Short-term power load forecasting is one of the most important issues in the economic and reliable operation of electricity power system. Taking the characteristics of randomness, tendency, and periodicity of short-term power load into account, a new method (SSA-AR model) which combines the univariate singular spectrum analysis and autoregressive model is proposed. Firstly, the singular spectrum analysis (SSA) is employed to decompose and reconstruct the original power load series. Secondly, the autoregressive (AR) model is used to forecast based on the reconstructed power load series. The employed data is the hourly power load series of the Mid-Atlantic region in PJM electricity market. Empirical analysis result shows that, compared with the single autoregressive model (AR), SSA-based linear recurrent method (SSA-LRF), and BPNN (backpropagation neural network) model, the proposed SSA-AR method has a better performance in terms of short-term power load forecasting.
Short-term power load forecasting is one of the most important issues in economic and reliable operation of power system. Many operating decisions related to electricity power system such as unit commitment, dispatch scheduling of the generating capacity, reliability analysis, security assessment, and maintenance scheduling of the generators are based on the short-term power load forecasting.
In recent years, domestic and foreign scholars have done many studies in the field of short-term power load forecasting. Currently, the short-term power load forecasting method can be divided into two categories, that is, load-series-based forecasting method and affecting-factors-based forecasting method. Although the power load shows the random and uncertain characteristic, it also has an apparent tendency. Therefore, the load-series-based forecasting method is based upon the internal structure of the short-term power load series, which includes ARIMA, ARMAX [
The singular spectrum analysis (SSA) technology is a typical time-series-based analysis method which has been used for industrial production forecasting [
From the perspective of the fluctuation characteristics the power load shows randomness, trend, and periodicity, which can be extracted by using the SSA method. That is to say, the stochastic noise components which influence the forecasting accuracy can be eliminated. And Afshar and Briceño have already applied this method to load forecasting with linear recurrent formulae (LRF) [
The rest of this paper is organized as follows. In Section
Singular spectrum analysis (SSA) method contains two phases named decomposition and reconstruction. The former phase arranges the original sequence in a form of time-delay matrix before decomposing the original time series. Then, the time series are reconstructed via grouping and diagonal averaging, which is called reconstruction. The reconstructed series is then used for forecasting the new data points.
Given a time series
In the second step, the
After decomposition, the time series is reconstructed via grouping and diagonal averaging. In grouping step, the indices
Let
Diagonal averaging transfers the matrix
The principle of autoregressive (AR) model is to use the current interference and the limited past observations to predict the present value. Given a time series
If the lag operator is defined as
Only the reciprocal of all roots of lag operator’s polynomial is less than one (both fall within the unit circle); the AR (
The application process of SSA-AR model for short-term power load forecasting is mainly divided into two steps: first, using the SSA method to decompose and reconstruct the original power load series and then using the AR model to forecast with the reconstructed sequences. The specific calculation procedure of SSA-AR model for short-term power load forecasting is shown in Figure
The specific flow chart of the SSA-AR model.
The hourly power load series of Mid-Atlantic region in the PJM electricity market from 18 June to 18 July 2013 containing 720 sample points is employed in the experimental data. The mentioned data versus time have been shown in Figure
The hourly load curve of the Mid-Atlantic region from June 18 to July 18 in 2013.
As mentioned earlier, the window length
Singular values curve (in descending order).
As shown in Figure
The first 30 singular values in descending order (taken to the base-10 logarithm).
Order |
|
Order |
|
Order |
|
---|---|---|---|---|---|
1 | 16.31 | 11 | 12.06 | 21 | 11.13 |
2 | 14.15 | 12 | 11.87 | 22 | 11.07 |
3 | 14.15 | 13 | 11.87 | 23 | 11.07 |
4 | 13.39 | 14 | 11.73 | 24 | 10.80 |
5 | 13.03 | 15 | 11.67 | 25 | 10.76 |
6 | 12.49 | 16 | 11.48 | 26 | 10.68 |
7 | 12.23 | 17 | 11.43 | 27 | 10.61 |
8 | 12.16 | 18 | 11.36 | 28 | 10.61 |
9 | 12.11 | 19 | 11.26 | 29 | 10.60 |
10 | 12.08 | 20 | 11.13 | 30 | 10.60 |
The decomposition helps us to make the proper groups to extract the trend, the harmonic component, and random noise component. Based on the relevant mathematical theory about the singular value decomposition, there are three basic conclusions:
Based on the above analysis, this paper tries to reconstruct the power load series using the first 30 Eigen values. Figure
The gap between the reconstructed power load series (RS) and the original power load series.
Figure
Because the AR model is only applicable to the stationary time series, the first thing that needs to be done is to examine the stationarity of the reconstructed power load series with the ADF unit root test. By employing the software
The result of ADF test on first order difference power load series.
Item | ADF statistics | Prob. |
---|---|---|
Original series | −2.77525 | 0.0623 |
First order difference | −3.72147 | 0.0040 |
Observe the autocorrelation coefficient and partial correlation coefficient of the first order difference sequence which are shown in Figure
The regression result of AR (
Variable | Coefficient | Std. error |
|
Prob. |
---|---|---|---|---|
AR ( |
2.067776 | 0.035534 | 58.19094 | 0.0000 |
AR ( |
−1.396917 | 0.066505 | −21.00462 | 0.0000 |
AR ( |
0.263791 | 0.035598 | 7.410290 | 0.0000 |
|
||||
Parament | Value | |||
|
||||
Adjusted |
0.986002 | |||
Schwarz criterion | 13.56932 | |||
Durbin-Watson stat | 1.967281 |
The autocorrelation and partial correlation of the reconstructed series.
From Figure
Unit circle test of the covariance stationarity.
Then, expand the sample size to 745 from 744, and use the AR
The AR model, SSA-LRF model, and BPNN model are selected as the comparative models. The AR model is applied to the original power load series without any treatment that can extract the main trend. Although AR model can well represent the whole tendency of the original series, the predicted value is not perfect in a way. The SSA-LRF model is the combination of SSA method and linear recurrent formula (LRF), in which LRF is a simple linear combination of the known data and its coefficients are determined by SSA method. BPNN (backpropagation neural network) is made of neuron. Not only are sufficient neurons connected to net properly, but the BPNN model should be trained appropriately before it can simulate all types of nonlinear characteristics. BPNN model is applied most widespread among all artificial neural network models, which has been widely applied to many fields related to forecasting. The predicted hourly power load results in 24 hours of Mid-Atlantic region in PJM market on July 19 by employing the above four forecasting methods are depicted in Figure
The comparison of prediction results of different models and the actual value.
In order to measure the performance of the three forecasting methods, two indices, that is, hourly mean error (HME) and the hourly peak error (HPE), have been employed in this paper. The HME and HPE are the well-known statistical indices for evaluating prediction methods defined as follows:
The calculation results of absolute error rate, HME, and HPE are shown in Figure
The forecasting error comparison of four methods: (a) absolute error rate; (b) HME and HPE.
In this paper, a hybrid short-term power load forecasting model based on the singular spectrum analysis (SSA) and autoregressive (AR) model is proposed. As we all know, the short-term power load forecasting is vital in the fundamental operational functions of electricity market, such as unit commitment, economic dispatch, interchange evaluation, scheduled maintenance, and security assessment. In this paper, the SSA-AR model has been employed as a tool for short-term power load forecasting. Firstly, the power load series is analyzed with the SSA method to obtain the effective and predictable components of power load series. Then, the AR method is used to forecast the future values of the power load series. The hybrid SSA-AR power load forecasting method is examined by using the experimental data of Mid-Atlantic region in PJM electricity market. The obtained results show that the proposed method has a good ability in the prediction of the desired power load series. However, there is one point that needs emphasis: this method does not take the factors influencing the power load fluctuation into account. Once the outside situation, such as political, economic, and climate condition, has a sudden change, this method may not work. Therefore, this method is applicable to the short-term power load forecasting without the tremendous changes of outside situation.
The authors declare that they have no conflict of interests.
This study is supported by the Beijing Philosophy and Social Science Planning Project (no. 11JGB070).