Non-cooperative differential game and feedback Nash equilibrium analysis for real-time electricity markets

• In conventional game theory, it is assumed that all players instantly respond the best actions to the market signal (e.g. price or demand). However, players cannot react so quickly. To better depict the economic behavior of players, we take the adjustment speed of market price into account and characterize the market price as the state variable using a sticky price model. • Compared with the day-ahead market, the real-time market is always in a disequilibrium process. Hence, Nash equilibrium cannot tell the players how to optimize their production in a disequilibrium market. Given that, we establish an N-player non-cooperative differential game model with finite-horizon is established to optimize the generation strategy of all generators before they reach an equilibrium. The units’ capacity and power flow constraints constitute the boundary of control variables. We construct a state-feedback information structure in which the existence of Markovian Nash equilibrium and the uniqueness of optimal state trajectory is proved in detail. It will help other scholars to build an effective non-cooperative differential game model in the real-time market. • Commonly, the diagonalization algorithm is used to calculate Nash equilibrium, but it needs to collect global information. Subjected to privacy protection issues, we proposed a distributed algorithm based on consensus theory to solve the Hamiltonian functions with coupled constraints. The whole solving process only needs local boundary information, which greatly protects the privacy of players. Based on the optimal control theory, we provide a new perspective on the study of the electricity market. Continuous liberalization of electricity markets makes a strong correlation between the economic behaviors of market participants and their profits. To guide the production of generators before the market reaches equilibrium, a new framework is proposed to model the real-time electricity market with help of optimal control theory. The market price is described as the dynamic state using a sticky price model. We establish an N-person non-cooperative differential game model, where all participants try to maximize their profits independently only by observing power prices. The existence of feedback Nash equilibrium and uniqueness of optimal price trajectory are proved in detail, which will help other scholars build an effective differential game model. To protect the privacy of all generators, a distributed algorithm is proposed based on neurodynamic and consensus theory, which only requires information exchange among neighboring participants. Furthermore, a special case of a duopoly power market is investigated in detail and we provide a feedback time-continuous solution for each generator. Compared with the commercial Cplex solver, the proposed distributed algorithm performs better in convergence speed.