The widespread use of optimization methods in the design phase of District Heating Networks is currently limited by the availability of scalable optimization approaches that accurately represent the network. In this paper, we compare and benchmark two different approaches to non-linear topology optimization of District Heating Networks in terms of computational cost and optimality gap. The first approach solves a mixed-integer non-linear optimization problem that resolves the binary constraints of pipe routing choices using a combinatorial optimization approach. The second approach solves a relaxed optimization problem using an adjoint optimization approach, and enforces a discrete network topology through penalization. Our benchmark shows that the relaxed penalized problem has a polynomial computational cost scaling, while the combinatorial solution scales exponentially, making it intractable for practical-sized networks. We also evaluate the optimality gap between the two approaches on two different District Heating Network optimization cases. We find that the mixed-integer approach outperforms the adjoint approach on a single-producer case, but the relaxed penalized problem is superior on a multi-producer case. Based on this study, we discuss the importance of initialization strategies for solving the optimal topology and design problem of District Heating Networks as a non-linear optimization problem.

翻译：优化方法在区域供热管网设计阶段的广泛应用目前受限于能够准确表征网络的可扩展优化方法的可用性。本文从计算成本和最优性差距两个维度，比较并基准测试了两种不同的区域供热管网非线性拓扑优化方法。第一种方法通过求解混合整数非线性优化问题，采用组合优化方式处理管道路径选择的二元约束。第二种方法采用伴随优化方法求解松弛后的优化问题，并通过惩罚项强制实现离散网络拓扑。基准测试表明，松弛的惩罚优化问题具有多项式计算复杂度，而组合优化方法的计算量呈指数级增长，导致其无法实际应用于工程规模管网。我们还在两个不同的区域供热管网优化案例中评估了两种方法的最优性差距。研究发现，在单热源案例中混合整数方法优于伴随方法，但在多热源案例中松弛惩罚问题表现更优。基于此研究，我们探讨了将区域供热管网最优拓扑与设计问题作为非线性优化问题求解时，初始化策略的重要性。