Mathematical optimization of unbalanced networks with smart grid devices
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Electric distribution networks should be prepared to provide an economic and reliable service to all customers, as well as to integrate technologies related to distributed generation, energy storage, and plug-in electric vehicles. A proper representation of the electric distribution network operation, taking into account smart grid technologies, is key to accomplish these goals. This chapter presents mathematical formulations for the steady-state operation of electric distribution networks, which consider the unbalance of three-phase grids. Mathematical models of the operation of smart grid related devices present in networks are discussed (e.g., volt-var control devices, energy storage systems, and plug-in electric vehicles). Furthermore, features related to the voltage dependency of loads, distributed generation, and voltage and thermal limits are also included. These formulations constitute a mathematical framework for optimization analysis of the network operation, which makes it possible to model decision-making processes. Different objectives related to technical and/or economic aspects can be pursued within the framework; in addition, the extension to multi-period and multi-scenario optimization is discussed. The presented models are built based on mixed integer linear programming formulations, avoiding the use of conventional mixed integer nonlinear formulations. The application of the presented framework is illustrated throughout control approaches for the voltage control and the plug-in electric vehicle charging coordination problems.