For implementation in low-level controllers with limited computational power, simplified optimal control formulations are highly desirable. For the case of a modulating heat pump system, however, the optimization problem is nonlinear due to the supply water temperature and load dependency of the heat pump coefficient of performance (COP). These nonlinearities do not only slow down computations but may also give rise to multiple local minima.
By neglecting these nonlinearities, e.g., by assuming a constant COP value, the optimal control problem becomes convex and fast convergence to the unique and global minimum is ensured. However, a performance loss due to the simplifications is expected.
In this paper this performance loss is quantified for the case of a modulating air-to-water heat pump connected to a floor heating system in a residential building.
Different levels of simplification are investigated. The most detailed heat pump model takes all the nonlinearities into account, while the simplest one assumes a constant COP. A first comparison suggests a significant impact of the heat pump model on the control performance. The formulations with the nonlinear models result in continuous heat pump operation at part load while the convex approximations give rise to large heat pump power fluctuations. The latter results in an energy cost increase of 7\% to 16\%. However, by penalizing power peaks in the cost function, the control performance obtained with the convex approximations is almost identical to the one obtained with the nonlinear models. Analysis of the different control trajectories and the resulting control performance reveals that the cost function is very flat near the optimum. For the investigated case, the savings in energy consumption and energy cost compared to a conventional heating curve control strategy amount to respectively 1% and 5% (in the case of a day/night tariff structure), suggesting only a minor potential for optimization-based control strategies for the investigated application.