- Email: [email protected]
Short-Term Forecasting of Electric Power System Load from a Weather-Dependent Model
IFAC Symposium 1977 Melbourne, 21-25 February 1977
Short-Term Forecasting of Electric Power System Load from a Weather-Dependent Model V. PANUSKA Associate Professor of Engineering, Concordia University, Montreal, Canada
SUMMARY A method for probabilistic hourly forecasts of electric power system lOad from a weather-dependent load model is presented. The forecasts and their corresponding probability limits are generated directly from the model equation which comprises a periodic nominal load component represented by a Fourier series and a residual component in the form of a linear difference equation driven by a temperature dependent variable and with an additional stochastic input to account for the uncertainty in the modelling process. The method has been tested on real load data. The model fitted to the data contains a delay of several hours between the temperature dependent input and the resulting load changes so that load forecasts several hours ahead do not require temperature forecasts. 1
Galiana (1971) and Galiana et al. (1974) with one important difference, which is the incorporation of a delay term for the temperature dependent input. The proposed forecasting procedure is based directlyon the difference equation of the model and is much simpler than the state-space approach of Galiana (1971).
INTRODUCTION
Load forecasting is gaining increasing importance in the aanagement of electric power system utilities. Short-term load forecasts with lead time from a few minutes to a few days are now being uaed for maintenance planning, the solution of economic scheduling problems, and in the real-time state monitoring and security programs. Many of the existing forecasting methods, however, do not take into account the inherent uncertainty in the load forecasting process and do not provide probability limits for the forecasts. Also, the characterization of the effects of the weather is frequently missing.
The structure of the paper is as follows. In Section 2 the load model is described. The forecasting algorithms based on this model are derived in Section 3. Experimental results using load data provided by Hydro-Quebec are presented in Section 4. 2
To describe briefly a few of the existing approaches, Farmer and Potton (1968) used the KarhunenLoeve expansion whose coefficients were identified by a linear estimation procedure. Christiaanse (1971) proposed a Fourier series model of the weekly load with the parameters updated by an exponential smoothing technique while Toyoda et al. (1970) and Gupta and Yamada (1972) made use of state space models. Galiana (1971) and Galiana et al. (1974) proposed to model the effect of temperature and uncertainty by a linear difference equation driyen by a temperature dependent and a noise input. Recently, Srinivasan and Pronovost (1975) used mUltiple correlation models and obtained load forecast as a combination of the model outputs. As far as the inclusion of the weather input is concerned, in Toyoda et al. (1970). Gupta and Yamada (1972); Galiana (1971); Dryar (1974); Heinemann et al. (1966); Stanton (1971); Davey et al. (1973); Lijensen and losing (1971); Galiana and Schweppe (1972) its use in advocated, while in Christiaanse (1971); Srinivasan and Pronovost (1975); Gupta (1971); Vermuri et al. (1973); Bachelet and Horlat (1966) the load is treated as a function of time with added uncertainty, part of which accounts for the exclusion of weather inputs. In this paper a weather dependent model is used for forecasting purposes since the analysis of the load data provided by a power utility has shown a pronounced temperature dependence. The model comprises a periodic nominal load component represented by a Fourier series and a residual component in the form of a linear stochastic difference equation. The structure of the model resembles the one used by
414
THE MODEL
The load model which will be used here and its identification from load data was described in detail in Galiana (1971) and Panuska and Koutchouk (1975). The model is of discrete type with time unit of one hour and the basic period of the nominal load component is 24 hours. To introduce the effect of temperature, it is argued (Galiana, 1971; Galiana et al., 1974) that only the difference of current tempe~ature T(t) and the average temperature profile T(t) should cause deviations in the load since the effect of f(t) can be thought of as to be incorporated in the periodic nominal load component. In addition, the dependence of the load on the temperature difference is obviously non-linear since both very cold weather (heating) and very hot weather (cooling) cause an increase in the load. A suitable non-linear function U - f(T,T) which can serve as the temperature dependent input for the dynamic residual model is shown in Fig. 1 (Galiana et aI, 1974).
I
T Cooling + H~ating ~ Cooling Cooling (T-2l)-(16-T) I U-T-21 I U"T-T o U~T+T-37 I I 21 C Heatins -------------------~---------~----------------I no effect I Cooli~g o U-(16-T) I U-O I U=2l-T _ 16 C HeatIni------------l-Heating--1-cooIIng-~Heat1Dl U=-(T-T) U--(T-16): (2l-T)-!T-16) I U=37-T-T
r
Figure 1
Non-linear Function ' U=f(T,T)
Let Zt denote the total load which is written as a sum of the periodic nominal load Y and the residual t load X ' t
~
m
,. 1,
... ,
L(degree of C(z-l»
1, ... , M(degree of D(z-l»
Substituting (4) and (5) into (1) gives the final expression for the total load.
(1)
The 2eriodic term is represented by a Fourier series. N N Y - Cl + L '1t sin kwt + L 8 cos kwt (2) t 0 k k=l k=l
d~
t-m
+ (7)
3
FORECASTING ALGORITHMS
where N is number of harmonics. 211 =-T
W
T is the period (24 hours)
To write the expression (2) in a comprehensive form, introduce the notation: {Pi } - ( Cl o ' {Cl k }, { 8k } ) { i , t } =(1, {sin kwt } , {cos kwt}) k i
a
1, 1,
.. ,
.. ,
N 2N+l
The expression (2) then becomes 2N+l Yt = L Pi i t i=l ' ~
(3)
It will be shown that the model (7) derived in the previous section can be directly used to forecast future load values and that a conversion to statespace form (Galiana, 1971) is not necessary. For for ecasting purposes it will be as sumed that the model parameters {p.}, {a. } , {b } , {c. } , {d. } and i l. l. l. l. the variance of the residuals 0 2 are known exactly. This is a simplifying assumptio~ but the experience shows (Panuska and Koutchouk, 1975; Galiana, 1971) that these parameters can be identified from given load-temperature data with fairly small estimation errors so that the load forecasts should not be seriously affected by using parameter estimates instead of their true values. Also, the temperature input U will be assumed to be available for any lead ti~e.
To further simplify the writing, a notation due to Einstein will be adopted:
3.1
The occurrence of a superscript index above a variable and the same subscript at another variable indicates summation over this index.
Denote by Z (q) the linear minimum forecast oft Z + made at origin t It is well knoSnq(Box and Jenkins, can be computed as the conditional
Forecast Based on both Periodic and Residual Models
Expression (3) is then written in the final form: Zt(q) = E [Z t +q
(4)
Iz t ,
Z t- 1 ' ' ' '
mean square error for lead time q. 1970) that Z(q) expectation
(8)
1
The following procedure for computing the forecast can now be outl~ed. The model equation (7) is first used to e~ress Z +q' Then taking the cont ditional expectation at time t gives
Z (q) - [Z ]=pi ~ -aj[X ]+bku t t+q ~ i,t+q t+q-j t+q-s-k
(5)
dm[Et+q_ml+c~ [Wt+q_~ l+[Wt+ql with the following definitions:
(9)
system output where the square brackets are used to denote the conditional expectation of the form shown on the right hand side of (8). The functions <1> . t+ and
system input (function of temperature defined above)
l.,
correlated noise white noise
not affected by the expectation operator. The remaining conditional expectations which occur on the right hand side of (9) are evaluated as follows. If j is a nonnegative integer
delay
s
-1 -1 -1 -1 A(z ), B(z ), C(z ), D(z ) are polynomials in the backward shift operator z-l with coefficients {ail, {b }, {c }, {d }· i i i
-ajx
Et
-d~
t
t-j
+ bkU t-s-k + E t
t-m
+ c Wt_ ~ +W t
~
[Xt - ]. ]
X
1 1 [E .1 t-]
x
[W
W . t-] 0
[X t +j [E t +j
Using again the same notation as in expression (4), a new form of equation (5) is obtained after a few algebraic manipulations. X
.] t-] [W t +j 1
(6)
j=O, 1,2,
~t-j
t
j=1,2,
(j)
j=O, 1,2,
Et (j)
j=l,2,
~t+j
Z
... ,
k
0,
... ,
t-j
...
...
...
- Zt-j-l (1) j=l,2,
. and E
t-]
I,
...
E
In other words, the X
with j
q
Ut + q-s- k in (9) are deterministic and are obviously
t-
(10) j=O, 1,2, ...
...
j (j=O,l, ••• ) which
are available at origin t are left unchanged. The X . and E . which have not yet happened, are ret+] t+] ~ placed by their forecasts Xt(j), Et(j) at origin t.
J(degree of A(z-l» K(degree of B(z-l»
415
A
Z
cast Zt(q) is given by
The forecasts (q), q=1,2, ••• , are then computed recursively usi~g equations (6) and (9).
;: 2 -X (q)] = E[Wt+q~lWt+q_l+ t 2 + ~q_lWt+l] 2 ] .:; Z = [1 + Y + , +;;,Z (14) q-l w l ~
V(q) = E[X
3.1.1 Mean square error of the forecast First note that the mean square error of the f0recast can be expressed as E[Z
t+q
t-q
..
- Zt(q)]2 where c: (ll)
2 w
~
E[W~].
3.l.Z Probability limits for the forecasts where the operator (.) denotes the excraction of the stochastic part (i.e. the periodic component Y t and the input Ut are forced to be zero).
Assuming that the residuals W are drawn from a t normal distribution, the derived variance V(q) of the forecast error can be used to obtain probability limits for the forecast. For example, the 95% probability limits Z + (=) are given by (Box and Jenkins, 1970) t q
It is now convenient to express X + in terms of the residuals W + ' W + l' ••• ,. t qFrom (5) t q
t q-
1 + C(z-l)
X
Z
(12)
W
t+q
t+q
3.2
+
t q
~t
(=) =
(q)' 1.96
IV'('q)
(15)
Forecasc Based on the Residual Model Only
which can be written in the form The load forecasts can also be obtained by the method which does not require the knowledge of the coefficiencs of periodic model (Box and Jenkins, 1970). The method can be briefly described as follows. First, the seasonal d1fferencing operator V is applied to both sides of equation (7) Z4 giving
oc
X
t+q
L j=O
(13)
'4i J' Wt+q-J'
where '4i =1 and the remaining '4i weights are obtained 0 e.g. by performing long division of the polynomials in (12) or recursively by the coefficient comparison method proposed in Box and Jenkins (1970).
.
Since the minimum mean square error forecast Xt(q) is of the form (Box and Jenkins, 1970) Xt(q)
= L
j =0
'4i
(16)
+. W .
q J
i.e. the periodic nominal load component has been eliminated and the simplified equation contains only the residual parameters. Equation (16) is then
t-J
then using (11) the mean square error V(q) the fore-
I
10
o INPUT U
[ctc]
r-J
1..
\
I
v,l
-;;jJ
I
\A
1
/.
~V
\
-10
/\
VI
,.~
/ \In I
rvv
100. ea
58.00
I
1
I
1
I ~
I
v ~
(
.. ~
~\l
-20
0.00
.
,
~
j
k
Zt- Zt- 24=-a J v24 Xt-J.+b v Z4 Ut-s- k -
-
V 250.00
200.00
150.00
J
9000
,
1\ 1\
8000
a
1\
1\
Il
/\
~
~
II
/\
fI
LOAD Z li1;~ 0 0 0
n
I 1
,
i'
A ,
'\,
T \,
\,j
rv
13.00
\ 50.00
Figure 2
1013.00
\
1513.00
r\ \ -. \
\
\
~
l 2aa.00
Temperature input Ut and load Zt
416
,1
r..
I"
'\
\
6000
1\
Al
1\ , 1\
1\
1\ f\ ,
i\
"
1\
~
1\
25a.ee
\I
rewritten in the form .
k
Zt-Zt_24-aJ(Xt_j-Xt_24_j)+b (Ut-s-k-Ut-24-s_k)- dm(E
-E )+c £ (W -W ) + t-m t-24-m t- ( t-24-£
(17)
5
which can be used as a starting point for forecasting purposes in a similar way as equation (7) in Section 3.1. 4
The actual and forecasted load are seen to be in good agreement. The actual load lies within the 95% one-step-ahead probability limits shown in Figure 3, with the exception of the region marked by an arrow where an anomaly of the actual load occurs. Even then the error is less than 3% of the peak load. CONCLUSIONS
A method for probabilistic hourly load forecast computed directly from the model equation has been proposed. The method is computationally much simpler than the state-spa ce approach of Galiana (1971). Compared with the method of Box and Jenkins (1970) described briefly in Section 3.2, it has the advantage that the forecast of the periodic nominal load term can be obtained separately. The method has been tested on real load data and the results indicate good agreement between the forecasted and actually realized loads.
EXPERIMENTAL RESULTS
To test the proposed forecasting method, hourly temperature and load data during three weeks in a winter month (January) were obtained together with the average temperature profile After the exclusion of weekends and ~ondays and the computation of the function U = f(T,T) according to Figure 1, 288 pairs of data were obtained and plotted in Figure 2, where the periodicity and temperature dependence are apparent.
T.
6
ACKNOWLE DGMENTS
This research was supported by the National Research Council of Canada, Gran t A-7393 and by the FCAC Grant of the Government of Quebec. The load data used in the experimental work were provided by the courtesy of Hydro-Quebec.
The first '240 pairs of data are now used to identify a model of the form (7) by a stochastic approximation method de scribed in Panuska and Koutchouk (1975). The forecasting algorithm of Section 3.1 based on this model then gene rates the forecast of the load in the remaining period, which is then plotted together with the actual load in Figure 3.
The author would like to acknowledge the contribution of Dr. J.P. Koutchouk in writing and running the identification and forecasting programs.
The identified model of the form (17) used in fore casting contains a third-order residual component and a delay s~7 hours, which can be attributed to the effect of the thermal insulation of the heated objects causing a time shift between the temperature changes outside and inside the objects. The estimated variance of the residuals is C =104.1. The periodic model is based on 8 harmoni~s, i.e. 17 co effi cients.
7
REFERENCES
a
BACHELET, D. and MORLAT, G. (1966). Modele deux aleas pour des chroniq ues ~conomiq ues. Revue Francaise de Recherche Operationelle, Vol. 40,
£.P..1.Z2. BOX, G.E.P. and JENKINS, G.M. (1970). Time series analysis - forecasting and contro l. San Francisco, Holden Day.
90 e::"
tOOG - '
:
."
=OC~1-------------~--------------r-------------~-------------T-----------240
25L
Figure 3
2f O
280
Actual and predicted load
417
tems. IEEE Trans. Power App. Syst., Vol. PAS-90, pp 824-832.
CHRlSTlAANSE, W.R. (1971). Short-term load forecasting using general exponential smoothing. IEEE Trans. Power App. Syst., Vol. PAS-89, pp 16781682.
GUPTA, P.C. and YAMADA, K. (1972). Adapti~e short term forecasting of hourly loads using weather information. IEEE Trans. Power App. Syst., Vol. PAS91, pp 2085-2094.
CLARKE, D.W. (1967). Generalized least squares estimation of the parameters of a dynamic model. IFAC Symposium on Identification in Automatic Control Systems, Prague.
HEINEMANN, G.T., NORDMAN, D.A., and PLANT, E.C. (1966). The relation between summer weather and summer loads - a regression analysis. IEEE Trans. Power App. Syst., Vol. PAS-85, pp 1144-1154.
DAVEY, J" SAACKS, J.J., CUNNIGHAM, G.W., and PRIEST, K.W. (1973). Practical application of weather sensitive load forecasting to system planning. IEEE Trans. Power App. and Syst., Vol. PAS92, pp 971-977.
LIJEENSEN, D.P. and ROSING, J. (1971). Adaptive forecasting of hourly load based load measurement and weather information. IEEE Trans. Power App. ~, Vol. PAS-90, pp 1757-1767.
DRYAR, H.A. (1944). The effect of weather on system load. Trans. Amer. Inst. Eng., Vol. 63, pp 1006-13.
PANUSKA, V. and KOUTCHOUK, J.P. (1975). Electrical power system load modelling by a two-stage stochastic approximation procedure. Preprints, 6th Int. Fed. Automat. Contr. World Congress, paper 31.4, Boston, Mass.
FARMER, E.D. and POTTON, M.J. (1968). Developing of on-line prediction technique with results from trials in the south west region of the CEGB. Proc. Inst. Elec. Eng., Vol. 15, No. 10.
SRINIVASAN, K. and PRONOVOST, R. (1975). Shortterm load forecasting using mUltiple correlation models. IEEE Trans. Power App. Syst., Vol. PAS-94 , pp 1854-1858.
GALIANA, F.D. (1971). An application of system identification and state prediction to electric load modelling and forecasting. Report, Elec. Power Syst. Lab., Mass. Inst. of Technol., Cambridge, Mass. GALIANA, F.D. and SCHWEPPE, F.C. (1972). A weatherdependent probabilistic model for short-term load forecasting. IEEE Conference Paper No. C72/7l-2, Winter Power Meeting. GALIANA, F.D., HANDSCHIN, E., and FIECHTER, A.E. (1974). Identification of stochastic electric load models from physical data. IEEE Trans. Automat. Contr., Vol. AC-19, pp 887-893.
STANTON, K.N. (1971). Medium range, weekly and seasonal peak demand forecasting by probability methods. IEEE Trans. Power App. Syst., Vol. PAS-90, pp 1183-1189. TOYODA, J., CHEN, M.S., and INOUE, Y. (1970). An application of state estimation to short-term load forecasting, Part I: Forecasting and modelling. IEEE Trans. Power App. Syst., Vol. PAS-89, pp 1678-1682. VEMURI, S., HILL, E.F., and BALASUBRAMANIAN, R. (1973). Load forecasting using stochastic models. 8th Power Industry Application Conference, Minneapolis, Minn.
GUPTA, P.C. (1971). A stochastic approach to peak power demand forecasting in electric utility sys-
418
Short-Term Forecasting of Electric Power System Load from a Weather-Dependent Model V. PANUSKA Associate Professor of Engineering, Concordia University, Montreal, Canada
SUMMARY A method for probabilistic hourly forecasts of electric power system lOad from a weather-dependent load model is presented. The forecasts and their corresponding probability limits are generated directly from the model equation which comprises a periodic nominal load component represented by a Fourier series and a residual component in the form of a linear difference equation driven by a temperature dependent variable and with an additional stochastic input to account for the uncertainty in the modelling process. The method has been tested on real load data. The model fitted to the data contains a delay of several hours between the temperature dependent input and the resulting load changes so that load forecasts several hours ahead do not require temperature forecasts. 1
Galiana (1971) and Galiana et al. (1974) with one important difference, which is the incorporation of a delay term for the temperature dependent input. The proposed forecasting procedure is based directlyon the difference equation of the model and is much simpler than the state-space approach of Galiana (1971).
INTRODUCTION
Load forecasting is gaining increasing importance in the aanagement of electric power system utilities. Short-term load forecasts with lead time from a few minutes to a few days are now being uaed for maintenance planning, the solution of economic scheduling problems, and in the real-time state monitoring and security programs. Many of the existing forecasting methods, however, do not take into account the inherent uncertainty in the load forecasting process and do not provide probability limits for the forecasts. Also, the characterization of the effects of the weather is frequently missing.
The structure of the paper is as follows. In Section 2 the load model is described. The forecasting algorithms based on this model are derived in Section 3. Experimental results using load data provided by Hydro-Quebec are presented in Section 4. 2
To describe briefly a few of the existing approaches, Farmer and Potton (1968) used the KarhunenLoeve expansion whose coefficients were identified by a linear estimation procedure. Christiaanse (1971) proposed a Fourier series model of the weekly load with the parameters updated by an exponential smoothing technique while Toyoda et al. (1970) and Gupta and Yamada (1972) made use of state space models. Galiana (1971) and Galiana et al. (1974) proposed to model the effect of temperature and uncertainty by a linear difference equation driyen by a temperature dependent and a noise input. Recently, Srinivasan and Pronovost (1975) used mUltiple correlation models and obtained load forecast as a combination of the model outputs. As far as the inclusion of the weather input is concerned, in Toyoda et al. (1970). Gupta and Yamada (1972); Galiana (1971); Dryar (1974); Heinemann et al. (1966); Stanton (1971); Davey et al. (1973); Lijensen and losing (1971); Galiana and Schweppe (1972) its use in advocated, while in Christiaanse (1971); Srinivasan and Pronovost (1975); Gupta (1971); Vermuri et al. (1973); Bachelet and Horlat (1966) the load is treated as a function of time with added uncertainty, part of which accounts for the exclusion of weather inputs. In this paper a weather dependent model is used for forecasting purposes since the analysis of the load data provided by a power utility has shown a pronounced temperature dependence. The model comprises a periodic nominal load component represented by a Fourier series and a residual component in the form of a linear stochastic difference equation. The structure of the model resembles the one used by
414
THE MODEL
The load model which will be used here and its identification from load data was described in detail in Galiana (1971) and Panuska and Koutchouk (1975). The model is of discrete type with time unit of one hour and the basic period of the nominal load component is 24 hours. To introduce the effect of temperature, it is argued (Galiana, 1971; Galiana et al., 1974) that only the difference of current tempe~ature T(t) and the average temperature profile T(t) should cause deviations in the load since the effect of f(t) can be thought of as to be incorporated in the periodic nominal load component. In addition, the dependence of the load on the temperature difference is obviously non-linear since both very cold weather (heating) and very hot weather (cooling) cause an increase in the load. A suitable non-linear function U - f(T,T) which can serve as the temperature dependent input for the dynamic residual model is shown in Fig. 1 (Galiana et aI, 1974).
I
T Cooling + H~ating ~ Cooling Cooling (T-2l)-(16-T) I U-T-21 I U"T-T o U~T+T-37 I I 21 C Heatins -------------------~---------~----------------I no effect I Cooli~g o U-(16-T) I U-O I U=2l-T _ 16 C HeatIni------------l-Heating--1-cooIIng-~Heat1Dl U=-(T-T) U--(T-16): (2l-T)-!T-16) I U=37-T-T
r
Figure 1
Non-linear Function ' U=f(T,T)
Let Zt denote the total load which is written as a sum of the periodic nominal load Y and the residual t load X ' t
~
m
,. 1,
... ,
L(degree of C(z-l»
1, ... , M(degree of D(z-l»
Substituting (4) and (5) into (1) gives the final expression for the total load.
(1)
The 2eriodic term is represented by a Fourier series. N N Y - Cl + L '1t sin kwt + L 8 cos kwt (2) t 0 k k=l k=l
d~
t-m
+ (7)
3
FORECASTING ALGORITHMS
where N is number of harmonics. 211 =-T
W
T is the period (24 hours)
To write the expression (2) in a comprehensive form, introduce the notation: {Pi } - ( Cl o ' {Cl k }, { 8k } ) { i , t } =(1, {sin kwt } , {cos kwt}) k i
a
1, 1,
.. ,
.. ,
N 2N+l
The expression (2) then becomes 2N+l Yt = L Pi i t i=l ' ~
(3)
It will be shown that the model (7) derived in the previous section can be directly used to forecast future load values and that a conversion to statespace form (Galiana, 1971) is not necessary. For for ecasting purposes it will be as sumed that the model parameters {p.}, {a. } , {b } , {c. } , {d. } and i l. l. l. l. the variance of the residuals 0 2 are known exactly. This is a simplifying assumptio~ but the experience shows (Panuska and Koutchouk, 1975; Galiana, 1971) that these parameters can be identified from given load-temperature data with fairly small estimation errors so that the load forecasts should not be seriously affected by using parameter estimates instead of their true values. Also, the temperature input U will be assumed to be available for any lead ti~e.
To further simplify the writing, a notation due to Einstein will be adopted:
3.1
The occurrence of a superscript index above a variable and the same subscript at another variable indicates summation over this index.
Denote by Z (q) the linear minimum forecast oft Z + made at origin t It is well knoSnq(Box and Jenkins, can be computed as the conditional
Forecast Based on both Periodic and Residual Models
Expression (3) is then written in the final form: Zt(q) = E [Z t +q
(4)
Iz t ,
Z t- 1 ' ' ' '
mean square error for lead time q. 1970) that Z(q) expectation
(8)
1
The following procedure for computing the forecast can now be outl~ed. The model equation (7) is first used to e~ress Z +q' Then taking the cont ditional expectation at time t gives
Z (q) - [Z ]=pi ~ -aj[X ]+bku t t+q ~ i,t+q t+q-j t+q-s-k
(5)
dm[Et+q_ml+c~ [Wt+q_~ l+[Wt+ql with the following definitions:
(9)
system output where the square brackets are used to denote the conditional expectation of the form shown on the right hand side of (8). The functions <1> . t+ and
system input (function of temperature defined above)
l.,
correlated noise white noise
not affected by the expectation operator. The remaining conditional expectations which occur on the right hand side of (9) are evaluated as follows. If j is a nonnegative integer
delay
s
-1 -1 -1 -1 A(z ), B(z ), C(z ), D(z ) are polynomials in the backward shift operator z-l with coefficients {ail, {b }, {c }, {d }· i i i
-ajx
Et
-d~
t
t-j
+ bkU t-s-k + E t
t-m
+ c Wt_ ~ +W t
~
[Xt - ]. ]
X
1 1 [E .1 t-]
x
[W
W . t-] 0
[X t +j [E t +j
Using again the same notation as in expression (4), a new form of equation (5) is obtained after a few algebraic manipulations. X
.] t-] [W t +j 1
(6)
j=O, 1,2,
~t-j
t
j=1,2,
(j)
j=O, 1,2,
Et (j)
j=l,2,
~t+j
Z
... ,
k
0,
... ,
t-j
...
...
...
- Zt-j-l (1) j=l,2,
. and E
t-]
I,
...
E
In other words, the X
with j
q
Ut + q-s- k in (9) are deterministic and are obviously
t-
(10) j=O, 1,2, ...
...
j (j=O,l, ••• ) which
are available at origin t are left unchanged. The X . and E . which have not yet happened, are ret+] t+] ~ placed by their forecasts Xt(j), Et(j) at origin t.
J(degree of A(z-l» K(degree of B(z-l»
415
A
Z
cast Zt(q) is given by
The forecasts (q), q=1,2, ••• , are then computed recursively usi~g equations (6) and (9).
;: 2 -X (q)] = E[Wt+q~lWt+q_l+ t 2 + ~q_lWt+l] 2 ] .:; Z = [1 + Y + , +;;,Z (14) q-l w l ~
V(q) = E[X
3.1.1 Mean square error of the forecast First note that the mean square error of the f0recast can be expressed as E[Z
t+q
t-q
..
- Zt(q)]2 where c: (ll)
2 w
~
E[W~].
3.l.Z Probability limits for the forecasts where the operator (.) denotes the excraction of the stochastic part (i.e. the periodic component Y t and the input Ut are forced to be zero).
Assuming that the residuals W are drawn from a t normal distribution, the derived variance V(q) of the forecast error can be used to obtain probability limits for the forecast. For example, the 95% probability limits Z + (=) are given by (Box and Jenkins, 1970) t q
It is now convenient to express X + in terms of the residuals W + ' W + l' ••• ,. t qFrom (5) t q
t q-
1 + C(z-l)
X
Z
(12)
W
t+q
t+q
3.2
+
t q
~t
(=) =
(q)' 1.96
IV'('q)
(15)
Forecasc Based on the Residual Model Only
which can be written in the form The load forecasts can also be obtained by the method which does not require the knowledge of the coefficiencs of periodic model (Box and Jenkins, 1970). The method can be briefly described as follows. First, the seasonal d1fferencing operator V is applied to both sides of equation (7) Z4 giving
oc
X
t+q
L j=O
(13)
'4i J' Wt+q-J'
where '4i =1 and the remaining '4i weights are obtained 0 e.g. by performing long division of the polynomials in (12) or recursively by the coefficient comparison method proposed in Box and Jenkins (1970).
.
Since the minimum mean square error forecast Xt(q) is of the form (Box and Jenkins, 1970) Xt(q)
= L
j =0
'4i
(16)
+. W .
q J
i.e. the periodic nominal load component has been eliminated and the simplified equation contains only the residual parameters. Equation (16) is then
t-J
then using (11) the mean square error V(q) the fore-
I
10
o INPUT U
[ctc]
r-J
1..
\
I
v,l
-;;jJ
I
\A
1
/.
~V
\
-10
/\
VI
,.~
/ \In I
rvv
100. ea
58.00
I
1
I
1
I ~
I
v ~
(
.. ~
~\l
-20
0.00
.
,
~
j
k
Zt- Zt- 24=-a J v24 Xt-J.+b v Z4 Ut-s- k -
-
V 250.00
200.00
150.00
J
9000
,
1\ 1\
8000
a
1\
1\
Il
/\
~
~
II
/\
fI
LOAD Z li1;~ 0 0 0
n
I 1
,
i'
A ,
'\,
T \,
\,j
rv
13.00
\ 50.00
Figure 2
1013.00
\
1513.00
r\ \ -. \
\
\
~
l 2aa.00
Temperature input Ut and load Zt
416
,1
r..
I"
'\
\
6000
1\
Al
1\ , 1\
1\
1\ f\ ,
i\
"
1\
~
1\
25a.ee
\I
rewritten in the form .
k
Zt-Zt_24-aJ(Xt_j-Xt_24_j)+b (Ut-s-k-Ut-24-s_k)- dm(E
-E )+c £ (W -W ) + t-m t-24-m t- ( t-24-£
(17)
5
which can be used as a starting point for forecasting purposes in a similar way as equation (7) in Section 3.1. 4
The actual and forecasted load are seen to be in good agreement. The actual load lies within the 95% one-step-ahead probability limits shown in Figure 3, with the exception of the region marked by an arrow where an anomaly of the actual load occurs. Even then the error is less than 3% of the peak load. CONCLUSIONS
A method for probabilistic hourly load forecast computed directly from the model equation has been proposed. The method is computationally much simpler than the state-spa ce approach of Galiana (1971). Compared with the method of Box and Jenkins (1970) described briefly in Section 3.2, it has the advantage that the forecast of the periodic nominal load term can be obtained separately. The method has been tested on real load data and the results indicate good agreement between the forecasted and actually realized loads.
EXPERIMENTAL RESULTS
To test the proposed forecasting method, hourly temperature and load data during three weeks in a winter month (January) were obtained together with the average temperature profile After the exclusion of weekends and ~ondays and the computation of the function U = f(T,T) according to Figure 1, 288 pairs of data were obtained and plotted in Figure 2, where the periodicity and temperature dependence are apparent.
T.
6
ACKNOWLE DGMENTS
This research was supported by the National Research Council of Canada, Gran t A-7393 and by the FCAC Grant of the Government of Quebec. The load data used in the experimental work were provided by the courtesy of Hydro-Quebec.
The first '240 pairs of data are now used to identify a model of the form (7) by a stochastic approximation method de scribed in Panuska and Koutchouk (1975). The forecasting algorithm of Section 3.1 based on this model then gene rates the forecast of the load in the remaining period, which is then plotted together with the actual load in Figure 3.
The author would like to acknowledge the contribution of Dr. J.P. Koutchouk in writing and running the identification and forecasting programs.
The identified model of the form (17) used in fore casting contains a third-order residual component and a delay s~7 hours, which can be attributed to the effect of the thermal insulation of the heated objects causing a time shift between the temperature changes outside and inside the objects. The estimated variance of the residuals is C =104.1. The periodic model is based on 8 harmoni~s, i.e. 17 co effi cients.
7
REFERENCES
a
BACHELET, D. and MORLAT, G. (1966). Modele deux aleas pour des chroniq ues ~conomiq ues. Revue Francaise de Recherche Operationelle, Vol. 40,
£.P..1.Z2. BOX, G.E.P. and JENKINS, G.M. (1970). Time series analysis - forecasting and contro l. San Francisco, Holden Day.
90 e::"
tOOG - '
:
."
=OC~1-------------~--------------r-------------~-------------T-----------240
25L
Figure 3
2f O
280
Actual and predicted load
417
tems. IEEE Trans. Power App. Syst., Vol. PAS-90, pp 824-832.
CHRlSTlAANSE, W.R. (1971). Short-term load forecasting using general exponential smoothing. IEEE Trans. Power App. Syst., Vol. PAS-89, pp 16781682.
GUPTA, P.C. and YAMADA, K. (1972). Adapti~e short term forecasting of hourly loads using weather information. IEEE Trans. Power App. Syst., Vol. PAS91, pp 2085-2094.
CLARKE, D.W. (1967). Generalized least squares estimation of the parameters of a dynamic model. IFAC Symposium on Identification in Automatic Control Systems, Prague.
HEINEMANN, G.T., NORDMAN, D.A., and PLANT, E.C. (1966). The relation between summer weather and summer loads - a regression analysis. IEEE Trans. Power App. Syst., Vol. PAS-85, pp 1144-1154.
DAVEY, J" SAACKS, J.J., CUNNIGHAM, G.W., and PRIEST, K.W. (1973). Practical application of weather sensitive load forecasting to system planning. IEEE Trans. Power App. and Syst., Vol. PAS92, pp 971-977.
LIJEENSEN, D.P. and ROSING, J. (1971). Adaptive forecasting of hourly load based load measurement and weather information. IEEE Trans. Power App. ~, Vol. PAS-90, pp 1757-1767.
DRYAR, H.A. (1944). The effect of weather on system load. Trans. Amer. Inst. Eng., Vol. 63, pp 1006-13.
PANUSKA, V. and KOUTCHOUK, J.P. (1975). Electrical power system load modelling by a two-stage stochastic approximation procedure. Preprints, 6th Int. Fed. Automat. Contr. World Congress, paper 31.4, Boston, Mass.
FARMER, E.D. and POTTON, M.J. (1968). Developing of on-line prediction technique with results from trials in the south west region of the CEGB. Proc. Inst. Elec. Eng., Vol. 15, No. 10.
SRINIVASAN, K. and PRONOVOST, R. (1975). Shortterm load forecasting using mUltiple correlation models. IEEE Trans. Power App. Syst., Vol. PAS-94 , pp 1854-1858.
GALIANA, F.D. (1971). An application of system identification and state prediction to electric load modelling and forecasting. Report, Elec. Power Syst. Lab., Mass. Inst. of Technol., Cambridge, Mass. GALIANA, F.D. and SCHWEPPE, F.C. (1972). A weatherdependent probabilistic model for short-term load forecasting. IEEE Conference Paper No. C72/7l-2, Winter Power Meeting. GALIANA, F.D., HANDSCHIN, E., and FIECHTER, A.E. (1974). Identification of stochastic electric load models from physical data. IEEE Trans. Automat. Contr., Vol. AC-19, pp 887-893.
STANTON, K.N. (1971). Medium range, weekly and seasonal peak demand forecasting by probability methods. IEEE Trans. Power App. Syst., Vol. PAS-90, pp 1183-1189. TOYODA, J., CHEN, M.S., and INOUE, Y. (1970). An application of state estimation to short-term load forecasting, Part I: Forecasting and modelling. IEEE Trans. Power App. Syst., Vol. PAS-89, pp 1678-1682. VEMURI, S., HILL, E.F., and BALASUBRAMANIAN, R. (1973). Load forecasting using stochastic models. 8th Power Industry Application Conference, Minneapolis, Minn.
GUPTA, P.C. (1971). A stochastic approach to peak power demand forecasting in electric utility sys-
418