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

Authors: Herui Cui; Xu Peng
Journal: Mathematical Problems in Engineering
 (2015)

---

## 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.


---

## Body

## 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 AR(p) or ARMA(p,q) [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(1)Sigmoid function:fNetkj=11+e-Netkj,where Netkj is the state of network unit uj. Consider(2)The output unit:okj=11+exp⁡-∑iωjioki-θj=11+e-Netkj.Through the operating of first-order derivative to output unit,f′(Netkj) is obtained as follows:(3)f′Netkj=δokjδNetkj=okj1-okj,Output layer unit:δkj=t∂j-okjokj1-okj,Hidden layer unit:δkj=okj1-okj∑mδkmωmj,Weight tuning function:Δωjit+1=ηδkjokj.The specific algorithm of Sigmoid-Function ANN model is shown as follows:(1)
The initial value of weight or threshold is defined asωji(0), while θj(0) is small random number.(2)
Training samples are input vectorXk, (k=1,2,…,P); expectation output dk, (k=1,2,…,P). Steps from (3) to (5) are carried out for each input sample.(3)
Computing actual output and the state of hidden units in network(4)okj=fj∑iωjioki+θj.(4)
Calculation training error(5)Output layer:δkj=t∂j-okjokj1-okj,Hidden layer:δkj=okj1-okj∑mδkmωmj.(5)
Correcting weights and thresholds(6)ωjit+1=ωjit+ηδjoki+αωjit-ωjit-1,θjit+1=θjt+ηδj+αθjt-θjt-1.(6)
When indexk is located at P, judging if E≤ε.


## 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 bookTime Series Analysis Forecasting and Control, in which it is called autoregressive moving average model (ARIMA). Some important concepts are displayed here.Stationarity. xt is set as time series, and m∈ȼ∗, (ȼ∗ is positive integer) t1,t2,…,tm∈T, ∀τ∈ȼ, (ȼ is integer), ∃Ft1,t2,…,tm(x1,x2,…,xm)=Ft1+τ,t2+τ,…,tm+τ(x1,x2,…,xm), named xt for strictly stationary time series.White Noise. Time series xt meet the condition(1)∀t∈T,∃EXt=μ  (2)  ∀t,s∈T; then(7)γt,s=σ2,t=s0,t≠snamed xt for white noise sequence or displayed as (8)Xt~WNμ,σ2.Definition 1.
The model is namedautoregressive moving average model, if it contains the following structures, abbreviated as ARMA(p,q):(9)xt=ϕ0+ϕ1xt-1+⋯+ϕpxt-p+εt-θ1εt-1-⋯-θqεt-q,ϕp≠0,θq≠0,Eεt=0,Varεt=σε2,Eεtεs=0,s≠t,Exsεt=0,∀s<t.Introduced delay operatorB, it can also be presented as (10)ΦBxt=ΘBεt.Cointegration Theory. The cointegration theory was put forward byEngle 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 EG test [18–20].Definition 2.
Supposing that the response variableyt and the input variable sequences x1,…,xk are all stationary, the regression model is established in response to the input variable sequences and response sequences:(11)yt=μ+∑i=1kΘiBΦiBBlixit+εt,εt=ΘBΦBat,abbreviated asARIMAXmodel.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,(12)ln⁡yt;ln⁡x1,…,ln⁡xk.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∇nln⁡xt,(13)∇nln⁡xit=ΘxiBΦxiBεxit.∇nln⁡xt are N-order difference stationary inputted sequences.(4)
Establishing the ARMA model about∇nln⁡yt, (14)∇nln⁡yit=ΘyiBΦyiBεyit.∇nlnyt are the N-order difference stationary response sequences.(5)
Exploring the correlation coefficient between the stationary N-order difference logarithmic sequences “∇nln⁡yt” and “∇nln⁡xt” 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:(15)yt∗=μ+∑i=1kΘiBΦiBBlixit∗+εt.(6)
Fitting residual sequenceεt(16)εt=ΘBΦBat.at 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 Figure1.Figure 1
ARIMAX modeling flowchart.


## 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,(12)ln⁡yt;ln⁡x1,…,ln⁡xk.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∇nln⁡xt,(13)∇nln⁡xit=ΘxiBΦxiBεxit.∇nln⁡xt are N-order difference stationary inputted sequences.(4)
Establishing the ARMA model about∇nln⁡yt, (14)∇nln⁡yit=ΘyiBΦyiBεyit.∇nlnyt are the N-order difference stationary response sequences.(5)
Exploring the correlation coefficient between the stationary N-order difference logarithmic sequences “∇nln⁡yt” and “∇nln⁡xt” 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:(15)yt∗=μ+∑i=1kΘiBΦiBBlixit∗+εt.(6)
Fitting residual sequenceεt(16)εt=ΘBΦBat.at 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 Figure1.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 Table12). 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 Figure2 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. Table1 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
∣        .                              ∣***********        .            ∣
0.397374At the same time, the partial autocorrelation table can be obtained. Table2 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, statisticP (probability) is less than 0.05; thus the sequence is nonwhite noise. Then, the AR(1) model is applied to forecast power load data. In residual autocorrelation coefficient test about AR(1) model, it is shown that statistic P is larger than 0.05; thus this model applies.After the SAS processing,AR(1) model can be presented as (17)xt=2895.82+εt1-0.90534∗B.In order to optimize the ARMA model, the minic option is used to detect the best order [30]. Setting ARMA(p,q) model as xt=μ+Θ(B)/Φ(B)εt, the option detects the ARMA(3,2). The ARMA(p,q) model is presented as (18)xt=2949.334+1-0.97485B+0.3208B21-1.97331B+1.49699B2-0.52368B3εt.


## 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 Table12). 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 Figure2 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. Table1 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
∣        .                              ∣***********        .            ∣
0.397374At the same time, the partial autocorrelation table can be obtained. Table2 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, statisticP (probability) is less than 0.05; thus the sequence is nonwhite noise. Then, the AR(1) model is applied to forecast power load data. In residual autocorrelation coefficient test about AR(1) model, it is shown that statistic P is larger than 0.05; thus this model applies.After the SAS processing,AR(1) model can be presented as (17)xt=2895.82+εt1-0.90534∗B.In order to optimize the ARMA model, the minic option is used to detect the best order [30]. Setting ARMA(p,q) model as xt=μ+Θ(B)/Φ(B)εt, the option detects the ARMA(3,2). The ARMA(p,q) model is presented as (18)xt=2949.334+1-0.97485B+0.3208B21-1.97331B+1.49699B2-0.52368B3εt.


## 6. Load Forecasting with Improved ARIMAX Model

### 6.1. Testing Statistics

Regardless of the kinds of models, theP value of test statistic τ 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,(19)x,y⟶ln⁡x,ln⁡y.Thus the sequences can meet the homogeneity of variance. The white noise test ofln⁡x sequence indicates that the ln⁡x sequence is a nonwhite noise sequence. Unit root test shows that the P value of τ statistic is significantly greater than 0.05. It is suggested that ln⁡x sequence is nonstationary. There is one unit root in ln⁡x sequence at least. And the analysis on ln⁡y sequence is similar to that of ln⁡x [23–25].Secondly, operating first-order differential operators toln⁡yt and ln⁡xt sequences to get stationary Δln⁡yt and Δln⁡xt,(20)ln⁡xt,ln⁡yt⟶Δln⁡xt,Δln⁡yt.Thirdly, operate stationary test and white noise test to logarithmic sequence after first-order differentialΔln⁡yt and Δln⁡xt. The test result shows that the P value of white noise test is greater than 0.05, which means that Δln⁡yt and Δln⁡xt sequences are pure random white noise sequences [26]. And the P value of τ statistic is less than 0.05, showing that Δln⁡yt and Δln⁡xt sequences are stationary series. Until now, the test analysis has been finished.


### 6.2. ComputingΔln⁡yt and Δln⁡xt Sequences Model

Firstly, theΔln⁡y model is established. The test shows that Δln⁡y is a stationary white noise sequence; thus the fitting model is(21)Δln⁡y=εt.Secondly, theΔln⁡x model is established (see Table 6). The test results obtained by SAS from Tables 3 to 5 show that Δln⁡x is a stationary white noise sequence (the P value in Table 3 is larger than 0.05, while the P value in Table 4 is smaller than 0.05). The best order for ARMA(p,q) model is [AR(0),MA(4)]. Therefore, the fitting model is ARMA(0,4) or MA(4) model. The constant term is not significant, using the noint option to remove the intercept. The final fitting model is shown as [31](22)Δln⁡xt=1-0.45098B4εt.Table 3
Autocorrelation check for white noise.
To lag
Pr > ChiSq
12
0.1066
18
0.2526Table 4
Augmented Dickey-Fuller unit root tests.
Type
Lags
Pr < Rho
Pr < Tau
Pr >F
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
0.001Table 5
Minimum information criterion.
Lags
MA(0)
MA(1)
MA(2)
MA(3)
MA(4)
MA(5)
AR(0)
−5.14349
−5.22355
−5.48448
−5.51557
−5.51716
−5.51704
AR(1)
−5.13228
−5.19252
−5.42776
−5.4607
−5.4677
−5.48567
AR(2)
−5.35327
−5.40243
−5.37285
−5.41553
−5.41986
−5.46426
AR(3)
−5.3958
−5.42964
−5.39755
−5.36473
−5.3641
−5.41041
AR(4)
−5.40213
−5.42436
−5.39489
−5.36661
−5.30905
−5.37159
AR(5)
−5.4457
−5.422
−5.40418
−5.3893
−5.33506
−5.32363
Minimum table value: BIC(0,4) = −5.51716Table 6
Fitting parameter ofΔln⁡x model.
Parameter
Estimate
Standard error
t-value
Approx. Pr >t
Lag
MA(1,1)
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 sequenceΔln⁡x1⋯Δln⁡xk and the response variable sequence Δln⁡yt. 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 Table7 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 ofΔln⁡y and Δln⁡x.
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 Table8 shows that the final regression coefficient is 0.37098.Table 8
Parameter estimates of REG procedure.
Variable
Dependent variable
Parameter estimates
Standard error
t-value
Pr >t
Δ
ln
⁡
x
Δ
ln
⁡
y
0.37098
0.07442
4.98
<0.0001The statistics test is carried out on residual sequence, showing that the residual sequence is stationary white noise sequence (Pr>0.05). The fitted model for residual sequence is εt=at, and at 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 Table7. The same period model is established between Δln⁡y and Δln⁡x, based on the parameter estimates in Table 8 and tests in Tables 9~10. The P value in Table 9 is larger than 0.05; thus the autocorrelation check of residuals shows that the model is effective for forecasting loads:(23)Δln⁡yt=0.37098Δln⁡xt+εt,εt=at.Table 9
Autocorrelation check of residuals.
To lag
Pr > ChiSq
6
0.0578
12
0.0687
18
0.1588Table 10
Crosscorrelation check of residuals with inputln⁡x.
To lag
Pr > ChiSq
5
0.0581
11
0.0634
17
0.0732The load from 15th to 31st is forecasted according to the improved ARIMAX model.By operating logarithm to the forecasting results “ln⁡y,” 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 ofAR(1)
Prediction value ofARMA(3,2)
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
0.0185Table 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
37.8MAE (Mean Absolute Error) is computed as follows:(24)MAE=∑ty^t-ytT,where y^t is the prediction value, yt is the actual value, and T is the sample size.It can be seen in Table11 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 Figure3 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 εt=at. There is second-order delay correlation between ln⁡y and ln⁡x. The final fitting model is(25)ln⁡yt=2.86647+1.41495ln⁡xt-2+εt,εt=at.Figure 3
Forecasting graphics.


## 6.1. Testing Statistics

Regardless of the kinds of models, theP value of test statistic τ 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,(19)x,y⟶ln⁡x,ln⁡y.Thus the sequences can meet the homogeneity of variance. The white noise test ofln⁡x sequence indicates that the ln⁡x sequence is a nonwhite noise sequence. Unit root test shows that the P value of τ statistic is significantly greater than 0.05. It is suggested that ln⁡x sequence is nonstationary. There is one unit root in ln⁡x sequence at least. And the analysis on ln⁡y sequence is similar to that of ln⁡x [23–25].Secondly, operating first-order differential operators toln⁡yt and ln⁡xt sequences to get stationary Δln⁡yt and Δln⁡xt,(20)ln⁡xt,ln⁡yt⟶Δln⁡xt,Δln⁡yt.Thirdly, operate stationary test and white noise test to logarithmic sequence after first-order differentialΔln⁡yt and Δln⁡xt. The test result shows that the P value of white noise test is greater than 0.05, which means that Δln⁡yt and Δln⁡xt sequences are pure random white noise sequences [26]. And the P value of τ statistic is less than 0.05, showing that Δln⁡yt and Δln⁡xt sequences are stationary series. Until now, the test analysis has been finished.


## 6.2. ComputingΔln⁡yt and Δln⁡xt Sequences Model

Firstly, theΔln⁡y model is established. The test shows that Δln⁡y is a stationary white noise sequence; thus the fitting model is(21)Δln⁡y=εt.Secondly, theΔln⁡x model is established (see Table 6). The test results obtained by SAS from Tables 3 to 5 show that Δln⁡x is a stationary white noise sequence (the P value in Table 3 is larger than 0.05, while the P value in Table 4 is smaller than 0.05). The best order for ARMA(p,q) model is [AR(0),MA(4)]. Therefore, the fitting model is ARMA(0,4) or MA(4) model. The constant term is not significant, using the noint option to remove the intercept. The final fitting model is shown as [31](22)Δln⁡xt=1-0.45098B4εt.Table 3
Autocorrelation check for white noise.
To lag
Pr > ChiSq
12
0.1066
18
0.2526Table 4
Augmented Dickey-Fuller unit root tests.
Type
Lags
Pr < Rho
Pr < Tau
Pr >F
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
0.001Table 5
Minimum information criterion.
Lags
MA(0)
MA(1)
MA(2)
MA(3)
MA(4)
MA(5)
AR(0)
−5.14349
−5.22355
−5.48448
−5.51557
−5.51716
−5.51704
AR(1)
−5.13228
−5.19252
−5.42776
−5.4607
−5.4677
−5.48567
AR(2)
−5.35327
−5.40243
−5.37285
−5.41553
−5.41986
−5.46426
AR(3)
−5.3958
−5.42964
−5.39755
−5.36473
−5.3641
−5.41041
AR(4)
−5.40213
−5.42436
−5.39489
−5.36661
−5.30905
−5.37159
AR(5)
−5.4457
−5.422
−5.40418
−5.3893
−5.33506
−5.32363
Minimum table value: BIC(0,4) = −5.51716Table 6
Fitting parameter ofΔln⁡x model.
Parameter
Estimate
Standard error
t-value
Approx. Pr >t
Lag
MA(1,1)
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 sequenceΔln⁡x1⋯Δln⁡xk and the response variable sequence Δln⁡yt. 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 Table7 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 ofΔln⁡y and Δln⁡x.
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 Table8 shows that the final regression coefficient is 0.37098.Table 8
Parameter estimates of REG procedure.
Variable
Dependent variable
Parameter estimates
Standard error
t-value
Pr >t
Δ
ln
⁡
x
Δ
ln
⁡
y
0.37098
0.07442
4.98
<0.0001The statistics test is carried out on residual sequence, showing that the residual sequence is stationary white noise sequence (Pr>0.05). The fitted model for residual sequence is εt=at, and at 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 Table7. The same period model is established between Δln⁡y and Δln⁡x, based on the parameter estimates in Table 8 and tests in Tables 9~10. The P value in Table 9 is larger than 0.05; thus the autocorrelation check of residuals shows that the model is effective for forecasting loads:(23)Δln⁡yt=0.37098Δln⁡xt+εt,εt=at.Table 9
Autocorrelation check of residuals.
To lag
Pr > ChiSq
6
0.0578
12
0.0687
18
0.1588Table 10
Crosscorrelation check of residuals with inputln⁡x.
To lag
Pr > ChiSq
5
0.0581
11
0.0634
17
0.0732The load from 15th to 31st is forecasted according to the improved ARIMAX model.By operating logarithm to the forecasting results “ln⁡y,” 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 ofAR(1)
Prediction value ofARMA(3,2)
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
0.0185Table 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
37.8MAE (Mean Absolute Error) is computed as follows:(24)MAE=∑ty^t-ytT,where y^t is the prediction value, yt is the actual value, and T is the sample size.It can be seen in Table11 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 Figure3 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 εt=at. There is second-order delay correlation between ln⁡y and ln⁡x. The final fitting model is(25)ln⁡yt=2.86647+1.41495ln⁡xt-2+εt,εt=at.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.



---

*Source: 589374-2015-07-14.xml*