Transmission-Constrained Residual Demand Derivative in Electricity Markets

The residual demand derivative plays a central role in constructing the best response to competitors´ strategies in widely used strategic models such as the Cournot model and the supply function model. In the absence of transportation or transmission constraints, the residual demand derivative is obtained straightforwardly by taking the derivative of the residual demand function with respect to price. However, in an electricity market, the market is embedded in a transmission network. When there is no transmission congestion, the residual demand derivative can be calculated as usual, but when there is transmission congestion, the residual demand derivative is more difficult to calculate. In this paper, we characterize the transmission-constrained residual demand derivative. We use the dc power flow model and characterize the residual demand derivative analytically. The residual demand derivative could also be obtained from the solution of a specific weighted least squares problem. Several properties of the residual demand derivative are implications of the weighted least squares theory. We also characterize the condition under which the residual demand derivative will be bounded or unbounded when there are perfectly elastic supplies/demands at some buses in the system. We verified our results in three examples: a two-bus system, a four-bus two-loop system, and a three-bus one-loop system with one perfectly elastic supply. The residual demand derivative characterization can be used to analyze the strategic behavior in both the Cournot model and the supply function model with transmission constraints, and it can be easily incorporated into sophisticated optimal strategy algorithms.