Optimal Capacitor Placement in Voltage-Dependent Distribution Networks Using Hybrid PSO–MILP: Case Study: Murang a 11 kV Distribution Feeder, Kenya

This paper presents a hybrid optimization methodology for optimal capacitor placement in radial distribution networks with voltage-dependent loads. The proposed approach employs Particle Swarm Optimization (PSO) for location selection and Mixed Integer Linear Programming (MILP) for discrete sizing, simultaneously minimizing active power losses, voltage deviation and equipment investment cost. The methodology incorporates the ZIP (constant impedance–constant current–constant power) load model and is validated against ETAP commercial software. Application to the Murang’a 11 kV distribution feeder in Kenya (253 buses, 69 km, 2,434 kW peak demand) identified a 450 kvar capacitor at Bus 115 as the optimal solution, reducing medium-voltage violations from 125 to 11 buses (91.2% reduction) while improving the power factor from 96.06% to 99.16%. The algorithm achieved convergence with an average execution time of 42.3 seconds on a standard desktop environment. When benchmarked against a standalone Genetic Algorithm, the hybrid PSO–MILP achieved superior solution quality (mean annual cost of USD 93,061 vs. USD 93,287), 38% faster convergence, and less than half the solution dispersion (standard deviation of USD 124.8 vs. USD 241.6). Statistical validation through 30 independent PSO runs confirmed algorithmic robustness with a coefficient of variation of 0.13%. The study quantifies the impact of voltage-dependent loads on capacitor performance and demonstrates the voltage compliance versus loss minimization trade-off characteristic of high R/X ratio feeders.