Complexity

Volume 2019, Article ID 1510257, 22 pages

https://doi.org/10.1155/2019/1510257

## A Novel Power-Driven Grey Model with Whale Optimization Algorithm and Its Application in Forecasting the Residential Energy Consumption in China

Correspondence should be addressed to Kun She; nc.ude.ctseu@nuk

Received 23 June 2019; Revised 6 September 2019; Accepted 16 September 2019; Published 6 November 2019

Academic Editor: Mohammad Hassan Khooban

Copyright © 2019 Peng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Along with the improvement of Chinese people’s living standard, the proportion of residential energy consumption in total energy consumption is rapidly increasing in China year by year. Accurately forecasting the residential energy consumption is conducive to making energy programming and supply plan for the administrative departments or energy companies. By improving the grey action quantity of traditional grey model with an exponential time term, a novel power-driven grey model is proposed to forecast energy consumption as reference data for decision makers. The nonlinear parameter of power-driven grey action quantity is a crucial factor to influence the prediction precision. To promote the prediction accuracy of the power-driven grey model, whale optimization algorithm is adopted to seek for the optimal value of the nonlinear parameter. Two validations on real-world datasets are conducted, and the results indicate that the power-driven grey model has significant advantages on the aspect of prediction performance compared with the other seven classical grey prediction methods. Finally, the power-driven grey model is applied in forecasting the total residential energy and the thermal energy consumption of China.

#### 1. Introduction

The residential energy, which includes the energy consumed by urban and rural residents and public facilities, accounts for a large percentage of total energy and continues to expand in China [1]. Meeting people’s residential energy demand is always an important part of the energy supply in China. Therefore, it is significantly crucial to predict the energy consumption accurately for energy programming and supply plan of governments or energy companies. Numerous studies have been conducted to predict total energy consumption or other various energy consumptions, e.g., natural gas consumption [2], oil consumption [3], electricity consumption [4], nuclear energy consumption [5], wind energy and renewable energy consumption prediction [6], and so on. For obtaining better prediction result, lots of conventional statistical models and machine learning models were adopted to predict energy consumption, such as ridge regression [7], autoregressive integrated moving average model (ARIMA) [8], support vector regression (SVR) [9], and artificial neural network (ANN) [10]. Unfortunately, machine learning models often need enough training samples to construct models, while the aforementioned statistical models require more available and reliable historical data. There are still difficulties to solve the prediction problems with poor information or small samples. Therefore, the grey prediction method (GM) becomes one of the inevitable choices to handle these problems.

The grey prediction theory was initially put forward to study the prediction problem with inadequate information or small samples by Deng in 1980s [11]. Traditional GM(1,1) model has exhibited excellent ability for homogeneous exponential datasets in the aspect of prediction. However, it cannot always provide a satisfactory result for inhomogeneous exponential sequence. For tackling the challenge, many scholars have engaged in optimization of the traditional grey prediction model. Wang et al. extended the existing grey model by using the exponential preprocessing method and applied it to forecast Beijing’s tertiary industry [12]. Xu et al. optimized the initial value of time response function to boost the stability of grey model and predict China’s electricity consumption [13]. Wang et al. analyzed the general analytic solution of the grey model’s whitening equation and presented an improved grey model by optimizing the initial condition which consisted of the first and last items of the accumulated generating sequence [14]. Meanwhile, optimization of the background value is another crucial aspect to boost the forecasting power of the classical grey prediction methods. Wang et al. utilized the finite integral of the accumulative generating sequence within the interval as background value to estimate the parameters of the grey model by the least-squares method [15]. Integrating the optimization of background value and the triangular whitening weight function, Ye et al. established a modified Grey-Markov model to handle the fluctuating sequences [16]. Zeng and Li modified the multivariate grey model based on dynamic background-value coefficient whose optimum value was sought out by PSO [17]. Chang et al. designed an adaptive grey prediction model to deal with the non-equigap sequence by optimizing background value coefficient [18]. The hybrid optimization is also an effective method to increase the grey model’s prediction performance. Li et al. enhanced the accuracy and application fields of the classical grey model by improving the grey model with the joint optimization of the initial condition and background values [19]. Besides, some scholars made efforts to enhance the adaptation ability of the grey model. Zeng et al. designed a grey predictive framework with a series of various grey structures which can intelligently select the most suitable model to predict the electricity consumption [20]. With the sum of weighted first-order accumulative generating values as an initial condition, Ding designed a self-adapting grey model called NSGM(1,1) to enhance the adaptation ability for various original sequences. Also, the tunable weighted parameters of NSGM(1,1) are automatically sought out by using the ant lion optimizer [21]. Zeng et al. optimized the structure compatibility of a multivariable grey model with adding a random term, a linear term, and a dependent variable lag term [22]. Besides, the fractional-order accumulation is also significantly valid for increasing the prediction capacity of the grey model [23]. Ma et al. presented a fractional time delayed grey model to boost the precision and applicability of the traditional fractional grey model [24]. At the same time, Ma et al. proposed an unbiased fractional discrete grey model (FDGM) in which the order was intelligently sought out by using the grey wolf optimizer to deal with the multivariate time series [25]. By eliminating the inconsistency between its grey difference equation for modeling and discrete function for forecasting, Ma and Liu designed an improved GMC(1,*n*) model to promote the accuracy of the classical GM(1,*n*) with convolution integral [26]. The optimization of the accumulated operator is also an effective measure to enhance the ability of the grey prediction model. Ma et al. presented a new fractional accumulated operator and designed a comfortable grey prediction model which obtained better accuracy than the classical fractional grey model [27]. Meanwhile, the optimization of the grey action quantity is also an important method used to boost the prediction performance and applicability of the grey model. Shaikh et al. utilized the GVM(1, 1) [28] and NGBM(1, 1) [29] models to handle the prediction problem of China’s natural gas demand with the characteristics of S-shaped data [30]. Li et al. proposed a full-order time power grey model to increase the structure adaptability of the grey model and adopted it to forecast the production of clean energy [31]. The grey action quantity affects the prediction performance and the applicability of the grey model. More details of the grey models with optimization of the grey action quantity is presented in Section 2. However, these improved grey models cannot solve all prediction problems. It is indispensable to continue improving the grey model and extending the application range of the grey model.

Therefore, a novel grey model is proposed to predict the residential energy consumption of China in this paper. There are two aspects of contribution as follows: (1) A novel power-driven grey model is proposed by optimizing the grey action quantity of traditional GM(1,1) model with an exponential term of time. The nonlinear parameter of the exponential term is determined by the whale optimization algorithm (WOA) to promote prediction accuracy. (2) The GM(1,1,) model is used to predict China’s total residential energy and thermal energy consumption, in which the prediction performance is significantly superior to the other seven contrast grey models.

The rest of the paper is organized as follows. Firstly, an overview of the traditional GM(1,1) model and its extension models is introduced in Section 2. Then, the power-driven grey model is proposed by substituting an exponential time grey action quantity for the constant grey input of traditional GM(1,1) in Section 3. In Section 4, a nonlinear programming problem with equality constraint is established to seek the optimum value of the nonlinear parameter by using the whale optimization algorithm. Meanwhile, the overall algorithm flowchart of GM(1,1,) is presented. In Section 5, the validations of GM(1,1,) is performed on two real-world datasets. Compared with the other seven existing grey prediction methods, the predicted results show that the proposed model has the most excellent prediction accuracy. In Section 6, the power-driven grey model is used to forecast China’s total residential energy and residential thermal energy consumption. At last, the conclusions are drawn in Section 7.

#### 2. GM(1,1) Model and Its Extension Models

The grey prediction method is one of the most popular and effective models to deal with time series prediction. It has been widely employed in many application areas and has obtained remarkable achievement in prediction problems with small samples. To boost the prediction accuracy of the classical grey prediction model, scholars have conducted numerous studies, such as improving the grey accumulation generation, optimizing grey background value, optimizing the initial condition, and so on. This section presents a survey of the classic grey prediction model and its extension models with optimizing grey action quantity.

*Definition 1. *Let be the raw sequence. The first-order accumulated sequence generated from the raw sequence is defined aswhere . The background value sequence generated from consecutive neighbors of first-order accumulated sequence is defined aswhere .

Then, the definition of the classical grey model for dealing with univariate time series prediction is presented as follows. The equationis the definition equation of GM(1,1), in which *a* and *b* are development coefficient and grey input of the grey model.

The differential equationis the whitening equation of GM(1,1).

In order to perform a prediction task, the key issue is to resolve the optimal value of linear parameters *a* and *b*. By using the least-squares method, the optimal parameters can be calculated as follows:whereOnce the linear parameters of the grey model are estimated, the time response function of the grey model can be obtained by solving the whitening equation (4). By substituting the initial condition into the solution of the whitening equation (4), the discrete time response sequence of the grey model is formulated asHowever, the sequence is not the final predicted result of the grey model. By using inverse accumulated generating operator, the restored value of the grey model is calculated asThen, the sequence produced by the grey model is represented aswhere , and .

By analyzing the above stored sequence (9), it can be noticed that the GM(1,1) model has an ideal performance for the univariate time series with homogeneous exponential characteristics. Nevertheless, there are many time series data with nonhomogeneous exponential characteristics. Many improved grey models have been studied and designed to enhance the prediction accuracy for the nonhomogeneous exponential data sequence. One of the boosting strategies is to optimize the grey action quantity. A series of grey models with optimization of grey action quantity were proposed as follows.

By replacing the grey input *b* of the original GM(1,1) model with the term , the NGM model [32] can be obtained with the following formula:By replacing the grey input *b* of the original GM(1,1) model with the term , the SAIGM model [20] can be obtained with the following formula:By replacing the grey input *b* of the whitening equation of the original GM(1,1) model with the term , the whitening equation of the NGBM model [29] can be obtained as follow:When , the NGBM model can be degenerated into the GVM(1,1) [28] model.

By replacing the grey input *b* of the original GM(1,1) model with the term , the FOTP-GM(1,1,*k*) model [31] can be obtained with the following equation:By replacing the grey input *b* of the whitening equation of the original GM(1,1) model with the term , GM(1,1,) is obtained with the whitening equation:By replacing the grey input *b* of the original GM(1,1) model with the term , the GM(1,*N*) model [33] is obtained with the following form:These research results show that the performance and accuracy of these grey models are significantly improved as well as the application range is also expanded into more fields by improving the grey action quality.

#### 3. The New Proposed Power-Driven Grey Model

Obviously, the optimization of the grey action quantity is an effective means to increase the performance and applicability of the grey model from the previous section. This section proposes a novel power-driven grey model in which a natural exponential function of time is considered as the grey action quantity.

##### 3.1. The Power-Driven Grey Model

*Definition 2. *Assume that are defined as the same in Definition 1. The differential equationis defined as the whitening equation of the power-driven grey model (GM(1,1,)). The parameter *a* denotes the development coefficient. The term denotes the power-driven grey input in which the coefficient *α* is a tunable parameter.

By integrating the both sides of whitening equation (16) within , the discrete formulation of the GM(1,1,) model can be represented as follows.

*Definition 3. *The grey differential equationis called the discrete form of the GM(1,1,) model.

##### 3.2. Parameter Estimation of the Power-Driven Grey Model

For the traditional GM(1,1) model, the parameters *a* and *b* can be directly estimated by using the least-squares method because they are linear parameters. From Definition 2, it can be clearly noticed that the parameters a, *b*, and *c* of the power-driven grey model are linear parameters while the parameter *α* is a nonlinear parameter. It is difficult to estimate the nonlinear parameter by using the least-squares method directly. A two-stage strategy is adopted to gain the optimum parameters of the proposed model. In the first stage, the equality equation between the linear parameters and nonlinear parameter is obtained by using the least-squares method under the hypothetical condition that the nonlinear parameter *α* is given. Then, the optimal nonlinear parameter *α* is determined by solving an established nonlinear programming problem with equality constraint by using an intelligence algorithm (e.g., whale optimization algorithm [34]). In the second stage, the linear parameters are determined by the least-squares method after seeking out the optimum value of the nonlinear parameter. The process of determining the nonlinear parameter is presented in Section 4, while the linear parameters are estimated as follows.

Assuming the nonlinear parameter *α* is given, the parameters of the power-driven grey model can be determined by employing the least-square method, and it satisfieswherein which and denotes the number of samples used for constructing model. The detailed proof process of the linear parameter estimation is omitted here because it is similar to the classical GM(1,1) model.

##### 3.3. The Time Response Function and Restored Response Sequence

After the parameters of the proposed grey model are determined, the time response and restored value sequence can be obtained as follows.

Theorem 1. *The time response function of the power-driven grey model is defined as*

The restored response sequence of the power-driven grey model can be obtained by

*Proof. *Assume that is an arbitrary function and satisfiesMultiply both sides of equation (16) by and obtainEquation (22) is substituted into equation (23) to get the following formula:Rearrange equation (24) and obtainIntegrate both sides of equation (25) and obtainSolving equation (22), the solution is obtained aswhere *k* is an arbitrary real number. Substitute equation (27) into equation (26) and obtainSubstituting the initial condition and the estimated parameters calculated by (18) into equation (28), the time response function is obtained asBy using inverse accumulation generating operator, the stored value can be calculated as follows:Then, the stored value is obtained asThis completes the proof.

According to Maclaurin’s formula, the expansion of the term can be obtained aswhere is known as the error term. If some higher-order term of equation (32) is ignored, the power-driven grey model can be degenerated into other existing grey models. If the higher-order terms other than first-order terms are ignored, the term can be obtained. Then, the GM(1,1,) model can be degenerated into the grey SAIGM [20] with whitening equation (11). In a similar way, the GM(1,1,) model can be degenerated to a kind of FOTP-GM(1,1,*k*) model [31] with special whitening equation (13) in which the parameter . When , the GM(1,1,) model can be degenerated into the traditional grey model with whitening equation (4).

#### 4. Determining the Nonlinear Parameter of the Power-Driven Grey Model with Whale Optimization Algorithm

From the previous section, the linear parameters of the proposed model are determined by using the least-squares approach under the assumption that the nonlinear parameter is given. However, the nonlinear parameter cannot be directly calculated by the ordinary least-squares method because it is an exponential coefficient of grey action quantity. In fact, the nonlinear parameter *α* plays an indispensable role in promoting the prediction performance of the power-driven grey model. In this section, an intelligent nature-inspired optimization method called whale optimization algorithm is employed to seek for the optimal value of nonlinear parameter *α*.

##### 4.1. Constructing the Optimization Problem for the Power-Driven Grey Model

Actually, an optimum value of nonlinear parameter *α* can make the power-driven grey model obtain the best prediction performance because the parameter not only directly affects the grey action quantity but also can control the development coefficient. Therefore, an optimization problem with constraint is built to obtain the optimum value of *α*, in which the objective function is to minimize the fit error of the power-driven grey prediction model. The equality constraints of the optimization problem are formulated in the previous modeling process. Mathematically, the optimization problem for seeking out optimal nonlinear parameter is formulated as follows:

In this paper, a different strategy which is similar to nest cross validation in machine learning [35] is utilized to seek the optimal value of the coefficient *α*.. During the simulation stage, the samples are partitioned into two subsets. The first set, including the first samples (from 1 to samples), is used for establishing the equality constraint equation (19) between the linear parameters and the nonlinear parameter. The second set, including the last samples (from to *n* samples), is utilized to compute the fitness value of the established optimization problem. The value of satisfies in Sections 5 and 6. The optimum value of nonlinear parameter *α* is sought out by solving the optimization problem equation (33). In the meanwhile, the linear parameters are also obtained when the optimal value *α* is substituted into equation (19). This strategy has been utilized to search for the optimum order of the fractional grey prediction model [25, 36].

##### 4.2. Whale Optimization Algorithm

Motivated by the social behavior of humpback group, an intelligent nature-based optimization approach called whale optimization algorithm (WOA) was originated by Mirijalili and Lewis in 2016 [34]. In recent years, WOA has been widely employed to settle the optimization problems in many fields such as image retrieval [37], classification [38], bioinformatics [39], feature selection [40], image processing [41], and so on. Meanwhile, it is also effective to solve the optimization problems like training multilayer perceptron neural network which involves a complex nonlinear optimization problem [42] and is more complicated than problem (33). This paper adopts the WOA algorithm to solve the nonlinear optimization problem (33). The main idea and model of WOA are mathematically described as follows.

The main idea of WOA is to imitate the predation behaviors of humpback group, for example, bubble-net feeding for catching fish. When the humpback whales catch fish, they usually encircle the fish school whose position is considered as the current best candidate target. Then, these whales update their positions based on the candidate target. Mathematically, the encircling behavior is represented as follows:where denotes the current position of the humpback whale, denotes the best current position of the humpback whale, the vector is randomly generated in the interval , and *T* denotes the maximum number of iterations. Furthermore, humpback whales move in spirals when they catch the prey. To simulate the helix-shaped movement, the spiral updating position is represented as follows:where the coefficient *l* is a stochastic number in the interval and *β* is an arbitrary constant which determines the shape of the spiral movement. However, encircling and spiral moving behaviors happen simultaneously in the real world. For keeping it simple in this model, the entire predation movement of humpbacks is mathematically represented as follows:where *ξ* is a probability to choose a movement strategy from encircling and spiral moving behaviors. When the norm of is greater than 1, the position of all whales is updated based on the position of a whale randomly selected, not on optimal ones. Mathematically, the model can be formulated as follows:where is the position of a whale randomly chosen from humpback group. Based on the principle of humpback’s predation behavior, it is iterative to update the position of each whale until the stop criteria are met.

##### 4.3. Implementation of WOA for Searching the Optimal Nonlinear Parameter

In the nonlinear programming problem (33), the main purpose is to find out the optimum value of the nonlinear parameter to obtain the highest performance of the proposed grey model. From section 4.2, it can be noticed that the original WOA is initially designed for unconstrained optimization and cannot directly solve the optimization problem with constraint. Therefore, the original WOA needs to be revised based on equation (33). Primarily, the fitness function needs to be established to calculate the fitness of each whale agent. According to equation (33), the fitness function can be represented as follows:

The revised WOA is presented in detail in Algorithm 1.