Analytic Modeling and Integral Control of Heterogeneous Thermostatically Controlled Load Populations

Thermostatically controlled loads (TCLs) account for approximately 50% of U.S. electricity consumption. Various techniques have been developed to model TCL populations. A Highfidelity analytical model of heterogeneous TCL populations facilitates the aggregate synthesis of power control in power networks. Such a model assists the utility manager to increase the stability margin of the network. The model, also, assists the customer to schedule his/her tasks in order to reduce his/her energy cost. We present a deterministic hybrid partial differential equation (PDE) model which accounts for heterogeneous populations of TCLs, 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 provide guidelines to maintain the numerical stability of the discretized model during computer simulations. Moreover, the proposed model is a close fit to design output feedback algorithms for power control purposes. Our integral output feedback control, designed using the comparison principle, guarantees fast and efficient power tracking for various real-world scenarios. We present simulation results to verify the effectiveness of the proposed modeling and control technique. INTRODUCTION Analytical and numerical models of thermostatically controlled loads (TCLs) including heating, ventilation, and air conditioning (HVAC) systems have been developed to study demand response in power networks [1–14]. One can refer to [12, 14] among the very first reports that used statistical and stochastic analysis to develop an aggregate model of TCLs. The effect of capital stock, lifestyle, usage response, and price impacts on power curves have been studied in [5, 13]. Mortensen and Haggerty [11] present a brief survey on five different TCL modeling techniques developed up to 1990. More recently TCL modeling has gained extensive attention [1–4, 6–10]. Coupled Fockker-Planck equations (CFPE), derived in [12], present statistical aggregate electrical dynamics for a homogeneous group of devices. A perturbation analysis yields the dynamics for a non homogeneous group. Equations (10)–(18) of [12] include CFPE, 4 algebraic boundary conditions, and 2 ordinary differential equations to guarantee probability conservation. Moreover, the expectation of the operating state of the homogeneous population is given by another ODE defined by Eqn. (43) in [12]. The proposed model does not provide direct access to manipulate the deadband and set-point temperature which makes the controller design process a hard task to achieve. An exact solution to the CFPE which describes the aggregate behavior of TCL populations is presented in [8]. Also, [8] demonstrates the potential to provide ancillary services by remotely manipulating thermostat set-points, particularly to balance fluctuations from intermittent renewable generators. Another statistical model based on the “state bin transition model” has been developed in [10], a formal abstraction of which is presented in [9] to relax some of the assumptions in [10]. These statistical group of models rely highly on the probability analysis and distribution functions of the TCL population. Proceedings of the ASME 2014 Dynamic Systems and Control Conference DSCC2014 October 22-24, 2014, San Antonio, TX, USA