# Combined Prediction of Clean Energy Consumption in China Based on the Nonlinear Programming Model

Authors: Shuanghua Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

With the increasing demand for energy and with the increasing awareness of environmental protection, clean energy will become the ultimate solution of energy in the near future. In order to meet the future demands of clean energy, it is important to accurately predict the consumption of clean energy. This paper is based on the total consumption data of clean energy in China from 2000 to 2019, including natural gas, hydropower, nuclear power, and wind power consumption. The combined model is used to predict the clean energy consumption and achieves the optimal prediction with minimum variance. The results show that compared to the GM(1,1) and ARIMA model, the combined forecasting model has lower relative error when fitting and predicting the consumption demand of clean energy. It is observed from the prediction results that the clean energy consumption would have a rapid growth tendency, and the growth rate will be about 8.6% in the next five years. The consumption would be about 1.7 billion tons standard coal in the year of 2024.


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## Body

## 1. Introduction

With the social progress and economic development, energy and environment have become the focus of attention of all countries in the world. Under the carbon neutral goal-directed, clean energy will be the development trend of future energy. Renewable energy is energy from sources that are naturally replenishing but flow-limited, such as hydroelectric power, wind power, solar energy, biological energy, geothermal energy, and tidal energy. Renewable energy does not exit the possibility of energy depletion; therefore, the development and utilization of renewable energy is becoming the center of attention for many countries. At present, the proportion of clean energy in China’s energy consumption is increasing year by year. According to the 2019 national energy consumption statistical results, the coal accounts for 57.7% of the total energy consumption and clean energy accounted for 23.4% of the national energy consumption, which is increased by 1.3% than in 2018. On the contrary, China’s renewable energy consumption utilization rate is also increasing year by year. At present, the utilization rate of solar energy water heater has reached the international advanced level. China’s solar water heater area has an accumulative total of 6, 5 million square meters and annual production capacity of 15 million square meters; the usage and production are the first in the world, which have accounted for 50% of global share and have a relatively complete industrial system. It has formed a complete upstream and downstream industrial chain and consumer chain radiating across the country [1]. In terms of investment in clean energy, China ranks first in the world. In the transition period of clean energy development, clean energy demand changes have not similar historical path reference; clean energy demand prediction is not easy, but, at the same time, the prediction results of clean energy dependence will increase; therefore, prediction of consumption of clean energy in future will play a significant role in increasing capacity of producing clean energy. Scientific analysis of the consumption trend of clean energy in China is of great significance to the formulation of energy development strategies and the optimization of energy consumption structure.Since the concept of clean energy was put forward, many scholars have devoted themselves to the research on the prediction of clean energy consumption trend, and many valuable results have been obtained. Wang [2] used the improved grey forecasting model to forecast residential solar energy consumption in the United States. Tong et al. [3] used the data of natural gas consumption in Beijing from 2013 to 2017. The grey fractional FGM(1,1) model is used to predict the natural gas consumption in Beijing. Cui et al. [4], based on China’s annual gas consumption datasets, such as using grey model and rolling grey model to predict the future gas consumption, analyzed and compared the two kinds of prediction accuracy of the model; the results show that the rolling grey model has better prediction ability. In [5],based on natural gas consumption and ground conventional meteorological observation data in Beijing in heating season from 2002 to 2018, as well as yearly social statistical information， the interannual variation characteristics of natural gas consumption in heating season in Beijing and its impact factors were analyzed by using empirical mode decomposition (EMD) and correlation analysis methods. On this basis, the forecast model of natural gas consumption in heating season was established by using backpropagation (BP) neural network method; furthermore, the model was tested and evaluated． Huang [6] applied the principle of BP neural network to predict China’s power demand. Paul Crompton [7] used the BVAR method to analyze China’s energy consumption and forecast China’s future energy consumption. Yin [8] used the principles of econometrics to make a dynamic analysis of new energy in China, established the ARIMA model, and gave the prediction method of new energy. Yang [9]used the GM (1,1) model to predict the development prospect of new energy in Hebei province.Through the analysis of relevant literature, it can be seen that the single method has certain limitations, which cannot systematically and truly predict the development potential of clean energy, resulting in a large deviation. The combined model [10–17] can make comprehensive use of the advantages of each single prediction model, improve the overall accuracy of prediction, and make the model fitting and prediction effect have strong stability. Ma [18] used the natural gas consumption in Chongqing from 2001 to 2014 as the original data and the data from 2015 to 2017 as the test data to predict the natural gas consumption in Chongqing based on the combination model of the DGM(1,1) model and the linear regression model, and the results showed that the deviation of the new combination prediction model was the least. In the framework of the basic GM(1,1) model, Zhang [19] modified the unbiased GM(1,1) model by re-accumulating and generating cumulative sequence predicted values and modified the pGM(1,1) model according to the weighted average background value reset. The energy consumption prediction of the combined BP neural network is incorporated with the mapping between various nonlinear parameters. The results show that the unbiased GM(1,1) and pGM(1,1) models can effectively reduce the average relative error of GM(1,1) prediction and then combine the prediction with BP neural network to form a better prediction accuracy of energy consumption. In this paper, the GM (1,1) and ARIMA(2,1,1) models are used to forecast the consumption of clean energy in China. There are great differences in the prediction accuracy, which illustrates the necessity of combined forecasting from the side. By building a nonlinear programming model, the weight of two single prediction models is given, which can effectively gather more useful information and give full play to the advantages of each prediction method, to improve the accuracy and effect of the prediction.


## 2. Model Construction and Analysis

The research data in this paper come from the National Bureau of Statistics, as shown in Table1. The total consumption of clean energy includes the consumption of natural gas, hydropower, nuclear power, and wind power. Firstly, the data were selected for modeling to observe the fitting effect of the model from 2000 to 2016. Then, the established model was used for prediction. The predicted value from 2017 to 2019 was compared with the real value to observe the prediction effect of the model.Table 1
Total clean energy consumption from 2000 to 2019 (10,000 tons of standard coal).
yearTotal clean energy consumption200013961.58200116799.08200217805.58200319117.05200422797.82200525614.17200628933.17200732701.41200837832.09200940335.12201048326.83201150315.59201258310.01201364621.52201472387.02201577382.9201684984.71201793294.0320181025442019113724To verify the model’s fitting effect and prediction effect, RMSE and MAPE are generally used to evaluate the model’s fitting effect and prediction effect. The smaller the RMSE and MAPE, the closer the predicted value to the actual value and the better the prediction effect:(1)RMSE=1n∑i=1nei2,MAPE=1n∑i=1neixi.


### 2.1. GM (1,1)

GM (1,1) [20–25] forecasting model has a low requirement on sample data and can get the prediction results reflecting the trend through a small amount of data. The modeling process is as follows.Assume a nonnegative sequence to be(2)X0=x01,x02,…,x0n.Tthe first-order accumulated generating operator sequence is given as(3)X1=x11,x12,⋯,x1n,where x1k=∑i=1kx0i,k=1,2,…,n,(4)Z1=z12,z13,⋯,z1n,where z1k=1/2x1k+x1k−1,k=2,3,…,n...With the help of the least-squares method, the model parameters can be computed as(5)a,bT=BTB−1BTY,where(6)Y=x02x03⋮x04,B=−z121−z131⋮⋮−z1n1.According to equation (6), it the following GM (1,1) model can be constructed:(7)x^1k+1=x01−bae−ak+ba,k=1,2,…,n,x^0k+1=x1k+1−x^1k=1−eax01−bae−ak,k=1,2,…,n.Through R4.0.2 language programming, we calculateda=−0.1091417 and b=14738.87. The grey forecast model of our country clean consumption total can be expressed as(8)x^1k+1=149005e0.1091417k−135043.5,k=1,2,…,n.The grey prediction model was tested, and the average relative error wasΔ¯=0.0335<0.05, and the simulation error was Δ17=0.0394<0.05, and the accuracy was the first level. The correlation degree is ε=0.9940>0.9, and the correlation degree is the first level; the mean square deviation ratio was C=S2/S1=0.0637<0.35, the mean square deviation ratio was the first level; small error probability p=1>0.95, small error probability is level 1. The model test result is excellent, so the grey prediction model equation (9) can be used to make the prediction. The specific fitting and prediction results are shown in Table 2.Table 2
Fitting and prediction results of total clean energy consumption in China (10,000 tons of standard coal).
yearThe actual valueGM(1,1)ARIMACombined modelPredictive valueThe relative error (%)Predictive valueThe relative error (%)Predictive valueThe relative error (%)200013961.58200116799.0817183.322.29200217805.5819164.97.63200319117.0521375.0111.8119476.281.8820186.605.59200422797.8223839.984.5722262.802.3522852.820.24200525614.1726589.223.8124081.685.9825019.752.32200628933.1729655.492.5027701.504.2628432.491.73200732701.4133075.381.1431883.472.5032329.361.14200837832.0936889.642.4936577.893.3236694.523.01200940335.1241143.772.0040810.651.1840935.271.49201048326.8345888.485.0548231.240.2047354.812.01201150315.5951180.361.7251318.561.9951266.861.89201258310.0157082.52.1160423.973.6359173.931.48201364621.5263665.271.4864725.560.1664328.910.45201472387.0271007.171.9170826.802.1670894.282.06201577382.979195.752.3475912.221.9077140.590.31201684984.7188328.633.9383860.281.3285531.890.64RMSE1517.00RMSE1098.64RMSE793.35MAPE3.55MAPE2.34MAPE1.74201793294.0398514.727.1590899.302.5793748.231.462018102544109875.487.7697907.144.52102384.491.082019113724122546.365.60105028.937.65111582.200.37RMSE7276.45RMSE5854.90RMSE1267.42MAPE6.83MAPE4.91MAPE0.842


### 2.2. ARIMA Model

ARIMA [26] is a time-series prediction model proposed by Box and Jenkins. The main idea is to transform the unstable time series into the stationary time series and then establish the model by parameter estimation. Where AR is autoregressive, p is the order of autoregressive, MA is the moving average, q is the order of moving average, and d is the number of difference when the time series becomes a stationary time series.Suppose that a zero-mean stationary sequence to bext and at to be a white noise; then, we obtain(9)xt−φ1xt−1−φ2xt−2−⋯−φpxt−p=θ1at−1−θ2at−2−⋯−θpat−p,where Extas=0t<s. [25].Furthermore, suppose that a nonstationary time sequence bezt; we obtain(10)ϕB1−Bdzt=θBat,where xt=∇dzt,t>d.The specific modeling process is as follows:(1)
Stationary test of time series: as can be seen from Figure1, the time series of China’s total clean energy consumption from 2000 to 2016 shows an exponential upward trend, which is intuitively nonstationary. To obtain a stationary time series, the first-order difference is used to obtain a new series Yt, and then, the ADF test is carried out. The test results obtained are shown in Table 3. It can be seen that the significance levels of 0.01, 0.05, and 0.1 are all greater than ADF statistics, indicating that the sequence Yt first-order difference is stable.The white noise test was further carried out on the data, and the test result is shown in Figure2. It can be seen that most of the right-most P values are greater than the confidence level of 0.05, so the sequence Yt after first-order difference can be regarded as a stable white noise sequence.(2)
Identify the order of the ARIMA model. The autocorrelation graph and partial autocorrelation graph of the treated time series are drawn to find the optimal combination of P and Q. Through Eviews7.2 analysis, the final established model is ARIMA (2,1,1).(3)
Parameter estimation and test are carried out.Figure 1
Time sequence diagram of total clean energy consumption in China from 2000 to 2016 (10,000 tons of standard coal).Table 3
ADF test results after first-order difference.
t-StatisticProbAugmented Dickey–Fuller test statistic254.712−9.33800.0000Test critical values1% level−4.72835% level−3.759710% level−3.3249Figure 2
White noise test.Table 4
Parameter estimation of ARIMA (2,1,1).
VariableCoefficientStd. errort-StatisticProbC8250.77221814.67894.54660.0008AR(2)0.80220.08829.08960.0000MA(1)−0.96710.0697−13.87300.0000Parameter estimation of the model obtained by least-squares estimation is shown in Table4.According to parameter estimation in Table3, the model Yt can be written as(11)Yt=8250.7722+1+0.9671B1−0.8022B2at.According to theT-statistic value and its P value of the model coefficient, the parameter estimates of all explanatory variables in the model are significant at a significant water level of 0.05. According to the analysis of Eviews7.2, F-statistic = 21.2339, Prob(F-statistic) = 0.0001, AIC = 17.1557, and D-W = 1.9756. The fitting effect of the whole equation is good.In addition, the residual white noise inspection according to theP value of the test statistics Q is greater than 0.05 (see Figure 3); there is no autocorrelation in the residual series, which is white noise, indicating that the model information is extracted sufficiently, and this model is a suitable model that can be used to forecast China's clean energy consumption. Static forecast values are used for in-sample (2000 to 2016) and dynamic forecast values are used for out-of-sample (2017 to 2019), and the specific fitting and forecasting results are shown in Table 2.Figure 3
Residual white noise test.


### 2.3. Combination Prediction Based on the Nonlinear Programming Model

The research of combination prediction is one of the hot issues in the field of prediction at present; its advantage is that the information provided by each prediction method can be used to improve the accuracy of prediction, and the key of the combination prediction method is to determine the weight of each single prediction method. For the total errorE of the combined prediction method, the following nonlinear optimization model is adopted:(12)minE=∑i=1nxj−fxij,lij2,s.t∑i=1nlij=1,lij≥0j=1,2,3,…,m.This is a nonlinear programming problem with only one constraint condition. The total error of the improved combined prediction method is(13)E∗=minE.A minimum variance after different combination forecast models is reached; the prediction formula of the combined model can be obtained as(14)x^j=0.3741x^1j+0.6259x^2j.


## 2.1. GM (1,1)

GM (1,1) [20–25] forecasting model has a low requirement on sample data and can get the prediction results reflecting the trend through a small amount of data. The modeling process is as follows.Assume a nonnegative sequence to be(2)X0=x01,x02,…,x0n.Tthe first-order accumulated generating operator sequence is given as(3)X1=x11,x12,⋯,x1n,where x1k=∑i=1kx0i,k=1,2,…,n,(4)Z1=z12,z13,⋯,z1n,where z1k=1/2x1k+x1k−1,k=2,3,…,n...With the help of the least-squares method, the model parameters can be computed as(5)a,bT=BTB−1BTY,where(6)Y=x02x03⋮x04,B=−z121−z131⋮⋮−z1n1.According to equation (6), it the following GM (1,1) model can be constructed:(7)x^1k+1=x01−bae−ak+ba,k=1,2,…,n,x^0k+1=x1k+1−x^1k=1−eax01−bae−ak,k=1,2,…,n.Through R4.0.2 language programming, we calculateda=−0.1091417 and b=14738.87. The grey forecast model of our country clean consumption total can be expressed as(8)x^1k+1=149005e0.1091417k−135043.5,k=1,2,…,n.The grey prediction model was tested, and the average relative error wasΔ¯=0.0335<0.05, and the simulation error was Δ17=0.0394<0.05, and the accuracy was the first level. The correlation degree is ε=0.9940>0.9, and the correlation degree is the first level; the mean square deviation ratio was C=S2/S1=0.0637<0.35, the mean square deviation ratio was the first level; small error probability p=1>0.95, small error probability is level 1. The model test result is excellent, so the grey prediction model equation (9) can be used to make the prediction. The specific fitting and prediction results are shown in Table 2.Table 2
Fitting and prediction results of total clean energy consumption in China (10,000 tons of standard coal).
yearThe actual valueGM(1,1)ARIMACombined modelPredictive valueThe relative error (%)Predictive valueThe relative error (%)Predictive valueThe relative error (%)200013961.58200116799.0817183.322.29200217805.5819164.97.63200319117.0521375.0111.8119476.281.8820186.605.59200422797.8223839.984.5722262.802.3522852.820.24200525614.1726589.223.8124081.685.9825019.752.32200628933.1729655.492.5027701.504.2628432.491.73200732701.4133075.381.1431883.472.5032329.361.14200837832.0936889.642.4936577.893.3236694.523.01200940335.1241143.772.0040810.651.1840935.271.49201048326.8345888.485.0548231.240.2047354.812.01201150315.5951180.361.7251318.561.9951266.861.89201258310.0157082.52.1160423.973.6359173.931.48201364621.5263665.271.4864725.560.1664328.910.45201472387.0271007.171.9170826.802.1670894.282.06201577382.979195.752.3475912.221.9077140.590.31201684984.7188328.633.9383860.281.3285531.890.64RMSE1517.00RMSE1098.64RMSE793.35MAPE3.55MAPE2.34MAPE1.74201793294.0398514.727.1590899.302.5793748.231.462018102544109875.487.7697907.144.52102384.491.082019113724122546.365.60105028.937.65111582.200.37RMSE7276.45RMSE5854.90RMSE1267.42MAPE6.83MAPE4.91MAPE0.842


## 2.2. ARIMA Model

ARIMA [26] is a time-series prediction model proposed by Box and Jenkins. The main idea is to transform the unstable time series into the stationary time series and then establish the model by parameter estimation. Where AR is autoregressive, p is the order of autoregressive, MA is the moving average, q is the order of moving average, and d is the number of difference when the time series becomes a stationary time series.Suppose that a zero-mean stationary sequence to bext and at to be a white noise; then, we obtain(9)xt−φ1xt−1−φ2xt−2−⋯−φpxt−p=θ1at−1−θ2at−2−⋯−θpat−p,where Extas=0t<s. [25].Furthermore, suppose that a nonstationary time sequence bezt; we obtain(10)ϕB1−Bdzt=θBat,where xt=∇dzt,t>d.The specific modeling process is as follows:(1)
Stationary test of time series: as can be seen from Figure1, the time series of China’s total clean energy consumption from 2000 to 2016 shows an exponential upward trend, which is intuitively nonstationary. To obtain a stationary time series, the first-order difference is used to obtain a new series Yt, and then, the ADF test is carried out. The test results obtained are shown in Table 3. It can be seen that the significance levels of 0.01, 0.05, and 0.1 are all greater than ADF statistics, indicating that the sequence Yt first-order difference is stable.The white noise test was further carried out on the data, and the test result is shown in Figure2. It can be seen that most of the right-most P values are greater than the confidence level of 0.05, so the sequence Yt after first-order difference can be regarded as a stable white noise sequence.(2)
Identify the order of the ARIMA model. The autocorrelation graph and partial autocorrelation graph of the treated time series are drawn to find the optimal combination of P and Q. Through Eviews7.2 analysis, the final established model is ARIMA (2,1,1).(3)
Parameter estimation and test are carried out.Figure 1
Time sequence diagram of total clean energy consumption in China from 2000 to 2016 (10,000 tons of standard coal).Table 3
ADF test results after first-order difference.
t-StatisticProbAugmented Dickey–Fuller test statistic254.712−9.33800.0000Test critical values1% level−4.72835% level−3.759710% level−3.3249Figure 2
White noise test.Table 4
Parameter estimation of ARIMA (2,1,1).
VariableCoefficientStd. errort-StatisticProbC8250.77221814.67894.54660.0008AR(2)0.80220.08829.08960.0000MA(1)−0.96710.0697−13.87300.0000Parameter estimation of the model obtained by least-squares estimation is shown in Table4.According to parameter estimation in Table3, the model Yt can be written as(11)Yt=8250.7722+1+0.9671B1−0.8022B2at.According to theT-statistic value and its P value of the model coefficient, the parameter estimates of all explanatory variables in the model are significant at a significant water level of 0.05. According to the analysis of Eviews7.2, F-statistic = 21.2339, Prob(F-statistic) = 0.0001, AIC = 17.1557, and D-W = 1.9756. The fitting effect of the whole equation is good.In addition, the residual white noise inspection according to theP value of the test statistics Q is greater than 0.05 (see Figure 3); there is no autocorrelation in the residual series, which is white noise, indicating that the model information is extracted sufficiently, and this model is a suitable model that can be used to forecast China's clean energy consumption. Static forecast values are used for in-sample (2000 to 2016) and dynamic forecast values are used for out-of-sample (2017 to 2019), and the specific fitting and forecasting results are shown in Table 2.Figure 3
Residual white noise test.


## 2.3. Combination Prediction Based on the Nonlinear Programming Model

The research of combination prediction is one of the hot issues in the field of prediction at present; its advantage is that the information provided by each prediction method can be used to improve the accuracy of prediction, and the key of the combination prediction method is to determine the weight of each single prediction method. For the total errorE of the combined prediction method, the following nonlinear optimization model is adopted:(12)minE=∑i=1nxj−fxij,lij2,s.t∑i=1nlij=1,lij≥0j=1,2,3,…,m.This is a nonlinear programming problem with only one constraint condition. The total error of the improved combined prediction method is(13)E∗=minE.A minimum variance after different combination forecast models is reached; the prediction formula of the combined model can be obtained as(14)x^j=0.3741x^1j+0.6259x^2j.


## 3. Model Comparison and Prediction Analysis

According to Table4, it can be seen that the average relative error of the fitting values obtained by the combination prediction method from 2000 to 2016 is 1.74%, which is less than the average relative errors of the GM(1,1) model and the ARIMA model in a single fitting which are 3.55% and 2.34%, indicating that the fitting accuracy of the combination model is relatively high. In addition, the average relative error of the predicted value of the combined model from 2017 to 2019 is 0.842%, which is the smallest, indicating that the model has good stability and strong generalization ability. The combination prediction method in this paper had better prediction effect. Compared with the combined prediction model in [27–29], the average relative error in this paper is also the smallest. Chong Wang [30] adopted the Shapley combination prediction method to get the average relative error was 3.05%, which was larger than that in this paper.Table5 gives the projected value of China’s clean energy consumption in 2020–2024. Firstly, the GM (1,1) model and ARIMA model are used to forecast by year, and then, the weight is calculated by Matlab R2016 A programming. Finally, the combined forecast is made to get the comprehensive predicted value.Table 5
2020–2024 comprehensive forecast results of China’s clean energy consumption demand (10,000 tons of standard coal).
YearGM (1,1)ArimaCombined predictive value2020136678.45112282.56122240.962021152440.26119627.61133021.742022170019.73127078.43144607.072023189626.46134602.60157063.342024211494.26142211.61170492.79According to the results in Table5, the change trend chart of clean energy consumption demand in 2020–2024 is drawn. As can be seen from Figure 4, the consumption demand for clean energy in China will continue to grow in the next five years, with the growth rate roughly stabilizing at 8.6%. By 2024, it is expected that the consumption demand for clean energy in China will reach about 1.7 billion tons of standard coal.Figure 4
2020–2024 forecast of China’s clean energy consumption demand trend (10,000 tons of standard coal).


## 4. The Effects of Sample Size of the Combination Model

GM(1, 1), i.e., the first-order grey model with one variable, can be constructed with at least four data points [22, 31]. It has been successfully used in forecasting problems in many disciplines [32–39]. Considering recent information first，using data modeling from 2013 to 2016, and then using data modeling from 2012 to 2016, add one sample data each time, and finally, make predictions from 2017 to 2019; the fitting accuracy and predicted accuracy change under different sample size conditions (see Figure 5). It can also be seen from Figure 5 that when the sample size is 6, the prediction effect of the model is relatively good. Therefore, it can be concluded that the sample size will have an impact on the prediction results, and the optimal sample size can be obtained through experimental testing for specific situations.Figure 5
Sample size effects on clean energy consumption prediction accuracy.


## 5. Conclusion

In this paper, based on two single forecasting methods and through the nonlinear programming model, a weighted combined forecasting model is constructed to forecast the consumption demand of clean energy in China. Through the comparison of the GM (1,1) model, the ARIMA model, and the combination prediction model, it can be concluded that the combination prediction model has higher fitting accuracy and prediction accuracy, which provides an innovative method for combination model prediction. The forecast results show that the consumption of clean energy in China will grow at a rate of about 8.6 percent, and demand will reach about 1.7 billion tons of standard coal by 2024. In the context of global energy network, the development prospect of clean energy is great [40], and clean energy will become a new trend of future energy development.In addition, the energy system is a complex time variation system, and it is difficult for a single forecasting method to achieve satisfactory results. How to select the combined model and how to construct the combined algorithm is still an important question.



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*Source: 5707174-2022-03-23.xml*