Applying artificial neural networks for modelling ship speed and fuel consumption

Ship owners and operators of different types of ships are interested in decreasing the costs related to the effectiveness of their operation. These costs are mainly associated with fuel consumption and operational losses, e.g. excessive travel time to the destination. In the case of ships equipped with a combustion engine (CE) coupled to a controllable pitch propeller (CPP), effectively managing both the fuel consumption and travel time to the destination is related to the optimal choice of commanded outputs determining the work of such a driveline system. This system generates thrust to move a ship across the water at the desired speed with different levels of fuel consumption for various combinations of the commanded outputs, namely the driveline shaft speed and the CPP pitch. The optimal combination of speed and pitch depends on several operational conditions and, therefore, must be subjected to dynamic optimization. For this purpose, the most operated ships used speed/pitch ratio controllers. In such a ship driveline system, the commanded torque is controlled to maintain a certain shaft speed.

However, there are ships which are not equipped with this kind of controller and the shaft speed or the CPP pitch ratio is used to control the propeller thrust indirectly. In such cases, due to variable environmental conditions (mainly weather conditions at sea), making decisions about setting the commanded outputs to ensure rational fuel use and the desired ship speed is extraordinarily difficult.

A literature review carried out in [1] showed that there are some methods that could support selection of the commanded outputs for a ship’s propulsion system equipped with the CPP. They are mainly based on models developed by use of polynomial or regression equations. As a rule, algorithms for solving of such equations in both types of models are too complex. For this reason, many assumptions are used that simplify these models and significantly decrease usefulness of these methods. Moreover, their disadvantages are the difficulties with estimating sea conditions. Nonetheless, knowledge of these conditions is essential to selecting the appropriate propeller pitches and engine rotational speeds.

For this reason, it would be useful to develop computer-based tools to support such decisions. The base of such tools should be mathematical models connecting fuel consumption and travel time to the destination with the commanded outputs and operational conditions subjected to the propulsion thrust. An analysis of bibliographic references concerning methods of setting the commanded outputs of this kind of ship drivelines was presented by Rudzki [1], who showed that the existing methods do not include models that allow formalizing the required heuristic knowledge. In our opinion, artificial neural network (ANN) techniques can be used for obtaining such models and can be used for better predicting both the fuel consumption and travel time to the destination for the selected commanded outputs and the observed parameters of operational conditions owing to their high accuracy, adequacy and quite promising applications in practice.

In this case, an important advantage of the ANN method is that it does not require mathematical relations of the input data and output data. Moreover, this technique allows to solve our problem that is not very well formulated formally.

This paper deals with the selected issues of developing ANN models combining the mentioned parameters. In our approach, to collect data needed for building ANN models, sea trials were conducted.

Two ANN models are needed to develop a computer-aided system supporting decision-making regarding setting the ship driveline commanded outputs to ensure rational fuel use and the desired ship speed. The first ANN model should connect a fuel consumption process to factors that influence this process, and the second ANN model connects ship speed to factors that influence this speed.

In a mathematical description of these kinds of phenomena, two fundamental approaches are used:

In practice, most developed models are obtained using gray box modelling that combines a partial theoretical structure with data to complete the models.

White box models, also called cause–effect models, deal with the variables impacting the distribution of a phenomenon and describe a physical process. They integrate existing knowledge about processes into a set of relationships (equations) for quantifying those processes. The most commonly used method to develop white box models of dynamic systems is the balance method. In systems where we must deal with physical quantities, balancing is carried out for parameters that are subordinate to the laws of conservation of energy and momentum.

Nevertheless, applying the balance method to modelling both fuel combustion and the ship motion processes to obtain a decision-making model supporting decision-making regarding the commanded outputs of the ship driveline system is practically impossible. This is because the equations describing these processes are so complex that for the given conditions of explicitness, they are not solvable. Moreover, processes running in the considered systems are dependent on many parameters at the same time and they influence the observed phenomenon in varying degrees (for example, meteorological conditions).

With some simplifying assumptions, we could try to bring these equations to more simple forms such as linear, parabolic or hyperbolic equations. Unfortunately, the equations describing the considered processes cannot be subjected to linearization procedures or they are simple enough for presentation in an unsophisticated mathematical form. In addition, analytical solutions of these equations are possible only by using such approximations, so that the results lose their practical utility.

Therefore, we should look for alternative modelling methods. For example, we could build the necessary models using the black box modelling method.

In general, black box models help us in understanding underlying processes. We can receive their descriptions without necessarily analysing or providing their causes. A procedure for setting up this kind of model is as follows:

If the conformity is not acceptable, we should take one or several steps back (we can also conduct additional measurements).

The starting point for using this procedure is to collect appropriate data by carrying out measurements or using historical records. As a rule, we should divide the collected data into those that will be estimated, which is dependent variables, and those that will be used as necessary conditions, which is independent variables.

There are a lot of scientific research papers that applied similar procedures in various areas of science. Most of them are based on new achievements in ANN, for example [2‐6], whereas some of them were applied in maritime industry, for example [7‐9].

In the classical approach for modelling similar phenomena, statistical models are most commonly used. In the field of ocean engineering, interpolating and predicting hull resistance from model experiments and tank testing have traditionally been done using statistical regression equations. For example, such models are used to determine parameters characterizing ship propulsion efficiency in the early stages of its design [10, 11]. Analysing application regression relationships for modelling the selection of optimal drive propulsion parameters was presented by Rudzki and Tarelko [12].

To receive models that could be used in the considered decision-making system, a black box model in the form of multiple regressions was developed [1]. The same data needed for building ANN models were used to build the regression models. These data directly connect both the fuel consumption and the travel time to the destination with the commanded outputs and parameters of operational conditions subjected to the propulsion thrust.

However, the obtained results show some limitations.

In contrast, ANN models enhance the generalizability and extrapolation capability and do not require a priori assumptions of function forms. Therefore, they can be used to develop a computer-aided system supporting decision-making regarding setting the ship driveline commanded outputs to ensure rational fuel use and the desired ship speed.

Applying ANN models for predicting ship fuel consumption or ship speed for various operational conditions is not presented in many publications.

Historical data acquired from ship logs (records of important events in the management, operation, and navigation of a ship) were used in all considered ANN models to predict ship fuel consumption.

In Arslan, Bal Beşikçi and Ölçer [13] and Bal Beşikçi, Arslan, Turan and Ölçer [14], decision support systems employing ANN based on fuel prediction models were developed. These models used operating data called ‘noon data’ obtained from ship noon reports.¹ The input parameters were seven variables, a ship’s speed, the rotational speed of the main engine, mean draft, trim, cargo quantity on board, wind and sea effects, and the output variable of the ANN model was ship fuel consumption. In both studies, neural network models were implemented using the Neural Network Toolbox in MATLAB 2010a. To design and construct the ANN, the data set derived from 233 [14] and 3646 (7 tanker ships) [13] noon reports was used. Initially, 70% of the noon reports were randomly selected for training, and the remaining 30% was used for validation. The modelling method for ANN was based on the backpropagation learning algorithm used in feedforward with one hidden layer. The learning algorithm used in these studies was Levenberg–Marquardt, the activation function was hyperbolic tangent sigmoid transfer functions, and the number of epochs was set to 10,000. Additionally, the performance of the developed ANN model was compared with the multiple regression model [13]. For both training and validation data, the correlation between the actual and predicted fuel consumption for the ANN model was shown to be much higher than for the linear regression model.

Pedersen and Larsen [15] presented a method using ANN to predict propulsive power from theoretical variables influencing ship resistance, such as a ship speed, relative wind speed and direction, air temperature and seawater temperature. Three data input sources were used to train and predict the propulsive power, onboard measured, noon report data and weather and sea state information based on the hindcast approach.² To design the ANN, a data set was derived from 323 samples of the noon reports. It was only trained for 5 and 20 hidden layers as these are the extremes as justified by Pedersen and Larsen. The ANN used noon report data to predict the specific fuel consumption with an accuracy of about 7%. It was noted that this accuracy was obtained using `time' as an input variable; this indicates that it is possible to detect a trend in fuel consumption over time.

Du and Meng [16, 17] proposed a model that complies with the fundamentals of ship propulsion and can precisely quantify the synergetic influence of several determinants on ship fuel efficiency. It also develops a ship fuel consumption management scheme based on tangible ANN models. They adopted a two-step procedure for ship fuel consumption assessment, estimating the ship engine rotational speed and then estimating engine power based on the obtained speed. The fuel consumption was calculated as a product of the estimated parameters. The ANN model input variables were speed, displacement, wind force, wind wave height, swell height, sea current factor and trim. The data set necessary to design and construct the ANN model was acquired from noon reports of 3 ships, 121, 160 and 153 reports, respectively. In conclusion, they stated that an ANN model with a simple model structure (1 hidden layer) exhibits the best (surely practically acceptable) fit performance.

Applying ANN models to predict ship speed for various operational conditions was not presented directly in the subject publications. In practice, the power required to move the ship hull at a given speed in the absence of propeller action can be determined from towing tank experiments at various speeds of a model ship. Therefore, we analysed ANN models that combine the ship resistance or power with parameters that influence these ship characteristics.

The applicability of ANN to the ship resistance prediction as an alternative to more traditional statistical regression models was investigated by Couser, Mason, Mason, Smith and von Konsky [18]. The ANN network was used as an interpolation tool to predict the residual resistance for a series of catamaran-type vessels. ANN was able to produce results of sufficient accuracy to be useful for the preliminary prediction of vessel resistance. To design and construct the ANN model, Couser et al. used a single hidden layer and 15 neurons in the hidden layer. In their opinion:

A similar approach to predict ship’s resistance using ANN is presented by Grabowska and Szczuko [19]. Their research used the parameters of 7 already built off shore vessels, with model parameters available as a result of tests conducted on a towing tank (a physical basin used to carry out hydrodynamic tests with ship models). The input and output layers were determined based on the input data dimensionality and required output; thus, the number of hidden layer neurons was estimated as a geometric mean using the formula proposed by Bishop [20]. The automatic network architecture exploration revealed that 24 neurons give the best accuracy. Therefore, several hidden layer configurations were tested to investigate the possible impact of architecture on the results. Grabowska and Szczuko evaluated seven training algorithms. They chose the Quick Propagation algorithm for further study because it gave the most promising results for correlation between target and output values, correlation coefficient (R²) and absolute validation error. To find the best network architecture, several cases were considered with 4, 6, 9, 12, 15, 18 and 24 neurons in a hidden layer; 24 neurons give the lowest absolute validation error; therefore, the network architecture 20-24-1 was used in the follow-up study. In conclusion, they stated that the proposed and trained network showed satisfactory accuracy compared with the results from the model tests. Nevertheless, the architecture might be improved, and further study on this approach should be undertaken.

In Mason, Couser, Mason, Smith and von Konsky [21], ANN was fitted directly to the original towing tank test data and different ANN architectures were investigated. The data used for the investigation originated from a series of tank tests using Holtrop and Mennen’s method [11]. The initial test network architectures were constrained to three layers, one each for input, hidden and output layers. Training runs used a quasi-Newton method for 50,000 iterations, and 10 retrains were performed for each network topology to minimize error. The range of architectures searched was from 4 inputs, 4 hidden layer neurons and 1 output to 4 inputs, 17 hidden layer neurons and 1 output. In Mason et al.’s opinion, the study demonstrated that

Ortigosa, López and García [22, 23] presented an ANN approach to predicting two resistance components. To provide estimates of the form coefficient and the wave’s coefficient as functions of ship hull geometry coefficients and the Froude number, the multilayer perceptron (MLP) was trained with generated data and experimental data. An empirical model was constructed using a neural network. The data necessary to train ANN were generated using Holtrop and Mennen’s method [11] that provides a prediction of the total resistance’s components. The MLP with a sigmoid hidden layer and two linear output layers was used. The selected training algorithm was the quasi-Newton method with Broyden–Fletcher–Goldfarb–Shanno train direction and Brent optimal train rate described by Bishop [20]. Different numbers of neurons in the hidden layer were tested, and the network architecture providing the best generalization properties for the validation data set was adopted. The optimal ANN architecture was 5 inputs, 9 hidden layer neurons and 2 outputs. The ANN results were compared against those provided by Holtrop and Mennen’s method for estimating the form factor and the wave’s coefficients, and it was found that the quality of the prediction with the ANN is improved over the entire range of data.

This analysis of references concerning the ANN approach to modelling ship fuel consumption and hull resistance shows us that.

Moreover, the major reference for designing and constructing ANNs was Bishop’s book [20].

In our approach, to design and construct more adequate ANN models enabling ship fuel consumption and speed prediction, we decided to plan a dedicated experiment. Such an experiment was conducted with different commanded outputs of the ship driveline system for various environmental conditions at sea.

Artificial neural network techniques were used to build models enabling ship fuel consumption and speed prediction, which are necessary to develop DSS. The input and output variables presented in Table 1 and their values acquired from the dedicated experiment conducted at open sea were used to design and construct these models. Generally, the design and construction of ANN models consist of the following main actions.

To verify the correctness of the presented approach, additional sea trials were conducted after developing the computer-aided system. The representative values of the input variables (observations) are shown in Table 2. To avoid excessive concentration of lines on graphs representing relationships between the ANN model variables, these graphs were constructed for decision-making variable values with steps of 200 [rpm] for X₁ and 2 [pitch scale] for X₂.

The impact of the decision-making variables X₁ ‘rotational speed of the engine’ and X₂ ‘pitch of the propeller’ on the first output variable Y₁ ‘hourly fuel consumption rate’ for the representative values of the uncontrollable variables X₃ – X₈ (Table 2) is shown in graphical form in Figs. 8 and 9, respectively. These same dependencies are collected together in a 3-D in Fig. 10.

An analysis of the constructed graphs shows us that the first developed ANN model that connects a fuel consumption process to factors that influence this process is a good representation of the ship operational practice. We notice very good smooth curves without any unexpected bends, bending or other undesirable distortions. However, we notice slight negative slopes of curves (Fig. 9c, d) for the minimum value of engine rotational speed (X₁= 100 rpm) in cases when wind affects in the direction of the vessel’s motion. This is consistent with observations during the operational practices of the tested ship, when the wind blowing from the stern drives provides additional thrust and simultaneously reduces the fuel consumption.

The received 2-D graphs (Figs. 8, 9) give us a basis to conclude that this representation

These conclusions are also confirmed by the three-dimensional visualization presented in Fig. 10, where any unexpected bends, bending or other undesirable distortion, called `sigmodal fluctuation' [24] do not appear.

Similar behaviour for the ANN model can be observed for the influence of the decision-making variables X₁ ‘rotational speed of the engine’ and X₂ ‘pitch of the propeller’ on the second output variable Y₂ ‘instantaneous speed over the ground’. These dependencies are presented in graphical form in Figs. 11 and 12, respectively, for the representative values of the uncontrollable variables X₃–X₈ (Table 2). Moreover, these same dependencies are collected together in the 3-D graphs presented in Fig. 13.

The second ANN model developed that links ship speed to factors that influence this speed is also a very good representation of the ship operational practice. Very good smooth curves without any unexpected bends, bending and negative slopes of curves characterize the presented graphs. Their courses also match the observations recorded during operation of the tested ship.

Based on the 2-D graphs (Figs. 11 and 12), we can conclude that this representation.

These conclusions are also confirmed by the three-dimensional visualization presented in Fig. 13, where any unexpected bends, bending or other undesirable distortion, called `sigmodal fluctuation' [24], also do not appear.

Analysis of the modelling results for each of the ANN networks by observing parameters that characterize the quality of their fitting to the data recorded during sea trials as well as analysis of graph curve courses (Figs. 8, 9, 11 and 12) for various values of the model input variables allows us to formulate the following statements:

Prediction of ship driveline system performance was carried out for various ranges of the decision-making variables (commanded outputs) for numerous of uncontrollable variables (mainly sea environmental conditions) and disturbances (quantity of fuel in tanks, etc.).

The results obtained for modelling ship speed fuel and consumption by applying ANN allow concluding that:

The developed two-objective optimization model provides different combinations of optimal commanded outputs, whereas expected values of the output variables (ship speed and fuel consumption) are similar. Therefore, future research should be conducted with additional optimization criteria, e.g. harmful air pollutants, including particulate matter (PM), sulphur dioxide (SO₂), nitrogen oxides (NO_x) emitted from ship engine.