Aditie Garg, Mana Jalali, Vassilis Kekatos, Nikolaos Gatsis
Distribution grids are currently challenged by frequent voltage excursions induced by intermittent solar generation. Smart inverters have been advocated as a fast-responding means to regulate voltage and minimize ohmic losses. Since optimal inverter coordination may be computationally challenging and preset local control rules are subpar, the approach of customized control rules designed in a quasi-static fashion features as a golden middle. Departing from affine control rules, this work puts forth non-linear inverter control policies. Drawing analogies to multi-task learning, reactive control is posed as a kernel-based regression task. Leveraging a linearized grid model and given anticipated data scenarios, inverter rules are jointly designed at the feeder level to minimize a convex combination of voltage deviations and ohmic losses via a linearly-constrained quadratic program. Numerical tests using real-world data on a benchmark feeder demonstrate that nonlinear control rules driven also by a few non-local readings can attain near-optimal performance.
Manish K. Singh, Vassilis Kekatos
Increasing concerns on the security and quality of water distribution systems (WDS), call for computational tools with performance guarantees. To this end, this work revisits the physical laws governing water flow and provides a hierarchy of solvers of complementary value. Given the water injection or pressure at each WDS node, finding the water flows within pipes and pumps along with the pressures at all WDS nodes constitutes the water flow (WF) problem. The latter entails solving a set of (non)-linear equations. We extend uniqueness claims on the solution to the WF equations in setups with multiple fixed-pressure nodes and detailed pump models. For networks without pumps, the WF solution is already known to be the minimizer of a convex function. The latter approach is extended to networks with pumps but not in cycles, through a stitching algorithm. For networks with non-overlapping cycles, a provably exact convex relaxation of the pressure drop equations yields a mixed-integer quadratically-constrained quadratic program (MI-QCQP) solver. A hybrid scheme combining the MI-QCQP with the stitching algorithm can handle WDS with overlapping cycles, but without pumps on them. Each solver is guaranteed to converge regardless of initialization, as numerically validated on a benchmark WDS.
Sina Taheri, Mana Jalali, Vassilis Kekatos, Lang Tong
Interconnection studies for distributed energy resources (DERs) can currently take months since they entail simulating a large number of power flow scenarios. If DERs are to be actively controlled, probabilistic hosting capacity analysis (PHCA) studies become more time-consuming since they require solving multiple optimal power flow (OPF) tasks. PHCA is expedited here by leveraging the powerful tool of multiparametric programming (MPP). Using an approximate grid model, optimal DER setpoints are decided by a quadratic program, which depends on analysis and uncertain parameters in a possibly nonlinear fashion. By reformulating this program, feasible and infeasible OPF instances alike are handled in a unified way to uniquely reveal the location, frequency, and severity of feeder constraint violations. The effect of voltage regulators is also captured by novel approximate models. Upon properly extending MPP to PHCA, we were able to find the exact minimizers for 518,400 OPF instances on the IEEE 123-bus feeder by solving only 6,905 of them, and 86,400 instances on a 1,160-bus feeder by solving only 2,111 instances. This accelerated PHCA by a factor of 10. Thus, a utility can promptly infer grid statistics using real-world data without a probabilistic characterization of uncertain parameters.
Sina Taheri, Vassilis Kekatos, Harsha Veeramachaneni
An investor has to carefully select the location and size of new generation units it intends to build, since adding capacity in a market affects the profit from units this investor may already own. To capture this closed-loop characteristic, strategic investment (SI) can be posed as a bilevel optimization. By analytically studying a small market, we first show that its objective function can be non-convex and discontinuous. Realizing that existing mixed-integer problem formulations become impractical for larger markets and increasing number of scenarios, this work put forth two SI solvers: a grid search to handle setups where the candidate investment locations are few, and a stochastic gradient descent approach for otherwise. Both solvers leverage the powerful toolbox of multiparametric programming (MPP), each in a unique way. The grid search entails finding the primal/dual solutions for a large number of optimal power flow (OPF) problems, which nonetheless can be efficiently computed several at once thanks to the properties of MPP. The same properties facilitate the rapid calculation of gradients in a mini-batch fashion, thus accelerating the implementation of a stochastic gradient descent search. Tests on the IEEE 118-bus system using real-world data corroborate the advantages of the novel MPP-aided solvers.
Sina Taheri, Vassilis Kekatos, Harsha Veeramachaneni, Baosen Zhang
Bundling a large number of distributed energy resources through a load aggregator has been advocated as an effective means to integrate such resources into whole-sale energy markets. To ease market clearing, system operators allow aggregators to submit bidding models of simple prespecified polytopic shapes. Aggregators need to carefully design and commit to a polytope that best captures their energy flexibility along a day-ahead scheduling horizon. This work puts forth a model-informed data-based optimal flexibility design for aggregators, which deals with the time-coupled, uncertain, and non-convex models of individual loads. The proposed solution first generates efficiently a labeled dataset of (non)-disaggregatable schedules. The feasible set of the aggregator is then approximated by an ellipsoid upon training a convex quadratic classifier using the labeled dataset. The ellipsoid is subsequently inner approximated by a polytope. Using Farkas lemma, the obtained polytope is finally inner approximated by the polytopic shape dictated by the market. Numerical tests show the effectiveness of the proposed flexibility design framework for designing the feasible sets of small- and large-sized aggregators coordinating solar photovoltaics, thermostatically-controlled loads, batteries, and electric vehicles. The tests further demonstrate that it is crucial for the aggregator to consider time-coupling and uncertainties in optimal flexibility design.
Manish K. Singh, Vassilis Kekatos
The dynamic response of power grids to small events or persistent stochastic disturbances influences their stable operation. Low-frequency inter-area oscillations are of particular concern due to insufficient damping. This paper studies the effect of the operating point on the linear time-invariant dynamics of power networks. A pertinent metric based on the frequency response of grid dynamics is proposed to quantify power system's stability against inter-area oscillations. We further put forth an optimal power flow formulation to yield a grid dispatch that optimizes this novel stability metric. A semidefinite program (SDP) relaxation is employed to yield a computationally tractable convex problem. Numerical tests on the IEEE-39 bus system demonstrate that the SDP relaxation is exact yielding a rank-1 solution. The relative trade-off of the proposed small-signal stability metric versus the generation cost is also studied.
Manish K. Singh, Vassilis Kekatos
The critical role of gas fired-plants to compensate renewable generation has increased the operational variability in natural gas networks (GN). Towards developing more reliable and efficient computational tools for GN monitoring, control, and planning, this work considers the task of solving the nonlinear equations governing steady-state flows and pressures in GNs. It is first shown that if the gas flow equations are feasible, they enjoy a unique solution. To the best of our knowledge, this is the first result proving uniqueness of the steady-state gas flow solution over the entire feasible domain of gas injections. To find this solution, we put forth a mixed-integer second-order cone program (MI-SOCP)-based solver relying on a relaxation of the gas flow equations. This relaxation is provably exact under specific network topologies. Unlike existing alternatives, the devised solver does not need proper initialization or knowing the gas flow directions beforehand, and can handle gas networks with compressors. Numerical tests on tree and meshed networks with random gas injections indicate that the relaxation is exact even when the derived conditions are not met.
Gang Wang, Vassilis Kekatos, Antonio J. Conejo, Georgios B. Giannakis
Contemporary electricity distribution systems are being challenged by the variability of renewable energy sources. Slow response times and long energy management periods cannot efficiently integrate intermittent renewable generation and demand. Yet stochasticity can be judiciously coupled with system flexibilities to enhance grid operation efficiency. Voltage magnitudes for instance can transiently exceed regulation limits, while smart inverters can be overloaded over short time intervals. To implement such a mode of operation, an ergodic energy management framework is developed here. Considering a distribution grid with distributed energy sources and a feed-in tariff program, active power curtailment and reactive power compensation are formulated as a stochastic optimization problem. Tighter operational constraints are enforced in an average sense, while looser margins are enforced to be satisfied at all times. Stochastic dual subgradient solvers are developed based on exact and approximate grid models of varying complexity. Numerical tests on a real-world 56-bus distribution grid and the IEEE 123-bus test feeder relying on both grid models corroborate the advantages of the novel schemes over their deterministic alternatives.
Sarthak Gupta, Vassilis Kekatos, Spyros Chatzivasileiadis
Distribution grids are challenged by rapid voltage fluctuations induced by variable power injections from distributed energy resources (DERs). To regulate voltage, the IEEE Standard 1547 recommends each DER inject reactive power according to piecewise-affine Volt/VAR control rules. Although the standard suggests a default shape, the rule can be customized per bus. This task of optimal rule design (ORD) is challenging as Volt/VAR rules introduce nonlinear dynamics, and lurk trade-offs between stability and steady-state voltage profiles. ORD is formulated as a mixed-integer nonlinear program (MINLP), but scales unfavorably with the problem size. Towards a more efficient solution, we reformulate ORD as a deep learning problem. The idea is to design a DNN that emulates Volt/VAR dynamics. The DNN takes grid scenarios as inputs, rule parameters as weights, and outputs equilibrium voltages. Optimal rule parameters can be found by training the DNN so its output approaches unity for various scenarios. The DNN is only used to optimize rules and is never employed in the field. While dealing with ORD, we also review and expand on stability conditions and convergence rates for Volt/VAR dynamics on single- and multi-phase feeders. Tests showcase the merit of DNN-based ORD by benchmarking it against its MINLP counterpart.
Ilgiz Murzakhanov, Sarthak Gupta, Spyros Chatzivasileiadis, Vassilis Kekatos
The IEEE 1547 Standard for the interconnection of distributed energy resources (DERs) to distribution grids provisions that smart inverters could be implementing Volt/VAR control rules among other options. Such rules enable DERs to respond autonomously in response to time-varying grid loading conditions. The rules comprise affine droop control augmented with a deadband and saturation regions. Nonetheless, selecting the shape of these rules is not an obvious task, and the default options may not be optimal or dynamically stable. To this end, this work develops a novel methodology for customizing Volt/VAR rules on a per-bus basis for a single-phase feeder. The rules are adjusted by the utility every few hours depending on anticipated demand and solar scenarios. Using a projected gradient descent-based algorithm, rules are designed to improve the feeder's voltage profile, comply with IEEE 1547 constraints, and guarantee stability of the underlying nonlinear grid dynamics. The stability region is inner approximated by a polytope and the rules are judiciously parameterized so their feasible set is convex. Numerical tests using real-world data on the IEEE 141-bus feeder corroborate the scalability of the methodology and its explore the trade-offs of Volt/VAR control with alternatives.
Sarthak Gupta, Sidhant Misra, Deepjyoti Deka, Vassilis Kekatos
A prominent challenge to the safe and optimal operation of the modern power grid arises due to growing uncertainties in loads and renewables. Stochastic optimal power flow (SOPF) formulations provide a mechanism to handle these uncertainties by computing dispatch decisions and control policies that maintain feasibility under uncertainty. Most SOPF formulations consider simple control policies such as affine policies that are mathematically simple and resemble many policies used in current practice. Motivated by the efficacy of machine learning (ML) algorithms and the potential benefits of general control policies for cost and constraint enforcement, we put forth a deep neural network (DNN)-based policy that predicts the generator dispatch decisions in real time in response to uncertainty. The weights of the DNN are learnt using stochastic primal-dual updates that solve the SOPF without the need for prior generation of training labels and can explicitly account for the feasibility constraints in the SOPF. The advantages of the DNN policy over simpler policies and their efficacy in enforcing safety limits and producing near optimal solutions are demonstrated in the context of a chance constrained formulation on a number of test cases.
Shaohui Liu, Hao Zhu, Vassilis Kekatos
Poorly damped oscillations pose threats to the stability and reliability of interconnected power systems. In this work, we propose a comprehensive data-driven framework for inferring the sources of forced oscillation (FO) using solely synchrophasor measurements. During normal grid operations, fast-rate ambient data are collected to recover the impulse responses in the small-signal regime, without requiring the system model. When FO events occur, the source is estimated based on the frequency domain analysis by fitting the least-squares (LS) error for the FO data using the impulse responses recovered previously. Although the proposed framework is purely data-driven, the result has been established theoretically via model-based analysis of linearized dynamics under a few realistic assumptions. Numerical validations demonstrate its applicability to realistic power systems including nonlinear, higher-order dynamics with control effects using the IEEE 68-bus system, and the 240-bus system from the IEEE-NASPI FO source location contest. The generalizability of the proposed methodology has been validated using different types of measurements and partial sensor coverage conditions.
Zain ul Abdeen, He Yin, Vassilis Kekatos, Ming Jin
In this paper, we examine an important problem of learning neural networks that certifiably meet certain specifications on input-output behaviors. Our strategy is to find an inner approximation of the set of admissible policy parameters, which is convex in a transformed space. To this end, we address the key technical challenge of convexifying the verification condition for neural networks, which is derived by abstracting the nonlinear specifications and activation functions with quadratic constraints. In particular, we propose a reparametrization scheme of the original neural network based on loop transformation, which leads to a convex condition that can be enforced during learning. This theoretical construction is validated in an experiment that specifies reachable sets for different regions of inputs.
Vassilis Kekatos, Georgios B. Giannakis
One of the key challenges in sensor networks is the extraction of information by fusing data from a multitude of distinct, but possibly unreliable sensors. Recovering information from the maximum number of dependable sensors while specifying the unreliable ones is critical for robust sensing. This sensing task is formulated here as that of finding the maximum number of feasible subsystems of linear equations, and proved to be NP-hard. Useful links are established with compressive sampling, which aims at recovering vectors that are sparse. In contrast, the signals here are not sparse, but give rise to sparse residuals. Capitalizing on this form of sparsity, four sensing schemes with complementary strengths are developed. The first scheme is a convex relaxation of the original problem expressed as a second-order cone program (SOCP). It is shown that when the involved sensing matrices are Gaussian and the reliable measurements are sufficiently many, the SOCP can recover the optimal solution with overwhelming probability. The second scheme is obtained by replacing the initial objective function with a concave one. The third and fourth schemes are tailored for noisy sensor data. The noisy case is cast as a combinatorial problem that is subsequently surrogated by a (weighted) SOCP. Interestingly, the derived cost functions fall into the framework of robust multivariate linear regression, while an efficient block-coordinate descent algorithm is developed for their minimization. The robust sensing capabilities of all schemes are verified by simulated tests.
Manish Kumar Singh, Vassilis Kekatos, Chen-Ching Liu
Increasing emphasis on reliability and resiliency call for advanced distribution system restoration (DSR). The integration of grid sensors, remote controls, and distributed generators (DG) brings about exciting opportunities in DSR. In this context, this work considers the task of single-step restoration of a single phase power distribution system. Different from existing works, the devised restoration scheme achieves optimal formation of islands without heuristically pre-identifying reference buses. It further facilitates multiple DGs running within the same island, and establishes a coordination hierarchy in terms of their PV/PQ operation modes. Generators without black-start capability are guaranteed to remain connected to a black-start DG or a substation. The proposed scheme models remotely-controlled voltage regulators exactly, and integrates them in the restoration process. Numerical tests on a modified IEEE 37-bus feeder demonstrate that the proposed mixed-integer linear program (MILP) takes less than four seconds to handle random outages of 1-5 lines. The scalability of this novel MILP formulation can be attributed to the unique use of cycles and paths on the grid infrastructure graph; the McCormick linearization technique; and an approximate power flow model.
Shaohui Liu, Hao Zhu, Vassilis Kekatos
Wide-area dynamic studies are of paramount importance to ensure the stability and reliability of power grids. The rising deployment synchrophasor and other sensing technologies has made data-driven modeling and analysis possible using the synchronized fast-rate dynamic measurements. This paper presents a general model-free framework of inferring the grid dynamic responses using the ubiquitous ambient data collected during normal grid operations. Building upon the second-order dynamic model, we have established the connection from the cross-correlation of various types of angle, frequency, and line flow data at any two locations, to their corresponding dynamic responses. The theoretical results enabled a fully data-driven framework for estimating the latter using real-time ambient data. Numerical results using the WSCC 9-bus system and a synthetic 2000-bus Texas system have demonstrated the effectiveness of proposed approaches for dynamic modeling of realistic power systems.
Sarthak Gupta, Vassilis Kekatos, Ming Jin
Coordinating inverters at scale under uncertainty is the desideratum for integrating renewables in distribution grids. Unless load demands and solar generation are telemetered frequently, controlling inverters given approximate grid conditions or proxies thereof becomes a key specification. Although deep neural networks (DNNs) can learn optimal inverter schedules, guaranteeing feasibility is largely elusive. Rather than training DNNs to imitate already computed optimal power flow (OPF) solutions, this work integrates DNN-based inverter policies into the OPF. The proposed DNNs are trained through two OPF alternatives that confine voltage deviations on the average and as a convex restriction of chance constraints. The trained DNNs can be driven by partial, noisy, or proxy descriptors of the current grid conditions. This is important when OPF has to be solved for an unobservable feeder. DNN weights are trained via back-propagation and upon differentiating the AC power flow equations assuming the network model is known. Otherwise, a gradient-free variant is put forth. The latter is relevant when inverters are controlled by an aggregator having access only to a power flow solver or a digital twin of the feeder. Numerical tests compare the DNN-based inverter control schemes with the optimal inverter setpoints in terms of optimality and feasibility.
Guido Cavraro, Vassilis Kekatos
Knowing the connectivity and line parameters of the underlying electric distribution network is a prerequisite for solving any grid optimization task. Although distribution grids lack observability and comprehensive metering, inverters with advanced cyber capabilities currently interface solar panels and energy storage devices to the grid. Smart inverters have been widely used for grid control and optimization, yet the fresh idea here is to engage them towards network topology inference. Being an electric circuit, a distribution grid can be intentionally probed by instantaneously perturbing inverter injections. Collecting and processing the incurred voltage deviations across nodes can potentially unveil the grid topology even without knowing loads. Using grid probing data and under an approximate grid model, the tasks of topology recovery and line status verification are posed respectively as non-convex estimation and detection problems. Leveraging the features of the Laplacian matrix of a tree graph, probing terminal nodes is analytically shown to be sufficient for exact topology recovery if voltage data are collected at all buses. The related non-convex problems are surrogated to convex ones, which are iteratively solved via closed-form updates based on the alternating direction method of multipliers and projected gradient descent. Numerical tests on benchmark feeders demonstrate that grid probing can yield line status error probabilities of 0.001 by probing 40% of the nodes.
Vassilis Kekatos, Gang Wang, Antonio J. Conejo, Georgios B. Giannakis
Distribution microgrids are being challenged by reverse power flows and voltage fluctuations due to renewable generation, demand response, and electric vehicles. Advances in photovoltaic (PV) inverters offer new opportunities for reactive power management provided PV owners have the right investment incentives. In this context, reactive power compensation is considered here as an ancillary service. Accounting for the increasing time-variability of distributed generation and demand, a stochastic reactive power compensation scheme is developed. Given uncertain active power injections, an online reactive control scheme is devised. This scheme is distribution-free and relies solely on power injection data. Reactive injections are updated using the Lagrange multipliers of a second-order cone program. Numerical tests on an industrial 47-bus microgrid and the residential IEEE 123-bus feeder corroborate the reactive power management efficiency of the novel stochastic scheme over its deterministic alternative, as well as its capability to track variations in solar generation and household demand.
Sarthak Gupta, Vassilis Kekatos, Walid Saad
With increasingly favorable economics and bundling of different grid services, energy storage systems (ESS) are expected to play a key role in integrating renewable generation. This work considers the coordination of ESS owned by customers located at different buses of a distribution grid. Customers participate in frequency regulation and experience energy prices that increase with the total demand. Charging decisions are coupled across time due to battery dynamics, as well as across network nodes due to competitive pricing and voltage regulation constraints. Maximizing the per-user economic benefit while maintaining voltage magnitudes within allowable limits is posed here as a network-constrained game. It is analytically shown that a generalized Nash equilibrium exists and can be expressed as the minimizer of a convex yet infinite-time horizon aggregate optimization problem. To obtain a practical solution, a Lyapunov optimization approach is adopted to design a real-time scheme offering feasible charging decisions with performance guarantees. The proposed method improves over the standard Lyapunov technique via a novel weighting of user costs. By judiciously exploiting the physical grid response, a distributed implementation of the real-time solver is also designed. The features of the novel algorithmic designs are validated using numerical tests on realistic datasets.