Modelling of domestic hot-water heaterload from online operating records and

some applicationsM.U. Kobe. B.E.. and A.C. Tsoi. Dip.Tech., M.Sc. Ph.D.. B.D.

Indexing terms: Modelling, Load and voltage regulation, Management

Abstract: In the paper, the problems involved in modelling domestic hot-water heater loads are discussed. Thehot-water heater load modelling problem is divided into two parts: load shedding and load restoration. Byconsidering the 'instantaneous' load recordings during normal day-to-day operations of the system, and themeasurements obtained from a small representative area, the parameters describing the load shed and loadrestore models are obtained. Two possible applications of the models are described. The first is to apply themodel for reconstructing the uncontrolled load. It is shown, by comparison of the reconstructed uncontrolledload and the uncontrolled load during days when no load shed actions are necessary, that the reconstructeduncontrolled load curve is reasonable. The second application is to use the model in actual load managementconsiderations.

1 Introduction

In New Zealand, electricity generation and supply are con-trolled by two separate groups of agencies. Electricity isgenerated by a central government agency known as theNew Zealand Electricity (NZE) Division of the Ministry ofEnergy. The supply of electricity to domestic and indus-trial consumers is controlled by 66 local-authority typeagencies, in the main known as Power Boards. These pur-chase power in bulk from the NZE and retail it to con-sumers in local districts.

The bulk tariff charged by the NZE includes a unitcharge and a maximum demand charge (for demand takenas the average load over a half-hour period). Themaximum demand charge is based on the average of thethree highest demand values throughout a chargeableperiod. Of the two chargeable periods per year one isApril to June, and the other July to March. Most PowerBoards have some form of load control mechanisminstalled — mainly ripple control of domestic hot-waterheater cylinders — which they can operate either manuallyor automatically. The aim of load shedding is to reduce thepeak demands during each chargeable period, to belowsome suitable 'maximum demand level'. A target value forthis level is generally set at the start of a chargeable period,but may need to be lifted if it is found that the load cannotbe kept below this target 'holding level' using the availableload shedding capacity. To limit the amount charged forthe maximum demand the Power Boards tend to exerciseactive load control over most of the winter months.

The domestic hot-water heater load is often arranged asa number of separately controllable channels, also knownas load channels. The number and capacity of the loadchannels varies for individual Power Boards, and the con-trollable hot-water heater load tends to constitute 10-20%of a Board's total load.

In the control room of most Power Boards, the oper-ating records kept routinely include:

{a) the average half-hourly load data (i.e. the supplied orcontrolled system load),

(b) the load shedding records (i.e. of the times load chan-nels were shed and restored)

Paper 4788C (P9, P5), first received 25th November 1985 and in revised form 30thApril 1986

Mr. Kobe is with the Department of Electrical and Electronic Engineering, Uni-versity of Auckland, Auckland, New Zealand. Dr. Tsoi is with the Department ofElectrical and Electronic Engineering, Faculty of Military studies, Royal MilitaryCollege, Duntroon, ACT 2600, Australia

336

(c) the average minute load data (most Power Boardswould have chart recorders recording the average oneminute or two minute load data, while others may haveautomatic data logging. Here it is assumed that oneminute average load values are available).

The controlled load values form the basis for billing pur-poses, whereas the load shedding records are kept mainlyfor operational reasons. The minute load values are oftennot kept permanently, but can be retrieved if needed eitherfrom chart recordings or computer files, respectively.

The problem this paper addresses is how to model theresponse of load channels to being shed and later restored.Such a model then makes it possible to try and reconstructwhat the system load would have been had there been noload control. This reconstructed uncontrolled load curvethen approximates the load demand on the system, in con-trast to the load actually supplied.

Knowing the uncontrolled load curve leads to a betterunderstanding of a system's loading. If the uncontrolledload curve were available over a number of years, it couldindicate the way the system load has changed, which isimportant for medium-term system planning. The recon-structed load curve is also of use if a Power Board intendsto investigate alternative load shed/restore strategies,which could result ultimately in an improved operation ofthe system. And even if the currently used method of per-forming load shedding is considered satisfactory, it may bean advantage to have a clear picture or model of howmuch load a load channel contributes to the system as it isshed or restored.

The reconstruction of the uncontrolled load curve fromthe controlled load and other measurements has been con-sidered by a number of utilities in the USA in recent years[1-6]. The growing interest comes in part from investiga-tions into the possibility and viability of active loadcontrol. (In contrast, direct hot-water heater load controlhas been a standard practice for many years in NewZealand).

Thus, in Reference 1, the authors describe how theydesigned load management experiments. Being in NorthAmerica, their deferrable load comprises both air-conditioning as well as the storage water heating, and theyalso consider the control of the service voltage. (Neitherair-conditioner nor service voltage-control are widely con-sidered as load control options in New Zealand.) Theauthors have not indicated how they obtained the seasonalwater-heater loading against time of day curves, nor

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

have they detailed their load channel model. Nevertheless,the hot-water usage pattern curve obtained in the presentstudy seems to agree with the ones they obtained. In Refer-ence 2, the authors considered hot-water heater channelmodelling. By using a set of 200 water heaters, theyobtained field measurements of typical hot-water usagepatterns. From these measurements they were also able todetermine the average net load that is restored when achannel is turned back on, as a function of the energydeferred while the channel was shed. The modelling report-ed in this paper is based more on operating records, andresults in a somewhat different load channel model. Refer-ence 3 gives an account of experience gained in using loadcontrol in Wisconsin. Although it is not reported explicitlyhow their channel model was constructed, both their hot-water usage curve, and their load channel model aresimilar in shape to the ones found in this study. References4 and 5 report further experiences in direct load controlfound in practice for US utilities and Reference 6 presentsa more theoretical simulation model for a hot-water heaterchannel.

The contributions of the present paper are that first, wedescribe how we obtain a daily channel loading patternand a load channel model from operating records and asmall set of specially performed experiments. By such amethod the expense involved in specially wiring-up hot-water heater cylinders for experimental purposes can beavoided. Secondly, by using the derived load channelmodel, an uncontrolled system load curve can be recon-structed. The reconstructed uncontrolled load is found tohave the same characteristic shape over a day as does theload measured on days with no load shedding. Thirdly, asa possible application of the load channel model thusderived, a set of guidelines is postulated that could help asystem load controller in manually managing the system'sload.

In Section 2, the difficulties involved in the reconstruc-tion of the uncontrolled load are described, and in Section3 the formulation of a load channel model is considered. InSection 4, an account is given of some preprocessing of theload records that is needed before the information theycontain can be used. Sections 5 and 6 consider how theparameters of the channel model can be evaluated, and inSection 7 the derived model is used to obtain reconstruct-ed uncontrolled load curves. Section 8 gives a descriptionof a possible application of the derived model.

2 Difficulties encountered in the reconstructionof the uncontrolled load curve

Many difficulties are encountered in the reconstruction ofthe uncontrolled load curve. The predominant difficulty isthe lack of detailed information on the system. Mostpower systems evolve over a long period of time. Becauseof different requirements over this period of time, the infor-mation that is kept concerning the system varies. In thesystem under study, domestic hot-water heaters arearranged in groups of separately controllable channels.However, the numbers and the mix of heating elementsizes that make up the load on each of these water heaterchannels are not known in detail. It would require amassive effort to collate this information now and furtherwork would need to go into keeping the information up todate. (In earlier years heating elements of 1 kW or lesswere installed. But as society has become more affluent,the requirement for more hot water has meant the install-ation of larger heater elements in renovated or newerhouses. In more recent years, the popularity of spa pools

has added to the installation of yet larger heating elements.Hence the numbers and the mix of heating element sizesare continuously changing quantities.)

This uncertainty only adds to the difficulties of model-ling the load channels accurately. The response of theentity 'hot-water heater channel' — the aggregate of largenumbers of hot-water heater elements each separatelyunder thermostatic control — to channel shedding(disconnection from the supply) or channel restoration(reconnection) is complex, and is far from just being a con-stant load that can be switched off and on instantaneously.At the time a channel is shed each individual heatingelement will either be on (water temperature in cylinderbelow thermostat set temperature, and water beingheated), or off (water temperature above thermostat settemperature, and no heating of water occurring). Duringthe period the channel remains disconnected no heating ofwater occurs, and the water temperature in all thechannel's cylinders drops. For cylinders where hot water isused during this period the water temperature drops by anamount dependent on the proportion of the cylinder's hotwater that has been drawn off. Some of the temperaturedrop will also be due to losses resulting from imperfectinsulation of the cylinder and the associated piping. Whenthe supply is then restored to the channel more cylinderswill have water temperatures below the thermostat setlevel than at the time of channel shedding. More load isrestored than was shed. A larger number of heating ele-ments are then on than the number that would undernormal diversity be on at that time of the day, on that dayof the week and in that season. This 'additional' load willremain on the system until such time as much of the wateris heated to above the thermostat set temperature again inindividual hot-water cylinders, and until natural diversityis restored. The time this will take for an individual cylin-der depends critically on that cylinder's capacity, heatingelement wattage and hot-water draw-off pattern during thechannel off time.

Thus it becomes apparent that the 'transient response'of the 'load channel', the aggregate of all the differentresponses of the individual cylinders, is highly complex,and difficult to model. This in turn renders the reconstruc-tion of the uncontrolled load curve a very difficult task.

3 Load channel modelling and the formulation ofthe problem

If a load channel could just be modelled as a constantload, only one value would need to be found to have amodel for the channel. Once the load of a channel wasknown, its response to switching on or off would also bedefined. Being a constant load, the amount of load re-stored would equal the amount shed. Also the amountshed would be the same whatever the time of day. Themodel would be that of Fig. 1. With such a model recons-tructing the uncontrolled load curve from the controlledload data would then simply be an exercise of looking atrecords of when a channel was shed, and when it wasagain restored, and adding that channel's load to the con-trolled load data over the corresponding period of time.Fig. 2 shows a controlled load data curve, and the type ofreconstructed uncontrolled load curve that would typicallyresult using a channel model such as the one describedabove.

However, a hot-water channel is not a constant load,and a model assuming this does not reflect the character-istics of a load channel's response to switching. As dis-cussed earlier more load is restored than is shed on a

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986 337

channel, and also 'additional' load remains on the systemuntil much of the water, the temperature of which dropped

MW

load shed

load restored

timeFig. 1 Load channel modelled as a constant load

MW

uncontrolled load

timeFig. 2 Reconstructed uncontrolled load curve if a constant load model isassumed

during the time the channel was shed, is reheated. In effect,when a channel is shed, the reheating of the water is justdeferred until the channel is restored. The energy not sup-plied during the shed period is supplied once the channel isrestored (or at least a considerable proportion of thatenergy). It is because of this that storage hot water heatingis classified as a (largely) deferrable load. The uncontrolledload curve of Fig. 2 which was reconstructed using thenaive 'constant load' model does not show this deferral ofload or paying back of energy process at all, thus indicat-ing the unsatisfactory nature of such a load channel model.

More in line with what a reconstructed uncontrolledload curve would be expected to look like, is the oneshown in Fig. 3, which does illustrate how load sheddingcan be used to 'shave peaks' and 'fill valleys' in the dailyload curve by the deferral of the supply of energy. It will beshown that this type of reconstructed curve can result fromthe more complex, but more realistic model of a loadchannel that is shown in Fig. 4. It incorporates the

MWuncontrolled load

timeFig. 3 Reconstructed uncontrolled load curve assuming a more realisticload channel model

observed characteristics of more load being restored thanwas shed, and shows the energy pay-back effect.

MW

LS

LR h K^< F ( °

T

Fig. 4time

Load channel modelled more realistically

The main difficulty with this more complete model isdetermining the parameter values, as few of the quantitiesare measurable with any degree of accuracy. So some com-ments are in order about the way the model parametershave been interpreted in this study, and how they weredetermined:

(a) In the system under study the domestic hot-waterheater load appears to be sufficiently equally dividedbetween the main hot-water heater channels, to justify thesame channel model being used for each of these channels.

(b) As indicated earlier, the hot water usage, and hencethe amount of load on a channel at any time, varies withthe time of day, and also from day to day. This can beincorporated into the model by making the parameter LSa time varying one. Although LS has been taken to be afunction of time of day in this present study, it has beenassumed that this function remains the same from day today. Determining the changes in LS from day to daynecessitates the availability of a large database stretchingover a number of years, and this is not available. But evenif it were, probably not much in the way of further accu-racy could be gained. In the present study it is assumedthat the LS function does not change significantly over theday in the 2-3 winter months during which load sheddingis used on the system.

(c) The most significant parameter to find is T, the lengthof time the channel is switched off. Load shedding recordsgive the times of shed and of restore. However, even heresome inaccuracy results because the switching of a hot-water heater channel is not instantaneous. From the startof, for example, the initiation of load shedding, it may be 2to 3 minutes before all addressed heating elements areactually disconnected.

{d) The LS, LR and hence h values (as a function of timeof day) were found by working from the minute to minutecontrolled load values over the winter months, and analys-ing how the system load was affected when channels wereshed or restored. (Sections 4 and 5 cover this in moredetail).

(e) The shape of F(t) was ascertained from load sheddingand load restorations performed especially for the purposein a typical residential district (and this is discussed furtherin Section 6).

4 Preliminary considerations of the data

To determine how much the load drops or increases whena load channel is shed or restored, respectively, 1-minutecontrolled load data records were examined. Those datasections of interest were the load values just prior to,

338 1EE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

during and just after a load shed or load restore action. Inthe process of shedding it often happens that different loadcontrol actions follow each other in close succession. Insuch cases it becomes impossible to determine to whatextent a variation in the load is due to one load controlaction as distinct from the next one. Thus of all the loadshed and load restore actions in the operating records onlythose were taken into consideration that did not followone another too closely. Looking at individual cases in the1-minute load records for the system studied it appearsthat within about 10 minutes of a switching action theload values have 'settled down' again. Thus going throughthe operating records over three winter months for oneyear around 70 'useful' load shed instances were found andaround 160 'useful' load restore ones, i.e. switching actionsseparated from others by 10 minutes or more.

However, even for these cases it was not directlyobvious how to assign a numerical value to the amount ofload shed (LS) or the amount of load restored (Li?). The1-minute load values do not form a smoothly varyingfunction. They fluctuate not only as a result of errorsintroduced by the metering method, but also because thesystem load varies with time. (The 1-minute values workedwith show a standard fluctuation of the order of ±5.5MW). By again looking at numerous individual cases ofswitching (a few of which are illustrated in Fig. 5), a stan-dardised method was arrived at allowing a computerworking through the 1-minute load records to calculateamounts of load shed or restored. The straight lines shownsuperimposed on the load plots of Fig. 5 are estimated pre-

load sheddingpre - load - shedd i ng i post-load-shedding

LS= 28MW

z L--

LS= 30 MW

L5= 42 MW

1 3 Atime, min

Fig. 5 Estimating from actual 1-minute system load values, the rate ofincrease of load just prior to and just after load is shed

MW

pre-load-shed ^ ^ .slope ^ ' ^ ^

^ \

loadshed

averageslope

averageslope

post-load-shedslope

time

Fig. 6 Further details of how amount of load shed is determined

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

shed and post-shed slopes. The average of the two is takento approximate the rate at which the load in the system isvarying at the time, and is used to estimate the amount ofload shed. This procedure is detailed further in Fig. 6. Inestimating the amount of load restored a similar approachis used, with the exception that pre- and post-restoreslopes are not averaged. The reason is that the post-restoreslope does not give a true picture of the natural rate atwhich the load on the system would be changing. (Thepost-restore slope also has superimposed the decayingeffect of the additional load due to energy pay back.) Themethod of finding LR is indicated in Fig. 7.

MW

pre-load-restore"1

slope

post -load-restoreslope

loadrestored

time

Fig. 7 Details of how amount of load restored is determined

By using this standardised method on the 'useful' loadshed and load restore cases, a database of LS and LRvalues, respectively, could be built up. The preprocessingmethods described here are specific to this present system.However, some form of preprocessing of this kind willalways be needed if LS and LR values are to be deter-mined from records of load values averaged over short timeintervals (whether 1-minute, 2-minute or even 5-minute).

5 LS as a function of the time of day

The values of LS found for the 70 or so instances of loadshedding taken from the 1-minute load records (and chan-nels are always dropped in pairs) are plotted against timeof day in Fig. 8. Those values shown by a A symbol indi-cate that that pair of channels had already been shed

20

10

A

••S A

AA

A

A

AA

ffA&A

A

A

«^A

\

A

A

A

AA

A AA

h

A

>A

r&A

A

A

A

A

A\' A A|A*^A

8 1810 12 U 16time of day

Fig. 8 Plot of amount of load shed against time of day

before in the proceeding 2.5 hours. (A is used where a pairof channels had not previously been shed in the proceeding2.5 hour period). In analysing these data, the A cases canbe considered directly. The A cases, however, would beartificially high (due to the channels not having had timesince they were last restored to their natural diversities).Thus the A points would need to be lowered before ana-lysing them together with the A points. With this in mind,the cubic spline curve of Fig. 8 which was fitted to only theA points, can be seen to follow the general pattern in thedata despite the wide scatter in the points. (It is thought

339

that much of this scatter can be attributed to the difficultyin determining values of LS from the 1-minute data withany degree of accuracy. Some of the variation is also dueto data from over two months being considered. The hot-water usage pattern would be expected to change some-what over the weeks; however, it is being assumed thatthese differences are not major, as both months consideredare winter months.)

The fitted curve appears reasonable in that the diversi-fied hot-water heater demand as a function of time of day(which is essentially what Fig. 8 represents), is shown topeak twice daily, once in the morning and once in theevening. This corresponds to the times when the highestdomestic hot-water use would be expected.

6 Model of the response of a channel onbeing restored

Modelling the restoration of a channel is more complexthan modelling its shedding. More factors influence theresponse, more parameters are involved in the model, andof these only x may be measured with any degree ofaccuracy.

LS values could be determined from inspecting the 1-minute load records. So also can values of LR for individ-ual load restoration actions. However, plotting LR valuesagainst time of day is not particularly useful. LR is depen-dent strongly on hot-water usage during the channel offtime, but this is not only a function of what time of day thechannel is restored, but also a function of how long thechannel has been shed (i.e. of T). (These two seem to be themain factors affecting the value of LR, but there are otherssuch as outside temperature and day of the week.) Oneparameter that incorporates the variations in water usageover the day, as well as the time T, is E, shown as theshaded area in Fig. 4. It corresponds to the amount ofenergy not supplied to the hot water heaters while thechannel was shed, i.e. the deferred energy. One mightexpect that LR values would be related to E values insome proportional manner.

In Fig. 9 the LR values found are plotted against corre-sponding E values. The curve superimposed is a best least-squares fitted cubic, with the curve made to cross the LR

40

20

400 800 1200 1600 2000 2400 2800 3200E. MWmin

Fig. 9 Plot of amount of load restored (LR) against the correspondingdeferred energy parameter value (E)

fitted cubic95% confidence limits

axis at the point (0 MW min, 13.5 MW) as shown (the jus-tification being that if the channel were shed and imme-diately restored, LR would be approximately equal to LS,and 13.5 MW is round the average LS value). Again thevalues for LR are widely scattered, and for the samereasons as the LS ones.

In modelling a load channel restore action, the pro-cedure used here will be to calculate E and to then deter-mine the corresponding estimate of LR from this best-fitcurve. From LR and the value of LS at the time of channelrestoration, h can be computed in each case.

To complete the model, the function F(t), describing theenergy pay-back pattern needs to be defined. The neces-sary information was obtained from a number of experi-ments performed by the supply authority especially for thispurpose. Test sheds and restores were carried out on amainly residential district (having little industrial or com-mercial load) in the Authority's supply area. The district issupplied from one point of supply, and the load variationscould therefore be monitored by measuring the currentbeing supplied at this point.

The tests consisted of measuring the effects on systemload of shedding the hot-water heater load at 8.30 a.m. onvarious days and leaving it shed for differing lengths oftime. The water heating load was taken to be the differenceof the readings just before and just after a load switchingaction. The values measured are shown in Fig. 10. (TheseFigures also show that further sheds and restores werecarried out at 45 minute intervals after the initial restore.)

MWhot-water «

= 122mins time cFig. 10 Results of different runs of test sheds performed to determine achannel's response to switching

Only the first shed and restore in each case were usedin the further analysis. However, it is interesting to notewith the subsequent sheds, that the amounts of loadrestored are larger than the amounts of load shed evensuch a short time as eight minutes earlier, although there isalso the added effect of the energy deferred from earliersheds not having been fully paid back yet.

The function F(t) appears from these graphs to beapproximately of an exponentially decaying form. Todetermine a suitable expression for F(t) to use in the modelof Fig. 4, the first restore actions from each of Figs. 10a, band c were considered. They have been redrawn on Fig.11, where the curves have been arranged relative to eachother in such a way that on the time axis the times of therestore coincide, and on the load axis the pre-load shed-ding loads (base values) coincide.

(In Fig. 10 the load value just before 8.30 a.m. has beendrawn in on each graph as a constant base value. Ideallythe base value should correspond to the natural diversity

340 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

hot-water heater load, i.e. the load, had there been no loadshedding. According to the results of Section 5 this tends

MW

hot-waterheater load

time

Fig. 11 Observed load drop offs after the channel is initially restoredshown in more detail

to drop off towards lunch time. Except when shed, the hot-water heater load should never be lower than this naturaldiversity load. It is, however, lower at times in Fig. 10 if thebase load is assumed to remain constant at the initialvalue. In particular, Fig. 10a points to a base load valuethat drops off markedly over the morning. The constant-value base load assumption gives less and less accurateestimates of 'additional load' on the system because ofenergy being 'paid back', the further removed the time isfrom 8.30 a.m. The decaying curves of Fig. 11, as drawn,therefore all drop off too much, and particularly curve a —from Fig. 10a.

A number of variables and ratios of variables for thethree cases have been tabulated in Table 1. The variablesh, A, I and T/3 are indicated for case a in Fig. 11.

Table

Case

abc

1 : Variables determined from the test shed results

E, MWmin

81611641464

h, MW

11.714.116.4

T, min

6897

122

/, MW

5.57.48.1

A, MWmin l/h

172323476

0.470.520.49

A/E

0.210.280.33

In trying to find a reasonable time function F(t), toapproximate the curves of Fig. 11, decaying exponentialfunctions, quadratics and hyperbolic functions were allexamined. For each type of function values of l/h and A/Ewere calculated for curves fitted to those of Fig. 11. Theseratio values were then compared with the ones in Table 1to see how closely they approach the tabulated ones. Bythis method the hyperbolic functions were found to followthe required shape more closely than the quadratics orexponentials.

In fitting a function F(t), two points that need to beobserved are that

(a) the initial value at t = tR has to be h, i.e. F(tR) = h

(b) the area under the F(t) curve (which corresponds tothe amount of energy paid back) should equal a propor-tion P of the deferred energy £, with P < 1. (Estimatedvalues of P reported in the literature [1, 4, 6] vary widelyover the range 0.7 < P < 1.0, and differences betweenvalues in winter and summer have also been observed.)

These two conditions could be satisfied by an equation ofthe form

F(t) =K

(t -tR) (1)

and they enable the constants K and D to be evaluated.Note that this choice of F(t) has one shortcoming, in thatwith this model it would take infinitely long for all energyto be paid back. Using more general rational functionssuch as

F(t) =K

(t -tR)+ C or F(t) =

g(t -tR)

C(t -tR)

would avoid this difficulty; however, more unknownswould need to be evaluated. In view of the sparsity of mea-surements and the uncertainty of the data, the use of themore complicated functions is not justified. To resolve thedifficulty of energy taking an infinite time to be paid back,eqn. 1 can be slightly modified, by assuming that the pro-portion P of the energy E is paid back in a time T, andsetting F{t) equal to zero for any time t > T + tR, i.e.

K(t -tR)

0

t < t(2)

t > tR + TUsing this modified form of F(t), the two conditions to beobserved become

(a)

tR+T

F(t) dt = PE=>K\nT + D

D= PE

(3)

(4)

The constants K and D can thus be evaluated for a givenload restore action, once E and h have been determinedand values given to P and T.

For the present study no specific information was avail-able about how much of the deferred energy is not paidback, so P = 1 is assumed. Also T cannot be measureddirectly. Experimenting with different values of T (T = 4hours, 5 hours, 6 hours), it was found that using a value ofT = 4 hours gives the best fitting curves; best fitting in thesense that the ratios l/h and A/E of the fitted curves wereclosest to those in Table 1.

The values in Table 2 are for fitted curves of eqn. 1 andusing P = 1.0 and T = 240 minutes and are comparablewith those of Table 1.

In Fig. 12 the fitted curves are drawn superimposed onthe experimental ones for comparison. (With reference tothe earlier comments that the experimental curves, andespecially curve a, drop off too rapidly, note that the fittedcurves do not drop-off as steeply.) With this the last of the

Table

Curve

abc

2: Variables determined from curves fitted to test shed

E, MWmin

81611641464

h, MW

11.714.116.4

D, min

33.044.551.5

K, MWmin

386.2627.4844.6

/, MW

6.98.29.2

results

A. MWmin l/h

202343492

0.590.580.56

A/E

0.250.290.34

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986 341

model parameters indicated in Figure 9 has been con-sidered, and a method of modelling a hot-water heaterchannel load developed.

MW

timeFig. 12 Fitted curves superimposed on the observed load drop off curves

measured values— O — fitted curves

It is of interest to compare F(t) of this paper with loadrestore transient models found in studies carried out byother power system operators.

One such work is reported in Reference 2. They fittedF( ) functions by least-square curve fitting methods todata collected from load control field tests. These consistedof individually monitoring the loads of 200 hot-waterheaters by means of magnetic tape recorders. Instead ofmaking F(-) a function of time, they expressed F ( ) as a setof different functions of E, the deferrable energy; the struc-ture of the function may be different for different times ofthe day.

Three comments are in order:(i) To build up a sophisticated model such as this

requires a considerable amount of data from load controltests

(ii) There is an initial capital outlay for the wiring of thehomes

(iii) The choice of a particular set of 200 homes fromthousands would need to be a representative cross-sectionof the society served.

Another model is reported in Reference 4. Water-heaterpay-back curves as measured on the Southern CaliforniaEdison System are plotted. Different curves are indicatedfor different off times of the heating elements. Thus, F ( ) isa function of time, but it may take on a different shapedepending on the shed duration T. The value of the defer-rable energy E is incorporated into the model by relatingthe initial value of F(t), i.e. at the time of the restoration,to E. This is similar to the model developed in the presentstudy. However, in the present study the shape of F(t)curves does not change with T. The information presentedas in Fig. 11 is insufficient to allow different functions to befitted. In Reference 4 no details are given of the way thecurves are obtained, hence it is not possible to commenton how closely their curve for a 3-hour off time actually

compares with the measured curves for different valuesof£.

A curious aspect of their work is that their model isbased on experiments covering ' . . . a predominantly all-electric retirement community with approximately 18 250middle and upper income senior citizens ...'. This couldeasily lead to some distortion in the load shed channelmodels thus developed as the measurements performedmay not be a representative cross-section of the com-munity served.

In contrast with References 2 and 4, our model is not assophisticated. However, our load channel tests (as shownin Fig. 11) are expected to be reasonably representative ofthe average domestic loading.

The exact F ( ) model chosen depends critically on theparticular system and to what extent information is avail-able about the transient response of the hot-water heaters.The present study, together with References 2 and 4,present three different approaches to the modellingproblem.

In deriving the channel model, the past 1-minute oper-ating records were considered. However, cases where achannel was repeatedly shed and restored were explicitlyexcluded (e.g. the A points in Fig. 8 were not considered).In the actual practice of load shedding it is, however, quitecommon for a channel to be shed again within a relativelyshort time of having been restored. In the following para-graphs it is described how the developed model may beextended to cover the multiple shed and restore case.

To obtain an accurate model for this case, extensiveexperimentation would again be required. The complexitystems from the large number of independent variablesinvolved. Even just to model a second shedding of achannel, experiments would need to be performed for thedifferent possible combinations of the three variables:duration of the first shed, duration of the channel restoredperiod, and the exact time of day of the shedding and therestoration. Such extensive experimentation would beexpensive and time consuming.

On the other hand, the developed load channel modelcan be used in the case of multiple shedding and res-toration, but the following 'features' of a re-shedding mustbe taken account of:

(a) If a channel is re-shed before all deferred energyfrom the previous shedding has been repaid, more loadwill be shed than if it had been the first shedding. This isbecause the channel will not yet have been given enoughtime to settle down to its natural diversity for that time ofday.

(b) The previous shedding action also affects the re-restoration. If a proportion of the energy deferred by theprevious shedding action has not been repaid, this corre-sponds to the water in a proportion of the hot-waterheaters not having had time to be reheated to above thethermostat temperature. When the channel is then re-shedthe remainder of the deferred energy can only be repaidafter the channel is again restored, and this deferred energywill be in addition to the energy deferred during thechannel off time of the re-shedding.

Both of these points can be incorporated logically into thedeveloped load channel model by using the followingadjustments:

(a) The additional load that is shed for a second or sub-sequent shedding (in excess of the LS value that wouldnormally be expected for that time of day) is taken to bethe value of F(t) (modelling the previous restore transient)at the re-shedding time

342 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

(b) For second or subsequent shed/restore actions, LR,h and F(t) parameters are found as previously, except thatthe E value is modified, by adding to it the amount ofdeferred energy not yet paid back from the previous loadshed/restore action.

By this method, the uncontrolled load curve wasreconstructed over a winter for the system under study.Fig. 14 shows the resulting reconstructed uncontrolledload curve for one day together with the controlled loadcurve. The pay back of energy can be clearly seen.

F , ( t )MW

time

Fig. 13 Load channel model — detailing the parameters defining thechannel's loading under multiple switching

Referring to Fig. 13 and using the two postulated adjust-ments, the parameters for the second shed would be foundas follows: the amount of load shed is the LS value at tS2

plus the value of F^t) at time tS2 (i.e. LS2 + Fl(tS2) MW);the amount of deferred energy from the first shed actionpaid back by time ts2 is ElR, where

'1R dt = Kx InD i MWmin

the portion not yetthen the adjusted

paid back is (Et — E1R) MWmin, andE value for the second shed/restore

action would be E'2 — E2+ (£ : — ElR) MWmin, and LR2,h2 and F2(t) should be found as a function of the adjustedE'2 energy value.

Similarly for theof load shed at tS3

the deferred energy

third shed/restore action, the amountis LS3 + F2(tS3) MW; the amount of

paid back by time tS3 is E'2R, where

E'2R =J2R F2(t) dt = K2 InD

MWmin

the portion not yet paid back is (£'? — E'2R) MWmin andthe adjusted energy value to use in calculating LR3, h3

and F3(t) would be E3 = E3 + (E'2 — E'2R) MWmin.Note that in the calculations of all of the F,(t), i = 1, 2,

3 , . . . the assumptions made are that(i) it takes four hours to pay back the amount of energy

deferred(ii) the whole of the deferred energy is paid back.

In particular, the second assumption becomes less andless tenable the longer the period becomes over which loadshedding occurs in a day. For example, non-temperature-sensitive users of hot water (e.g. some automatic dish-washers and washing machines) use water from the hotwater tank at whatever temperature it happens to be at.With the heaters off for considerable stretches of time, thelukewarm water in the tanks will be used by these users,and the energy, that would have gone to make this waterhot, will not be repaid later.

7 Application of the toad channel model in thereconstruction of the uncontrolled load

Once the response of a hot-water heater channel to switch-ing can be modelled, it becomes possible to attempt toreconstruct what the load curve would have been hadthere been no shedding. Using the above modellingmethod the uncontrolled load curve can be reconstructed.The load shed values (negative values in the model of Fig.9), and the load restore values (positive values) need to besubtracted from the controlled load values.

MWreconstructed

uncontrolled load

U 8 12 16 20time of day

Fig. 14 Uncontrolled load curve reconstructed using the postulatedmodel

MW

8 12 16time of day

20

Fig. 15 Uncontrolled load curves (both measured and reconstructed) fora number of days

In Fig. 15, the uncontrolled load curves for a number ofMondays are shown. Those of weeks 15 and 17 and arereconstructed ones, as load shedding was used on thosedays. It is observed that the shape of the curves are fairlysimilar. There was no direct way to determine how accu-rately this method reconstructs the uncontrolled load;however, the similarity in load shapes was taken as confir-mation that the reconstruction method has some validity.

8 Application of the load channel model in theactual process of controlling the system load

The load channel model derived can be useful in situationsarising from controlling the system load. These include:

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986 343

(a) In the shedding of a channel: Once the decision istaken that load shedding is necessary to keep the half-houraverage demand below a preset holding level value, theload channel model may be used ao determine by howmuch the half-hour average demand can be lowered, andthis as a function of how soon after the start of the currenthalf-hour period the channel is shed.

From Fig. 8 it is possible to read off LS, the amount ofload on a channel in MW, at the particular time of daybeing considered. If a channel is shed tx minutes into thathalf-hour period, the half-hour average demand will belowered by

From knowing Lo it can then be checked whether shed-ding a channel at tt is expected to drop the half-hourdemand to below the preset value as required.

(b) In the restoration of a channel: Once again it willhave been decided that the system load has dropped suffi-ciently to make it safe to restore a previously shed channelwithout causing the half-hour average demand to over-shoot the preset target load. The load channel model canthen be used to determine by how much the current andthe subsequent half-hour average demand values would beraised as a result of a given channel being restored.

The first thing to be determined would be a value for E,the amount of energy deferred for the channel. This isfound from knowing the channel off time T and calculatingthe area under the LS against time-of-day curve of Fig. 8,over the appropriate period T (i.e. the shaded area shownin Fig. 16).

8 10 12 14time of day

Fig. 16 Illustration of how a value for E, the amount of energy deferredfor a channel, can be determined

From Fig. 10 the corresponding instantaneous load onrestoration, LR, may then be obtained, and from this thevalue for h from h = LR — LS. (Note that h is thus a func-tion of the time of day, as well as being one of E.) Fromthe values of E and h the unknowns K and D of the func-tion F(t) may be found by solving the pair of equations

and

E = K InT + D

D

with T given some predetermined value.F(t) models the transient response of the channel's

loading (over and above what it would have been had thechannel not been shed previously) on being restored. OnceF(t) is known it is possible to compute the additional half-hourly average demand for the current half-hour thatwould result from restoring the channel. If the channel isrestored t2 minutes before the end of the current half-hour,the additional half-hour demand (in excess of the value

had the channel not been shed previously) would be

D

InD

Similarly, the additional half-hour demands (in excess ofthe values had the channel not been previously shed) forsubsequent half-hour periods would be

ln

F(t) dt = -

t2 + 30 + D

D

lnD

30 12 + 30

t2 + 60 + D

E_ " \t2 + 30 + D30

lnT + D

D

and so on. (Note that the model assumes that all deferrableenergy E is paid back within a time T. Therefore in findingthe L; values, the upper limit integration may not exceed avalue of T.) From F(t) it is also straightforward to calcu-late what proportion, R, of the total deferred energy willhave been paid back by tr minutes after channelrestoration:

, n , ^

lnT + D~D

(c) In the controlling of more than one channel at thesame time: Where a number of channels are being simulta-neously controlled, the effects of the different shed/restoreactions on the half-hour average demand values are addi-tive, as would be expected considering that the load chan-nels are independent of each other.

(d) Where a channel is shed and restored a number oftimes in succession: As described in the final portion ofSection 6, an extension of the simple load channel modelallows this case to be dealt with also. The multiple shed-ding of a channel is more complex to model, but the levelof increased complexity is not unmanageable. The effect onhalf-hour demand values can be found by modifying theprocedures of (a) and (b) so they reflect the modificationsto the model.

Although the computations involved are of no difficulty toa computer, the information and the number of steps maybe overwhelming for a human controller. However, it maybe possible to tailor the above techniques for applicationto a particular situation in such a way, that the informa-tion presented can still be grasped by the human mind.For example, the information could be suitably presentedin the form of a set of graphs, and a programmable deskcalculator could be of use in performing the calculations.

9 Conclusions

This paper reports on a methodology used to reconstructan uncontrolled load curve, from the operating records ofa power system. It has been shown that by going throughthe 1-minute load records, and the hot-water heater shed-ding and restoration records for a supply system, it is pos-sible to find the approximate loading of a hot-water heater

344 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

load channel at a given time of day. (From this the averagedaily hot-water usage pattern may also be inferred.) Froma small set of experiments performed on a residential dis-trict with little industrial or commercial load, the typicalloading pattern (for the system under study) of a hotwaterheater channel in response to switching could be found, inparticular, the loading pattern on restoration of thechannel as the deferred energy is paid back.

These results enabled the modelling of a load channel'sresponse to switching, which is the basic requirement if theuncontrolled load curve is to be reconstructed from thesystem controlled load values.

The uncontrolled load curves reconstructed using thismodelling technique were found to have much the samecharacteristic shape as the load curves for the days whenload shedding was not necessary. This is taken as at least apartial validation of the model developed.

In addition, some possible applications of the channelmodels for load management considerations are discussed.It is shown how some simple computations may be per-formed to ascertain the amount of load shed and restoredin each situation. It is also noted that in the case of loadmanagement the usefulness of the computations showndepends critically on the ability to forecast the load a shorttime ahead accurately.

10 Acknowledgments

The authors wish to acknowledge the financial and otherassistance given by the Auckland Electric Power Board incarrying out the studies reported in this paper. In addition,the authors wish to acknowledge the constructive criti-cisms offered by anonymous referees on an earlier versionof this paper.

11 References

1 DAVIS, M.W., KRUPA, T.J., and DIEDZIC, M.J.: 'The economics ofdirect control of residential loads on the design and operation of thedistribution system', IEEE Trans., 1983, PAS-102, (3), pp. 654-665

2 LEE, S.H., and WILKINS, C.L.: 'A practical approach to applianceload control analysis: a water heater case study', ibid., 1983, PAS-102,(4), pp. 1007-1013

3 ABDOO, R.A., LOKKEN, G., and BISCHKE, R.F.: 'Load manage-ment implementation: decisions, opportunities and operation', ibid.,1982, PAS-101, (10), pp. 3902-3906

4 DAVIS, E.J.: 'Impact of several major load management projects', ibid.,1982, PAS-101, (10), pp. 3885-3891

5 HASTINGS, B.F.: Ten years of operating experience with a remotecontrolled water heater load mangement system at Detroit Edison',ibid., 1980, PAS-99, (4), pp. 1437-1441

6 RAU, N.S., and GRAHAM, R.W.: 'Analysis and simulation of theeffects of controlled water heaters in a winter peaking system', ibid.,1979, PAS-98, (2), pp. 458^*64

Abstracts of papers published in other Parts of the IEE PROCEEDINGSThe following papers of interest to readers of IEE Proceedings Part C, Generation, Transmission & Distribution have.appeared in other Parts of the IEE Proceedings:

Thyristor controlled reactors as harmonic sources in HVDCconvertor stations and AC systemsR. YACAMINI and J.W. RESENDE

IEE Proc. B, Electr. Power AppL, 1986, 133, (4), pp.263-269

Thyristor controlled reactors (TCRs) are a relatively newsource of harmonic distortion in power systems. Thesteady-state balanced (or characteristic) harmonics areeasily calculated and are well known. Other non-characteristic harmonics can, however, be generated byTCRs. A detailed representation is therefore necessarywhich will consider all types of imbalance, be it in the TCRor in other parts of the total system to which they areconnected. It is also important to consider the effect ofconnecting TCRs to systems which do not have a highfault level (weak AC systems) and to consider the effect ofcontrol feedback on harmonic generation. The methods bywhich these factors can be included in calculations of non-

ideal systems are the subject of this paper. The methodsare illustrated by example.

Novel type of high-frequency link inverter for photovoltaicresidential applicationsP. SAVARY, M. NAKAOKA and T. MARUHASHI

IEE Proc. B, Electr. Power AppL, 1986, 133, (4), pp.279-284

The paper demonstrates the possibility of utilising reson-ant convertor technology in the high-frequency linkinverter configuration. In this system, an amplitude modu-lated high-frequency sinusoidal waveform is generated by anovel type of series resonant inverter allowing electric iso-lation through a high-frequency transformer. A completedescription of the system is presented along with itscontrol technique for interfacing a solar photovoltaic arraywith the utility line. Experimental results are included.

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986 345