Mathematical methods are extensively employed in simulation and optimisation models to solve present and future problems pertaining to the efficient and sustainable supply of energy. In particular, MILP modelling is a common practice in the field of industrial and municipal energy systems, with the objective of minimising operating costs or greenhouse gas emissions. However, MILP modelling techniques have limitations when it comes to temperature-sensitive systems where variable efficiencies or temperature constraints are a crucial part, as is the case for most low-temperature heat generation facilities. State-of-the-art approaches have a tendency to rely on assumptions and simplifications because nonlinear formulations are generally avoided due to concerns about computational time and convergence. The present study focuses on an efficient consideration of the nonlinearities. The model under investigation includes a sector-coupled energy system for a small industrial site: A photovoltaic-thermal (PVT) module supports the electricity supply and cools the PV panels via thermal heat exchange to enhance electrical efficiency. This heat is also used to drive a heat pump to provide building and process heat. Nonlinearities of critical importance, including the coefficient of performance (COP) of the heat pump depending on temperature levels, the PVT’s electrical efficiency depending on cell temperature, and pump performance depending on mass flow, are formulated as quadratic constraints resulting in an MIQCP. This approach offers the advantages of reduced computational time and stable convergence behaviour. The current work presents a model that performs within an acceptable computational time, which is comparable to the range typically required by an MILP. This model is employed to ascertain the optimal operating strategy for the drive heat of the heat pump. While a higher supply temperature of the PVT cell leads to improved efficiency of the heat pump, this is only achieved if less drive energy is extracted from the environment. The model presented here is capable of optimally exploiting this trade-off. The utilisation of such models enables the reliable quantification of the potential of PVT systems and the selection of appropriate sizes, thereby facilitating the advancement of this technology.