Continuous-Time Locational Marginal Price of Electricity

Modern electricity markets with large penetration of renewable energy resources require fair and accurate pricing methods to elicit generation flexibility and foster competition in electricity markets. This paper proposes the fundamental theory and closed-form formulas for continuous-time locational marginal price (LMP) of electricity, which more accurately integrates the spatio-temporal variations of load and operational constraints of power systems in the electricity price calculation. This paper first formulates the network-constrained generation scheduling and pricing problems as continuous-time optimal control problems using two methods for modeling transmission network, i.e., Theta and generation shift factor (GSF) methods. The continuous-time network-constrained scheduling and pricing problems minimize in their objective functional the total operation cost of power systems over the scheduling horizon subject to generation and transmission constraints. The closed-form continuous-time LMP formulas are derived for each transmission network model, which explicitly include terms that reflect the simultaneous spatio-temporal impacts of transmission flow limits and intertemporal generation ramping constraints in LMP formation. A scalable and computationally efficient function space solution method is proposed that converts the continuous-time problems into mixed integer linear programming problems with finite-dimensional decision space. The proposed solution method enables high-fidelity solution of transmission-constrained scheduling and pricing problems in higher dimensional spaces, while including as a special case the current discrete-time solutions. The proposed models are implemented on a three-bus system and the IEEE reliability test system, where the proposed models showcase more accuracy in reflecting the impacts of fast net-load variations over discrete-time counterparts.