Robustness of flow networks against cascading failures under partial load redistribution

TL;DR

This paper analyzes the robustness of flow networks under partial load redistribution, deriving the final network state after attacks, and shows that equal free-space allocation maximizes robustness, with implications for power grid design.

Contribution

It provides a comprehensive analytical framework for understanding how partial load redistribution affects network robustness and identifies optimal free-space allocation strategies.

Findings

Derived the final fraction of surviving lines for all attack sizes and redistribution parameters.

Showed that partial redistribution can change the transition order at the critical attack size.

Confirmed that equal free-space allocation maximizes network robustness on real power grid data.

Abstract

We study the robustness of flow networks against cascading failures under a partial load redistribution model. In particular, we consider a flow network of $N$ lines with initial loads $L_1, \ldots, L_N$ and free-spaces (i.e., redundant space) $S_1, \ldots, S_N$ that are independent and identically distributed with joint distribution $P_{LS}(x,y)=\mathbb{P}(L \leq x, S \leq y)$. The capacity $C_i$ is the maximum load allowed on line $i$, and is given by $C_i=L_i + S_i$. When a line fails due to overloading, it is removed from the system and $(1-\varepsilon)$-fraction of the load it was carrying (at the moment of failing) gets redistributed equally among all remaining lines in the system; hence we refer to this as the {\it partial} load redistribution model. The rest (i.e., $\varepsilon$-fraction) of the load is assumed to be lost or absorbed, e.g., due to advanced circuitry disconnecting overloaded power lines or an inter-connected network/material absorbing a fraction of the flow from overloaded lines. We analyze the robustness of this flow network against random attacks that remove a $p$-fraction of the lines. Our contributions include (i) deriving the final fraction of alive lines $n_{\infty}(p,\varepsilon)$ for all $p, \varepsilon \in (0,1)$ and confirming the results via extensive simulations; (ii) showing that partial redistribution might lead to (depending on the parameter $0<\varepsilon \leq 1$) the order of transition at the critical attack size $p^{*}$ changing from first to second-order; and (iii) proving analytically that flow networks achieve maximum robustness (quantified by the area $\int_{0}^{1} n_{\infty}(p,\varepsilon) \mathrm{d}p$) when all lines have the same free-space regardless of their initial load. The optimality of equal free-space allocation is also confirmed on real-world data from the UK National Power Grid.