Multi-Step Ahead Natural Gas Consumption Forecasting Based on a Hybrid Model: Case Studies in The Netherlands and the United Kingdom

Abstract

With worldwide activities of carbon neutrality, clean energy is playing an important role these days. Natural gas (NG) is one of the most efficient clean energies with less harmful emissions and abundant reservoirs. This work aims at developing a swarm intelligence-based tool for NG forecasting to make more convincing projections of future energy consumption, combining Extreme Gradient Boosting (XGBoost) and the Salp Swarm Algorithm (SSA). The XGBoost is used as the core model in a nonlinear auto-regression procedure to make multi-step ahead forecasting. A cross-validation scheme is adopted to build a nonlinear programming problem for optimizing the most sensitive hyperparameters of the XGBoost, and then the nonlinear optimization is solved by the SSA. Case studies of forecasting the Natural gas consumption (NGC) in the United Kingdom (UK) and Netherlands are presented to illustrate the performance of the proposed hybrid model in comparison with five other intelligence optimization algorithms and two other decision tree-based models (15 hybrid schemes in total) in 6 subcases with different forecasting steps and time lags. The results show that the SSA outperforms the other 5 algorithms in searching the optimal parameters of XGBoost and the hybrid model outperforms all the other 15 hybrid models in all the subcases with average MAPE 4.9828% in NGC forecasting of UK and 9.0547% in NGC forecasting of Netherlands, respectively. Detailed analysis of the performance and properties of the proposed model is also summarized in this work, which indicates it has high potential in NGC forecasting and can be expected to be used in a wider range of applications in the future.

1. Introduction

1.1. Background

Since the Industrial Revolution, the continuous progress of human civilization and science and technology, the intensification of industrialization, and the acceleration of urbanization, the global demand for energy consumption has continued to increase, bringing a large number of ecological problems such as atmospheric pollution, global warming, and water pollution [1], which have had a significant impact on economic development and society. In order to better cope with the severe energy crisis and environmental problems, developing green energy with less pollution will become a trend in the energy field. The pressures of increasing pollution and reducing energy supplies are also driving society to develop in the direction of renewable energy [2].

NG is a clean and environmentally friendly high-quality energy that contains almost no harmful substances such as sulfur and dust. Compared with other fossil fuels, its combustion produces less carbon dioxide, effectively reducing the greenhouse effect and fundamentally improving environmental quality. Especially in recent years, NG as a vehicle fuel has received widespread attention [3]. The U.S. Energy Information Administration (EIA) predicts in its latest report International Energy Outlook 2019 that global NGC will grow by more than 40% from 2018 to 2050. By 2050, the total global NGC will reach nearly 200 Giga British Thermal Units. Therefore, NG is highly critical to the country’s development, so the accurate forecasting of NGC is of vital strategic significance.

The UK is one of the the earliest capitalist industrialized countries and was once the most powerful countries in the world. After World War II, air pollution in the UK continued due to the large-scale mining and use of coal in industrial production. The ‘Great Smog’ incident in December 1952 [4] directly caused about 150,000 people to be taken to hospital with respiratory problems, and eventually, 4000 people died. This pollution incident made the British government start the energy transition and gradually give up the use of coal to generate power. In 2017, the UK’s energy consumption structure is still dominated by fossil energy, with NGC accounting for 39.02% of the total and coal consumption accounting for only 5.26%. The Netherlands is the EU’s largest NG exporter [5]. Since the discovery of the Groningen gas field in 1959, the Netherlands has become an important European gas transportation and trade hub with its strategic geographical location and well-developed pipeline network facilities. According to official data from the Netherlands, from 1963 to 2018, NG contributed approximately forty-one point seven billion euros in revenue to the Netherlands. The proportion of NG in the energy structure of the Netherlands is also much higher than that of other European countries, which is approximately 40% [6].

The UK and the Netherlands are two important western European countries. In 2020, the gross domestic product (GDP) of the UK and the Netherlands ranked second and sixth among all European countries, respectively. According to the UK Public Sector Information website and CBS statistics (Figure 1), NG is the most used fossil energy source in the UK, with a total NGC of 495,400 (GWH) in 2017, accounting for 39.02% of all energy consumption. The Netherlands is an essential European exporter of NG, producing 86 billion cubic meters per year, accounting for 2.5% of total world gas production. Both countries experienced a downward trend in NG production, notably since 2013, when Netherlands NG production steadily declined. NGC in both countries has been more stable with fewer fluctuations. NG imports have increased in both countries, but the UK began a slow decline in importance in 2010, with The Netherlands’ imports increasing. In addition, gas exports from the UK have been low and steady, but exports from the Netherlands have been significantly declining in recent years. In addition, with the outbreak of war between Russia and Ukraine, a prolonged and complete shutdown of Russian gas to the whole of Europe could interact with infrastructure bottlenecks. In some countries, gas has become very expensive and in severely short supply [7].

1.2. Related Work

NGC data are characterized by volatility and nonlinear time series, and datasets from different sources may vary significantly. Therefore, the primary issue considered by researchers is how to effectively improve the forecasting accuracy and generalization performance of the models. To date, researchers have performed a lot of research in NGC forecasting and achieved many satisfactory results, whose models mainly include machine learning models, economic models, grey system models, deep learning models, time series models, and other models. The detailed results of the literature study are presented in

Table 1. Brief summary of literature NG forecasting.

Model Type	Reference	Model	Conclusions
grey model	[8]	DFGM(1,1,$t^{\alpha }$)	It is expected that China’s NGC will maintain an upward trend, reaching 439.14 billion cubic meters in 2025.
	[9]	GPRM	GPRM method has high forecasting accuracy and applicability in the presence of limited data.
	[10]	TDPGM(1,1)	The annual NGC will exceed 5000 (${10}^9$ m${}^3$) by 2020.
	[11]	A fractional-order incomplete gamma grey model	The model has excellent predictive performance in predicting NGC and can be generalized to more energy forecasting problems.
	[12]	Fractional time delayed grey model optimized by GWO	Compared with some traditional grey models, the new model has better forecasting performance and generalization.
	[13]	PFSM(1,1)	This model is suitable for data with significant seasonal variation.
	[14]	FPDGM(1,1)	The novel model has better predictive performance than the other models in all cases.
	[15]	DGMNF(1,1)	The forecasting results obtained by the new model are more accurate and reliable than other models, and the forecasting error is smaller.
	[16]	CFNHGBM(1,1,k)	The novel model outperforms other competitors.
	[17]	A novel self-adapting intelligent grey model	By 2020, China’s NG demand will exceed 340 billion m${}^3$.
Machine learning model	[18]	GB, GB-PCA, ANN-CG-PCA	The forecasting accuracy of the combined model is better than that of the individual models, and the MAPE of the combined model is about 15% smaller than that of the individual models.
	[19]	ANN	The system can be used to monitor the necessary gas flow and predict demand changes due to internal and external temperature, which can effectively reduce costs.
	[20]
	[21]	MLR, SVR, ANN	The forecasting performance of SVR is much better than that of ANN, and it has a smaller forecasting error in NGC forecasting, and can provide more reliable and accurate results.
	[22]	GRA-LSSVM	The proposed model has a better generalization ability, and compared with the PSO algorithm and the SecPSO algorithm, the GRA-LSSVM model optimized by WASecPSO has higher forecasting accuracy.
Deep learning model	[23]	A hybrid model based on wavelet transform	Wavelet transform can effectively improve the forecasting accuracy of the model, and the forecasting accuracy of the hybrid model exceeds that of other AI models.
	[24]	MLP, LSTM	A two-stage FM-MLP method is proposed.
	[25]	LMD-WTD-LSTM	When the forecasting time length is 20 days, this method has the best forecasting performance among the five methods, and its MAPE is 11.63%.
	[26]	ISSA-LSTM	ISSA-LSTM is the best performing model among all the forecasting models. In its forecasts in the four cities of London, Melbourne, Karditsa, and Hong Kong, the obtained MAPE values are 4.68%, 5.72%, 5.76%, and 14.10%, respectively.
Time series	[27]	An integrated genetic-ARMA approach	The proposed hybrid model has more stable forecasting performance, outperforming the classic ARMA model in terms of MAPE.
Other model	[28]	Integrated assessment models	China’s total primary energy consumption in 2060 will be smaller than in 2019 and NGC will account for approximately 6% of China’s total primary energy consumption in 2060
	[29]	MARS, CMARS	MARS and CMRS outperform NN and LR in all evaluation metrics.

.

The grey system model is one of the most popular models in energy forecasting, and many scholars have worked to develop a forecasting model with strong predictive ability and good generalization performance [8,9,10,11]. More notably, many scholars have combined intelligent optimization algorithms with grey models to select the grey model parameters by optimization algorithms and achieved satisfactory results [12,13,14,15,16,17]. Machine learning models are widely used in various fields due to their powerful predictive capabilities. Neural networks [18,19,20] and support vector machines [21,22] are two of the most used models in NGC forecasting. Notably, Yong-Hong and WuHuiShen proposed the least squares support vector machine model (GRA-LSSVM) based on grey correlation analysis in 2018 and designed a weighted adaptive second-order particle swarm optimization algorithm (WASecPSO) to optimize the model parameters. The results show that the GRA-LSSVM has a better generalization ability and training effect, and the model optimized by the WASecPSO algorithm has higher forecasting accuracy [22]. This work illustrates that an approach based on intelligent optimization algorithms for tuning the hyperparameters of machine learning models is fully feasible. Deep learning is the most popular method today, and it has also been widely used in energy forecasting with very satisfactory results [23,24,25,26]. It is evident from recent studies that multi-model fusion has become the preferred choice of forecasting method by a wide range of scholars. In addition, econometric and other models are occasionally applied to NG forecasting [27,28,29].

The literature review shows that the grey system, machine learning, and deep learning models are the three most widely used models for current NGC forecasting. Many scholars have contributed to developing models with solid forecasting capability, but the forecasting of single model suffers from the problem of poor stability. In terms of machine learning, the forecasting ability and generalization performance of a model often depends on the choice of hyperparameters, and many scholars adjust the model’s hyperparameters based on their own experience, and such approaches have sinificant subjectivity and often fail to achieve satisfactory results. In this work we choose to use the XGBoost, one of the state-of-the-art tree-based models, to build nonlinear auto-regressive models for time series forecasting. And then the Salp Swarm Algorithm (SSA) is adopted to optimizing the hyperparameters of these models by solving a nonlinear programming problem with respect to the hyperparameters to achieve better global convergence and take most advantages of the XGBoost model. The real-world case studies along with comprehensive discussions are also presented.

2. XGBoost and Its Nonlinear Auto-Regressive Formulation

2.1. Extreme Gradient Boosting

The XGBoost is essentially one kind of gradient boosting descision trees [30], which can be used for classification and regression problems proposed by Chen et al. [31]. Regularization and parallel computing are used in XGBoost, which make it generally more stable and faster than most conventional tree-based models. A brief overview of the main steps of the XGBoost will be summarized in this subsection.

Define $D=\left\{(x_i,y_i)\right\},\left(\right|D|=n)$ as a sample set of $n\times m$, where n is the number of samples and m is the number of features. The output function of the XGBoost model is shown in Equation (1):

$${\widehat{y}}_i=\phi \left(x_i\right)=\sum _{t=1}^T{f}_t\left(x_i\right)$$

(1)

where $\mathcal{T}=f_t\left(x_i\right)={\omega }_{q\left(x\right)}$ represents the space of a regression tree, and q represents the structure of each tree when the samples are mapped to the leaves. T represents the number of leaves in the tree. Furthermore, each $f_k$ contains a separate q and $\omega$.

The optimization problem for constructing the XGBoost model is formulated as:

$$\begin{array}{c}\hfill obj=\sum _{i}l\left({\widehat{y}}_i,y_i\right)+\sum _k\mathsf{\Omega }\left(f_k\right)\\ \hfill \mathrm{where}\mathsf{\Omega }\left(f\right)=\gamma T+\frac{1}{2}{\lambda \left|\right|w\left|\right|}^2\end{array}$$

(2)

The objective function of XGBoost is mainly divided into two parts: the loss function and the regularization term. Here, $l\left({\widehat{y}}_i,y_i\right)$ is a differentiable convex loss function, which is the first part of the objective function, and is used to calculate the error between the forecasted value ${\widehat{y}}_i$ and the target value $y_i$. The $\omega$ represents the second part of the objective function: the regularization term. The regularization term encourages the use of simpler models, which makes the forecast of the final model more stable and easy not to over-fitting.

According to the main idea of XGBoost, the iterative process of residual fitting is as follows:

$$\begin{array}{cc}\hfill & {\widehat{y}}_i^{\left(0\right)}=0\hfill \\ \hfill & {\widehat{y}}_i^{\left(1\right)}=f_1\left(x_i\right)={\widehat{y}}_i^{\left(0\right)}+f_1\left(x_i\right)\hfill \\ \hfill & {\widehat{y}}_i^{\left(2\right)}=f_1\left(x_i\right)+f_2\left(x_i\right)={\widehat{y}}_i^{\left(1\right)}+f_2\left(x_i\right)\hfill \\ \hfill ⋮\\ \hfill & {\widehat{y}}_i^{\left(t\right)}=\sum _{k=1}^t{f}_k\left(x_i\right)={\widehat{y}}_i^{(t-1)}+f_t\left(x_i\right)\hfill \end{array}$$

(3)

In the above formula, ${\widehat{y}}_i^{\left(t\right)}$ represents the forecasting result of the t-th round model, and ${\widehat{y}}_i^{(t-1)}$ is the forecasting result of the $t-1$ round model. Additionally, $f_t\left(x_i\right)$ is the newly added function.

According to the above iterative process, the objective function can be rewritten as:

$$\begin{array}{cc}\hfill ob{j}^{\left(t\right)}=& \sum _{i=1}^{n}l\left(y_i,{\widehat{y}}_i^{\left(t\right)}\right)+\sum _{i=1}^t\mathsf{\Omega }\left(f_i\right)\hfill \\ \hfill =& \sum _{i=1}^{n}l\left(y_i,{\widehat{y}}_i^{(t-1)}+f\left(x_i\right)\right)+\mathsf{\Omega }\left(f_t\right)+\mathrm{constant}\hfill \end{array}$$

(4)

Obviously, our goal is to select an optimal $f_t\left(x_i\right)$ in each iteration to minimize the objective function.

For Equation (4), approximate the objective function using the second-order expansion of the Taylor formula, and remove the constant term:

$$\begin{array}{cc}\hfill & ob{j}^{\left(t\right)}=\sum _{i=1}^n\left[g_i{f}_t\left(x_i\right)+h_i{f}_t^2\left(x_i\right)\right]+\mathsf{\Omega }f\left(t\right)\hfill \\ \hfill & where{g}_i=\partial {\widehat{y}}^{(t-1)}l\left(y_i,{\widehat{y}}_i^{(t-1)}\right),h_i={\partial }^2{\widehat{y}}^{(t-1)}l\left(y_i{\widehat{y}}_i^{(t-1)}\right)\hfill \end{array}$$

(5)

Define $I_j=\left\{i\right|q\left(x_i\right)=j\}$ as the set of subscripts of the samples on each leaf node j, so that each sample value can be mapped to the tree through the function $q\left(x_i\right)$ on a leaf node. Combining the above conclusions, we expand $\omega$ and rewrite Equation (5):

$$\begin{array}{cc}\hfill ob{j}^{\left(t\right)}& =\sum _{i=1}^n\left[g_i{f}_t\left(x_i\right)+h_i{f}_t^2\left(x_i\right)\right]+\mathsf{\Omega }f\left(t\right)\hfill \\ \hfill & =\sum _{i=1}^n\left[g_i{\omega }_{q\left(x\right)}+h_i{\omega }_{q\left(x\right)}^2\right]+\gamma T+\frac{\lambda }{2}\sum _{j=1}^T{\omega }_j^2\hfill \\ \hfill & =\sum _{j=1}^T\left[\left(\sum _{i\in I_j}{g}_i\right){\omega }_j+\frac{1}{2}\left(\sum _{i\in I_j}{h}_i+\lambda \right){\omega }_j^2\right]+\gamma T\hfill \end{array}$$

(6)

Define $G_j={\sum }_{i\in I_j}{g}_i,H_j={\sum }_{i\in I_j}{h}_i$:

$$ob{j}^{\left(t\right)}=\sum _{j=1}^T\left[G_j{\omega }_j+\frac{1}{2}\left(H_j+\lambda \right){\omega }_j^2\right]+\gamma T$$

(7)

Since the structure $q\left(x\right)$ of each tree has been determined, the optimal ${\omega }_j^{*}$ of the leaf can be obtained by the following formula:

$${\omega }_j^{*}=-\frac{G_j}{H_j+\lambda }$$

(8)

and the corresponding optimal objective function value is calculated by the following formula:

$$ob{j}^{\left(t\right)}=-\frac{1}{2}\sum _{j=1}^T\frac{G_j}{H_j+\lambda }+\gamma T$$

(9)

The result of Equation (9) is used to evaluate the quality of the tree structure. The smaller the result, the better the tree structure.

2.2. Data Division Method and Forecasting Method

In order to obtain better generalization of the forecasting model and avoid overfitting, in this study, the "hold-out" method is adopted to divide the dataset used to adjust the parameters [32]. The idea of this method is to successively divide the raw data into two parts for fitting and forecasting, and then the data for fitting is devided into two parts in the same way, of which the first part is used to train the model, and the other part is used as a validation set for the search to obtain the optimal hyperparameters of the model. Performance of the models will be evaluated on the data used for forecasting. The process is shown in Figure 2.

As NGC is generally highly cyclical, it is often difficult for traditional forecasting methods to obtain satisfactory results. Therefore, a novel multi-step ahead forecasting scheme is adopted in this study. The main idea of the multi-step ahead forecasting scheme is to gradually predict the future values of the time series based on the historical values of the time series [33].

Suppose an original sequence of length $\tau$: $X=\{x_{t-1+\tau },\dots ,x_{t-1},x_t\}$, and a general model $x=f\left(x\right)$. We first use the previous $\tau$ values to forecast ${\widehat{x}}_{t+1}$:

$${\widehat{x}}_{t+1}=f(x_{t-1+\tau },\dots ,x_{t-1},x_t)$$

(10)

Then, we forecast ${\widehat{x}}_{t+2}$ based on its previous $\tau$ values, which include the forecasted value for ${\widehat{x}}_{t+1}$:

$${\widehat{x}}_{t2}=f(x_{t+\tau },\dots ,x_t,{\widehat{x}}_{t+1})$$

(11)

Finally, we repeat the process until the last value is predicted. The entire process is shown in Figure 3.

3. The Overall Computational Steps

3.1. Structure of the Optimization Problem

Hyperparameters optimization is one of the most vital problems in machine learning. The performance of the model heavily depends on the choice of hyperparameters. Choosing appropriate method to search for the model’s hyperparameters is also challenging. Traditional optimization methods often take too much time when the amount of data and features are large. In recent years, with the popularity of meta-heuristic algorithms, more and more scholars have begun to use meta-heuristic algorithms in the hyperparameter adjustment of machine learning models and have achieved satisfactory results. For instance, Qiu et al. (2021) adopted WOA, GWO, and BO algorithms to optimize the XGBoost model [34], Abbasi et al. adopted the HHO and PSO algorithms to optimize hybrid support vector regression to predict meteorological drought [35].

In this work, we transform the hyperparameter tuning into an optimization problem and solve the optimization problem through a meta-heuristic algorithm. The main computational steps can be summarized as follows:

Step 1 Initialize the parameters of the algorithm and determine the hyperparameters that the model needs to tune.

Step 2 Construct the optimization problem:

$$\begin{array}{c}min\mathrm{Obj}=\frac{1}{d}\sum _{i=n+1}^{n+d}\left|\frac{{\widehat{y}}_i-y_i}{y_i}\right|\times 100\%\\ \hfill s.t.\left\{\begin{array}{c}\mathrm{Training}\mathrm{XGBoost}\mathrm{by}\mathrm{the}\mathrm{data}\mathrm{for}\mathrm{fitting}(\mathrm{from}\mathrm{first}\mathrm{sample}\mathrm{to}\mathrm{the}n-\mathrm{th}\mathrm{sample}),\\ \mathrm{Computing}{\widehat{y}}_i\mathrm{by}i\mathrm{from}n+1\mathrm{to}n+d\mathrm{using}\mathrm{XGBoost}\mathrm{using}\mathrm{the}\mathrm{process}\mathrm{in}\mathrm{Figure\; 3}\end{array}\right.\end{array}$$

(12)

where d represents the number of samples for validation.

Step 3 Start iteration by some updating rule(s).

Step 4 If the maximum iteration is reached, continue the iteration. Otherwise, return Step 3.

Step 5 Output the parameters of the optimal model and use the optimal model to make forecasting.

3.2. Salp Swarm Algorithm

The SSA is a novel heuristic algorithm proposed by Seyedali Mirjalili et al. in 2017 [36]. This algorithm simulates salp’s swarming behaviors [37] of hunting in the deep ocean. The key steps and settings are presented in the following content.

3.2.1. Leaders’ Position Update

The search space is defined as a $K\times I$ Euclidean space, where K represents spatial dimension, and I represents the population of individuals. The positions of all salps in the space are all stored in a two-dimensional matrix X, where food is the target of the entire population, and F represents its position.

The position update of the leader is performed by the following equation:

$$x_k^1=\left\{\begin{array}{c}{F}_k+r_1\left(\left(u{b}_k-l{b}_k\right)r_2+l{b}_k\right),r_3\ge 0\\ F_k-r_1\left(\left(u{b}_k-l{b}_k\right)r_2+l{b}_k\right),r_3<0\end{array}.\right.$$

(13)

$x_k^1$ and $F_k$ are the position of the first salp (leader) and food in the $k-$th dimension, respectively; $l{b}_k$ and $u{b}_k$ are the corresponding upper and smaller bounds, respectively. $r_1$, $r_2$, and $r_3$ are control parameters, which are three randomly generated numbers. Equation (13) shows that the leader’s position update is only related to the position of the food.

$r_1$ is the convergence factor in the optimization algorithm, which balances global exploration and local development and is the most crucial of all control parameters of SSA. The expression for $r_1$ is:

$$r_1=2e^{-{\left(\frac{4l}{L}\right)}^2}$$

(14)

In the above formula, l and L represent the current and total number of iterations, respectively. $r_1$ is a function whose values decreases from 2 to 0. The control parameters $r_2$ and $r_3$ are randomly generated numbers in $[0,1]$. $r_2$ and $r_3$ are used to enhance the randomness of $x_k^1$, thereby improving the global search ability of the chain groups and increasing the diversity of individuals. Additionally, these two parameters also determine whether the next position in the dimension should be in the positive or negative direction as well as determine the forecasting step.

3.2.2. Folsmallers’ Position Update

During the movement and foraging of the salp chains, the folsmallers move forward in a chain-like manner through the mutual influence between the front and rear individuals. Update the position of the folsmaller according to Newton’s law of motion formula:

$$\begin{array}{c}\hfill x_k^i=\frac{1}{2}a{t}^2+v_{0}t\end{array}$$

(15)

where $i\ge 2$, $x_k^i$ represents the position of the k dimension, t, $v_0$, and a are the time, initial velocity, and acceleration, respectively, and $a=\frac{v_{final}}{v_0}$ where $v=\frac{x-x_0}{t}$. Because the optimization time varies with the number of iterations and takes into account $v_0=0$, therefore, the position of the folsmaller can be described by the following equation:

$$\begin{array}{c}\hfill x_k^i=\frac{1}{2}\left(x_k^i+x_k^{i-1}\right)\end{array}$$

(16)

where $i\ge 2$, and $x_k^i$ and $x_k^{i-1}$ are the positions of two salps that are next to each other in the $k-$th dimension.

3.3. Hyperparameter Optimization of XGBoost by SSA

There are many hyperparameters (described in Section 3) in XGBoost need to be optimized. These hyperparameters will greatly affect the model results because SSA has strong global exploration and local development capabilities, making it better global convergence in practice. Within such merits, we use SSA to search the optimal hyperparameters of XGBoost. Combining the above descriptions, the overall procedure of optimizing the hyperparameters of XGBoost by SSA can be summarized in Figure 4. The forecsating model obtained by this procedure can be finally used for multi-step ahead forecasting. As it combines the XGBoost and SSA, the proposed model is abbreviated as SSA-XGBoost in the rest of this paper.

4. Application

In this section the SSA-XGBoost is used to forecast the NGC of the UK and the Netherlands. Moreover, to evaluate the performance of this hybrid model, fifteen hybrid models will be used for comparison. The complete modelling and comparison procedures are summarized in Figure 5.

4.1. Experimental Settings

Information of the other 15 hybrid models in comparison with the proposed SSA-XGBoost is listed in

Table 2. Hybrid model for comparison.

Algorithm	Model	Hybrid Mode
SSA [36]	XGBoost	SSA-XGBoost
	Random Forest	SSA-RF [38]
	LightGBM (LGB)	SSA-LGB
PSO [39]	XGBoost	PSO-XGBoost [40]
	Random Forest	PSO-RF [41]
	LightGBM	PSO-LGB [42]
GWO [43]	XGBoost	GWO-XGBoost [44]
	Random Forest	GWO-RF [45]
	LightGBM	GWO-LGB
HHO [46]	XGBoost	HHO-XGBoost
	Random Forest	HHO-RF [47]
	LightGBM	HHO-LGB
MVO [48]	XGBoost	MVO-XGBoost
	Random Forest	MVO-RF [49]
	LightGBM	MVO-LGB
WOA [50]	XGBoost	WOA-XGBoost [34]
	Random Forest	WOA-RF [51]
	LightGBM	WOA-LGB

.

The above hybrid models shwon in Table 2 often have good predictive performance in existing studies. Zhang et al. (2020) [40] adopted the PSO-XGBoost to predict explosion-induced peak particle velocity. Yosra Grichi (2018) [41] used RF and PSO to perform obsolescence forecasting. To more accurately perform intrusion detection, Liu et al. (2021) [42] proposed a new hybrid model based on particle swarm optimization. Wang et al. (2021) [45] applied the GWO-RF hybrid model to the power field to achieve an efficient malfunction diagnosis of transformers. Lu et al. (2020) [44] and Yan et al. (2020) [34], respectively, used GWO-XGBoost and WOA-XGB hybrid models to predict reference evaporation. Yu et al. (2020) [47] used a RF and HHO algorithm to forecast, analyze, and control ground vibrations caused by explosions. Liu et al. (2019) [51] proposed an evaluation model based on the WOA algorithm and an improved RF to resolve ambiguity in resilience evaluation.

MAPE is one of the most commonly used evaluation metrics in regression tasks, and it is mainly used to measure the forecasting accuracy of statistical forecasting methods. In this study, we primarily adopted MAPE to evaluate the forecasting performance of the SSA-XGBoost and other models. Root mean square error (RMSE) is also commonly used to measure the forecasting results of machine learning models. It measures the mean errors between the observed value and the actual value. These two evaluation metrics are calculated by the formulae shown in

Table 3. Metrics for evaluating the accuracy of forecasting models.

Metrics	Equation	Remark
MAPE	$\mathrm{MAPE}=\frac{100\%}{n}{\sum }_{t-1}^n\left|\frac{A_t-F_t}{A_l}\right|$	The smaller, the better
RMSE	$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\sum }_{t-1}^n{\left(\frac{A_t-F_t}{A_t}\right)}^2}$	The smaller, the better

. Furthermore, in this study, all models and algorithms are implemented in Python3.8.5, hardware mainly includes Intel(R) Xeon(R) W-2123 CPU @ 3.60 GHzand and 62.5 GiB RAM, and the operation system is Linux Ubuntu 20.04.1. The Python packages include scikit-learn0.23.2, xgboost1.3.3, numpy1.19.2, and pandas1.1.3.

4.2. Information of the Medels and Algorithms for Comparison

• Random Forest (RF). Two American statisticians, Leo Breiman and Adele Culter, first proposed RF in 2001. It is an ensemble learning algorithm that combines classification and regression trees [52]. The RF algorithm mainly consists of the following three steps: (a) First, through the method of simple random sampling with replacement, n samples are randomly selected from the sample set; (b) Arbitrarily choose k features from all features, and use these features and the samples from $step1$ to build a decision tree; (c) Repeat the above steps continuously until multiple decision trees are generated to form a RF.
• Light Gradient Boosting Machine (LightGBM). LightGBM is a distributed gradient boosting framework based on the decision tree algorithm [53]. Its essence is an integrated learning algorithm that promotes weak learners to strong learners. Specifically, it combines many tree models with smaller accuracy. In each iteration, the loss function is made smaller and smaller by moving to the negative gradient direction of the loss function, and finally, a better tree is obtained. Compared with the traditional GBDT model, LightGBM has several very prominent advantages: smaller memory consumption, faster training speed, higher accuracy, and fast processing of massive data.
• Particle swarm optimization (PSO). Particle swarm optimization is a swarm intelligence algorithm designed to simulate the predation behavior of birds. It was first proposed by American scientists Eberhart and Kennedy [39] in 1995. Particle swarm optimization mimics the behavior of groups of insects, birds, fish, etc., which cooperatively search for food. Each group member continually changes its search patterns by learning from their own and other members’ experiences.
• Grey wolf optimization (GWO). The GWO is an algorithm with a simple structure small number of parameters and is easy to implement. Since it was first proposed by Mirjalili et al. in 2014 [43], it has been rapidly and widely used in parameter optimization and image classification. Since the algorithm is inspired by the predation behavior of the grey wolf, it is named the grey wolf optimization algorithm.
• Harris hawk optimization (HHO). HHO is an intelligent optimization algorithm simulating the predation behavior of the Harris Eagle, which was proposed by Heidari et al. in 2019 [46]. The HHO algorithm comprises three stages: search, search and development conversion, and development. The algorithm has excellent global search ability and requires the adjustment of few parameters.
• Multi-Verse Optimizer (MVO), The MVO algorithm is inspired by the motion behavior of multiverse populations under the combined action of white holes, black holes, and wormholes. Seyedali Mirjalili first proposed the algorithm in 2016 [48]. In the MVO algorithm, the main performance parameters are the wormhole existence probability and the wormhole travel distance rate. The parameters are relatively few, and the low-dimensional numerical experiments show fairly excellent performance.
• Whale optimization algorithm (WOA). Mirjalili, a researcher at Griffith University in Australia, proposed a novel swarm intelligence optimization algorithm called Whale optimization algorithm in 2016 [50]. The algorithm is a novel swarm optimization algorithm imitating the hunting behavior of humpback whales. Its advantages are simple operation, few parameters, and the fact it does not easy fall into local optimum.

4.3. Data Description

The quarterly NGC of the UK from first quarter of 1998 to the first quarter of 2021 were collected from the UK office for national statistics (https://www.ons.gov.uk/ (accessed on 10 July 2021)), and the monthly NGC of the Netherlands from January 1998 to April 2021 was collected from the CBS-Statistics Netherlands (https://www.cbs.nl/ (accessed on 10 July 2021)).

Table 4. Statistical features of the dataset.

Dataset	Interval	Mean	Maximum	Minimum	Standard Deviation	Numbers	Units
NGC (UK)	Quarterly	146,035.137	257,790.410	51,914.490	63,678.848	93	Gigawatt–hour
NGC (Netherlands)	Monthly	3682.230	7079.000	1696.000	1223.898	472	million m${}^3$

shows some statistical characteristics of the two datasets.

In Figure 6, the geographical location information of the UK and Netherlands and the autocorrelation plots of the two datasets are shown. It can be seen that both countries have significant strategic locations, and they promote the development of trade and transportation in the whole of western Europe. Both datasets are highly cyclical, which is very consistent with the correlation characteristics of NGC data.

In this study, we divide the two datasets according to 8:1:1(sample sizes for fitting, validation and forecasting) using the data division method proposed in Section 3. In the case for UK the training set contains 75 samples, the validation set contains 9 samples, and the test set contains 9 samples. In the case for Netherlands, the training set contains 378 samples, the validation set contains 47 samples, and the test set contains 47 samples.

4.4. Forecasting Results

In this subsection the proposed SSA-XGBoost hybrid model is use to forecasting the NGC of the UK and the Netherlands. In the forecasting process for both examples, we set up three different scenarios with $\tau =6$, $\tau =9$ and $\tau =12$, respectively. We perform a 5-step ahead forecasting in each scenario. In addition, to demonstrate the intense competitiveness of the proposed model, we also introduced the fifteen hybrid models used in Section 4.1 to compare with SSA-XGBoost in the actual case.

Table 5. MAPEs of multi-step ahead forecasting of the NGC of UK.

Indices		XGBoost						Random Forest						LightGBM
		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step
$\tau =6$	SSA	14.124	4.428	4.453	4.304	4.298		9.726	9.750	10.757	11.476	11.894		6.389	6.184	6.280	5.723	5.492
	PSO	5.355	5.414	5.081	4.809	4.592		7.529	7.368	6.727	6.343	5.988		7.551	7.453	7.687	7.263	7.263
	GWO	6.102	6.310	6.699	6.886	6.728		6.032	7.698	9.089	9.380	10.071		7.196	7.050	7.117	6.510	6.517
	WOA	6.740	6.479	6.175	5.508	5.349		9.573	11.518	13.638	15.049	15.191		6.980	6.925	7.351	7.038	7.042
	HHO	5.907	6.114	5.857	5.497	5.290		8.867	9.703	10.497	10.602	10.879		6.902	6.734	6.864	6.324	6.124
	MVO	5.673	5.645	5.528	5.401	5.362		9.096	9.604	9.485	10.001	10.460		7.201	7.053	7.120	6.511	6.512
$\tau =9$	SSA	5.839	5.809	5.791	5.515	5.300		7.020	6.528	6.329	6.102	5.855		9.749	10.273	10.415	10.119	9.977
	PSO	5.798	6.106	6.831	7.355	7.619		8.428	8.668	9.324	9.497	9.666		9.960	10.472	10.643	10.335	10.204
	GWO	6.380	6.591	7.250	7.474	7.787		12.457	13.163	13.652	13.694	13.847		9.767	10.292	10.437	10.141	9.999
	WOA	7.397	7.630	7.764	7.698	7.618		9.424	9.632	9.754	9.283	8.961		7.995	8.751	8.960	8.612	8.566
	HHO	6.087	6.168	6.251	6.264	6.215		10.011	9.695	9.867	9.594	9.729		9.729	10.236	10.327	10.009	9.864
	MVO	6.928	7.117	6.920	6.616	6.442		8.378	8.486	8.599	8.705	8.908		9.755	10.279	10.422	10.126	9.984
$\tau =12$	SSA	4.490	4.734	5.182	5.287	5.188		8.009	7.755	7.767	7.489	7.328		15.684	15.966	16.248	16.315	16.657
	PSO	7.150	6.938	7.366	7.666	7.835		7.147	7.212	7.493	7.421	7.458		15.684	15.966	16.248	16.315	16.657
	GWO	6.736	6.724	6.877	7.152	7.375		7.390	7.798	7.936	7.585	7.418		15.681	15.964	16.246	16.311	16.652
	WOA	6.731	6.477	6.594	6.577	6.786		6.888	6.947	7.848	7.977	8.152		15.684	15.966	16.248	16.315	16.657
	HHO	6.072	6.542	6.780	6.864	7.070		10.649	10.477	10.847	10.481	10.456		15.518	15.872	16.118	15.987	16.172
	MVO	5.490	5.606	5.852	5.929	5.981		8.045	8.043	8.045	8.070	7.783		15.650	16.043	16.337	16.314	16.564

¹ All the bold numbers represent the best (smallest) metrics. And it is the same case in Table 6, Table 7 and Table 8.

and

Table 6. MAPEs of multi-step ahead forecasting of the NGC of Netherlands.

Indices		XGBoost						Random Forest						LightGBM
		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step
$\tau =6$	SSA	8.889	9.528	9.966	10.445	10.693		11.290	12.557	13.414	13.894	14.003		13.605	15.782	16.697	17.521	17.880
	PSO	9.483	10.247	10.595	10.765	10.980		10.990	12.239	13.229	13.821	14.100		12.733	14.154	15.291	16.136	16.519
	GWO	10.052	10.974	11.619	11.898	12.050		11.227	12.319	13.568	14.737	15.569		12.820	14.294	15.519	16.395	16.848
	WOA	10.157	11.057	11.559	11.713	11.945		11.526	12.984	14.076	14.639	14.852		11.646	13.209	14.435	15.533	16.265
	HHO	9.656	10.650	11.120	11.444	11.549		10.816	12.195	13.339	13.927	14.157		12.563	14.026	15.127	15.819	16.318
	MVO	9.366	10.233	10.631	10.826	11.149		11.939	12.329	13.629	14.468	14.703		9.366	10.233	10.631	10.826	11.149
$\tau =9$	SSA	9.365	10.058	10.136	10.235	10.326		11.272	11.975	12.487	12.734	12.906		12.085	13.298	13.679	13.897	13.866
	PSO	9.663	10.600	10.945	11.026	11.105		11.319	12.347	13.048	13.415	13.637		11.943	13.185	13.785	14.140	14.135
	GWO	9.687	10.438	10.817	10.775	10.637		12.292	13.222	13.773	13.985	14.125		11.466	12.514	13.002	13.185	13.067
	WOA	10.024	10.892	11.226	11.277	11.326		11.111	12.415	13.088	13.410	13.610		10.835	11.974	12.579	12.923	13.106
	HHO	10.019	10.949	11.375	11.547	11.669		11.621	12.766	13.384	13.571	13.577		10.850	12.112	12.864	13.250	13.410
	MVO	9.552	10.100	10.334	10.392	10.374		11.430	12.612	13.429	13.745	13.886		9.552	10.100	10.334	10.392	10.374
$\tau =12$	SSA	7.238	7.175	7.211	7.248	7.307		8.507	8.755	8.916	9.047	9.101		9.317	9.952	10.153	10.154	10.188
	PSO	7.117	7.565	7.712	7.871	7.908		8.599	8.966	9.182	9.304	9.374		9.477	9.963	10.118	10.158	10.148
	GWO	7.609	7.846	7.872	7.879	7.935		9.271	9.597	9.811	9.974	10.000		8.777	9.040	9.196	9.297	9.278
	WOA	7.722	7.969	8.348	8.525	8.532		8.970	9.485	9.741	9.883	9.898		9.486	9.956	10.052	10.165	10.110
	HHO	7.583	7.765	7.871	7.937	7.996		8.857	9.247	9.496	9.656	9.729		9.662	10.118	10.121	10.116	10.003
	MVO	7.909	7.853	7.922	7.995	8.014		9.035	9.351	9.551	9.759	9.836		7.909	7.853	7.922	7.995	8.014

show the MAPE of the predicted results for the UK and the Netherlands, respectively. From the first table, we can observe that the values of SSA-XGBoost MAPE are less than 6% for all lag and step scenarios. However, in the second table, the value of SSA-XGBoost MAPE increases, but it is still less than 11%. Moreover, it should be mentioned that SSA-XGBoost consistently has the smallest MAPE values compared to the other comparable models, which indicates excellent predictive performance and high competitivity of this hybrid model.

Table 7. RMSEs of multi-step ahead forecasting of the NGC of UK.

Indices		XGBoost						Random Forest						LightGBM
		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step
$\tau$ = 6	SSA	7083.789	7284.455	7360.300	7397.350	7625.696		11,224.130	10,725.460	12,734.978	14,494.509	15,601.880		9130.401	8297.729	8132.263	7825.397	7720.972
	PSO	7697.141	7677.178	7379.942	7294.624	7278.992		12,259.029	11,750.404	11,103.528	10,886.963	10,557.026		10,049.930	9488.347	9467.416	9325.154	9472.552
	GWO	8676.628	8805.201	8986.479	9205.312	8921.273		10,619.739	12,073.005	12,891.806	13,235.472	15,042.777		9849.726	9167.982	9060.385	8792.475	9404.556
	WOA	12,758.536	10,620.400	9585.015	8832.159	9084.639		22,079.333	25,402.291	27,561.550	29,142.259	28,812.118		8774.836	8307.744	8472.084	8394.022	8518.347
	HHO	8257.153	10,896.835	11,838.193	12,340.044	13,144.967		10,944.117	11486.998	12403.300	12,759.115	13,076.733		9513.738	8768.511	8650.105	8384.835	8318.968
	MVO	10,667.405	10,626.755	9712.374	9297.356	9600.802		11,110.344	11,430.623	11,080.077	11,936.990	12,432.405		9870.943	9186.526	9078.698	8808.895	9413.312
$\tau$ = 9	SSA	7365.363	7028.222	6955.842	6905.631	6727.568		12,405.341	15,353.614	16,598.810	16,810.806	16,476.152		13,307.409	13,521.694	13,707.760	13,881.910	14,007.213
	PSO	12,801.210	13,169.912	13,894.771	14,508.964	14,837.154		11,696.771	11,844.595	12,250.961	12,589.016	12,784.163		13,340.530	13,525.821	13,718.938	13,881.420	14,018.135
	GWO	10,107.088	10,325.109	10,860.832	11,271.831	11,819.062		24,427.493	25,134.852	25,866.764	26,620.196	27,366.568		13,316.945	13,533.257	13,720.707	13,895.577	14,019.931
	WOA	11,756.110	11,999.820	12,274.938	12,575.960	12,688.952		10,208.066	10,170.301	10,240.144	10,148.747	9890.040		10,213.108	10,747.026	10,982.412	11,076.597	11,284.956
	HHO	8503.489	8498.406	8755.758	9022.435	9012.168		11,593.676	11,088.775	11,133.853	11,124.317	11,365.227		13,122.186	13,342.096	13,490.780	13,642.937	13,772.342
	MVO	13,149.165	13,448.241	13,482.311	13,670.174	13,938.481		11,991.064	11,988.071	12,168.603	12,449.394	12,785.796		13,310.375	13,525.289	13,711.784	13,886.158	14,011.166
$\tau$ = 12	SSA	8379.824	8621.319	8902.975	9156.637	8951.197		11,384.764	10,343.044	10,099.481	10,007.199	9918.844		30,666.753	31,186.128	31,952.261	32,858.987	35,050.104
	PSO	16,009.912	15,760.478	16,094.090	16,413.965	16670.100		9697.053	9716.762	9935.180	10,124.934	10,312.847		30,666.757	31,186.132	31,952.265	32,858.991	35,050.108
	GWO	13932.088	14115.732	14451.148	14859.809	15245.074		8995.554	9254.092	9414.683	9450.739	9505.770		30,654.423	31,174.299	31,940.251	32,846.564	35,038.724
	WOA	9890.686	9565.611	9614.929	9758.711	10,058.354		15,973.339	16,225.195	16,731.542	17,185.036	17,673.006		30,666.736	31,186.112	31,952.245	32,858.970	35,050.089
	HHO	12,460.225	12,891.544	13,278.332	13,685.566	14,288.333		11,864.541	11,211.504	11,193.034	11,045.136	11,083.211		29,638.496	30,214.224	30,988.834	31,862.582	34,091.278
	MVO	12,045.233	12,294.365	12,637.379	13,012.692	13,337.793		17,104.638	17,226.866	17,563.214	18,024.568	17,815.142		30,137.679	30,755.022	31,546.292	32,447.906	34,541.608

and

Table 8. RMSEs of multi-step ahead forecasting of the NGC of UK Netherlands.

Indices		XGBoost						Random Forest						LightGBM
		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step		1-Step	2-Step	3-Step	4-Step	5-Step
$\tau =6$	SSA	356.238	381.175	401.012	417.665	430.288		472.075	532.859	576.005	599.399	597.645		546.790	667.395	716.870	760.507	773.509
	PSO	373.979	400.275	420.036	427.138	433.865		471.816	524.954	567.269	594.772	598.810		524.414	616.136	675.307	717.661	730.802
	GWO	407.599	442.148	467.035	483.877	488.219		493.738	537.951	594.793	645.197	668.219		535.987	623.412	678.135	715.468	732.752
	WOA	408.537	442.148	467.193	477.913	481.535		493.536	560.126	604.106	628.672	633.016		498.255	575.298	620.185	668.858	698.847
	HHO	406.515	435.701	458.468	471.961	470.648		461.945	524.314	575.269	602.803	608.983		540.492	621.025	672.508	707.921	728.220
	MVO	388.870	418.748	435.887	444.497	451.226		509.677	536.868	581.791	614.797	628.032		524.460	616.209	675.389	717.747	730.876
$\tau =9$	SSA	384.360	408.778	418.485	425.234	425.147		479.534	508.496	532.474	546.459	554.607		511.036	563.624	583.815	593.139	591.898
	PSO	403.497	432.150	446.117	450.034	448.399		482.550	533.534	561.512	577.765	586.517		496.293	554.076	580.429	599.770	602.119
	GWO	384.848	406.469	428.627	429.633	425.440		535.503	571.591	589.594	593.953	597.448		485.978	525.657	545.414	556.849	553.014
	WOA	412.235	442.998	459.982	464.254	463.782		450.004	511.061	546.656	560.572	566.189		483.814	535.551	563.002	578.734	586.113
	HHO	428.424	456.915	474.113	479.481	480.546		476.351	527.784	559.621	569.669	569.772		480.889	531.030	560.691	578.016	585.106
	MVO	389.824	409.517	420.104	422.943	419.757		471.128	527.959	566.788	581.958	588.160		516.489	558.465	585.009	600.780	600.495
$\tau =12$	SSA	311.836	313.310	316.684	320.091	323.343		359.217	367.487	378.066	384.599	388.549		401.884	426.376	436.110	441.555	445.513
	PSO	308.835	329.864	335.774	341.866	342.365		376.505	389.588	400.976	408.514	411.706		407.908	430.400	433.532	439.031	442.448
	GWO	331.762	343.138	344.481	345.524	347.518		398.146	415.099	423.329	430.700	431.755		399.393	415.227	420.986	425.814	425.697
	WOA	340.117	343.945	361.685	370.629	371.046		388.525	412.406	423.497	430.657	431.623		422.878	444.544	449.673	454.801	452.427
	HHO	327.979	332.795	337.025	340.658	342.889		385.326	395.703	407.409	414.682	418.466		421.483	437.880	440.650	442.834	442.810
	MVO	335.774	338.018	341.628	345.156	346.876		378.489	394.041	405.139	415.111	418.640		424.321	435.284	439.702	444.536	441.716

show the RMSE of the forecasting results for the gas consumption dataset in the UK and the Netherlands, respectively. XGBoost remains the best performer among all the models involved in the forecasting task. However, in the second dataset $\tau =9$, both MVO-XGBoost and SSA-XGBoost have outstanding performance, but compared to MVO-XGBoost, SSA-XGBoost has more dominant forecasting results in the first three steps.

The above results illustrate that SSA-XGBoost has the most powerful forecasting performance among all the compared models. In addition, it is worth noting that the statistical properties of the two datasets are very different, but this does not affect the final forecasting results, which is sufficient to show that SSA-XGBoost has good generalization performance. Another point is that SSA does not perform the best among all intelligent optimization algorithms when optimizing the parameters of RF and LightGBM. Moreover, the predictive performance and generality of the hybrid models combined with other optimization algorithms with XGBoost are not all better than the models constructed by different optimization algorithms with RF and LightGBM.

The convergnce curves are plotted in Figure 7, Figure 8 and Figure 9, which illustrated the searching process of the optimization algorithms for the XGBoost in different cases. The results listed in the Table 5 and Table 6 are produced by the optimal models obtained in these processes.

We first observe the results related for the UK. It can be seen from the Figure 7, Figure 8 and Figure 9 that SSA-XGBoost reaches the minimum error after approximately four hundred iterations in all three different $\tau$ scenarios. In addition, GWO, MVO, and WOA cannot find the optimal combination of the hyperparameters of XGBoost using only five hundred iterations. In contrast, the GWO and PSO algorithms have excellent results in parameter selection for RF. SSA-LGB fails to obtain the optimal parameter combination when $\tau =3$ and the maximum number of iterations is reached. SSA-RF does not obtain the optimal parameter combination in all cases. We then observe the convergence curves of all the hybrid models in the case of the Netherlands. In each subcase, SSA-XGBoost has reached the minimal error in the validation set when the number of iterations is only three hundred. However, SSA-LGB and SSA-RF did not find the optimal combination of parameters for the model in all cases. These two figures fully illustrate that the SSA performs much better in hyperparameter optimization for XGBoost than other algorithms. And on the other hand, it is much easier for the optimization algorithms to optimizing the XGBoost than the other models.

In Figure 10 and Figure 11, the values of MAPE and RMSE of the best five hybrid model are shown. It can be seen that SSA-XGBoost has MAPE values as 4.9828% and 9.0547% for all $\tau$ and forecasting steps, respectively, and the values of RMSE are 7716.411 and 375.576, respectively, which are the best among all the hybrid models. More intuitively, the convergence curves of the most important parameters ($n\_estimators,max\_depth,learning\_rate$) of the SSA-XGBoost model during the iterations are plotted in Figure 12. It can be observed that when $\tau =6$ and the number of iterations reaches four hundred, the three parameters do not change significantly, and when $\tau =9$, the values of the three parameters stabilize when the number of iterations reaches three hundred, while when $\tau =12$, only about two hundred iterations are needed to find the optimal combination of parameters. In all cases, the depths of the trees is less than twenty, which is a relatively simple tree structure, so the SSA-XGBoost model has good stability and is not overfitted in these cases.

Moreover, from Figure 7, Figure 8 and Figure 9 we observe that the MAPE of the other algorithms in the validation decreases in the first few iterations. In contrast, only the MAPE of the SSA algorithm has significant volatility. This is because the decision tree-based model is very sensitive to the and the hyperparameters. From Figure 12, we can see that when the number of iterations is less than two hundred, the parameter values fluctuate strongly, bringing large variance of the model’s error on the validation set.

In summary, the forecasting results of the SSA-XGBoost model are the best in both cases. The comparison with the results of fifteen different forecasting models shows that SSA-XGBoost has the best forecasting performance among all forecasting models. Although there are considerable differences in the original data of the two applications, the forecasting results are the same. The MAPE and RMSE of SSA-XGBoost are the smallest among all models in terms of average values and detailed results for different scenarios.

4.5. Discussions

4.5.1. Boosting Effect of the Same Algorithm for Different Models

Figure 13 and Figure 14 show the MAPE values of NGC forecasting results of the UK and the Netherlands, and Figure 15 and Figure 16 show the RMSE values of NGC forecasting results of the UK and the Netherlands. The discussions will be presented in two points of view.

In the first point of view, we compare the results with the same forecasting steps. It is noticed that SSA-XGBoost has the smallest MPAE and RMSE values in most cases comparing with other models with the same step. Therefore, it can be concluded that XGBoost outperforms RF and LightGBM in the forecasting results with all steps, regardless of the algorithm used for hyperparameter selection.

In the second point of view, we compare the results with the same time lag $\tau$ and the summarized results are plotted in Figure 17. Combining the results shwon in Figure 16 and Figure 17, it can be seen that the MAPE values of XGBoost are always the smallest among the three models in both datasets, regardless of the variation of $\tau$ values. Moreover, XGBoost always has a smaller RMSE when $\tau$ is different.

It is worth mentioning that when values of $\tau$ change, the structure of the entire training data changes significantly. In addition, the principles of different optimization algorithms are very different, and each algorithm’s optimal combination of model parameters may be different. In this case, XGBoost still outperforms random forest and LightGBM; in some cases, the MAPE and RMSE values are much smaller than the other models. This implies that the proposed optimization scheme can build a stronger learner based on a learner with a weak generalization performance. XGBoost expands the loss function, taking into account the second-order derivatives, so XGBoost has a higher forecasting accuracy. Our results also show that the XGBoost model has satisfactory forecasting accuracy and generalization performance. In summary, the XGBoost model consistently has the best forecasting performance among the three based on the decision tree model when using the same algorithm.

4.5.2. Optimization Effect of Different Algorithms on XGBoost

In this study, six different metaheuristic algorithms were used. Since each metaheuristic algorithm principle has its own characteristics and advantages, effect of different algorithms on the models may not be the same. In the discussion in the previous subsection, it was seen that XGBoost was the best performer among the three basic models. In this subsection, we discuss the effect of different optimization algorithms on the enhancement of forecasting results based on the XGBoost model.

Figure 18 and Figure 19 show the MAPE values and RMSE values of the forecasting results by XGBoost optimized by the six algorithms. It can be seen from the figures that SSA has the best performance of optimizing the XGBoost among all metaheuristics. From 2-step to 5-step forecasting, SSA has the smallest MAPE value regardless of $\tau$. Moreover, SSA also outperforms the other algorithms in terms of RMSE values of the forecasting results with both datasets. Furthermore, it is worth noting that the performance of other optimization algorithms is unstable, and the optimization effect of the algorithm changes significantly when $\tau$ changes.

4.5.3. Sensitive Analysis of Time Lags and Forecasting Steps of the Proposed Method

This subsection will analyze the sensitivity of time lag $\tau$ and the forecasting steps on the proposed model. The results of MAPEs with different time lag and forecating stpes in different views are shown in Figure 20 and Figure 21.

From the Figure 20 we can see that the MAPE increases with the increase of forecasting step when $\tau =6$ and $\tau =9$. When $\tau =12$, the MAPEs of PSO-XGBoost, GWO-XGBoost, HHO-XGBoost, and WOA-XGBoost also increase with the increase in the forecasting step. Only one exception is when $\tau =12$ the MAPEs of the 1-step forecasting errors of PSO-XGBoost and SSA-XGBoost are not all smaller than the 2-step forecasting results. The Figure 21 presents another view of the results. We can observe that the RMSE values of all hybrid models increase as the forecasting step increases. The results of the above two figures show that the forecasting accuracy decreases as the forecasting step increases, and the values of MAPE and RMSE become larger as the forecasting step increases. This is coincidence with the common sense that the forecasting errors will accumulated with larger forecasting steps.

In addition, another insteresting phenomenon is that the MAPE and RMSE values of the forecasting results tend to increase when $\tau$ increases. This is because the number of features of the data increases as $\tau$ increases, leading to more sufficient information for training the models. This implies that machine learning algorithms are more suitable for forecasting tasks with large-scale datasets and may perform better in large-sample forecasting tasks.

4.5.4. The Effect of Different Time Scales on the Forecasting Results

In this study, we selected two quarterly (UK) and monthly (Netherlands) datasets with different time scales. By analyzing Table 5, Table 6, Table 7 and Table 8 as well as Figure 10 and Figure 11, we can find that the errors differ significantly between the datasets with different time scales. The MAPE values are two times higher for the monthly data than the quarterly data in average.

The main reason is that monthly gas consumption data are more volatile and often have more noises, while quarterly data often have smoother trend. The smaller the dimension, the greater the uncertainty, which is reflected in Figure 22, where the black boxes show all the places with substantial volatility. It also indicates that our model is more suitable for time series forecasting with less volatility, while poorer results may be obtained for more volatile time series data.

5. Conclusions

In this work, we proposed a novel SSA-XGBoost model by adopting the SSA to optimize the hyperparameters of the XGBoost. To demonstrate the predictive ability and generalization of the hybrid model, we compared the forecasting results of SSA-XGBoost with another fifteen similar hybrid models in two different cases. The results show that the SSA-XGBoost model is very competitive in various datasets. The main findings can be summarized as: (a) The SSA-XGBoost has the best performance in various datasets with different time scales, which shows that the SSA-XGBoost model has great potential and hope in NG forecasting; (b) The XGBoost model, and SSA algorithm performed the best among all the compared models and algorithms; (c) As the forecasting step increases, the forecasting error will generally increase. This is because the error of the previous time step will be propagated to future forecasting, leading to enlarge the errors with long forecasting.

Limitations and perspectives can be summarized as: (a) Both datasets used in this study are small, which may lead to more uncertainty of the results. Future works may be conducted with larger data sets, such as the daily NGC; (b) Although the optimization algorithm tuning model hyperparameters can significantly improve the model’s forecasting performance, it is still time-consumption. When the time scale becomes large, tuning parameters will become more complex. Some specific techniques may be considered to improve the time efficiency of the proposed model in the future work.