Approximating Economic Dispatch by Linearizing Transmission Losses

We study a non-convex Transmission-Constrained Economic Dispatch (TCED) problem that uses the traditional linear DC model of the transmission system together with a non-linear representation of losses. This problem is typically approximated by the convex problem obtained by linearizing the constraints around some base-case state. Electricity prices and dispatch decisions are then chosen based on the resulting linearly-constrained economic dispatch (LCED) problem. Different LCED problems have been suggested in the literature and they are all derived using one of two linearization techniques, which we call direct and indirect linearization, respectively. An LCED problem often used in practice that uses Loss Distribution Factors (LDFs) is derived using indirect linearization and is termed the common LCED problem. This paper studies the assumptions required to recover the optimal dispatch of the non-convex TCED problem from the solution of the common LCED problem. We show that the common LCED problem may have multiple minimizers, in which case small perturbations of the base-case state may result in large dispatch approximation error. Furthermore, even if the base-case state matches a minimizer of the non-convex TCED problem, it is proven that there does not always exist a choice of LDFs such that the optimal dispatch of the TCED problem is also optimal for the common LCED problem. On the other hand, such LDFs do exist and are identified for the special case where no line limits are binding.