Optimal Control of a Stochastic Power System -- Algorithms and Mathematical Analysis

TL;DR

This paper develops algorithms and provides mathematical analysis for optimally controlling a stochastic power system to minimize the risk of instability, using a nonconvex, nondifferentiable objective function.

Contribution

It introduces a two-step approximation procedure with convergence guarantees for a complex stochastic control problem in power systems.

Findings

Existence of a minimum in the control objective function.

Development of a projected subgradient method for initial approximation.

Convergence theorems ensuring local minima approximation.

Abstract

The considered optimal control problem of a stochastic power system, is to select the set of power supply vectors which infimizes the probability that the phase-angle differences of any power flow of the network, endangers the transient stability of the power system by leaving a critical subset. The set of control laws is restricted to be a periodically recomputed set of fixed power supply vectors based on predictions of power demand for the next short horizon. Neither state feedback nor output feedback is used. The associated control objective function is Lipschitz continuous, nondifferentiable, and nonconvex. The results of the paper include that a minimum exists in the value range of the control objective function. Furthermore, it includes a two-step procedure to compute an approximate minimizer based on two key methods: (1) a projected generalized subgradient method for computing an initial vector, and (2) a steepest descent method for approximating a local minimizer. Finally, it includes two convergence theorems that an approximation sequence converges to a local minimum.