Steven H. Low

Masoud Farivar, Steven H. Low

We propose a branch flow model for the anal- ysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

Enrique Mallada, Changhong Zhao, Steven H. Low

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. In particular, 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 the currently implemented frequency control), 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. We illustrate its effectiveness through simulations.

Ying Jun, Zhang, Changhong Zhao et al.

We consider a two-level profit-maximizing strategy, including planning and control, for battery energy storage system (BESS) owners that participate in the primary frequency control (PFC) market. Specifically, the optimal BESS control minimizes the operating cost by keeping the state of charge (SoC) in an optimal range. Through rigorous analysis, we prove that the optimal BESS control is a "state-invariant" strategy in the sense that the optimal SoC range does not vary with the state of the system. As such, the optimal control strategy can be computed offline once and for all with very low complexity. Regarding the BESS planning, we prove that the the minimum operating cost is a decreasing convex function of the BESS energy capacity. This leads to the optimal BESS sizing that strikes a balance between the capital investment and operating cost. Our work here provides a useful theoretical framework for understanding the planning and control strategies that maximize the economic benefits of BESSs in ancillary service markets.

Zhaojian Wang, Feng Liu, Steven H. Low et al.

This paper addresses the distributed optimal frequency control of multi-area power system with operational constraints, including the regulation capacity of individual control area and the power limits on tie-lines. Both generators and controllable loads are utilized to recover nominal frequencies while minimizing regulation cost. We study two control modes:the per-node balance mode and the network balance mode. In Part I of the paper, we only consider the per-node balance case, where we derive a completely decentralized strategy without the need for communication between control areas. It can adapt to unknown load disturbance. The tie-line powers are restored after load disturbance, while the regulation capacity constraints are satisfied both at equilibrium and during transient. We show that the closed-loop systems with the proposed control strategies carry out primal-dual updates for solving the associated centralized frequency optimization problems. We further prove the closed-loop systems are asymptotically stable and converge to the unique optimal solution of the centralized frequency optimization problems and their duals. Finally, we present simulation results to demonstrate the effectiveness of our design. In Part II of the paper, we address the network power balance case, where transmission congestions are managed continuously.

Zhaojian Wang, Feng Liu, Steven H. Low et al.

In Part I of this paper we propose a decentralized optimal frequency control of multi-area power system with operational constraints, where the tie-line powers remain unchanged in the steady state and the power mismatch is balanced within individual control areas. In Part II of the paper, we propose a distributed controller for optimal frequency control in the network power balance case, where the power mismatch is balanced over the whole system. With the proposed controller, the tie-line powers remain within the acceptable range at equilibrium, while the regulation capacity constraints are satisfied both at equilibrium and during transient. It is revealed that the closed-loop system with the proposed controller carries out primal-dual updates with saturation for solving an associated optimization problem. To cope with discontinuous dynamics of the closed-loop system, we deploy the invariance principle for nonpathological Lyapunov function to prove its asymptotic stability. Simulation results are provided to show the effectiveness of our controller.

Linqi Guo, Changhong Zhao, Steven H. Low

We present a framework based on spectral graph theory that captures the interplay among network topology, system inertia, and generator and load damping in determining the overall grid behavior and performance. Specifically, we show that the impact of network topology on a power system can be quantified through the network Laplacian eigenvalues, and such eigenvalues determine the grid robustness against low frequency disturbances. Moreover, we can explicitly decompose the frequency signal along scaled Laplacian eigenvectors when damping-inertia ratios are uniform across buses. The insight revealed by this framework partially explains why load-side participation in frequency regulation not only makes the system respond faster, but also helps lower the system nadir after a disturbance. Finally, by presenting a new controller specifically tailored to suppress high frequency disturbances, we demonstrate that our results can provide useful guidelines in the controller design for load-side primary frequency regulation. This improved controller is simulated on the IEEE 39-bus New England interconnection system to illustrate its robustness against high frequency oscillations compared to both the conventional droop control and a recent controller design.

Zhaojian Wang, Feng Liu, John Z. F. Pang et al.

This paper addresses the distributed optimal frequency control of power systems considering a network-preserving model with nonlinear power flows and excitation voltage dynamics. Salient features of the proposed distributed control strategy are fourfold: i) nonlinearity is considered to cope with large disturbances; ii) only a part of generators are controllable; iii) no load measurement is required; iv) communication connectivity is required only for the controllable generators. To this end, benefiting from the concept of 'virtual load demand', we first design the distributed controller for the controllable generators by leveraging the primal-dual decomposition technique. We then propose a method to estimate the virtual load demand of each controllable generator based on local frequencies. We derive incremental passivity conditions for the uncontrollable generators. Finally, we prove that the closed-loop system is asymptotically stable and its equilibrium attains the optimal solution to the associated economic dispatch problem. Simulations, including small and large-disturbance scenarios, are carried on the New England system, demonstrating the effectiveness of our design.

Zhaojian Wang, Shengwei Mei, Feng Liu et al.

This paper addresses the distributed frequency control problem in a multi-area power system taking into account of unknown time-varying power imbalance. Particularly, fast controllable loads are utilized to restore system frequency under changing power imbalance in an optimal manner. The imbalanced power causing frequency deviation is decomposed into three parts: a known constant part, an unknown low-frequency variation and a high-frequency residual. The known steady part is usually the prediction of power imbalance. The variation may result from the fluctuation of renewable resources, electric vehicle charging, etc., which is usually unknown to operators. The high-frequency residual is usually unknown and treated as an external disturbance. Correspondingly, in this paper, we resolve the following three problems in different timescales: 1) allocate the steady part of power imbalance economically; 2) mitigate the effect of unknown low-frequency power variation locally; 3) attenuate unknown high-frequency disturbances. To this end, a distributed controller combining consensus method with adaptive internal model control is proposed. We first prove that the closed-loop system is asymptotically stable and converges to the optimal solution of an optimization problem if the external disturbance is not included. We then prove that the power variation can be mitigated accurately. Furthermore, we show that the closed-loop system is robust against both parameter uncertainty and external disturbances. The New England system is used to verify the efficacy of our design.

Yuxiao Chen, James Anderson, Karan Kalsi et al.

This paper presents an assume-guarantee reasoning approach to the computation of robust invariant sets for network systems. Parameterized signal temporal logic (pSTL) is used to formally describe the behaviors of the subsystems, which we use as the template for the contract. We show that set invariance can be proved with a valid assume-guarantee contract by reasoning about individual subsystems. If a valid assume-guarantee contract with monotonic pSTL template is known, it can be further refined by value iteration. When such a contract is not known, an epigraph method is proposed to solve for a contract that is valid, ---an approach that has linear complexity for a sparse network. A microgrid example is used to demonstrate the proposed method. The simulation result shows that together with control barrier functions, the states of all the subsystems can be bounded inside the individual robust invariant sets.

Linqi Guo, Chen Liang, Alessandro Zocca et al.

Cascading failures in power systems propagate non-locally, making the control and mitigation of outages extremely hard. In this work, we use the emerging concept of the tree partition of transmission networks to provide an analytical characterization of line failure localizability in transmission systems. Our results rigorously establish the well perceived intuition in power community that failures cannot cross bridges, and reveal a finer-grained concept that encodes more precise information on failure propagations within tree-partition regions. Specifically, when a non-bridge line is tripped, the impact of this failure only propagates within well-defined components, which we refer to as cells, of the tree partition defined by the bridges. In contrast, when a bridge line is tripped, the impact of this failure propagates globally across the network, affecting the power flow on all remaining transmission lines. This characterization suggests that it is possible to improve the system robustness by temporarily switching off certain transmission lines, so as to create more, smaller components in the tree partition; thus spatially localizing line failures and making the grid less vulnerable to large-scale outages. We illustrate this approach using the IEEE 118-bus test system and demonstrate that switching off a negligible portion of transmission lines allows the impact of line failures to be significantly more localized without substantial changes in line congestion.

Xiang Pan, Wanjun Huang, Minghua Chen et al.

The existence of multiple load-solution mappings of non-convex AC-OPF problems poses a fundamental challenge to deep neural network (DNN) schemes. As the training dataset may contain a mixture of data points corresponding to different load-solution mappings, the DNN can fail to learn a legitimate mapping and generate inferior solutions. We propose DeepOPF-AL as an augmented-learning approach to tackle this issue. The idea is to train a DNN to learn a unique mapping from an augmented input, i.e., (load, initial point), to the solution generated by an iterative OPF solver with the load and initial point as intake. We then apply the learned augmented mapping to solve AC-OPF problems much faster than conventional solvers. Simulation results over IEEE test cases show that DeepOPF-AL achieves noticeably better optimality and similar feasibility and speedup performance, as compared to a recent DNN scheme, with the same DNN size yet elevated training complexity.

Tongxin Li, Ruixiao Yang, Guannan Qu et al.

Machine-learned black-box policies are ubiquitous for nonlinear control problems. Meanwhile, crude model information is often available for these problems from, e.g., linear approximations of nonlinear dynamics. We study the problem of equipping a black-box control policy with model-based advice for nonlinear control on a single trajectory. We first show a general negative result that a naive convex combination of a black-box policy and a linear model-based policy can lead to instability, even if the two policies are both stabilizing. We then propose an adaptive $λ$-confident policy, with a coefficient $λ$ indicating the confidence in a black-box policy, and prove its stability. With bounded nonlinearity, in addition, we show that the adaptive $λ$-confident policy achieves a bounded competitive ratio when a black-box policy is near-optimal. Finally, we propose an online learning approach to implement the adaptive $λ$-confident policy and verify its efficacy in case studies about the CartPole problem and a real-world electric vehicle (EV) charging problem with data bias due to COVID-19.

Yiheng Xie, Lucien Werner, Kaibo Chen et al.

We provide an open-access dataset of phasor & waveform measurement units (PMUs/WMUs) of a real-world electrical distribution network. The network consists of diverse sets of generation resources (including solar panels, fuel cells, natural gas generators, and utility interconnections), loads (including large-scale electric vehicle charging, data centers, central cooling, offices), topology changes (such as line outages and load transfers), as well as a mixture of single- and three-phase networks. We describe a densely deployed PMU sensor network in a distribution grid, in which all buses with non-zero power injections are measured. This approach enables a range of applications such as state estimation, system identification, power flow optimization, and feedback control, several of which are discussed in this paper. Additionally, we provide a synchronized waveform dataset which allows the analysis of harmonics, transient events, dynamic grid impedance, and stability. Data collection started in 2023 while new data is generated continuously and made available online. A characterization of measurement error is provided. Finally, we provide circuit topology and parameters as a part of the dataset. Together, the circuit and timeseries data offer an opportunity for researchers to develop and test algorithms on a real-world system.

Yiwei Dong, Wenqi Cui, Han Xu et al.

Power distribution systems are increasingly exposed to large voltage fluctuations driven by intermittent renewable generation and time varying loads (e.g., electric vehicles and storage). To address this challenge, a number of advanced controllers have been proposed for voltage regulation. However, these controllers typically rely on fixed linear approximations of voltage dynamics. As a result, the solutions may become infeasible when applied to the actual voltage behavior governed by nonlinear power flow equations, particularly under heavy power injection from distributed energy resources. This paper proposes a data-driven successive linearization approach for voltage control under nonlinear power flow constraints. By leveraging the fact that the deviation between the nonlinear power flow solution and its linearization is bounded by the distance from the operating point, we perform data-driven linearization around the most recent operating point. Convergence of the proposed method to a neighborhood of KKT points is established by exploiting the convexity of the objective function and structural properties of the nonlinear constraints. Case studies show that the proposed approach achieves fast convergence and adapts quickly to changes in net load.

Tianyu Zhao, Xiang Pan, Minghua Chen et al.

Ensuring solution feasibility is a key challenge in developing Deep Neural Network (DNN) schemes for solving constrained optimization problems, due to inherent DNN prediction errors. In this paper, we propose a ``preventive learning'' framework to guarantee DNN solution feasibility for problems with convex constraints and general objective functions without post-processing, upon satisfying a mild condition on constraint calibration. Without loss of generality, we focus on problems with only inequality constraints. We systematically calibrate inequality constraints used in DNN training, thereby anticipating prediction errors and ensuring the resulting solutions remain feasible. We characterize the calibration magnitudes and the DNN size sufficient for ensuring universal feasibility. We propose a new Adversarial-Sample Aware training algorithm to improve DNN's optimality performance without sacrificing feasibility guarantee. Overall, the framework provides two DNNs. The first one from characterizing the sufficient DNN size can guarantee universal feasibility while the other from the proposed training algorithm further improves optimality and maintains DNN's universal feasibility simultaneously. We apply the framework to develop DeepOPF+ for solving essential DC optimal power flow problems in grid operation. Simulation results over IEEE test cases show that it outperforms existing strong DNN baselines in ensuring 100% feasibility and attaining consistent optimality loss ($<$0.19%) and speedup (up to $\times$228) in both light-load and heavy-load regimes, as compared to a state-of-the-art solver. We also apply our framework to a non-convex problem and show its performance advantage over existing schemes.

Wanjun Huang, Xiang Pan, Minghua Chen et al.

AC optimal power flow (AC-OPF) problems need to be solved more frequently in the future to maintain stable and economic power system operation. To tackle this challenge, a deep neural network-based voltage-constrained approach (DeepOPF-V) is proposed to solve AC-OPF problems with high computational efficiency. Its unique design predicts voltages of all buses and then uses them to reconstruct the remaining variables without solving non-linear AC power flow equations. A fast post-processing process is developed to enforce the box constraints. The effectiveness of DeepOPF-V is validated by simulations on IEEE 118/300-bus systems and a 2000-bus test system. Compared with existing studies, DeepOPF-V achieves decent computation speedup up to four orders of magnitude and comparable performance in optimality gap and preserving the feasibility of the solution.

Xin Chen, Guannan Qu, Yujie Tang et al.

With large-scale integration of renewable generation and distributed energy resources, modern power systems are confronted with new operational challenges, such as growing complexity, increasing uncertainty, and aggravating volatility. Meanwhile, more and more data are becoming available owing to the widespread deployment of smart meters, smart sensors, and upgraded communication networks. As a result, data-driven control techniques, especially reinforcement learning (RL), have attracted surging attention in recent years. This paper provides a comprehensive review of various RL techniques and how they can be applied to decision-making and control in power systems. In particular, we select three key applications, i.e., frequency regulation, voltage control, and energy management, as examples to illustrate RL-based models and solutions. We then present the critical issues in the application of RL, i.e., safety, robustness, scalability, and data. Several potential future directions are discussed as well.

Xiang Pan, Minghua Chen, Tianyu Zhao et al.

High percentage penetrations of renewable energy generations introduce significant uncertainty into power systems. It requires grid operators to solve alternative current optimal power flow (AC-OPF) problems more frequently for economical and reliable operation in both transmission and distribution grids. In this paper, we develop a Deep Neural Network (DNN) approach, called DeepOPF, for solving AC-OPF problems in a fraction of the time used by conventional solvers. A key difficulty for applying machine learning techniques for solving AC-OPF problems lies in ensuring that the obtained solutions respect the equality and inequality physical and operational constraints. Generalized the 2-stage procedure in [1], [2], DeepOPF first trains a DNN model to predict a set of independent operating variables and then directly compute the remaining dependable ones by solving power flow equations. Such an approach not only preserves the power-flow balance equality constraints but also reduces the number of variables to predict by the DNN, cutting down the number of neurons and training data needed. DeepOPF then employs a penalty approach with a zero-order gradient estimation technique in the training process to preserve the remaining inequality constraints. As another contribution, we drive a condition for tuning the size of the DNN according to the desired approximation accuracy, which measures the DNN generalization capability. It provides theoretical justification for using DNN to solve the AC-OPF problem. Simulation results of IEEE 30/118/300-bus and a synthetic 2000-bus test cases show that DeepOPF speeds up the computing time by up to two orders of magnitude as compared to a state-of-the-art solver, at the expense of $<$0.1% cost difference.

Andreas Venzke, Guannan Qu, Steven Low et al.

This paper introduces for the first time a framework to obtain provable worst-case guarantees for neural network performance, using learning for optimal power flow (OPF) problems as a guiding example. Neural networks have the potential to substantially reduce the computing time of OPF solutions. However, the lack of guarantees for their worst-case performance remains a major barrier for their adoption in practice. This work aims to remove this barrier. We formulate mixed-integer linear programs to obtain worst-case guarantees for neural network predictions related to (i) maximum constraint violations, (ii) maximum distances between predicted and optimal decision variables, and (iii) maximum sub-optimality. We demonstrate our methods on a range of PGLib-OPF networks up to 300 buses. We show that the worst-case guarantees can be up to one order of magnitude larger than the empirical lower bounds calculated with conventional methods. More importantly, we show that the worst-case predictions appear at the boundaries of the training input domain, and we demonstrate how we can systematically reduce the worst-case guarantees by training on a larger input domain than the domain they are evaluated on.

Guannan Qu, Chenkai Yu, Steven Low et al.

Model-free learning-based control methods have seen great success recently. However, such methods typically suffer from poor sample complexity and limited convergence guarantees. This is in sharp contrast to classical model-based control, which has a rich theory but typically requires strong modeling assumptions. In this paper, we combine the two approaches to achieve the best of both worlds. We consider a dynamical system with both linear and non-linear components and develop a novel approach to use the linear model to define a warm start for a model-free, policy gradient method. We show this hybrid approach outperforms the model-based controller while avoiding the convergence issues associated with model-free approaches via both numerical experiments and theoretical analyses, in which we derive sufficient conditions on the non-linear component such that our approach is guaranteed to converge to the (nearly) global optimal controller.

Fengyu Zhou, James Anderson, Steven H. Low

We consider the problem of privately releasing aggregated network statistics obtained from solving a DC optimal power flow (OPF) problem. It is shown that the mechanism that determines the noise distribution parameters are linked to the topology of the power system and the monotonicity of the network. We derive a measure of "almost" monotonicity and show how it can be used in conjunction with a linear program in order to release aggregated OPF data using the differential privacy framework.

Linqi Guo, Chen Liang, Alessandro Zocca et al.

Cascading failures in power systems exhibit non-local propagation patterns which make the analysis and mitigation of failures difficult. In this work, we propose a distributed control framework inspired by the recently proposed concepts of unified controller and network tree-partition that offers strong guarantees in both the mitigation and localization of cascading failures in power systems. In this framework, the transmission network is partitioned into several control areas which are connected in a tree structure, and the unified controller is adopted by generators or controllable loads for fast timescale disturbance response. After an initial failure, the proposed strategy always prevents successive failures from happening, and regulates the system to the desired steady state where the impact of initial failures are localized as much as possible. For extreme failures that cannot be localized, the proposed framework has a configurable design, that progressively involves and coordinates more control areas for failure mitigation and, as a last resort, imposes minimal load shedding. We compare the proposed control framework with Automatic Generation Control (AGC) on the IEEE 118-bus test system. Simulation results show that our novel framework greatly improves the system robustness in terms of the N-1 security standard, and localizes the impact of initial failures in majority of the load profiles that are examined. Moreover, the proposed framework incurs significantly less load loss, if any, compared to AGC, in all of our case studies.

James Anderson, John C. Doyle, Steven Low et al.

This article surveys the System Level Synthesis framework, which presents a novel perspective on constrained robust and optimal controller synthesis for linear systems. We show how SLS shifts the controller synthesis task from the design of a controller to the design of the entire closed loop system, and highlight the benefits of this approach in terms of scalability and transparency. We emphasize two particular applications of SLS, namely large-scale distributed optimal control and robust control. In the case of distributed control, we show how SLS allows for localized controllers to be computed, extending robust and optimal control methods to large-scale systems under practical and realistic assumptions. In the case of robust control, we show how SLS allows for novel design methodologies that, for the first time, quantify the degradation in performance of a robust controller due to model uncertainty -- such transparency is key in allowing robust control methods to interact, in a principled way, with modern techniques from machine learning and statistical inference. Throughout, we emphasize practical and efficient computational solutions, and demonstrate our methods on easy to understand case studies.

Zhaojian Wang, Feng Liu, Ying Chen et al.

Stand-alone direct current (DC) microgrids may belong to different owners and adopt various control strategies. This brings great challenge to its optimal operation due to the difficulty of implementing a unified control. This paper addresses the distributed optimal control of DC microgrids, which intends to break the restriction of diversity to some extent. Firstly, we formulate the optimal power flow (OPF) problem of stand-alone DC microgrids as an exact second order cone program (SOCP) and prove the uniqueness of the optimal solution. Then a dynamic solving algorithm based on primal-dual decomposition method is proposed, the convergence of which is proved theoretically as well as the optimality of its equilibrium point. It should be stressed that the algorithm can provide control commands for the three types of microgrids: (i) power control, (ii) voltage control and (iii) droop control. This implies that each microgrid does not need to change its original control strategy in practice, which is less influenced by the diversity of microgrids. Moreover, the control commands for power controlled and voltage controlled microgrids satisfy generation limits and voltage limits in both transient process and steady state. Finally, a six-microgrid DC system based on the microgrid benchmark is adopted to validate the effectiveness and plug-n-play property of our designs.

Krishnamurthy Dvijotham, Michael Chertkov, Steven Low

The AC power flow equations are fundamental in all aspects of power systems planning and operations. They are routinely solved using Newton-Raphson like methods. However, there is little theoretical understanding of when these algorithms are guaranteed to find a solution of the power flow equations or how long they may take to converge. Further, it is known that in general these equations have multiple solutions and can exhibit chaotic behavior. In this paper, we show that the power flow equations can be solved efficiently provided that the solution lies in a certain set. We introduce a family of convex domains, characterized by Linear Matrix Inequalities, in the space of voltages such that there is at most one power flow solution in each of these domains. Further, if a solution exists in one of these domains, it can be found efficiently, and if one does not exist, a certificate of non-existence can also be obtained efficiently. The approach is based on the theory of monotone operators and related algorithms for solving variational inequalities involving monotone operators. We validate our approach on IEEE test networks and show that practical power flow solutions lie within an appropriately chosen convex domain.

Krishnamurthy Dvijotham, Steven Low, Michael Chertkov

The AC power flow equations underlie all operational aspects of power systems. They are solved routinely in operational practice using the Newton-Raphson method and its variants. These methods work well given a good initial "guess" for the solution, which is always available in normal system operations. However, with the increase in levels of intermittent generation, the assumption of a good initial guess always being available is no longer valid. In this paper, we solve this problem using the theory of monotone operators. We show that it is possible to compute (using an offline optimization) a "monotonicity domain" in the space of voltage phasors. Given this domain, there is a simple efficient algorithm that will either find a solution in the domain, or provably certify that no solutions exist in it. We validate the approach on several IEEE test cases and demonstrate that the offline optimization can be performed tractably and the computed "monotonicity domain" includes all practically relevant power flow solutions.