Influence of linear, mixed-integer linear, piecewise-linear, and nonlinear thermal power flow models on district heating network design

Optimization enables cost-effective district heating network design, but scalability constraints limit its applicability to large urban districts. Although many mixed-integer linear programming models rely on linearized thermal power flow models, no study has systematically investigated how regression model choices affect computational scalability and network design quality. Existing benchmarks compare entire optimization frameworks without isolating the impact of the underlying thermal flow model, since the chosen regression directly determines the achievable district size and the design reliability. Thus, this paper systematically compares six regression models for pipe investment costs and heat losses. Piecewise-linear regressions achieve the highest accuracy for investment costs, while logarithmic models achieve the highest accuracy for heat losses. Seven linear, mixed-integer linear, and nonlinear programming formulations based on these regressions are benchmarked across 56 real-world districts with up to 10,000 heat sinks, spanning forced and economic designs, single and multiple design periods, and single and multiple heat sources. A novel hybrid thermal power flow model is introduced, and a nonlinear power flow model incorporating economic connection decisions for heat sinks is implemented. Results reveal that purely linear models offer the shortest solving times but usually yield inferior network designs. Piecewise-linear models achieve the best objective values yet suffer from limited scalability for districts. Affine and hybrid models deliver near-optimal solutions with significantly improved scalability, while constant models sacrifice minimal optimality for substantial gains in solvable district size. For multi-period and economic optimizations, only linear formulations achieve optimal designs for large districts with more than 2250 heat sinks, whereas hybrid and constant models best balance solution quality and computational efficiency for medium-scale systems.