Optimization of District Heating Networks : Differential Algebraic Equations, Hyperbolic Systems and Receding Horizon Control

Abstract

Many optimization problems arising in applications are constrained by ordinary differential equations (ODEs), partial differential equations (PDEs) or differential algebraic equations (DAEs). Among these, DAEs present a distinctive challenge because they combine both differential and algebraic constraints in an implicit manner. Optimal control problems involving DAEs aim to find the control inputs that optimize a specific objective function while satisfying these complex constraints. This thesis examines the challenging analysis of an optimal control problem for a system of DAEs inspired by the dynamics of district heating networks. DAEs appear in many application areas such as multibody dynamics, electrical circuits, and district heating networks. The underlying concept is to model systems combining both differential and algebraic equations, addressing complex dynamics and constraints. To classify DAEs several index concepts are used. A well-known concept is the differentiation index. Utilizing Caratheodory's theory, we establish the existence and uniqueness of a solution for the DAE system in the cases of index-1 and index-2 by using the properties inherent in the district heating network model. Modeling district heating networks involves capturing the dynamics of heat transfer and distribution within networks. These models include various elements such as pipes, consumers, and power plants. The heat distribution is determined by the temperature, velocity and pressure. After semi-discretization of the convective heat equation and implementing coupling conditions at the network nodes, a system of DAEs emerges. In this setting graph theory is used to reproduce the network structure. By neglecting the velocity derivative in the pipe equations, numerical solutions remain largely unchanged, resulting in DAEs with index-1. Based on this, we consider optimal control problems for a system of DAEs with index-1 and index-2. We show that the optimal control problem possesses optimal solutions by leveraging specific properties of the district heating network model. Dispensing with the semi-discretization of the convective heat equation and decoupling the system results in a hydrodynamic and a temperature part. The hydrodynamic part is described by a DAE system, while the temperature part is modeled by linear hyperbolic equations with non-homogeneous boundary conditions. In this context, we are concerned with boundary control systems. Solvability results for a specific structure of linear hyperbolic equations with non-homogeneous boundary conditions motivated by district heating networks are presented based on the theory of semigroups of bounded linear operators. Considering an optimal control problem with linear hyperbolic equations over an infinite time horizon, one method to solve this is the Receding Horizon Framework, also known as Model Predictive Control (MPC). In this approach, the optimal control problem with an infinite time horizon is approximated by a sequence of optimal control problems with a finite time horizon. However, stability is not guaranteed in this method. Therefore, in the final part of this thesis, we deal with the stabilizability of boundary control systems by Receding Horizon Control (RHC).