A Convex-Variational Analysis Approach for Optimal Power Flow Control Under Renewable Energy Uncertainty

Abstract

The emergence of solar and wind energy introduces much uncertainty in modern transmission and distribution networks. The classical non-convex OPF problem does not ensure rational dispatch decisions under such uncertainties. To address this gap, the paper proposes a convex-variational approach that reformulates the AC OPF problem as an SOCP problem using a combined variational inequality formulation of the KKT optimality conditions, enhanced with chance constraints that limit renewable forecast errors at a specified risk level. The approach is based on the ADMM method that breaks down the convex feasible region into several zones and performs parallel dispatch calculations based on preserving global feasibility. The method has been successfully implemented on IEEE 33-bus, 57-bus, and 118-bus benchmark systems with synthetic wind and solar profiles that cover renewable penetration levels from 20% to 45%. The results of the study demonstrated that the proposed convex-variational algorithm produced energy at 6.6% lower cost compared to DC-OPF approach and standard SOCP relaxation, had 59% less voltage deviation compared to DC-OPF, reduced constraint violation from 4.3% to 0.6% and minimized renewable curtailment from 8.9% to 3.1% while at the same time converging in 41% fewer iterations compared to the classical AC OPF approach. Moreover, the RMSE between the relaxation and the real AC solution was equal to 0.42%, which indicates that the convex relaxation has performed with high accuracy across all test cases. Thus, the results reveal that the introduction of convex relaxation combined with variational inequality-based optimality conditions allows developing a promising and efficient way of controlling grids with a high share of renewables.