Proportional-Integral Extremum Seeking for Optimizing Power of Vapor Compression Systems

While traditional perturbation-based extremum seeking controllers (ESC) for vapor compression systems have proven effective at optimizing 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. 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 an improved gradient estimation routine previously developed by the authors but also modifies the control law part of the algorithm to include terms proportional to the estimated gradient. PI-ESC is applied to the problem of compressor discharge temperature selection for a vapor compression system so that power consumption is minimized. We test the performance of this method using a custom-developed model of a vapor compression system written in the Modelica object-oriented modeling language. We compare the convergence times of PI-ESC to our previously developed time-varying ESC method and the conventional perturbation-based ESC method. For the conditions tested, PI-ESC is shown to converge to the optimum in about 15 minutes, whereas TV-ESC converges in 45 minutes and perturbation ESC requires more than 7,000 minutes due to its inefficient estimation of the gradient. Because of the improved convergence properties of PI-ESC, self-optimization algorithms for HVAC equipment can be deployed into situations where previous methods have failed.