TR2014-072

Realtime Optimization of MPC Setpoints using Time-Varying Extremum Seeking Control for Vapor Compression Machines



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A heat load increase of 400 W is modeled at 30, 000 s. As the cost surface changes due to the new heat load of 2300 W, the time-varying extremum seeking algorithm converges to 62.5C and 463.6 W, a region near the true optimum at 58C and 456 W.

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.

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 in application 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.