The real-time pricing optimization model of smart grid based on the utility function of the logistic function

Highlights: • A new utility function is constructed by the Logistic function. • By the KKT conditions and FB function, the problem is transformed into an equation. • The nonsingularity of Jacobian matrix and the global convergence are proved. • The new function is superior to quadratic and logarithmic functions in tests. • The higher smoothing function's approximation, the better the convergence effect. The utility function is very significant for solving the real-time pricing problem of smart grid. Based on the Logistic function, a new utility function is constructed to satisfy four properties of the utility function. In addition, from the perspective of social welfare, the real-time pricing optimization model of smart grid is established. By using the KKT conditions and the improved Fischer-Burmerister smoothing function, the optimization model is transformed into a smoothing equations problem and the smoothing Newton algorithm is used to obtain the optimal solution of the problem. The nonsingularity of the Jacobian matrix and the global convergence of the algorithm are proved. The simulation results show that, compared with previous quadratic and logarithmic utility functions, the new utility function can not only reduce the user's electricity consumption and the supplier's cost can but also improve the user's utility and the total social welfare, which also indicates that the new utility function is effective in establishing the real-time pricing model of smart grid. Furthermore, the iteration times of several algorithms to solve the real-time pricing problem of smart grid are compared, which showed that the convergence rate of the smoothing Newton algorithm is very fast.