Nonlinear topology optimization of District Heating Networks: A benchmark of a mixed-integer and a density-based approach

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 benchmark two different approaches to nonlinear topology optimization of District Heating Networks in terms of computational cost and optimality gap. We compare a combinatorial approach, which directly solves a mixed-integer nonlinear program, against a density-based approach. This density-based approach relaxes the integer constraint on pipe placement and ensures near-discrete topologies through penalization. The benchmark shows subquadratic scaling of the computational cost for the density-based approach, making it tractable for large problems, while the combinatorial approach scales exponentially. The combinatorial approach took 29 h to optimize a network for a neighborhood of about 600 streets, compared to 35 min for the density-based method. Optimality gap analysis indicates that resolving the integer constraint on pipe placement does not necessarily lead to a superior design, while making the optimization of large practical problems intractable. In contrast, the scaling of the density-based approach remains tractable for large problems. Further study of the optimality gap highlights the importance of consciously choosing initialization strategies when deciding to solve the nonlinear topology optimization problem.