Control of the Power Flows of a Stochastic Power System

TL;DR

The paper develops a framework to control power flows in a stochastic power system by restricting the probability that any line’s phase-angle difference exits a safe set $(-\pi/2,\pi/2)$ over short horizons. It linearizes the nonlinear AC model around a strictly-stable synchronous state and analyzes the invariant Gaussian distribution of line flows under Brownian disturbances, deriving an upper bound on the exit probability. A nonconvex, nondifferentiable control objective is defined to minimize the maximum, over all lines, of the mean phase-angle difference plus a multiple of the line-variance, Subject to a compact polyhedral feasible set for the next-horizon power supplies; a generalized-subgradient method yields good local minima. Numerical results on an eight-node network, a twelve-node ring, and a Manhattan-grid network demonstrate improved transient stability and identify how network structure and line vulnerabilities shape control decisions. The approach provides a practical, short-horizon control mechanism that complements existing frequency-control and SC-OPF frameworks, and points to extensions for probabilistic SC-OPF and low-inertia, high-renewable power systems.

Abstract

How to determine the vector of power supplies of a stochastic power system for the next short horizon, such that the probability is less than a prespecified value that any phase-angle difference of a power line of the power network exits from a safe set? The power system is modelled such that the differential equation of each frequency is affected by a Brownian motion process. A safe set can be selected to be any subset of the interval $(-π/2, ~ +π/2)$, which is a sufficient condition for not losing synchronization. That the controlled system has an improved performance is shown by numerical results of three academic examples including a particular eight-node academic network, a twelve-node ring network, and a Manhattan-grid network.

Figures (9)

• Figure 1: Two stable points $x_{s,a}$ and $x_{s,b}$ with their domain of attraction and a saddle point $x_{s,c}$
• Figure 2: The probability density function of the phase-angle difference $(\theta_{i_k}-\theta_{j_k})$ of the power flow in power line $k=(i_k,j_k)\in \mathcal{\mathcal{E}}$; and two red bars for the probabilities that the phase-angle difference is larger than $+\pi/2$ or less than $-\pi/2$. The parameters of the probability density function displayed are $(m, ~ \sigma) = (0.4995, ~ 0.3344)$ which values are chosen identical to those of Fig. \ref{['fig:ringnetcontrolwithoutwith']} for 'without control'.
• Figure 3: A particular eight-node academic example
• Figure 4: The outputs of the particular eight-node academic network
• Figure 5: A ring network

Theorems & Definitions (17)

• Remark 2.1
• Theorem 2.3
• Remark 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Proposition 2.11
• Proposition 2.12