Non-parametric Joint Chance-Constrained OPF via Maximum Mean Discrepancy Penalization

The chance-constrained optimal power flow (CC-OPF) has gained prominence due to increased uncertainty in the power system. However, solving CC-OPF for general uncertainty distribution classes is challenging due to lack of analytical formulation of probabilistic constraints and cost-complexity trade-off issues. This work proposes a novel joint chance-constrained optimal power flow (JCC-OPF) via maximum mean discrepancy (MMD) penalization to obtain a probabilistically feasible low-cost solution. The idea is to view the JCC-OPF problem as a distribution matching problem. The MMD quantifies the distance between two probability distributions embedded into reproducing kernel Hilbert space (RKHS) and thus provides an efficient way to minimize the distance between distributions. The RKHS embedding, also called kernel mean embedding (KME), is a non-parametric method that does not require any information about the random injection's distribution while performing the embedding. Furthermore, the proposed method is based on a point-wise evaluation of the constraint functions and has the same complexity as a deterministic OPF problem. The proposed penalization-based formulation handles JCC directly and does not require the conversion of joint chance constraints into individual ones. Simulations on IEEE 24-Bus, 30-Bus, and 57-Bus systems validate the proposed method's non-parametric nature and ability to obtain a probabilistically feasible solution. Benchmarking results against existing approaches indicate the better computational performance of the proposed method.