Modeling, Control, and Stability Analysis of Heterogeneous Thermostatically Controlled Load Populations Using Partial Differential Equations

Abstract

Thermostatically controlled loads (TCLs) account for more than one-third of the U.S. electricity consumption. Various techniques have been used to model TCL populations. A high-fidelity analytical model of heterogeneous TCL (HrTCL) populations is of special interest for both utility managers and customers (that facilitates the aggregate synthesis of power control in power networks). We present a deterministic hybrid partial differential equation (PDE) model which accounts for HrTCL populations and facilitates analysis of common scenarios like cold load pick up, cycling, and daily and/or seasonal temperature changes to estimate the aggregate performance of the system. The proposed technique is flexible in terms of parameter selection and ease of changing the set-point temperature and deadband width all over the TCL units. We investigate the stability of the proposed model along with presenting guidelines to maintain the numerical stability of the discretized model during computer simulations. Moreover, the proposed model is a close fit to design feedback algorithms for power control purposes. Hence, we present output- and state-feedback control algorithms, designed using the comparison principle and Lyapunov analysis, respectively. We conduct various simulations to verify the effectiveness of the proposed modeling and control techniques.

The Green’s Type Matrices for Consumption Reduction in a Heterogeneous Population Model of TCLs with Diffusion

“…In this section, we consider the problem of controlling the amount of the aggregate power consumption (7) by choosing the quantities Θ sp and σ accordingly. Such a problem for the simplest model described by (4) has been considered, e.g., in [? ].…”

Controlling Power Consumption in a Heterogeneous Population Model of TCLs with Diffusion: The Green’s Function Approach

“…TCLs account for more than one-third of electricity consumption in the United States [4]. Nowadays, the model of TCLs has gained extensive attention.…”

Robust H∞ control for uncertain switched nonlinear polynomial systems: Parameterization of controller approach

“…The robust H ∞ control problem for the nonlinear systems including switched nonlinear systems has attracted much attention of researchers in recent years. It should be pointed out that, although many achievements on robust H ∞ control for classes of nonlinear systems have been made, there are still many problems needed to be solved . The ADT, piecewise Lyapunov function approach was utilized in the work of Niu and Zhao and the multiple Lyapunov functions approach was applied in the works of Long and Zhao and Zhao and Hill .…”

“…It should be pointed out that, although many achievements on robust H ∞ control for classes of nonlinear systems have been made, there are still many problems needed to be solved. [11][12][13][14][15][16][17][18] The ADT, piecewise Lyapunov function approach was utilized in the work of Niu and Zhao 11 and the multiple Lyapunov functions approach was applied in the works of Long and Zhao 12 and Zhao and Hill. 13 However, in the results of the aforementioned works, [11][12][13] it needs to solve some class of partial differential inequalities that are difficult to solve.…”

“…17 In the work of Ghaffari et al, 18 a deterministic hybrid partial differential equation model was presented. It should be noted that the nonlinearities of the systems of the related works [14][15][16][17] are restricted by constraint conditions and some boundary conditions are needed for the sake of solving some dynamic equations in the work of Ghaffari et al 18 Besides the general nonlinear systems, there exists an important class of nonlinear systems, ie, polynomial nonlinear systems. Recently, along with the research progress on the control of nonlinear polynomial systems, [19][20][21] several sum of squares-based approaches are presented to solve the H ∞ control problem of nonlinear polynomial systems.…”