Resistance distance criterion for optimal slack bus selection

We investigate the dependence of transmission losses on the choice of a slack bus in high voltage AC transmission networks. We formulate a transmission loss minimization problem in terms of slack variables representing the additional power injection that each generator provides to compensate the transmission losses. We show analytically that for transmission lines having small, homogeneous resistance over reactance ratios $r\mathrm{/}x\ll 1$, transmission losses are generically minimal in the case of a unique $\mathit{\text{slack}}\mathit{\text{bus}}$ instead of a distributed slack bus. For the unique slack bus scenario, to lowest order in $r\mathrm{/}x$, transmission losses depend linearly on a resistance distance based indicator measuring the separation of the slack bus candidate from the rest of the network. We confirm these results numerically for several IEEE and Pegase testcases, and show that our predictions qualitatively hold also in the case of lines having inhomogeneous $r\mathrm{/}x$ ratios, with optimal slack bus choices reducing transmission losses by $10\mathrm{\%}$ typically.