IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 3, MAY 2012 581 A Stochastic Approach to “Dynamic-Demand” Refrigerator Control David Angeli, Senior Member, IEEE, and Panagiotis-Aristidis Kountouriotis Abstract—Dynamic demand management is a very promising In order for such (supply) regulation to be possible, however, research direction for improving power system resilience. This it is required that “frequency response services”, as well as suffi- paper considers the problem of managing power consumption by cient reserves, are included in the system.1 This is essential not means of “smart” thermostatic control of domestic refrigerators. In this approach, the operating temperature of these appliances only for instantaneous frequency balancing, but, more impor- and thus their energy consumption, is modified dynamically, tantly, for the ability to respond to sudden power plant failures, within a safe range, in response to mains frequency fluctuations. which would otherwise lead to severe blackouts. Previous research has highlighted the potential of this idea for From an economic perspective, frequency response services responding to sudden power plant outages. However, deterministic and reserve power are costly and any method which manages control schemes have proved inadequate as individual appliances tend to “synchronize” with each other, leading to unacceptable to reduce the magnitude of these services, without sacrificing levels of overshoot in energy demand, when they “recover” their system stability, is of significant importance [16]. In recent steady-state operating cycles. In this paper we design decentral- years, research has been initiated on the possibility of using ized random controllers that are able to respond to sudden plant frequency responsive loads, commonly referred to as “dynamic outages and which avoid the instability phenomena associated demand control”, so as to reduce the amount of frequency with other feedback strategies. Stochasticity is used to achieve desynchronization of individual refrigerators while keeping response and reserve services that are required, potentially overall power consumption tightly regulated. leading to significant reductions in overall system costs [6]. The principal idea of these methods is to shift part of the Index Terms—Dynamic demand management, jump linear sys- tems, load balancing, power system stability, refrigerators. regulation burden to the consumer side, by employing the use of intelligent domestic appliances that can defer their energy consumption in such a way so as not to apply excessive stress I. INTRODUCTION on the grid when the need arises. Such ideas have been ex- plored, for example, in [5], in the context of domestic air condi- tioning devices and [7] for water heater load control. There are YNAMIC demand management (also known as demand- two principal alternatives that present themselves for the con- D side management) is a very promising research direction trol task, which differ in their complexity and, therefore, im- for improving power system resilience [5], [6]. In a power grid, plementation costs: the loads can either react to changes in the the system frequency (mains frequency) is an indicator of the supply in a completely autonomous (i.e., decentralized) fashion, balance between demand (load) and supply (generation), with by (individually) monitoring the overall system frequency, or the nominal frequency of 50 Hz corresponding to perfect bal- they can respond to external requests made, for example, by ance between the two. When demand levels exceed the available the grid operator [18]. An intermediate solution is for the con- supply, the frequency drops below 50 Hz, while in the case of sumer appliances to respond to (dynamic) price information excess (with respect to system load) generation, the frequency originating from the utility company and decide on their op- rises above 50 Hz. As a result, system frequency continuously erating schedule accordingly; this approach, however, still re- fluctuates around the nominal level and the system operator en- quires the availability of a communications infrastructure, as sures that the balance between demand and supply is continu- does the method of externally supervised control which was dis- ously maintained, stabilizing the frequency within narrow bands cussed above. around 50 Hz, by regulating the available supply. In this paper, we consider the problem of managing power de- mand by means of “smart” thermostatic control of domestic re- Manuscript received November 25, 2010; accepted April 01, 2011. Manu- frigerators. In this approach, the operating temperature of these script received in final form April 06, 2011. Date of publication May 10, 2011; appliances and thus their energy consumption, is modified dy- date of current version April 11, 2012. Recommended by Associate Editor Z. namically, within a safe range, in response to mains frequency Wang. This work was supported by the EPSRC Grant “Control For Energy and Sustainability”, Grant reference EP/G066477/1. fluctuations. Previous research [2], [9], [16] has shown that this D. Angeli is with the Department of Electrical and Electronic Engineering, is an effective way to respond to sudden power plant outages, Imperial College, London SW7 2AZ, U.K., and also with the Department of reducing the cost of reserve power required to deal with such Systems and Information, University of Florence, 50139 Florence, Italy (e-mail: [email protected]). events. The feasibility of the approach for demand management P.-A. Kountouriotis is with the Department of Electrical and Electronic En- stems from the large number of domestic refrigerators that are gineering, Imperial College, London SW7 2AZ, U.K. (e-mail: pk201@impe- rial.ac.uk). 1Frequency response services are provided by synchronized generators, run- Color versions of one or more of the figures in this paper are available online ning only part-loaded (and hence not at maximum efficiency), as well as from at http://ieeexplore.ieee.org. industrial customers [2]. Reserve power is identified with slower, part-loaded Digital Object Identifier 10.1109/TCST.2011.2141994 plants and generation units that can start producing at short notice. 1063-6536/$26.00 © 2011 IEEE 582 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 3, MAY 2012 in use (around 40 million appliances are estimated to operate vices and so each device has to act in an autonomous setting. in the U.K. [2]). Similar control schemes can also be employed While this is a severe constraint and complicates the problem, for any other types of appliances, both domestic and industrial, we note that the quantity of interest is the temperature distri- that exhibit energy storage in the form of heat, such as freezers, bution of the whole population of appliances at each particular water heaters, etc. [16], greatly expanding the potential applica- time point. We therefore pose the problem in a probabilistic tions. framework, in which we try to find control schemes that steer In particular, [9] and [16] investigate the potential of de- the probability densities involved towards desired distributions. centralized dynamic demand control of domestic refrigerators, The advantage of this approach is that it greatly reduces the di- when the thermostat’s temperature thresholds (and, thus, the mensionality of the original problem, while it allows for simple, duty cycle of appliances) are varied as linear functions of yet successful solutions. mains frequency deviation from its nominal value, while they A viable control scheme in this setting, is the replacement also perform an assessment of the control method in scenarios of classical hysterisis-based controllers with controls that ran- with significant supply variability, due to power generated by domly jump between the “on” and “off” states of the appliances. wind turbines. In both cases, their results demonstrate that Careful selection of the jump propensities allows for the de- the amount of standing reserve required by the power system centralized control of individual appliances’ duty cycles (and, can be safely reduced. A similar approach is followed in [2], therefore, power consumption), while, during “recovery”, the where the economic impacts of such control strategies are also “population” of refrigerators is sufficiently diversified (mixed) quantified depending on the types of generation units in the with respect to temperature, thereby avoiding undesirable over- system (nuclear plants, coal plants, combined cycle gas turbine shoot phenomena. plants, etc.). The probabilistic description of the problem allows for the The problem of dynamic demand management of refriger- derivation of closed-form expressions for the first two moments ator appliances is also addressed in [18] and [19], in the context of the temperature distributions involved, in terms of the afore- of (centralized) model predictive control (MPC). In this case, mentioned jump propensities, so that the latter can be selected the appliances are assumed to be connected to a communica- in order to control these quantities of interest. The resulting tions network and are able to receive and execute commands that closed-loop system can be shown to exhibit properties of local are generated by a central processing node. The approach is ap- asymptotic stability, regardless of parameter values, as well as plied to problems in which there is considerable supply variation boundedness of solutions for all initial conditions. due to the significant