# Forecasting of Wastewater Discharge and the Energy Consumption in China Based on Grey Model

Authors: ZhenHua Li; ZhiHong Zou; Yang Yu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Water pollution control is currently an important development strategy for China. Analysis and forecasting of wastewater discharge and the energy consumption are critical for this strategy. In this paper, we proposed the two time series models, namely, improved GM(0,n) model (IGM(0,n)) and optimized FGM(1,1) model (OFGM(1,1)). Two numerical experiments were used to test the validity of these models. It was indicated that the forecasting ability of two models had remarkably improved, compared to traditional FGM(1,1) and GM(0,n), respectively. We applied the two models to forecast wastewater discharge and energy consumption in China. The forecasting results have shown that the current growth rate in treated wastewater could meet the planning target of the 13th Five-Year Plan. The wastewater discharge and the energy consumption would reach 77.39 billion tons and 22.75 billion kWh, respectively, in 2020.


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## Body

## 1. Introduction

With the rapid development of the Chinese economy, wastewater discharge has increased dramatically during the past decades, and China is currently the largest sewage producer. Since 2005 to 2015 the wastewater discharge has up to 73.53 billion tons from 52.45 billion tons. A large amount of wastewater has negative effects on the environment and human health [1–3]. The government attaches great importance to water pollution. The daily wastewater treatment capacity of China increased dramatically during the past decade. The administration of water pollution control and energy conservation is a significant development strategy for China [4, 5]. As of September 30, 2016, China had built 3552 WWTPs, with a total capacity of 1.7946×108m3/d [6]. About 72.39% of wastewater was treated. A large amount of wastewater would be treated in the future. Wastewater treatment plant (WWTP) is one of the energy-intensive industries [7, 8]. The energy demand for wastewater treatment would increase substantially. Therefore, accurate prediction of the wastewater discharge and energy consumption constitutes a vital part of water pollution control policy of China.Several studies focused on wastewater discharge and energy demand in China. Wang et al. applied an optimized NGBM(1,1) model for predicting the qualified discharge rate of industrial wastewater in China [9]. Liu et al. employed an indicator system to forecast the amount of domestic wastewater discharge in Jiangsu Province [10]. Li et al. used stochastic gradient regression to predict wastewater discharge of Tianjin and analyze its influence factors. The results showed that the model has higher precision than the support vector machine method and adaptive regression splines method [11]. Chen et al. investigated the overall status of wastewater discharge in China to analyze its driving factors [12]. Yin et al. developed a neural network model to forecast and analyze urban water-energy demand in China [13]. Xie and Wang examined the challenge of energy consumption in wastewater treatment plants [14]. Jian et al. applied the various methods to survey the energy consumption status and its influencing factors in 1441 municipal wastewater treatment plants [15]. Currently, the efforts to improve wastewater treatment are not keeping pace with economic development. At the same time, the average electricity consumption per cubic meter for wastewater was higher than developed country [14, 15].The present work attempted to forecast wastewater discharge and energy consumption in China. Various prediction methods including ARIMA, artificial neural network (ANN), and nonparametric regression used to predict the system’s future behavior [11, 16–18]. The forecasting results of ARIMA, ANN, and nonparametric regression depend on the number of training data and the availability of data. These limitations have not yet been overcome. The advantage of grey forecasting model is that we only need small amounts of data to predict the system’s future behavior [19]. It can overcome the problem that the observed data available are very small. Just a few data are sufficient to characterize the system’s behavior. It is more suitable for short-term prediction than a medium-term and long-term prediction. Different grey forecasting models were suited for data sequence with different characteristics. The grey Bernoulli model was suitable for fluctuant data. Grey Verhulst model was suitable for the historical data with ‘S’ distributing. Grey GM(1,1) model was suitable for historical data with exponential distributing. Grey forecasting model has been widely used in economics, finance, and other fields and has achieved lots of good results [20–25]. A grey forecasting model may yield large forecasting errors. Types of research concentrated on grey forecasting models to improve the forecasting precision. Xu et al. presented an approach to the least squares to grey Verhulst model. The model achieves reliable and precise results [26]. Jin et al. proposed a new grey model to avoid the error amplification resulting from the improper choice of the initial condition [27]. Wang developed a grey dynamic model of GMC (1,n) and proposed the Nash equilibrium idea-based optimization method to solve the model parameters. The results showed that the model was with higher accuracy than traditional model [28]. Wang and Hao constructed a nonlinear optimization model of GMC (1,n) and optimized the parameters of background. The results showed that the method could minimize the modeling error [29]. Wang et al. proposed an approach called DGGM (1,1) to predict quarterly hydropower production in China. The results indicated that DGGM (1,1) has a higher prediction accuracy than traditional GM (1,1) and SARIMA models [30]. Hsu and Wang presented Bayesian grey forecast models and used the technique of Markov Chain Monte Carlo to estimate the parameters for grey differential function [31]. Chen et al. improved nonlinear grey Bernoulli model (NGBM) by the Nash equilibrium concept. They found that the prediction by the method was better than the NGBM [32]. Lee and Tong combined residual modification with genetic programming sign estimation to improve the grey forecasting model [33].In this paper, we presented two grey forecasting models an optimized FGM(1,1) (OFGM(1,1)) model and improved GM(0,n) (IGM(0,n)) to forecast wastewater discharge and energy consumption in China. The GM(0,n) model was a special form of GM(1,n) model with no derivatives [34, 35]. A new method was used to improve the accuracy of the traditional GM(0,n) model. The empirical results showed that IGM(0,n) model has a better performance than the traditional model. Tien proposed the FGM(1,1) model and showed that the model had higher prediction accuracy than GM(1,1) model [36]. To improve the forecasting accuracy of FGM(1,1) model, an optimized FGM(1,1) model was proposed. The particle swarm optimization algorithm was used to optimize the generating coefficient of the model. FGM(1,1) model and GM(1,1) were employed to compare the effectiveness of OFGM(1,1) model. The mean absolute percentage error (MAPE) was chosen to minimize the error between actual values and forecasting values [37, 38]. The results demonstrate that OFGM(1,1) model had higher forecasting accuracy than the other forecasting model. So, the paper adopted IGM(0,n) model and OFGM(1,1) model to forecast the wastewater discharge and energy consumption in China. It would provide a profitable reference for the economic and environmental benefits of sewage treatment.


## 2. Methodology and Data

### 2.1. The OFGM(1,1) Model

GM(1,1) model is a single-variable grey forecasting model. To improve its performance, Tien put forward a new prediction model FGM(1,1) model. For the GM(1,1) model, the forecasting values of this model are independent of the first data of time series. So an arbitrary number could be inserted in front of the time series [34]. By doing so, it only requires three data values to forecast the system’s future behavior. Empirical results show that this model has a higher prediction accuracy than GM(1,1) model. The existing model FGM(1,1) has several defects. It is inaccurate to set generating coefficient as a constant. The constant could affect the prediction accuracy of the model. To improve its performance, the article introduces an unknown interpolation coefficient γ to calculate the generating coefficient of FGM(1,1).The procedures of the new method can be concluded as follows.An arbitrary number would be inserted in front of seriesX0=(x0(1),x0(2),…,x0(n)). The new series is(1)X^0=(x00,x01,x02,…,x0nwherex0(0) is arbitrary number; to keep life simple, we assume x00=0. Consequently, the 1-GAO of X^0 is given by(2)x1k=∑i=0kx0kThe new series is written as(3)X^1=x10,x11,x12,…,x1nHence, grey model FGM(1,1) based on seriesX^1 can be written as(4)x0k+az1k=bwhere z1k=0.5x1k+0.5x1k-1(k=1,2,3,…,n).The grey differential equation of Eq. (4) is(5)dx1dt+ax1=bwhere a and b are the interim parameters. The parameters are(6)ab=BTB-1BTYwhere(7)B=-γx10+1-γx111-γx11+1-γx121⋮⋮-γx1n-1+1-γx10n1,Y=x01x02⋮x0nIf γ=0.5, the model is transformed into FGM(1,1) model.The time response function is(8)x^1k=x01-baek-1-bak=1,2,3,…,nApplying the 1-GAOto Eq. (8), we have the modeling value x0k as forecasts(9)x^0k=x01-ba1-eae-ak-1k=1,2,3,…,n


### 2.2. The Improved Grey Multivariable Model IGM(0,n)

The GM(0,n) is a multivariable grey forecasting model, which is a particular case of the GM(1,n) model [29]. To improve the modeling and forecasting precision of the system, we used the secondary approximation model to enhance the adaptability of IGM(0,n) on real data. The procedures of IGM(0,n) could be indicated as follows.Suppose the set of a primitive data sequence is(10)X10,X20,…,Xn0where Xk0=(xk01,xk02,…,xk(0)(m)) and Xj0i is the jth value of Xj(0) at equispaced interval of i time.X10 is the feature sequence of system, and the others are the influence factors sequences of system.The first order accumulation generation operator (1-GAO) forXi(0) is given as(11)X11,X21,…,Xn1where Xk1=(xk11,xk12,…,xk(1)(m)) and xk1i=∑j=1ixk0j,i=1,2,…,m.Then, the GM(0,n) model is written as(12)x11k=a+b2x21k+b3x31k+⋯+bnxn1kk=2,⋯,mThe least-squares solution of the parameters of GM(0,n) is(13)b^=BTB-1BTYwhereb^=(a,b2,…,bn)T, Y=(xk12,…,xk1m)T, and(14)B=1x212…xn-112xn121x213…xn-113xn13⋮⋮⋱⋮⋮1x21m…xn-11mxn1m.The time response of Eq. (12) is(15)x^11k=a+b2x21k+b3x31k+⋯+bnxn1kAfter inversely accumulating generation operation from Eq. (11), the grey forecasting model of the original sequence is defined as(16)x^10k=x^11k-x^11k-1k=2,3,…,n.To improve the modelling and prediction precision of the GM(0,n), this paper inputs quadratic polynomials of the influence factors sequence in GM(0,n). On this basis, the improved GM(0,n) (IGM(0,n)) is defined as(17)x11k=a+b2x21k+b21x21k2+⋯+bnxn1k+bn1xn1k2The least-squares solution of the parameters of IGM(0,n) is(18)b~=B~TB~TYwhere b~=(a,b2,b21,…,bn,bn1)T, Y=(xk12,…,xk1m)T, and(19)B~=1x212x2122⋯xn12xn1221x213x2132⋯xn13xn132⋮⋮⋮⋱⋮⋮1x21mx21m2⋯xn1mxn1m2.The time response of Eq. (17) is(20)x~11k=a+b2x21k+b21x21k2+⋯+bnxn1k+bn1xn1k2By performing the inverse accumulation generation operator from Eq. (10), the grey forecasting model of the primitive data sequences is finally obtained as(21)x~10k=x~11k-x~11k-1k=2,3,…,n.


### 2.3. Particle Swarm Optimization (PSO) Algorithm

PSO algorithm was proposed by Kennedy and Eberhart [39]. As one of the intelligent optimization algorithms, the method was simple implementation [40, 41]. It had better efficiency than other evolutionary computations to obtain the optimal solution. To solve the generating coefficient problem, PSO was employed to calculate the generating coefficient of FGM(1,1).In this article, the velocity and particle iteration followed the equations(22)vijk+1=ωvijk+c1r1pijk-xijk+c2r2gijk-xijk(23)xijk+1=xijk+vijkj=1,2,⋯,40where inertia weight ω was 1; c1 and c2 were acceleration factors that were 0.85; r1 and r2 were random numbers in the interval between 0 and 1; maximum number of iterations was 200; population was D=40.


### 2.4. Assessment of the Modelling and Forecasting Precision

Mean absolute percentage error (MAPE), mean absolute error (MAE), and mean square error (MSE) were used to test the accuracy of the forecast models. Generally, the lower deviation indexes meant higher accuracy. The calculation formulas were defined as(24)MAPE=1k∑i=1kx^1i-x0ix0i×100%(25)MAE=1k∑i=1kx^1i-x0i(26)MSE=1k∑i=1kx^1i-x0i2The fitness function was defined as(27)minMAPE=1k∑i=1kx^1i-x0ix0i×100%where x0i and x^1i were the ith actual value and prediction value, respectively, and k was the number of predictions.


### 2.5. Validation of the OFGM(1, 1) Model and IGM(0,n) Model

#### 2.5.1. OFGM(1, 1) Model

In this section, data sequence A was used to verify the performance of OFGM(1,1) model [39]. For this data sequence, the first five data were used for training data, while the other data were used to evaluate the forecasting performance. The evolution of fitness of the OFGM(1,1) model was shown in Figure 1. It can be seen in Figure 1 that the fitness value converges very fast to the stagnation point. This algorithm is very effective in solving the parameter optimization problem.Figure 1
Evolution of fitness of OFGM(1,1) model.The actual and forecast values of data sequence A using FGM(1,1) and OFGM(1,1) were shown in Table1 and Figure 2. There deviation indexes were shown in Tables 2 and 3. It can be seen from Table 2 that the training accuracies of FGM(1,1) and OFGM(1,1) were 2.42% and 1.92%, respectively. The other deviation indexes also showed that he optimized FGM(1,1) had higher training accuracy than FGM(1,1). According to Table 3, deviation indexes of OFGM(1,1) were the smallest. Results showed that OFGM(1,1) model had a better performance than FGM(1,1) model. The OFGM(1,1) model would be used to predict the energy consumption in China.Table 1
Actual and forecast values of data sequence A.
SequenceX1 FGM(1,1) model OFGM(1, 1) model Model value Relative error (%) Model value Relative error (%) 9.4 9.4 0 9.4 0 12.5 11.6 6.9 11.77 5.80 14.0 13.70 2.16 13.88 0.84 15.9 16.12 1.39 16.37 2.95 19.3 18.97 1.68 19.30 0 24.1 22.33 7.33 22.76 5.57 25.8 26.29 1.89 26.83 4.00 28.7 30.94 7.81 31.64 10.23 39.6 36.42 8.03 37.30 5.80 42.2 42.87 15.82 43.98 4.22 58.3 50.46 13.45 51.86 11.05 77.5 59.39 23.37 61.15 21.10 89.6 69.90 21.98 72.09 19.54 98.0 82.28 16.04 85.01 13.26 106.4 96.84 8.99 100.23 5.80Table 2
Deviation indexes of simulation results for different forecasting methods.
Method MAPE(%) MAE MSE FGM(1,1) model 2.42 0.3500 0.2115 OFGM(1, 1) model 1.92 0.2640 0.1536Table 3
Deviation indexes of forecasting results for different forecasting methods.
Method MAPE(%) MAE MSE FGM(1,1) model 11.05 7.928 113.49 OFGM(1, 1) model 10.58 6.8850 84.2164Figure 2
Forecasting values by FGM(1,1) and of OFGM(1,1).


#### 2.5.2. IGM(0,n) Model

In this section, the data sequencesX10 and X20 [30] were used to evaluate the performance of improved IGM(0,2) model. The simulation results of these data sequences using GM(0,2) model and IGM(0,2) model were depicted in Table 4 and Figure 3. To show the superiority of IGM(0,2) model, the deviation indexes of GM(0,2) and IGM(0,2) model were compared in Table 5. As can be seen in Tables 4 and 5, the relative errors of GM(0,2) were ranging from 0.7% to 3.8%. The relative errors of IGM(0,2) were ranging from 0.4% to 2.5%. The MAPEs for GM(0,2) and IGM(0,2) were 2.48% and 1.53%, respectively. Meanwhile, the MAE and MSE of IGM(0,2) model were smaller than the GM(0,2) model. The performance comparisons showed that the IGM(0,2) had the best fitting effect with minimum deviation indexes. The IGM(0,2) model would be used to forecast the wastewater discharge of China.Table 4
Forecasting values by GM(0,2) and IGM(0,2).
Sequence number Actual value GM(0,2) model IGM(0, 2) model Model value Relative error (%) Model value Relative error (%) 2 3.278 3.153 3.8 3.293 0.4 3 3.307 3.331 0.7 3.248 1.7 4 3.390 3.518 3.8 3.476 2.5 5 3.679 3.619 1.6 3.623 1.5Table 5
Deviation indexes of forecasting results for different forecasting methods.
Method MAPE(%) MAE MSE GM(0,2) model 2.48 0.0842 0.0540 IGM(0, 2) model 1.53 0.0090 0.0036Figure 3
Forecasting values by GM(0,2) and IGM(0,2).


### 2.6. Data Sources

This article collected annual data on wastewater discharged and the energy consumption in China for 2005-2015. The IGM(0,2) was used to predict wastewater discharge. The OFGM(1,1) method was employed to forecast energy consumption. The data sequences of wastewater discharge and real GDP were feature sequence and influence factor sequence of a system, respectively.Water pollution and energy shortage would become tumbling block of China’s economic growth. With the accelerated industrialization and urbanization, wastewater discharge would continue to increase in the future. From 2005 to 2015, real GDP had risen from 18.7319 trillion yuan to 68.9052 yuan. Based on economic development target, the country’s per capita GDP will probably reach 20 thousand yuan in 2020, assuming that China’s average annual economic growth rate is 7% in 2016-2020 [42]. Treated wastewater had up to 72.39% from 60.8% since 2012 to 2015. Energy demand for sewer had increased remarkably. So far, the average electricity consumption per cubic meter for wastewater was 0.2927kWh/m3 in China [14, 15] and would not be changed in the short term. It means that the energy consumption of wastewater treatment system had up to 15.58 billion kWh in 2015. Energy demand for wastewater in the recent year was listed in Table 6 (2012 to 2015).Table 6
The wastewater discharge and energy consumption in China (2012 to 2015).
Year 2012 2013 2014 2015 Wastewater discharge (billion tons) 68.43 69.49 71.56 73.53 Treated wastewater (billion tons) 41.62 45.61 49.43 53.23 Energy consumption (billion kWh) 12.18 13.35 14.47 15.58


## 2.1. The OFGM(1,1) Model

GM(1,1) model is a single-variable grey forecasting model. To improve its performance, Tien put forward a new prediction model FGM(1,1) model. For the GM(1,1) model, the forecasting values of this model are independent of the first data of time series. So an arbitrary number could be inserted in front of the time series [34]. By doing so, it only requires three data values to forecast the system’s future behavior. Empirical results show that this model has a higher prediction accuracy than GM(1,1) model. The existing model FGM(1,1) has several defects. It is inaccurate to set generating coefficient as a constant. The constant could affect the prediction accuracy of the model. To improve its performance, the article introduces an unknown interpolation coefficient γ to calculate the generating coefficient of FGM(1,1).The procedures of the new method can be concluded as follows.An arbitrary number would be inserted in front of seriesX0=(x0(1),x0(2),…,x0(n)). The new series is(1)X^0=(x00,x01,x02,…,x0nwherex0(0) is arbitrary number; to keep life simple, we assume x00=0. Consequently, the 1-GAO of X^0 is given by(2)x1k=∑i=0kx0kThe new series is written as(3)X^1=x10,x11,x12,…,x1nHence, grey model FGM(1,1) based on seriesX^1 can be written as(4)x0k+az1k=bwhere z1k=0.5x1k+0.5x1k-1(k=1,2,3,…,n).The grey differential equation of Eq. (4) is(5)dx1dt+ax1=bwhere a and b are the interim parameters. The parameters are(6)ab=BTB-1BTYwhere(7)B=-γx10+1-γx111-γx11+1-γx121⋮⋮-γx1n-1+1-γx10n1,Y=x01x02⋮x0nIf γ=0.5, the model is transformed into FGM(1,1) model.The time response function is(8)x^1k=x01-baek-1-bak=1,2,3,…,nApplying the 1-GAOto Eq. (8), we have the modeling value x0k as forecasts(9)x^0k=x01-ba1-eae-ak-1k=1,2,3,…,n


## 2.2. The Improved Grey Multivariable Model IGM(0,n)

The GM(0,n) is a multivariable grey forecasting model, which is a particular case of the GM(1,n) model [29]. To improve the modeling and forecasting precision of the system, we used the secondary approximation model to enhance the adaptability of IGM(0,n) on real data. The procedures of IGM(0,n) could be indicated as follows.Suppose the set of a primitive data sequence is(10)X10,X20,…,Xn0where Xk0=(xk01,xk02,…,xk(0)(m)) and Xj0i is the jth value of Xj(0) at equispaced interval of i time.X10 is the feature sequence of system, and the others are the influence factors sequences of system.The first order accumulation generation operator (1-GAO) forXi(0) is given as(11)X11,X21,…,Xn1where Xk1=(xk11,xk12,…,xk(1)(m)) and xk1i=∑j=1ixk0j,i=1,2,…,m.Then, the GM(0,n) model is written as(12)x11k=a+b2x21k+b3x31k+⋯+bnxn1kk=2,⋯,mThe least-squares solution of the parameters of GM(0,n) is(13)b^=BTB-1BTYwhereb^=(a,b2,…,bn)T, Y=(xk12,…,xk1m)T, and(14)B=1x212…xn-112xn121x213…xn-113xn13⋮⋮⋱⋮⋮1x21m…xn-11mxn1m.The time response of Eq. (12) is(15)x^11k=a+b2x21k+b3x31k+⋯+bnxn1kAfter inversely accumulating generation operation from Eq. (11), the grey forecasting model of the original sequence is defined as(16)x^10k=x^11k-x^11k-1k=2,3,…,n.To improve the modelling and prediction precision of the GM(0,n), this paper inputs quadratic polynomials of the influence factors sequence in GM(0,n). On this basis, the improved GM(0,n) (IGM(0,n)) is defined as(17)x11k=a+b2x21k+b21x21k2+⋯+bnxn1k+bn1xn1k2The least-squares solution of the parameters of IGM(0,n) is(18)b~=B~TB~TYwhere b~=(a,b2,b21,…,bn,bn1)T, Y=(xk12,…,xk1m)T, and(19)B~=1x212x2122⋯xn12xn1221x213x2132⋯xn13xn132⋮⋮⋮⋱⋮⋮1x21mx21m2⋯xn1mxn1m2.The time response of Eq. (17) is(20)x~11k=a+b2x21k+b21x21k2+⋯+bnxn1k+bn1xn1k2By performing the inverse accumulation generation operator from Eq. (10), the grey forecasting model of the primitive data sequences is finally obtained as(21)x~10k=x~11k-x~11k-1k=2,3,…,n.


## 2.3. Particle Swarm Optimization (PSO) Algorithm

PSO algorithm was proposed by Kennedy and Eberhart [39]. As one of the intelligent optimization algorithms, the method was simple implementation [40, 41]. It had better efficiency than other evolutionary computations to obtain the optimal solution. To solve the generating coefficient problem, PSO was employed to calculate the generating coefficient of FGM(1,1).In this article, the velocity and particle iteration followed the equations(22)vijk+1=ωvijk+c1r1pijk-xijk+c2r2gijk-xijk(23)xijk+1=xijk+vijkj=1,2,⋯,40where inertia weight ω was 1; c1 and c2 were acceleration factors that were 0.85; r1 and r2 were random numbers in the interval between 0 and 1; maximum number of iterations was 200; population was D=40.


## 2.4. Assessment of the Modelling and Forecasting Precision

Mean absolute percentage error (MAPE), mean absolute error (MAE), and mean square error (MSE) were used to test the accuracy of the forecast models. Generally, the lower deviation indexes meant higher accuracy. The calculation formulas were defined as(24)MAPE=1k∑i=1kx^1i-x0ix0i×100%(25)MAE=1k∑i=1kx^1i-x0i(26)MSE=1k∑i=1kx^1i-x0i2The fitness function was defined as(27)minMAPE=1k∑i=1kx^1i-x0ix0i×100%where x0i and x^1i were the ith actual value and prediction value, respectively, and k was the number of predictions.


## 2.5. Validation of the OFGM(1, 1) Model and IGM(0,n) Model

### 2.5.1. OFGM(1, 1) Model

In this section, data sequence A was used to verify the performance of OFGM(1,1) model [39]. For this data sequence, the first five data were used for training data, while the other data were used to evaluate the forecasting performance. The evolution of fitness of the OFGM(1,1) model was shown in Figure 1. It can be seen in Figure 1 that the fitness value converges very fast to the stagnation point. This algorithm is very effective in solving the parameter optimization problem.Figure 1
Evolution of fitness of OFGM(1,1) model.The actual and forecast values of data sequence A using FGM(1,1) and OFGM(1,1) were shown in Table1 and Figure 2. There deviation indexes were shown in Tables 2 and 3. It can be seen from Table 2 that the training accuracies of FGM(1,1) and OFGM(1,1) were 2.42% and 1.92%, respectively. The other deviation indexes also showed that he optimized FGM(1,1) had higher training accuracy than FGM(1,1). According to Table 3, deviation indexes of OFGM(1,1) were the smallest. Results showed that OFGM(1,1) model had a better performance than FGM(1,1) model. The OFGM(1,1) model would be used to predict the energy consumption in China.Table 1
Actual and forecast values of data sequence A.
SequenceX1 FGM(1,1) model OFGM(1, 1) model Model value Relative error (%) Model value Relative error (%) 9.4 9.4 0 9.4 0 12.5 11.6 6.9 11.77 5.80 14.0 13.70 2.16 13.88 0.84 15.9 16.12 1.39 16.37 2.95 19.3 18.97 1.68 19.30 0 24.1 22.33 7.33 22.76 5.57 25.8 26.29 1.89 26.83 4.00 28.7 30.94 7.81 31.64 10.23 39.6 36.42 8.03 37.30 5.80 42.2 42.87 15.82 43.98 4.22 58.3 50.46 13.45 51.86 11.05 77.5 59.39 23.37 61.15 21.10 89.6 69.90 21.98 72.09 19.54 98.0 82.28 16.04 85.01 13.26 106.4 96.84 8.99 100.23 5.80Table 2
Deviation indexes of simulation results for different forecasting methods.
Method MAPE(%) MAE MSE FGM(1,1) model 2.42 0.3500 0.2115 OFGM(1, 1) model 1.92 0.2640 0.1536Table 3
Deviation indexes of forecasting results for different forecasting methods.
Method MAPE(%) MAE MSE FGM(1,1) model 11.05 7.928 113.49 OFGM(1, 1) model 10.58 6.8850 84.2164Figure 2
Forecasting values by FGM(1,1) and of OFGM(1,1).


### 2.5.2. IGM(0,n) Model

In this section, the data sequencesX10 and X20 [30] were used to evaluate the performance of improved IGM(0,2) model. The simulation results of these data sequences using GM(0,2) model and IGM(0,2) model were depicted in Table 4 and Figure 3. To show the superiority of IGM(0,2) model, the deviation indexes of GM(0,2) and IGM(0,2) model were compared in Table 5. As can be seen in Tables 4 and 5, the relative errors of GM(0,2) were ranging from 0.7% to 3.8%. The relative errors of IGM(0,2) were ranging from 0.4% to 2.5%. The MAPEs for GM(0,2) and IGM(0,2) were 2.48% and 1.53%, respectively. Meanwhile, the MAE and MSE of IGM(0,2) model were smaller than the GM(0,2) model. The performance comparisons showed that the IGM(0,2) had the best fitting effect with minimum deviation indexes. The IGM(0,2) model would be used to forecast the wastewater discharge of China.Table 4
Forecasting values by GM(0,2) and IGM(0,2).
Sequence number Actual value GM(0,2) model IGM(0, 2) model Model value Relative error (%) Model value Relative error (%) 2 3.278 3.153 3.8 3.293 0.4 3 3.307 3.331 0.7 3.248 1.7 4 3.390 3.518 3.8 3.476 2.5 5 3.679 3.619 1.6 3.623 1.5Table 5
Deviation indexes of forecasting results for different forecasting methods.
Method MAPE(%) MAE MSE GM(0,2) model 2.48 0.0842 0.0540 IGM(0, 2) model 1.53 0.0090 0.0036Figure 3
Forecasting values by GM(0,2) and IGM(0,2).


## 2.5.1. OFGM(1, 1) Model

In this section, data sequence A was used to verify the performance of OFGM(1,1) model [39]. For this data sequence, the first five data were used for training data, while the other data were used to evaluate the forecasting performance. The evolution of fitness of the OFGM(1,1) model was shown in Figure 1. It can be seen in Figure 1 that the fitness value converges very fast to the stagnation point. This algorithm is very effective in solving the parameter optimization problem.Figure 1
Evolution of fitness of OFGM(1,1) model.The actual and forecast values of data sequence A using FGM(1,1) and OFGM(1,1) were shown in Table1 and Figure 2. There deviation indexes were shown in Tables 2 and 3. It can be seen from Table 2 that the training accuracies of FGM(1,1) and OFGM(1,1) were 2.42% and 1.92%, respectively. The other deviation indexes also showed that he optimized FGM(1,1) had higher training accuracy than FGM(1,1). According to Table 3, deviation indexes of OFGM(1,1) were the smallest. Results showed that OFGM(1,1) model had a better performance than FGM(1,1) model. The OFGM(1,1) model would be used to predict the energy consumption in China.Table 1
Actual and forecast values of data sequence A.
SequenceX1 FGM(1,1) model OFGM(1, 1) model Model value Relative error (%) Model value Relative error (%) 9.4 9.4 0 9.4 0 12.5 11.6 6.9 11.77 5.80 14.0 13.70 2.16 13.88 0.84 15.9 16.12 1.39 16.37 2.95 19.3 18.97 1.68 19.30 0 24.1 22.33 7.33 22.76 5.57 25.8 26.29 1.89 26.83 4.00 28.7 30.94 7.81 31.64 10.23 39.6 36.42 8.03 37.30 5.80 42.2 42.87 15.82 43.98 4.22 58.3 50.46 13.45 51.86 11.05 77.5 59.39 23.37 61.15 21.10 89.6 69.90 21.98 72.09 19.54 98.0 82.28 16.04 85.01 13.26 106.4 96.84 8.99 100.23 5.80Table 2
Deviation indexes of simulation results for different forecasting methods.
Method MAPE(%) MAE MSE FGM(1,1) model 2.42 0.3500 0.2115 OFGM(1, 1) model 1.92 0.2640 0.1536Table 3
Deviation indexes of forecasting results for different forecasting methods.
Method MAPE(%) MAE MSE FGM(1,1) model 11.05 7.928 113.49 OFGM(1, 1) model 10.58 6.8850 84.2164Figure 2
Forecasting values by FGM(1,1) and of OFGM(1,1).


## 2.5.2. IGM(0,n) Model

In this section, the data sequencesX10 and X20 [30] were used to evaluate the performance of improved IGM(0,2) model. The simulation results of these data sequences using GM(0,2) model and IGM(0,2) model were depicted in Table 4 and Figure 3. To show the superiority of IGM(0,2) model, the deviation indexes of GM(0,2) and IGM(0,2) model were compared in Table 5. As can be seen in Tables 4 and 5, the relative errors of GM(0,2) were ranging from 0.7% to 3.8%. The relative errors of IGM(0,2) were ranging from 0.4% to 2.5%. The MAPEs for GM(0,2) and IGM(0,2) were 2.48% and 1.53%, respectively. Meanwhile, the MAE and MSE of IGM(0,2) model were smaller than the GM(0,2) model. The performance comparisons showed that the IGM(0,2) had the best fitting effect with minimum deviation indexes. The IGM(0,2) model would be used to forecast the wastewater discharge of China.Table 4
Forecasting values by GM(0,2) and IGM(0,2).
Sequence number Actual value GM(0,2) model IGM(0, 2) model Model value Relative error (%) Model value Relative error (%) 2 3.278 3.153 3.8 3.293 0.4 3 3.307 3.331 0.7 3.248 1.7 4 3.390 3.518 3.8 3.476 2.5 5 3.679 3.619 1.6 3.623 1.5Table 5
Deviation indexes of forecasting results for different forecasting methods.
Method MAPE(%) MAE MSE GM(0,2) model 2.48 0.0842 0.0540 IGM(0, 2) model 1.53 0.0090 0.0036Figure 3
Forecasting values by GM(0,2) and IGM(0,2).


## 2.6. Data Sources

This article collected annual data on wastewater discharged and the energy consumption in China for 2005-2015. The IGM(0,2) was used to predict wastewater discharge. The OFGM(1,1) method was employed to forecast energy consumption. The data sequences of wastewater discharge and real GDP were feature sequence and influence factor sequence of a system, respectively.Water pollution and energy shortage would become tumbling block of China’s economic growth. With the accelerated industrialization and urbanization, wastewater discharge would continue to increase in the future. From 2005 to 2015, real GDP had risen from 18.7319 trillion yuan to 68.9052 yuan. Based on economic development target, the country’s per capita GDP will probably reach 20 thousand yuan in 2020, assuming that China’s average annual economic growth rate is 7% in 2016-2020 [42]. Treated wastewater had up to 72.39% from 60.8% since 2012 to 2015. Energy demand for sewer had increased remarkably. So far, the average electricity consumption per cubic meter for wastewater was 0.2927kWh/m3 in China [14, 15] and would not be changed in the short term. It means that the energy consumption of wastewater treatment system had up to 15.58 billion kWh in 2015. Energy demand for wastewater in the recent year was listed in Table 6 (2012 to 2015).Table 6
The wastewater discharge and energy consumption in China (2012 to 2015).
Year 2012 2013 2014 2015 Wastewater discharge (billion tons) 68.43 69.49 71.56 73.53 Treated wastewater (billion tons) 41.62 45.61 49.43 53.23 Energy consumption (billion kWh) 12.18 13.35 14.47 15.58


## 3. Results and Discussion

Wastewater discharge and energy consumption in China were chosen as the goals of this study. As the trend of each series may be changing quickly over time, the data from 2011 to 2015 were applied for in-sample simulation, while data from 2016 to 2020 were utilized for out-of-sample prediction. The data were collected from the published documents and the Ministry of Environmental Protection of the People’s Republic of China.


### 3.1. Forecasting Energy Consumption for Wastewater Treatment

The forecasting results of wastewater discharge using OFGM(1,1) were depicted in Table7 and Figure 4. According to Table 7, the wastewater discharge in China would increase to 78.02 billion tons in 2019, before decreasing to 77.39 in 2020. All the wastewater would be treated in 2020. According to the planning target of the Chinese waste-water treatment industry in the period of the 13th Five-Year Plan, more than 95% wastewater would be treated by the end of 2020. These results showed that the current growth rate in wastewater treatment could meet the planning target. The government should formulate more effective incentive and policies for water efficiency to reduce sewage discharge.Table 7
Forecasting of wastewater discharge, treated wastewater, and energy consumption in China.
Year 2016 2017 2018 2019 2020 Total wastewater discharge (billion tons) 75.87 77.13 77.89 78.02 77.39 Treated wastewater (billion tons) 58.01 62.93 68.27 74.06 77.39 Power consumption (billion kWh) 17.04 18.49 20.06 21.77 22.75Figure 4
Simulation and forecasting of treated wastewater and wastewater discharge in China (2012 to 2020).The forecasting results of energy consumption using IGM(0,2) were depicted in Table7 and Figure 5. According to Table 7, the energy consumption for wastewater treatment is expected to increase to 22.75 billion kWh in 2020. Currently, low energy efficiency is one of the significant issues faced by most WWTPs in China. The energy efficiency measure and treatment process modification could significantly reduce energy consumption. In addition, energy recovery from wastewater treatment could offset the electricity consumption for the wastewater treatment. Therefore, the government should increase investment into energy efficiency measure to reduce the energy consumption in the coming year.Figure 5
Simulation and forecasting energy demand in China.


## 3.1. Forecasting Energy Consumption for Wastewater Treatment

The forecasting results of wastewater discharge using OFGM(1,1) were depicted in Table7 and Figure 4. According to Table 7, the wastewater discharge in China would increase to 78.02 billion tons in 2019, before decreasing to 77.39 in 2020. All the wastewater would be treated in 2020. According to the planning target of the Chinese waste-water treatment industry in the period of the 13th Five-Year Plan, more than 95% wastewater would be treated by the end of 2020. These results showed that the current growth rate in wastewater treatment could meet the planning target. The government should formulate more effective incentive and policies for water efficiency to reduce sewage discharge.Table 7
Forecasting of wastewater discharge, treated wastewater, and energy consumption in China.
Year 2016 2017 2018 2019 2020 Total wastewater discharge (billion tons) 75.87 77.13 77.89 78.02 77.39 Treated wastewater (billion tons) 58.01 62.93 68.27 74.06 77.39 Power consumption (billion kWh) 17.04 18.49 20.06 21.77 22.75Figure 4
Simulation and forecasting of treated wastewater and wastewater discharge in China (2012 to 2020).The forecasting results of energy consumption using IGM(0,2) were depicted in Table7 and Figure 5. According to Table 7, the energy consumption for wastewater treatment is expected to increase to 22.75 billion kWh in 2020. Currently, low energy efficiency is one of the significant issues faced by most WWTPs in China. The energy efficiency measure and treatment process modification could significantly reduce energy consumption. In addition, energy recovery from wastewater treatment could offset the electricity consumption for the wastewater treatment. Therefore, the government should increase investment into energy efficiency measure to reduce the energy consumption in the coming year.Figure 5
Simulation and forecasting energy demand in China.


## 4. Conclusions

In this article, we attempted to model and predict wastewater discharge and energy consumption in China based on the grey model. Using the historical data, the OFGM(1,1) model obtained excellent results regarding MAPE, when compared with FGM(1,1) and GM(1,1). The MAPEs of the OFGM(1,1) for training data and test predictions are 1.92% and 10.58%. Actual examples clearly show that the new forecasting model has the higher forecasting precision than traditional grey forecasting model FGM(1,1). The model was used to forecast the energy consumption in wastewater treatment. On the other hand, IGM(0,n) model was employed to forecast the wastewater discharge in China. The numerical experiment indicates that the model has a higher accuracy than GM(0,n) model. The MAPEs of this model are 1.85%. Performance assessment results clearly indicate that IGM(0,n) model can be employed for future projection of the wastewater discharge. The results show that China’s wastewater discharge will increase 78.02 billion tons in 2019, before decreasing to 77.39 in 2020. The energy consumption for wastewater treatment is expected to increase to 22.75 billion kWh in 2020. These results show that the current growth rate in treated wastewater could meet the target of the Chinese wastewater treatment industry in the period of the 13th Five-Year Plan.However, with the strict effluent limitations, wastewater treatment plants are likely to become more energy-intensive. Energy demand for WWTPs will increase significantly in the future. The government should provide more support in finance and credit policies for improving the processing and general techniques in wastewater treatment and encourage technological innovations to recover energy from the WWTPs. The appropriate use of electricity production from sludge incineration and anaerobic digester could transform from considering wastewater as an issue to considering it as a renewable energy resource. The results shown here could offer references to policy makers in setting energy strategies and protecting water resource.



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*Source: 4141085-2019-03-18.xml*