We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in Grimm et al. (2019) that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of Grimm et al. (2019) by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.

我们考虑在一个多层次的框架中结合使用网络设计和图分区模型，以确定最优的网络扩展和区域定价电力市场的最佳区域配置，这是Grimm等人讨论的模型的扩展。（2019），其中不包含网络设计问题。网络设计和图形划分的两个经典离散优化问题以及由于经济建模导致的非线性产生了极富挑战性的混合整数非线性多级模型，为此我们开发了两种针对问题的求解技术。第一种方法依赖于等效的双层结构及其标准的KKT转换，包括新颖的原始-对偶束紧技术，而第二种方法则是量身定制的广义Benders分解。对于后者，我们加强了Grimm等人的Benders切割。（2019）通过使用新引入的网络设计子问题的结构。我们证明这两种方法都能产生全局最优解。之后，我们在数值研究中比较了这些方法，并表明量身定制的Benders方法明显优于标准KKT变换。最后，我们提供了一个案例研究，说明了我们模型中捕获的经济影响。