Robustness of power systems under a democratic fiber bundle-like model

TL;DR

This paper analyzes the robustness of power systems with line loads following various distributions under random failures, revealing conditions for abrupt breakdowns and identifying the most resilient load distribution.

Contribution

It provides an analytical framework for understanding power system robustness under a fiber bundle-like model, including explicit critical attack sizes and effects of load distribution heterogeneity.

Findings

System always fails via a first-order transition at a critical attack size.

Pareto load distribution is the most vulnerable, Weibull is the most robust among tested distributions.

Maximum robustness occurs when all lines carry equal load, with critical attack size rac{7}{7+1}.

Abstract

We consider a power system with $N$ transmission lines whose initial loads (i.e., power flows) $L_1, \ldots, L_N$ are independent and identically distributed with $P_L(x)$. The capacity $C_i$ defines the maximum flow allowed on line $i$, and is assumed to be given by $C_i=(1+\alpha)L_i$, with $\alpha>0$. We study the robustness of this power system against random attacks (or, failures) that target a $p$-{\em fraction} of the lines, under a democratic fiber bundle-like model. Namely, when a line fails, the load it was carrying is redistributed equally among the remaining lines. Our contributions are as follows: i) we show analytically that the final breakdown of the system always takes place through a first-order transition at the critical attack size $p^{\star}=1-\frac{E[L]}{\max\{P(L>x)(\alpha x + E[L ~|~ L>x])\}}~~~$; ii) we derive conditions on the distribution $P_L(x)$ for which the first order break down of the system occurs abruptly without any preceding diverging rate of failure; iii) we provide a detailed analysis of the robustness of the system under three specific load distributions: Uniform, Pareto, and Weibull, showing that with the minimum load $L_{\textrm{min}}$ and mean load $E[L]$ fixed, Pareto distribution is the worst (in terms of robustness) among the three, whereas Weibull distribution is the best with shape parameter selected relatively large; iv) we provide numerical results that confirm our mean-field analysis; and v) we show that $p^{\star}$ is maximized when the load distribution is a Dirac delta function centered at $E[L]$, i.e., when all lines carry the same load; we also show that optimal $p^{\star}$ equals $\frac{\alpha}{\alpha+1}$. This last finding is particularly surprising given that heterogeneity is known to lead to high robustness against random failures in many other systems.