Levenberg-Marquardt method for smart grid with controllable supply
Tian Jing, Shouqiang Du, Yuanyuan Chen · Pacific Journal of Optimization · 2023
(Communicated by Donghui Li) Traditional power systems face unprecedented challenges, such as energy, environment, and economy. At the same time, fossil fuels used for traditional thermal power generation are in- creasingly depleted. Currently, in the context of carbon reduction and harmonious advance in the environment, environment-friendly smart grid is developing rapidly. A strong smart grid is based on a strong grid structure, supported by communication information platforms, and con- trolled by intelligent means. It includes various links of power generation, power transmission, power transformation, power distribution, power consumption, and power dispatching in the smart grid. Smart grid is the inevitable result of economic and technological development, which specif- ically employs advanced technique to increase the performance for electricity power system in power utilization, power supply quality and reliability. The foundation of smart grid is divided into data transferring, calculation and control technique between several electricity providing units. Smart grid has excellent features of energy-saving, reliable, self-healing, fully controllable and asset efficient. So in recent years, the theory and method of smart grid with the character- istics of high-quality, reliability, self-healing, interaction, security and environmental protection have been studied. There exist two models for solving uncertain optimization systems: stochastic optimization and fuzzy optimization. In stochastic optimization problems, we need to know the distribu- tion function of system parameters. Similarly, in fuzzy optimization problems, the membership function of system parameters is required to be known. However, in practical problems, the distribution function and membership function are not always easy to obtain. In most cases, we only know the coefficients of the system at a certain point. The interval optimization model is more practical and easy to handle when there exist changes within an interval. Research on interval optimization has aroused widespread interest and achieved rich theoretical and practi- cal results. In interval optimization methods, it is not necessary to know the specific numerical values of parameters. It operates in the form of intervals and only needs to know their range of variation, i.e. the upper and lower boundaries, without the need for precise numerical val- ues. This greatly simplifies the preliminary data processing work. Interval optimization has been widely studied, especially its optimality conditions. Interval optimization is divided into single- objective interval optimization and multi-objective interval optimization. Interval optimization is increasingly used in practice such as economic planning, energy development, engineering design, environmental protection and other fields. Among the existing smart grid models, there exists little research about the smart grid with controllable supply, so this paper aims to study the problem of smart grid with controllable supply. Therefore, the interval optimization theory is applied to the real-time pricing problem of smart grid for maximizing social welfare. The intelli- gent control of electricity generation in the model has interval change in reality, so we combine the interval optimization with the analysis of the real-time pricing problem of smart grid based on social welfare maximization, and add the change in electricity generation to the objective function of the smart grid model, which becomes an interval objective function. We propose the model of smart grid with controllable supply based on maximizing social welfare. The model is transformed into a real-time pricing problem for smart grid with interval change. We transform the interval optimization problem into a real-valued optimization problem and solve it based on Karush-Kuhn-Tuker(KKT) conditions. The KKT conditions for smart grid with controllable supply based on social welfare maximization are also given. In smart grid, there are currently three forms of pricing mechanisms: time of use pricing mechanism, critical peak load pricing mechanism, and real-time pricing mechanism. Unlike the time of use pricing mechanism and critical peak load pricing mechanism, real-time pricing is not pre-set, but fluctuates continuously every day, directly reflecting the relationship between market price and market electricity cost. It is an ideal pricing mechanism that can encourage users to consume more wisely and effectively. Therefore, real-time pricing mechanisms have become a current research hotspot. The KKT conditions, which are transformed into a nonsmooth equation system. And we introduce the value function to transform it into an unconstrained optimization problem. The real-time price of smart grid with controllable supply can be obtained by the KKT conditions of interval optimization. Levenberg-Marquardt method is a type of optimization method. Its application fields are very wide, such as economics, management optimization, network analysis, optimal design, me- chanical or electronic design. In this paper, Levenberg-Marquardt method is applied to solve the transformed problem of smart grid with controllable supply. We give the convergence analysis of Levenberg-Marquardt method under mild conditions. Finally, related numerical experiments have shown that Levenberg-Marquardt method can effectively solve the real-time price problem of smart grid. The real-time price obtained meets the effect of peak-shaving and valley-filling, which indicates that Levenberg-Marquardt method can effectively get real-time price. The re- search on this kind of smart grid problem further enriches the research work in the field of real-time price of smart grid based on social welfare maximization.