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Monthly electric energy consumption forecasting is important for electricity production planning and electric power engineering decision making. Multiwindow moving average algorithm is proposed to decompose the monthly electric energy consumption time series into several periodic waves and a long-term approximately exponential increasing trend. Radial basis function (RBF) artificial neural network (ANN) models are used to forecast the extracted periodic waves. A novel hybrid growth model, which includes a constant term, a linear term, and an exponential term, is proposed to forecast the extracted increasing trend. The forecasting results of the monthly electric energy consumption can be obtained by adding the forecasting values of each model. To test the performance by comparison, the proposed and other three models are used to forecast China's monthly electric energy consumption from January 2011 to December 2012. Results show that the proposed model exhibited the best performance in terms of mean absolute percentage error (MAPE) and maximal absolute percentage error (MaxAPE).

Electric energy consumption is essential for promoting economic development and raising people’s living standard [

Researchers have developed two ideas for mid-/long-term electric energy consumption forecasting. One idea focuses on the relationship between electric energy consumption and its influencing factors. Models originating from this idea usually formulate an equation to quantitatively simulate such relationship. These models obtain the forecasting results by inputting the values of the influencing factors during the forecasting period into the equation [

The difficulty in forecasting monthly electric energy consumption by means of trend extrapolation lies in the complexity of the consumption curve. The consumption curve generally includes two subtrends. First, a long-term increasing trend typically exists in most cases. Basic electric energy consumption usually increases with the development of social economy. Second, periodic waves always exist, as a result of the effects of people’s living habits and work styles according to alternating seasons. Therefore, the presence of both a long-term increasing trend and periodic waves with different frequencies and amplitudes increases the difficulties in trend extrapolation.

The artificial neural network (ANN) is a traditional nonlinear trend extrapolation model. The primary advantage of ANN is its capability of modeling nonlinear relations without the supervision of human experts [

Zhao and Wei [

Meng et al. [

Nevertheless, the forecasting model proposed by Meng et al. [

Therefore, the present study proposes a multiwindow moving average algorithm to obtain accurate extracted results of the wave curves. This algorithm can extract wave curves under all the possible periods. Furthermore, a hybrid growth model is proposed to improve the forecasting results of the extracted long-term increasing curve. A constant term is added to the forecasting equation of GM (1, 1), as well as a linear term, because the approximate exponential trend of the extracted long-term increasing curve is unstable and a linear trend may appear in special periods.

The moving average algorithm is written as follows:

Let

Figure

Process of the multiwindow moving average algorithm.

The left branch of Figure

As the OTS are decomposed twice, R2, R3, …, R12 and S12 are multiplied by 1/2 and renamed as AR2, AR3, …, AR12 and AS12 to ensure that the summation of all the extracted time series is equal to the OTS. Furthermore, although the left and right branches can both obtain AS12 and AR12, the two AS12 and two AR12 should be summed separately. Accordingly, the results are named as SAS12 and SAR12, respectively.

The extracted time series are AR2, AR3, AR4, AR6, SAR12, and SAS12. The sum of all the extracted time series is equal to the OTS. As the maximum period of all the waves in monthly electric energy consumption time series is 12, time series SAS12 should have no periodic waves and should thus be a long-term increasing trend. Other time series are wave trends with different periods.

The forecasting/simulation equation of GM (1, 1) is usually written as follows:

Clearly, (

Equation (

If

The hybrid growth equation, which includes a constant term, a linear term, and an exponential term, can then be written as follows:

Given that

If

To obtain the optimal parameter estimations, the residual sum of squares must be minimized.

Let

To obtain the minimum of

Then,

Up to this point, the optimal parameter estimations of (

However, (

The optimal estimation of

Equation (

Thus, the hybrid growth model, which is used to forecast the extracted long-term increasing trend, is completed.

The above models can be used for both one-step and multistep ahead forecastings. As the former is used more often, and especially useful for the production operation management of the power-generation companies, this paper designs the one-step ahead forecasting process as follows.

The radial basis function (RBF) ANN model is used to forecast the extracted wave curves. Similar to [

At present, China is the largest electric energy consumer in the world. In 2012, its global consumption share reached 21.94% [^{8} kW

Monthly electric energy consumption curve of China.

As shown in Figure

The monthly electric energy consumption curve of China was decomposed to several subcurves by using the aforementioned multiwindow moving average algorithm. Figure

Trends of extracted curves from January 1993 to December 2012.

The SAS12 in Figure

To evaluate the performance by comparison, the forecasting model proposed in this paper (denoted by M1), model used in [

Real data and forecasting results.

Month | RD | M1 | M2 | M3 | M4 |
---|---|---|---|---|---|

January 2011 | 3672.1 | 3811.3 | 3402 | 3677.8 | 3642.3 |

February | 3100.8 | 3545.7 | 3347.4 | 3672.1 | 3672.5 |

March | 3830.1 | 3722 | 3863.9 | 3100.8 | 3277.6 |

April | 3663.8 | 3817.8 | 3826.2 | 3830.1 | 3838.4 |

May | 3775.4 | 3841.6 | 3837 | 3663.8 | 3465.2 |

June | 3968.2 | 3860 | 3863.5 | 3775.4 | 3818 |

July | 4251.5 | 4062.2 | 4106.7 | 3968.2 | 3875.7 |

August | 4260.4 | 4066.6 | 4099.7 | 4251.5 | 4116.1 |

September | 3860.6 | 3694.3 | 3783.7 | 4260.4 | 4244.3 |

October | 3640.4 | 3669.4 | 3636.1 | 3860.6 | 4124.5 |

November | 3713 | 3886.3 | 3724.6 | 3640.4 | 3904.2 |

December | 4038.1 | 3967.6 | 3939.6 | 3713 | 3793.3 |

January 2012 | 3485 | 3963 | 3909 | 4038.1 | 3983.8 |

February | 3702 | 3454.6 | 3597.3 | 3485 | 3624.5 |

March | 4109 | 3966 | 3882.4 | 3702 | 3822.5 |

April | 3718 | 3953.7 | 4174.9 | 4109 | 3913.2 |

May | 3898 | 3999.8 | 4013 | 3718 | 3720 |

June | 3934 | 4004.5 | 4071.4 | 3898 | 3990.4 |

July | 4351 | 4175 | 4415.3 | 3934 | 3869.8 |

August | 4373 | 4222.6 | 4607.5 | 4351 | 4188.2 |

September | 3907.3 | 4158.1 | 4438.2 | 4373 | 4285.1 |

October | 3897.7 | 3972.3 | 4403.3 | 3907.3 | 4194.5 |

November | 4010.5 | 4137.2 | 4613 | 3897.7 | 4123.7 |

December | 4327.2 | 4277.4 | 4682.8 | 4010.5 | 4015.4 |

To evaluate the forecasting performance of each model by comparison, the mean absolute percentage error (MAPE, in %) and the maximal absolute percentage error (MaxAPE, in %) [

Despite having similar functions of determining which is the best model, these two error analysis indicators have fine distinctions. MAPE is the most widely used indicator of accuracy, because this indicator reflects the general closeness of the forecasting results to RD. MaxAPE generates the worst forecasting result and reflects the maximal forecasting risk.

By using the data in Table

Forecasting errors of each model.

M1 | M2 | M3 | M4 | |
---|---|---|---|---|

MAPE | 4.37 | 5.53 | 6.81 | 7.26 |

MaxAPE | 14.35 | 15.02 | 19.04 | 18.44 |

On the whole, as the trend extraction algorithms simplified the forecasting curves, M1 and M2 are better than M3 and M4 for each indicator. As the period of the extracted waves of M1 is more reasonable than M2 and the hybrid growth model is more applicable to the extracted long-term increasing curve than GM (1, 1), M1 is always better than M2 for each indicator. Especially for MAPE, which is usually considered the most important indicator, M1 (4.37%) is much smaller than M2 (5.53%).

Monthly electric energy consumption time series usually has two characteristics. First, the main trend of the said time series presents an approximately exponential increasing feature. Second, the period of the wave trends can only be 12, 6, 4, 3, or 2 months. Thus, this study proposed a multiwindow moving average algorithm to decompose the said time series and a novel hybrid growth model to forecast the extracted long-term increasing trend. To evaluate the performance by comparison, the proposed and other three models were used to forecast China’s monthly electric energy consumption from January 2011 to December 2012. Empirical results show that, because of the aforementioned innovations, the MAPE and MaxAPE of the proposed model are smaller than those of the other three models.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This study was supported by the National Natural Science Foundation of China (NSFC) (71201057 and 71071052) and the Fundamental Research Funds for the Central Universities (2014MS148).

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