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We consider the problem of identifying the admittance matrix of a three-phase radial network from voltage and current measurements at a subset of nodes. These measurements are used to estimate a virtual network represented by the Kron reduction (Schur complement) of the full admittance matrix. We focus on recovering exactly the full admittance matrix from its Kron reduction, i.e., computing the inverse of Schur complement. The key idea is to decompose Kron reduction into a sequence of iterations that maintains an invariance structure, and exploit this structure to reverse each step of the iterative Kron reduction.

Deep reinforcement learning has been recognized as a promising tool to address the challenges in real-time control of power systems. However, its deployment in real-world power systems has been hindered by a lack of explicit stability and safety guarantees. In this paper, we propose a stability-constrained reinforcement learning (RL) method for real-time voltage control, that guarantees system stability both during policy learning and deployment of the learned policy. The key idea underlying our approach is an explicitly constructed Lyapunov function that leads to a sufficient structural condition for stabilizing policies, i.e., monotonically decreasing policies guarantee stability. We incorporate this structural constraint with RL, by parameterizing each local voltage controller using a monotone neural network, thus ensuring the stability constraint is satisfied by design. We demonstrate the effectiveness of our approach in both single-phase and three-phase IEEE test feeders, where the proposed method can reduce the transient control cost by more than 25% and shorten the voltage recovery time by 21.5% on average compared to the widely used linear policy, while always achieving voltage stability. In contrast, standard RL methods often fail to achieve voltage stability., Comment: This paper is accepted by TCNS. arXiv admin note: text overlap with arXiv:2109.14854

First we present an approach to formulate unbalanced three-phase power flow problems for general networks that explicitly separates device models and network models. A device model consists of (i) an internal model and (ii) a conversion rule. The conversion rule relates the internal variables (voltage, current, and power) of a device to its terminal variables through a conversion matrix {\Gamma} and these terminal variables are related by network equations. Second we apply this approach to balanced three-phase networks to formalize per-phase analysis and prove its validity for general networks using the spectral property of the conversion matrix {\Gamma}., Comment: In proceedings of the 11th Bulk Power Systems Dynamics and Control Symposium (IREP 2022), July 25-30, 2022, Banff, Canada

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., Comment: 3 pages,2 figures

We study the problem of phase optimization for electric-vehicle (EV) charging. We formulate our problem as a non-convex mixed-integer programming problem whose objective is to minimize the charging loss. Despite the hardness of directly solving this non-convex problem, we solve a relaxation of the original problem by proposing the PXA algorithm where "P", "X", and "A" stand for three variable matrices in the formed phase optimization problems. We show that under certain conditions, the solution is given by the PXA precisely converges to the global optimum. In addition, using the idea of model predictive control (MPC), we design the {PXA-MPC}, which is an online implementation of the PXA. Compared to other empirical phase balancing strategies, the PXA algorithm significantly improves the charging performance by maximizing energy delivery, minimizing charging price, and assisting future energy planning. The efficacy of our algorithm is demonstrated using data collected from a real-world adaptive EV charging network (ACN)., Comment: 8 pages, 6 figures, accepted by PSCC 2022

Transmission power systems usually consist of interconnected sub-grids that are operated relatively independently. When a failure happens, it is desirable to localize its impact within the sub-grid where the failure occurs. This paper introduces three interface networks to connect sub-grids, achieving better failure localization while maintaining robust network connectivity. The proposed interface networks are validated with numerical experiments on the IEEE 118-bus test network under both DC and AC power flow models., Comment: Accepted to the 2022 American Control Conference (ACC 2022)

As the number of prosumers with distributed energy resources (DERs) grows, the conventional centralized operation scheme may suffer from conflicting interests, privacy concerns, and incentive inadequacy. In this paper, we propose an energy sharing mechanism to address the above challenges. It takes into account network constraints and fairness among prosumers. In the proposed energy sharing market, all prosumers play a generalized Nash game. The market equilibrium is proved to have nice features in a large market or when it is a variational equilibrium. To deal with the possible market failure, inefficiency, or instability in general cases, we introduce a price regulation policy to avoid market power exploitation. The improved energy sharing mechanism with price regulation can guarantee existence and uniqueness of a socially near-optimal market equilibrium. Some advantageous properties are proved, such as prosumer's individual rationality, a sharing price structure similar to the locational marginal price, and the tendency towards social optimum with an increasing number of prosumers. For implementation, a practical bidding algorithm is developed with convergence condition. Experimental results validate the theoretical outcomes and show the practicability of our model and method., Comment: 23 pages, 14 figures

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., Comment: 62pages, 10 figures. This work is accepted in The Eleventh International Conference on Learning Representations (ICLR2023) as Oral - top 25% accepted

We study the problem of learning-augmented predictive linear quadratic control. Our goal is to design a controller that balances \textit{"consistency"}, which measures the competitive ratio when predictions are accurate, and \textit{"robustness"}, which bounds the competitive ratio when predictions are inaccurate. We propose a novel $\lambda$-confident policy and provide a competitive ratio upper bound that depends on a trust parameter $\lambda\in [0,1]$ set based on the confidence in the predictions and some prediction error $\varepsilon$. Motivated by online learning methods, we design a self-tuning policy that adaptively learns the trust parameter $\lambda$ with a competitive ratio that depends on $\varepsilon$ and the variation of system perturbations and predictions. We show that its competitive ratio is bounded from above by $ 1+{O(\varepsilon)}/({{\Theta(1)+\Theta(\varepsilon)}})+O(\mu_{\mathsf{Var}})$ where $\mu_\mathsf{Var}$ measures the variation of perturbations and predictions. It implies that when the variations of perturbations and predictions are small, by automatically adjusting the trust parameter online, the self-tuning scheme ensures a competitive ratio that does not scale up with the prediction error $\varepsilon$., Comment: 34 pages, 8 figures, ACM SIGMETRICS 2022

Transmission line failures in power systems propagate and cascade non-locally. This well-known yet counter-intuitive feature makes it even more challenging to optimally and reliably operate these complex networks. In this work we present a comprehensive framework based on spectral graph theory that fully and rigorously captures how multiple simultaneous line failures propagate, distinguishing between non-cut and cut set outages. Using this spectral representation of power systems, we identify the crucial graph sub-structure that ensures line failure localization -- the network bridge-block decomposition. Leveraging this theory, we propose an adaptive network topology reconfiguration paradigm that uses a two-stage algorithm where the first stage aims to identify optimal clusters using the notion of network modularity and the second stage refines the clusters by means of optimal line switching actions. Our proposed methodology is illustrated using extensive numerical examples on standard IEEE networks and we discussed several extensions and variants of the proposed algorithm., Comment: 45 pages, 7 figures

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., Comment: 4 pages, 1 figure

Non-convex optimization problems can be approximately solved via relaxation or local algorithms. For many practical problems such as optimal power flow (OPF) problems, both approaches tend to succeed in the sense that relaxation is usually exact and local algorithms usually converge to a global optimum. In this paper, we study conditions which are sufficient or necessary for such non-convex problems to simultaneously have exact relaxation and no spurious local optima. Those conditions help us explain the widespread empirical experience that local algorithms for OPF problems often work extremely well., Comment: 14 pages, 2 figures

Adaptive charging can charge electric vehicles (EVs) at scale cost effectively, despite the uncertainty in EV arrivals. We formulate adaptive EV charging as a feasibility problem that meets all EVs' energy demands before their deadlines while satisfying constraints in charging rate and total charging power. We propose an online algorithm, smoothed least-laxity-first (sLLF), that decides the current charging rates without the knowledge of future arrivals and demands. We characterize the performance of the sLLF algorithm analytically and numerically. Numerical experiments with real-world data show that it has a significantly higher rate of feasible EV charging than several other existing EV charging algorithms. Resource augmentation framework is employed to assess the feasibility condition of the algorithm. The assessment shows that the sLLF algorithm achieves perfect feasibility with only a 0.07 increase in resources., Comment: 14 pages, 4 figures

Aggregators have emerged as crucial tools for the coordination of distributed, controllable loads. To be used effectively, an aggregator must be able to communicate the available flexibility of the loads they control, as known as the aggregate flexibility to a system operator. However, most of existing aggregate flexibility measures often are slow-timescale estimations and much less attention has been paid to real-time coordination between an aggregator and an operator. In this paper, we consider solving an online optimization in a closed-loop system and present a design of real-time aggregate flexibility feedback, termed the maximum entropy feedback (MEF). In addition to deriving analytic properties of the MEF, combining learning and control, we show that it can be approximated using reinforcement learning and used as a penalty term in a novel control algorithm -- the penalized predictive control (PPC), which modifies vanilla model predictive control (MPC). The benefits of our scheme are (1). Efficient Communication. An operator running PPC does not need to know the exact states and constraints of the loads, but only the MEF. (2). Fast Computation. The PPC often has much less number of variables than an MPC formulation. (3). Lower Costs. We show that under certain regularity assumptions, the PPC is optimal. We illustrate the efficacy of the PPC using a dataset from an adaptive electric vehicle charging network and show that PPC outperforms classical MPC., Comment: 13 pages, 5 figures, extension of arXiv:2006.13814

ACN-Sim is a data-driven, open-source simulation environment designed to accelerate research in the field of smart electric vehicle (EV) charging. It fills the need in this community for a widely available, realistic simulation environment in which researchers can evaluate algorithms and test assumptions. ACN-Sim provides a modular, extensible architecture, which models the complexity of real charging systems, including battery charging behavior and unbalanced three-phase infrastructure. It also integrates with a broader ecosystem of research tools. These include ACN-Data, an open dataset of EV charging sessions, which provides realistic simulation scenarios and ACN-Live, a framework for field-testing charging algorithms. It also integrates with grid simulators like MATPOWER, PandaPower and OpenDSS, and OpenAI Gym for training reinforcement learning agents., Comment: 9 pages, 8 figures. [v2] Update timezone issue with Fig. 8 where x-axis and background load was shifted by 3 hours

We describe the architecture and algorithms of the Adaptive Charging Network (ACN), which was first deployed on the Caltech campus in early 2016 and is currently operating at over 100 other sites in the United States. The architecture enables real-time monitoring and control and supports electric vehicle (EV) charging at scale. The ACN adopts a flexible Adaptive Scheduling Algorithm based on convex optimization and model predictive control and allows for significant over-subscription of electrical infrastructure. We describe some of the practical challenges in real-world charging systems, including unbalanced three-phase infrastructure, non-ideal battery charging behavior, and quantized control signals. We demonstrate how the Adaptive Scheduling Algorithm handles these challenges, and compare its performance against baseline algorithms from the deadline scheduling literature using real workloads recorded from the Caltech ACN and accurate system models. We find that in these realistic settings, our scheduling algorithm can improve operator profit by 3.4 times over uncontrolled charging and consistently outperforms baseline algorithms when delivering energy in highly congested systems., Comment: 11 pages, 8 figures

Deep Neural Networks (DNNs) approaches for the Optimal Power Flow (OPF) problem received considerable attention recently. A key challenge of these approaches lies in ensuring the feasibility of the predicted solutions to physical system constraints. Due to the inherent approximation errors, the solutions predicted by DNNs may violate the operating constraints, e.g., the transmission line capacities, limiting their applicability in practice. To address this challenge, we develop DeepOPF+ as a DNN approach based on the so-called "preventive" framework. Specifically, we calibrate the generation and transmission line limits used in the DNN training, thereby anticipating approximation errors and ensuring that the resulting predicted solutions remain feasible. We theoretically characterize the calibration magnitude necessary for ensuring universal feasibility. Our DeepOPF+ approach improves over existing DNN-based schemes in that it ensures feasibility and achieves a consistent speed up performance in both light-load and heavy-load regimes. Detailed simulation results on a range of test instances show that the proposed DeepOPF+ generates 100% feasible solutions with minor optimality loss. Meanwhile, it achieves a computational speedup of two orders of magnitude compared to state-of-the-art solvers., Comment: 7 pages, SmarGridComm2020

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., Comment: 13 pages, 2 figures

Aggregators have emerged as crucial tools for the coordination of distributed, controllable loads. However, to be used effectively, aggregators must be able to communicate the available flexibility of the loads they control to the system operator in a manner that is both (i) concise enough to be scalable to aggregators governing hundreds or even thousands of loads and (ii) informative enough to allow the system operator to send control signals to the aggregator that lead to optimization of system-level objectives, such as cost minimization, and do not violate private constraints of the loads, such as satisfying specific load demands. In this paper, we present the design of a real-time flexibility feedback signal based on maximization of entropy. The design provides a concise and informative signal that can be used by the system operator to perform online cost minimization and real-time capacity estimation, while provably satisfying the private constraints of the loads. In addition to deriving analytic properties of the design, we illustrate the effectiveness of the design using a dataset from an adaptive electric vehicle charging network., Comment: The Eleventh ACM International Conference on Future Energy Systems (e-Energy'20)

With the advent of prosumers, the traditional centralized operation may become impracticable due to computational burden, privacy concerns, and conflicting interests. In this paper, an energy sharing mechanism is proposed to accommodate prosumers' strategic decision-making on their self-production and demand in the presence of capacity constraints. Under this setting, prosumers play a generalized Nash game. We prove main properties of the game: an equilibrium exists and is partially unique; no prosumer is worse off by energy sharing and the price-of-anarchy is 1-O(1/I) where I is the number of prosumers. In particular, the PoA tends to 1 with a growing number of prosumers, meaning that the resulting total cost under the proposed energy sharing approaches social optimum. We prove that the corresponding prosumers' strategies converge to the social optimal solution as well. Finally we propose a bidding process and prove that it converges to the energy sharing equilibrium under mild conditions. Illustrative examples are provided to validate the results., Comment: 15 pages, 8 figures; accepted by IEEE Transactions on Smart Grid

Transmission line failure in power systems prop-agate non-locally, making the control of the resulting outages extremely difficult. In Part II of this paper, we continue the study of line failure localizability in transmission networks and characterize the impact of cut set outages. We establish a Simple Path Criterion, showing that the propagation pattern due to bridge outages, a special case of cut set failures, are fully determined by the positions in the network of the buses that participate in load balancing. We then extend our results to general cut set outages. In contrast to non-cut outages discussed in Part I whose subsequent line failures are contained within the original blocks, cut set outages typically impact the whole network, affecting the power flows on all remaining lines. We corroborate our analytical results in both parts using the IEEE 118-bus test system, in which the failure propagation patterns exhibit a clear block-diagonal structure predicted by our theory, even when using full AC power flow equations., Comment: arXiv admin note: text overlap with arXiv:1803.08551

Transmission line failures in power systems propagate and cascade non-locally. In this work, we propose an adaptive control strategy that offers strong guarantees in both the mitigation and localization of line failures. Specifically, we leverage the properties of network bridge-block decomposition and a frequency regulation method called the unified control. If the balancing areas over which the unified control operates coincide with the bridge-blocks of the network, the proposed strategy drives the post-contingency system to a steady state where the impact of initial line outages is localized within the areas where they occurred whenever possible, stopping the cascading process. When the initial line outages cannot be localized, the proposed control strategy provides a configurable design that progressively involves and coordinates more balancing areas. We compare the proposed control strategy with the classical Automatic Generation Control (AGC) on the IEEE 118-bus and 2736-bus test networks. Simulation results show that our strategy greatly improves overall reliability in terms of the N-k security standard, and localizes the impact of initial failures in the majority of the simulated contingencies. Moreover, the proposed framework incurs significantly less load loss, if any, compared to AGC, in all our case studies., Comment: Accepted to IEEE Transactions on Control of Network Systems. arXiv admin note: text overlap with arXiv:1904.05461

Transmission line failures in power systems propagate non-locally, making the control of the resulting outages extremely difficult. In this work, we establish a mathematical theory that characterizes the patterns of line failure propagation and localization in terms of network graph structure. It provides a novel perspective on distribution factors that precisely captures Kirchhoff's Law in terms of topological structures. Our results show that the distribution of specific collections of subtrees of the transmission network plays a critical role on the patterns of power redistribution, and motivates the block decomposition of the transmission network as a structure to understand long-distance propagation of disturbances. In Part I of this paper, we present the case when the post-contingency network remains connected after an initial set of lines are disconnected simultaneously. In Part II, we present the case when an outage separates the network into multiple islands.

Simulations have shown that while semi-definite relaxations of AC optimal power flow (AC-OPF) on three-phase radial networks with only wye connections tend to be exact, the presence of delta connections seem to render them inexact. This paper shows that such inexactness originates from the non-uniqueness of relaxation solutions and numerical errors amplified by the non-uniqueness. This finding motivates two algorithms to recover the exact solution of AC-OPF in the presence of delta connections. In simulations using IEEE 13, 37 and 123-bus systems, the proposed algorithms provide exact optimal solutions up to numerical precision., Comment: 10 pages, 1 figure

The transmission grid is often comprised of several control areas that are connected by multiple tie lines in a mesh structure for reliability. It is also well-known that line failures can propagate non-locally and redundancy can exacerbate cascading. In this paper, we propose an integrated approach to grid reliability that (i) judiciously switches off a small number of tie lines so that the control areas are connected in a tree structure; and (ii) leverages a unified frequency control paradigm to provide congestion management in real time. Even though the proposed topology reduces redundancy, the integration of tree structure at regional level and real-time congestion management can provide stronger guarantees on failure localization and mitigation. We illustrate our approach on the IEEE 39-bus network and evaluate its performance on the IEEE 118-bus, 179-bus, 200-bus and 240-bus networks with various network congestion conditions. Simulations show that, compared with the traditional approach, our approach not only prevents load shedding in more failure scenarios, but also incurs smaller amounts of load loss in scenarios where load shedding is inevitable. Moreover, generators under our approach adjust their operations more actively and efficiently in a local manner., Comment: Accepted to the 21st Power Systems Computation Conference (PSCC 2020)

In this paper we consider the problem of analyzing the effect a change in the load vector can have on the optimal power generation in a DC power flow model. The methodology is based upon the recently introduced concept of the $\mathcal{OPF}$ operator. It is shown that for general network topologies computing the worst-case sensitivities is computationally intractable. However, we show that certain problems involving the $\mathcal{OPF}$ operator can be equivalently converted to a graphical discrete optimization problem. Using the discrete formulation, we provide a decomposition algorithm that reduces the computational cost of computing the worst-case sensitivity. A 27-bus numerical example is used to illustrate our results.

When the shunt elements in the Pi circuit line model are assumed zero, it has been proved that branch flow models are equivalent to bus injection models and that the second-order cone relaxation of optimal power flow problems on a radial network is exact under certain conditions. In this note we propose a branch flow model that includes nonzero line shunts and prove that the equivalence and the exactness of relaxation continue to hold under essentially the same conditions as for zero shunt elements., Comment: 6 pages, 1 figure

This paper aims at the safety-critical control synthesis of network systems such that the satisfaction of the safety constraints can be guaranteed. To handle the large state dimension of such systems, an assume-guarantee contract is used to break the large synthesis problem into smaller subproblems. 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 robust control invariant sets (RCIs) for the subsystems can be composed to form a robust control invariant set for the whole network system under a valid assume-guarantee contract. An epigraph algorithm is proposed to solve for a contract that is valid, ---an approach that has linear complexity for sparse networks, which leads to a robust control invariant set for the whole network system. Implemented with control barrier function (CBF), the state of each subsystem is guaranteed to stay within the safe set. Furthermore, we propose a contingency tube Model Predictive Control approach based on the RCI, which is capable of handling severe contingencies, including topology changes of the network. A power grid example is used to demonstrate the proposed method. The simulation result includes both set point control and contingency recovery, and the safety constraint is always satisfied., Comment: arXiv admin note: text overlap with arXiv:1810.10636

This paper proves that in an unbalanced multi-phase network with a tree topology, the semidefinite programming relaxation of optimal power flow problems is exact when critical buses are not adjacent to each other. Here a critical bus either contributes directly to the cost function or is where an injection constraint is tight at optimality. Our result generalizes a sufficient condition for exact relaxation in single-phase tree networks to tree networks with arbitrary number of phases., Comment: Accepted by 2019 Conference on Decision and Control (CDC)

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental trade-offs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdos-Renyi (n,p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomial-time (in n) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction.

Optimal power flow problems (OPFs) are mathematical programs used to determine how to distribute power over networks subject to network operation constraints and the physics of power flows. In this work, we take the view of treating an OPF problem as an operator which maps user demand to generated power, and allow the network parameters (such as generator and power flow limits) to take values in some admissible set. The contributions of this paper are to formalize this operator theoretic approach, define and characterize restricted parameter sets under which the mapping has a singleton output, independent binding constraints, and is differentiable. In contrast to related results in the optimization literature, we do not rely on introducing auxiliary slack variables. Indeed, our approach provides results that have a clear interpretation with respect to the power network under study. We further provide a closed-form expression for the Jacobian matrix of the OPF operator and describe how various derivatives can be computed using a recently proposed scheme based on homogenous self-dual embedding. Our framework of treating a mathematical program as an operator allows us to pose sensitivity and robustness questions from a completely different mathematical perspective and provide new insights into well studied problems., Comment: 16 pages, 7 figures

The advent of intelligent agents who produce and consume energy by themselves has led the smart grid into the era of "prosumer", offering the energy system and customers a unique opportunity to revaluate/trade their spot energy via a sharing initiative. To this end, designing an appropriate sharing mechanism is an issue with crucial importance and has captured great attention. This paper addresses the prosumers' demand response problem via energy sharing. Under a general supply-demand function bidding scheme, a sharing market clearing procedure considering network constraints is proposed, which gives rise to a generalized Nash game. The existence and uniqueness of market equilibrium are proved in non-congested cases. When congestion occurs, infinitely much equilibrium may exist because the strategy spaces of prosumers are correlated. A price-regulation procedure is introduced in the sharing mechanism, which outcomes a unique equilibrium that is fair to all participants. Properties of the improved sharing mechanism, including the individual rational behaviors of prosumers and the components of sharing price, are revealed. When the number of prosumers increases, the proposed sharing mechanism approaches social optimum. Even with fixed number of resources, introducing competition can result in a decreasing social cost. Illustrative examples validate the theoretical results and provide more insights for the energy sharing research., Comment: 13 pages, 7 figures

This paper proposes a novel energy sharing mechanism for prosumers who can produce and consume. Different from most existing works, the role of individual prosumer as a seller or buyer in our model is endogenously determined. Several desirable properties of the proposed mechanism are proved based on a generalized game-theoretic model. We show that the Nash equilibrium exists and is the unique solution of an equivalent convex optimization problem. The sharing price at the Nash equilibrium equals to the average marginal disutility of all prosumers. We also prove that every prosumer has the incentive to participate in the sharing market, and prosumers' total cost decreases with increasing absolute value of price sensitivity. Furthermore, the Nash equilibrium approaches the social optimal as the number of prosumers grows, and competition can improve social welfare., Comment: 16 pages, 7 figures

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.

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., Comment: Accepted by 2019 American Control Conference (ACC)

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., Comment: Submitted to 2019 American Control Conference

This paper analyses stability of the Unified Controller (UC) that combines frequency control and congestion management and therefore makes it possible to move from preventive to corrective power system control. Earlier work by the authors of UC proved asymptotic stability of the methodology but the proof was based on a simplified first-order model of the turbine and turbine governor. We show that a higher order model of the turbine governor leads to eigenvalues with small but positive real parts. Consequently, we develop a modification of the methodology that decouples the physical and control systems and therefore results in all the eigenvalues having negative real parts. We illustrate the effectiveness of the modification on a realistic model of 39-bus model of New England power system implemented in Power System Toolbox (PST).

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.

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.

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.

Large-scale integration of distributed energy resources into residential distribution feeders necessitates careful control of their operation through power flow analysis. While the knowledge of the distribution system model is crucial for this type of analysis, it is often unavailable or outdated. The recent introduction of synchrophasor technology in low-voltage distribution grids has created an unprecedented opportunity to learn this model from high-precision, time-synchronized measurements of voltage and current phasors at various locations. This paper focuses on joint estimation of model parameters (admittance values) and operational structure of a poly-phase distribution network from the available telemetry data via the lasso, a method for regression shrinkage and selection. We propose tractable convex programs capable of tackling the low rank structure of the distribution system and develop an online algorithm for early detection and localization of critical events that induce a change in the admittance matrix. The efficacy of these techniques is corroborated through power flow studies on four three-phase radial distribution systems serving real household demands.

Direct-current microgrids (DC-MGs) can operate in either grid-connected or stand-alone mode. In particular, stand-alone DC-MG has many distinct applications. However, the optimal power flow problem of a stand-alone DC-MG is inherently non-convex. In this paper, the optimal power flow (OPF) problem of DC-MG is investigated considering convex relaxation based on second-order cone programming (SOCP). Mild assumptions are proposed to guarantee the exactness of relaxation, which only require uniform nodal voltage upper bounds and positive network loss. Furthermore, it is revealed that the exactness of SOCP relaxation of DC-MGs does not rely on either topology or operating mode of DC-MGs, and an optimal solution must be unique if it exists. If line constraints are considered, the exactness of SOCP relaxation may not hold. In this regard, two heuristic methods are proposed to give approximate solutions. Simulations are conducted to confirm the theoretic results.

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.

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., Comment: 12 pages, 5 figures

In Part I of this paper we formulate an optimal scheduling problem for battery swapping that assigns to each electric vehicle (EV) a best station to swap its depleted battery based on its current location and state of charge. The schedule aims to minimize a weighted sum of total travel distance and generation cost over both station assignments and power flow variables, subject to EV range constraints, grid operational constraints and AC power flow equations. We propose there a centralized solution based on the second-order cone programming (SOCP) relaxation of optimal power flow (OPF) and generalized Benders decomposition that is suitable when global information is available. In this paper we propose two distributed solutions based on the alternating direction method of multipliers (ADMM) and dual decomposition respectively that are suitable for cases where the distribution grid, battery stations and EVs are managed by separate entities. Our algorithms allow these entities to make individual decisions but coordinate through privacy-preserving information exchanges to jointly solve an approximate version of the joint battery swapping scheduling and OPF problem. We evaluate our algorithms through simulations.

We formulate an optimal scheduling problem for battery swapping that assigns to each electric vehicle (EV) a best station to swap its depleted battery based on its current location and state of charge. The schedule aims to minimize total travel distance and generation cost over both station assignments and power flow variables, subject to EV range constraints, grid operational constraints and AC power flow equations. To deal with the nonconvexity of power flow equations and the binary nature of station assignments, we propose a solution based on second-order cone programming (SOCP) relaxation of optimal power flow (OPF) and generalized Benders decomposition. When the SOCP relaxation is exact, this approach computes a globally optimal solution. We evaluate the performance of the proposed algorithm through simulations. The algorithm requires global information and is suitable for cases where the distribution network, stations, and EVs are managed centrally by the same operator. In Part II of the paper, we develop distributed solutions for cases where they are operated by different organizations that do not share private information.

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., Comment: 12 pages

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., Comment: Under revision

Distribution networks are usually multiphase and radial. To facilitate power flow computation and optimization, two semidefinite programming (SDP) relaxations of the optimal power flow problem and a linear approximation of the power flow are proposed. We prove that the first SDP relaxation is exact if and only if the second one is exact. Case studies show that the second SDP relaxation is numerically exact and that the linear approximation obtains voltages within 0.0016 per unit of their true values for the IEEE 13, 34, 37, 123-bus networks and a real-world 2065-bus network., Comment: 9 pages, 2 figures, 3 tables, accepted by Power System Computational Conference