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We consider the problem of controlling the frequency of low-inertia power systems via inverter-based resources (IBRs) that are weakly connected to the grid. We propose a novel grid-forming control strategy, the so-called frequency shaping control, that aims to shape the frequency response of synchronous generators (SGs) to load perturbations so as to efficiently arrest sudden frequency drops. Our solution relaxes several existing assumptions in the literature and is able to navigate tradeoffs between peak power requirements and maximum frequency deviations. Finally, we analyze the robustness to imperfect knowledge of network parameters, while particularly highlighting the importance of accurate estimation of these parameters.

This paper considers a nonconvex optimization problem that evolves over time, and addresses the synthesis and analysis of regularized primal-dual gradient methods to track a Karush-Kuhn-Tucker (KKT) trajectory. The proposed regularized primal-dual gradient methods are implemented in a running fashion, in the sense that the underlying optimization problem changes during the iterations of the algorithms. For a problem with twice continuously differentiable cost and constraints, and under a generalization of the Mangasarian-Fromovitz constraint qualification, sufficient conditions are derived for the running algorithm to track a KKT trajectory. Further, asymptotic bounds for the tracking error (as a function of the time-variability of a KKT trajectory) are obtained. A continuous-time version of the algorithm, framed as a system of differential inclusions, is also considered and analytical convergence results are derived. For the continuous-time setting, a set of sufficient conditions for the KKT trajectories not to bifurcate or merge is proposed. Illustrative numerical results inspired by a real-world application are provided.

This paper develops an algorithmic framework for real-time optimization of distribution-level distributed energy resources (DERs). The proposed framework optimizes the operation of both DERs that are individually controllable and groups of DERs (i.e., aggregations) at an electrical point of connection that are jointly controlled. From an electrical standpoint, wye and delta single- and multi-phase connections are accounted for. The algorithm enables (groups of) DERs to pursue given performance objectives, while adjusting their (aggregate) powers to respond to services requested by grid operators and to maintain electrical quantities within engineering limits. The design of the algorithm leverages a time-varying bi-level problem formulation capturing various performance objectives and engineering constraints, and an online implementation of primal-dual projected-gradient methods. The gradient steps are suitably modified to accommodate appropriate measurements from the distribution network and the DERs. By virtue of this approach, the resultant algorithm can cope with inaccuracies in the distribution-system modeling, it avoids pervasive metering to gather the state of non-controllable resources, and it naturally lends itself to a distributed implementation. Analytical stability and convergence claims are established in terms of tracking of the solution of the formulated time-varying optimization problem. The proposed method is tested in a realistic distribution system with real data.

The ODE method has been a workhorse for algorithm design and analysis since the introduction of the stochastic approximation. It is now understood that convergence theory amounts to establishing robustness of Euler approximations for ODEs, while theory of rates of convergence requires finer analysis. This paper sets out to extend this theory to quasi-stochastic approximation, based on algorithms in which the "noise" is based on deterministic signals. The main results are obtained under minimal assumptions: the usual Lipschitz conditions for ODE vector fields, and it is assumed that there is a well defined linearization near the optimal parameter $\theta^*$, with Hurwitz linearization matrix $A^*$. The main contributions are summarized as follows: (i) If the algorithm gain is $a_t=g/(1+t)^\rho$ with $g>0$ and $\rho\in(0,1)$, then the rate of convergence of the algorithm is $1/t^\rho$. There is also a well defined "finite-$t$" approximation: \[ a_t^{-1}\{\Theta_t-\theta^*\}=\bar{Y}+\Xi^{\mathrm{I}}_t+o(1) \] where $\bar{Y}\in\mathbb{R}^d$ is a vector identified in the paper, and $\{\Xi^{\mathrm{I}}_t\}$ is bounded with zero temporal mean. (ii) With gain $a_t = g/(1+t)$ the results are not as sharp: the rate of convergence $1/t$ holds only if $I + g A^*$ is Hurwitz. (iii) Based on the Ruppert-Polyak averaging of stochastic approximation, one would expect that a convergence rate of $1/t$ can be obtained by averaging: \[ \Theta^{\text{RP}}_T=\frac{1}{T}\int_{0}^T \Theta_t\,dt \] where the estimates $\{\Theta_t\}$ are obtained using the gain in (i). The preceding sharp bounds imply that averaging results in $1/t$ convergence rate if and only if $\bar{Y}=\sf 0$. This condition holds if the noise is additive, but appears to fail in general. (iv) The theory is illustrated with applications to gradient-free optimization and policy gradient algorithms for reinforcement learning.

Distributed control agents have been advocated as an effective means for improving the resiliency of our physical infrastructures under unexpected events. Purely local control has been shown to be insufficient, centralized optimal resource allocation approaches can be slow. In this context, we put forth a hybrid low-communication saturation-driven protocol for the coordination of control agents that are distributed over a physical system and are allowed to communicate with peers over a "hotline" communication network. According to this protocol, agents act on local readings unless their control resources have been depleted, in which case they send a beacon for assistance to peer agents. Our ripple-type scheme triggers communication locally only for the agents with saturated resources and it is proved to converge. Moreover, under a monotonicity assumption on the underlying physical law coupling control outputs to inputs, the devised control is proved to converge to a configuration satisfying safe operational constraints. The assumption is shown to hold for voltage control in electric power systems and pressure control in water distribution networks. Numerical tests corroborate the efficacy of the novel scheme., Comment: Accepted for presentation at the American Control Conference (ACC) 2021

The increasing penetration of distributed energy resources (DERs) in the distribution networks has turned the conventionally passive load buses into active buses that can provide grid services for the transmission system. To take advantage of the DERs in the distribution networks, this letter formulates a transmission-and-distribution (T&D) systems co-optimization problem that achieves economic dispatch at the transmission level and optimal voltage regulation at the distribution level by leveraging large generators and DERs. A primal-dual gradient algorithm is proposed to solve this optimization problem jointly for T&D systems, and a distributed market-based equivalent of the gradient algorithm is used for practical implementation. The results are corroborated by numerical examples with the IEEE 39-Bus system connected with 7 different distribution networks.

This paper examines the problem of real-time optimization of networked systems and develops online algorithms that steer the system towards the optimal trajectory without explicit knowledge of the system model. The problem is modeled as a dynamic optimization problem with time-varying performance objectives and engineering constraints. The design of the algorithms leverages the online zero-order primal-dual projected-gradient method. In particular, the primal step that involves the gradient of the objective function (and hence requires networked systems model) is replaced by its zero-order approximation with two function evaluations using a deterministic perturbation signal. The evaluations are performed using the measurements of the system output, hence giving rise to a feedback interconnection, with the optimization algorithm serving as a feedback controller. The paper provides some insights on the stability and tracking properties of this interconnection. Finally, the paper applies this methodology to a real-time optimal power flow problem in power systems, and shows its efficacy on the IEEE 37-node distribution test feeder for reference power tracking and voltage regulation., Comment: 10 pages, 7 figures

As electric grids experience high penetration levels of renewable generation, fundamental changes are required to address real-time situational awareness. This paper uses unique traits of tensors to devise a model-free situational awareness and energy forecasting framework for distribution networks. This work formulates the state of the network at multiple time instants as a three-way tensor; hence, recovering full state information of the network is tantamount to estimating all the values of the tensor. Given measurements received from $\mu$phasor measurement units and/or smart meters, the recovery of unobserved quantities is carried out using the low-rank canonical polyadic decomposition of the state tensor---that is, the state estimation task is posed as a tensor imputation problem utilizing observed patterns in measured quantities. Two structured sampling schemes are considered: slab sampling and fiber sampling. For both schemes, we present sufficient conditions on the number of sampled slabs and fibers that guarantee identifiability of the factors of the state tensor. Numerical results demonstrate the ability of the proposed framework to achieve high estimation accuracy in multiple sampling scenarios., Comment: 8 pages, 7 figures, 5 tables

We develop an online data-enabled predictive (ODeePC) control method for optimal control of unknown systems, building on the recently proposed DeePC [1]. Our proposed ODeePC method leverages a primal-dual algorithm with real-time measurement feedback to iteratively compute the corresponding real-time optimal control policy as system conditions change. The proposed ODeePC conceptual-wise resembles standard adaptive system identification and model predictive control (MPC), but it provides a new alternative for the standard methods. ODeePC is enabled by computationally efficient methods that exploit the special structure of the Hankel matrices in the context of DeePC with Fast Fourier Transform (FFT) and primal-dual algorithm. We provide theoretical guarantees regarding the asymptotic behavior of ODeePC, and we demonstrate its performance through numerical examples.

We present explicit sufficient conditions that guarantee the existence and uniqueness of the feasible load-flow solution for distribution networks with a generic topology (radial or meshed) modeled with positive sequence equivalents. In the problem, we also account for the presence of shunt elements. The conditions have low computational complexity and thus can be efficiently verified in a real system. Once the conditions are satisfied, the unique load-flow solution can be reached by a given fixed point iteration method of approximately linear complexity. Therefore, the proposed approach is of particular interest for modern active distribution network (ADN) setup in the context of real-time control. The theory has been confirmed through numerical experiments.

The paper investigates the problem of estimating the state of a time-varying system with a linear measurement model; in particular, the paper considers the case where the number of measurements available can be smaller than the number of states. In lieu of a batch linear least-squares (LS) approach -- well suited for static networks, where a sufficient number of measurements could be collected to obtain a full-rank design matrix -- the paper proposes an online algorithm to estimate the possibly time-varying state by processing measurements as and when available. The design of the algorithm hinges on a generalized LS cost augmented with a proximal-point-type regularization. With the solution of the regularized LS problem available in closed-form, the online algorithm is written as a linear dynamical system where the state is updated based on the previous estimate and based on the new available measurements. Conditions under which the algorithmic steps are in fact a contractive mapping are shown, and bounds on the estimation error are derived for different noise models. Numerical simulations are provided to corroborate the analytical findings.

Increasing integration of distributed energy resources (DERs) within distribution feeders provides unprecedented flexibility at the distribution-transmission interconnection. To exploit this flexibility and to use the capacity potential of aggregate DERs, feasible substation power injection trajectories need to be efficiently characterized. This paper provides an ellipsoidal inner approximation of the set of feasible power injection trajectories at the substation such that for any point in the set, there exists a feasible disaggregation strategy of DERs for any load uncertainty realization. The problem is formulated as one of finding the robust maximum volume ellipsoid inside the flexibility region under uncertainty. Though the problem is NP-hard even in the deterministic case, this paper derives novel approximations of the resulting adaptive robust optimization problem based on optimal second-stage policies. The proposed approach yields less conservative flexibility characterization than existing flexibility region approximation formulations. The efficacy of the proposed method is demonstrated on a realistic distribution feeder., Comment: 6 pages, 1 figure, to be published in IEEE Control Systems Letters

Deep learning for distribution grid optimization can be advocated as a promising solution for near-optimal yet timely inverter dispatch. The principle is to train a deep neural network (DNN) to predict the solutions of an optimal power flow (OPF), thus shifting the computational effort from real-time to offline. Nonetheless, before training this DNN, one has to solve a large number of OPFs to create a labeled dataset. Granted the latter step can still be prohibitive in time-critical applications, this work puts forth an original technique for improving the prediction accuracy of DNNs by taking into account the sensitivities of the OPF minimizers with respect to the OPF parameters. By expanding on multiparametric programming, it is shown that although inverter control problems may exhibit dual degeneracy, the required sensitivities do exist in general and can be computed readily using the output of any standard quadratic program (QP) solver. Numerical tests showcase that sensitivity-informed deep learning can enhance prediction accuracy in terms of mean square error (MSE) by 2-3 orders of magnitude at minimal computational overhead. Improvements are more significant in the small-data regime, where a DNN has to learn to optimize using a few examples. Beyond multiparametric QPs, the approach is currently being generalized to parametric (non)-convex optimization problems., Comment: Manuscript under review

Operations of substation circuit breakers are important for maintenance needs and topology reconfiguration in power systems. Bus splitting is one type of topology change where the two bus bars at a substation can become electrically disconnected under certain actions of circuit breakers. Because these events involve detailed substation modeling, they are typically not considered in routine power system operation and control. In this paper, an improved substation-level topology optimization framework is developed by expanding traditional line switching decisions by breaker-level bus splitting, which can further reduce grid congestion and generation costs. A tight McCormick relaxation is proposed to reformulate the bilinear terms in the resultant optimization problem to linear inequality constraints. Thus, a tractable mixed-integer linear program reformulation is attained that allows for efficient solutions in real-time operations. Numerical studies on the IEEE 14-bus and 118-bus systems demonstrate the computational performance and economic benefits of the proposed topology optimization approach.

Feedback-based online optimization algorithms have gained traction in recent years because of their simple implementation, their ability to reject disturbances in real time, and their increased robustness to model mismatch. While the robustness properties have been observed both in simulation and experimental results, the theoretical analysis in the literature is mostly limited to nominal conditions. In this work, we propose a framework to systematically assess the robust stability of feedback-based online optimization algorithms. We leverage tools from monotone operator theory, variational inequalities and classical robust control to obtain tractable numerical tests that guarantee robust convergence properties of online algorithms in feedback with a physical system, even in the presence of disturbances and model uncertainty. The results are illustrated via an academic example and a case study of a power distribution system.

In this paper, we propose an analytical framework to quantify the amount of data samples needed to obtain accurate state estimation in a power system - a problem known as sample complexity analysis in computer science. Motivated by the increasing adoption of distributed energy resources into the distribution-level grids, it becomes imperative to estimate the state of distribution grids in order to ensure stable operation. Traditional power system state estimation techniques mainly focus on the transmission network which involve solving an overdetermined system and eliminating bad data. However, distribution networks are typically underdetermined due to the large number of connection points and high cost of pervasive installation of measurement devices. In this paper, we consider the recently proposed state-estimation method for underdetermined systems that is based on matrix completion. In particular, a constrained matrix completion algorithm was proposed, wherein the standard matrix completion problem is augmented with additional equality constraints representing the physics (namely power-flow constraints). We analyze the sample complexity of this general method by proving an upper bound on the sample complexity that depends directly on the properties of these constraints that can lower number of needed samples as compared to the unconstrained problem. To demonstrate the improvement that the constraints add to distribution state estimation, we test the method on a 141-bus distribution network case study and compare it to the traditional least squares minimization state estimation method.

This paper develops and compares algorithms to compute inner approximations of the Minkowski sum of convex polytopes. As an application, the paper considers the computation of the feasibility set of aggregations of distributed energy resources (DERs), such as solar photovoltaic inverters, controllable loads, and storage devices. To fully account for the heterogeneity in the DERs while ensuring an acceptable approximation accuracy, the paper leverages a union-based computation and advocates homothet-based polytope decompositions. However, union-based approaches can in general lead to high-dimensionality concerns; to alleviate this issue, this paper shows how to define candidate sets to reduce the computational complexity. Accuracy and trade-offs are analyzed through numerical simulations for illustrative examples., Accepted for presentation at the 57th IEEE Conference on Decision and Control

This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. This paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power rebalancing and tracking of the DC optimal power flow points.

This paper addresses the design and analysis of feedback-based online algorithms to control systems or networked systems based on performance objectives and engineering constraints that may evolve over time. The emerging time-varying convex optimization formalism is leveraged to model optimal operational trajectories of the systems, as well as explicit local and network-level operational constraints. Departing from existing batch and feed-forward optimization approaches, the design of the algorithms capitalizes on an online implementation of primal-dual projected-gradient methods; the gradient steps are, however, suitably modified to accommodate feedback from the system in the form of measurements - hence, the term "online optimization with feedback." By virtue of this approach, the resultant algorithms can cope with model mismatches in the algebraic representation of the system states and outputs, they avoid pervasive measurements of exogenous inputs, and they naturally lend themselves to a distributed implementation. Under suitable assumptions, analytical convergence claims are established in terms of dynamic regret. Furthermore, when the synthesis of the feedback-based online algorithms is based on a regularized Lagrangian function, Q-linear convergence to solutions of the time-varying optimization problem is shown., Alphabetical order, authors contributed equally. Old title: Online Optimization with Feedback

With the rising penetration of distributed energy resources, distribution system control and enabling techniques such as state estimation have become essential to distribution system operation. However, traditional state estimation techniques have difficulty coping with the low-observability conditions often present on the distribution system due to the paucity of sensors and heterogeneity of measurements. To address these limitations, we propose a distribution system state estimation algorithm that employs matrix completion (a tool for estimating missing values in low-rank matrices) augmented with noise-resilient power flow constraints. This method operates under low-observability conditions where standard least-squares-based methods cannot operate, and flexibly incorporates any network quantities measured in the field. We empirically evaluate our method on the IEEE 33- and 123-bus test systems, and find that it provides near-perfect state estimation performance (within 1% mean absolute percent error) across many low-observability data availability regimes.

This paper considers distribution systems with a high penetration of distributed, renewable generation and addresses the problem of incorporating the associated uncertainty into the optimal operation of these networks. Joint chance constraints, which satisfy multiple constraints simultaneously with a prescribed probability, are one way to incorporate uncertainty across sets of constraints, leading to a chance-constrained optimal power flow problem. Departing from the computationally-heavy scenario-based approaches or approximations that transform the joint constraint into conservative deterministic constraints, this paper develops a scalable, data-driven approach which learns operational trends in a power network, eliminates zero-probability events (e.g., inactive constraints), and accurately and efficiently approximates bounds on the joint chance constraint iteratively. In particular, the proposed framework improves upon the classic methods based on the union bound (or Boole's inequality) by generating a much less conservative set of single chance constraints that also guarantees the satisfaction of the original joint constraint. The proposed framework is evaluated numerically using the IEEE 37-node test feeder, focusing on the problem of voltage regulation in distribution grids., Comment: 10 pages

This paper examines distribution systems that have a high integration level of distributed energy resources (DERs), and addresses the design of local control methods for real-time voltage regulation. Particularly, the paper focuses on proportional control strategies wherein the active and reactive power output of DERs are adjusted in response to (and proportionally to) local changes in voltage levels. The design of the voltage-active power and voltage-reactive power characteristics leverages suitable linear approximations of the ac power flow equations and is network-cognizant; that is, the coefficients of the controllers embed information on the location of the DERs and forecasted noncontrollable loads/injections and, consequently, on the effect of DERs power adjustments on the overall voltage profile. A robust approach is pursued to cope with uncertainty in the forecasted noncontrollable loads/power injections. The stability of the proposed local controllers is analytically assessed and numerically corroborated.