基于变分不等式理论的电力–交通耦合网络均衡状态研究

Research on Coupled Power-transportation Network Equilibrium State Based on Variational Inequality Theory

随着能源网络的发展,电力系统与交通系统逐步呈现出深度融合的形式。为了研究耦合两网相互作用机理并描述其达到的平衡状态,该文基于变分不等式理论,提出研究电力–交通耦合网络均衡描述的数学理论框架。针对数学建模方面,首先定义交通网侧的混合用户均衡状态。其次,提出含二阶锥约束的变分不等式形式以描述配电网最优潮流问题。在此基础上,建立变分不等式数学框架来描述耦合电力–交通系统的网络均衡状态。进一步地,为了对耦合网络均衡解进行定性分析,依托变分不等式框架给出研究均衡解存在性和唯一性的理论方法。最后,为有效求解复杂的网络均衡问题,设计基于松弛改进的投影–收缩算法,并借助变分不等式有关性质给出算法的最优性和收敛性证明。仿真算例验证了所提模型、框架及求解算法的有效性。

With the development of energy networks,power grid and transportation systems have gradually integrated. To study the interaction mechanism of the coupled networks and describe the equilibrium state achieved by their interaction, a unified mathematical theoretical framework for the research of the equilibrium description of the coupled power-traffic networks was proposed based on the theory of variational inequality in a creative way. In terms of mathematical modeling, the mixed user equilibrium state at the traffic network side was defined firstly. Secondly, a variational inequality with second-order conical constraints was proposed to describe the optimal power flow problem of the distribution network. On this basis, a unified variational inequality mathematical framework was established for describing the network equilibrium state of coupled power-traffic networks. In order to qualitatively analyze the equilibrium solution of coupled networks, a theoretical method for studying the existence and uniqueness of the equilibrium solution was given based on the variational inequality framework. Finally, to solve the complicated network equilibrium problem effectively, a projection-contraction algorithm based on relaxation improvement was designed, and the optimality and convergence of the algorithm were proved by the properties of variational inequality. The effectiveness of the proposed models, frameworks and algorithms has been verified by the simulation method.