Design optimization of a heating network with multiple heat pumps using mixed integer quadratically constrained programming

District heating is a state of the art technology for efficient supply of heat. Modern 4th generation and 5th generation district heating networks can be used to integrate sources of waste heat, which allows efficient operation. The design of such heating networks is subject of many optimization models. Most optimization models focus on energy flows and result in Mixed Integer Linear Programs. This requires simplifications, where temperatures and mass flow rates are neglected or simplified. This work presents a Mixed Integer Quadratically Constrained Program with temperature constraints. A case study is presented, where the integration of low temperature waste heat in a district heating network is optimized. In this case study the positioning of heat pumps at the supply or at the consumers influences network operation. The results show a trade-off between economical and ecological optimal solutions with a range of total annualized costs from 120,000 EUR/a to 307,000 EUR/a and a range of CO2-Emissions from 193 t/a to 605 t/a. Furthermore, the influence of design decisions on the optimal operation is demonstrated. All in all, the quadratic model formulation stresses the influence of temperatures on the optimization outcome and offers pareto optimal solutions for the design of the presented case study