Mathematical Modeling of the Influence of External Environmental Factors on the Performance of Turbocharger Marine Propulsion Station

Marine diesel propulsion stations represent the most widely used propulsion systems in modern maritime transport. Their operational efficiency directly influences ship performance, fuel consumption, and operating costs. Environmental conditions such as atmospheric temperature, pressure, humidity, sea state, and wind characteristics significantly affect the operational parameters of marine engines and propulsion systems.

Determining the influence of these environmental parameters through full-scale experimental studies requires considerable financial and technical resources, including fuel consumption, lubrication oil usage, and reduction of engine lifetime. Therefore, mathematical modeling methods are widely used for analyzing the interaction between propulsion systems and external environmental conditions.

In this research, the performance of the integrated system consisting of the external environment – marine propulsion complex – main engine – charging system is investigated using a comprehensive mathematical modeling approach [1, 2].

The issues raised in this research were addressed in two ways:

1. Mathematical modeling;

2. Verification of the results of mathematical modeling through practical experiments.

3.1 Mathematical model of the marine propulsion station

The mathematical model of the marine propulsion station developed in this research can solve the following problems:

1. Determining the maximum and minimum speeds of the vessel under varying external environmental parameters or operating conditions;

2. Determining the hourly fuel consumption of the main engines and the entire propulsion station at any vessel speed;

3. Determining the cost per hour of cruising from the propulsion station's operating expenses at any vessel speed;

4. Determining the vessel's cruising range at any vessel speed.

3.2 Preparing the initial data matrix

The mathematical model of the external environment of the Eastern Mediterranean region and the performance of the shipborne power plants enable the implementation of various tests for the "External Environment - Offshore Propulsion Complex" system. To obtain a stable mathematical prediction of the values of the indicators (parameters) of the effectiveness of the shipborne propulsion complex, the number of tests to be conducted must be determined. It is suggested that the number of tests should fall within a certain range [4], while another criterion is proposed for determining the number of tests through the stability of the mathematical prediction in the form of the relationship [5]:

$U=\frac{1}{M_1(Z)}\left[M_1(Z)-M_2(Z)\right]$             (1)

where, $M_1(Z)$ and $M_2(Z)$ are the mathematical predictions of the Z function corresponding to two equal sizes of statistical trial matrices, each of which is determined by the relation:

$M(Z)=\frac{1}{N} \sum_{n=1}^N Z_n$        (2)

The stability of the mathematical prediction $(U \approx 0)$ occurs rapidly, and in the marine power station problems addressed in this research, stability occurs at $(N=70 / 150)$. In the experiments of this research, the number was adopted $(N=100)$. For each ship model, independently and at a specific step, the possible speed spectrum from $v_{\text {min}}$ to $v_{\text {max}}$ is assumed, and then the mathematical prediction and dispersion (squared deviation) are calculated for the values of three parameters: hourly fuel consumption $B$, cost per hour of sailing $C_{\text {rhe}}$, and sailing range $\left(S_v\right)$. For fast ships, three sets of statistical data were obtained, two of which were for individual propeller working cases (Table 1).

Table 1. Studied speed ranges and operating conditions of the main propulsion station on high-speed vessels

Statistical data were obtained, two of which were for individual propeller working cases (Table 1).

For ship voyages ($k_N$) and time elapsed after berthing ($T_{\text {exp}}$) during two different climatic seasons, both regression and correlational (reciprocal) analysis were used. For example, for high-speed vessels operating in independent propulsion groups, the regression relationships for hourly fuel consumption at the power station $(B[k g / h])$, ship voyage range $\left(S_v[\right.$mile$\left.]\right)$, and hourly voyage cost $\left(C_{\text {rhe}}[u . e / h]\right)$ are as follows:

${January}(S-1) B=2491.1 \cdot \bar{v}_i-94.62 \cdot K_N+44.11 \cdot T_{{exp}}$       (3)

${August}(S-2) B=2500 \cdot \bar{v}_i-121.4 \cdot K_N+34.6 \cdot T_{\exp}$          (4)

${January}(S-1) S_v=1765.4-60.97 \cdot T N \bar{v}_{i_{{exp}}}$           (5)

August $(S-2) S_v=2151-57.6 \cdot T N \bar{v}_{i_{{exp}}}$           (6)

${January}(S-1) C_{r h e}=597.9 \cdot \bar{v}_i-22.74 \cdot K_N+10.59 \cdot T_{e x p}$             (7)

${August}(S-2) C_{r h e}=600 \cdot \bar{v}_i-29.15 \cdot K_N+8.3 \cdot T_{e x p}$             (8)

To assess the quality of the choice of return relations (functions), the correlation constant is used, which expresses the strength of the linear relationship of the variable $y_i\left(B, C_{\text {rhe}}, S_v\right)$ to the set of variables $X_i\left(v_i, T N_{\text {exp}}\right)$, according to the relationship:

$R=\sqrt{1-\left(\sum e_i^2 / \sum\left(Y_i-\bar{Y}\right)^2\right)}$            (9)

where, $\sum e_i^2$ - the sum of the squares of the actual deviations of the values (of the observations) from the return curve; $\sum\left(Y_i-\bar{Y}\right)^2$ - the sum of the squares of the actual deviations of the values (of the observations) from the mathematical expectation.

In this research, calculations were performed using the software package as follows: for a range of diverse marine propulsion systems; for all use cases of each system studied; for all parameter ($B, S_v, C_{r h e}$) return relationships, within $95 \%$ certainty ranges, within which the expected value of each parameter $\left(B, S_v, C_{r h e}\right)$ can be guaranteed. Based on initial statistical data and using the Math Cad software package to determine the values of: hourly fuel consumption, vessel cruising range, and hourly cruising cost according to nonlinear return relationships, the relationships (3/8) were transformed into the following form:

January $(S-1) B=-533+2726 \cdot \bar{v}_i-242 \cdot \bar{v}_i^2-92 \cdot K_N+44 \cdot T_{{exp}}$          (10)

August $(S-2) B=-624+2460 \cdot \bar{v}_i+44 \cdot \bar{v}_i^2-122 \cdot K_N+35 \cdot T_{{exp}}$         (11)

January $(S-1) S_v=2315-3860 \cdot \bar{v}_i+2616 \cdot \bar{v}_i^2+187 \cdot K_N-58 \cdot T_{\exp}$      (12)

August $(S-2) S_v=2430-4140 \cdot \bar{v}_i+2470 \cdot \bar{v}_i^2+283 \cdot K_N-56 \cdot T_{{exp}}$        (13)

January $(S-1) C_{r h e}=-127.9+654.2 \cdot \bar{v}_i-58.2 \cdot \bar{v}_i^2-22.1 \cdot K_N+10.53 \cdot T_{{exp}}$     (14)

August $(S-2) C_{r h e}=-149.7+589 \cdot \bar{v}_i+10.5 \cdot \bar{v}_i^2-29.2 \cdot K_N+8.31 \cdot T_{{exp}}$             (15)

A mathematical model representing the operational behavior of a marine propulsion station was developed. The model enables the determination of several operational parameters, including [6]:

• Maximum and minimum ship speeds
• Hourly fuel consumption of main engines
• Total fuel consumption of the propulsion station
• Navigation hour cost
• Ship navigation range

Figure 1 illustrates the overall structure of the mathematical model that represents the performance (work) of a marine propulsion station.

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Figure 1. General diagram of the mathematical representational model for the operation of a marine propulsion station (algorithm)

The model was developed based on operational data collected from ships operating in the Eastern Mediterranean region. This mathematical model allows for the identification of two types of specifications [4]:

I. First Type: A set of indicators that allows comparison between the specifications of the ship propulsion station, such as: the maximum speed $\left(v_{\max}\right)$ and minimum speed $\left(v_{\min}\right)$ of the ship, total fuel consumption $(B)$, the required power from the main propulsion station $\left(P_{e \Sigma}\right)$, and electrical loads $\left(P_{e D G}\right), \ldots$, etc.

II. Second Type: A set of indicators that can be used as criteria to evaluate the performance efficiency of the ship propulsion station, such as: the cost per hour of navigation, the amount of metal constituting the propulsion station $\left(\gamma_{S P P}\right)$ and the ship as a whole $\left(\gamma_{\text {Ship}}\right)$, fuel consumption per nautical mile traveled by the ship $\left(b_{\text {mile}}\right)$, hourly fuel consumption $\left(b_h\right)$, the annual investment $\operatorname{cost}\left(C_y\right)$, and the total cost over the entire service life $\left(C_{\Sigma}\right)$.

Marine propulsion stations are characterized by thermodynamic indicators, power indicators, mass indicators, cost indicators, and others. The random numbers required to determine the inverse integrals of the distribution functions of external variables are obtained using the standard instruction (subroutine) URAND, block (2).

The subroutine KRITER, block (3), is considered central, and its main function is to determine the performance (efficiency) criteria of the ship power plant, as well as the criteria for its comparison with other possible configurations (cases) of power plants.

In general, by means of block 3 (KRITER), matrices of parameter values describing the various operational characteristics of the marine propulsion station are obtained. These matrices represent a set of statistical values generated by the computer and describe different characteristics of the ship and its onboard power plant, such as: absolute costs $(C)$ and relative costs $\left(C^{\prime}\right)$, average economic speed $\left(v_s\right)$, fuel consumption $(B)$, ship hull mass $\left(G_{\text {ship}}\right)$, and propulsion station mass $\left(G_{\text {SPP}}\right)$, onboard fuel reserve $\left(Q_{f i}\right)$, ship displacement $\left(D_i\right)$, costs allocated for intermediate overhaul ($B_p$), other types of maintenance and repair costs ($\mathrm{B}^{\prime} p$), service costs over a specified standard period or over the entire investment (service) cycle $\left(T_{\text {cycle}}\right)$, ship construction costs $\left(C_b\right)$, positioning costs ($C_{\text {base}}$) costs of routine repairs and inspections ($C_r$), costs of replacing certain equipment ($C_{r p}$), fuel cost $\left(C_f\right)$, oil cost $\left(C_{\text {oil}}\right)$, crew expenses $\left(C_{\text {pers}}\right)$, and other total investment costs $\left(C_{\Sigma}\right)$, fuel consumption per hour $\left(b_h\right)$ and per mile $\left(b_{\text {mile}}\right)$, the amount of metal constituting the propulsion station $\left(\gamma_{\text {SPP}}\right)$, Total Investment Costs $\left(C_{\text {exp}}^a\right)$ and average annual costs $\left(C_y\right)$.

The criteria (C) and (C') are calculated and determined for all propulsion station configurations for each of the studied ships over the entire investment spectrum. The optimal propulsion station configurations are those characterized by minimum values of the parameters (C) and (C').

A set of calculations is performed in block 3 with the assistance of the subroutine RASP (block 4) and the subroutine TOREC (block 5). The latter subroutine TOREC, in turn, interacts with the subroutine GIDRA and the subroutine RUN A_v (block 6).

The subroutine TOREC is dedicated to determining (calculating) the operational (investment) characteristics of the ship propulsion station installed on board the vessel. The hourly fuel consumption is calculated independently (separately) for each of the Main Engines $\left(B_{M E}\right)$, the diesel engines used for electrical power generation (Diesel Generator) $\left(B_{D G}\right)$, and auxiliary steam generators, if present (auxiliary steam generator) $\left(B_{A S G}\right)$, on a monthly and yearly basis. These values are then aggregated while considering the duration of navigation periods and the number of operating main and auxiliary engines required to ensure the assumed ship speed.

To determine the influence of environmental factors (external conditions), the subroutine TOREC refers (returns) to the subroutines GIDRA and RUN A_v. During this process, the subroutine GIDRA determines the hydrometeorological parameters, while the subroutine RUN A_v calculates the values of the power correction coefficients $A_v$, which are considered random variables that allow, with good accuracy, the determination of the expected ship speed, the load level of the main and auxiliary engines and auxiliary boilers, as well as the necessity and method of applying individual operating modes for the main engines.

The subroutine RUN analyzes the changes in the propulsion system characteristics and selects the appropriate operating modes for the ship power plant.

According to hydrometeorological conditions, the voyage route (L) is divided into four segments of different lengths (L₁), (L₂), (L₃), (L₄). During this process, the economic speed of the ship is calculated for each segment while taking into account the spectrum of operating modes.

The list of calculated specifications includes:

• Propulsion station operating conditions and modes:
• Calculated, maximum, and minimum ship speeds for each operating mode (i);
• Average ship speed $v_{s r}$;
• Propeller rotational speed under nominal (rated) conditions $n_p$;
• Main engine power under mode (i), according to the limiting curve and the propeller operating curve;
• Ship range at any possible speed $S_v$;
• Verification of the values of both the navigation load coefficient $K_N$ and the voyage time $T_h$ [5];

Using the subroutine GIDRA, one of the possible configurations of hydrometeorological factors prevailing in the Eastern Mediterranean region has been determined, consisting of:

• Ambient temperature $\left(T_a\right)$, pressure $\left(p_a\right)$, and humidity $\left(\phi_a\right)$;
• Ship course angle relative to wind direction $\left(q_w\right)$, and wind speed $\left(v_w\right)$;
• Wave height exceeding 3% (critical limit), i.e., $h_{(3 \%)}$;
• Lengths of voyage segments under varying hydrometeorological conditions $L_1, L_2, L_3, L_4$ [6];

3.3 Environmental parameters modeling

The model incorporates a set of environmental variables representing hydrometeorological conditions, including [7, 8]:

• Atmospheric temperature
• Atmospheric pressure
• Air humidity
• Wind speed and direction
• Sea wave height

These parameters were modeled using statistical distributions derived from real environmental data recorded in the Eastern Mediterranean maritime region [9].

The mathematical model of the external environment of the Eastern Mediterranean region and the performance of the shipborne propulsion station enable the implementation of various tests for the "external environment-marine propulsion system." To obtain a stable mathematical prediction of the values (parameters) of the effectiveness of the shipborne propulsion system investment, the number of tests to be conducted must be determined. It is suggested that the number of tests should fall within the range of (50/100) [10], while another criterion is proposed for determining the number of tests based on the stability of the mathematical prediction [11].

For each vessel type, independently and at a specific step, the possible speed spectrum from $v_{\min}$ to $v_{\max}$ is assumed. Then, the mathematical prediction and dispersion (squared deviation) are calculated for the values of three parameters: hourly fuel consumption (B), cost per hour of sailing ($C_{{rhe}}$), and cruising range ($S_v$). For high-speed vessels, three sets of statistical data were obtained, two of which were for individual propeller operating conditions, as shown in Table 1 [12].

Statistical data were obtained during simulated experiments on the "external environment - marine propulsion system" on fast vessels using a computer program presented in tabular form. This data was collected to study the impact of climatic conditions and operational factors on the performance of the propulsion system. Table 2 includes some of this data for fast vessels operating with only two propellers $\left(z_p=2\right)$.

Table 2. Some statistical data obtained from experiments conducted on the mathematical model “External Medium – Marine Propulsion station” for high-speed vessels at (zₚ = 2)

3.4 Simulation procedure

The developed mathematical model was implemented using computational tools including [13, 14]:

Simulink/MATLAB

Microsoft Excel

Mathcad

The simulation model integrates several computational subroutines responsible for evaluating propulsion performance, generating environmental parameters, and determining operational performance indicators [15].

3.5 Experimental validation

The selection and rationale for the operation of diesel propulsion station used on ships begins with formulating a system matrix that considers the expected sailing conditions and the anticipated technical condition of the ship and all its components. Initially, the following preliminary data are assumed: sailing speed or speed range; external environmental factors (climatic and operational factors); and constraints (precautions, standards, efficiency constants, service life, etc.).

When determining (assuming) the required speed spectrum, the formation of the work systems matrix and the solution of the example problem are done according to the algorithm shown in Figure 2, Case No. 1.

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Figure 2. Algorithm for forming a matrix of working systems to solve the example problem (Case No. 1)

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Figure 3. Algorithm for forming a matrix of working systems to solve the example problem (Case No. 2)

When a specific range of required speeds exists, the problem is solved according to the sequence shown in Figure 3, Case No. 2 [16, 17].

To verify the accuracy of the developed model, a series of experimental studies were conducted on a marine propulsion station installed on operational ships [18].

Statistical experiments were carried out to analyze the interaction between environmental conditions and propulsion system performance. The results obtained were processed using regression and correlation analysis methods to determine the relationships between ship speed, fuel consumption, navigation cost, and navigation range.

The diagram of the method used to justify the systems suitable for the use of main power stations on ships is shown in Figure 4.

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Figure 4. Diagram of the method used to justify and select suitable systems for use in main power stations on ships

The research resulted in the following main conclusions:

1. A mathematical model representing the performance of a marine diesel propulsion station equipped with fixed-pitch propellers was successfully developed.
2. The developed model accounts for the combined influence of environmental conditions and operational factors on marine propulsion performance.
3. Simulation studies demonstrated that environmental parameters significantly affect ship propulsion efficiency and fuel consumption.
4. A methodology for selecting optimal operating modes of marine propulsion stations under varying environmental conditions was proposed.

The developed model can be used as an effective tool for analyzing and optimizing marine propulsion systems during both design and operational stages.