A Weighted Mean of Vectors–Based Mathematical Optimization Framework for PV-STATCOM Deployment in Distribution Systems under Time-Varying Load Conditions

The increasing penetration of photovoltaic (PV) systems in distribution networks has introduced new challenges in voltage regulation and energy loss mitigation, particularly under time-varying loading conditions. This paper presents a constrained multi-objective mathematical optimization framework for the optimal allocation and sizing of PV-STATCOM devices in radial distribution systems. The problem is formulated as a nonlinear optimization model that simultaneously minimizes daily energy losses and voltage deviation indices over a 24-hour operating horizon while satisfying network operational constraints, inverter capacity limits, and renewable penetration restrictions. To efficiently solve the resulting non-convex optimization problem, a metaheuristic algorithm based on the Weighted Mean of Vectors (WMV) is employed. The WMV method integrates wavelet-based weighting mechanisms, mean-driven update rules, vector combination strategies, and a local refinement operator to balance global exploration and local exploitation within the feasible search domain. Constraint violations are handled through a penalty-based mathematical transformation of the objective function. The proposed framework is validated on the IEEE 33-bus and IEEE 69-bus distribution systems under realistic daily load variations. Numerical results demonstrate significant reductions in daily energy losses and voltage profile deviations compared to differential evolution, particle swarm optimization, artificial rabbits optimization, and golden search optimization algorithms. Furthermore, convergence analysis confirms the robustness and computational efficiency of the WMV approach in solving large-scale constrained power system optimization problems.