Recently, model predictive control (MPC) has received increased attention in the HVAC community, largely due to its ability to systematically manage constraints while optimally regulating signals of interest to setpoints. For example, in a common formulation of an MPC control problem for variable compressor speed vapor compression machines, the setpoints often include the zone temperature and the evaporator superheat temperature. However, the energy consumption of vapor compression systems has been shown to be sensitive to these setpoints. Further, while superheat temperature is often preferred because it can be easily correlated to heat exchanger efficiency (and therefore cycle efficiency), direct measurement of superheat is not always available. Therefore, identifying alternate signals in the control of vapor compression machines that correlate to efficiency is desired. Conventionally, methods for maximizing the energy efficiency rely on the use of mathematical models of the physics of vapor compression systems. These model-based approaches attempt to describe the influence of commanded inputs on the thermodynamic behavior of the system and the consumed electrical energy, and they are used to predict the combination of inputs that both meet the heat load requirements and minimize energy consumption. However, these models of vapor compression systems rely on simplifying assumptions in order to produce a mathematically tractable representation. Further, they are difficult to derive and calibrate, and often do not describe variations over long time scales, such as those due to refrigerant losses or accumulation of debris on the heat exchangers. In this paper, we consider a model-free extremum seeking algorithm that adjusts setpoints provided to a model predictive controller. While perturbation-based extremum seeking methods have been known for some time, they suffer from slow convergence rates---a problem emphasized by the long time constants associated with thermal systems. Our method uses a new algorithm (time-varying extremum seeking), which has dramatically faster and more reliable convergence properties. In particular, we regulate the compressor discharge temperature using an MPC controller with setpoints selected from a model-free time-varying extremum seeking algorithm. We show that the relationship between compressor discharge temperature and power consumption is convex (a requirement for this class of realtime optimization), and use time-varying extremum seeking to drive these setpoints to values that minimize power. The results are compared to the traditional perturbation-based extremum seeking approach. Further, because the required cooling capacity (and therefore compressor speed) is a function of measured and unmeasured disturbances, the optimal compressor discharge temperature setpoint must vary according to these conditions. We show that the energy optimal discharge temperature is tracked with the time-varying extremum seeking algorithm in the presence of disturbances
