optimization, control, heat pump, extremum seeking
As the vapor compression machine has become more sophisticated (for example, through the adoption of variable speed compressor technology, electronic expansion valves and variable speed fans), the opportunities to improve efficiency are increasingly realized through the control algorithms that operate machine actuators. However, designing control algorithms that minimize energy consumption is not straightforward: the heat load disturbances to be rejected are not measured, the governing dynamics are nonlinear and interactive, and the machine exhibits strong coupling between the multivariate inputs and outputs. Further, many heat pumps must also operate in cooling mode, forcing compromises in sensor locations and actuator selection. This paper compares two controllers for realtime (online) energy optimization of heat pumps. The first energy-optimizing controller is model-based. A custom multi-physical model of the dynamics of a heat pump is developed in the Modelica modeling language and used to obtain the relationship between control inputs and power consumption as a function of the operating conditions. The gradient of this relationship is computed symbolically and used to derive a gradient descent control law that is shown to drive actuator inputs such that the system power consumption is minimized. To address the concern of modeling error on optimization performance, the controller based on a model of a heat pump will be tested on a physical system in an experimental setting for the submitted paper. We expect the convergence rate to be exponential, and will quantify the sensitivity between modeling errors and the non-optimality of the stabilized system. The second approach is model-free and based on the authors' time-varying (TV) and proportional-integral (PI) extremum seeking control (ESC) algorithms. Briefly, extremum seeking controllers use an estimate of the gradient between a plant's manipulated inputs and an objective signal (i.e., power consumption) to steer the system toward an optimum operating point, under the assumption that this relationship is convex. Whereas traditional ESC methods exhibit slow and non-robust convergence, our TV-ESC and PI-ESC methods have demonstrated higher performance due to the estimation routine that tracks the gradient as a time-varying parameter. We expect this algorithm to converge faster than transitional perturbation-based ESC methods (as we have previously demonstrated), but perhaps slower than the model-based approach. However, we expect this controller to converge to a neighborhood around the true optimum since modeling errors are not applicable in this model-free algorithm. The final paper will compare convergence properties of these two methods through experiments obtained on a commercial four-zone heat pump installed in calorimetric-style testing chambers, and the resultant coefficients-of-performance (COPs) will be measured.