Modelling nonlinear heat pump part-load efficiency for optimal investment planning

Heat pumps are a key technology for decarbonising the heating sector and achieving net-zero targets. In practice, however, their performance characteristics are inherently nonlinear, depending on factors such as minimum load constraints and part-load efficiency, particularly for inverter-driven systems. While advanced optimisation methods exist for operational scheduling (e.g., in model predictive control), investment planning models typically neglect part-load behaviour due to the computational challenges associated with solving nonlinear mixed-integer problems. Instead, simplified linear approximations are commonly employed. This work addresses this gap by presenting an integrated optimisation framework for heat pump investment planning that explicitly accounts for nonlinear part-load efficiency. The framework targets heat supply facilities at the building or building-complex level, modelled as a single-node energy hub that may comprise multiple heat pumps and additional generation technologies. To manage the resulting computational complexity, we develop both a relaxed piecewise-linear formulation and a novel second-order cone relaxation. The proposed approaches are evaluated using a real-world case study. Incorporating part-load efficiency into investment planning leads to structurally different capacity decisions compared to conventional formulations that neglect such effects. The resulting system design reduces electricity consumption by 4.5% and annual costs by 2.9%, confirmed in an ex-post validation on the full, unaggregated time horizon. Furthermore, computational performance analysis shows a 10.3-fold speed-up for the piecewise-linear relaxation and a 4.3-fold speed-up for the conic relaxation, measured in deterministic work units, paving the way for their application to larger energy hub and multi-technology investment problems. Overall, the proposed methods facilitate more accurate and computationally tractable investment decisions, thereby supporting informed decision-making in the transition towards clean and affordable heat.