Complementarity system-based equilibrium characterization for optimal energy coordination in renewable integrated smart grid networks

Abstract

The popularization of advanced smart grid technology combined with the increased prevalence of distributed renewable energy sources are two key factors that create new strategic conflicts of a multi-agent framework. This paper considers interaction in the microgrid under the paradigm of non-cooperative game theory. In this framework, prosumers are engaged in the optimization of their quadratic cost function subject to the constraints related to their own operating conditions and a single transmission capacity. The entire problem is written down as Generalized Nash Equilibrium Problem (GNEP) according to some Karush-Kuhn-Tucker conditions used to formulate its solution. Notably, the whole system of original first-order conditions of the agents reduces to a single multivariate mixed complementarity problem of the form 0 <= F(z) perp z >= 0 which is equivalent to a variational inequality VI(K,F) over a non-empty closed and convex strategy set K. Finally, the equilibrium existence is proven through the use of Kakutani’s fixed-point theorem, while it is shown that equilibrium is unique through proving that the Jacobian of the system F is positive definite (thus proving that F is strongly monotone on K). In order to solve the mixed complementarity problem resulting from the analysis, the author uses the PATH solver based on Newton’s method. It is presumed that the computational complexity of the algorithm is equal to O(|N|^3) for a network with |N| agents. Numerical tests on an IEEE 33-bus test network where levels of renewable energy interruption range from 0% to 80% indicate that PATH converges in 11 iterations and 0.84 seconds compared to the ADMM-based solver requiring 34 iterations and 3.62 seconds, and the penalty-function method that uses 47 iterations and 5.91 seconds while showing a residual norm less than 1e-4 in all the cases. Finally, sensitivity analysis indicates that the shadow price of the equilibrium shifts from 0.5/MWh to 46.1/MWh as the network capacity limit decreases from 140 MW to 60 MW.