Capacitor placement and real time control in large-scale unbalanced distribution systems: loss reduction formula, problem formulation, solution methodology and mathematical justification

A comprehensive study of capacitor placements and real-time control in general unbalanced distribution systems is undertaken. New developments in a loss reduction formula, problem formulations, solution methodology and mathematical justification are presented. The problem is decoupled into two subproblems: the capacitor placement subproblem and the real-time control subproblem. An effective solution algorithm for placing capacitors and determining their real-time control schemes for general unbalanced distribution systems is proposed. To meet the need for efficient implementation in real-time environments, a fast pseudo gradient type mechanism for deriving capacitor control settings is incorporated into the solution algorithms. A quadratic integer programming based approach is proposed to determine the number, locations and sizing of capacitors to be placed in the distribution systems. The subproblem of determining capacitor control settings under varying loading conditions is formulated as another quadratic integer programming problem. Mathematical justification is provided to show that the proposed algorithms are guaranteed to yield local optimal solutions. Numerical studies on the determination of capacitor number, size, location and real-time control schemes on large-scale unbalanced distribution systems under nonlinear load variations are presented in great detail in the companion paper [17].