  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.
