Short-term load forecasting using threshold autoregressive models

S.R. Huang

Indexing terms: Loud jbrecusting, Threshold regressive models, Autoregressive models

Abstract: The author presents a method of forecasting the hourly load demand on a power system. The forecasting method uses threshold autoregressive models with the stratification rule. With the proposed threshold model algorithm, fewer parameters arc required to capture the random component in load dynamics. The techniques employed herein are the determination of an optimum threshold number and the construction of the threshold. The optimum stratification rule attempts not only to remove any judgmental input, but also to render the threshold process entirely mechanistic. Hence, the results demonstrate the proposed method’s effectiveness in terms of improving precision and reliability.

1 Introduction

Depending on the time period of interest, the load fore- casting domain can be divided into three types: long- term forecasts ( 5 years to 10 years), medium-term fore- casts (several months to 5 years) and short-term fore- casts (several hours to several weeks). Long-term forecasting is primarily used to evaluate the trends for the future generation expansion program. Medium- term forecasting is used in the maintenance scheduling of system generating units and planning of energy transactions. Finally, short-term forecasting is prima- rily used for the economic dispatch and the costlbenefit of a load management programme, fuel purchase scheduling. Hence, precise load forecasting not only increases economic benefits, but also maintains the reli- ability of power supply.

Extensive methods involving short-term load fore- casting have been developed in recent decades. The forecasting methodologies can be categorised as fol- lows: (1) Multiple linear regression method [l]: This method utilises the relations between the hourly system load and weather variation not only to search for its multi- ple linear regression model, but also to solve the regres-

0 TEE, 1997 IEE Proceedings online no. 19971144 Paper first received 23rd September 1996 and in revised form 8th January 1997 The author is with the Department of Electrical Engineering, Feng Chia University Taichung, 100 Wen Hwa Road, Taichung, Taiwan, Republic of China

IEE Proc-Gener. Trunsm. Distnb., Vol. 144, No. 5, September 1997

sion coefficients by the least square based method. This approach requires performing long offline analysis and accurate results depend heavily on the model assumed at the beginning. The method has been extensively used in medium- and long-term load forecasting. (2) Stochastic time series method [2-41: This method is one of the most accurate load forecasting methodolo- gies. The stochastic time series approach generally refers to the Box-Jenkines transfer function method. The method is not only well documented in previous literature, but also easy to understand and apply. A major limitation of this approach is that human deci- sions must be made in both data transformation and model identification. Available autoregressive inte- grated moving-average models are insufficiently flexible to handle load forecasting modelling requests; of par- ticular concern are modelling holidays and other spe- cific events. Moreover, the order of the model is frequently too high. (3) State space method [5]: In this method, the load is modelled as a state variable using a state-space formu- lation designated by two sets of equations: the system state equations and the measurement equations. The state-space method is highly attractive for online pre- diction owing to the recursive property of the Kalman filter. However, the forecast depends heavily on the assumed model, which must be known prior to using the filter. Estimating the noise covariance matrices at the model updating stage is computationally expensive. (4) General exponential smoothing method [6]: The method of applying the least square method to solve the optimal solution of smoothing constant is the most feasible approach. To yield accurate results, correlating the algorithm selected with the historical data is more important than theory would suggest. However, a larger smoothing constant is required if the variation of the load curve is too complex. (5) Expert system method [7]: This method corresponds to the rules based on the experts’ experience to accu- rately predict the load variations. As a rule-based approach, it is frequently regarded as an important complementary method capable of coping with sudden changes when most data-driven approaches fail due to a lack of data. (6) Artificial neural network [8, 91: An advantage of this methodology is the ability to understand the com- plex relationships between input and output vectors while the input and output vectors are extremely diffi- cult to embody in conventional models. As classifying design procedures are currently unavailable, the input point of the model’s artificial neural network architec-

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ture is constructed entirely by trial and error based on engineering judgement. However, in adding a new group of load data to the original data, the artificial neural network requires further disciplines. Hence, it is not economical in real applications.

This study also confirms the theoretical limitations of most conventional statistical methods: a) the nonlinear relationships of the input and output variables are diffi- cult to capture; b) the models cannot adapt to the rapid changes of system loading; and c) the order of the models to be identified is frequently too high. To either enhance the precision or reduce the order of models, a threshold model algorithm [lo, 111 which adopts a stratification technique [12, 131 is proposed herein.

2 models

Basic concept of threshold autoregressive

First, let us consider an autoregressive (AR) process of order k related by

xt + q X t - 1 + . . . + akXt-k = E t

where N observations of signal X, are available. To estimate the unknown parameters a', a2, ..., ak from the observation data, eqn. 1 can be rewritten as

and express eqn. 2 for k = 1, 2, ..., N. The basic concept of threshold autoregressive mod-

els, as introduced by Tong and Lim (1980) [ll], can be described as follows. Given N observations (Xl, X,, ..., XN) from a series { X t } , a self-exciting threshold autore- gressive model is composed of, say, L submodels which can be modelled separately to the appropriate L subsets of the observations, For instance, a kth-order 'thresh- old autoregressive model' (TAR(k)) can typically be of the form

(1)

Xt = -alXt-l - ( . . - U k X t - k + E t ( 2 )

k ,

xt = at' + c ajJ)Xt-? + 6:) z = 1

if Xt-d E J = 1,. . . , L, d = 0,1,2,. . .,IC, ( 3 )

where { E ~ ~ ) are each strict white noise processes, {a,") are constants, and the set (Rb))&l are called the 'L- threshold'. ..., are given subsets of the real line R', which define a partition of R' into disjoint intervals (-00, rl], (rl, r2], ..., (rL-l, +CO], with R(') denoting the interval (-CO, r ] and the interval (rG1, -1.

By considering thej th component model of eqn. 3, the coefficients alO, a20, ..., a$) can be estimated by the standard least-squares method as in the case of linear AR model fitting according to the observed subset {Xt-d) E RO. Estimating of the coefficients thus poses little difficultly. However, determining the structural parameters, i.e. the delay parameter d, the threshold region { RO}, and the individual model orders, k l , k2, ..., kL, is a more difficult task by previous literature [l 11. Their algorithms are generally effective, but there are two limitations when applying to a real utility system. First, their threshold region is based on engineering judgment, which has difficulty in applying to a real utility system. Secondly, their estimation by convention threshold models requires evaluation of threshold number L.

To overcome the above limitations, a new algorithm, comprised of the following two advanced elements, is

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proposed: (i) a mathematically proven threshold region, and (ii) estimation of the L-threshold number by the stratification rule.

3 Optimum stratification

Following the stratified rule [12, 131, N observations {Il, X2, ..., 1,) are divided into L nonoverlapping subpopulations and are called strata. Assume that N observations {XI, X,, ..., X,} are a heterogeneous pop- ulation. N observations of signal X , are stratified into strata, each being homogeneous internally. Our pro- posed stratification process comprises of two major steps: (1) constructing the L-threshold, and (2) determining threshold number L.

3. I To construct the threshold for N observations {XI, X,, ..., X,} from a series (X,} , one needs to arrange variable X,, i = 1, 2, ..., N into the ascending order. The cum3@ rule, a natural extension of the cum3df rule, is adopted in our proposed algorithm due to Singh [13]. Given the frequency distribution of load {X,} , denoted by AX,), the cum3d' rule forms the cumulative of 'dr(XJ1 rule and selects Xt so that they create equal intervals on the cum3df scale. This example data consisted of a total of five days (i.e. N = 5 days * 24hrs = 120 examples). Referring to Table 1, Xl = 9523.4MW and X , = 17246MW. Assume that L = 3 and following the cum3dfrule, an equal interval on the cum3dm)] scale can be calculated as 21.900513 = 7.3001, which yields the threshold region in Fig. 1.

Table 1: Stratification by Cum 3df rule applied to exam- ple problem

Load IO3 x MW

Construction of the L-threshold

36f Cum36f

4 1.5874 1.5874 10.303-1 1.075 15 2.4662 4.0536 11.075-11.846 13 2.3513 6.4049

11.846-12.618 4 1.5874 7.9923

12.61 8-1 3.389 4 1.5874 9.5797

13.389-14.160 6 1.8171 11.3968

14.160-14.932 15 2.4662 13.8630 14.932-15.703 27 3.0000 16.8630

15.703-16.474 15 2.4662 19.3292 16.474-17.246 17 2.5713 21.9005

Frequency

9.5324-1 0.303

3.2 Optimum number of thresholds The number of thresholds can be obtained by observ- ing the reduction in variance, as affected by adding another threshold. That is, the variance from threshold L is compared with the variance resulting from L - 1 threshold ( V ( X t ) J V ( Z f ) J . Cochran [ 121 suggests that the number of threshold (L) be selected at the lowest decreasing rate of V(x,) . Herein, L is selected in our algorithm [12] at the gradual smooth values of the VDRL defined as:

L L-1

(4) 3=1 3=1

where V J is the variance of the population in threshold j ; W, = N,IN is the weight of threshold j ; and NI is the

IEE Proc -Gener Transm Dlsmb, Vol 144, No 5, September 1997

threshold sizes of threshold j . The optimum number of thresholds from eqn. 4 can be determined by finding the values of y Y / E G / y y from N observa- tions of signal X,. In our afgorithm, VDR , i s evaluated iteratively, i.e. L = 2 at the first iteration and L = L + 1 in the subsequent iterations. As Table 1 indicates, the selected L by evaluating VDR, is 3, where X>l M( 5 (32 x 3.4 x lo5 + 29 x 8.201 x lo5 + 59 x 4.791 x lo5)/ 120 (see Table 2). On solving eqn. 4, the optimum number of thresholds is found to be 3 (see Fig. 1). Thus, by taking equal intervals (22.1524/3 = 7.3841) on the cum3v‘’in Table 1, the threshold region is obtained as rl = 11846 MW, r2 = 14932MW.

180001

16000

3 ILOOO r - 12000 U 0

10000 v 8000 I,-7 0 LO 80 120

time,h Fig. 1 lem

Select threshold region by Cim ’@rule applied to example prob-

Table 2: Select number of threshold by threshold region variances of the example problem

Variance decreasing Number of Weighting

threshold (L) variance (2) rate (VDRL)

1 5144166.81 - 2 861682.50 5.9698 3 524398.58 1.6432 4 316496.48 1.6568 5 218368.89 1.4494

4 Sequential description of proposed algorithm

Given N load values from a series {X,}, a threshold model can be established by corresponding each of the L component models separately to the appropriate subset of the load values. The algorithm proceeds as follows: Step 1: Arrange X,s into an ascending order Step 2: Construct threshold by the cum3d’ rule pre-

sented in Section 3. Step 3: Find variance decreasing rate VDRL by eqn. 4. Step 4: If VDRL < VDRL-,, let L = L + 1 and go to

step 3; otherwise, go to step 5. Our proposed algorithm suggests that number of thresholds (L) be selected at the gradual smooth values of the VDR,.

Step 5: Find delay parameter k of individual threshold AR(K) models by inspecting the graphs of partial autocorrelation function (PACF) of the series.

Step 6: Estimate the coefficients aoO, nlG), ..., akjG) in eqn. 3 by using the standard least-squares method.

Table 3: Stratification by Cum 3df rule applied to load for July of 1993 ~ ~~ ~ ~ ~

3df Cum3v’f Frequency Load IO3 x M W

97574-9.63 13 9.6313-10.205

10.205-10.779 10.779-1 1.352 11.352-1 1.926 11.926-12.500 12.500- 13.074 13.074-1 3.648 13.648- 14.222 14.222- 14.796 14.796- 15.370

15.370-15.944 15.944-1 6.518

16.518-17.092 17.092- 17.666

6 12

53 50 37 18 17 13 31 45 87 93

58 47

33

1.7100 2.2894

3.7563 3.6840 3.3322 2.6207 2.5713 2.3513 3.1414 3.5569 4.4310 4.5307

3.8709 3.6088

3.2075

1.7100 3.9974

7.7557 11.4397 14.77 19 17.3926 19.9639 22.3152 25.4566 29.0135 33.4445 37.9752

41.8461 45.4549 48.6624

Table 4: Select number of threshold by threshold region variances of the load for July of 1993

Number of Weighting Variance decreasing

threshold (L) variance (2) rate (VDRL)

1 5160866.81 -

2 902222.50 5.7201 3 541648.36 1.6657 4 328384.28 1.6494 5 205524.72 1.5978

5 Numerical tests

5.1 Results of proposed algorithm In this case study, actual data from the Taipower utility was used to verify the algorithm’s effectiveness in terms of short-term load forecasting. The data consisted of 25 days from the 1st to the 31st July (i.e. 25 days * 24hrs = 600 examples) of 1993 after deleting weekends. This month was characterised by rapidly variant weather (particularly for the l l th , 12th and 29th July). Above data were used to forecast the load demand from the 2nd to the 6th August individually. The computation results are summarised below: Stage 1 : Construction of the L-threshold-Tables 3

and 4 summarise the sequential results. As those Tables reveal, the selected number of thresholds (L) is 3.

Stage 2: Following the cum3@ rule, an equal interval on the cum3dflXt)] scale can be calculated as 48.662413 = 16.2208, which yields the thresh- old region in Table 3.

Stage 3: Table 5 lists the coefficients aoG), alh), ..., ak,O.

Stage 4: Table 6 displays their forecasting accuracies for those five days.

479 IEE Proc-Gener. Transm. Distrib., Vol. 144, No. 5, September 1997

Table 5: Parameter estimation of threshold AR(k) models for the month of July of 1993

Threshold Terms Parameter code i values ak

1 x(t-I)

x(t-2)

x(t-5)

2 x(t-2)

x(t-5)

3 x(t-1)

x(t-5)

~ ( t - 6 )

x(t-12)

~ ( t -13 )

0.617324

-0.539256

0.332346

-0.21 3627

0.204671

0.5872 14

-0.21 3467

-0.201472

0.326421

0.306147

Table 6: Peak/hourly load forecast comparison results August 2-6 of 1993

Threshold AR models

Peak forecasts Day type

Hourly forecasts

Absolute error >300MW

Absolute error (MW)

Absolute error (MW) (times)

Monday 204.81 1 228.83

Tuesday 140.32 1 204.46

Wednesday 114.75 1 208.66

Thursday 85.18 0 186.11

Friday 85.79 0 188.60

Averagehotal 126.17 3 203.33

The results for the proposed algorithm were obtained by backcasting the system peak and hourly loads for the 2nd to the 6th August 1993. Table 6 lists the results derived by the threshold AR models as broken down in the five day types starting from Monday. The first column represents the average absolute peak load forecast error, the second column the number of large peak load forecast errors and the third column the average absolute hourly load forecast error. As this table reveals, the threshold AR models forecasts corresponded to the actual load sample more precisely and reliably, on the average absolute peak load, throughout the forecasting period (only three times the error was larger than 300MW). Similar precision improvements were achieved in the hourly forecasts as well. However, as expected, the hourly load forecasts are less precise than the peak load forecasts using the same model. This same Table also indicates that the peak load forecast errors are less than the hourly load forecast errors. The average absolute hourly load forecast error for all day types is 203.33MW, while the average absolute peak load forecast error is 126.17MW, which amounts to a decrease of 61.16 present.

5.2 Comparison with AR model The proposed algorithm is tested for short-term load forecasting using actual data from Taipower utility. The algorithm is also compared with AR model the results applied to the same data. Both models of errors should be described when assessing the achievements of

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a short-term load forecasting algorithm. However, from a real custom of view, large errors weaken a system operator’s confidence in a short-term load forecasting algorithm. The Taipower utility operations personnel consider a 300MW peak load forecast error to be the experience rule distinguish index between accurate and inaccurate forecasts. Restated, when the error exceeds this index, 300 MW, substantial problems arise in load management program, reserve portion, and security evaluation. For illustrative purposes, the proposed algorithm and AR(24) model were obtained by backcasting the system peak loads for August 1993 in Fig. 2. As this Figure reveals, the AR(24) model did not perform adequately under these case (fifteen times the error was larger than 300MW). However, the forecasts of the threshold models corresponded to the actual load sample more precisely and reliably, on the average, throughout the entire month (only three times the error was large than 300MW). Such a high achievement is due primarily to (a) the more accurate AR modelling in threshold models and (b) the threshold AR models’ capability to respond rapidly to sudden changes.

300

fidence region

-300

-800/1 0 5 10 15 20 25

time, h Fig.2

~ threshold model _ ~ _ _ AR(24) model

Peak load forecast errors for August of 1993

Similar accuracies and reliability improvements were also achieved in the hourly and peak loads forecasts for 1993. The data consisted of a total of 261 days from the 1st January to the 31st December (i.e. 261 days * 24hrs = 6264 examples) of 1992 after deleting weekends. The parameters were regularly updated at the end of each month and the new model used for forecasting the following month. The constructing threshold and determining threshold number were regularly updated at the end of each season and the proposed model was used for forecasting the following season. Table 7 summarises the results derived by the two models broken down in the five day types starting from Monday. As this table indicates, the achievement was improved for all day types. For the Thursday day type, for instance, the average absolute peak load forecast error for 1993 was reduced from 372.36MW to 75.18MW, which amounts to an improvement of 395.29%. Moreover, the number of large peak load forecast errors (large than 300MW) was reduced from 34 to 2, which amounts to an improvement of 1600%.

IEE Proc -Gener Transm Distrib , Vol 144, No 5, Septembev 1997

Table 7: 1993 Peak/hourly load forecast comparison results

Threshold AR models AR(24)

Peak forecasts Day type

Hourly forecasts

Peak forecasts Hourly forecasts

Absolute error Absolute error Absolute Absolute error (MW) error (MW) >300MW Absolute

error (MW) >300MW Absolute error (MW) (times) (times)

Monday 192.85 7 208.83 482.19 37 348.22

Tuesday 112.32 6 202.46 206.36 23 307.18

Wednesday 104.75 7 192.66 399.58 39 336.52

Thursday 75.18 2 176.11 372.36 34 310.26

Friday 80.79 2 178.60 467.35 45 362.10

Averagehotal 113.18 24 191.73 385.57 178 332.86

6 Conclusions

The short-term load forecasting model is an essential decision support tool for operating the power system economically and securely. The proposed algorithm combines the use of a threshold model via the stratification rule. According to simulation results, we can conclude the following: (1) The optimum stratification rule attempts to remove any judgmental input and to render the threshold proc- ess entirely mechanistic. (2) The simplicity of the proposed threshold autore- gressive model varies under different perspectives, such as the piecewise linear algorithms and the threshold procedures of the stratification to effectively handle nonstationarity. Therefore, the simplicity consists of finding architectures which are autoregressive to model the nonlinearities of the series, and economical in terms of parameters. (3) Results obtained from an AR model yield an aver- age absolute peak load forecast error that is fairly high, even through the AR can model the nonlinearity of the data. The reason for this, as explained above, is that this load data is nonstationary, thereby making it extremely difficult to model. (4) Such a high level of achievement is due primarily to a more accurate AR modelling in a threshold model, and the threshold AR model’s ability to respond rap- idly to sudden changes.

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