A Large-Deviation-Based Splitting Estimation of Power Flow Reliability

Abstract

Given the continued integration of intermittent renewable generators in electrical power grids, connection overloads are of increasing concern for grid operators. The risk of an overload due to injection variability can be described mathematically as a barrier crossing probability of a function of a multidimensional stochastic process. Crude Monte Carlo is a wellknown technique to estimate probabilities, but it may be computationally too intensive in this case as typical modern power grids rarely exhibit connection overloads. In this paper we derive an approximate rate function for the overload probability using results from large deviations theory. Based on this large deviations approximation, we apply a rare event simulation technique called splitting to estimate overload probabilities more efficiently than Crude Monte Carlo simulation.We show on example power grids with up to eleven stochastic power injections that for a fixed accuracy Crude Monte Carlo would require tens to millions as many samples than the proposed splitting technique required. We investigate the balance between accuracy and workload of three splitting schemes, each based on a different approximation of the rate function. We justify the workload increase of large deviations based splitting compared to naive splitting -that is, splitting based on merely the Euclidean distance to the rare event set. For a fixed accuracy naive splitting requires over 60 times as much CPU time as large deviation based splitting, illustrating its computational advantage. In these examples naive splitting -unlike large deviations based splitting -requires even more CPU time than CMC simulation, illustrating its pitfall.

Optimal Inflow Control Penalizing Undersupply in Transport Systems with Uncertain Demands

“…As a simplification, here, we restrict our network to |V in | = |V out | = 1. Note however that an extension to several inflow nodes (|V in | > 1), and outflow nodes (|V out | > 1) is straightforward by considering a vector-valued inflow control, and a multivariate Ornstein-Uhlenbeck process as used in [48]. The inflow control u(t) acts on the vertex v in ∈ V in , and the demand Y t is realized at the vertex v d ∈ V out .…”

A computational method for optimizing storage placement to maximize power network reliability

“…The values of these terms are determined later in section 6. Modelling power injections as O-U processes have been implemented by Wadman et al in [12] and [20].…”

“…A theoretical analysis would be interesting but is beyond the scope of the paper. Note that this is non-trivial since the power injections in the networks are not diffusion processes due to the buffers (storage devices), and as such, our setting does not fit in the framework of Wadman et al in [20].…”

Chance‐constrained optimal inflow control in hyperbolic supply systems with uncertain demand

“…This is also a popular choice to model demand in various fields (see, eg, Reference 25 in the context of electricity). In References 26,27, they use a multivariate Ornstein‐Uhlenbeck process from the supply‐side point of view. They model uncertain injections into the power network and assess the probability of outages.…”