Demonstration of the equivalence between two district heating networks optimization problems

This paper presents a demonstration of the equivalence between the optimization problem minimizing the supply cost of $p\in \mathbb{N}$ independent District Heating Networks (DHN) and the optimization problem minimizing the supply cost of a single aggregated DHN. This demonstration is presented in three stages. First, we demonstrate the equivalence in the case of two DHNs, each one composed of two plants. In the second stage, we enlarge the demonstration for two DHNs with $n$ plants. Finally, the general demonstration ($p$ DHNs and $n$ plants) is presented. Although the last demonstration includes the previous cases, the two first sections are necessary to understand the general demonstration.