Managing Load Uncertainty in Distribution Network Capacitor Planning: A Master–Slave Stochastic Optimization Framework

Abstract

This paper presents a novel master–slave stochastic optimization framework for the optimal siting and sizing of fixed-step capacitor banks in medium-voltage distribution networks, explicitly addressing the inherent variability of load demand that is typically neglected in conventional deterministic approaches. The proposed methodology integrates a scenario-based stochastic optimization model with a Chu and Beasley genetic algorithm (CBGA) as the master stage, which handles discrete placement decisions, and a successive-approximation power flow method (SAPF) as the slave stage, which evaluates the technical and economic performance of each candidate solution under multiple load scenarios. To capture demand uncertainties, 365 daily load realizations are generated using independent Gaussian noise with a relative standard deviation of 10% applied to each load point. These are subsequently reduced to ten representative scenarios via k-means clustering, reducing the number of power-flow evaluations per candidate solution from 365 to 10 (a 36.5-fold reduction); the reduced scenarios exhibit a low mean absolute error (MAE: <2%) with respect to the original mean, indicating faithful representation of the average load behavior, while the silhouette score is modest (approximately 0.25), consistent with the unimodal nature of the generated data and implying that the clusters are not well separated. Extensive simulations on a 33-bus test feeder considering three energy-cost-escalation scenarios (0%, 10%, and 20%) over a 20-year planning horizon demonstrate that both the deterministic and stochastic approaches reduce the total net present cost by 16.52% to 17.34% compared to the uncompensated network; the stochastic approach consistently delivers solutions that are either superior or comparable to deterministic planning (yielding up to approximately 0.16% additional cost reduction) while offering enhanced robustness against load variability. The stochastic framework offers distinct advantages, including robust solutions across a wide range of operating conditions, an inherent ability to adjust investment levels in response to probabilistic load distributions, and the ability to quantify uncertainty in decision making, with the most significant benefits observed when energy costs are low and load variability is high. The convergence of both approaches at a 20% escalation level further validates the reliability of high-resolution deterministic modeling when economic factors strongly dominate the optimization objective. This study underscores the importance of probabilistic modeling for modern distribution network planning, providing a practical and computationally efficient decision-support tool for utility planners to enhance grid resilience and operational efficiency in the context of increasing demand variability and renewable energy integration.