A Comparative Study to Estimate Fuel Consumption: A Simplified Physical Approach against a Data-Driven Model

Abstract

Two methods were compared to predict a ship’s fuel consumption: the simplified naval architecture method (SNAM) and the deep neural network (DNN) method. The SNAM relied on limited operational data and employed a simplified technique to estimate a ship’s required power by determining its resistance in calm water. Here, the Holtrop–Mennen technique obtained hydrostatic information for each selected voyage, the added resistance in the encountered natural seaways, and the brake power required for each scenario. Additional characteristics, such as efficiency factors, were derived from literature surveys and from assumed working hypotheses. The DNN method comprised multiple fully connected layers with the nonlinear activation function rectified linear unit (ReLU). This machine-learning-based method was trained on more than 12,000 sample voyages, and the tested data were validated against realistic operational data. Our results demonstrated that, for some ship topologies (general cargo and containerships), the physical models predicted more accurately the realistic data than the machine learning approach despite the lack of relevant operational parameters. Nevertheless, the DNN method was generally capable of yielding reasonably accurate predictions of fuel consumption for oil tankers, bulk carriers, and RoRo ships.

1. Introduction

1.1. The Problem of Reducing Ship Fuel Consumption

In the shipping industry, assessing a ship’s fuel consumption has attained paramount importance. This is because various aspects need to be considered, for instance, environmental protection, energy management in maritime operations, and sustainability of concepts. To reduce fuel consumption, all stakeholders are working on defining and understanding which principal parameters contribute most to increasing fuel consumption during a voyage. Consequently, multiple methods have been proposed in the past as well as recently to predict fuel consumption. Indeed, the International Maritime Organization (IMO) [1] developed the Ship Energy Efficiency Management Plan (SEEMP Part III). Coming into force in 2023, this mandatory measure documents how ships of 5000 gross tonnage (GT) and above achieve their carbon intensity indicator (CII) targets.

Ship weather routing is the process of choosing the optimum route while considering weather forecasts, a ship’s unique characteristics, and sea conditions along the intended voyage [2]. Weather routing has a high potential for efficiency savings on specific voyages, inducing fuel reduction up to 3% apart from time savings [3]. By simulating a ship’s operational profile over one month, Takashima [4] demonstrated that the so-called minimum fuel route (MFR) model, based on forecasted environmental data and the propulsive performance of the ship in the encountered seaways, enables saving relatively large amounts of fuel. Specifically, the average savings for the two ships he examined amounted to 2.4 and 18.4%, and the maximum savings were 6.9 and 22.1%, respectively.

The fuel consumption can be considered approximately proportional to the third power of the sailing speed. However, this is not always valid under all operational conditions, especially in slow steaming [5]. Consequently, measures to optimize ship speed have been widely considered by many operators, even in compliance with the required Energy Efficiency Existing Ship Index (EEXI) of IMO (International Maritime Organization) resolution MEPC.334(76) [6]. With this in mind, Kim et al. [7] studied multiple time windows for each port of call and formulated the problem as a nonlinear mixed integer program, with the objective of minimizing the total fuel consumption. Ronen [8] also conducted groundbreaking studies on identifying the ideal ship speed by weighing the trade-off between fuel savings from slow steaming and income losses caused by the extended journey.

Hull roughness is another parameter affecting a ship’s fuel consumption as it increases ship resistance and, consequently, the power required to maintain speed. Adland [9] confirmed the effects of hull cleaning on fuel usage in terms of hull maintenance. His research led to three main conclusions. First, regular hull cleaning significantly lowers daily fuel consumption; second, dry-docking reduces fuel consumption more dramatically and more significantly than underwater hull cleaning (about 17 versus 9%); third, the energy efficiency benefit from hull cleaning is greater when the ship is loaded as opposed to in ballast. In addition to reducing the hull’s roughness, cleaning and polishing the propeller may significantly increase fuel efficiency [1], and this process is valid for any kind of vessel. A ship operator can induce this either when the ship is in dry dock or while it is berthed and loading cargo. The ship’s reduced power loss will increase propeller efficiency, leading to lower fuel costs for the ship. Wilkinson [10] demonstrated that frequent polishing to lessen propeller roughness reduced fuel consumption and that the fuel savings alone more than covered the costs.

Ballast and trim optimizations are also mean to enhance fuel saving. Braidotti et al. [11] proposed a reliable technique for an optimal ballast water distribution to ensure minimizing fuel consumption. This may be advantageous, especially considering the impact of added resistance in waves. This technique was applied on a crude-oil carrier of 179,500 ton and sailing in ballast condition at the service speed of 13 kn. The result obtained for such optimized conditions yielded a fuel saving of 0.26 t/h compared to a standard ballast condition.

The installation of energy saving devices (ESDs) may also reduce fuel consumption, largely by improving the overall propulsive efficiency. A representative example of this kind of device (often used to improve the EEXI value) is the so-called Mewis Duct (MD), developed in 2009. Installing an MD may reduce not only the rotational losses in the fins’ slipstream, but also the wake losses in the propellers’ duct. Mewis [12] carried out over 35 model tests to demonstrate this. He obtained an average 6.5% power reduction in the ballast and laden conditions.

1.2. Studies on Ship Fuel Consumption

Numerous studies have been performed to assess fuel consumption of ships. Fan et al. [13], for instance, carried out a literature review of analytical models capable of determining ship fuel consumption. They subdivided the current ship fuel consumption models into three categories: so-called white, black, and gray boxes, which mainly depend on the nature of the algorithm applied (i.e., artificial intelligence or physical model).

Kim et al. [14] proposed the improved method of ISO 15016:2015 to estimate a ship’s fuel oil consumption, which was based on the older method of ISO 15016:2002. They demonstrated the increased accuracy of the improved method by comparing predictions from a gray box model based on operational data.

Applying a regression analysis technique often yielded good results. Bialystocki and Konovessis [15] proposed an operational approach based on polynomial regressions capable of estimating fuel consumption and speed based on a variety of influential factors, such as ship draft, displacement, wind speed and direction, and hull and propeller roughness.

Wang et al. [16] and Bocchetti et al. [17] used this kind of mathematical technique with the intention of considering historical fuel consumption events and linking these to operational and environmental factors.

Machine learning approaches to predict fuel consumption of ships are widely applied. Due to the advances of machine learning techniques, they have been increasingly used to predict a ship’s power. Jeon et al. [18] proposed a regression model employing artificial neural networks (ANNs) to better predict the main engine’s fuel consumption, thereby demonstrating their advantage over the polynomial regression and support vector machine. Tarelko [19] developed an ANN model to predict fuel consumption and the associated travel time to a destination for selected outputs, such as the ship’s driveline shaft speed and the pitch of its propeller, all of which are parameters specified by the ship operator. Petersen et al. [20], by combining ship service metrics (rotations per minute, average draft, trim, cargo capacity) and wind and wave influence as input parameters, created an ANN to estimate fuel usage. To evaluate fuel consumption, Luan Thanh Le et al. [21] described its architecture via a multilayer perceptron (MLP), i.e., via a fully connected feed-forward artificial network (ANN). They also examined the effectiveness of two ANN models and two multiple-regression models, using operational data obtained from 100 to 143 containerships, to estimate fuel consumption for five differently grouped containership sizes. In addition, Kim et al. [22] presented statistical and domain-knowledge methods to choose the appropriate input variables for their ANN models, which estimate fuel usage using in-service data gathered from a 13,000 TEU (Twenty-foot Equivalent Unit) containership. Their technique presents a practical application to avoid overfitting and multi-collinear treatment.

The purpose of our study was to compare using our proposed SNAM (simplified naval architecture method) and the deep neural network (DNN) method to predict the fuel consumption for several ship topologies. The physically based SNAM is a fast method that relies on a limited number of operational parameters, whereas the DNN method applies a data-driven model. For us, it was of paramount importance to supply evidence that our proposed physically based model relies on commonly known working hypotheses and simplifying assumptions, because often sufficient in-service data are lacking to reliably predict fuel consumption.

As with every white box model, most of the parameters are generally established in advance. This means that the values inside the data-driven model cannot be adjusted during a sample voyage [13]. Additionally, parameters to model the resistance components are typically separate and do not account for possible interactions of system components, and this leads to approximate modeling for this kind of physically based approach [23]. Furthermore, aspects linked to various effects of environmental conditions (wind and waves) and ship age depended on simplifying assumptions.

Esmailian [24] performed a similar study by comparing the power prediction obtained with an ANN model and a numerical model, albeit only for one general cargo ship en route between Italy and Norway.

2. The SNAM

The SNAM (simplified naval architecture method) is a fast approach capable of approximating a ship’s fuel consumption via a limited number of parameters (Figure 1). The main steps characterizing this process are the following:

(1)

Determination of calm water resistance via the ship’s approximate draft and the available information regarding the cargo being transported.

(2)

Determination of the approximated added resistance caused by waves and wind.

(3)

Approximation of the efficiency factors representing the overall propulsion chain.

(4)

Determination of the brake power via the total resistance obtained from step 1, the efficiency factor obtained from step 3, and the ship’s voyage speed.

(5)

Estimation of fuel consumption via the product of the hours underway, the brake power obtained from step 4, and the approximated specific fuel oil consumption (SFOC) based on the ship’s principal particulars.

Based on fundamental principles of naval architecture, the developed SNAM employs a numerical technique or regression-based formula to first calculate a ship’s hydrostatic characteristics in calm water by applying the well-established method of Holtrop and Mennen [25]. Their technique represents a statistical approach, resolved through a regression analysis of randomly chosen model test experiments and full-scale data available at the Netherlands Ship Model Basin (MARIN).

At MARIN, a variety of calm water model resistance tests have been carried out. An example is the widely applied method of Guldhammer and Harvald [26], applicable to single- and twin-screw ships. Another example is the lesser used technique of Hollenbach [27], established for resistance and powering estimates of single- and twin-screw ships. This technique appears to yield more trustworthy predictions, particularly for twin-screw ships.

For our purpose, we subdivided a ship’s total resistance R_total into the following components:

$$R_{total}=R_F\left(1+k_1\right)+R_{APP}+R_W+R_B+R_{TR}+R_A+R_{wind}$$

(1)

where:

• R_F is the frictional resistance;
• k₁ is the form factor describing the viscous resistance of the hull;
• R_APP is the appendage resistance;
• Rw is the added resistance in waves;
• R_B is the additional pressure resistance of a bulbous bow;
• R_TR is the additional pressure resistance of an immersed transom;
• R_A is the model ship correlation resistance;
• R_wind is the wind resistance.

According to Perez [28], considering the added resistance in waves may increase overall total resistance between 15 and 30% compared to calm water resistance [28]. Several methods do exist to more accurately calculate this added resistance. The ISO Guidelines 15016:2015 [29] recommend determining this added resistance in waves by using one of the three methods documented by Kim and Roh [14], namely, STAWAVE-1, STAWAVE-2, and their theoretical method, or by performing seakeeping model tests.

Among these, STAWAVE-1 is the simplest method. It is mainly applied for predictions under head sea conditions, where pitch and heave motions are small. However, it does yield acceptable predictions for ships under relatively moderate weather conditions. The added resistance in waves R_AWL is here expressed as follows:

$$R_{AWL}=\frac{\rho ·g·H_{\frac{W1}{3}}{}^2}{16}B\sqrt{\frac{B}{L_{BWL}}}$$

(2)

where:

• B is the beam of the ship;
• ρ is the water density;
• $H_{W1/3}$ is the significant wave height;
• $L_{BWL}$ is the length of the bow on the waterline at 95% of B.

To account for the effects of wind, we increased the total resistance for oil tankers, bulk carriers, general cargo, and RoRo vessels by 2%, whereas for other ship topologies, such as containerships with containers stacked on deck, we increased the total resistance by 10% [30].

The following operational input parameters are considered for each specific voyage:

• Hours underway;
• Cargo weight transported;
• Total voyage distance.

The flowchart in Figure 1 presents an overview of the steps involved in the SNAM. This method takes into account a ship’s main particulars, its maximum deadweight DWT at summer load conditions, and the corresponding draft $T_D$. First, the hull’s wetted surface S_wet is calculated as follows:

$$S_{wet}=L_{bp}·\left(2T_{DD}+B\right)·\sqrt{C_M}·\left(0.453+0.4425C_B-0.2862C_B-\frac{0.003467B}{T_{DD}}+0.3696·C_{WP}\right)+2.38\frac{A_{BT}}{C_B}$$

(3)

where:

• L_bp is the length between perpendiculars;
• C_M is the midship section coefficient for the sampled voyage;
• C_B is the block coefficient for the sampled voyage;
• C_WP is the water plane area coefficient for the sampled voyage;
• T_DD is the draft for the sampled voyage;
• A_BT is the transverse sectional area of the bulb at the position where the calm water surface intersects the stem.

2.1. Draft Determination

As the ship’s draft information is generally not available at first, the following simple formula [31] was applied to determine it, T_b at ballast:

$$T_b=\left(D_P+e+0.02L\right)·0.5$$

(4)

where:

• D_P is the assumed propeller diameter, approximated as 0.65 of design draft;
• e is the distance of lower extremity of the propeller blades to the base;
• L is the ship’s length.

The corresponding block coefficient $C_{Bb}$ is determined as follows:

$$C_{Bb}=C_{BD}-C\frac{T_D-T_b}{T_D}·\left(1-C_{BD}\right)$$

(5)

where:

• C = 0.4;
• C_BD is the block coefficient at design draft;
• T_D is the design draft.

The draft T_L and block coefficient $C_{BL}$ under laden conditions during the voyage are calculated as follows:

$$T_L=T_D·{\left(\frac{∆}{{∆}_O}\right)}^{\left(\frac{C_{BD}}{C_{WD}}\right)}$$

(6)

$$C_{BL}=\frac{C_{Bb}-C_{BD}}{T_b-T_D}·T_L+\frac{C_{BD}·T_{BL}-C_{Bb}·T_D}{T_b-T_D}$$

(7)

where:

• ∆ is the approximated displacement during the voyage;
• C_WD is the water plane area coefficient at design draft;
• ∆_O is the approximated displacement of design draft.

2.2. Brake Power

Generally, fuel consumption depends on the fuel’s calorific value, the powering system’s effectiveness, and the main engine’s required power output [32]. Typically, the specific fuel oil consumption (SFOC) is expressed as a function of the main engine’s revolutions per minute (RPM). However, we considered the RPM to be constant.

Table 1. SFOC according to the IMO Greenhouse Gas Study 2009.

Engine Age	Power Output above 15,000 kW	Power Output between 15,000 and 5000 kW
Before 1983	205	215
1984–2000	185	195
2001–2007	175	185

lists the SFOC according to the Second IMO Greenhouse Gas Study 2009 [33] for ships built before 1983, between 1984 and 2000, and between 2001 and 2007.

Although system effectiveness varies with the main engine’s power output, we estimated the assumed break power P_B from the efficiency factors of [30,34,35].

Table 2. Assumed efficiency factors.

Efficiency Factor	Assumed Value	Literature
${\mathsf{\eta }}_{\mathrm{R}}$ (relative rotative)	(single screw) 1.00–1.07	[34]
	(twin screw) 0.98	[30]
${\mathsf{\eta }}_{\mathrm{H}}$ (hull)	1.10–1.30	[30]
${\mathsf{\eta }}_{\mathrm{S}}$ (shaft)	0.95–0.99	[30]
${\mathsf{\eta }}_{\mathrm{O}}$ (propeller—open water)	0.55–0.70	[35]

lists the range values which might be applicable. In more detail, we considered the lower limit value for each factor to simulate a more degraded propulsion system.

Specifying ship speed as $v_s$ and the total resistance as $R_{total}$, we approximated the final brake power P_B as follows:

$$P_B=\frac{R_{total}·v_s}{{\mathsf{\eta }}_{tot}·0.85}$$

(8)

3. The DNN Approach

Among machine learning models, artificial neural networks (ANNs) are most prevalent.

These models include a process of learning and training by using input parameters and understanding their initial/output values. The weight of each input parameter is then changed to update the connections between input and output. A major drawback of most of these models is the difficulty of identifying their internal operation. However, methods to specify the internal process do exist, and these are known as explainable artificial intelligence (XAI) techniques. Such techniques are already being used in the field of medicine, where they help support human decisions to triage and treat patients across a growing number of health care issues.

Most models using ANNs are called post hoc methods because they work after the neural network has learned how to achieve the proposed task. Various methods were developed, and they provided a simple and clear, but logically explainable model describing their internal functions [36]. The typical model consists of layers of input data, one or more hidden layers of artificial neurons, and one or more output layers [37]. One of the most frequently used models relies on reinitializing a fully connected layer. The associated output is then described by a linear combination of an adjustable input and a bias expressed as follows:

$$y=x{W}^T+b$$

(9)

where y is the output vector, x is the input vector, b is the bias vector, and W is the weight matrix comprising the input’s weighing factors; T represents transform. Our neural network comprised a feed-forward network of two layers. One layer consisted of 8192 and the other layer of 4096 activation units.

According to the universal approximation theorem [38], a neural network with a single hidden layer and its units (corresponding to the number of neurons in the weight matrix), is generally activated by a nonlinear activation function, thus sending the resulting output to the next hidden layer or to the node output values. This procedure enables introducing nonlinearities in the neural network. This is a fundamental operation because, usually, these data are nonlinearly separable, and this helps to obtain a projection of the transformed data in a nonlinear space that usually is higher than the previous one. In addition, this operation separates the data in the new space, where they have a higher probability to be linearly separable. It allows one to more easily perform certain operations, such as classifying data or carrying out a regression analysis.

In the hidden and output layers, various activation functions may be used, such as a hyperbolic tangent, a sigmoid, or a rectified linear unit (ReLU) and its evolutions. As an alternative to the ReLU, we applied the nonlinear swish activation function [39]:

$$f\left(x\right)=x·sigmoid\left(\beta x\right)=\frac{x}{1+e^{-\beta x}}$$

(10)

This function, derived from the sigmoid-weighted linear units, is a continuous smooth function that permits a few negligibly small weights to be propagated in the network. The use of an activation function aided the projection of linear data obtained from the fully connected layer onto a nonlinear space.

One of the main issues in training a DNN approach was to determine the optimized neuron weight values. For this, we used a back propagation technique to minimize backward propagation of errors. Specifically, this process updated and optimized the weights $w_n{}^{*}$ via a descending gradient to determine the minimum of the associated error function E as follows:

$$w^{*}{}_n=w_n-a\left(\frac{\partial E}{\partial w_n}\right)$$

(11)

To determine an updated (optimized) weight $w_n{}^{*}$, the partial derivative of the error function E with respect to $w_n$, multiplied by a specified learning rate a, is subtracted from the current weight ${\mathrm{w}}_{\mathrm{n}}$. These more representative weights enabled the neural network to comprehend the training data more quickly.

We regressed the ship fuel consumption data from the earlier nonlinear projections in the output layer. We employed the learned weights’ linear combination, resulting from deactivating the output layer, as an estimating method. We then developed the network using Google Colab and TensorFlow [40], a deep learning computational toolkit, and trained it on an Nvidia Tesla K80 GPU (graphics processing unit). For this, we selected the algorithm optimizer ADAM [41], whereby we lowered its learning rate of 0.001 to obtain an exponential decay rate [42] of 0.96 over every 10 epochs.

The DNN approach was chosen because compared with ANN, this approach consisted of an increased number of hidden layers located between the input and the output layers, thereby possibly improving its learning ability. Thus, it was specifically designed as an optimization strategy, such ADAM.

Furthermore, compared to other machine learning techniques, the typical DNN method rapidly produced results that were less prone to overfitting, and the computing times were shorter compared to those of the SVM (Support Vector Machine), which was inadequate for a large data set [13]. The DNN as a subset of an ANN had the ability to handle not only noise, but also highly accurate continuous and discrete data [12].

For a total number of 12,563 sample voyages,

Table 3. Ship topologies considered to generate the training set and the associated number of ships and voyages.

Ship Topology	Number of Ships	Number of Voyages
General cargo ship	61	433
Oil tanker	52	324
Containership	300	10,805
RoRo ship	2	236
Bulk carrier	147	765

lists the five ship topologies chosen to generate the training data set. Table 3 also lists the associated number of ships as well as the number of voyages for each ship topology. The total amount of voyage samples in European waters was selected from one of the authors’ databases and from a publicly available emissions report portal (Thetis) to ensure reaching a minimum threshold number of 10,000 samples needed for a consistent data set.

Figure 2 displays the network architecture of our DNN approach. We chose the following input variables to train the network for each scenario:

• Gross tonnage (GT);
• Maximum deadweight (Dwt);
• Efficiency value (EIV);
• Hours underway (H);
• Cargo transported (Cgo);
• Ship type (ST);
• Distance (Dis);
• Engine total output power (ETP);
• Draft (T).

The draft T of each voyage was estimated by considering the same approach used for the SNAM, thus providing an additional physical input parameter to the DNN approach for the determination of fuel consumption (FC).

Initially, we employed sigmoid functions; however, we replaced them by more sophisticated nonlinear functions.

We also extended our network architecture by adding additional layers to obtain the most feasible approximation. In this way, our neural networks attained their alleged “depth.”

Although several parameters could have been applied to understand the performance of the DNN approach, we used the regression objective functions SmoothL1 and SmoothL2 [43]. These functions were minimized while learning from the data set. They yielded the two standard loss functions’ mean squared error (MSE) and mean absolute error (MAE), expressed as follows:

$$MSE=\frac{1}{\mathrm{N}}·{\displaystyle \sum }_{i=1}^N{(y_i-\overbrace{y_i})}^2$$

(12)

$$MAE=\frac{1}{\mathrm{N}}·{\displaystyle \sum }_{i=1}^N|y_i-\overbrace{y_i}|$$

(13)

where $y_i$ is fuel consumption from the data set, $\overbrace{y_i}$ is the estimated fuel consumption using the neural network, and N is the number of samples in the data set. Fewer oscillations characterized the MSE during the updates when values were small, while steady gradients, typical for large values, characterized the MAE, which introduced a certain scarcity in the model [44]. After completing the training, the neural network was tested with completely new scenarios not previously known to the trained neural network. Therefore, we used our DNN approach to predict fuel consumption for a fewer number of selected test scenarios. For each ship topology,

Table 4. Ship topologies selected to generate the training set for the DNN approach and the associated number of ships and voyages.

Ship Topology	Number of Ships	Number of Voyages
General cargo ship	14	100
Oil tanker	9	100
Containership	2	100
RoRo ship	4	100
Bulk carrier	9	70

lists these fewer number of ships as well as the associated number of ships and voyages.

4. Results and Discussion

Figure 3 plots predicted values of fuel consumption obtained with both methods together with the actually reported fuel consumptions. The vertical axis represents fuel consumption expressed in tons; the horizontal axis, the number of sample voyages for the five ship topologies listed in Table 4. A continuous blue line connects fuel consumption obtained from the SNAM; a dashed red line, fuel consumption obtained from the DNN approach; a dashed black line, the actually reported fuel consumption.

Figure 3a plots predictions for bulk carriers. As seen at voyage sample numbers 6, 30, 44, and 57, the SNAM yielded significantly overestimated values compared to the corresponding reported predictions, whereas the DNN approach predicted values that generally compared favorably to reported values.

Beginning at sample number 58, predictions from the DNN approach were nearly identical to the reported values, although at sample numbers 63 and 69, they slightly exceeded the reported values.

Figure 3b plots predictions for oil tankers. Here, the DNN approach also delivered predictions that were closer to reported values than those from the SNAM.

At sample numbers 43 and 77, the SNAM method strongly overestimated the reported values. This could have been due to false draft estimates.

Figure 3c plots predictions for containerships. At sample numbers up to 15, the SNAM and the DNN approach yielded predictions that generally compared favorably to reported values. However, at sample numbers between 16 and 60, only the SNAM predictions were close to reported values, whereas the DNN approach had difficulties predicting fuel consumption, as it yielded overestimated values. Starting at sample number 61, again only the SNAM predictions compared favorably to reported values, whereas the DNN approach obtained underestimated values. Thus, the trend of DNN predictions for the containerships differed from the trend of DNN predictions for bulk carriers and oil tankers. For containerships, the SNAM provided more reliable predictions.

Figure 3d plots predictions for general cargo ships. In general, both methods provided predictions that compared favorably to reported values. Only at some sample numbers, notably 23 and 50, did the DNN approach yield predictions slightly less than those from the SNAM.

Figure 3e plots predictions for RoRo ships. As seen, for sample voyages with fuel consumption below 200 tons, both methods yielded similar predictions, and they both compared favorably to reported fuel consumptions. Relevant deviations occurred at sample numbers 7 and 22.

Here, the SNAM overestimated and the DNN approach underestimated the reported fuel consumption, whereas at sample number 24, both methods drastically underestimated the reported fuel consumption.

Figure 4 plots the so-called loss value versus the number of epochs of the network’s training and test processes needed for our developed DNN approach. The solid curve refers to the network’s training process; the dashed curve, to the network’s test process. These fuel losses represented the difference between predicted and reported fuel consumptions. The DNN approach was considered to be an optimization process. Therefore, the better the predicted fuel consumption agrees with the reported fuel consumption, the closer to zero will be the fuel loss.

The horizontal axis represents the number of epochs, which is an important parameter for the training process of the model as it indicates the number of times the learning algorithm performed its computations using the provided input training data set [44]. As seen, as the number of epochs increased, the loss value decreased until the desired output was obtained. To obtain a minimum loss value, the network had to be trained several times using the same amount of data. Generally, our network was trained over about 45 epochs; however, to validate our approach, we conducted training over additional epochs to obtain the lowest possible loss value. The trend of both curves demonstrates that the learning process converged. Hence, no overfitting and underfitting was necessary in the process. The validation process’s minimum training loss of 0.0132 occurred at epoch number 44, whereas the minimum loss of 0.0143 for the entire training phase test occurred at epoch number 22. This indicated that the network learned to estimate fuel consumption while considering the proposed specifications.

Additionally, these low loss values demonstrated that the process was able to deliver reliable predictions.

Table 5. Normalized RMSEs for the SNAM and the DNN method and the associated loss values.

Ship Topology	Normalized RMSE (SNAM)	Normalized RMSE (DNN Method)	Loss Value
Bulk carrier	0.062	0.032	0.0067059
Oil tanker	0.105	0.050	0.0373286
Containership	0.086	0.267	0.0590038
General cargo ship	0.062	0.082	0.0212367
RoRo ship	0.061	0.052	0.0060399

summarizes results for the five considered ship topologies in terms of normalized root mean squared errors (RMSEs) and loss values.

Compared to the DNN approach, the SNAM predicted fuel consumptions for the general cargo ship and containership topologies that agreed more closely with reported fuel consumptions. As seen in Table 5, for these two ship topologies, the relatively higher loss values turned out to be 0.0212 and 0.0590, respectively. Despite the extensive amount of data used for training the containership typology, the network’s predictions were not as satisfactory as those for the other ship topologies.

Once the network’s training process was completed, the computer time to determine fuel consumption for the new scenarios (wild data) was minimal. Indeed, such predictions were obtained within a few milliseconds.

5. Conclusions

The simplified method SNAM, based on principles of naval architecture, and the deep neural network DNN method, based on a machine learning approach, were developed to predict a ship’s fuel consumption. The initially required source data of input samples were gathered for more than 560 ships of five different typologies: bulk carriers, oil tankers, containerships, general cargo ships, and RoRo ships.

Comparing predicted fuel consumption from both methods demonstrated that, generally, the DNN approach delivered predictions that agreed more favorably with actually reported fuel consumptions. The corresponding loss values obtained from the DNN approach were generally satisfying. Nevertheless, further investigations would have been necessary for some scenarios where fuel consumptions were consistently overestimated. On the other hand, for the general cargo ship and the containership topologies, predictions from the SNAM compared more favorably to reported fuel consumptions than predictions form the DNN approach. This was likely due to the limited amount of operational input data for the DNN approach.

In addition, the SNAM model did not estimate with the same level of accuracy all vessels’ scenarios. Therefore, for determined vessel topologies (oil tanker, bulk carrier) more specific data such as RPM, SFOC, and propeller details could have yielded additional valuable information to the model.

Although, from a practical point of view, the relatively simple SNAM provided reliable estimates of a ship’s fuel consumption, predictions from the DNN approach were based on a sounder theoretical basis.