastic coupling strength,” in Information Theory and Applications Workshop (ITA), 2015, pp. 291-300. doi:10.1109/ITA.2015.7309003
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In this article we study synchronization of systems of homogeneous phase-coupled oscillators with plastic coupling strengths and arbitrary underlying topology. The dynamics of the coupling strength between two oscillators is governed by the phase difference between these oscillators. We show that, under mild assumptions, such systems are gradient systems, and always achieve frequency synchronization. Furthermore, we provide sufficient stability and instability conditions that are based on results from algebraic graph theory. For a special case when underlying topology is a tree, we formulate a criterion (necessary and sufficient condition) of stability of equilibria. For both, tree and arbitrary topologies, we provide sufficient conditions for phase-locking, i.e. convergence to a stable equilibrium almost surely. We additionally find conditions when the system possesses a unique stable equilibrium, and thus, almost global stability follows. Several examples are used to demonstrate variety of equilibria the system has, their dependence on system’s parameters, and to illustrate differences in behavior of systems with constant and plastic coupling strengths.
@inproceedings{gmt2015ita, abstract = {In this article we study synchronization of systems of homogeneous phase-coupled oscillators with plastic coupling strengths and arbitrary underlying topology. The dynamics of the coupling strength between two oscillators is governed by the phase difference between these oscillators. We show that, under mild assumptions, such systems are gradient systems, and always achieve frequency synchronization. Furthermore, we provide sufficient stability and instability conditions that are based on results from algebraic graph theory. For a special case when underlying topology is a tree, we formulate a criterion (necessary and sufficient condition) of stability of equilibria. For both, tree and arbitrary topologies, we provide sufficient conditions for phase-locking, i.e. convergence to a stable equilibrium almost surely. We additionally find conditions when the system possesses a unique stable equilibrium, and thus, almost global stability follows. Several examples are used to demonstrate variety of equilibria the system has, their dependence on system's parameters, and to illustrate differences in behavior of systems with constant and plastic coupling strengths.}, author = {Gushchin, Andrey and Mallada, Enrique and Tang, Ao}, booktitle = {Information Theory and Applications Workshop (ITA)}, doi = {10.1109/ITA.2015.7309003}, keywords = {Synchronization}, month = {02}, pages = {291-300}, title = {Synchronization of phase-coupled oscillators with plastic coupling strength}, url = {https://mallada.ece.jhu.edu/pubs/2015-ITA-GMT.pdf}, year = {2015} } - E. Mallada, C. Zhao, and S. H. Low, “Optimal load-side control for frequency regulation in smart grids,” in 52nd Allerton Conference on Communication, Control, and Computing, 2014, p. 731—738. doi:10.1109/ALLERTON.2014.7028527
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Frequency control rebalances supply and demand while maintaining the network state within operational margins. It is implemented using fast ramping reserves that are expensive and non-renewable, and which are expected to grow with the increasing penetration of renewables. The most promising solution to this problem is the use of demand response, i.e. load participation in frequency control. Yet it is still unclear how to efficiently integrate load participation without introducing instabilities and violating operational constraints. In this paper we present a comprehensive load-side frequency control mechanism that can maintain the grid within operational constraints. Our controllers can rebalance supply and demand after disturbances, restore the frequency to its nominal value and preserve inter-area power flows. Furthermore, our controllers are distributed (unlike generation-side), fair among participating loads, and can further maintain line flows within thermal limits. We prove that such a distributed load-side control is globally asymptotically stable and illustrate its convergence with simulation.
@inproceedings{mzl2014allerton, abstract = {Frequency control rebalances supply and demand while maintaining the network state within operational margins. It is implemented using fast ramping reserves that are expensive and non-renewable, and which are expected to grow with the increasing penetration of renewables. The most promising solution to this problem is the use of demand response, i.e. load participation in frequency control. Yet it is still unclear how to efficiently integrate load participation without introducing instabilities and violating operational constraints. In this paper we present a comprehensive load-side frequency control mechanism that can maintain the grid within operational constraints. Our controllers can rebalance supply and demand after disturbances, restore the frequency to its nominal value and preserve inter-area power flows. Furthermore, our controllers are distributed (unlike generation-side), fair among participating loads, and can further maintain line flows within thermal limits. We prove that such a distributed load-side control is globally asymptotically stable and illustrate its convergence with simulation.}, author = {Mallada, Enrique and Zhao, Changhong and Low, Steven H}, booktitle = {52nd Allerton Conference on Communication, Control, and Computing}, doi = {10.1109/ALLERTON.2014.7028527}, keywords = {Power Networks}, month = {10}, pages = {731---738}, title = {Optimal load-side control for frequency regulation in smart grids}, url = {https://mallada.ece.jhu.edu/pubs/2014-Allerton-MZL.pdf}, web = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=7028527}, year = {2014} } - E. Mallada and S. H. Low, “Distributed frequency-preserving optimal load control,” in IFAC World Congress, 2014, pp. 5411-5418. doi:10.3182/20140824-6-ZA-1003.02012
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Frequency control is traditionally done on the generation side. Recently we have formulated an optimal load control (OLC) problem and derived a load-side primary frequency control as a primal-dual solution to OLC. The load-side control rebalances power and resynchronize frequencies after a disturbance but cannot restore the nominal frequency. In this paper we modify OLC and derive from it a frequency-preserving load-side control that rebalances power and restores the nominal frequency after a disturbance. Unlike the generation-side secondary frequency control that is centralized, our load-side control only requires each bus to communicate a Lagrange multiplier with its neighbors. We prove that such a distributed load-side control is globally asymptotically stable and illustrate its convergence with simulation.
@inproceedings{ml2014ifac-wc, abstract = {Frequency control is traditionally done on the generation side. Recently we have formulated an optimal load control (OLC) problem and derived a load-side primary frequency control as a primal-dual solution to OLC. The load-side control rebalances power and resynchronize frequencies after a disturbance but cannot restore the nominal frequency. In this paper we modify OLC and derive from it a frequency-preserving load-side control that rebalances power and restores the nominal frequency after a disturbance. Unlike the generation-side secondary frequency control that is centralized, our load-side control only requires each bus to communicate a Lagrange multiplier with its neighbors. We prove that such a distributed load-side control is globally asymptotically stable and illustrate its convergence with simulation.}, author = {Mallada, Enrique and Low, Steven H}, booktitle = {IFAC World Congress}, doi = {10.3182/20140824-6-ZA-1003.02012}, keywords = {Power Networks; Optimization}, month = {08}, pages = {5411--5418}, title = {Distributed frequency-preserving optimal load control}, url = {https://mallada.ece.jhu.edu/pubs/2014-IFAC-WC-ML.pdf}, year = {2014} } - E. Mallada and A. Tang, “Dynamics-aware optimal power flow,” in 52nd IEEE Conference on Decision and Control (CDC), 2013. doi:10.1109/CDC.2013.6760118
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The development of open electricity markets has led to a decoupling between the market clearing procedure that defines the power dispatch and the security analysis that enforces predefined stability margins. This gap results in market inefficiencies introduced by corrections to the market solution to accommodate stability requirements. In this paper we present an optimal power flow formulation that aims to close this gap. First, we show that the pseudospectral abscissa can be used as a unifying stability measure to characterize both poorly damped oscillations and voltage stability margins. This leads to two novel optimization problems that can find operation points which minimize oscillations or maximize voltage stability margins, and make apparent the implicit tradeoff between these two stability requirements. Finally, we combine these optimization problems to generate a dynamics-aware optimal power flow formulation that provides voltage as well as small signal stability guarantees.
@inproceedings{mt2013cdc, abstract = {The development of open electricity markets has led to a decoupling between the market clearing procedure that defines the power dispatch and the security analysis that enforces predefined stability margins. This gap results in market inefficiencies introduced by corrections to the market solution to accommodate stability requirements. In this paper we present an optimal power flow formulation that aims to close this gap. First, we show that the pseudospectral abscissa can be used as a unifying stability measure to characterize both poorly damped oscillations and voltage stability margins. This leads to two novel optimization problems that can find operation points which minimize oscillations or maximize voltage stability margins, and make apparent the implicit tradeoff between these two stability requirements. Finally, we combine these optimization problems to generate a dynamics-aware optimal power flow formulation that provides voltage as well as small signal stability guarantees.}, author = {Mallada, Enrique and Tang, Ao}, booktitle = {52nd IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2013.6760118}, keywords = {Power Networks; Optimization}, month = {12}, title = {Dynamics-aware optimal power flow}, url = {https://mallada.ece.jhu.edu/pubs/2013-CDC-MT.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6760118}, year = {2013} } - E. Mallada, X. Meng, M. Hack, L. Zhang, and A. Tang, “Skewless network clock synchronization,” in 21st IEEE International Conference on Network Protocols (ICNP), 2013, pp. 1-10. doi:10.1109/ICNP.2013.6733612
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This paper examines synchronization of computer clocks connected via a data network and proposes a skewless algorithm to synchronize them. Unlike existing solutions, which either estimate and compensate the frequency difference (skew) among clocks or introduce offset corrections that can generate jitter and possibly even backward jumps, our algorithm achieves synchronization without these problems. We first analyze the convergence property of the algorithm and provide necessary and sufficient conditions on the parameters to guarantee synchronization. We then implement our solution on a cluster of IBM BladeCenter servers running Linux and study its performance. In particular, both analytically and experimentally, we show that our algorithm can converge in the presence of timing loops. This marks a clear contrast with current standards such as NTP and PTP, where timing loops are specifically avoided. Furthermore, timing loops can even be beneficial in our scheme. For example, it is demonstrated that highly connected subnetworks can collectively outperform individual clients when the time source has large jitter. It is also experimentally demonstrated that our algorithm outperforms other well-established software-based solutions such as the NTPv4 and IBM Coordinated Cluster Time (IBM CCT).
@inproceedings{mmhzt2013icnp, abstract = {This paper examines synchronization of computer clocks connected via a data network and proposes a skewless algorithm to synchronize them. Unlike existing solutions, which either estimate and compensate the frequency difference (skew) among clocks or introduce offset corrections that can generate jitter and possibly even backward jumps, our algorithm achieves synchronization without these problems. We first analyze the convergence property of the algorithm and provide necessary and sufficient conditions on the parameters to guarantee synchronization. We then implement our solution on a cluster of IBM BladeCenter servers running Linux and study its performance. In particular, both analytically and experimentally, we show that our algorithm can converge in the presence of timing loops. This marks a clear contrast with current standards such as NTP and PTP, where timing loops are specifically avoided. Furthermore, timing loops can even be beneficial in our scheme. For example, it is demonstrated that highly connected subnetworks can collectively outperform individual clients when the time source has large jitter. It is also experimentally demonstrated that our algorithm outperforms other well-established software-based solutions such as the NTPv4 and IBM Coordinated Cluster Time (IBM CCT).}, author = {Mallada, Enrique and Meng, Xiaoqiao and Hack, Michel and Zhang, Li and Tang, Ao}, booktitle = {21st IEEE International Conference on Network Protocols (ICNP)}, doi = {10.1109/ICNP.2013.6733612}, keywords = {Networking; Synchronization}, month = {10}, pages = {1--10}, title = {Skewless network clock synchronization}, url = {https://mallada.ece.jhu.edu/pubs/2013-ICNP-MMHZT.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6733612}, year = {2013} } - M. Wang, W. Xu, E. Mallada, and A. Tang, “Sparse recovery with graph constraints: Fundamental limits and measurement construction,” in Proceedings of IEEE Infocom, 2012, pp. 1871-1879. doi:10.1109/INFCOM.2012.6195562
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This paper addresses the problem of sparse recovery with graph constraints in the sense that we can take additive measurements over nodes only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs. A general measurement construction algorithm is also proposed and evaluated. For any given graph G with n nodes, we derive order optimal upper bounds of the minimum number of measurements needed to recover any k-sparse vector over $G (M_k,n^G)$. Our study suggests that $M_k,n^G$ may serve as a graph connectivity metric.
@inproceedings{wxmt2012infocom, abstract = {This paper addresses the problem of sparse recovery with graph constraints in the sense that we can take additive measurements over nodes only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs. A general measurement construction algorithm is also proposed and evaluated. For any given graph G with n nodes, we derive order optimal upper bounds of the minimum number of measurements needed to recover any k-sparse vector over $G (M_k,n^G)$. Our study suggests that $M_k,n^G$ may serve as a graph connectivity metric.}, author = {Wang, Meng and Xu, Weiyu and Mallada, Enrique and Tang, Ao}, booktitle = {Proceedings of IEEE Infocom}, doi = {10.1109/INFCOM.2012.6195562}, keywords = {Sparse Recovery}, month = {03}, pages = {1871--1879}, title = {Sparse recovery with graph constraints: Fundamental limits and measurement construction}, url = {https://mallada.ece.jhu.edu/pubs/2012-Infocom-WXMT.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6195562}, year = {2012} } - E. Mallada and A. Tang, “Improving damping of power networks: Power scheduling and impedance adaptation,” in 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011, pp. 7729-7734. doi:10.1109/CDC.2011.6161287
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This paper studies the effect of power scheduling and line impedances on the damping of a power network. We relate the damping of a network with the algebraic connectivity of a state dependent Laplacian. Via implicit function theorem, we further characterize its dependence on network parameters. This allows us to derive several updating directions that can locally improve the damping. The analysis also provides some interesting insight. For example, improving connectivity, by adding lines for instance, may not be beneficial in terms of damping.
@inproceedings{mt2011cdcb, abstract = {This paper studies the effect of power scheduling and line impedances on the damping of a power network. We relate the damping of a network with the algebraic connectivity of a state dependent Laplacian. Via implicit function theorem, we further characterize its dependence on network parameters. This allows us to derive several updating directions that can locally improve the damping. The analysis also provides some interesting insight. For example, improving connectivity, by adding lines for instance, may not be beneficial in terms of damping.}, author = {Mallada, Enrique and Tang, Ao}, booktitle = {50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC)}, doi = {10.1109/CDC.2011.6161287}, keywords = {Power Networks; Optimization}, month = {12}, pages = {7729--7734}, title = {Improving damping of power networks: Power scheduling and impedance adaptation}, url = {https://mallada.ece.jhu.edu/pubs/2011-CDCb-MT.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6161287}, year = {2011} } - E. Mallada and A. Tang, “Distributed clock synchronization: Joint frequency and phase consensus,” in 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011, pp. 6742-6747. doi:10.1109/CDC.2011.6161231
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Distributed synchronization has gradually gained importance over the last two decades. The ad-hoc nature of new applications has increased the need for robust and scalable distributed algorithms that are capable of generating high precision timing information. However, current solutions usually produce phase errors when the frequencies are heterogeneous. This paper proposes a distributed synchronization procedure that can achieve consensus in both frequency and phase. The algorithm uses only local information and is robust to frequency heterogeneity and network topology. A sufficient condition for global convergence is shown by leveraging recent results on coupled oscillators. We further characterize an invariant constant of the algorithm that relates the limiting frequency $ømega^*$ with the harmonic mean of the clocks’ natural frequencies. Simulations are provided to illustrate and verify these properties.
@inproceedings{mt2011cdca, abstract = {Distributed synchronization has gradually gained importance over the last two decades. The ad-hoc nature of new applications has increased the need for robust and scalable distributed algorithms that are capable of generating high precision timing information. However, current solutions usually produce phase errors when the frequencies are heterogeneous. This paper proposes a distributed synchronization procedure that can achieve consensus in both frequency and phase. The algorithm uses only local information and is robust to frequency heterogeneity and network topology. A sufficient condition for global convergence is shown by leveraging recent results on coupled oscillators. We further characterize an invariant constant of the algorithm that relates the limiting frequency $ømega^*$ with the harmonic mean of the clocks' natural frequencies. Simulations are provided to illustrate and verify these properties.}, author = {Mallada, Enrique and Tang, Ao}, booktitle = {50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC)}, doi = {10.1109/CDC.2011.6161231}, keywords = {Coupled Oscillators; Synchronization}, month = {12}, pages = {6742--6747}, title = {Distributed clock synchronization: Joint frequency and phase consensus}, url = {https://mallada.ece.jhu.edu/pubs/2011-CDCa-MT.pdf}, web = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6161231}, year = {2011} } - W. Xu, M. Wang, E. Mallada, and A. Tang, “Recent results on sparse recovery over graphs,” in Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), 2011, pp. 413-417. doi:10.1109/ACSSC.2011.6190031
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In this paper, we review our recent results on sparse recovery over graphs, which was motivated by network tomography problems. Our finding has made a new connection between coding theory and graph theory. We also discuss robustness of our proposed measurement construction.
@inproceedings{xwmt2011asilomar, abstract = {In this paper, we review our recent results on sparse recovery over graphs, which was motivated by network tomography problems. Our finding has made a new connection between coding theory and graph theory. We also discuss robustness of our proposed measurement construction.}, author = {Xu, Weiyu and Wang, Meng and Mallada, Enrique and Tang, Ao}, booktitle = {Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)}, doi = {10.1109/ACSSC.2011.6190031}, keywords = {Sparse Recovery}, month = {11}, pages = {413--417}, title = {Recent results on sparse recovery over graphs}, url = {https://mallada.ece.jhu.edu/pubs/2011-Asilomar-XWMT.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6190031}, year = {2011} } - W. Xu, E. Mallada, and A. Tang, “Compressive sensing over graphs,” in Proceeding of IEEE Infocom, 2011, pp. 2087-2095. doi:10.1109/INFCOM.2011.5935018
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In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with n nodes, it is shown that, using O(k log(n)) path measurements, we are able to recover any k-sparse link vector (with no more than k nonzero elements), even though the measurements have to follow the graph path constraints. We mainly show that the computationally efficient ℓ1 minimization can provide theoretical guarantees for inferring such k-sparse vectors with O(k log(n)) path measurements from the graph.
@inproceedings{xmt2011infocom, abstract = {In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with n nodes, it is shown that, using O(k log(n)) path measurements, we are able to recover any k-sparse link vector (with no more than k nonzero elements), even though the measurements have to follow the graph path constraints. We mainly show that the computationally efficient ℓ1 minimization can provide theoretical guarantees for inferring such k-sparse vectors with O(k log(n)) path measurements from the graph.}, author = {Xu, Weiyu and Mallada, Enrique and Tang, Ao}, booktitle = {Proceeding of IEEE Infocom}, doi = {10.1109/INFCOM.2011.5935018}, keywords = {Sparse Recovery}, month = {03}, organization = {IEEE}, pages = {2087--2095}, title = {Compressive sensing over graphs}, url = {https://mallada.ece.jhu.edu/pubs/2011-Infocom-XMT.pdf}, year = {2011} } - E. Mallada and A. Tang, “Weakly pulse-coupled oscillators: Heterogeneous delays lead to homogeneous phase,” in 49th IEEE Conference on Decision and Control (CDC), 2010, pp. 992-997. doi:10.1109/CDC.2010.5717864
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This paper studies the effect of heterogeneous delays in networks of weakly pulse-coupled identical oscillators. We develop a new framework to study them by constructing a non-delayed phase model that is equivalent to the original one in the continuum limit. Using existing results for non-delayed phase-coupled oscillators we analyze the delayed system and show how its stability properties depend on the delay distribution. In particular, we show that in some scenarios, heterogeneity, i.e. wider delay distribution, can help reach in-phase synchronization.
@inproceedings{mt2010cdc, abstract = {This paper studies the effect of heterogeneous delays in networks of weakly pulse-coupled identical oscillators. We develop a new framework to study them by constructing a non-delayed phase model that is equivalent to the original one in the continuum limit. Using existing results for non-delayed phase-coupled oscillators we analyze the delayed system and show how its stability properties depend on the delay distribution. In particular, we show that in some scenarios, heterogeneity, i.e. wider delay distribution, can help reach in-phase synchronization.}, author = {Mallada, Enrique and Tang, Ao}, booktitle = {49th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2010.5717864}, keywords = {Coupled Oscillators; Synchronization}, month = {12}, pages = {992--997}, title = {Weakly pulse-coupled oscillators: Heterogeneous delays lead to homogeneous phase}, url = {https://mallada.ece.jhu.edu/pubs/2010-CDC-MT.pdf}, web = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=5717864}, year = {2010} } - E. Mallada and A. Tang, “Synchronization of phase-coupled oscillators with arbitrary topology,” in 2010 American Control Conference (ACC), 2010, pp. 1777-1782. doi:10.1109/ACC.2010.5531468
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This paper studies networks of identical phase-coupled oscillators with arbitrary underlying connected graph. By using results from algebraic graph theory, a sufficient condition is obtained which can be used to check equilibrium stability. This condition generalizes existing results and can solve some previously unsolved cases. It also leads to the first sufficient condition on the coupling function with which the system is guaranteed to reach synchronization. Throughout the paper, several examples are used to verify and illustrate the theory. We also correct some mistakes in the existing literature.
@inproceedings{mt2010acc, abstract = {This paper studies networks of identical phase-coupled oscillators with arbitrary underlying connected graph. By using results from algebraic graph theory, a sufficient condition is obtained which can be used to check equilibrium stability. This condition generalizes existing results and can solve some previously unsolved cases. It also leads to the first sufficient condition on the coupling function with which the system is guaranteed to reach synchronization. Throughout the paper, several examples are used to verify and illustrate the theory. We also correct some mistakes in the existing literature.}, author = {Mallada, Enrique and Tang, Ao}, booktitle = {2010 American Control Conference (ACC)}, doi = {10.1109/ACC.2010.5531468}, keywords = {Coupled Oscillators; Synchronization}, month = {06}, pages = {1777--1782}, title = {Synchronization of phase-coupled oscillators with arbitrary topology}, url = {https://mallada.ece.jhu.edu/pubs/2010-ACC-MT.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5531468}, year = {2010} } - E. Mallada and F. Paganini, “Stability of node-based multipath routing and dual congestion control,” in 47th IEEE Conference on Decision and Control (CDC), 2008, pp. 1398-1403. doi:10.1109/CDC.2008.4739209
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This paper considers a network flow control problem where routing and input rates are controlled in a decentralized way across a network, to optimize a global welfare objective. We build on our recent work which combines “dual” congestion control for the traffic sources, with multipath routing at the router nodes, controlling the traffic split among outgoing links based on downstream congestion prices. The challenge is to obtain stabilization of the optimum point; in fact, controlling the split fractions following the price gradient has the correct equilibrium, but can lead to oscillatory instabilities. This suggests the use of derivative action to damp such oscillations. We study two alternatives in this regard; either anticipatory control of routing splits, which yields local stability in an arbitrary network topology, or anticipatory price generation, which yields a global result for the case of a network of parallel links. Proofs are based on a Lyapunov argument. Results are illustrated through simulations.
@inproceedings{mp2008cdc, abstract = {This paper considers a network flow control problem where routing and input rates are controlled in a decentralized way across a network, to optimize a global welfare objective. We build on our recent work which combines ``dual'' congestion control for the traffic sources, with multipath routing at the router nodes, controlling the traffic split among outgoing links based on downstream congestion prices. The challenge is to obtain stabilization of the optimum point; in fact, controlling the split fractions following the price gradient has the correct equilibrium, but can lead to oscillatory instabilities. This suggests the use of derivative action to damp such oscillations. We study two alternatives in this regard; either anticipatory control of routing splits, which yields local stability in an arbitrary network topology, or anticipatory price generation, which yields a global result for the case of a network of parallel links. Proofs are based on a Lyapunov argument. Results are illustrated through simulations.}, author = {Mallada, Enrique and Paganini, Fernando}, booktitle = {47th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC.2008.4739209}, keywords = {Networking}, month = {12}, pages = {1398--1403}, title = {Stability of node-based multipath routing and dual congestion control}, url = {https://mallada.ece.jhu.edu/pubs/2008-CDC-MP.pdf}, web = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4739209&tag=1}, year = {2008} } - E. Mallada and F. Paganini, “Optimal congestion control with multipath routing using TCP-FAST and a variant of RIP,” in NET-COOP, 2007, pp. 205-214. doi:10.1007/978-3-540-72709-5_22
[BibTeX] [Abstract] [Download PDF]
This paper discusses an optimization-based approach for congestion control together with multipath routing in a TCP/IP network. In recent research we have shown how natural optimization problems for resource allocation can be solved in decentralized way across a network, by traffic sources adapting their rates and routers adapting their traffic splits, all using a common congestion measure. We present here a concrete implementation of such algorithms, based on queueing delay as congestion price. We use TCP-FAST for congestion control, and develop a multipath variant of the distance vector routing protocol RIP. We demonstrate through ns2-simulations the collective behavior of the system, in particular that it reaches the higher transfer rates available through multiple routes.
@inproceedings{mp2007netcoop, abstract = {This paper discusses an optimization-based approach for congestion control together with multipath routing in a TCP/IP network. In recent research we have shown how natural optimization problems for resource allocation can be solved in decentralized way across a network, by traffic sources adapting their rates and routers adapting their traffic splits, all using a common congestion measure. We present here a concrete implementation of such algorithms, based on queueing delay as congestion price. We use TCP-FAST for congestion control, and develop a multipath variant of the distance vector routing protocol RIP. We demonstrate through ns2-simulations the collective behavior of the system, in particular that it reaches the higher transfer rates available through multiple routes.}, author = {Mallada, Enrique and Paganini, Fernando}, booktitle = {NET-COOP}, doi = {10.1007/978-3-540-72709-5_22}, keywords = {Networking}, month = {06}, note = {also in Lecture Notes in Computer Science 4465, Springer-Verlag, Berlin-Heidelberg}, pages = {205--214}, title = {Optimal congestion control with multipath routing using TCP-FAST and a variant of RIP}, url = {https://mallada.ece.jhu.edu/pubs/2007-NETCOOP-MP.pdf}, year = {2007} } - F. Paganini and E. Mallada, “Congestion pricing for flow control and multipath routing in TCP/IP networks,” in Congreso Latinoamericano de Investigación Operativa, 2006.
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We consider a TCP/IP style network, in which end-systems control their traffic rates based on congestion feedback, and routing is performed at intermediate nodes on a perdestination basis; extending standard IP, routers are allowed to use multiple outgoing links per destination. We pose two optimization problems, that generalize and combine those in congestion control and traffic engineering, with variables which are local to either sources or routers. We obtain decentralized algorithms by propagating price variables and using them to control source rates and router traffic splits; we give conditions for convergence of these algorithms to optimal points. Keywords: congestion control, routing, TCP/IP, optimization, pricing.
@inproceedings{pm2006claio, abstract = {We consider a TCP/IP style network, in which end-systems control their traffic rates based on congestion feedback, and routing is performed at intermediate nodes on a perdestination basis; extending standard IP, routers are allowed to use multiple outgoing links per destination. We pose two optimization problems, that generalize and combine those in congestion control and traffic engineering, with variables which are local to either sources or routers. We obtain decentralized algorithms by propagating price variables and using them to control source rates and router traffic splits; we give conditions for convergence of these algorithms to optimal points. Keywords: congestion control, routing, TCP/IP, optimization, pricing.}, author = {Paganini, Fernando and Mallada, Enrique}, booktitle = {Congreso Latinoamericano de Investigación Operativa}, keywords = {Networking}, month = {11}, title = {Congestion pricing for flow control and multipath routing in TCP/IP networks}, url = {https://mallada.ece.jhu.edu/pubs/2006-CLAIO-MP.pdf}, year = {2006} }
Abstracts
- C. Ji, M. H. Hajiesmaili, D. F. Gayme, and E. Mallada, Coordinating Distribution System Resources for Co-optimized Participation in Energy and Ancillary Service Transmission System Markets, 2018.
[BibTeX] [Download PDF]@misc{jhgm2018ferc-taic, author = {Ji, Chengda and Hajiesmaili, Mohammad H. and Gayme, Dennice F. and Mallada, Enrique}, grants = {CAREER-1752362, ENERGISE-DE-EE0008006, EPCN-1711188}, howpublished = {FERC Trans-Atlantic Infraday Workshop}, month = {11}, title = {Coordinating Distribution System Resources for Co-optimized Participation in Energy and Ancillary Service Transmission System Markets}, url = {https://mallada.ece.jhu.edu/pubs/2018-FERC-TAIC-JHGM.pdf}, year = {2018} }
Group Theses
- Y. Shen, “Learning Safe Regions in High-dimensional Dynamical Systems via Recurrent Sets,” PhD Thesis, 2024.
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Safety certification in dynamical systems has relied heavily on the identification of an invariant set that strictly requires trajectories to always lie within it. This foundational approach addresses two key safety objectives: stability and avoidance. To this end, techniques such as the Lyapunov method and the barrier function method that are capable of characterizing an invariant set have been pivotal. However, the invariance requirement poses strict constraints on the learning outcome. Therefore, as the system dimension increases, directly characterizing invariant sets or identifying functions that define invariant sets typically demands considerable domain-specific knowledge or extensive computational resources. This thesis seeks to develop new data-driven methodologies that facilitate the verification of stability and avoidance in dynamical systems, without relying on the identification of invariant sets. A key innovation of this thesis is the application of the concept of recurrence to relax the stringent constraints imposed by invariance. Specifically, a set is recurrent if trajectories originating from it return to it infinitely often. By leveraging recurrence, safety can be characterized with enhanced efficiency and accuracy. This thesis theoretically establishes necessary and sufficient conditions for using recurrence to characterize safety, offering a deeper understanding of how recurrence can serve as a reliable proxy for invariance. Practically, it introduces practical, data-driven algorithms that utilize only a finite number of finite-length sampled trajectories to determine safe regions within a dynamical system. Optimized for computational efficiency, these algorithms can be implemented on parallel processing units, making them highly applicable in real-world scenarios where rapid and reliable safety verification is crucial.
@phdthesis{s2023phd-thesis, abstract = {Safety certification in dynamical systems has relied heavily on the identification of an invariant set that strictly requires trajectories to always lie within it. This foundational approach addresses two key safety objectives: stability and avoidance. To this end, techniques such as the Lyapunov method and the barrier function method that are capable of characterizing an invariant set have been pivotal. However, the invariance requirement poses strict constraints on the learning outcome. Therefore, as the system dimension increases, directly characterizing invariant sets or identifying functions that define invariant sets typically demands considerable domain-specific knowledge or extensive computational resources. This thesis seeks to develop new data-driven methodologies that facilitate the verification of stability and avoidance in dynamical systems, without relying on the identification of invariant sets. A key innovation of this thesis is the application of the concept of recurrence to relax the stringent constraints imposed by invariance. Specifically, a set is recurrent if trajectories originating from it return to it infinitely often. By leveraging recurrence, safety can be characterized with enhanced efficiency and accuracy. This thesis theoretically establishes necessary and sufficient conditions for using recurrence to characterize safety, offering a deeper understanding of how recurrence can serve as a reliable proxy for invariance. Practically, it introduces practical, data-driven algorithms that utilize only a finite number of finite-length sampled trajectories to determine safe regions within a dynamical system. Optimized for computational efficiency, these algorithms can be implemented on parallel processing units, making them highly applicable in real-world scenarios where rapid and reliable safety verification is crucial.}, author = {Shen, Yue}, month = {9}, school = {Electrical and Computer Engineering, Johns Hopkins University}, title = {Learning Safe Regions in High-dimensional Dynamical Systems via Recurrent Sets}, url = {https://mallada.ece.jhu.edu/pubs/2024-Thesis-Shen.pdf}, year = {2024} } - R. K. Bansal, “Efficiency and Market Power in Electricity Markets with Inelastic Demand, Energy Storage, and Hybrid Energy Resources,” PhD Thesis, 2023.
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This thesis studies electricity markets and analyzes market mechanisms – operation rules for participants in achieving supply-demand balance, to understand how competition between resources, e.g., conventional generators, inelastic loads, and new market participants, such as energy storage and hybrid resources, affects market efficiency. We first consider the participation of conventional resources in a two-stage market, i.e., day-ahead and real-time settlement. Although designed to allocate resources efficiently and prevent speculation, price manipulation by strategic participants can undermine these goals. To address price manipulation, some markets have proposed system-level market power mitigation (MPM) policies, which substitute noncompetitive bids with default bids based on estimated generation costs. Using equilibrium analysis, we illustrate that such a policy in the day-ahead stage is more robust to price manipulations than in real-time, which may lead to non-equilibrium solutions. Despite being inelastic, loads can shift their allocation between the two stages to manipulate prices and reduce their payments. Further, heterogeneity in cost coefficients, estimation of dispatch cost in excess, and demand uncertainty tend to benefit generators. Together, system-level MPM policies can have unintended consequences when implemented without accounting for the conflicting interests of participants. We then study how integrating energy storage affects market efficiency. Our analysis indicates that the existing participation mechanism, where energy storage bids power in a market, may diminish market benefits due to its unique operational characteristics, e.g., the operating cost depends on charge-discharge cycles, unlike conventional generators. We propose a novel market mechanism based on an energy-cycling function that maps cycle depth to per-cycle prices. An equilibrium analysis illustrates the efficient competitive equilibrium that aligns with the social planner problem, i.e., net surplus maximization. However, at the Nash equilibrium, storage incurs reductions in profit relative to the competitive equilibrium due to price manipulation by strategic generators. Finally, we study the participation of hybrid resources combining energy storage and renewable energy sources. We use the New York zonal model as an example of a large-scale electricity market to benchmark the performance of the following two types of market models. We consider (i) a granular model, where the market operator manages the operation of constituent energy storage, and (ii) an integrated model, where the owner manages the storage operation. Our analysis shows that granular models lead to lower operating costs but add computational complexity, which may not be desirable from the operator’s perspective. Though less computationally intensive, integrated models result in more intervals violating the physical limits of constituent energy storage.
@phdthesis{b2023phd-thesis, abstract = {This thesis studies electricity markets and analyzes market mechanisms - operation rules for participants in achieving supply-demand balance, to understand how competition between resources, e.g., conventional generators, inelastic loads, and new market participants, such as energy storage and hybrid resources, affects market efficiency. We first consider the participation of conventional resources in a two-stage market, i.e., day-ahead and real-time settlement. Although designed to allocate resources efficiently and prevent speculation, price manipulation by strategic participants can undermine these goals. To address price manipulation, some markets have proposed system-level market power mitigation (MPM) policies, which substitute noncompetitive bids with default bids based on estimated generation costs. Using equilibrium analysis, we illustrate that such a policy in the day-ahead stage is more robust to price manipulations than in real-time, which may lead to non-equilibrium solutions. Despite being inelastic, loads can shift their allocation between the two stages to manipulate prices and reduce their payments. Further, heterogeneity in cost coefficients, estimation of dispatch cost in excess, and demand uncertainty tend to benefit generators. Together, system-level MPM policies can have unintended consequences when implemented without accounting for the conflicting interests of participants. We then study how integrating energy storage affects market efficiency. Our analysis indicates that the existing participation mechanism, where energy storage bids power in a market, may diminish market benefits due to its unique operational characteristics, e.g., the operating cost depends on charge-discharge cycles, unlike conventional generators. We propose a novel market mechanism based on an energy-cycling function that maps cycle depth to per-cycle prices. An equilibrium analysis illustrates the efficient competitive equilibrium that aligns with the social planner problem, i.e., net surplus maximization. However, at the Nash equilibrium, storage incurs reductions in profit relative to the competitive equilibrium due to price manipulation by strategic generators. Finally, we study the participation of hybrid resources combining energy storage and renewable energy sources. We use the New York zonal model as an example of a large-scale electricity market to benchmark the performance of the following two types of market models. We consider (i) a granular model, where the market operator manages the operation of constituent energy storage, and (ii) an integrated model, where the owner manages the storage operation. Our analysis shows that granular models lead to lower operating costs but add computational complexity, which may not be desirable from the operator's perspective. Though less computationally intensive, integrated models result in more intervals violating the physical limits of constituent energy storage.}, author = {Bansal, Rajni Kant}, month = {9}, school = {Mechanical Engineering, Johns Hopkins University}, title = {Efficiency and Market Power in Electricity Markets with Inelastic Demand, Energy Storage, and Hybrid Energy Resources}, url = {https://mallada.ece.jhu.edu/pubs/2023-Thesis-Bansal.pdf}, year = {2023} } - H. Min, “Exploiting Structural Properties in the Analysis of High-dimensional Dynamical Systems,” PhD Thesis, 2023.
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The physical and cyber domains with which we interact are filled with high-dimensional dynamical systems. In machine learning, for instance, the evolution of overparametrized neural networks can be seen as a dynamical system. In networked systems, numerous agents or nodes dynamically interact with each other. A deep understanding of these systems can enable us to predict their behavior, identify potential pitfalls, and devise effective solutions for optimal outcomes. In this dissertation, we will discuss two classes of high-dimensional dynamical systems with specific structural properties that aid in understanding their dynamic behavior. In the first scenario, we consider the training dynamics of multi-layer neural networks. The high dimensionality comes from overparametrization: a typical network has a large depth and hidden layer width. We are interested in the following question regarding convergence: Do network weights converge to an equilibrium point corresponding to a global minimum of our training loss, and how fast is the convergence rate? The key to those questions is the symmetry of the weights, a critical property induced by the multi-layer architecture. Such symmetry leads to a set of time-invariant quantities, called weight imbalance, that restrict the training trajectory to a low-dimensional manifold defined by the weight initialization. A tailored convergence analysis is developed over this low-dimensional manifold, showing improved rate bounds for several multi-layer network models studied in the literature, leading to novel characterizations of the effect of weight imbalance on the convergence rate. In the second scenario, we consider large-scale networked systems with multiple weakly-connected groups. Such a multi-cluster structure leads to a time-scale separation between the fast intra-group interaction due to high intra-group connectivity, and the slow inter-group oscillation, due to the weak inter-group connection. We develop a novel frequency-domain network coherence analysis that captures both the coherent behavior within each group, and the dynamical interaction between groups, leading to a structure-preserving model-reduction methodology for large-scale dynamic networks with multiple clusters under general node dynamics assumptions.
@phdthesis{m2023phd-thesis, abstract = {The physical and cyber domains with which we interact are filled with high-dimensional dynamical systems. In machine learning, for instance, the evolution of overparametrized neural networks can be seen as a dynamical system. In networked systems, numerous agents or nodes dynamically interact with each other. A deep understanding of these systems can enable us to predict their behavior, identify potential pitfalls, and devise effective solutions for optimal outcomes. In this dissertation, we will discuss two classes of high-dimensional dynamical systems with specific structural properties that aid in understanding their dynamic behavior. In the first scenario, we consider the training dynamics of multi-layer neural networks. The high dimensionality comes from overparametrization: a typical network has a large depth and hidden layer width. We are interested in the following question regarding convergence: Do network weights converge to an equilibrium point corresponding to a global minimum of our training loss, and how fast is the convergence rate? The key to those questions is the symmetry of the weights, a critical property induced by the multi-layer architecture. Such symmetry leads to a set of time-invariant quantities, called weight imbalance, that restrict the training trajectory to a low-dimensional manifold defined by the weight initialization. A tailored convergence analysis is developed over this low-dimensional manifold, showing improved rate bounds for several multi-layer network models studied in the literature, leading to novel characterizations of the effect of weight imbalance on the convergence rate. In the second scenario, we consider large-scale networked systems with multiple weakly-connected groups. Such a multi-cluster structure leads to a time-scale separation between the fast intra-group interaction due to high intra-group connectivity, and the slow inter-group oscillation, due to the weak inter-group connection. We develop a novel frequency-domain network coherence analysis that captures both the coherent behavior within each group, and the dynamical interaction between groups, leading to a structure-preserving model-reduction methodology for large-scale dynamic networks with multiple clusters under general node dynamics assumptions.}, author = {Min, Hancheng}, month = {8}, school = {Electrical and Computer Engineering, Johns Hopkins University}, title = {Exploiting Structural Properties in the Analysis of High-dimensional Dynamical Systems}, url = {https://mallada.ece.jhu.edu/pubs/2023-Thesis-Min.pdf}, year = {2023} } - T. Zheng, “Online Decision Making for Dynamical Systems: Model-Based and Data-Driven Approaches,” PhD Thesis, 2023.
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The widespread availability of data sources and their increased speed compared to the past decade has created both new opportunities and challenges for developing decision-making algorithms for data streams. The ability to process data streams and make real-time decisions that align with system dynamics is a crucial aspect in the development of online decision-making algorithms. This thesis leverages tools from control theory, optimization, and learning to address the problem of online decision-making for dynamical systems, considering streaming data and dynamically changing information. Two online decision-making frameworks are presented in this thesis, depending on the availability of system dynamic information. In the first scenario, where the system can be represented by ordinary differential equations using a state-space model, a time-varying convex optimization framework is introduced. This framework combines motion planning and control to design control signals that lead the dynamical system to asymptotically track optimal trajectories implicitly defined through constrained time-varying optimization problems. Consequently, the nonlinear dynamical system is effectively transformed into an optimization algorithm that seeks the optimal solution to the optimization problem. Global asymptotic convergence of the optimization dynamics to the minimizer of the time-varying optimization problem is proven under sufficient regularity assumptions. In the second scenario, when system dynamics are not available, a data-driven approach called constrained reinforcement learning is adopted. Constrained reinforcement learning deals with sequential decision-making problems where an agent aims to maximize its expected total reward while interacting with an unknown environment and receiving sequentially available information over time. The constrained reinforcement learning framework further includes safety constraints or conflicting requirements during the learning process through secondary expected cumulative rewards. To address the limitations of the learning process in constrained reinforcement learning problems, a novel first-order stochastic gradient descent-ascent (GDA) algorithm is proposed: the stochastic dissipative GDA algorithm. This algorithm almost surely converges to the optimal occupancy measure and optimal policy, overcoming the issue of policy oscillation and convergence to suboptimal policies often encountered in C-RL problems. ii
@phdthesis{z2023phd-thesis, abstract = {The widespread availability of data sources and their increased speed compared to the past decade has created both new opportunities and challenges for developing decision-making algorithms for data streams. The ability to process data streams and make real-time decisions that align with system dynamics is a crucial aspect in the development of online decision-making algorithms. This thesis leverages tools from control theory, optimization, and learning to address the problem of online decision-making for dynamical systems, considering streaming data and dynamically changing information. Two online decision-making frameworks are presented in this thesis, depending on the availability of system dynamic information. In the first scenario, where the system can be represented by ordinary differential equations using a state-space model, a time-varying convex optimization framework is introduced. This framework combines motion planning and control to design control signals that lead the dynamical system to asymptotically track optimal trajectories implicitly defined through constrained time-varying optimization problems. Consequently, the nonlinear dynamical system is effectively transformed into an optimization algorithm that seeks the optimal solution to the optimization problem. Global asymptotic convergence of the optimization dynamics to the minimizer of the time-varying optimization problem is proven under sufficient regularity assumptions. In the second scenario, when system dynamics are not available, a data-driven approach called constrained reinforcement learning is adopted. Constrained reinforcement learning deals with sequential decision-making problems where an agent aims to maximize its expected total reward while interacting with an unknown environment and receiving sequentially available information over time. The constrained reinforcement learning framework further includes safety constraints or conflicting requirements during the learning process through secondary expected cumulative rewards. To address the limitations of the learning process in constrained reinforcement learning problems, a novel first-order stochastic gradient descent-ascent (GDA) algorithm is proposed: the stochastic dissipative GDA algorithm. This algorithm almost surely converges to the optimal occupancy measure and optimal policy, overcoming the issue of policy oscillation and convergence to suboptimal policies often encountered in C-RL problems. ii}, author = {Zheng, Tianqi}, month = {9}, school = {Electrical and Computer Engineering, Johns Hopkins University}, title = {Online Decision Making for Dynamical Systems: Model-Based and Data-Driven Approaches}, url = {https://mallada.ece.jhu.edu/pubs/2023-Thesis-Zheng.pdf}, year = {2023} } - J. Guthrie, “Novel Representations of Semialgebraic Sets Arising in Planning and Control,” PhD Thesis, 2022.
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The mathematical notion of a set arises frequently in planning and control of autonomous systems. A common challenge is how to best represent a given set in a manner that is efficient, accurate, and amenable to computational tools of interest. For example, ensuring a vehicle does not collide with an obstacle can be generically posed in multiple ways using techniques from optimization or computational geometry. However these representations generally rely on executing algorithms instead of evaluating closed-form expressions. This presents an issue when we wish to represent an obstacle avoidance condition within a larger motion planning problem which is solved using nonlinear optimization. These tools generally can only accept smooth, closed-form expressions. As such our available representations of obstacle avoidance conditions, while accurate, are not amenable to the relevant tools. A related problem is how to represent a set in a compact form without sacrificing accuracy. For example, we may be presented with point-cloud data representing the boundary of an object that our vehicle must avoid. Using the obstacle avoidance conditions directly on the point-cloud data would require performing these calculations with respect to each point individually. A more efficient approach is to first approximate the data with simple geometric shapes and perform later analysis with the approximation. Common shapes include bounding boxes, ellipsoids, and superquadrics. These shapes are convenient in that they have a compact representation and we have good heuristic objectives for fitting the data. However, their primitive nature means accuracy of representation may suffer. Most notably, their inherent symmetry makes them ill-suited for representing asymmetric shapes. In theory we could consider more complicated shapes given by an implicit function $S = \x | f(x) łeq 1\$. However we lack reliable methods for ensuring a good fit. This thesis proposes novel approaches to these problems. Throughout, the sets of interest are described by polynomial inequalities, making them semialgebraic.
@phdthesis{g2022phd-thesis, abstract = {The mathematical notion of a set arises frequently in planning and control of autonomous systems. A common challenge is how to best represent a given set in a manner that is efficient, accurate, and amenable to computational tools of interest. For example, ensuring a vehicle does not collide with an obstacle can be generically posed in multiple ways using techniques from optimization or computational geometry. However these representations generally rely on executing algorithms instead of evaluating closed-form expressions. This presents an issue when we wish to represent an obstacle avoidance condition within a larger motion planning problem which is solved using nonlinear optimization. These tools generally can only accept smooth, closed-form expressions. As such our available representations of obstacle avoidance conditions, while accurate, are not amenable to the relevant tools. A related problem is how to represent a set in a compact form without sacrificing accuracy. For example, we may be presented with point-cloud data representing the boundary of an object that our vehicle must avoid. Using the obstacle avoidance conditions directly on the point-cloud data would require performing these calculations with respect to each point individually. A more efficient approach is to first approximate the data with simple geometric shapes and perform later analysis with the approximation. Common shapes include bounding boxes, ellipsoids, and superquadrics. These shapes are convenient in that they have a compact representation and we have good heuristic objectives for fitting the data. However, their primitive nature means accuracy of representation may suffer. Most notably, their inherent symmetry makes them ill-suited for representing asymmetric shapes. In theory we could consider more complicated shapes given by an implicit function $S = \x | f(x) łeq 1\$. However we lack reliable methods for ensuring a good fit. This thesis proposes novel approaches to these problems. Throughout, the sets of interest are described by polynomial inequalities, making them semialgebraic. }, author = {Guthrie, James}, month = {10}, school = {Electrical and Computer Engineering, Johns Hopkins University}, title = {Novel Representations of Semialgebraic Sets Arising in Planning and Control}, url = {https://mallada.ece.jhu.edu/pubs/2022-Thesis-Guthrie.pdf}, year = {2022} } - C. Avraam, “Designing Resilient Interdependent Infrastructures Across Spatial and Temporal Scales,” PhD Thesis, 2021.
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Modern infrastructures are comprised of networked, interdependent systems that deliver critical services. A failure in an infrastructure’s component can propagate through the network and a ect interconnected assets. Addressing the challenges faced by modern infrastructures requires understanding disruptions across regional and temporal scales, and how disruptions propagate to other infrastructures through their interdependencies. This dissertation contributes to the design of resilient electricity, natural gas, and food infrastructures across spatial and temporal scales. In the electricity sector, we develop a controller of exible loads that enhances power systems resilience by preventing voltage collapse. Voltage collapse is a blackoutinducing dynamic instability that occurs when power demand exceeds the maximum transferable power through a network. We formulate the dynamic voltage stability problem for a star direct current network and introduce a voltage collapse stabilization controller that stabilizes power supply at the maximum transferable power under extreme loading conditions, for given potential of exible loads. The controller allocates total load shedding proportionally among exible loads. In the natural gas sector, we assess the resilience of North American natural gas infrastructure to long-term changes in resource availability, global crude oil prices, and technological progress, and identify vulnerable regional assets in the integrated Canada-U.S.-Mexico markets. Natural gas demand in North America depends on gas- red power generation that can change due to policy shifts. We study the resilience of North American natural gas infrastructure to more stringent Renewable Portfolio Standards and greater inter-regional coordination of renewable policies through 2050. We distinguish between three levels of regional renewable policies coordination regional, national, and international and establish the relationship between renewable policy coordination and regional natural gas prices. In the food sector, we assess the resilience of international livestock infrastructures to global coordination policies that aim to curb antimicrobial use. We propose an integrated framework that is an advancement over existing statistical methods by endogenously accounting for competing agents in global meat markets, antimicrobial use in livestock production, and inter-country trade. We formulate three global coordination scenarios in order to understand the tradeo between national antimicrobial use and international trade. We identify vulnerabilities of Low-and-Middle-Income- Countries and derive policy insights.
@phdthesis{a2021phd-thesis, abstract = {Modern infrastructures are comprised of networked, interdependent systems that deliver critical services. A failure in an infrastructure's component can propagate through the network and a ect interconnected assets. Addressing the challenges faced by modern infrastructures requires understanding disruptions across regional and temporal scales, and how disruptions propagate to other infrastructures through their interdependencies. This dissertation contributes to the design of resilient electricity, natural gas, and food infrastructures across spatial and temporal scales. In the electricity sector, we develop a controller of exible loads that enhances power systems resilience by preventing voltage collapse. Voltage collapse is a blackoutinducing dynamic instability that occurs when power demand exceeds the maximum transferable power through a network. We formulate the dynamic voltage stability problem for a star direct current network and introduce a voltage collapse stabilization controller that stabilizes power supply at the maximum transferable power under extreme loading conditions, for given potential of exible loads. The controller allocates total load shedding proportionally among exible loads. In the natural gas sector, we assess the resilience of North American natural gas infrastructure to long-term changes in resource availability, global crude oil prices, and technological progress, and identify vulnerable regional assets in the integrated Canada-U.S.-Mexico markets. Natural gas demand in North America depends on gas- red power generation that can change due to policy shifts. We study the resilience of North American natural gas infrastructure to more stringent Renewable Portfolio Standards and greater inter-regional coordination of renewable policies through 2050. We distinguish between three levels of regional renewable policies coordination regional, national, and international and establish the relationship between renewable policy coordination and regional natural gas prices. In the food sector, we assess the resilience of international livestock infrastructures to global coordination policies that aim to curb antimicrobial use. We propose an integrated framework that is an advancement over existing statistical methods by endogenously accounting for competing agents in global meat markets, antimicrobial use in livestock production, and inter-country trade. We formulate three global coordination scenarios in order to understand the tradeo between national antimicrobial use and international trade. We identify vulnerabilities of Low-and-Middle-Income- Countries and derive policy insights.}, author = {Avraam, Charalampos}, month = {07}, school = {Civil Engineering, Johns Hopkins University}, title = {Designing Resilient Interdependent Infrastructures Across Spatial and Temporal Scales}, url = {https://mallada.ece.jhu.edu/pubs/2021-Thesis-Avraam.pdf}, year = {2021} } - Y. Jiang, “Leveraging Inverter-Interfaced Energy Storage for Frequency Control in Low-Inertia Power Systems,” PhD Thesis, 2021.
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The shift from conventional synchronous generation to renewable inverter-interfaced sources has led to a noticeable degradation of frequency dynamics in power systems, mainly due to a loss of inertia. Fortunately, the recent technology advancement and cost reduction in energy storage facilitate the potential for higher renewable energy penetration via inverter-interfaced energy storage. With proper control laws imposed on inverters, the rapid power-frequency response from energy storage contributes to mitigating the degradation. A straightforward choice is to emulate the droop response and/or inertial response of synchronous generators through droop control (DC) or virtual inertia (VI), yet they do not necessarily fully exploit the benefits of inverter-interfaced energy storage. This thesis thus seeks to challenge this naive choice of mimicking synchronous generator characteristics by advocating for a principled control design perspective. To achieve this goal, we build an analysis framework for quantifying the performance of power systems using signal and system norms, within which we perform a systematic study to evaluate the effect of different control laws on both frequency response metrics and storage economic metrics. More precisely, under a mild yet insightful proportionality assumption, we are able to perform a modal decomposition which allows us to get closed-form expressions or conditions for synchronous frequency, Nadir, rate of change of frequency (RoCoF), synchronization cost, frequency variance, and steadystate effort share. All of them pave the way for a better understanding of the sensitivities of various performance metrics to different control laws. Our analysis unveils several limitations of traditional control laws, such as the inability of DC to improve the dynamic performance without sacrificing the steady-state performance and the unbounded frequency variance introduced by VI in the presence of frequency measurement noise. Therefore, rather than clinging to the idea of imitating synchronous generator behavior via inverter-interfaced energy storage, we prefer searching for better solutions. We first propose dynam-i-c Droop control (iDroop)–inspired by the classical lead/lag compensator–which is proved to enjoy many good properties. First of all, the added degrees of freedom in iDroop allow to decouple the dynamic performance improvement from the steady-state performance. In addition, the lead/lag property of iDroop makes it less sensitive to stochastic power fluctuations and frequency measurement noise. Last but not least, iDroop can also be tuned either to achieve the zero synchronization cost or to achieve Nadir elimination, by which we mean to remove the overshoot in the transient system frequency. Particularly, the Nadir elimination tuning of iDroop exhibits the potential for a balance among various performance metrics in practice. However, iDroop has no control over the RoCoF, which is undesirable in low-inertia power systems for the risk of falsely triggering protections. We then propose frequency shaping control (FS)–an extension of iDroop– whose most outstanding feature is its ability to shape the system frequency dynamics following a sudden power imbalance into a first-order one with the specified synchronous frequency and RoCoF by adjusting two independent control parameters. We finally validate theoretical results through extensive numerical experiments performed on a more realistic power system test case that violates the proportionality assumption, which clearly confirms that our proposed control laws outperform the traditional ones in an overall sense.
@phdthesis{j2021phd-thesis, abstract = {The shift from conventional synchronous generation to renewable inverter-interfaced sources has led to a noticeable degradation of frequency dynamics in power systems, mainly due to a loss of inertia. Fortunately, the recent technology advancement and cost reduction in energy storage facilitate the potential for higher renewable energy penetration via inverter-interfaced energy storage. With proper control laws imposed on inverters, the rapid power-frequency response from energy storage contributes to mitigating the degradation. A straightforward choice is to emulate the droop response and/or inertial response of synchronous generators through droop control (DC) or virtual inertia (VI), yet they do not necessarily fully exploit the benefits of inverter-interfaced energy storage. This thesis thus seeks to challenge this naive choice of mimicking synchronous generator characteristics by advocating for a principled control design perspective. To achieve this goal, we build an analysis framework for quantifying the performance of power systems using signal and system norms, within which we perform a systematic study to evaluate the effect of different control laws on both frequency response metrics and storage economic metrics. More precisely, under a mild yet insightful proportionality assumption, we are able to perform a modal decomposition which allows us to get closed-form expressions or conditions for synchronous frequency, Nadir, rate of change of frequency (RoCoF), synchronization cost, frequency variance, and steadystate effort share. All of them pave the way for a better understanding of the sensitivities of various performance metrics to different control laws. Our analysis unveils several limitations of traditional control laws, such as the inability of DC to improve the dynamic performance without sacrificing the steady-state performance and the unbounded frequency variance introduced by VI in the presence of frequency measurement noise. Therefore, rather than clinging to the idea of imitating synchronous generator behavior via inverter-interfaced energy storage, we prefer searching for better solutions. We first propose dynam-i-c Droop control (iDroop)--inspired by the classical lead/lag compensator--which is proved to enjoy many good properties. First of all, the added degrees of freedom in iDroop allow to decouple the dynamic performance improvement from the steady-state performance. In addition, the lead/lag property of iDroop makes it less sensitive to stochastic power fluctuations and frequency measurement noise. Last but not least, iDroop can also be tuned either to achieve the zero synchronization cost or to achieve Nadir elimination, by which we mean to remove the overshoot in the transient system frequency. Particularly, the Nadir elimination tuning of iDroop exhibits the potential for a balance among various performance metrics in practice. However, iDroop has no control over the RoCoF, which is undesirable in low-inertia power systems for the risk of falsely triggering protections. We then propose frequency shaping control (FS)--an extension of iDroop-- whose most outstanding feature is its ability to shape the system frequency dynamics following a sudden power imbalance into a first-order one with the specified synchronous frequency and RoCoF by adjusting two independent control parameters. We finally validate theoretical results through extensive numerical experiments performed on a more realistic power system test case that violates the proportionality assumption, which clearly confirms that our proposed control laws outperform the traditional ones in an overall sense.}, author = {Jiang, Yan}, month = {07}, school = {Electrical and Computer Engineering, Johns Hopkins University}, title = {Leveraging Inverter-Interfaced Energy Storage for Frequency Control in Low-Inertia Power Systems}, url = {https://mallada.ece.jhu.edu/pubs/2021-Thesis-Jiang.pdf}, year = {2021} } - E. Mallada, “Distributed synchronization in engineering networks: The Internet and electric power grids,” PhD Thesis, 2014.
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Synchronization is a fundamental requirement of most networked engineering applications. It enables the necessary coordination among agents required to implement several communication systems as well as network protocols. Despite the great recent advances in understanding synchronization, a complete synchronization theory is yet to be developed. This thesis presents a systematic study of synchronization on distributed systems that covers theoretical guarantees for synchronization, performance analysis and optimization, as well as design and implementation of algorithms. We first present several theoretical results that deepen the understanding of how coupling, delay and topology affect the behavior of a system of coupled oscillators. We obtain a sufficient condition that can be used to check limit cycle stability, and use it to characterize a family of coupling functions guaranteeing convergence to in-phase synchronization (phase consensus). The effect of heterogeneous delay is then investigated by developing a new framework that unveils the dependence of the orbit’s stability on the delay distribution. Finally, we consider the effect of frequency heterogeneity. While coupled oscillators with heterogeneous frequency cannot achieve phase consensus, we show that a second order version of the system can achieve synchronization for arbitrary natural frequencies and we relate the limiting frequency of the system to the harmonic mean of the natural frequencies. Based on the insight provided by our theoretical results, we then focus on more practical aspects of synchronization in two particular areas: information networks and power networks. Within information networks, we examine the synchronization of computer clocks connected via a data network and propose a discrete algorithm to synchronize them. Unlike current solutions, which either estimate and compensate the frequency difference (skew) among clocks or introduce offset corrections that can generate jitter and possibly even backward jumps, this algorithm achieves synchronization without any of these problems. We present a detailed convergence analysis together with a characterization of the parameter values that guarantee convergence. We then study and optimize the effect of noisy measurements and clock wander on the system performance using a parameter dependent H2 norm. In particular, we show that the frequency of the system drifts away from its theoretical value in the absence of a leader. We implement the algorithm on a cluster of IBM BladeCenter servers running Linux and we experimentally verify that our algorithm outperforms the well-established solution. We also show that the optimal parameter values depend on the network conditions and topology. Finally, we study synchronization on power networks. By relating the dynamics of power networks to the dynamics of coupled oscillators, we can gain insight into how different network parameters affect performance. We show that the rate of convergence of networks is related to the algebraic connectivity of a state dependent Laplacian which varies with the network power scheduling and line impedances. This provides a novel method to change the voltage stability margins by updating the power scheduling or line impedances. Unfortunately, there exists a decoupling between the market clearing procedure used to dispatch power and the security analysis of the network, that prevents the direct use of this solution. Furthermore, focusing on voltage stability may generate other types of instabilities such as larger transient oscillations. This motivates the use of a unifying stability measure that can minimize oscillations or maximize voltage stability margins, and can be readily combined with current dispatch mechanisms generating a dynamics-aware optimal power flow formulation.
@phdthesis{m2014phd-thesis, abstract = {Synchronization is a fundamental requirement of most networked engineering applications. It enables the necessary coordination among agents required to implement several communication systems as well as network protocols. Despite the great recent advances in understanding synchronization, a complete synchronization theory is yet to be developed. This thesis presents a systematic study of synchronization on distributed systems that covers theoretical guarantees for synchronization, performance analysis and optimization, as well as design and implementation of algorithms. We first present several theoretical results that deepen the understanding of how coupling, delay and topology affect the behavior of a system of coupled oscillators. We obtain a sufficient condition that can be used to check limit cycle stability, and use it to characterize a family of coupling functions guaranteeing convergence to in-phase synchronization (phase consensus). The effect of heterogeneous delay is then investigated by developing a new framework that unveils the dependence of the orbit's stability on the delay distribution. Finally, we consider the effect of frequency heterogeneity. While coupled oscillators with heterogeneous frequency cannot achieve phase consensus, we show that a second order version of the system can achieve synchronization for arbitrary natural frequencies and we relate the limiting frequency of the system to the harmonic mean of the natural frequencies. Based on the insight provided by our theoretical results, we then focus on more practical aspects of synchronization in two particular areas: information networks and power networks. Within information networks, we examine the synchronization of computer clocks connected via a data network and propose a discrete algorithm to synchronize them. Unlike current solutions, which either estimate and compensate the frequency difference (skew) among clocks or introduce offset corrections that can generate jitter and possibly even backward jumps, this algorithm achieves synchronization without any of these problems. We present a detailed convergence analysis together with a characterization of the parameter values that guarantee convergence. We then study and optimize the effect of noisy measurements and clock wander on the system performance using a parameter dependent H2 norm. In particular, we show that the frequency of the system drifts away from its theoretical value in the absence of a leader. We implement the algorithm on a cluster of IBM BladeCenter servers running Linux and we experimentally verify that our algorithm outperforms the well-established solution. We also show that the optimal parameter values depend on the network conditions and topology. Finally, we study synchronization on power networks. By relating the dynamics of power networks to the dynamics of coupled oscillators, we can gain insight into how different network parameters affect performance. We show that the rate of convergence of networks is related to the algebraic connectivity of a state dependent Laplacian which varies with the network power scheduling and line impedances. This provides a novel method to change the voltage stability margins by updating the power scheduling or line impedances. Unfortunately, there exists a decoupling between the market clearing procedure used to dispatch power and the security analysis of the network, that prevents the direct use of this solution. Furthermore, focusing on voltage stability may generate other types of instabilities such as larger transient oscillations. This motivates the use of a unifying stability measure that can minimize oscillations or maximize voltage stability margins, and can be readily combined with current dispatch mechanisms generating a dynamics-aware optimal power flow formulation.}, author = {Mallada, Enrique}, keywords = {Coupled Oscillators; Networking; Power Networks; Synchronization}, month = {01}, school = {Electrical and Computer Engineering, Cornell University}, title = {Distributed synchronization in engineering networks: The Internet and electric power grids}, url = {https://mallada.ece.jhu.edu/pubs/2014-Thesis-M.pdf}, web = {http://ecommons.library.cornell.edu/handle/1813/36073}, year = {2014} }