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Mixed-integer nonlinear optimization for district heating network expansion

Gemischt-ganzzahlige nichtlineare Optimierung für den Ausbau von Fernwärmenetzen
Marius Roland and Martin Schmidt


We present a mixed-integer nonlinear optimization model for computing the optimal expansion of an existing tree-shaped district heating network given a number of potential new consumers. To this end, we state a stationary and nonlinear model of all hydraulic and thermal effects in the pipeline network as well as nonlinear models for consumers and the network’s depot. For the former, we consider the Euler momentum and the thermal energy equation. The thermal aspects are especially challenging. Here, we develop a novel polynomial approximation that we use in the optimization model. The expansion decisions are modeled by binary variables for which we derive additional valid inequalities that greatly help to solve the highly challenging problem. Finally, we present a case study in which we identify three major aspects that strongly influence investment decisions: the estimated average power demand of potentially new consumers, the distance between the existing network and the new consumers, and thermal losses in the network.


Wir präsentieren ein gemischt-ganzzahliges und nichtlineares Optimierungsmodell zur Berechnung des optimalen Ausbaus eines bestehenden und baumförmigen Fernwärmenetzes für eine Menge an gegebenen, potenziellen neuen Verbrauchern. Zu diesem Zweck stellen wir ein stationäres und nichtlineares Modell aller hydraulischen und thermischen Effekte im Leitungsnetz sowie nichtlineare Modelle für die Verbraucher und den Betriebshof vor. Hierfür betrachten wir zunächst die Eulergleichungen und die Energiegleichung. Die thermischen Aspekte stellen eine besondere Herausforderung dar, für die wir eine neuartige polynomielle Approximation entwickeln, die wir im Optimierungsmodell verwenden. Die Ausbauentscheidungen werden durch binäre Variablen modelliert, für die wir zusätzliche gültige Ungleichungen herleiten, die die Lösung der äußerst anspruchsvollen Probleme beschleunigen. Schließlich analysieren wir eine Fallstudie und identifizieren drei Hauptaspekte, die die Investitionsentscheidungen stark beeinflussen: der geschätzte durchschnittliche Leistungsbedarf neuer Verbraucher, die Entfernung zwischen dem bestehenden Netz und den neuen Verbrauchern und die thermischen Verluste im Netz.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 154

Funding source: Bundesministerium für Bildung und Forschung

Award Identifier / Grant number: EiFer

Funding statement: The second author thanks the Deutsche Forschungsgemeinschaft for their support within projects A05 and B08 in the Sonderforschungsbereich/Transregio 154 “Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks”. Both authors acknowledge the support by the German Bundesministerium für Bildung und Forschung within the project “EiFer”.


We are very grateful to all the colleagues within the EiFer consortium for many fruitful discussions on the topics of this paper and for providing the data.

Appendix A Proof of Lemma 1

Two cases are considered for solving the ODE. First, we consider va=0. Then, (5) reduces to


This obviously leads to the first case in (6).

Next, we consider the case va>0. In this case, (5) can be rewritten as


We know from classical ODE theory, see, e. g., [13], that the first-order ODE of the form


with constants a and b has the solution


with C being another constant. This then allows to derive the second case of (6).

To prove continuity and since both cases in (6) are composed of continuous functions it suffices to see that


for all x[0,La]. This concludes the proof.


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Received: 2020-04-23
Accepted: 2020-09-15
Published Online: 2020-11-27
Published in Print: 2020-11-18

© 2020 Walter de Gruyter GmbH, Berlin/Boston