Realtime optimization, extremum seeking control, model-free optimization
Conventionally, online methods for minimizing power consumption of vapor compression systems rely on the use of physical models. These model-based approaches attempt to describe the influence of commanded inputs, disturbances and setpoints on the thermodynamic behavior of the system and the resultant consumed electrical power. These models are then used online to predict the combination of inputs for a measured set of thermodynamic conditions that both meets the heat load and minimizes power consumption. However, these models of vapor compression systems must contain nonlinear terms of sufficient complexity in order to accurately describe the region near the optimum operating point(s), but also must rely on simplifying assumptions in order to produce a mathematically tractable representation. For these reasons, model-based online optimization of vapor compression machines have not gained traction in application, and have created an opportunity for model-free techniques such as extremum seeking control, which is gradient descent optimization implemented as a feedback controller. While traditional perturbation-based extremum seeking controllers for vapor compression systems have proven effective at minimizing power without requiring a process model, the algorithm's requirement for multiple distinct timescales has limited the applicability of this method to laboratory tests where boundary conditions can be carefully controlled, or simulation studies with unrealistic convergence times. Perturbation-based extremum seeking requires that the control input be manipulated with a time constant approximately two orders of magnitude slower than the slowest vapor compression system dynamics, otherwise instabilities in the closed loop system occur. As a result, convergence to the optimum for slow processes such as thermal systems is restrictive due to inefficient estimation of the gradient, and slow (integral-action dominated) adaptation in the extremum seeking control law. In order to address this timescale separation issue, we have previously developed an algorithm called ``time-varying extremum seeking" that more efficiently estimates the gradient of the performance metric and applied this algorithm to the problem of setpoint optimization for compressor temperatures. That algorithm improved the convergence rate to one timescale slower than the vapor compression machine dynamics. In this paper, we optimize power consumption through the application of a newly-developed proportional--integral extremum seeking controller (PI-ESC) that converges at the same timescale as the process. This method uses the improved gradient estimation routines of time-varying extremum seeking but also modifies the control law to include terms proportional to the estimated gradient. This modification of the control law, in turn, requires a revision to the gradient estimator in order to avoid bias. PI-ESC is applied to the problem of compressor discharge temperature selection for a vapor compression system so that power consumption is minimized. Because of the improved convergence properties of PI-ESC, we show that optimum values of discharge temperature can be tracked in the presence of realistic disturbances such as variation in the outdoor air temperature---enabling application of extremum seeking control to vapor compression systems in environments where previous methods have failed. The method is demonstrated experimentally on a 2.8 kW split ductless room air conditioner and in simulation using a custom-developed Modelica model.