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This article proposes a data-driven control framework to regulate an unknown stochastic linear dynamical system to the solution of a stochastic convex optimization problem. Despite the centrality of this problem, most of the available methods critically rely on a precise knowledge of the system dynamics, thus requiring offline system identification. To solve the control problem, we first show that the steady-state gain of the transfer function of a linear system can be computed directly from historical data generated by the open-loop system, thus overcoming the need to first identify the full system dynamics. We leverage this data-driven representation of the steady-state gain to design a controller, which is inspired by stochastic gradient descent methods, to regulate the system to the solution of the prescribed optimization problem. A distinguishing feature of our method is that it does not require any knowledge of the system dynamics or of the possibly time-varying disturbances affecting them (or their distributions). Our technical analysis combines concepts from behavioral system theory, stochastic optimization with decision-dependent distributions, and Lyapunov stability. We illustrate the applicability of the framework in a case study for mobility-on-demand ride service scheduling in Manhattan.

The analysis and application of three different optimization algorithms are presented for tunable RF systems along with a comparison of the different tuning control schemes. These approaches are explored both theoretically and in measurement and are applied to a tunable matching network (TMN) to demonstrate how they address dynamic loading conditions in RF systems. Specifically, we present a data-driven (zeroth-order) projected gradient descent (ZO-PGD) algorithm, a feedforward neural network (FNN), and a novel hybrid approach that combines ZO-PGD and an FNN. The techniques are compared in the measurement of the representative TMN system for a variety of different load impedance trajectories and at different rates of change, and the relative merits of each technique are discussed.

Motivated by perception-based and sensing-based control problems in autonomous systems, this paper addresses the problem of developing feedback controllers to regulate the inputs and the states of a dynamical system to optimal solutions of an optimization problem when one has no access to exact measurements of the system states. In particular, we consider the case where the states need to be estimated from high-dimensional sensory data received only at some time instants. We develop a sampled-data feedback controller that is based on adaptations of a projected gradient descent method and includes neural networks as integral components to estimate the state of the system from perceptual information. We derive sufficient conditions to guarantee (local) input-to-state stability of the control loop. Moreover, we show that the interconnected system tracks the solution trajectory of the underlying optimization problem up to an error that depends on the approximation errors of the neural network and on the time-variability of the optimization problem; the latter originates from time-varying safety and performance objectives, input constraints, and unknown disturbances.

This article is a reprint from B. Kroposki et al., "Autonomous Energy Grids: Controlling the Future Grid With Large Amounts of Distributed Energy Resources," in IEEE Power and Energy Magazine, vol. 18, no. 6, pp. 37-46, Nov.-Dec. 2020, doi: 10.1109/MPE.2020.3014540. Please access the original article on IEEE Xplore.

This article considers a time-varying optimization problem associated with a network of systems, with each of the systems shared by (and affecting) a number of individuals. The objective is to minimize cost functions associated with the individuals' preferences, which are unknown, subject to time-varying constraints that capture physical or operational limits of the network. To this end, this article develops a distributed online optimization algorithm with concurrent learning of the cost functions. The cost functions are learned on-the-fly based on the users' feedback (provided at irregular intervals) by leveraging tools from the shape-constrained Gaussian process. The online algorithm is based on a primal–dual method and acts effectively in a closed-loop fashion where: First, users' feedback is utilized to estimate the cost, and second, measurements from the network are utilized in the algorithmic steps to bypass the need for sensing of (unknown) exogenous inputs of the network. The performance of the algorithm is analyzed in terms of dynamic network regret and constraint violation. Numerical examples are presented in the context of real-time optimization of distributed energy resources.

This article investigates the problem of regulating, at every time, a linear dynamical system to the solution trajectory of a time-varying constrained convex optimization problem. The proposed feedback controller is based on an adaptation of the saddle-flow dynamics, modified to take into account projections on constraint sets and output feedback from the plant. We derive sufficient conditions on the tunable parameters of the controller (inherently related to the time-scale separation between plant and controller dynamics) to guarantee exponential input-to-state stability of the closed-loop system. The analysis is tailored to the case of time-varying strongly convex cost functions and polytopic output constraints. The theoretical results are further validated in a ramp metering control problem in a network of traffic highways.

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 develops an algorithmic framework for real-time optimization of distribution-level distributed energy resources (DERs). The framework optimizes the operation of DERs that are individually controllable as well as groups of DERs that are jointly controlled at an electrical point of connection. From an electrical standpoint, wye and delta single-phase and multiphase connections are accounted for. The algorithm enables DERs to pursue given performance objectives, while adjusting their 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 bilevel 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. The resulting algorithm can cope with inaccuracies in the distribution-system modeling; moreover, it avoids pervasive metering to gather the state of noncontrollable resources, and it naturally lends itself to a distributed implementation. Analytical stability and convergence claims are established in terms of tracking the solution of the formulated time-varying optimization problem. The proposed method is tested in a realistic distribution system by using real data.

This paper formalizes an optimal water–power flow (OWPF) problem to optimize the use of controllable assets across power and water systems while accounting for the couplings between the two infrastructures. Tanks and pumps are optimally managed to satisfy water demand while improving power grid operations; for the power network, an ac optimal power-flow formulation is augmented to accommodate the controllability of water pumps. Unfortunately, the physics governing the operation of the two infrastructures and coupling constraints leads to a nonconvex (and, in fact, NP-hard) problem; however, after reformulating OWPF as a nonconvex, quadratically constrained quadratic problem, a feasible point pursuit-successive convex approximation approach is used to identify feasible and optimal solutions. In addition, a distributed solver based on the alternating direction method of multipliers enables water and power operators to pursue individual objectives while respecting the couplings between the two networks. The merits of the proposed approach are demonstrated for the case of a distribution feeder coupled with a municipal water distribution network.

This paper focuses on the design of online algorithms based on prediction-correction steps to track the optimal solution of a time-varying constrained problem. Existing prediction-correction methods have been shown to work well for unconstrained convex problems and for settings where obtaining the inverse of the Hessian of the cost function can be computationally affordable. The prediction-correction algorithm proposed in this paper addresses the limitations of existing methods by tackling constrained problems and by designing a first-order prediction step that relies on the Hessian of the cost function. Analytical results are established to quantify the tracking error. Numerical simulations corroborate the analytical results and showcase the performance and benefits of the algorithms.

Efforts to ensure reliable operation of existing low-voltage distribution systems with high photovoltaic (PV) generation have focused on the possibility of inverters providing ancillary services such as active power curtailment and reactive power compensation. Major benefits include the possibility of averting overvoltages, which may otherwise be experienced when PV generation exceeds the demand. This paper deals with ancillary service procurement in the face of solar irradiance forecasting errors. In particular, assuming that forecasted PV irradiance can be described by a random variable with known (empirical) distribution, the proposed uncertainty-aware optimal inverter dispatch (OID) framework indicates which inverters should provide ancillary services with a guaranteed a priori risk level of PV generation surplus. To capture forecasting errors and strike a balance between risk of overvoltages and (re)active power reserves, the concept of conditional value-at-risk is advocated. Due to AC power balance equations and binary inverter selection variables, the formulated OID involves the solution of a nonconvex mixed-integer nonlinear program. However, a computationally affordable convex relaxation is derived by leveraging sparsity-promoting regularization approaches and semidefinite relaxation techniques.

We study the stability properties of a closed-loop system composed of a dynamical plant and a feedback controller, the latter generating control signals that can be compromised by a malicious attacker. We consider two classes of feedback controllers: a static output-feedback controller, and a dynamical gradient-flow controller that seeks to steer the output of the plant towards the solution of a convex optimization problem. In both cases, we analyze the stability properties of the closed-loop system under a class of switching attacks that persistently modify the control inputs generated by the controllers. Our stability analysis leverages the framework of hybrid dynamical systems, Lyapunov-based arguments for switching systems with unstable modes, and singular perturbation theory. Our results reveal that, under a suitable time-scale separation between plant and controllers, the stability of the interconnected system can be preserved when the attack occurs with “sufficiently low frequency” in any bounded time interval. We present simulation results in a power-grid example that corroborate the technical findings.

We consider time-varying nonconvex optimization problems where the objective function and the feasible set vary over discrete time. This sequence of optimization problems induces a trajectory of Karush-Kuhn-Tucker (KKT) points. We present a class of regularized primal-dual gradient algorithms that track the KKT trajectory. These algorithms are feedback-based algorithms, where analytical models for system state or constraints are replaced with actual measurements. We present conditions for the proposed algorithms to achieve bounded tracking error when the cost and constraint functions are twice continuously differentiable. We discuss their practical implications and illustrate their applications in power systems through numerical simulations.