Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 589374, 10 pages
http://dx.doi.org/10.1155/2015/589374

Research Article

Short-Term City Electric Load Forecasting with Considering Temperature Effects: An Improved ARIMAX Model

Herui Cui and Xu Peng

School of Economics and Management, North China Electric Power University, Baoding 071003, China

Received 13 April 2015; Revised 9 June 2015; Accepted 21 June 2015

Academic Editor: Marco Mussetta

Copyright © 2015 Herui Cui and Xu Peng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Short-term electric load is significantly affected by weather, especially the temperature effects in summer. External factors can result in mutation structures in load data. Under the influence of the external temperature factors, city electric load cannot be easily forecasted as usual. This research analyzes the relationship between electricity load and daily temperature in city. An improved ARIMAX model is proposed in this paper to deal with the mutation data structures. It is found that information amount of the improved ARIMAX model is smaller than that of the classic method and its relative error is less than AR, ARMA and Sigmoid-Function ANN models. The forecasting results are more accurately fitted. This improved model is highly valuable when dealing with mutation data structure in the field of load forecasting. And it is also an effective technique in forecasting electric load with temperature effects.

1. Introduction

Short-term load forecasting (STLF) is mainly used to forecast the power load for the next few days or week [1–3]. It plays an important role in the modern electricity Demand Side Management (DSM), as its accuracy directly affects the economic cost of operators in the electricity market. Accurate load forecasting is helpful for security, stability, and economic operation in power grid. It is also advantageous in making reasonable arrangements for maintenance plan. Meanwhile, power load forecasting can optimize power system dispatch and reduce production cost.

Short-term daily peak power load in summer fluctuates regularly, showing an obvious periodical characteristic. It is greatly affected by temperature, wind, precipitation, and other meteorological factors. There are significant mutation structures in load data [4–6]. There are traditional methods in power load forecasting, such as regression model, gray model, support vector machines, neural networks, and time series. Ramón Cancelo et al. [7] used Red Eléctrica de España (REE) to forecast the electricity load from a day to a week ahead. Hipperta et al. [8] adapted large neural networks in electricity load forecasting to handle nonlinear time series data. Felipe Amarala and Castro Souza [9] used smooth transition periodic autoregressive (STPAR) models for short-term load forecasting. Amjady and Keynia [10] proposed a new neural network learning algorithm based on a new modified harmony search technique. This learning algorithm is widely used to search the solution space in various directions, by which overfitting problem and trapping in local minima and dead bands can be avoided. Wangdi et al. [11] adapted ARIMAX model to determine predictors of malaria for the subsequent month. And the test showed that prediction accuracy has been greatly improved. Chadsuthi et al. [12] studied seasonal leptospirosis transmission and the association with rainfall and temperature by using ARIMAX model showing that factoring in rainfall (with an 8-month lag) yields the best model for the northern region. The above forecasting methods are obviously effective in dealing with mutation structures and intelligent algorithms. However, they are not ideal in practical operation due to the limitation of data and laboratory equipment. The generalization capability is also weak. Traditional time series forecasting methods highlight the time role, without considering the external factor effects. Thus, the forecasting accuracy of time series methods is poor, with obvious defect [13, 14]. Based on the above research, an improved ARIMAX model is proposed here by combining the traditional time series with regression analysis to forecast short-term electric load, which has a strong practice value in the short-term power load forecasting field. This model fills the gaps of external effects on electric load. The prediction result showed that the improved ARIMAX model has a smaller model information amount than or [15, 16].

2. Sigmoid-Function ANN Model

ANN (Artificial Neural Network) is very practical forecasting technology in short-term electric load forecasting fields, especially for those nonlinear data. The basic concepts about ANN are shown as

where is the state of network unit . Consider

Through the operating of first-order derivative to output unit, is obtained as follows:

The specific algorithm of Sigmoid-Function ANN model is shown as follows:

(1)The initial value of weight or threshold is defined as , while is small random number.(2)Training samples are input vector , ; expectation output , . Steps from to are carried out for each input sample.(3)Computing actual output and the state of hidden units in network

(4)Calculation training error

(5)Correcting weights and thresholds

(6)When index is located at , judging if .

3. Time Series Theory

Time series is a typical time-domain analysis method. It can be used to reveal the internal laws of the sequences from the perspective of autocorrelation.

Typical time-domain analysis steps are the following:

(1)Observing sequence features.(2)Selecting the appropriate fitted model according to the features computed by SAS.(3)Model testing and optimization process.(4)Using fitted model to infer the nature of sequence.

Core contents of time series analysis method are proposed by American statistician George E. P. Box and United Kingdom statistician Gwilym M. Jenkins in their book Time Series Analysis Forecasting and Control, in which it is called autoregressive moving average model (ARIMA). Some important concepts are displayed here.

Stationarity. is set as time series, and , ( is positive integer) , , ( is integer), , named for strictly stationary time series.

White Noise. Time series meet the condition∀     ; then

named for white noise sequence or displayed as

Definition 1. The model is named autoregressive moving average model, if it contains the following structures, abbreviated as

Introduced delay operator , it can also be presented as

Cointegration Theory. The cointegration theory was put forward by Engle and Granger in 2001 [17]. Model can be calculated without the requirement that all sequences are stationary, if the cointegration relationship is obvious. The typical cointegration test is test [18–20].

Definition 2. Supposing that the response variable and the input variable sequences are all stationary, the regression model is established in response to the input variable sequences and response sequences:

In the actual modeling process, an improved ARIMAX model is proposed to forecast the short-term electric load. The specific process is displayed below.

4. The Improved ARIMAX Modeling Process

4.1. Modeling Steps

These are as follows

(1)Perform logarithmic transformation on the original response sequence and the inputted sequences in order to meet the homogeneity of variance assumption.(2)Checking the stationarity of logarithmic transformation sequences,

 If the sequences are stationary, move on to the next step; if not, conduct differential operation to the logarithmic sequences and testing stationarity again; then execute the second-order differential operation until the stationarity is satisfied.(3)Establishing the ARMA model about ,

  are N-order difference stationary inputted sequences.(4)Establishing the ARMA model about ,

  are the N-order difference stationary response sequences.(5)Exploring the correlation coefficient between the stationary N-order difference logarithmic sequences “” and “” to determine the structure of improved ARIMAX model. This step is the improved part for traditional ARIMAX model. Therefore, the revised ARIMAX model can be calculated as follows:

(6)Fitting residual sequence

  is a zero mean white noise sequence.

Based on the above steps the improved ARIMAX model can be applied into load forecasting process.

4.2. Modeling Flowchart

See Figure 1.

Figure 1: ARIMAX modeling flowchart.

5. Load Forecasting with ARMA Model

5.1. Load Data

The table in the appendix shows the daily maximum power load data and the maximum temperatures in a city from 1st June to 14th August (see Table 12). In this paper, the data is used to explore the classical time series models and ANN models are used to firstly forecast load. In Section 6, an improved ARIMAX model is established to compare the prediction accuracy [21–23].

It can be seen in Figure 2 that load data has an upward trend and clear cyclical fluctuations by observing the sequences, showing that the sequences are nonstationary [24, 25].

Figure 2: Load and temperature figure.

5.2. Establishing ARMA Model

After the time series analysis on the load data by SAS software, autocorrelation table is previously mentioned. Table 1 shows that the autocorrelation coefficients of the sequences are always positive [26–28]. It can be inferred that daily peak power load data is nonstationary series with a monotonic trend, which is shown in Figure 2.

Table 1: Autocorrelation table.

Lag
Correlation
−1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
Std. error

0
1
∣                                        ∣********************∣
0

1
0.93898
∣                              .        ∣******************* ∣
0.114708

2
0.86458
∣                        .              ∣*****************      ∣
0.190683

3
0.81703
∣                      .                ∣****************        ∣
0.236709

4
0.78057
∣                  .                    ∣****************        ∣
0.271289

5
0.74394
∣                .                      ∣***************            ∣
0.299386

6
0.70116
∣              .                        ∣**************             ∣
0.322794

7
0.66555
∣            .                          ∣*************.                 ∣
0.342248

8
0.63503
∣            .                          ∣*************.                 ∣
0.358874

9
0.60586
∣          .                            ∣************    .              ∣
0.373367

10
0.5797
∣          .                            ∣************    .              ∣
0.386086

11
0.54107
∣        .                              ∣***********        .            ∣
0.397374

At the same time, the partial autocorrelation table can be obtained. Table 2 shows that only the first-order partial autocorrelation coefficient is significantly greater than two-time standard errors [29]. The rest partial autocorrelation coefficients rapidly decline to zero, making random fluctuations within two-time standard deviation ranges. Thus it can be regarded as the first-order truncation.

Table 2: Partial autocorrelation table.

Lag
Correlation
−1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

1
0.93898
∣                              .        ∣*******************∣

2
−0.14458
∣                              . ***∣          .                                      ∣

3
0.2036
∣                              .        ∣****.                                      ∣

4
0.01297
∣                              .        ∣          .                                    ∣

5
0.01204
∣                              .        ∣          .                                    ∣

6
−0.05103
∣                              .      *∣          .                                      ∣

7
0.06133
∣                              .        ∣*        .                                      ∣

According to white noise test, statistic (probability) is less than 0.05; thus the sequence is nonwhite noise. Then, the model is applied to forecast power load data. In residual autocorrelation coefficient test about model, it is shown that statistic is larger than 0.05; thus this model applies.

After the SAS processing, model can be presented as

In order to optimize the ARMA model, the minic option is used to detect the best order [30]. Setting model as , the option detects the . The model is presented as

6. Load Forecasting with Improved ARIMAX Model

6.1. Testing Statistics

Regardless of the kinds of models, the value of test statistic is significantly greater than 0.05 by ADF test. Daily maximum power load data series are markedly nonstationary. Therefore, the following analysis is conducted on nonstationary data sequence.

Firstly, performing logarithmic transformation on the original sequence,

Thus the sequences can meet the homogeneity of variance. The white noise test of sequence indicates that the sequence is a nonwhite noise sequence. Unit root test shows that the value of statistic is significantly greater than 0.05. It is suggested that sequence is nonstationary. There is one unit root in sequence at least. And the analysis on sequence is similar to that of [23–25].

Secondly, operating first-order differential operators to and sequences to get stationary and ,

Thirdly, operate stationary test and white noise test to logarithmic sequence after first-order differential and . The test result shows that the value of white noise test is greater than 0.05, which means that and sequences are pure random white noise sequences [26]. And the value of statistic is less than 0.05, showing that and sequences are stationary series. Until now, the test analysis has been finished.

6.2. Computing and Sequences Model

Firstly, the model is established. The test shows that is a stationary white noise sequence; thus the fitting model is

Secondly, the model is established (see Table 6). The test results obtained by SAS from Tables 3 to 5 show that is a stationary white noise sequence (the value in Table 3 is larger than 0.05, while the value in Table 4 is smaller than 0.05). The best order for model is . Therefore, the fitting model is or model. The constant term is not significant, using the noint option to remove the intercept. The final fitting model is shown as [31]

Table 3: Autocorrelation check for white noise.

To lag
Pr > ChiSq

12
0.1066

18
0.2526

Table 4: Augmented Dickey-Fuller unit root tests.

Type
Lags
Pr < Rho
Pr < Tau
Pr >

Zero means
0
<0.0001
<0.0001

1
0.0001
<0.0001

2
0.0001
<0.0001

Single mean
0
0.0007
0.0001
0.001

1
0.0001
0.0001
0.001

2
0.0002
0.0001
0.001

Table 5: Minimum information criterion.

Lags
MA()
MA()
MA()
MA()
MA()
MA()

AR()
−5.14349
−5.22355
−5.48448
−5.51557
−5.51716
−5.51704

AR()
−5.13228
−5.19252
−5.42776
−5.4607
−5.4677
−5.48567

AR()
−5.35327
−5.40243
−5.37285
−5.41553
−5.41986
−5.46426

AR()
−5.3958
−5.42964
−5.39755
−5.36473
−5.3641
−5.41041

AR()
−5.40213
−5.42436
−5.39489
−5.36661
−5.30905
−5.37159

AR()
−5.4457
−5.422
−5.40418
−5.3893
−5.33506
−5.32363

Minimum table value: BIC(0,4) = −5.51716

Table 6: Fitting parameter of model.

Parameter
Estimate
Standard error
-value
Approx. Pr >
Lag

MA()
0.45098
0.10373
4.35
<0.0001
1

6.3. Computing Load Data with Improved ARIMAX Model

The above model is used to filter input variable sequence and the response variable sequence . The mutual relationships numbers between the independent variables and the response variable are calculated after filtration by ARIMA analysis process.

It can be found in Table 7 that only the 0-order delay mutual relationship number is significantly nonzero, which means that there is no hysteretic effect between response sequence and input sequences. Therefore, the model should be treated in the same period.

Table 7: Relationship number table of and .

Lag
Correlation
−1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

−3
0.05145
∣                              .           ∣*      .                                      ∣

−2
−0.24894
∣                            *****∣        .                                      ∣

−1
−0.17628
∣                              .****  ∣        .                                      ∣

0
0.49787
∣                              .           ∣**********                      ∣

1
0.15016
∣                              .           ∣*** .                                      ∣

2
−0.02468
∣                              .           ∣        .                                      ∣

3
−0.05311
∣                              .       *∣        .                                      ∣

The regression analysis in Table 8 shows that the final regression coefficient is 0.37098.

Table 8: Parameter estimates of REG procedure.

Variable
Dependent variable
Parameter estimates
Standard error
-value
Pr >

0.37098
0.07442
4.98
<0.0001

The statistics test is carried out on residual sequence, showing that the residual sequence is stationary white noise sequence (). The fitted model for residual sequence is , and is zero mean white noise sequence [32–35].

It is known that there is significant correlation in the zero-order between the two sequences in Table 7. The same period model is established between and , based on the parameter estimates in Table 8 and tests in Tables 9~10. The value in Table 9 is larger than 0.05; thus the autocorrelation check of residuals shows that the model is effective for forecasting loads:

Table 9: Autocorrelation check of residuals.

To lag
Pr > ChiSq

6
0.0578

12
0.0687

18
0.1588

Table 10: Crosscorrelation check of residuals with input .

To lag
Pr > ChiSq

5
0.0581

11
0.0634

17
0.0732

The load from 15th to 31st is forecasted according to the improved ARIMAX model.

By operating logarithm to the forecasting results “,” the next 15-day maximum load can be obtained, which is shown in Table 11.

Table 11: Predictions of different models.

Obs.
Actual power load data
Prediction value of improved ARIMAX
Prediction value of
Prediction value of
Prediction value of Sigmoid-Function ANN

16
2685
2678
2648
2699
2698

17
2833
2820
2810
2855
2844

18
2924
2930
2917
2934
2934

19
3112
3122
3128
3130
3128

20
2393
2400
2428
2441
2412

21
2570
2600
2650
2661
2598

22
2689
2709
2723
2741
2699

23
2730
2788
2796
2801
2812

24
2597
2638
2654
2687
2611

25
2216
2219
2225
2258
2215

26
1935
1929
1971
1995
1952

27
1826
1822
1910
1984
1857

28
1897
1850
1927
1968
1904

29
1945
1926
2018
2100
1965

30
2244
2230
2101
2154
2264

31
2261
2258
2177
2137
2274

AIC Criterion
−72.7
381.4
433.8
78.6

SBC Criterion
−71.5
383.9
436.6
79.2

MAE
0.0037
0.0389
0.0494
0.0185

Table 12: Daily maximum power load and temperature data in some city in 2010.

Time
Peak power load
Maximum temperature
Time
Peak power load
Maximum temperature

1st Jun.
1787
24.9
9th Jul.
2662
33.3

2nd Jun.
1752
23.5
10th Jul.
2679
35.6

3rd Jun.
1804
26.3
11th Jul.
2383
32.6

4th Jun.
1858
28.7
12th Jul.
2473
32.2

5th Jun.
1804
30.2
13th Jul.
2596
35.6

6th Jun.
1810
31.1
14th Jul.
2429
31.1

7th Jun.
1790
25.9
15th Jul.
2585
32.8

8th Jun.
1788
23.8
16th Jul.
2558
34.5

9th Jun.
1739
24.2
17th Jul.
2628
35.2

10th Jun.
1791
25.5
18th Jul.
2739
35.1

11th Jun.
1822
28.2
19th Jul.
2883
35.8

12th Jun.
1848
28.9
20th Jul.
2896
36.5

13th Jun.
1811
22.8
21st Jul.
2829
36.8

14th Jun.
1746
28
22nd Jul.
2812
36

15th Jun.
1652
32.1
23rd Jul.
2841
35.6

16th Jun.
1698
33.2
24th Jul.
2746
34.8

17th Jun.
1983
29.9
25th Jul.
2672
32.9

18th Jun.
2106
33.4
26th Jul.
2642
32.6

19th Jun.
1881
27.7
27th Jul.
2791
35.3

20th Jun.
1869
29.7
28th Jul.
3026
36.4

21st Jun.
2069
32.2
29th Jul.
3105
36.8

22nd Jun.
2109
30.3
30th Jul.
3054
37

23rd Jun.
2007
26.9
31st Jul.
2870
36.2

24th Jun.
1869
27.5
1st Aug.
2999
36.7

25th Jun.
1994
30.3
2nd Aug.
3256
38

26th Jun.
2014
31.4
3rd Aug.
3286
38.7

27th Jun.
2059
32
4th Aug.
3404
39.5

28th Jun.
2163
29.1
5th Aug.
3464
40.5

29th Jun.
2237
32
6th Aug.
2766
32.9

30th Jun.
2456
35.8
7th Aug.
2646
33.9

1st Jul.
2708
35.9
8th Aug.
2847
35.7

2nd Jul.
2760
35.9
9th Aug.
3094
36.8

3rd Jul.
2829
35.6
10th Aug.
3310
38.8

4th Jul.
2972
37
11th Aug.
3348
39

5th Jul.
2829
32.1
12th Aug.
3357
37.5

6th Jul.
2387
28.5
13th Aug.
3290
37.8

7th Jul.
2528
33.7
14th Aug.
3265
37.8

MAE (Mean Absolute Error) is computed as follows:

where is the prediction value, is the actual value, and is the sample size.

It can be seen in Table 11 that the MAE of the improved ARIMAX model is the minimum, Sigmoid-Function ANN ranked second small, followed by AR model, and ARMA model is the maximum. It means that the improved ARIMAX model is better than S-ANN, AR, or ARMA model according to the AIC and SBC Criterion in Table 11. The revised ARIMAX model is more effective, by which more accurate load results can be obtained.

It can be seen in Figure 3 that the blue line is the actual daily maximum power load data, while the red line is the forecasting data of improved ARIMAX. The difference between improved ARIMAX model and actual power load data is the minimum among these models. Residual stationarity and white noise test show that the residual is stationary white noise sequence, showing that . There is second-order delay correlation between and . The final fitting model is

Figure 3: Forecasting graphics.

7. Conclusion

Based on the above analysis, the improved ARIMAX model can effectively dig up self-related information of load data. As an effective method for short-term load forecasting, the model can get a more accurate prediction result than traditional time series models. Prediction accuracy of this model is greatly improved, which is of high value in engineering application area. It is verified by relative error analysis of ARMA and the improved ARIMAX that the revised model has higher prediction accuracy than usual forms.

Notation

: | The state of network unit
: | Output unit
: | Output (hidden) layer unit
: | Initial value of weight or threshold
: | Training samples
: | Accuracy requirements
: | Time series
: | Mean of time series
: | Autoregressive coefficient
: | Random interference coefficient
: | -order moving average coefficient polynomials
: | Residual sequence moving average coefficient polynomials
: | Delay operator
: | Standard deviation.

Conflict of Interests

The authors declare no conflict of interests.

Acknowledgment

The authors gratefully acknowledge the financial support from the National Natural Science Fund of China (no. 71471061).

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