The real-time optimization of district heating (DH) networks is computationally demanding due to the strong interdependence between their state variables. This thesis investigates whether a Reduced Order Model (ROM) can improve the computational efficiency of state estimation in DH networks. Existing ROM approaches typically rely on simplified physical assumptions or require manual, topology-dependent modifications to the model architecture, limiting their applicability and preventing fully automated retraining.

To address these limitations, this study proposes a method based exclusively on Full Order Model (FOM) data that follows a uniform training pipeline applicable to any DH network without requiring manual analysis of the network topology or case-specific architectural adjustments. A hybrid framework based on Proper Orthogonal Decomposition (POD) is developed, in which POD extracts dominant spatial modes from high-dimensional FOM data, while a feedforward neural network predicts the corresponding temporal coefficients from compressed input features. The ROM output is subsequently used as an initial guess for the FOM state iteration procedure, thereby preserving physical consistency.

The approach is evaluated on two realistic DH networks of different scales. In both cases, the ROM achieves total relative reconstruction errors below 5% (4.8% for the smaller network and 3.6% for the larger network), with prediction times below 0.1 seconds compared to approximately 100 seconds for a single FOM iteration. For the smaller network, integrating the ROM into the optimization workflow results in a 1.17× speed-up while producing decision variables nearly identical to those obtained with the FOM. This improvement arises from skipping the first FOM iteration, reducing the number of iterations required for convergence, and updating fewer time steps per iteration. For the larger network, the ROM maintains high predictive accuracy but performs less reliably during optimization, likely due to limited training data for rarely activated backup sources. Overall, the results demonstrate that hybrid POD-based ROMs can significantly improve the computational efficiency of DH network state estimation and optimization, provided that the training dataset adequately represents all relevant operational regimes.

This thesis considers solutions to the discrete Nagumo equation u˙ n = d(un−1 − 2un + un+1) + f(un), n ∈ Z. For sufficiently large d, the solutions are of the form un(t) = U(n + ct) with c > 0. This thesis contains the proof of existence of traveling wave solutions of the discrete Nagumo equations and originates from Bertram Zinner’s article ”Existence of Traveling Wavefront Solutions for the Discrete Nagumo equation” [Zin90]. In the first chapter, all the prerequisite knowledge needed to understand the proof, such as Brouwer’s fixed point theorem, is presented. The proof starts by considering f(un) as a linear function and thus simplifying the problem. The simplified problem is converted into a fixed point problem by considering a Poincar´e map which can be solved using Brouwer’s fixed point theorem. Finally, the proof ends by confirming that the solutions of the approximated, simplified problem have a limit point which corresponds to the traveling wave solutions of the discrete Nagumo equation