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1 paper accepted to IEEE TEMPR

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Our paper on market power mitigation in two-stage markets [1] has been accepted to Transactions on Energy Markets, Policy, and Regulation!

[1] 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, pp. 1-10, 2023.
[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},
  grants = {CAREER-1752362, CPS-2136324, EPICS-2330450},
  journal = {IEEE Transactions on Energy Markets, Policy and Regulation},
  month = {9},
  pages = {1-10},
  pubstate = {accepted},
  record = {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},
  year = {2023}
}

Rajni defended his dissertation

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Rajni Kant Bansal, a MechE Ph.D. student in our lab, defended his dissertation entitled “Efficiency and Market Power in Electricity Markets with Inelastic Demand, Energy Storage, and Hybrid Energy Resources” on Wednesday, August 30th. Congratulations!

2 papers accpeted to CDC

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Our paper on non-monotonic Lyapunov functions [1] and our paper on simultaneous state and sparse input recovery [2] have been accepted to the Conference on Decision and Control 2023!

[1] 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.
[Bibtex] [Abstract] [Download PDF]

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)},
  grants = {CPS-2136324, CAREER-1752362, EPICS-2330450},
  month = {12},
  pubstate = {to appear, 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}
}
[2] 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.
[Bibtex] [Abstract] [Download PDF]

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)},
  grants = {CPS-2136324,CAREER-1752362},
  month = {12},
  pubstate = {to appear, 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}
}

1 paper accepted to ICML

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Our paper on convergence of gradient flow on multi-layer linear networks [1] has been accepted to International Conference on Machine Learning! Congrats Hancheng!

[1] 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.
[Bibtex] [Abstract] [Download PDF]

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}
}