A physically-based computer model of aggregate ... - IEEE Xplore

IEEE Transactions on Power Systems, Vol. 9, No. 3, August 1994

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A PHYSICALLY-BASED COMPUTER MODEL OF AGGREGATE ELECTRIC WATER HEATING LOADS J.C Laurent

R.P Malhame

h o l e Polytechnique de Montreal C.P 6079 SUCC.A, Montreal, PQ H3C CANADA

models that could derive from individual physically-based models would inherit physical meaning and with it a better prediction Abstract capability. Updating such models as new physical information (weather for example or new device statistics) becomes available would Electric water heaters have been the focus also be greatly facilitated. We can identify of several previous studies because of their three problem areas that could directly benefit pervasiveness in power systems and their from such models : energy conservation consequent potential importance when analysis, construction of load management considering conservation through more efficient strategies and analysis of cold load pickup; design and operation of the heaters. Also, the latter is in view of the fact that electric because such devices are associated with an water heaters (as well as air conditioners or energy storage capability, they are often electric space heaters) tend to be responsible considered within load management by direct for prolonged system load transients upon power device control programs. Finally, they tend to reconnection after a power failure. In this paper, a physically-based computer be responsible for persistent system load transients in a cold load pickup situation. model of aggregate electric water heating loads Understanding of the above issues can be is reported. The model has been first derived greatly enhanced with the availability of a and partially analyzed by Malhame elsewhere in computer model of aggregate electric water the mathematical literature [12]. However, a heating loads. A physically-based such model is complete understanding of its dynamics can only presented and its dynamic properties are be obtained through numerical simulation. This investigated via numerical simulation under is our objective here. The model is derived various operating conditions and parameter following a statistical end use load configurations. The results are analyzed in the aggregation methodology first proposed in [8], paper. and subsequently refined and articulated in [ 9 ] . In this load modeling methodology, loads are first separated into groups which are I. Introduction fairly homogeneous from the points of view of Electric water heating loads have been the their nature (refrigerators, air conditioners focus of several previous studies where the . . ) , size, user class (residential, commercial water heaters are viewed either as an energy ..), geography (same weather conditions, consuming device with energy conservation i>:ientation . . ) and usage statistics (simil objectives in mind [ l 3!, or as an energy usage levels at given time of the day). In each storage device to be considered within peak- homogeneous group of loads, a so-called shaving by direct device control programs [ 4 - elemental load model which is constructed from 61. While we have discovered in [l-31 a wealth physical considerations is specified. This of useful information on the physical analysis model is typically stochastic so as to account of individual electric water heaters, the for the random nature of device usage. The nature and statistics of residential demand for elemental load model acts as a representative hot water, as well as in [ 4 - 7 ] very interesting for the homogeneous group of loads it belongs ideas on how to organize the control of such to (mathematically, it is a representative of devices for optimal load management purposes an equivalence class). Subsequently, the assuming their aggregate dynamic behaviour is aggregate behaviour of the loads in the well understood, we find a paucity of homogeneous group is obtained by using the analytical results concerning precisely this theory of stochastic processes to propagate the aggregate behaviour; this is particularly true mean behaviour of the particular stochastic when it comes to relating aggregate electric process corresponding to the elemental load water heating load behaviour to the parameters dynamics. Physical meaning at the aggregate governing individual electric water heating model level is thus preserved. load dynamics about which much is known. This modelling methodology has been Clearly aggregate electric water heating load successfully applied to heating and cooling loads (see [ll] and [12]). For related work, see [13] and [14]. 93 SM 496-0 PWRS A paper recommended and approved The paper is organized as follows. In section 11, we review a stochastic model of by the IEEE Power System Engineering Committee of individual electric water heating loads first the IEEE Power Engineering Society for presentation proposed by Chong and Debs in [8]. In section at the IEEE/PES 1993 Summer Meeting, Vancouver, B.C., Canada, July 18-22, 1993. Manuscript submitted Sept. 111, the solution of the aggregation problem 1, 1992; made available for printing May 7, 1993. for a large number of similar devices as first presented in [12] is reported. It consists of a system of coupled partial differential PRINTED IN USA equations characterizing the distribution of temperatures within the aggregated devices as a function of time. This is our aggregate model. .~ . 0885-8950/94/$04.00 0 1993 IEEE Keywords : water heaters, statistical load modeling,load management, cold load pickup

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Further analytical results obtained in [12] are summarized in section IV. In section VI the issue of identifying model parameters from physical data is considered. In section VI, using a numerical scheme reported in the appendix, power failures are simulated as well as post-interruption dynamics for various parameter values. Numerical experiments are accordingly designed to study the effects of each parameter separately. The effect of parameter spread in non homogeneous groups of devices is also studied. Section VI1 is the conclusion. 11. The Elemental Electric Water Heating Load Model

The following elemental electric water heating load model was first proposed by Chong and Debs in [ 8 ] . It is based on a linearized energy balance analysis. The following assumptions are made : - The water in the water heater is perfectly mixed at all times and thus it can be characterized dynamically by a single temperature variable. - Only one thermostatic element is active (out of possibly two). This is typically the thermostat located in the lower part of the water heater; note that in the very comprehensive Gilbert study [4] it has been reported that, under normal operating conditions, the upper thermostatic element is activated no more than 5% of the time. Both assumptions can be waived at the expense of a significant increase in complexity of the aggregate model. The necessity of that appears highly questionable at least at this stage of our analysis. The model comprises a continuous state x(t) (tank water temperature at time t) and a discrete binary state m(t) to account for thermostatic action. Continuous-state dvnamics :

setting x- and deadband A=(x+-x-). m(t) switches from 1 to 0 when m(t) reaches x+ and from 0 to 1 when m(t) reaches x-. No switching occurs otherwise. The modeling of the customer-driven hot water demand process qf(t) is a crucial but difficult step. It is clearly a non-stationary random process. In [8], piecewise-constant demand processes with random amplitudes and random switching times have been proposed. However, and to preserve the tractability of the model, a two state Markov chain with a possibly time varying amplitude was chosen in [12]. In this case, division of (1) by C yields

where q(t) is the \\normalizedI1 (1-0) hot water demand jump process and a, A(t), R are respectively, the thermal resistance of tank walls, the rate of energy extraction when water is in demand, and the power rating of the heating element, all expressed per unit of tank thermal capacity. The switching of q(t) is characterized by the following (possibly timevarying) transition probabilities Pr ( q (t+h)= l / q (t )=O) -a,$ (3) Pr ( q (t+h)= O / q ( t)=1)-a,h

where a i , i=O,l are positive and infinitesimal time increment.

(4)

h

is an

Equations (2)- ( 4 ) define our elemental electric water heating stochastic model. We now turn to the aggregation problem. 111. Aggregation of Elemental Loads

The aggregation problem for a homogeneous M-member control group of devices (identical statistics, identical parameters, identical control for all M water heaters in the group) can be stated as follows : U to describe approximately the dynamics of the quantity M

mi ( t)

P ( t)=b ( t)R' The variables in (1) are defined as follows

c: X,(t)

:

Xd:

a' : m(t) : b(t) : q'(t) : CIe:

RI :

tank thermal capacity (Joules / 'C.) ambient temperature (*c.)at time t desired outlet water femperature (depends on customer, in C . ) thermal resistance of tank walls (a function of water heater insulation in Watts/ C. ) thermostat binary state (1 for on, 0 for off) at time t the on-off control applied by the utility within a load management program (1 for on, 0 for off) hot water rate of extraction (m3/sec) at time t specific ?eat constant for water (1 joule/ m3 C) power rating of the heating element (Watts)

Discrete state dvnamics : The evolution of the discrete state m(t) is governed by a thermostat with temperature

(5)

i -1

where mi(t) is the operating state of the device i in the group and P(t) is the total electricity demand of the group at any time t For M large enough (over lo4 guarantees 1% accuracy), and under appropriate assumptions of independence of demand for water processes, one has approximately: -

In ( 6 ) , m(t) is the so-called 8\aggregate'f operating state and physically represents the average fraction of devices in the \'on' I state. In the homogeneous case, the average value of p(t) is proportional to fii(t). E,(t) represents the expectation operator conditional on weather information. Relation (6) viewed as an equality constitutes the essence of our aggregate load modelling methodology. In [12], it is shown that the evolution of the average fraction of on devices is governed by the interaction of two systems of coupled partial differential equations involving functions fij(x,t), i=O,l, j=o,l, hereby defined

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fi,(I,t)dI=Pr[Atx(t)~I+dI,m(t) = i , q ( t ) = j l( 7 ) i.e f..(I,t)dA represents the probability of a givenl'water heater to have its water in the temperature range (I,I+dI), its thermostat in state i (1 or 0) and the demand for water in state j (1 for demand present, 0 for demand absent). The probability density functions (p.d.f's) f..(I,t) satisfy the following coupled system of pe(rtia1 differential equations (geometrically represented in Figure 1 of [12])

for i=l, in the temperature regions I