Power Flows with Flat Voltage Profiles: an Exact Approach

TL;DR

This paper presents an exact solution to the AC power flow problem under flat voltage profiles, extending the classic DC power flow model to include network resistances and deriving explicit formulae for power flows, phase angles, and losses.

Contribution

It introduces a tractable, exact analytical approach for power flows with resistances under flat voltage assumptions, generalizing the traditional lossless model.

Findings

Including resistance increases phase angle differences beyond the lossless relation.

Circulating active power flows are always accompanied by reactive power flows in resistive networks.

Explicit formulae relate impedance, power flows, and voltage phase displacements.

Abstract

This note outlines the exact solution to the power flow problem in AC electrical networks under the assumption of 'flat' or uniform voltage profiles. This solution generalises the common 'DC power flow' approach to electrical network problems where the detail of the voltage profile is disregarded and the focus is on the distribution of active power flows and voltage phase angles. In the usual approach to the problem, network resistance is ignored and simplified power-angle relations are used based on branch reactances alone. The purpose of this note is to describe a tractable generalisation of this approach to account for arbitrary network resistances. The solution is worked in detail for a single network branch with resistance $R \geq 0$ and reactance $X > 0$, with explicit formulae derived linking branch impedance, active and reactive power flows, voltage phase displacement and losses under the 'flat' voltage profile assumption. These formulae have a convenient expression in terms of a 'coefficient of support' which relates the assumed active power flow to the corresponding reactive power consumed on the branch, and encapsulates much of the mathematical technicalities in the exact solution. It is shown that when network resistance is taken into account, the angle displacement $\delta_j - \delta_k$ between adjacent bus voltages is always larger than given by the lossless power-angle relation $\sin(\delta_j - \delta_k) = XP$. The approach is applied to the analysis of circulating power flows in large networks, building on the classic work of Janssens and Kamagate (2003). It is seen that in networks with nonzero branch resistances, a circulating flow of active power is always accompanied by a counter-circulating flow of reactive power.