Application of Regression Analysis Using Broad Learning System for Time-Series Forecast of Ship Fuel Consumption

Abstract

Accurately forecasting the fuel consumption of ships is critical for improving their energy efficiency. However, the environmental factors that affect ship fuel consumption have not been researched comprehensively, and most of the relevant studies continue to present efficiency and accuracy issues. In view of such problems, a time-series forecasting model of ship fuel consumption based on a novel regression analysis using broad learning system (BLS) was developed in this study. The BLS was compared to a diverse set of fuel consumption forecasting models based on time-series analyses and machine learning techniques, including autoregressive integrated moving average model with exogenous inputs (ARIMAX), support vector regression (SVR), recurrent neural network (RNN), long short-term memory network (LSTM), and extreme learning machines (ELM). In the experiment, two types of passenger roll-on roll-off (ro-ro) ship and liquefied petroleum gas (LPG) carrierwere used as research objects to verify the proposed method’s generalizability, with data divided among two groups (R_M, R_B). The experimental results showed that the BLS model is the best choice to forecast fuel consumption in actual navigation, with mean absolute error (MAE) values of 0.0140 and 0.0115 on R_M and R_B, respectively. For the LPG carrier, it has also been proven that the forecast effect is improved when factoring the sea condition, with MAE reaching 0.0108 and 0.0142 under ballast and laden conditions, respectively. Furthermore, the BLS features the advantages of low computing complexity and short forecast time, making it more suitable for real-world applications. The results of this study can therefore effectively improve the energy efficiency of ships by reducing operating costs and emissions.

1. Introduction

With the recently growing demand and promotion of shipping transportation, the carbon emissions of ships have increased significantly, and the sustainability of the shipping industry has attracted considerable attention from public and the governmental entities [1]. Subsequently, few shipping corporations have also focused on this issue to protect the environment with the objective of carbon neutrality, and promoting green development within the shipping industry to minimize pollution [2]. To develop an effective method to improve energy efficiency and reduce emissions reductions in the shipping industry, accurate model must be designed to forecast ship fuel consumption. In particular, it may prove very helpful to discover the principal factors affecting fuel consumption in existing ships, and extract corresponding relationships with fuel cost [3].

Many studies have been devoted to the use of time-series analyses and machine learning techniques to forecast ship fuel consumption. In a regression analysis of ship fuel consumption, Wang et al. considered the convexity, non-negativity, and univariate properties of the fuel speed function, and proposed an exact outer-approximation algorithm to address the speed optimization problem by obtaining an ε-optimal solution [4]. Gao et al. considered the influence of ship speed and ship payload on fuel consumption, and then built a mixed integer nonlinear rule for container transshipment [5]. An increasing number of factors related to fuel consumption have been collected. To effectively utilize these factors, Erto et al. proposed a method based on the multiple linear regression model to monitor fuel consumption, producing superior results to those obtained by the traditional speed–power curves method [6]. Wang et al. proposed an estimation model for fuel consumption based on the least absolute shrinkage and selection operator (LASSO). This model, which can integrate onboard selection for feature variables, was compared with several existing approaches for fuel consumption forecasting [7]. Given the increasing volume of fuel consumption-related data, as well as problems such as nonlinearity and discreteness, the linear regression (LR) approach cannot meet the demand for accurate forecasting. Accordingly, Zhu et al. proposed an integrated model and compared the performance of various machine learning methods —including LR, support vector regression (SVR), and artificial neural networks (ANN)—to obtain accurate fuel consumption forecasting results [8]. Likewise, Peng et al. compared five machine learning models—including gradient boosting regression, LR, k-nearest neighbor regression and ANN—to forecast the energy consumption of ships [9]. Yuan et al. constructed a forecast model of fuel consumption rate and speed based on long short-term memory (LSTM) to optimize resource cost [10] (Appendix A,

Table A1. Summary of this paper and related literature review.

Literature	Case Ship	Data Source	Forecast Model	Measuring Criterion
[4]	Container ships	Ship operational data	LR	$R^2$
[6]	Cruise ship	Navigation data, engine data	MLR	$R^2$
[7]	Container ship	The dataset derived from the fleet management system	LASSO	MAE, RMSD, CS
[8]	Passenger ro-ro ship	Ship operational data	LR, SVR, ANN	RMSE
[9]	More than 30 types of ships	Collected by Jingtang Port	GBR, RF, ANN, LR, KNN	EVS, MAE, MSE, $R^2$
[10]	Cargo ship	The monitoring data of ship sailing	ANN, RNN, LSTM	RMSE, MAE, $R^2$
This paper	Passenger ro-ro ship, LPG carrier	Onboard data, available oceanographic and meteorological data	ARIMAX, SVR, RNN, LSTM, ELM, BLS	MAE, Time

). Although existing studies have yielded impressive results, many issues for expansion in terms of effectiveness and efficiency remain. Considering that the collected data represent time-series data, the quantity and quality of the data have a significant impact on forecasting performance. Therefore, an appropriate model must be developed for the characteristics of time-series data. Furthermore, most prior studies only considered onboard fuel consumption impact variables, resulting in insufficient data volume and sources. Given the complexity of the shipping process, more consideration should be given to the marine environment and weather conditions.

In response to the aforementioned problems, this study was conducted to examine passenger roll-on roll-off (ro-ro) ship and liquefied petroleum gas (LPG) carrier as experimental targets, consider of real navigating conditions, and builds a forecast model of fuel consumption based on a novel regression analysis using a broad learning system (BLS) [11]. This study fills a gap in the application of BLS to time-series analyses for fuel consumption. To determine a model that demands minimal time and is more accurate with different structures and complexity, BLS was compared with autoregressive integrated moving average model with exogenous inputs (ARIMAX) [12], SVR [8], recurrent neural network (RNN), LSTM [10] and extreme learning machines (ELM) [13], which are models based on time-series analyses and machine learning techniques. According to the characteristics of different models to demonstrated the applicability and accuracy by systematic verification analysis. Two types of ships were considered in this study to verify the models’ generalizability. Furthermore, to enable the forecasting model’s operation in the real navigation environment and adaptability to possible changes, this study considered the internal and external factors relating to fuel consumption. According to the sea state and weather conditions, the data of the two ships are divided into two groups for separate discussion (Appendix A,

Table A2. Summary of the academic contributions of this paper.

Highlights	Academic Contributions
1	The application of emerging BLS model in the field of time series forecast for ship fuel consumption is expanded.
2	Using two different types of case ships as research objects to verify the generalization of the results.
3	According to the sea state and weather conditions, the data of the two case ships are divided into two groups for separate discussion, so as to restore the actual navigation environment to the maximum extent and improve the reliability of the forecast model.
4	The model which consume less time and more accurately is obtained by systematic verification analysis on the applicability and accuracy of different time series analysis and machine learning techniques for ship fuel consumption forecast.

). Passenger ro-ro ship data were allocated among two different groups—$R_M$ and $R_B$—according to the route for analysis. The two groups obtained mean absolute error (MAE) values of 0.0140 and 0.0115, respectively. When the BLS model was applied to the LPG carrier under consideration of sea conditions ($C_1$), the MAE under ballast and laden reached values 0.0108 and 0.0142, respectively. These values represent decreases of 28.00% and 32.70% compared with the corresponding MAE results when sea conditions were not considered ($C_2$). We therefore conclude that the sea conditions must be considered for LPG carriers in either state. Furthermore, abundant experimental results indicate that all the forecast consumption times incurred by the BLS model for both cases were superior to those obtained by other models.

The remainder of this paper is organized as follows. Section 2 briefly introduces ship fuel consumption forecasting based on time-series analyses and machine learning techniques. The proposed approach and corresponding data are presented in Section 3. Comparative experimental results are discussed in Section 4. Finally, Section 5 presents our conclusions.

2. Literature Review of Forecasting Methods

2.1. Time-Series Analyses

The development of fuel consumption forecast models based on time-series analyses uses historical data as a forecasting tool through the inherent correlation between time- and fuel consumption-related data in various dimensions. Traditional approaches primarily solve the model parameters by determining the time-series parameter model, and uses the solved model to complete the forecast. Conventional models, including the autoregressive average (AR), moving average (MA), autoregressive integrated moving average (ARIMA) [14], and ARIMAX models, are introduced in this section.

Suppose that at time $t$, $x_t$ represents the fuel consumption of the time series, and $u_t$ and ${\epsilon }_t$ denote the error term and the white noise, respectively.

The AR model is a linear model. When $u_t={\epsilon }_t$, $p$ represents the dimension of the time window and indicates that $x_t$ is forecasted using the past fuel consumption $x_{t-j}$ and ${\epsilon }_t$ of period $p$. This relation is expressed in Equation (1), where $c$ and ${\varphi }_j$ denote the model parameter and weight coefficient, respectively [14].

$$x_t=c+{\displaystyle \sum }_{j=1}^p{\varphi }_j{x}_{t-j}+{\epsilon }_t$$

(1)

When $x_t$ depends only on the linear combination of historical white noise, it can be expressed as $x_t=u_t$. From this, the formula for MA is as shown in Equation (2), where ${\theta }_h$ denotes the weight coefficient.

$$x_t=c+{\displaystyle \sum }_{h=1}^q{\theta }_h{\epsilon }_{t-h}+{\epsilon }_t$$

(2)

The ARIMA model was obtained by combining AR and MA using the difference method, as expressed in Equation (3).

$$x_t=\mu +{\displaystyle \sum }_{j=1}^p{\varphi }_j{x}_{t-j}+{\displaystyle \sum }_{h=1}^q{\theta }_h{\epsilon }_{t-h}+{\epsilon }_t$$

(3)

For multivariate time-series forecasting, the ARIMAX model is a generalization of ARIMA model, which is capable of incorporating an exogenous input variable [12], as shown in Equation (4).

$$y_t=\mu +{\displaystyle \sum }_{i=1}^k\frac{{\mathsf{\Theta }}_i\left(B\right)}{{\mathsf{\Phi }}_i\left(B\right)}{B}^{l_i}{x}_{ti}+{\epsilon }_t$$

(4)

where $y_t$ and $x_{ti}$ are the output variable and the $ith$ input variable at time $t$, respectively; $B$ is the backshift operator; ${\mathsf{\Phi }}_i\left(B\right)$ is the autoregressive coefficient polynomial of the $ith$ input variable; ${\mathsf{\Theta }}_i\left(B\right)$ is the moving average coefficient polynomial of the $ith$ input variable; $l_i$ is the delay order of the $ith$ input variable.

2.2. Support Vector Regression

As branch of support vector machines (SVM), SVR is a machine learning method based on statistical theory that can solve regression problems [8]. In SVR model, the ship fuel consumption variable $X$ is mapped to a higher-dimensional space using a kernel function φ for LR estimation, as shown in Equation (5) [15]. $Y$ and $Y^{\prime }$ represent real and forecasted fuel consumption, respectively. Some common kernel functions are the linear kernel, polynomial (Poly) kernel and radial basis function (RBF).

$$Y^{\prime }=W\phi \left(X\right)+b$$

(5)

In the high-dimensional space, a hyperplane that minimizes the maximum distance between the mapping points and hyperplane—which is guaranteed to be less than $\epsilon$—determined using the model parameters $W$ and $b$. The optimization objective of SVR is shown in Equation (6) and Figure 1 [15], where $f\left(x\right)=y\prime$,$y_i\in Y$, $y_i^{\prime }\in Y^{\prime }$, $\forall i,i=1,2,\dots ,m$. Model training is guided by the error between the forecasted and real fuel consumption values.

$$\{\begin{array}{c}\underset{W,b}{\min }\frac{1}{2}{\Vert W\Vert }^2\\ s.t.\left|y_i-y_i^{\prime }\right|\le \epsilon \end{array}$$

(6)

2.3. Recurrent Neural Network

RNN is a type of recursive neural network, which is one of the most commonly used models when handling time-series problems based on machine learning methods [10]. The structure of an RNN-based fuel consumption model is illustrated in Figure 2.

As shown in the figure, the left side is folded whereas the right side is unfolded. Dashed arrows represent recurrences in the model structure embodied in the hidden layers. The activation process from the input layer to the hidden layer at time $t$ can be expressed by Equation (7).

$$h_t=f_1\left(U{x}_t+W{h}_{t-1}\right)$$

(7)

where $h_t$ represents the output of neurons in the hidden layer at the time $t$; $U$ is the connection weight matrix from the input layer or previous hidden layer to the current hidden layer; $W$ is the connection weight matrix between the previous and current time-series hidden layers; and $f_1\left(U{x}_t+W{h}_{t-1}\right)$ is the activation function of the hidden layer, generally set to $tanh$ function.

The activation process from the hidden layer to the output layer is expressed by Equation (8).

$$o_t=f_2\left(V{h}_t\right)$$

(8)

where $o_t$ represents the output, $V$ is the connection weight matrix from the hidden layer to the output layer, and $f_2\left(x\right)$ is the activation function of the output layer, generally set to the sigmoid function.

The RNN uses the hidden node state $h_{t-1}$ at the last time point as the input to the neural network unit. In other words, weight connections are established between different layers’ neurons. This may explain the excellent performance exhibited by RNNs when handling time-series data. Neurons in the hidden layer also have weights, with corresponding weight connections $W, U, V$ [16].

2.4. Long Short-Term Memory

LSTM is a special type of RNN that incorporates a memory cell state to memorize the long-term state of information [10]. The underlying process of an LSTM-based fuel consumption model is illustrated in Figure 3.

As shown in the figure, LSTM controls the circulation and loss of features through a gate mechanism.

For the forget gate $f_t$, the cell state can be represented as $c_{t-1}$, $c_t$, $c_{t+1}$. It combines the state value $h_{t-1}$ of the previous hidden layer with the current input $x_t$, and outputs an $f_t$ through the $sigmoid$ function $\sigma$ to determine what information to discard from the cell state $c_{t-1}$. The process is expressed by Equation (9), where $\sigma$ is the activation function, and $W_f$ and $b_f$ are the weight and bias, respectively.

$$f_t=\sigma \left(W_f\cdot \left[h_{t-1}, x_t\right]+b_f\right)$$

(9)

Next, the input gate $i_t$ and $tanh$ function determine the new information that can be added to the cell state from the activation value $h_{t-1}$ of the hidden layer at the previous moment and current fuel consumption input variable $x_t$, thereby obtaining the candidate value $\widetilde{c_t}$. This procedure is expressed by Equations (10) and (11):

$$i_t=\sigma \left(W_i\cdot \left[h_{t-1}, x_t\right]+b_i\right)$$

(10)

$$\widetilde{c_t}=tanh\left(W_c\cdot \left[h_{t-1}, x_t\right]+b_c\right)$$

(11)

The next step updates the cell state. As shown in Equation (12), the forget gate $f_t$ and the input gate $i_t$ are combined, and appropriate information is saved and discarded to obtain the cell state $c_t$ at the current moment.

$$c_t=f_t\cdot c_{t-1}+i_t\cdot \widetilde{c_t}$$

(12)

Finally, the output gate $o_t$ combines with $tanh$ to determine which information of $h_{t-1}$, $x_t$, $c_t$ is the output to the hidden layer state $h_t$ at this moment. The specific method is expressed by Equations (13) and (14), wherein the $\sigma$ activation function is used to determine the part of the cell state that needs to be output, and the $tanh$ function is used to obtain the backup output 16.

$$o_t=\sigma \left(W_o\cdot \left[h_{t-1}, x_t\right]+b_o\right)$$

(13)

$$h_t=o_t\cdot tanh\left(c_t\right)$$

(14)

2.5. Extreme Learning Machine

The ELM is a special single-hidden layer feedforward neural networks (SLFNs) proposed by Huang et al., comprising an input layer, hidden layer and output layer [13]. In the ELM training process, the weight and bias of the hidden layer are generated randomly and do not require updating. The weight of the output layer is calculated to complete the training process. Consequently, ELM is characterized by a short learning time and superior generalization performance. The output function of ELM is given by Equation (15).

$$f_L\left(x\right)={{\displaystyle \sum }}_{i=1}^L{\beta }_{i}G\left(w_i,b_i,x\right)$$

(15)

where $L$ indicates the number of hidden layer nodes and ${\beta }_i$ is the connection weight vector between the $ith$ hidden layer neuron and the output neuron. $G\left(w_i,b_i,x\right)$ represents the output of the $ith$ hidden layer neuron corresponding to the input $x$.

$$G\left(w_i,b_i,x\right)=g\left(w_i·x+b_i\right)$$

(16)

where $g$ represents the activation function, $w_i$ is the weight vector between the $ith$ hidden layer neuron and input neuron, and $b_i$ is the bias of the $ith$ hidden layer neuron. Although the overall training process of ELM is similar to that of the traditional back-propagation neural network, the weight matrix between the hidden layer and the output is a Moore–Penrose generalized inverse matrix. Equation (17) can be abbreviated as

$$T=H\beta$$

(17)

where $H$ denotes the hidden layer output matrix, and T denotes the training data target matrix. The next step is to calculate the weight $\beta$ by minimizing the square approximation error, as shown in Equation (18).

$$\underset{\beta }{\min }{\Vert H\beta -T\Vert }^2$$

(18)

The following appropriate solution can be obtained for Equation (18), where $H^{†}$ is the Moore–Penrose generalized inverse matrix of $H$ [17].

$${\beta }^{*}=H^{†}T$$

(19)

3. Material and Method

3.1. Data Source

3.1.1. Passenger Ro-Ro Ship

All passenger ro-ro ship data used in this study were provided by the Danish University of Technology, with main specifications listed in

Table 1. Principal particulars of passenger ro-ro ship.

Parameters	Value
Length Overall	135 m
Breadth Molded	22.7 m
Depth to Main Deck	8.1 m
Designed Speed	21 knots
Main Engine Power	3360 kW

[18]. The collected raw dataset was recorded for two consecutive months from February–April 2010, and includes the main route ($R_M$) and the backup route ($R_B$) adopted under bad weather conditions for $R_M$. The following sections discuss the data characteristics of $R_M$ and $R_B$ individually.

Following preprocessing and feature selection, the number of data points for $R_M$ and $R_B$ were reduced to approximately 19,200 and 5700, respectively. Fuel consumption was designated as the output, and 11 feature variables related to fuel consumption were designated as the input. More information on the feature variables for the passenger ro-ro ship can be seen in

Table A3. The description about the feature variables for the passenger ro-ro ship.

Feature Variables	Description
SOG	SOG is the speed of the ship relative to the ground or any other fixed object, such as island or buoy.
Trim	The trim indicates its floating position in the direction of length, i.e., whether the bow or stern is deeper submerged into the water.
Pitch	The distance traveled by the propeller in one rotation.
Draught	The draught describes the vertical distance between the waterline and the bottom of the hull.
Headwind	The headwind means the direction of travel is the opposite direction of the wind.
Crosswind	The crosswind means the direction of travel is perpendicular to the wind direction.
Rudder	The rudder is a piece of blade installed aft in the submerged part of the ship to provide directional control for ship.

of Appendix A. For convenience, the names of the feature variables are abbreviated according to

Table 2. Feature variables of passenger ro-ro ship.

No	Feature Variables	Abbreviation
1	ME Total Fuel Consumption	METF
2	Speed Over Ground	SOG
3	Trim	Trim
4	Port Pitch	PPitch
5	Starboard Pitch	SPitch
6	Port Draught	PDraught
7	Starboard Draught	SDraught
8	Headwind	Headwind
9	Crosswind	Crosswind
10	Port Rudder	PRudder
11	Starboard Rudder	SRudder

.

3.1.2. Liquefied Petroleum Gas Carrier

All LPG carrier data used in this study were obtained from onboard data, as well as open real-time oceanographic and meteorological data, with principal particulars listed in

Table 3. Principal particulars of LPG carrier.

Parameters	Value
Capacity	54340 DWT
Length Overall	225 m
Breadth Molded	37 m
Designed Speed	16.8 knots
Main Engine Power	12,400 kW
Propeller Diameter	7400 mm
Propeller Pitch	5971.06 mm

[19]. The raw dataset comprises data recorded over seven consecutive months from December 2016 to July 2017, including internal and external factors related to the ship. Onboard data sources include noon reports, the ship automation system (SAS), and the electronic chart display and information system (ECDIS).

The draught of the LPG carrier did not change continuously. Specifically, it can be considered as binary, alternating between ballast or laden. Accordingly, the following chapters discuss the ballast and laden data separately.

After preprocessing and feature selection, the number of data points for the ballast and laden sets were reduced to approximately 9900 and 7400, respectively. Fuel consumption was designated as the output, and 17 feature variables related to fuel consumption were designated as the input. More information on the feature variables for the LPG carrier can be seen in

Table A4. The description about the feature variables for $R_B$.

Feature Variables	Description
RPM	RPM is the number of complete revolutions or rotations of a propeller in one minute.
COG	COG refers to the actual direction of progress of the ship, between two points, relative to the surface of the earth.
TL	TL represents the need for liquefaction of cargo on the case of LPG.
List	List refers to a floating state in which the ship tilts to starboard or port with unequal draft on the starboard or port sides.
Uwind	Uwind is the component of wind speed in the horizontal direction.
Vwind	Vwind is the vertical component of wind speed
WG	WG means a brief and sudden increase in speed of the wind.
SWH	SWH is the actual wave height measured according to certain rules.
WD	The line along which a wave travels with respect to its compass heading.
WP	WP is the time for a particle on a medium to make one complete vibrational cycle.
SD	SD indicates the direction they are headed for or where the current is flowing towards.

of Appendix A. The names of all feature variables were abbreviated, as listed in

Table 4. Feature variables of the LPG carrier.

No	Feature Variables	Abbreviation
1	ME Total Fuel Consumption	METF
2	Revolution Per Minute	RPM
3	Speed Over Ground	SOG
4	Course Over Ground	COG
5	Total Load	TL
6	Ambient Air Temperature	AT
7	Sea Water Temperature	SWT
8	Trim	Trim
9	List	List
10	Uwind	Uwind
11	Vwind	Vwind
12	Wind Gust	WG
13	Significant Wave Height	SWH
14	Wave Direction	WD
15	Wave Period	WP
16	Sea Direction	SD
17	Sea Current Speed	SCS

. In addition, the other acronyms used in the paper are shown in

Table A5. The list of acronyms used in the paper.

Explanation	Abbreviation	Explanation	Abbreviation
Broad Learning System	BLS	Autoregressive Integrated Moving Average Model with Exogenous Inputs	ARIMAX
Support Vector Regression	SVR	Recurrent Neural Network	RNN
Long Short-term Memory Network	LSTM	Extreme Learning Machines	ELM
Roll-on Roll-off	Ro-Ro	Liquefied Petroleum Gas	LPG
Least Absolute Shrinkage and Selection Operator	LASSO	Linear Regression	LR
Artificial Neural Networks	ANN	Autoregressive Average	AR
Moving Average	MA	Autoregressive Integrated Moving Average	ARIMA
Support Vector Machines	SVM	Polynomial	Poly
Radial Basis Function	RBF	Single-hidden Layer Feedforward Neural Networks	SLFNs
Ship Automation System	SAS	Electronic Chart Display and Information System	ECDIS
Mean Absolute Error	MAE	Multiple linear regression	MLR
Gradient Boosting Regression	GBR	Random Forest Regression	RF
K-Nearest Neighbor Regression	KNN	Explained Variance Score	EVS
Mean Squared Error	MSE	Cumulative Score	CS
Root Mean Square Deviation	RMSD

of Appendix A.

3.2. Data Structure

Data corresponding to both case ships required normalization prior to the ship fuel consumption model training. Normalization essentially represents data feature scaling. In this study, the min–max scaling method was employed for data transformation. Let $x_{i_{Min}}^{\prime }$ and $x_{i_{Max}}^{\prime }$ be the minimum and maximum values of the $ith$ feature variable, respectively. When $1\le i\le n$, $n$ denotes the number of feature variables. The original value $x_{ti}^{\prime }$ of the $ith$ feature variable at time $t$ is mapped to the [0, 1] interval to obtain the normalized value $x_{ti}$, as shown in Equation (20), where $1\le t\le m$, and $m$ is the total number of samples at all time points. The probability density distribution of the normalized datasets for the two ships are shown in Figure 4 and Figure 5.

$$x_{ti}=\frac{x_{ti}^{\prime }-x_{i_{Min}}^{\prime }}{x_{i_{Max}}^{\prime }-x_{i_{Min}}^{\prime }}$$

(20)

Consequently, the input matrix $X\in {ℝ}^{m\times n}$ and output matrix $Y\in {ℝ}^{m\times 1}$ of the ship fuel consumption forecast model are given by Equation (21).

$$X=\left[\begin{array}{c}{X}_1\\ X_2\\ ⋮\\ X_t\\ ⋮\\ X_m\end{array}\right]=\left[\begin{array}{c}{x}_{11}\\ x_{21}\\ ⋮\\ \cdots \\ ⋮\\ x_{m1}\end{array}\begin{array}{c}{x}_{12}\\ x_{22}\\ ⋮\\ \cdots \\ ⋮\\ x_{m2}\end{array}\begin{array}{c}\cdots \\ \cdots \\ ⋮\\ x_{ti}\\ ⋮\\ \cdots \end{array}\begin{array}{c}{x}_{1n}\\ x_{2n}\\ ⋮\\ \cdots \\ ⋮\\ x_{mn}\end{array}\right],=\left[\begin{array}{c}{y}_{1+k}\\ y_{2+k}\\ ⋮\\ y_{\mathrm{t}+k}\\ ⋮\\ y_{\mathrm{m}+k}\end{array}\right]$$

(21)

where $X_t$ represents the total number $n$ of features required to calculate ship fuel consumption at time $t$, $x_{ti}$ represents the normalized value of the $ith$ feature variable at time $t$, and $y_{t+k}$ denotes the real value of fuel consumption at time $t+k$. Note that $t=1, 2, \dots , m$, $i=1, 2, \dots , n$, and $k=1, 2, \dots , 10$. Both case ships’ data were allocated among training (70%), validation (10%) and test (20%) sets by using a randomly sampling approach. For Ro-Ro ship, the training set, validation set and test set contain about 13,440, 1920 and 3840 data points respectively on $R_M$, and 3990, 570 and 1140 data points on $R_B$; For LPG carrier, training set, validation set and test set contain about 6930, 990 and 1980 data points respectively under ballast and 5180, 740 and 1480 data points under laden.

3.3. Broad Learning System

3.3.1. Basic Concepts of BLS

Whereas all aforementioned models are constructed in the depth direction, BLS is employed to construct its network in the breadth direction. That is, the entire network is scaled horizontally, which effectively reduces network complexity [11]. The BLS model consists of four components, as shown in Figure 6: an input layer, a feature mapping layer, an enhancement node layer, and an output layer.

The specific algorithm flow corresponding to each BLS layer is as follows:

In the input layer, the input matrix $X$ is entered into the model, and the BLS network performs feature mapping. Through the feature mapping function ${\varphi }_g(*)$, $X$ is mapped to the mapping nodes of $t$ groups of different features, and each group of mappings is set to generate $k$ nodes; the calculation formula of the $gth$ group feature map node $Z_g$ is given by

$$Z_g={\varphi }_g\left(X{W}_{e_g}+{\beta }_{e_g}\right)$$

(22)

The parallel representation of $Z_g$ yields $Z^j=\left(Z_1,Z_2,\dots ,Z_j\right)$, where $Z^j\in {ℝ}^{m\times jk}$. The mapped features are regarded as a single layer; that is, the feature-mapping layer. Here ${\varphi }_g(*)$ is a linear mapping function, $W_{e_g}\in {ℝ}^{n\times k}$ is a randomly generated weight matrix, and ${\beta }_{e_g}$ is the corresponding bias term.

The $t$ group feature map node $Z_g$ is connected to the enhancement node layer. The weight coefficient matrix $W_{h_s}$ and bias term ${\beta }_{h_s}$ are generated randomly, and $Z^j$ is mapped to the $p$ group of randomly generated weight enhancement nodes through the nonlinear function ${\xi }_s(*)$, expressed as $H^p=\left(H_1,H_2,\dots ,H_p\right)$. Here, $H_p\in {ℝ}^{m\times pq}$ constitutes the enhanced node layer. Let each group of mappings generate $q$ nodes and $H_s$ represents the $sth$ group of enhanced nodes, as expressed in Equation (23).

$$H_s={\xi }_s\left(Z^j{W}_{h_s}+{\beta }_{h_s}\right)$$

(23)

The parallel connection between $Z^j$ and $H^p$, expressed as $A$, is directly connected to the output terminal, also known as the output layer. Therefore, the representation of the output matrix is given by Equation (24):

$$Y=\left(Z^j|H^p\right)W=AW$$

(24)

Here, $W$ is the connection weight matrix of the output layer, calculated according to Equation (25).

$$W=A^{-1}Y$$

(25)

where $A^{-1}$, representing the pseudoinverse matrix of matrix $A$, is calculated using the approximate solution of the ridge regression of $A^{-1}$. Subsequently, the connection weight matrix $W$ is obtained, and model training is completed. This is expressed in Equation (26), where $I$ represents the identity matrix and $A^T$ is the transpose matrix of $A$ [11].

$$A^{-1}=\underset{\lambda \to 0}{\lim }{\left(\lambda I+A^{T}A\right)}^{-1}{A}^T$$

(26)

3.3.2. Regression Analysis using BLS

As the most popular machine learning model among multiple fields, BLS has recently attracted significant attention from academia and industry. Owing to its efficiency and effectiveness, BLS has numerous applications in classification and time-series forecasting [20]. The application of BLS in classification achieved good results, such as hyperspectral imagery classification [21], zero-shot image classification [22] and event-based object classification [23]. As for time-series forecasting, Han et al. developed a maximum information exploitation BLS to solve the issue of extreme information utilization for large-scale chaotic time-series. The chaotic systems linear information is captured through an improved leaky integrator dynamical reservoir. In addition, the structural characteristics of BLS can be combined to effectively utilize nonlinear information. Finally, comparative experiments verified that MIE-BLS achieve better results in large-scale dynamical system modeling [24]. Liu et al. applied BLS in the study of a fast architecture for real-time training of the traffic predictor. Compared with LASSO regression, RNN and other machine learning models, BLS was concluded to significantly improve the forecast speed [25]. Yang et al. proposed a novel time-series model based on BLS to forecast port throughput, which exhibited superior performance to that of classical-time series models [26].

In addition to the aforementioned applications, BLS also can also be utilized to address problems related to time series across different domains, as in [27,28,29,30]. Owing to the impressive generalizability and remarkable efficiency, the present study extended the applicability of emerging BLS models to the field of fuel consumption forecasting.

4. Numerical Experiments

4.1. Measuring Critera

The MAE, one of the most common evaluation metrics for model performance, represents the mean of the absolute error between the forecasted and real fuel consumption values, as expressed in Equation (27).

$$MAE=\frac{1}{m}{\displaystyle \sum }_{i=1}^m\left|y_i^{\prime }-y_i\right|$$

(27)

where $m$ denotes the volume of test data. To efficiently forecast fuel consumption in practice, this study also considers the time consumed by the model. Experimental results obtained in this study are uniformly compared and analyzed in Section 4.3.

4.2. Comparison with Baseline Methods

As described in Section 3.1, the fuel consumption variable data were allocated among training and testing sets. Each forecast model was trained with the former, and subsequently verified with the latter. To objectively compare forecast performance between different models, default hyperparameters in the Python programming language were employed for simulation training. The hyperparameters settings of the ARIMAX, SVR, RNN, LSTM, ELM and BLS models are presented in

Table 5. The model hyperparameter settings.

Forecast Model	Hyperparameter Setting
ARIMAX	p = 1
	q = 1
	d = ’None’
SVR	C = 1.0
	Kernel = ‘Linear’
RNN	Number of Hidden Layers = 1
	Number of Neurons = 100
	Activation Function = Tanh
	Batch Size = 1
	Dropout = 0.2
LSTM	Number of Hidden Layers = 1
	Number of Neurons = 100
	Activation Function = Tanh
	Batch Size = 1
	Dropout = 0.2
ELM	Number of Hidden Layer Nodes = 50
BLS	N1 = 10
	N2 = 10
	N3 = 10

. In addition, the time step of the SVR, RNN, LSTM, ELM, and BLS models were set to 5.

The real and forecast fuel consumption values were compared for each model. Figure 7 and Figure 8 represent comparative test results for the passenger ro-ro ship on $R_M$ and $R_B$, respectively. Figure 9 and Figure 10 correspond to comparative test results for the LPG carrier under the ballast and laden states, respectively. The red curve in each figure represents real fuel consumption values, and the other curves measure the fuel consumption values obtained by different forecast models.

As shown in Figure 7, the real fuel consumption value was distributed in stages. The data difference between each stage was large, and the data difference within each stage was small. Before the 700th data point, the prediction deviation of each model was relatively large. However, from the overall point of view, the prediction deviation of each model was small. The forecast result of BLS model was the closest to reality, and the MAE of the BLS on $R_M$ reached 0.0345, while the deviations of the ARIMAX and SVR models were relatively large.

As shown in Figure 8, the real fuel consumption had great overall fluctuation. In particular, after the 450th data point, the data fluctuated greatly and the forecast deviation was large. The BLS model had the best forecast result, and the MAE of the BLS on $R_B$ reached 0.0397. The results of the ARIMAX and SVR models were relatively poor.

As shown in Figure 9, the data as a whole were less volatile. On the whole, the forecast of the region showed a stable trend after the 1000th data point and was relatively accurate, while the forecast of the region before the 1000th data point showed large fluctuation and error was relatively large. The forecast result of BLS was closer to the real value than other models, and the MAE of BLS reached 0.0272 under ballast. In addition, ARIMAX failed to perform well in the forecast.

It can be seen from Figure 10 that the real value of fuel consumption was less volatile overall. The forecast error of the regions with large fluctuations showed a trend in each model that was relatively large before the 400th data point and after the 1200th data point, while the forecast was relatively close to the real value between the 400th and 1200th data points. The forecasted result of the BLS model was closest to the real value, and the MAE of the BLS under laden reached 0.0759, while the overall deviations of the SVR and ARIMAX models were large.

4.3. Optimization of Parameters and Discussion

4.3.1. Optimization of Parameters

According to the observed data, each hyperparameter has a relatively suitable value. Furthermore, the hyperparameters will have an impact on the forecast accuracy of the model. Hence, a series of hyperparameters need to be tuned. However, iterated experiments for all possible hyperparameter combinations would incur a long computation time. Therefore, this study only considered several important hyperparameters. The optimal setting for each hyperparameter was determined according to the MAE on the test set. All parameters were adjusted according to the regression principles of ARIMAX, SVR, RNN, LSTM, ELM and BLS, as well as prior studies. The results of each model were the average results on 100 runs to avoid overtraining and validate generalization.

For the ARIMAX model, $d$ was set to the default value following the stability test. The influence of $p$ and $q$ on forecast accuracy is discussed later in the section [12]. Results of four groups of cases corresponding to the two case ships are shown in Figure 11. It can be seen from observation that the optimal parameters ($p, q$) corresponding to Figure 11a–d are (1, 3), (2, 5), (3, 4), and (2, 1).

For the SVR model, four commonly used kernel functions were compared in this section: linear, RBF, sigmoid and poly [8]. The experimental results shown in Figure 12a that the sigmoid and RBF kernels produced the best performance for $R_M$ and $R_B$, respectively. In the case of the LPG carrier, the poly and linear kernel functions yielded the best respective performance under state of ballast and laden states. The optimal parameters obtained for each dataset were used to adjust the time step from 1 to 10. As shown in Figure 12b, the optimal time step parameters of $R_M$, $R_B$, ballast and laden were 1, 4, 4 and 5, respectively.

For the RNN and LSTM models, we primarily considered the number of hidden layers and the time step [10]. The number of neurons in the first and second hidden layers was set to 100 and 80, respectively. Starting from the third hidden layer, the number of neurons in each layer was set to half of the previous layer. In all datasets, the optimal parameters determined in the final parameter adjustment process were selected, and the time-step parameters were set between 1–10 for optimization. Early stopping method was employed to avoid overfitting by terminating training when the training loss did not fall within 50 epochs.

Figure 13 and Figure 14 show the forecasted MAE results using RNN and LSTM with different numbers of hidden layers for the two case ships, respectively. For $R_M$ and $R_B$ of the passenger ro-ro ship, optimal performances was obtained under hidden layer parameters of (100,80). Furthermore, RNN yielded its best performance at 8 time step, whereas LSTM produced its best performance at 4 time step. For the ballast and laden states of the LPG carrier, the optimal hidden layer parameters corresponding to the RNN were (100,80), whereas those corresponding to LSTM were (100,80,40). Likewise, the best performance time step parameters of the RNN and LSTM models were 7 and 3, respectively.

For the ELM model, the parameter that had the greatest influence on model performance was the number of hidden layer nodes, which was adjusted in a range between 10 and 100 for optimization [17]. As shown in Figure 15a, optimal performance was achieved for the passenger ro-ro ship with 80 and 90 hidden layer nodes on $R_M$ and $R_B$, respectively. For the LPG carrier, performance under ballast and laden states was optimized with 70 and 40 hidden layer nodes, respectively. The best parameters for each dataset were subsequently used to adjust the time step from 1 to 10. As shown in Figure 15b, the optimal time step parameters of $R_M$, $R_B$, ballast and laden were 2, 2, 5 and 8, respectively.

In the BLS forecast model, the most significant parameters are the number of feature nodes N1, number of feature windows N2, and number of enhanced nodes N3. Therefore, a grid search method was applied to traverse these parameters. Specifically, N1 was first optimized with N2, N3 to 10. To optimize N2, N1 was subsequently set to the optimal parameter while N3 remained at 10. Finally, both N1 and N2 was subsequently set to the optimal parameter while N3 remained at 10. Finally, both N1 and N2 were set to their respective optimal values to optimize N3. The results are shown in Figure 16.

Figure 16a shows that the best performance parameters of the BLS model on $R_M$ and $R_B$ were (N1 = 13, N2 = 19, N3 = 16) and (N1 = 3, N2 = 7, N3 = 3), respectively. Likewise, Figure 16b indicates that the best performance parameters of the BLS model under the ballast and laden states were (N1 = 6, N2 = 5, N3 = 2) and (N1 = 19, N2 = 9, N3 = 19), respectively. The optimal parameters for each dataset were used to adjust the time step from 1 to 10. As shown in Figure 16c, the optimal time step parameters of $R_M$, $R_B$, ballast, and laden were 2, 4, 2 and 6, respectively.

4.3.2. Discussion of Passenger Ro-Ro Ship

The figures in the previous section reflect the MAE changes of each fuel consumption forecast model on the four datasets during the parameter optimization. It is still necessary to compare the test sets accuracy obtained by the models using these optimized parameters. Two evaluation indicators—MAE and time—were considered in the comparative analysis.

For passenger ro-ro ships, the comparative experimental results obtained by each model with optimized hyperparameters are listed in

Table 6. Comparison of various fuel consumption forecast models (passenger ro-ro ship).

Dataset	Forecast Model	MAE	Time(s)
$R_M$	ARIMAX	0.0343	70.0374
	SVR	0.0312	2.9718
	RNN	0.0152	100.3603
	LSTM	0.0149	174.8111
	ELM	0.0173	1.3166
	BLS	0.014	1.2559
$R_B$	ARIMAX	0.1165	41.6143
	SVR	0.0265	0.1039
	RNN	0.0174	41.9314
	LSTM	0.0148	58.6204
	ELM	0.0149	0.4194
	BLS	0.0115	0.0967

. It is apparent that the BLS model obtained the highest accuracy for both $R_M$ and $R_B$, with the MAE reaching 0.0140 and 0.0115, respectively. These figures represent a decrease by 59.42% and 71.03%, respectively, compared with the results obtained under default parameters. Furthermore, the MAE of BLS on $R_M$ was 59.18%, 55.13%, 7.89%, 6.04%, and 19.08% lower than those of ARIMAX, SVR, RNN, LSTM and ELM, respectively. For $R_B$, the MAE likewise decreased by 90.12%, 56.60%, 33.91%, 22.29%, and 22.82%, respectively. It can be concluded that BLS yields a significant improvement over the baseline models in forecast performance for the passenger ro-ro ship. In terms of time consumption, the result obtained by BLS on $R_M$ is better than that of ELM, and approximately half that of SVR. Meanwhile, BLS has absolute advantage over the ARIMAX, RNN and LSTM models, with approximately 100-fold decreases in time consumption. On $R_B$, BLS is faster than SVR, approximately four times as fast as ELM, and hundreds of times faster than ARIMAX, RNN and LSTM. Meanwhile, BLS has absolute advantage over the ARIMAX, RNN, and LSTM models, with approximately 100-fold decreases in time consumption. On $R_B$, BLS is faster than SVR, approximately four times as fast as ELM, and hundreds of times faster than ARIMAX, RNN, and LSTM.

4.3.3. Discussion of the Liquefied Petroleum Gas Carrier

The following section discusses the impact of sea conditions on the fuel consumption forecast for the LPG carrier. Data corresponding to the LPG carrier were allocated between a dataset considering sea condition ($C_1$), and a dataset without considering sea condition ($C_2$) under the two loading situations. $C_1$ comprises the feature variables listed in Table 4, whereas $C_2$ omitted the SWH, WD, WP, SD and SCS variables. Under the ballast and laden conditions, $C_1$ and $C_2$ were forecasted using six models: ARIMAX, SVR, RNN, LSTM, ELM, and BLS. The experimental results are listed in

Table 7. Comparison of various fuel consumption forecast models (LPG carrier).

Dataset	Forecast Model	MAE	Time(s)
Ballast-$C_1$	ARIMAX	0.1295	28.1713
	SVR	0.0438	0.2886
	RNN	0.0248	65.7348
	LSTM	0.0172	111.9017
	ELM	0.0213	0.4667
	BLS	0.0108	0.1716
Ballast-$C_2$	ARIMAX	0.2945	25.0112
	SVR	0.0728	0.2398
	RNN	0.0356	65.737
	LSTM	0.0221	112.4083
	ELM	0.0259	0.4633
	BLS	0.015	0.1574
Laden-$C_1$	ARIMAX	0.2722	15.0424
	SVR	0.0415	2.3295
	RNN	0.0401	67.2778
	LSTM	0.0255	88.4812
	ELM	0.0279	0.4832
	BLS	0.0142	0.4638
Laden-$C_2$	ARIMAX	0.7434	13.0898
	SVR	0.0446	1.3082
	RNN	0.0497	54.3062
	LSTM	0.027	89.382
	ELM	0.041	0.449
	BLS	0.0211	0.434

.

For all datasets in Table 7, it is apparent that the BLS model achieved the highest forecast accuracy, with the MAE of $C_1$ under the ballast and laden conditions decreasing by 60.29% and 81.29%, respectively, compared with that under the default parameters. Moreover, the MAE of BLS for $C_1$ and $C_2$ under ballast and laden can reach 0.0108, 0.0150, 0.0142 and 0.0211, respectively. The MAE of $C_1$ under the ballast state decreased by 28.00% compared with that of $C_2$, while the MAE of $C_1$ under the laden state was 32.70% lower than that of $C_2$. In terms of time consumption, BLS performed better than SVR under the ballast state, and is four times faster than ELM. Furthermore, BLS performs better than ELM under laden state, and is five times and three times faster than SVR of $C_1$ and $C_2$, respectively. In addition, BLS performs significantly better than ARIMAX, RNN and LSTM, with 10-fold to 100-fold improvements in speed for all datasets.

In summary, the BLS model obtained the optimal practical fuel consumption forecasting results for both ships. Furthermore, the forecast performance of all six models was better under $C_1$ than under $C_2$. Therefore, the sea condition factors must be considered to achieve optimal forecast performance.

5. Conclusions

The key to maximizing the energy efficiency and minimizing the consumption costs associated with shipping is the accurate forecasting of fuel consumption. Currently, the accuracy and time performance of existing ship fuel consumption forecast models require improvement. In view of this problem, this paper presents a study comparing BLS with the ARIMAX, SVR, RNN LSTM and ELM models with actual navigation data obtained from two types of ships.

For the LPG carrier, data were further allocated among the $C_1$ and $C_2$ datasets to examine the significance of sea conditions on fuel consumption. Finally, conclusions were drawn based on an analysis of experimental results. Specifically, the BLS model achieved optimal forecasting performance for both ships. These results correlate with the unique characteristics of BLS, which includes its simple structure and short training time, making it suitable for applications in real-time navigation scenarios. Furthermore, results on the LPG carrier data indicate that sea conditions must be accounted for to maximize forecast performance.

The optimal time-series forecasting model based on a novel regression analysis using BLS in this study can be employed for the real-time monitoring of fuel consumption, thereby enabling energy- and environmentally friendly shipping conditions. This is of great significance in the problems of route, trim and ship speed optimization. Furthermore, additional sailing data corresponding to different ships will be collected in the future to verify the model’s generalizability.