Online optimization in closed loop on the power flow manifold

Subotić

2018 IEEE Conference on Decision and Control (CDC)

et al. 2018

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This paper is concerned with the study of continuous-time, non-smooth dynamical systems which arise in the context of time-varying non-convex optimization problems, as for example the feedback-based optimization of power systems. We generalize the notion of projected dynamical systems to time-varying, possibly non-regular, domains and derive conditions for the existence of so-called Krasovskii solutions. The key insight is that for trajectories to exist, informally, the time-varying domain can only contract at a bounded rate whereas it may expand discontinuously. This condition is met, in particular, by feasible sets delimited via piecewise differentiable functions under appropriate constraint qualifications. To illustrate the necessity and usefulness of such a general framework, we consider a simple yet insightful power system example, and we discuss the implications of the proposed conditions for the design of feedback optimization schemes.Index Terms-Non-smooth analysis, nonlinear dynamical systems, power systems. arXiv:1809.07288v2 [math.OC]

Section: Discussionmentioning

confidence: 97%

Time-varying Projected Dynamical Systems with Applications to Feedback Optimization of Power Systems

Subotić

2018 IEEE Conference on Decision and Control (CDC)

et al. 2018

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“…can be interpreted as continuous-time limit of Nesterov's accelerated gradient descent. As before, we can derive a feedback controller from (22). Strictly speaking, Theorem IV.2 does not apply to this type of time-varying control, but an extension is possible.…”

Section: B Non-example: Accelerated Gradient Descentmentioning

confidence: 99%

Timescale Separation in Autonomous Optimization

IEEE Trans. Automat. Contr.

Hug

et al. 2021

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Autonomous optimization refers to the design of feedback controllers that steer a physical system to a steady state that solves a predefined, possibly constrained, optimization problem. As such, no exogenous control inputs such as setpoints or trajectories are required. Instead, these controllers are modeled after optimization algorithms that take the form of dynamical systems. The interconnection of this type of optimization dynamics with a physical system is however not guaranteed to be stable unless both dynamics act on sufficiently different timescales. In this paper, we quantify the required timescale separation and give prescriptions that can be directly used in the design of this type of feedback controllers. Using ideas from singular perturbation analysis we derive stability bounds for different feedback optimization schemes that are based on common continuoustime optimization schemes. In particular, we consider gradient descent and its variations, including projected gradient, and Newton gradient. We further give stability bounds for momentum methods and saddle-point flows interconnected with dynamical systems. Finally, we discuss how optimization algorithms like subgradient and accelerated gradient descent, while well-behaved in offline settings, are unsuitable for autonomous optimization due to their general lack of robustness.

“…Previous works on real-time feedback-based optimization schemes [6], [7], [4] have considered only the interconnection with a steady-state map and have shown that feedback controllers can reliably track the solution of a time-varying OPF problem, even for nonlinear setups. Hence, a second insight of our simulations is that the interconnection of a gradient-based controller with a dynamical system instead of an algebraic steady-state map does not significantly deteriorate the long-term tracking performance.…”

Section: Simulation Resultsmentioning

confidence: 99%

“…1 to provide fast grid operation in order to accommodate fluctuating renewable energy sources, and to yield increased efficiency, safety, and robustness of the system trajectories, even in the presence of a substantial model mismatch. For example, feedback optimization has been proposed to drive the state of the power grid to the solution of an AC optimal power flow (OPF) program [3] while enforcing operational limits (either as soft constraints [4] or hard ones [5], [6]) and even in the case of time-varying problem parameters [7], [8].…”

Section: Introductionmentioning

confidence: 99%

Stability of Dynamic Feedback optimization with Applications to Power Systems

Menta

2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

et al. 2018

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We consider the problem of optimizing the steady state of a dynamical system in closed loop. Conventionally, the design of feedback optimization control laws assumes that the system is stationary. However, in reality, the dynamics of the (slow) iterative optimization routines can interfere with the (fast) system dynamics. We provide a study of the stability and convergence of these feedback optimization setups in closed loop with the underlying plant, via a custom-tailored singular perturbation analysis result. Our study is particularly geared towards applications in power systems and the question whether recently developed online optimization schemes can be deployed without jeopardizing dynamic system stability.