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.

我们为区域供热网络的时间相关控制开发了互补约束非线性优化模型。这些网络中水和热流的主要物理方面由非线性和双曲的一维偏微分方程控制。另外，在网络的节点处需要池化混合模型来处理不同水温的混合。可以使用适当的互补性约束来重铸此混合模型。由此产生的问题是具有互补约束的数学程序，该程序受描述物理的非线性偏微分方程的约束。为了解决一个棘手的问题，我们在时空上应用了适当的离散化，导致具有互补约束的有限维优化问题，为此，我们开发了一种具有改进的约束规则性的合适公式。此外，我们提出了一种离散问题的瞬时控制方法，讨论了实用的惩罚公式，并提出了用于简化网络节点混合模型的预处理技术。最后，我们使用所有这些技术来解决现实情况。我们的数值结果表明了我们的技术在实践中的适用性。我们使用所有这些技术来解决现实情况。我们的数值结果表明了我们的技术在实践中的适用性。我们使用所有这些技术来解决现实情况。我们的数值结果表明了我们的技术在实践中的适用性。