- Email: [email protected]
Development of a Monte Carlo based aggregate model for residential electric water heater loads
ELSEVIER
Electric
Power
Research 36 (1996)
Systems
29-35
Development of a Monte Carlo based aggregate model for residential electric water heater loads P.S. Dolan’, De,~arruwt~r
of’ Elecrrictrl
M.H. Nehrir,
Engineering, Received
Montana
16 May
Slate
1995: accepted
V. Gerez
Chiversiry, 3 August
Bownan,
XI T 59 71 7, l’SA
1995
Abstract A model for a residential electric water heater load is developed using an energy flow analysis of the water heater tank. An aggregate model for residential electric water heater loads is then developed using a rejection type Monte Carlo simulation technique. The resultant aggregate model is then used to assess the effectiveness of several demand-side management strategies using the water heater load profile presented in a previous analysis. Kqxmds:
Water
heater
modeling:
Monte Carlo simulation; Demand-side management
1. Introduction Due to the increased interest of utilities in load management strategies for such things as load peak shaving, valley filling, and cold load pickup prediction, there has been an increased emphasis on residential load (appliance) modeling. One of the appliances which has been of particular interest to utilities and researchers alike is the residential electric water heater. Utilities have historically relied upon pilot testing programs to assess the impacts of water heater demandside management (DSM) strategies. While the pilot testing programs have provided invaluable data on the aggregate response of water heater loads, performing such testing is expensive and time consuming. Pilot testing could be avoided by using an aggregate water heater load model to assess the load response to DSM strategies which manipulate water heater operation. In the literature. what is typically referred to as an aggregate electric water heater load mode1 is an attempt to mathematically describe the operation of the water heater loads in a particular system by statistical curve fitting techniques [I -41. Another approach for developing an aggregate load mode1 for water heater operation would be to mode1 individual water heaters and add all ’ Presently with SSR Engineers. Bismarck, ND 58501, USA.
Inc..
037%7796~96~S15.00
Science
SSDI
037%7796(95)OlOl
P 1996 Elsevier I-B
4205
State
Street.
S.A. All rights
reserved
of their power demand profiles to obtain the overall system water heater load profile. This approach is similar to that used by the Electric Power Research Institute (EPRI) in its WATSIM program which allows the user to aggregate up to 300 water heaters using 300 stored water demand profiles. Using this kind of approach, the control of the heating elements of the individual water heaters being modeled may be manipulated in any manner desired to simulate various DSM strategies. In this paper we propose using a discontinuous random process for the hot water use of each of the modeled water heaters instead of predetermined water use profiles. This approach results in an aggregate water heater response that closely matches the overall system water heater load profile. While the approach does not correctly represent the actual load profile of each water heater on the system. it does provide a close representation of the system’s water heater load profile. In order to obtain the aggregate water heater load for a power system, it is not necessary to accurately model every single water heater in a particular system as the individual water heater characteristics tend to be canceled out when applied in an aggregate manner. All that is required is that the genera1 characteristics of the water heaters in the system be modeled. It is possible to statistically represent the overall mode1 parameters of the water heaters in the system as being normally
distributed with associated means and standard deviations. Once the means and standard deviations of the various water heater parameters such as size, element rating, and insulation level are known for a particular system, a rejection type Monte Carlo method [5] may be used to generate the model parameters for any number of water heaters. With this information and an acceptable water heater model, it is easy to develop a computer model to simulate the aggregate load profile of a large number of water heaters. This paper discusses the development of a single electric water heater model as well as an aggregate model for water heater loads.
Ci;,(t)
= - SA(lIR)[T,(t)
- T,,,]
- 8.3 ~n(~K’,[T,(f) where C is the equivalent the tank given by
C= 8.3 x (no. of gallons) x CP Letting G= SA(l/R) and B(t) = lVD(t) x 8.3C,, differential equation becomes C’fdt)
= - G[T,(t)
- Tc,“J - B(t)[TH(t)
WliRU’df)
- r,,,l
(1)
where SA ii R
T”(I) T ““t
tank surface area, SA = 2r(r+h) (ft’) tank radius (ft) tank height (ft) thermal resistance of tank insulation (h ft2 “F/Btu) mean tank water temperature at time (“F) outside ambient temperature (“F) t
The water demand losses (in Btu/h) as given by Ref. [7] are 8.3 WDif)Cp[TH(f)
-
Tin]
(2)
where 8.3 is the weight, in lb, of 1 USgal. of water, and PVD(t) water demand (USgal/h) specific heat of water, 1 Btu/(‘F lb) CP temperature of incoming make-up water (“F) T,, The power input to the tank (in Btu/h) is Q(t) = 3.4121 x 10” x (element kW rating) The general differential model is
(3)
equation of the water heater
th.:
- TiJ + Q(f) (51
It can be shown that a solution of Eq. (5) is
+ [GR’T,,,, + B(r)R’T,,
By summing the energy sources and losses in the water heater, one can develop a general differential equation describing the mean or average thermal response of the water in the tank. As indicated in Ref. [6], water heaters have two sources of energy losses: (i) thermal convection or ‘standby losses’ where heat is lost through the tank’s exterior walls to the surrounding environment; (ii) ‘water demand energy losses’ associated with the use of the hot water stored in the tank. The energy supplied to the tank comes from the heating elements. The convection losses (in Btuih) as given by Ref. [6] are
(4)
thermal mass (in Btu/“F) OF
T,,(t) = T,(s) exp[ - (l/R’C)(r 2. Water heater model development
- T,l + Q(t)
x (1 -exp[-(l/R’C)(t-t)]}
- r)] + Q(t)R’]
(6 ’
where R’ = l/[G+ B(t)]. At this point it should be noted that the terms Q(t) and B(t) in Eq. (6) are piecewise continuous [8,9]. Recall that Q(t) is controlled by the water heate,, thermostat and only becomes nonzero when the tern. perature falls below the lower thermostat setpoint. Af. ter the temperature of the water surrounding the thermostat reaches the thermostat’s upper temperature: setpoint, the heating element is shut off and Q(t) gee:; to zero. The B(t) term, which is associated with the water demand at any time t, is only nonzero when there is a demand for hot water. Because of the piecewisc! continuous nature of Q(t) and B(t), every time either one changes from one state to another, the initial condition term TH(r) in Eq. (6) must be modified fol. the new condition. Also, r must be set to the time t a. which the change of state occurred. In order to simulate the load profile of a large number of water heaters, it is necessary to obtain thts parameters describing each of the water heaters to be simulated. As previously stated, it is not necessary tcl know the actual parameters for each of the water heaters on the system to simulate the system. All that i:. needed are the mean and standard deviations associatec with a few of the key parameters presented in the single water heater model to describe the operational charac teristics of the aggregate water heater system. Once the means and standard deviations are known, a Monte Carlo rejection type method may be used to genera& the parameters needed in order to run the aggregate water heater simulation. The means and standard devi. ations are needed for the following parameters associ ated with the water heaters on the system to be simulated: Tank capacity (USgal) Thermostat setpoint (“F) Thermostat deadband (“F) Hot water usage rate (USgallh)
P.S.
Dolun
et al. : Ektric
Po~vrr
Thermal resistance R of tank insulation (h ft’ “F/Btu) Heating element rating (kW) Temperature of incoming make-up water (“F) Ambient temperature of tank location (“F) The lower thermostat setpoint is obtained by subtracting the thermostat deadband from the upper setpoint. The initial water temperature in the water heater tanks is generated as a uniformly distributed random variable between the thermostat’s upper and lower temperature setpoints. If we assumethe water heater tanks on the system are cylindrical in shape and that the radius of the tank is one fifth of the tank height (which is a good assumption), we can write the following: h = (0.13368/0.04)“’ SA = 0.48/z’
(ft)
(7)
(ft’)
(8)
3. The Monte Carlo rejection method
Slsrtws
Reseuwll
36 (1996)
29-35
31
water demand profile in USgal/h for each water heater being modeled. The random process developed will result in an aggregate water heater load that closely matches that of the system. For an accurate match of the system water heater load profile, a good knowledge of the aggregate water heater load profile in kW is needed. In general, knowledge of the system water heater load profile will not be available. Obtaining accurate information about the system water heater load profile is only possible through monitoring the power demand of each water heater. However, the random process may be generated using an approximation or best guess as to what the water heater load profile is. It should be noted that the more accurate the information on the system water heater load profile, the better the random process generated will allow the simulation to mimic the system water heater load profile. Once the system water heater steady-state load profile is known, the steps used to develop the random process describing the hot water demand profile are straightforward and can be summarized as follows.
The steps required for using the Monte Carlo rejection method for generating water heater model parameters with a normal distribution function with a mean .? and a standard deviation cr are given below. The reader is referred to Ref. [5] for a formal mathematical definition of the process. Step 1. Generate a random number x which is uniformly distributed in the interval .f - 3a < x < R + 3a. Step 2. Form f(x) = (l,!a&)exp[
-(x
- X-)2/2cr’]
(9)
Step 3. Generate a second random number which is uniformly distributed in the interval 0 y, then keep x becausex lies under the normal distribution curve with mean .? and standard deviation cr. If f(x)
I
4. Developing the hot water use random process Here we develop a random process describing the hot
Fig.
I. Flowchart
describing
End
the Monte
Carlo
rejection
method.
Fig. 2. Water heater experiment [IO].
operation
curve
from
Athens
load
control
Step 1. Make a plot of the percentage of water heaters on versus time of day by dividing the system water heater load by the product of the mean water heater element kW rating and the number of water heaters on the system (Fig. 2). Step 2. Fit simple geometric shapes such as parabolas, ramps, and straight lines to the percentage of water heaters on versus time of day (water heater operation curve of Fig. 2) to create an approximate ‘smooth’ representation of the curve as shown in Fig. 3. Note that we are only concerned with the portions of the geometric-shape curves that fit the water heater operation curve; there could be more than one set of geometric curves that fit the operation curve. Statistical curve fitting techniques can also be used to obtain equations describing different portions of the operation curve. However, considering the random nature of hot water usage, the approximate geometric-shape curve fitting is used here.
Step 3. Obtain the portion of the geometric share over each of the various time intervals which best fils the curve. Note the time intervals associated with each portion of geometric shape obtained above along with the peak percentage of water heaters on for each tim’e interval. Step 4. Normalize the equation for each of the geometric shapes to the peak percentage of water heaters on during their respective time intervals. The equations for each of the dominant geometric shapes should be written to be active for the time of day that they are valid. During the simulation process for a particular water heater, a uniform random number is generated in the interval between zero and one. The random number is then compared with the peak percentage of water heaters on during the time interval in question. If th\: random number is less than or equal to the percentagof water heaters on during that time interval, then th.: hot water usage rate for that particular water heater is multiplied by the value obtained from the active gee.metric equation to obtain the instantaneous water demand rate. If the random number is greater than th.: percentage of water heaters on for that time interval, then no hot water is being drawn. While this method OI determining the water demand rate would in no wa:: accurately model the hot water use by any given household, it is extremely effective in giving a close represen., tation of the aggregate water heater load on the system To further clarify the steps associated in developing the random hot water demand process, we will follov., through with an example. Fig. 2 shows the percentage of water heaters on versus time of day for the Athen load control experiment [lo]. We will now proceed tc develop the random hot water demand process for thL figure.
I---
ST
ea?!3
1a-
P ?II-
1
23
4
Rs-1
Fig. 3. Simple
geometric
shapes fitted
to the curve
of Fig. 2
Fig. 4. Dominant geometric for the curve of Fig. 2.
curves
over their respective
time intervalr
P.S. Dolan Table 1 Normalized Curve
equations
for the curves
no.
Equation
I 2 3 4 5
y(t) y(r)
y(t) y(t) y(t)=
= = = =
et al. 1 Electric,
Power
Table Water
2 heaters
of curve
Curve
no.
1 1 -(1:2)(r-7)” 1 -(l/30)(/-9)’ 1.7107-0.06607t 1 -(1:36)(r-20)’
IF 6.00
1 2 3 4 5
End IF
The remainder of the process consists of generating a uniform random number in the interval between zero and one. The random number is then compared with the peak percentage of water heaters on (in decimal form). If the random number is less than or equal to the peak percentage of water heaters on for the time interval under consideration, the instantaneous hot water demand of each water heater is given by multiplying the hot water usage rate for the water heater by the value obtained from the active geometric equation. Otherwise, the water demand is set to zero.
simulation
Resrurcll
of Fig. 4
We fit simple geometric shapes to the percentage of water heaters on versus time of day curve. In this case the curve of Fig. 2 can be fitted well with three inverted parabolas, a horizontal line, and a sloping line, as shown in Fig. 3. Fig. 4 shows the dominant portions of the curves over their respective time intervals. The normalized equations for the curves are shown in Table 1. The peak percentage of water heaters on in each interval shown in Fig. 4 is given in Table 2. Next, we write the equations for the geometric shapes so that they are activated only during the time of day that they are valid. One way to do this is to imbed them into IF.. .THEN type statements such as:
5. Computer
Systems
and results
In this section the components of the aggregate water heater load model discussed above will be put together to form the algorithm used to generate the aggregate water heater load profile. The steps outlining the algorithm are as follows. Step 1. Enter the initial time the simulation begins, length of simulation, number of water heaters simulated, and the means and standard deviations of the various model parameters. Step 2. Convert the simulation start and end times to the time of day in hours and tenths of hours. Step 3. For each water heater, generate the model parameters using the Monte Carlo rejection method outlined in Section 3.
36 (1996)
29-35
on in each interval Time
interval
1:OO-6:00 6:00-7:30 7:30- 11 :oo 11 :OO- 16:30 16:30-1:00
33
of Fig. 4 Peak % on 8 32 27 I9 22
Step 4. Form the remaining water heater model parameters from those generated in the Monte Carlo method. Step 5. Loop through Steps 6- 16 to simulate the operation of each water heater. Step 6. Loop through Steps 7- 15 for the simulation time interval. Step 7. Find the instantaneous hot water demand for the water heater being simulated using the method of Section 4. Step 8. Check the instantaneous hot water demand to see if it has changed from the previous hot water demand. If it is different, then update the water temperature initial condition term and set it equal to the simulation time. Step 9. Update the parameters B and R’ in Eq. (6). Step 10. Compare the tank water temperature with the thermostat’s lower temperature setpoint. If the water temperature is lower than the lower temperature setpoint, then activate (turn on) the heating element. Also, update the water temperature initial condition term and set it equal to the simulation time. Step II. Compare the tank water temperature to the thermostat’s upper temperature setpoint. If the water temperature is higher than the upper temperature setpoint, then disable (turn off) the heating element. Also, update the water temperature initial condition term and set it equal to the simulation time. Step 12. Find the new tank water temperature using Eq. (6). Step 13. Record the heating element power demand in kW. Step 14. Compare the simulation time with the simulation end time. If the simulation time is equal to the end time, go to Step 16. Step 15. Increment the simulation time and go to Step 7. Step 16. Check to see if the operation of all of the water heaters has been simulated. If it has not, then update the water heater parameters to those of the next water heater and go to Step 6. Step 17. Loop through all the time increments and add the power demand of all the water heaters simulated for each respective time interval. Step 18. End.
34
P.S.
0:0
2
Fig. 5. Simulation water heaters.
4
6
6 10 12 Tme~lDay(24H01~~)
of the diversified
14
daily
Dolan
16
et ul. / Electric
1s
20
operation
i2
Power
24
400 360
Fig. ments.
U
6. Aggregated
4
6
electric
8 10 12 14 Tu-cadDay(24i-kurCkdc)
water
response
16
i6
36 (1996)
.,
;
29-35
a
, 10
....
9. keofDay(24HarC!&o
11
12
of 100 electric
450
2
Research
2Q6
A computer program was written using the above algorithm to simulate an aggregate water heater load for a system comprising 100 water heaters. The program can be run on a 386 or better IBM-compatible personal computer. The random hot water demand process used in the simulation was that developed in Section 4 for the Athens load control experiment profile [lo]. Fig. 5 shows the simulation result for the percentage of water heaters on at steady state, which closely matches that shown in Fig. 2. The aggregate water heater load was then simulated for a condition in which all of the heating elements in the water heater system were disabled for one hour during the morning and evening peak load times. The system power demand can be seen in Fig. 6. The peaks of the payback curve in the simulation have the same shape as those seen in Ref. [lo], but are higher in amplitude. One explanation for this discrepancy is that when there is a drastic reduction in water temperature, people typically stop using hot water. This modification of hot water use behavior can be incorporated into the water heater model by adding a term which sets the instantaneous hot water demand term to zero when the water temperature falls below a certain level. Several demand-side management strategies were simulated using the aggregate water heater load model
Oo
Systems
ti
to disabling
22
24
heating
ele-
8
9
10
Trc.adDay(24Hwaoa))
‘lo! ,..,,
20‘ 6
7
9
,,,,,,,_,_,,. ...,.,,,, ,,,,,,,,,,,,,,,,, 11
.._. j 12
TkMhy(24Hour&,
Fig. 7. Aggregated electric water heater response to cycling 20% of the water heaters off (a) every 10 min, (b) every 20 min, (c) every 30 min.
to simulate direct water heater load control for reducing peak load. Different percentages of water heaters were disabled for different time intervals and simulation results were compared. Among the DSM strategies simulated, 20% of the water heaters’ heating elements were disabled for lo-, 20-, and 30-minute time intervals. After the time interval specified had elapsed, the heating elements of the water heaters that were disabled were allowed to activate normally and a new set of water heaters’ heating elements were disabled. This process was repeated for the entire simulation time. Simulation results for this control strategy are shown in Fig. 7. In this figure the total water heater power demand when all the water heaters are in operation is compared with that when the DSM strategy (water
heater cycling) is in effect. As can be seen from the simulation results, a proper selection of the percentage of water heaters to be cycled and a proper cycling time can reduce the peak load. As shown in Fig. 7(b), cycling 20% of the water heaters (under study) off every 20 minutes reduced the peak power demand by about 25%. However, no improvement of the ability of the DSM strategy to reduce the peak load is seen once a certain cycling time is exceeded for each of the percentage cycling blocks (see Fig. 7(c)). In fact, the DSM strategy can cause an increase in the peak load observed for the larger off-time intervals. This is due to the larger payback curves associated with larger offtime intervals.
6. Conclusions A model for simulating aggregate water heater loads was developed in this paper. A methodology was presented for generating the model parameters for a large number of water heaters to fit the statistical characteristics of the water heaters in any system. In addition, a method for generating a random process for the instantaneous hot water demand for the water heaters being simulated was presented. The model developed was then evaluated using computer simulations. The usefulness of the aggregate water heater load model developed was shown in evaluating a rotational type of demand-side management strategy for water heater loads for a system of 100 water heaters. While this was the only DSM strategy assessed, the aggregate water heater load model should be effective in evaluating any water heater DSM strategy without conducting
a full-scale pilot program. This would result in considerable savings in cost and time for utilities considering implementing a water heater load management program.
References [I] J.C. Laurent and R.P. Malhamt, Physically based computer model of aggregate electric water heating loads, IEEE Trans. Pmter Sysr.. Y (3) (1994). [2] E. Hirst, R. Goeltz and M. Hubbard. Determinants of electricity use for residential water heating: the Hood River conservation project. EnrrgJ Consew. Manage.. 27 (2) (1987) I71 -178. [3] R.F. Bischke and R.A. Sella, Design and controlled use of water heater load management. IEEE Truns. Pouw Appar. Syst., PAS/04(19X5) 1290 1293. [4] S.H. Lee and C.L. Wilkins, PI practical approach to appliance load control analysis: a water heater case study, IEEE Trans. Power Appur. Sysr.. P.4S-102 (1982) 1007-1013. [5] H. Neiderreiter. Run&m Number Getwrution and Quasi-Monte Crrrlo ,Vefhorls. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. PA, 1992. pp. 1644165. [6] M.W. Gustafson, J.S. Baylor and G. Epstein. Direct water heater load control ~~ estimating program effectiveness using an engineering model, IEEE Trans. Power S~sst.. 8 (I) (1993) I37 143. [7] J. Woodard, Electric load modeling, Ph.D. Tllesis, Massachusetts Institute of Technology, Cambridge, MA, 1974. pp. 2755286. [X] C.Y. Chong and A.S. Debs. Statistical synthesis of power system functional load models. Pror,. IEEE Conf: Decision and Contra/, Fort Lumlwdule. FL. USA, 1979. IEEE, New York, pp. 264269. [9] C. Alvarez. R.P. Malhame and A. Gabaldon, A class of models for load management application and evaluation revisited, IEEE Truns. Power SJY/.. 7 (4) (1992) 1435- 1443. [IO] J. Thompson, R. Broadwater and A. Chandrasekaran, Analysis of water heater data from Athens load control experiment, IEEE Truru. Poller Deliwry, 4 (2) (1989) 1232 1237.
Electric
Power
Research 36 (1996)
Systems
29-35
Development of a Monte Carlo based aggregate model for residential electric water heater loads P.S. Dolan’, De,~arruwt~r
of’ Elecrrictrl
M.H. Nehrir,
Engineering, Received
Montana
16 May
Slate
1995: accepted
V. Gerez
Chiversiry, 3 August
Bownan,
XI T 59 71 7, l’SA
1995
Abstract A model for a residential electric water heater load is developed using an energy flow analysis of the water heater tank. An aggregate model for residential electric water heater loads is then developed using a rejection type Monte Carlo simulation technique. The resultant aggregate model is then used to assess the effectiveness of several demand-side management strategies using the water heater load profile presented in a previous analysis. Kqxmds:
Water
heater
modeling:
Monte Carlo simulation; Demand-side management
1. Introduction Due to the increased interest of utilities in load management strategies for such things as load peak shaving, valley filling, and cold load pickup prediction, there has been an increased emphasis on residential load (appliance) modeling. One of the appliances which has been of particular interest to utilities and researchers alike is the residential electric water heater. Utilities have historically relied upon pilot testing programs to assess the impacts of water heater demandside management (DSM) strategies. While the pilot testing programs have provided invaluable data on the aggregate response of water heater loads, performing such testing is expensive and time consuming. Pilot testing could be avoided by using an aggregate water heater load model to assess the load response to DSM strategies which manipulate water heater operation. In the literature. what is typically referred to as an aggregate electric water heater load mode1 is an attempt to mathematically describe the operation of the water heater loads in a particular system by statistical curve fitting techniques [I -41. Another approach for developing an aggregate load mode1 for water heater operation would be to mode1 individual water heaters and add all ’ Presently with SSR Engineers. Bismarck, ND 58501, USA.
Inc..
037%7796~96~S15.00
Science
SSDI
037%7796(95)OlOl
P 1996 Elsevier I-B
4205
State
Street.
S.A. All rights
reserved
of their power demand profiles to obtain the overall system water heater load profile. This approach is similar to that used by the Electric Power Research Institute (EPRI) in its WATSIM program which allows the user to aggregate up to 300 water heaters using 300 stored water demand profiles. Using this kind of approach, the control of the heating elements of the individual water heaters being modeled may be manipulated in any manner desired to simulate various DSM strategies. In this paper we propose using a discontinuous random process for the hot water use of each of the modeled water heaters instead of predetermined water use profiles. This approach results in an aggregate water heater response that closely matches the overall system water heater load profile. While the approach does not correctly represent the actual load profile of each water heater on the system. it does provide a close representation of the system’s water heater load profile. In order to obtain the aggregate water heater load for a power system, it is not necessary to accurately model every single water heater in a particular system as the individual water heater characteristics tend to be canceled out when applied in an aggregate manner. All that is required is that the genera1 characteristics of the water heaters in the system be modeled. It is possible to statistically represent the overall mode1 parameters of the water heaters in the system as being normally
distributed with associated means and standard deviations. Once the means and standard deviations of the various water heater parameters such as size, element rating, and insulation level are known for a particular system, a rejection type Monte Carlo method [5] may be used to generate the model parameters for any number of water heaters. With this information and an acceptable water heater model, it is easy to develop a computer model to simulate the aggregate load profile of a large number of water heaters. This paper discusses the development of a single electric water heater model as well as an aggregate model for water heater loads.
Ci;,(t)
= - SA(lIR)[T,(t)
- T,,,]
- 8.3 ~n(~K’,[T,(f) where C is the equivalent the tank given by
C= 8.3 x (no. of gallons) x CP Letting G= SA(l/R) and B(t) = lVD(t) x 8.3C,, differential equation becomes C’fdt)
= - G[T,(t)
- Tc,“J - B(t)[TH(t)
WliRU’df)
- r,,,l
(1)
where SA ii R
T”(I) T ““t
tank surface area, SA = 2r(r+h) (ft’) tank radius (ft) tank height (ft) thermal resistance of tank insulation (h ft2 “F/Btu) mean tank water temperature at time (“F) outside ambient temperature (“F) t
The water demand losses (in Btu/h) as given by Ref. [7] are 8.3 WDif)Cp[TH(f)
-
Tin]
(2)
where 8.3 is the weight, in lb, of 1 USgal. of water, and PVD(t) water demand (USgal/h) specific heat of water, 1 Btu/(‘F lb) CP temperature of incoming make-up water (“F) T,, The power input to the tank (in Btu/h) is Q(t) = 3.4121 x 10” x (element kW rating) The general differential model is
(3)
equation of the water heater
th.:
- TiJ + Q(f) (51
It can be shown that a solution of Eq. (5) is
+ [GR’T,,,, + B(r)R’T,,
By summing the energy sources and losses in the water heater, one can develop a general differential equation describing the mean or average thermal response of the water in the tank. As indicated in Ref. [6], water heaters have two sources of energy losses: (i) thermal convection or ‘standby losses’ where heat is lost through the tank’s exterior walls to the surrounding environment; (ii) ‘water demand energy losses’ associated with the use of the hot water stored in the tank. The energy supplied to the tank comes from the heating elements. The convection losses (in Btuih) as given by Ref. [6] are
(4)
thermal mass (in Btu/“F) OF
T,,(t) = T,(s) exp[ - (l/R’C)(r 2. Water heater model development
- T,l + Q(t)
x (1 -exp[-(l/R’C)(t-t)]}
- r)] + Q(t)R’]
(6 ’
where R’ = l/[G+ B(t)]. At this point it should be noted that the terms Q(t) and B(t) in Eq. (6) are piecewise continuous [8,9]. Recall that Q(t) is controlled by the water heate,, thermostat and only becomes nonzero when the tern. perature falls below the lower thermostat setpoint. Af. ter the temperature of the water surrounding the thermostat reaches the thermostat’s upper temperature: setpoint, the heating element is shut off and Q(t) gee:; to zero. The B(t) term, which is associated with the water demand at any time t, is only nonzero when there is a demand for hot water. Because of the piecewisc! continuous nature of Q(t) and B(t), every time either one changes from one state to another, the initial condition term TH(r) in Eq. (6) must be modified fol. the new condition. Also, r must be set to the time t a. which the change of state occurred. In order to simulate the load profile of a large number of water heaters, it is necessary to obtain thts parameters describing each of the water heaters to be simulated. As previously stated, it is not necessary tcl know the actual parameters for each of the water heaters on the system to simulate the system. All that i:. needed are the mean and standard deviations associatec with a few of the key parameters presented in the single water heater model to describe the operational charac teristics of the aggregate water heater system. Once the means and standard deviations are known, a Monte Carlo rejection type method may be used to genera& the parameters needed in order to run the aggregate water heater simulation. The means and standard devi. ations are needed for the following parameters associ ated with the water heaters on the system to be simulated: Tank capacity (USgal) Thermostat setpoint (“F) Thermostat deadband (“F) Hot water usage rate (USgallh)
P.S.
Dolun
et al. : Ektric
Po~vrr
Thermal resistance R of tank insulation (h ft’ “F/Btu) Heating element rating (kW) Temperature of incoming make-up water (“F) Ambient temperature of tank location (“F) The lower thermostat setpoint is obtained by subtracting the thermostat deadband from the upper setpoint. The initial water temperature in the water heater tanks is generated as a uniformly distributed random variable between the thermostat’s upper and lower temperature setpoints. If we assumethe water heater tanks on the system are cylindrical in shape and that the radius of the tank is one fifth of the tank height (which is a good assumption), we can write the following: h = (0.13368/0.04)“’ SA = 0.48/z’
(ft)
(7)
(ft’)
(8)
3. The Monte Carlo rejection method
Slsrtws
Reseuwll
36 (1996)
29-35
31
water demand profile in USgal/h for each water heater being modeled. The random process developed will result in an aggregate water heater load that closely matches that of the system. For an accurate match of the system water heater load profile, a good knowledge of the aggregate water heater load profile in kW is needed. In general, knowledge of the system water heater load profile will not be available. Obtaining accurate information about the system water heater load profile is only possible through monitoring the power demand of each water heater. However, the random process may be generated using an approximation or best guess as to what the water heater load profile is. It should be noted that the more accurate the information on the system water heater load profile, the better the random process generated will allow the simulation to mimic the system water heater load profile. Once the system water heater steady-state load profile is known, the steps used to develop the random process describing the hot water demand profile are straightforward and can be summarized as follows.
The steps required for using the Monte Carlo rejection method for generating water heater model parameters with a normal distribution function with a mean .? and a standard deviation cr are given below. The reader is referred to Ref. [5] for a formal mathematical definition of the process. Step 1. Generate a random number x which is uniformly distributed in the interval .f - 3a < x < R + 3a. Step 2. Form f(x) = (l,!a&)exp[
-(x
- X-)2/2cr’]
(9)
Step 3. Generate a second random number which is uniformly distributed in the interval 0
I
4. Developing the hot water use random process Here we develop a random process describing the hot
Fig.
I. Flowchart
describing
End
the Monte
Carlo
rejection
method.
Fig. 2. Water heater experiment [IO].
operation
curve
from
Athens
load
control
Step 1. Make a plot of the percentage of water heaters on versus time of day by dividing the system water heater load by the product of the mean water heater element kW rating and the number of water heaters on the system (Fig. 2). Step 2. Fit simple geometric shapes such as parabolas, ramps, and straight lines to the percentage of water heaters on versus time of day (water heater operation curve of Fig. 2) to create an approximate ‘smooth’ representation of the curve as shown in Fig. 3. Note that we are only concerned with the portions of the geometric-shape curves that fit the water heater operation curve; there could be more than one set of geometric curves that fit the operation curve. Statistical curve fitting techniques can also be used to obtain equations describing different portions of the operation curve. However, considering the random nature of hot water usage, the approximate geometric-shape curve fitting is used here.
Step 3. Obtain the portion of the geometric share over each of the various time intervals which best fils the curve. Note the time intervals associated with each portion of geometric shape obtained above along with the peak percentage of water heaters on for each tim’e interval. Step 4. Normalize the equation for each of the geometric shapes to the peak percentage of water heaters on during their respective time intervals. The equations for each of the dominant geometric shapes should be written to be active for the time of day that they are valid. During the simulation process for a particular water heater, a uniform random number is generated in the interval between zero and one. The random number is then compared with the peak percentage of water heaters on during the time interval in question. If th\: random number is less than or equal to the percentagof water heaters on during that time interval, then th.: hot water usage rate for that particular water heater is multiplied by the value obtained from the active gee.metric equation to obtain the instantaneous water demand rate. If the random number is greater than th.: percentage of water heaters on for that time interval, then no hot water is being drawn. While this method OI determining the water demand rate would in no wa:: accurately model the hot water use by any given household, it is extremely effective in giving a close represen., tation of the aggregate water heater load on the system To further clarify the steps associated in developing the random hot water demand process, we will follov., through with an example. Fig. 2 shows the percentage of water heaters on versus time of day for the Athen load control experiment [lo]. We will now proceed tc develop the random hot water demand process for thL figure.
I---
ST
ea?!3
1a-
P ?II-
1
23
4
Rs-1
Fig. 3. Simple
geometric
shapes fitted
to the curve
of Fig. 2
Fig. 4. Dominant geometric for the curve of Fig. 2.
curves
over their respective
time intervalr
P.S. Dolan Table 1 Normalized Curve
equations
for the curves
no.
Equation
I 2 3 4 5
y(t) y(r)
y(t) y(t) y(t)=
= = = =
et al. 1 Electric,
Power
Table Water
2 heaters
of curve
Curve
no.
1 1 -(1:2)(r-7)” 1 -(l/30)(/-9)’ 1.7107-0.06607t 1 -(1:36)(r-20)’
IF 6.00
1 2 3 4 5
End IF
The remainder of the process consists of generating a uniform random number in the interval between zero and one. The random number is then compared with the peak percentage of water heaters on (in decimal form). If the random number is less than or equal to the peak percentage of water heaters on for the time interval under consideration, the instantaneous hot water demand of each water heater is given by multiplying the hot water usage rate for the water heater by the value obtained from the active geometric equation. Otherwise, the water demand is set to zero.
simulation
Resrurcll
of Fig. 4
We fit simple geometric shapes to the percentage of water heaters on versus time of day curve. In this case the curve of Fig. 2 can be fitted well with three inverted parabolas, a horizontal line, and a sloping line, as shown in Fig. 3. Fig. 4 shows the dominant portions of the curves over their respective time intervals. The normalized equations for the curves are shown in Table 1. The peak percentage of water heaters on in each interval shown in Fig. 4 is given in Table 2. Next, we write the equations for the geometric shapes so that they are activated only during the time of day that they are valid. One way to do this is to imbed them into IF.. .THEN type statements such as:
5. Computer
Systems
and results
In this section the components of the aggregate water heater load model discussed above will be put together to form the algorithm used to generate the aggregate water heater load profile. The steps outlining the algorithm are as follows. Step 1. Enter the initial time the simulation begins, length of simulation, number of water heaters simulated, and the means and standard deviations of the various model parameters. Step 2. Convert the simulation start and end times to the time of day in hours and tenths of hours. Step 3. For each water heater, generate the model parameters using the Monte Carlo rejection method outlined in Section 3.
36 (1996)
29-35
on in each interval Time
interval
1:OO-6:00 6:00-7:30 7:30- 11 :oo 11 :OO- 16:30 16:30-1:00
33
of Fig. 4 Peak % on 8 32 27 I9 22
Step 4. Form the remaining water heater model parameters from those generated in the Monte Carlo method. Step 5. Loop through Steps 6- 16 to simulate the operation of each water heater. Step 6. Loop through Steps 7- 15 for the simulation time interval. Step 7. Find the instantaneous hot water demand for the water heater being simulated using the method of Section 4. Step 8. Check the instantaneous hot water demand to see if it has changed from the previous hot water demand. If it is different, then update the water temperature initial condition term and set it equal to the simulation time. Step 9. Update the parameters B and R’ in Eq. (6). Step 10. Compare the tank water temperature with the thermostat’s lower temperature setpoint. If the water temperature is lower than the lower temperature setpoint, then activate (turn on) the heating element. Also, update the water temperature initial condition term and set it equal to the simulation time. Step II. Compare the tank water temperature to the thermostat’s upper temperature setpoint. If the water temperature is higher than the upper temperature setpoint, then disable (turn off) the heating element. Also, update the water temperature initial condition term and set it equal to the simulation time. Step 12. Find the new tank water temperature using Eq. (6). Step 13. Record the heating element power demand in kW. Step 14. Compare the simulation time with the simulation end time. If the simulation time is equal to the end time, go to Step 16. Step 15. Increment the simulation time and go to Step 7. Step 16. Check to see if the operation of all of the water heaters has been simulated. If it has not, then update the water heater parameters to those of the next water heater and go to Step 6. Step 17. Loop through all the time increments and add the power demand of all the water heaters simulated for each respective time interval. Step 18. End.
34
P.S.
0:0
2
Fig. 5. Simulation water heaters.
4
6
6 10 12 Tme~lDay(24H01~~)
of the diversified
14
daily
Dolan
16
et ul. / Electric
1s
20
operation
i2
Power
24
400 360
Fig. ments.
U
6. Aggregated
4
6
electric
8 10 12 14 Tu-cadDay(24i-kurCkdc)
water
response
16
i6
36 (1996)
.,
;
29-35
a
, 10
....
9. keofDay(24HarC!&o
11
12
of 100 electric
450
2
Research
2Q6
A computer program was written using the above algorithm to simulate an aggregate water heater load for a system comprising 100 water heaters. The program can be run on a 386 or better IBM-compatible personal computer. The random hot water demand process used in the simulation was that developed in Section 4 for the Athens load control experiment profile [lo]. Fig. 5 shows the simulation result for the percentage of water heaters on at steady state, which closely matches that shown in Fig. 2. The aggregate water heater load was then simulated for a condition in which all of the heating elements in the water heater system were disabled for one hour during the morning and evening peak load times. The system power demand can be seen in Fig. 6. The peaks of the payback curve in the simulation have the same shape as those seen in Ref. [lo], but are higher in amplitude. One explanation for this discrepancy is that when there is a drastic reduction in water temperature, people typically stop using hot water. This modification of hot water use behavior can be incorporated into the water heater model by adding a term which sets the instantaneous hot water demand term to zero when the water temperature falls below a certain level. Several demand-side management strategies were simulated using the aggregate water heater load model
Oo
Systems
ti
to disabling
22
24
heating
ele-
8
9
10
Trc.adDay(24Hwaoa))
‘lo! ,..,,
20‘ 6
7
9
,,,,,,,_,_,,. ...,.,,,, ,,,,,,,,,,,,,,,,, 11
.._. j 12
TkMhy(24Hour&,
Fig. 7. Aggregated electric water heater response to cycling 20% of the water heaters off (a) every 10 min, (b) every 20 min, (c) every 30 min.
to simulate direct water heater load control for reducing peak load. Different percentages of water heaters were disabled for different time intervals and simulation results were compared. Among the DSM strategies simulated, 20% of the water heaters’ heating elements were disabled for lo-, 20-, and 30-minute time intervals. After the time interval specified had elapsed, the heating elements of the water heaters that were disabled were allowed to activate normally and a new set of water heaters’ heating elements were disabled. This process was repeated for the entire simulation time. Simulation results for this control strategy are shown in Fig. 7. In this figure the total water heater power demand when all the water heaters are in operation is compared with that when the DSM strategy (water
heater cycling) is in effect. As can be seen from the simulation results, a proper selection of the percentage of water heaters to be cycled and a proper cycling time can reduce the peak load. As shown in Fig. 7(b), cycling 20% of the water heaters (under study) off every 20 minutes reduced the peak power demand by about 25%. However, no improvement of the ability of the DSM strategy to reduce the peak load is seen once a certain cycling time is exceeded for each of the percentage cycling blocks (see Fig. 7(c)). In fact, the DSM strategy can cause an increase in the peak load observed for the larger off-time intervals. This is due to the larger payback curves associated with larger offtime intervals.
6. Conclusions A model for simulating aggregate water heater loads was developed in this paper. A methodology was presented for generating the model parameters for a large number of water heaters to fit the statistical characteristics of the water heaters in any system. In addition, a method for generating a random process for the instantaneous hot water demand for the water heaters being simulated was presented. The model developed was then evaluated using computer simulations. The usefulness of the aggregate water heater load model developed was shown in evaluating a rotational type of demand-side management strategy for water heater loads for a system of 100 water heaters. While this was the only DSM strategy assessed, the aggregate water heater load model should be effective in evaluating any water heater DSM strategy without conducting
a full-scale pilot program. This would result in considerable savings in cost and time for utilities considering implementing a water heater load management program.
References [I] J.C. Laurent and R.P. Malhamt, Physically based computer model of aggregate electric water heating loads, IEEE Trans. Pmter Sysr.. Y (3) (1994). [2] E. Hirst, R. Goeltz and M. Hubbard. Determinants of electricity use for residential water heating: the Hood River conservation project. EnrrgJ Consew. Manage.. 27 (2) (1987) I71 -178. [3] R.F. Bischke and R.A. Sella, Design and controlled use of water heater load management. IEEE Truns. Pouw Appar. Syst., PAS/04(19X5) 1290 1293. [4] S.H. Lee and C.L. Wilkins, PI practical approach to appliance load control analysis: a water heater case study, IEEE Trans. Power Appur. Sysr.. P.4S-102 (1982) 1007-1013. [5] H. Neiderreiter. Run&m Number Getwrution and Quasi-Monte Crrrlo ,Vefhorls. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. PA, 1992. pp. 1644165. [6] M.W. Gustafson, J.S. Baylor and G. Epstein. Direct water heater load control ~~ estimating program effectiveness using an engineering model, IEEE Trans. Power S~sst.. 8 (I) (1993) I37 143. [7] J. Woodard, Electric load modeling, Ph.D. Tllesis, Massachusetts Institute of Technology, Cambridge, MA, 1974. pp. 2755286. [X] C.Y. Chong and A.S. Debs. Statistical synthesis of power system functional load models. Pror,. IEEE Conf: Decision and Contra/, Fort Lumlwdule. FL. USA, 1979. IEEE, New York, pp. 264269. [9] C. Alvarez. R.P. Malhame and A. Gabaldon, A class of models for load management application and evaluation revisited, IEEE Truns. Power SJY/.. 7 (4) (1992) 1435- 1443. [IO] J. Thompson, R. Broadwater and A. Chandrasekaran, Analysis of water heater data from Athens load control experiment, IEEE Truru. Poller Deliwry, 4 (2) (1989) 1232 1237.