As is well known, the utility function is significant for solving the real-time pricing problem of smart grids. Based on a new utility function, the social welfare maximization model is considered in this paper. First, we transform the social welfare maximization model into a smooth system of equations using Krush-Kuhn-Tucker (KKT) conditions, then propose a two-step smoothing Levenberg-Marquardt method with global convergence, where an LM step and an approximate LM step are computed at every iteration. The local convergence of the algorithm is cubic under the local error bound condition, which is weaker than the nonsingularity. The simulation results show that, the algorithm can not only reduce the user's electricity consumption but also improve the total social welfare at the most time when compared with the fixed pricing method. Additionally, when different values of the approximating parameter are adopted in a smoothing quasi-Newton method, the price tends to that obtained by the present algorithm. Furthermore, the CPU time of the one-step smoothing Levenberg-Marquardt algorithm and the proposed algorithm are also listed.

With the proposal of the new power system, power supply from renewable energy sources and traditional power supply have emerged as the future development directions of the power grid, while the traditional pricing mechanisms are facing new challenges. Considering the different characteristics of renewable energy power supply and traditional power supply, such as being clean and sustainable, but unstable, for renewable energy power supply, and being stable and technologically mature, but causing significant pollution, for traditional power supply, a multi-price model with the cost of pollution treatment under the multi-energy electricity generation was established in this paper. A distributed algorithm with the non-dominated sorting genetic algorithm (NSGA-Ⅱ) was also proposed. In the model, the power supply side includes traditional energy generation, renewable energy generation, and the energy storage device. The proposed algorithm was designed using Lagrangian duality theory, and the multi-price is obtained by solving the different lagrange multipliers. Finally, the numerical results show that the model is reasonable when comparing the obtained price and social welfare with that untreated-pollution model, as well as a single supply model. Also, the proposed algorithm always has better computational efficiency, when compared with PSO, HS, and GA algorithms. The proposed model and algorithm provide a new idea and method for the optimal scheduling of a smart grid.

In smart grids, the interactions among diverse participants significantly affect system efficiency and social welfare. Considering that the real-time electricity pricing (RTP) mechanism under traditional social welfare maximization fails to account for the independent interests of various entities in the power system, a two-level load-utility balancing model for real-time pricing in smart grids is proposed in this paper, where the welfare of the demand side and the multi-energy supply side is collaboratively optimized and balanced. Furthermore, a multi-agent reinforcement learning (MARL) algorithm based on the centralized training and decentralized execution (CTDE) framework is designed for this model, and a multi-agent electricity market environment is constructed accordingly, comprising users, an aggregated power supplier, and a power market scheduling center (PMSC). The user agent is modeled with a heterogeneous utility function, the supplier agent is modeled with a profit function coordinating both traditional and renewable energy, while the PMSC agent is responsible for real-time pricing and cross-agent welfare balance coordination. Finally, simulation results show the effectiveness of the proposed model and algorithm in achieving welfare balance between the supplier and users. Compared with the pricing scheme without a welfare-balancing mechanism, the proposed model reduces the welfare gap between the supplier and users by approximately 46.9%. Compared with the non-dominated sorting genetic algorithm II (NSGA-II), the proposed method can achieve a comparable level of total social welfare.