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

TL;DR

This work develops a rigorous framework for the optimal control of a stochastic power system, formulating a constrained nonlinear, nondifferentiable objective that minimizes the probability of transients destabilizing the network. The control objective combines a differentiable mean-angle term $f_{as,k}=\arcsin(|(A p_s+b)_k|)$ with a nondifferentiable, stochastic component $r\sigma_k(p_s)$, yielding $f(p_s)=\|f_k(p_s)\|_{\infty}$ over a polyhedral domain $P^+$. The authors prove the existence of a minimizer, analyze continuity and directional derivatives, and propose a two-step optimization algorithm: a projected generalized subgradient method to obtain a good initial vector, followed by a steepest-descent procedure with convergence guarantees to a local minimum. Numerical experiments on a multi-bus network demonstrate the method, uncovering multiple local minima and illustrating computational feasibility with a polynomial complexity around $O(n^5)$. The results provide a practical, theoretically grounded approach for transient-stability-oriented control in stochastic power systems and open avenues for improved convergence rates and global methods.

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.

Figures (2)

• Figure 1: A connected network with two subrings
• Figure 2: Figure (3.a) depicts the initial iteration of the proposed algorithm using the projected generalized subgradient method with the starting vector set as $[23,19,24]$. The x-axis represents the iteration number, and the y-axis reflects the value of the control objective function multiplied by a factor of 100. Figure (3.b) shows the second iteration of the proposed algorithm using the projected generalized subgradient method. The initial vector is the optimal vector computed in (3.a). The axes are similar to (3.a). Figure (3.c) displays the iteration of the proposed algorithm using the steepest descent method. The initial vector for this iteration is the optimal vector obtained in (4.b). Here, the x-axis corresponds to the iteration number, and the y-axis displays the control objective function value.

Theorems & Definitions (44)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Remark 2.4
• Theorem 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Lemma 3.3
• Proof 3