Random Perturbations of a Three-Machine Power System Network

Abstract

This paper develops an asymptotic method based on averaging and large deviations to study the transient stability of a noisy three-machine power system network. We study the dynamics of these nonlinear oscillators (swing equations) as random perturbations of two-dimensional periodically driven Hamiltonian systems. The phase space for periodically driven nonlinear oscillators consists of many resonance zones. It is well known that, as the strengths of periodic excitation and damping go to zero, the measure of the set of initial conditions which lead to capture in a resonance zone goes to zero. In this paper we study the effect of weak noise on the escape from a resonance zone and obtain the large-deviation rate function for the escape. The primary goal is to show that the behavior of oscillators in the resonance zone can be adequately described by the (slow) evolution of the Hamiltonian, for which simple analytical results can be obtained, and then apply these results to study the transient stability margin of power system with stochastic loads. The classical swing equations of a power system of three interconnected generators with non-zero damping and small noise is considered as a nontrivial example to derive the “exit time” analytically. This work may play an important role in designing and upgrading existing electrical power system networks.

Appendix: Calculation of \(J_s\) and \(J_c\) in (19)–(21)

The reduced order system with $\varepsilon =0$ with $c=1$ are the equations for a non-linear pendulum. The pendulum has two modes of motion dependent on total system energy. When $H\in (-1,1)$ the system is described by oscillatory solutions. Denoting k as the elliptic modulus we have [5]

$$\begin{aligned} H=2k^2-1 \end{aligned}$$

With $K=K(k)$ and $E=E(k)$ being complete elliptic integrals of first and second kind respectively,

$$\begin{aligned} I(k)=[\frac{8}{\pi }[E-(k^2-1)K], \quad \varOmega ={\pi }{2K}, \quad \dot{\varphi }=\varOmega \end{aligned}$$

(41)

The oscillating displacement and velocity in terms of the angle variable $\varphi$ are

$$\begin{aligned} \delta (\varphi )=2\arcsin (kSn(\frac{2K\varphi }{\pi })), \quad \omega (\varphi )=2kCn(\frac{2K\varphi }{\pi })\cdot \frac{2K}{\pi } \end{aligned}$$

We have that

$$\begin{aligned} \mathfrak {F}(I,\varphi ,\frac{m}{n}(\psi -\varphi ))=(\alpha -\beta \omega -\mu _3 c_2 (q_1(\varphi )+r)\cos (q_1(\varphi )+\tau _1)+\\ \quad c_1\sin (\frac{m}{n}(\psi -\varphi ))\cos (q_1(\varphi )+\tau _2)- \mu _1 c_2 \sin (\frac{m}{n}(\psi -\varphi ))\cos (q_1(\varphi )+\tau _1))q_2(\varphi ). \end{aligned}$$

Noting that $\frac{1}{2m\pi }\int _0^{2m\pi } \mathfrak {F}(I,\psi +\frac{n}{m}\theta ,\theta ) d\theta = \frac{1}{2n\pi }\int _0^{2n\pi } \mathfrak {F}(I,\varphi ,\frac{m}{n}(\psi -\varphi )) d\varphi$ due to the resonance condition. Even though it is natural to choose $\theta _t$ as the fast variable for multi-phase averaging, in order to simplify the averaging of certain elliptic functions in the expressions $\mathfrak {F}$ and $\mathfrak {G}$ ${\varphi }$ is used as the fast angle for multi-phase averaging. We can evaluate the more tractable form $\frac{1}{2n\pi }\int _0^{2n\pi } \mathfrak {F}(I,\varphi ,\frac{m}{n}(\psi -\varphi )) d\varphi$ which gives

$$\begin{aligned} \bigg \langle \mathfrak {F}(I_r,\varphi ,\frac{m}{n}(\psi -\varphi ))\bigg \rangle&= \beta \, I_{r} +A_1\left( \frac{\pi ^3}{K^3k}\right) \frac{q^{\frac{m}{2n}}}{1+q^{\frac{m}{n}}}\sin \left( \frac{m}{n}\psi \right) \mathbf {1}_{\left\{ \frac{m}{n}\in 2Z^{+}+1\right\} }\\&\quad -A_2\left( \frac{\pi ^3}{K^3k}\right) \left( \frac{m}{n}\right) ^2\frac{q^{\frac{m}{n}}}{1-q^{\frac{2m}{n}}}\cos \left( \frac{m}{n}\psi \right) \mathbf {1}_{\left\{ \frac{m}{n}\in Z^{+}\right\} } \\&\quad -\tilde{A}_1\left( \frac{\pi ^3}{K^3k}\right) \frac{q^{\frac{m}{2n}}}{1+q^{\frac{m}{n}}}\sin \left( \frac{m}{n}\psi \right) \mathbf {1}_{\left\{ \frac{m}{n}\in 2Z^{+}+1\right\} } \\&\quad +\tilde{A}_2\left( \frac{\pi ^3}{K^3k}\right) \left( \frac{m}{n}\right) ^2\frac{q^{\frac{m}{n}}}{1-q^{\frac{2m}{n}}}\cos \left( \frac{m}{n}\psi \right) \mathbf {1}_{\left\{ \frac{m}{n}\in Z^{+}\right\} } + \mathscr {C}, \end{aligned}$$

where $I_{r}\buildrel \mathrm{def}\over =\frac{8}{\pi n} \left( (k^2-1)K+E\right)$ is the resonant value of the action, $\mathscr {C}$ represents the contribution due to the term $\langle q_1(\varphi ) \cos (q_1(\varphi )+\tau _1)q_2(\varphi ) \rangle$, which can be argued to be negligible, and

$\mathbf {J_c}$ and $\mathbf {J_s}$ are the coefficients of the $\cos (\frac{m}{n}\psi )$ and $\sin (\frac{m}{n}\psi )$ terms. Neglecting the $\sin (\frac{m}{n}\psi )$ terms means m:n is even, this can be done without loss of generality.