  This chapter presents the development and the analysis of a scheme for
aggregate power tracking control of heterogeneous populations of
thermostatically controlled loads (TCLs) based on partial differential
equations (PDEs) control theory and techniques. By employing a
thermostat--based deadband control with forced switching in the operation of
individual TCLs, the aggregated dynamics of TCL populations are governed by a
pair of Fokker--Planck equations coupled via the actions located both on the
boundaries and in the domain. The technique of input-output feedback
linearization is used for the design of aggregate power tracking control, which
results in a nonlinear system in a closed loop. As the considered setting is a
problem with time-varying boundaries, well-posedness assessment and stability
analysis are carried out to confirm the validity of the developed control
scheme. A simulation study is contacted to evaluate the effectiveness and the
performance of the proposed approach.
