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Swarm intelligence (SI) is widely and successfully applied in the engineering field to solve practical optimization problems because various hybrid models, which are based on the SI algorithm and statistical models, are developed to further improve the predictive abilities. In this paper, hybrid intelligent forecasting models based on the cuckoo search (CS) as well as the singular spectrum analysis (SSA), time series, and machine learning methods are proposed to conduct short-term power load prediction. The forecasting performance of the proposed models is augmented by a rolling multistep strategy over the prediction horizon. The test results are representative of the out-performance of the SSA and CS in tuning the seasonal autoregressive integrated moving average (SARIMA) and support vector regression (SVR) in improving load forecasting, which indicates that both the SSA-based data denoising and SI-based intelligent optimization strategy can effectively improve the model’s predictive performance. Additionally, the proposed CS-SSA-SARIMA and CS-SSA-SVR models provide very impressive forecasting results, demonstrating their strong robustness and universal forecasting capacities in terms of short-term power load prediction 24 hours in advance.

With the additional types of energy integration into the power grid and the development of generation technologies, power utilities are going through a crucial challenge stemming from maintaining the desired security and reliability of the electricity supply [

Therefore, according to the aforementioned analysis, the power load prediction should be urgently conducted with high accuracy to guarantee the operational performance of the power system. Over the past few decades, large efforts have been devoted to improving the power load forecasting accuracy. The various methods utilized for load prediction range from the traditional statistical models to the complicated artificial intelligence-based models [

The cuckoo search (CS) algorithm is a new optimization metaheuristic algorithm [

The most popular and classic statistical models include the linear or nonlinear regression models, time series models, state estimation, and Kalman filtering technology [

The SSA [

If the original load data are directly applied to a train model without eliminating noise, the high-frequency components may disturb the forecasted load patterns [

This paper starts with a brief description of the related methodology in Section

In this section, SSA, SARIMA, the SVR model, the SI algorithm (CS and PSO), and the design of the proposed hybrid SI-based predictive models are summarized as the foundation to construct the proposed hybrid model.

The SSA technique, a well-developed method of time series analysis, can extract major information from a time series, such as the trend and periodicities components without prior knowledge regarding the trend as well as period values [

The basic SSA consists of two complementary parts: decomposition and reconstruction. For the decomposition part, it comprises two steps: embedding and singular values decomposition. For the reconstruction part, two steps are also involved, which are the Eigentriple grouping and diagonal averaging [

Consider the following.

Both the columns and rows of

The trajectory matrix

basic:

Toeplitz:

In both cases, the eigenvectors are ordered, which can thereby guarantee that the corresponding eigenvalues are placed in decreasing order.

Note that the Case (B) version is only suitable for the analysis of the stationary time series with mean value zero.

Note also that the Case (A) version corresponds to the SVD of

Consider the following.

The reconstructed series produced by the elementary grouping is called the elementary reconstructed series.

The SSA is a data-driven technique that can extract information from a short and noisy time series without prior knowledge of the dynamics affecting the time series. A significant characteristic of the SSA is that trend patterns obtained in this way are not necessarily linear [

The time series that mainly contains the periodic and stochastic components can be forecasted by the SARIMA model, which is the most popular linear model for a seasonal time series and has achieved great success in both academic research and industrial applications over the past few decades [

A time series

When fitting a SARIMA model, the following four steps are involved [

The SVR is an adaptation of a recently developed machine learning theory (MLT) known as the support vector machine (SVM) proposed by Vapnik et al. [

The

The first term in (

In recent years, the metaheuristic optimization algorithms and evolutionary computation have been a noticeable part for solving real mathematics and engineering problems [

The CS algorithm, inspired by the breeding behavior of cuckoos, is a recently developed metaheuristic algorithm by Yang and Deb [

Figure

The basic steps of the CS algorithm and Lévy flight together with a detailed immigration of a cuckoo toward a goal habitat [

The PSO algorithm, inspired by the social behaviors of animal movements, investigates the search space by applying a flock of potential solutions named particle swarms, characterized by their corresponding position and velocity [

The original power load data have some noisy information. If not eliminating noise and directly training and building models, the high frequency components may disturb the forecasted load patterns [

The structure of the proposed SI-based hybrid models.

The determination of which prediction model outperforms the other models is of prime concern. In most study cases, model performance is evaluated by numerous error evaluation criteria that can be classified into two main types: absolute error and relative error. For the absolute error, there are the mean absolute error (MAE) and root mean square error (RMSE). For the relative error, there are the mean absolute percentage error (MAPE) and symmetrical mean absolute percentage error (SMPAE). All of them are commonly used to evaluate the accuracy. In this paper, the mean absolute error (MAE) and mean absolute percentage error (MAPE) are used to measure the prediction accuracy of these models. The smaller these values are, the better the predictive performance is.

The MAE can be defined as

The MAPE can be defined as

As described previously, the SSA technique is widely used in extracting principal information; that is, its trend and oscillation components, which are then effectively used for time series forecasting. Understanding that extracting leading information by the SSA is also a procedure of denoising such as wavelet denoising, in this work, we decompose the power load time series and then reconstruct the principal components into a smoother time series. To demonstrate the performance of the SSA denoising used in the power load, in Section

We choose two samples of a half-hour power load, each of which contains 336 training data and 48 test data from April to May in NSW of Australia.

The power load time series in Case 1 is chosen as an example to show the detailed process of SSA-based denoising. The data in Case 2 are denoised by the same procedure as described below. SSA-based denoising has two main stages: the 1st stage is to extract the trend; the 2nd stage is to extract the seasonal components from the residuals from the first step and then reconstruct it.

Six leading eigenvectors are displayed in Figure

1st stage: eigenvectors (

1st stage: elementary reconstructed series (

1st stage: initial series, estimated trend, and residuals (

To properly identify the sine waves, we use the graph of eigenvalues (Figure

2nd stage: eigenvalues of residuals (

2nd stage: scatterplot for eigenvector pairs (

2nd stage: periodogram of the series (i.e., of the residual at the 1st stage).

2nd stage:

2nd stages: original series and its trend-periodic-residuals decomposition.

Reconstruction series and original series.

In this section, we employ SARIMA and SVR to build the forecasting models after denoising the useless information in the power load by SSA. To further improve the accuracy of forecasting, we optimize the parameters of SAMRIA and SVR by the CS and PSO.

The proposed hybrid models are applied for short-term (half an hour) load forecasting with a 48-step ahead of NSW in Australia over a prediction of one day in two cases. The performance of the proposed methods is marked as CS-SSA-SARIMA and CS-SSA-SVR. The comparisons of the power load forecasting results are intuitively shown in Figures

The forecasting results of the proposed PSO-SSA-SARIMA, CS-SSA-SARIMA, PSO-SSA-SVR, and CS-SSA-SVR models for Case 1 ((b, e):

The forecasting results of the proposed PSO-SSA-SARIMA, CS-SSA-SARIMA, PSO-SSA-SVR, and CS-SSA-SVR models for Case 2 ((b, e):

It can be seen from Figures

In Figures

Figure

Statistical error measures’ comparison between different models.

Model | MAPE (%) | MAE | ||
---|---|---|---|---|

Case 1 | Case 2 | Case 1 | Case 2 | |

SARIMA | 4.22 | 6.87 | 344.32 | 613.62 |

SSA-SARIMA | 2.37 | 4.60 | 190.68 | 404.10 |

PSO-SSA-SARIMA | 2.37 | 4.47 | 190.79 | 392.06 |

CS-SSA-SARIMA | 1.38 | 3.82 | 113.72 | 333.12 |

SVR | 8.75 | 6.64 | 734.83 | 580.10 |

SSA-SVR | 5.53 | 5.76 | 451.26 | 516.21 |

PSO-SSA-SVR | 4.96 | 4.68 | 404.26 | 392.42 |

CS-SSA-SVR | 3.42 | 3.73 | 278.86 | 305.88 |

CS and PSO convergence procedure during the training for the NSW load.

To evaluate the forecasting model quantitatively, the statistical errors are computed in testing datasets over the forecasting horizon. Table

Statistical error measures’ comparison between different models.

This paper presents hybrid swarm intelligent forecasting model strategies to accurately predict the short-term power load. The results obtained in this study illustrate that the SSA technique can be successfully used as a noise eliminating technique for time series similar to the short-term power load time series used here. The SSA-based denoising technique is capable of extracting important trend and seasonal components and then reconstructing it into smooth data to enhance the forecasting accuracy for SARIMA and SVR. In addition, the good noise eliminating ability via SSA could make these characteristics more obvious when modeling and could provide a more accurate forecast by SARIMA and SVR.

The CS algorithm is a recently developed metaheuristic artificial intelligence algorithm of parameter optimization. It has the ability to search parameters outstrips, that of the maximum likelihood estimation method and that of the traditional optimization algorithm (PSO) when estimating the parameters of SARIMA. Similarly, its capability of expanding the scope of the search intelligently provides an optimization that is more effective and efficient than a grid search when searching the optimal hyperparameters in SVR. Although PSO has a faster convergence velocity, it appears to have an overfitting problem. In our cases, it is revealed that by optimizing the parameters of the SSA-SARIMA and SSA-SVR models, the CS algorithm can further enhance the accuracy of prediction in short-term power loads and obtains a higher precision than PSO. The proposed hybrid swarm intelligent forecasting model could predict the short-term power load in a real-world scenario, which helps to enhance the predictive accuracy of the power system.

In this paper, our contribution is that an SI-based forecasting model is proposed to highly increase the accuracy. However, we did not sufficiently compare other feasible forecasting models, data preprocessing methods, and AI algorithms, such as BP, autoregressive integrated moving average (ARIMA), wavelet analysis, and GA. A more detailed comparison between the proposed method and other feasible forecasting models, data preprocessing methods, and SI optimizations is required. This is a very heavy workload but is very meaningful research; thus, it is necessary to perform additional research in future work.

For future work, we outline four directions. The first direction is to study the use of the other feasible forecasting models mentioned above within our framework. The second direction is to study in detail the other feasible SI optimization algorithms mentioned in this paper to search parameters of various forecasting models. The third direction is to study the use of other feasible data preprocessing methods, including the methods of not only denoising useless information but also removing outliers. The fourth direction is to explore the ability of different SI optimization algorithms to search certain parameters of certain forecasting models.

A novel swarm intelligence-based hybrid approach is proposed for short-term load forecasting.

The proposed approach consists of three steps to increase the forecasting accuracy.

SSA is used for removing noised information in the first step.

SARIMA and SVR are used for forecasting in the second step.

CS is employed to optimize the parameters of SARIMA and SVR.

The proposed approach can improve the forecasting accuracy.

The authors declare that they have no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 7117110).