A Three-Stage Mixed-Integer Convex Optimization Framework for Capacitor Bank Allocation in Distribution Systems

Abstract

This paper presents a mixed-integer convex optimization framework for the optimal placement and sizing of capacitor banks in radial distribution networks. The proposed formulation employs a linearized power flow representation with a convex quadratic objective and linear constraints in the complex domain, implemented through a three-stage methodology: initial exact power flow, convex optimization for capacitor placement, and exact validation via power flow. The framework is evaluated on the 24-bus, 33-bus, and 69-bus test systems, demonstrating loss reductions of up to 179.49 kW (26.8%) for the 24-bus system, 72.65 kW (34.4%) for the 33-bus system, and 79.73 kW (35.4%) for the 69-bus system. The proposed method consistently outperforms state-of-the-art metaheuristic approaches including the gravitational search algorithm, the two-stage method, the flower pollination algorithm, and the discrete vortex search algorithm. In all test systems, voltage profiles are improved to satisfy the 7% regulation criterion, with minimum voltages raised above 0.93 p.u. All cases are solved in less than 10 seconds, confirming the computational efficiency of the approach.