On the convexity of the system loss function

We show that the system loss function in a power network is bounded below by any number of supporting hyper-planes in the space of generalized injections. A supporting hyper-plane is defined by the linear Taylor series expansion of the system loss function around a given operating point. The supporting hyperplane property is valid provided that the expansion point is sufficiently near the flat-voltage profile. We have assessed experimentally the range of validity of this assumption, called the range of hyper-plane support (RHS), showing that for typical networks, the RHS is broad, particularly when the bus voltages are controlled near 1 per unit. The supporting hyper-plane model was also tested as part of an economic dispatch with transmission losses to demonstrate that this linear model provides the same results as when the losses are treated as a nonlinear function.