Nonlinear optimization of district heating networks
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Nonlinear optimization of district heating networks Richard Krug1 · Volker Mehrmann2 · Martin Schmidt3 Received: 6 October 2019 / Revised: 7 July 2020 / Accepted: 30 July 2020 © The Author(s) 2020
Abstract We develop a complementarity-constrained nonlinear optimization model for the time-dependent control of district heating networks. The main physical aspects of water and heat flow in these networks are governed by nonlinear and hyperbolic 1d partial differential equations. In addition, a pooling-type mixing model is required at the nodes of the network to treat the mixing of different water temperatures. This mixing model can be recast using suitable complementarity constraints. The resulting problem is a mathematical program with complementarity constraints subject to nonlinear partial differential equations describing the physics. In order to obtain a tractable problem, we apply suitable discretizations in space and time, resulting in a finite-dimensional optimization problem with complementarity constraints for which we develop a suitable reformulation with improved constraint regularity. Moreover, we propose an instantaneous control approach for the discretized problem, discuss practically relevant penalty formulations, and present preprocessing techniques that are used to simplify the mixing model at the nodes of the network. Finally, we use all these techniques to solve realistic instances. Our numerical results show the applicability of our techniques in practice. Keywords District heating networks · Nonlinear optimization · Euler equations · Differential-algebraic equations · Mixing · Complementarity constraints Mathematics Subject Classification 90-XX · 90Cxx · 90C30 · 90C35 · 90C90
* Martin Schmidt martin.schmidt@uni‑trier.de Richard Krug [email protected] Volker Mehrmann [email protected]‑berlin.de 1
Friedrich-Alexander-Universität Erlangen-Nürnberg, Discrete Optimization, Cauerstr. 11, 91058 Erlangen, Germany
2
Institute for Mathematics, MA 4‑5, TU Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
3
Department of Mathematics, Trier University, Universitätsring 15, 54296 Trier, Germany
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1 Introduction Many countries in the world are striving to make a transition towards an energy system that is mainly based on using energy from renewable sources like wind and solar power, complemented by classical energy sources like gas, oil, coal, or waste incineration. The increasing use of highly fluctuating renewable energy sources leads to many challenging problems from the engineering, mathematical, and economic point of view. A key to the success of this energy transition is the efficient and intelligent coupling of the energy resources and the optimal operation of the energy networks and energy storage. In this direction, district heating networks play an important role, since they can be used as energy storage, e.g., to balance fluctuations at the electricity exchange. To this end, district heating networks need to be operated efficiently so that n
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