# Analytic Loss Minimization: A Proof

Abstract

Loss minimizing generator dispatch profiles for power systems are usually derived using optimization techniques. However, by partitioning and manipulating the $$Y_{bus}$$ matrix, it is possible to derive the $$K_{GL}$$ matrix, which can be used to analytically determine a loss minimizing dispatch for generators. This letter draws on recent research on the characterization of transmission system losses to demonstrate how the $$K_{GL}$$ matrix achieves this impressive function. A new proof of the observed zero row summation property of the $$Y_{GGM}$$ matrix is provided to this end.

# Introduction

Various works, such as (Visakha 2004, Thukaram 2009, Thukaram 2009a) have noted that the $$K_{GL}$$ matrix allows the direct calculation of a generator dispatch profile that appears to minimize active power losses. The literature offers scant explanation for how this impressive result is achieved. The recent work of Abdelkader et al. (Abdelkader 2014, Abdelkader 2011, Abdelkader 2008), which draws on the $$Y_{bus}$$ partitioning approach introduced in (Kessel 1986), offers useful insight here. Crucially, (Abdelkader 2011) shows how transmission losses can be separated into three distinct components. One of these loss components arises solely because of mismatched generator voltages, which cause circulating currents to flow through the system (Abdelkader 2014). This letter demonstrates how the $$K_{GL}$$ matrix can give a generator dispatch profile which equalizes generator complex voltages, so no circulating currents will flow and this loss component is nullified.