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.