# Model Predictive Control of a μ-CHP System under Uncertainty

This work aims at developing a new control methodology to minimize operational cost in presence of variable energy prices and energy demand of residential energy use, while meeting technical constraints and operational limits. Model Predictive Control (MPC) is applied here to determine the operation strategies of μ-CHP systems; and Polynomial Chaos Theory (PCT) is adopted to quantify uncertainties of variable energy costs of a μ-CHP system.

Copyright: RWTH AachenPervasive distributed generation (DG) at the residential level implies that its potential to contribute to overall energy

efficiency, environmental and economic benefits becomes significant. Very small combined heat and power units (μ-CHP), among the DG devices, are the focus of this work. The μ-CHP is an energy conversion unit with a capacity below 15 kW that can simultaneously generate heat (usually as main product) and electrical power. Instead of following the heat demand (or electricity demand) unilaterally, the μ-CHP is, in principle, able to operate flexibly in heat driven or electricity driven mode, thus responding for example to energy price variations with the most economically convenient operation for the homeowner. However, μ-CHPs are affected by various sources of uncertainty from economical, technological, or regulatory sides that affect investment decisions and operational decisions. In this research focus is on the latter, and in particular of the effects of uncertainty in variable energy price and demand.

The operation of the μ-CHP system is then formalized as a constrained, stochastic, optimal control problem, the control being realized through Model Predictive Control (MPC). MPC, one of the major recent advances in controls, is based on a receding horizon philosophy, and numerical optimization algorithms. The MPC, as compared to the classical controllers, leverages on future information, handles input and state constraints, and is applicable to multi-input multioutput systems.

The accuracy of the model significantly affects the MPC algorithm. Thus, it is necessary to quantify uncertainties on parameters and initial conditions of the system. However, most of the research on stochastic MPC adopts sampling based methods, or mean-variance approaches to propagate uncertainty sources. Both methods have drawbacks, the former ones have slow convergence rate while the latter ones do not offer complete statistical information. Polynomial Chaos theory (PCT) is a non-sampling based method to determine propagation of stochastic uncertainty in dynamic systems. Integrating this method with MPC algorithm yields the potential to reduce the computational burden with a desired accuracy in the design of a robust controller, allowing for the estimation of probability density functions of the quantities of interest. In order to take advantage of different realizations of the PCT implementation, the Galerkin projection and the non-intrusive probabilistic collocation methods are considered.