A conditional Monte Carlo method for estimating the failure probability of a distribution network with random demands

Abstract

We consider a model of an irreducible network in which each node is subjected to a random demand, where the demands are jointly normally distributed. Each node has a given supply that it uses to try to meet its demand; if it cannot, the node distributes its unserved demand equally to its neighbors, which in turn do the same. The equilibrium is determined by solving a linear program (LP) to minimize the sum of the unserved demands across the nodes in the network. One possible application of the model might be the distribution of electricity in an electric power grid. This paper considers estimating the probability that the optimal objective function value of the LP exceeds a large threshold, which is a rare event. We develop a conditional Monte Carlo algorithm for estimating this probability, and we provide simulation results indicating that our method can significantly improve statistical efficiency.

Rare-Event Simulation for Distribution Networks

“…Some of the results regarding CMC previously appeared in a conference version of this paper ( [8]). Our conference paper restricted the LP's objective function to be the sum of the unserved demands, and we now prove its invariance, as described in contribution 1), which greatly expands the applicability of our approach.…”

“…Regarding contribution 3), we develop an importance sampling algorithm which is not studied in the conference version, and we provide a proof of asymptotic optimality and algorithm implementation. As for contribution 4), although in the conference version, we have studied the CMC algorithm and its implementation (see [8], Section 4.3), no mathematical proof is provided regarding the asymptotic optimality of this algorithm. Here, in the journal version, we prove it rigorously.…”

“…Note that while in [8], we assume that the unserved demands are equally distributed to neighbors, here we make a small but important extension. We allow the proportions to be any non-negative numbers.…”

“…Let α(k) represent this failure probability, and L(D) denote the optimal value of the dual when the demand vector is D. As discussed in [8],…”

“…Here we use the same basis for comparing the estimators using different simulation algorithms as in [8]. Suppose we want to estimate α = E[X], and X 1 , X 2 , .…”