A nonsmooth Levenberg–Marquardt method based on KKT conditions for real-time pricing in smart grid

The real-time pricing of smart grid, which is an important mean of demand side management, is an ideal method to adjust the power balance between the supply and demand in smart grid system. By transforming the social welfare maximization model into a nonsmooth system of equations based on KKT conditions and investigating the related nonsmooth-function’s generalized Jacobi, a nonsmooth Levenberg–Marquardt method with global convergence is proposed in this paper. Also, the local convergence rate is obtained under the local error bound, which is weaker than the nonsingular condition. Finally, numerical simulations show that the smaller the approximate parameter in smoothing Newton’s method, the closer the electricity price is to the value obtained by the proposed algorithm. And the price of the multi price based on the method is lower than that of the single price algorithm that does not consider user types. Meanwhile, compared with the distributed algorithm with different scales, it is found that the price is lower and the social welfare is higher, which shows that the algorithm is robust and effective. • In recent years, the subgradient algorithms, the dual methods, and the smart algorithms are popular for solving the real-time pricing of smart grid. However, the subgradient algorithms always have slow convergence, the shadow price, namely the real-time price in dual methods is actually obtained as a by-product and its accuracy is not high, and some of the smart algorithms may be nonconvergent although its speed is fast. • Considering that, we proposed a nonsmooth Levenberg Marquardt method with global convergence by transforming the social welfare maximization model into a nonsmooth system of equations based on KKT conditions in this paper, where the real-time price is a product, not a by-product. • The contributions of the research are as follows. (1) Consider the multiple price and utility functions for different types of users.(2)Investigate the computation of the corresponding vector-function’s B-differential. (3)Obtain the locally Lipschitzian continuous property of the corresponding function and the local quadratic convergence of the proposed algorithm.(4) A nonsmooth algorithm is first applied into the RTP as far as we know.