Modeling And Algorithm Analysis Of Power Markets Based On Semi-smooth Theory

The system of nonlinear equations has been widely raised in practical problems, a typical of which is to solve the system of semi-smooth equations. However, so far, there are few works related to this theory and methods to practical engineering applications. According to the development of research for this problem, our aim is to study some applications such as optimal power flows, bidding of power market based on the semi-smooth theory and semi-smooth Newton methods. The primary contents are as follows:In the first section, we mainly introduce the methods for solving a system of nonlinear equations by semi-smooth theory, research state of a semi-infinite programming (SIP) and its application in optimal power flows with transient stability constraints (OTS), bilevel programmings and the application in power market. In addition, our research works of this paper are also briefly introduced.In the second section, we present a smoothing quasi-Newton method for solving the semi-infinite programming (SIP). Based on the nonlinear complementary problem (NCP) function, the KKT (Karush Kuhn Tucker) system of the SIP problem is reformulated to a system of nonsmooth equations. A smoothing quasi-Newton method is designed to solve this system. The remarkable characteristic of the method is that it only solves a system of linear equations at each iteration, and it enjoys nice global and locally superlinear convergence. Finally, the algorithm is used to solve the optimal power flows with transient stability constraints (OTS) in power market with spot price. Numerical results show the feasibility of the proposed approach.In the third section, we present a dynamic bidding model of power markets based on the supply function and the semi-smooth theory in which the transmission constraints of the network are involved. The new model is composed of difference dynamic system and semi-smooth equations reformulated by NCP. As an example, three buses and five buses of a power market are as numerical examples to test the dynamic models, the Nash equilibrium and its stability are analyzed with different market parameters and different operational conditions of transmission network, i.e. congestion and non-congestion; By using the numerical simulations, the effect of different market parameters to the dynamic behaviors and stability of markets is studied. The simulating results show that the new model is valid.