Trim and Engine Power Joint Optimization of a Ship Based on Minimum Energy Consumption over a Whole Voyage

Abstract

Trim optimization is an available approach for the energy saving and emission reduction of a ship. As a ship sails on the water, the draft and trim undergo constant changes due to the consumption of fuel oil and other consumables. As a result, the selection of the initial trim is important if ballasting or shifting liquid among the tanks is not considered during a voyage. According to the characteristics of ship navigation and maneuvering, a practical trim optimization method is proposed to identify the Optimal Trim over a Whole Voyage (OTWV) which makes the fuel consumption of the voyage minimum. The calculations of speed vs. draft and trim surfaces are created according to hull resistance data generated by CFD, model tests, or real ship measurements, and these surfaces are used to calculate the OTWV. Ultimately, a trim and Main Engine (ME) power joint optimization method is developed based on the OTWV to make the total fuel consumption minimum for a voyage with a fixed length and travel time. A 307000 DWT VLCC is taken as an example to validate the practicality and effect of the two proposed optimization methods. The trim optimization example indicates that the OTWV could save up to 1.2% of the total fuel consumption compared to the Optimal Trim at Initial Draft (OTID). The trim and ME power joint optimization results show that the proposed method could steadily find the optimal trim and ME power combination, and the OTWV could save up to 1.0% fuel consumption compared to the OTID in this case.

1. Introduction

As the primary means of transportation in global trade, ships undertake approximately 80% of the global trade volume; meanwhile, they emit about 3% of the total amount of global anthropogenic greenhouse gas [1]. Therefore, the shipping industry is an important factor in global climate issues such as global warming and air pollution. As a result, research on the energy saving and emission reduction of ships is of great significance for the development of a low-carbon economy.

Generally speaking, there are four main categories of methods for the energy saving and emission reduction of ships, which are as follows: optimal design of ship form [2], application of green energy [3], waste heat recovery [4], and operation optimization of ships, etc. Modern ships typically first collect ship performance data [5], then employ one or more of the aforementioned methods to reduce the fuel consumption for better economy and to cut carbon emissions in order to meet the increasingly strict standards, such as the Energy Efficiency Operational Indicator (EEOI) [6] and the Energy Efficiency Design Index (EEDI) [7], for cargo ships. Among these methods, operation optimization is an efficient but low-cost way to reduce energy consumption for a ship, by means of speed optimization [8,9], navigation scheduling optimization [10], route optimization [11], and trim optimization, etc. For a ship sailing on the sea, trim is an important factor that influences the hull resistance. Therefore, it is an efficient way to reduce fuel consumption by ballasting or load distribution which makes the ship sail with a low-resistance trim. Trim optimization does not increase the design or construction costs of a ship, and it has a very small effect on operation costs, but it could efficiently cut the carbon emissions and improve operational economics [12]. As a result, trim optimization is an efficient method for energy saving and emission reduction, and it has become a research focus in recent years.

Lokukaluge et al. [13] studied the identification of the optimal trim with ship performance and navigation data, and concluded that selecting proper trim conditions could improve the log speed or reduce the fuel consumption of a ship. Sun et al. [14] conducted a trim optimization study for a container ship using CFD and towing tank experiments, which demonstrated the significant impact of trim adjustments on reducing ship resistance and fuel consumption, offering an effective strategy for energy saving and emission reduction in real ship operations. Coraddu et al. [15] researched the problems of accurate fuel consumption prediction based on data measured by onboard automation systems, and made a detailed comparison of three fuel consumption predicting models, and finally came to the conclusion that the gray box model is the best solution to obtain the optimal trim for a ship in real operations. Islam et al. [16] made an investigation into the resistance prediction of a container ship at different drafts and Froude numbers, and revealed that trim optimization could increase the propulsive efficiency and reduce fuel consumption. They also proved that the optimal trim varies with the draft and speed, and that the variation does not follow any particular trend. Du et al. [17] constructed a neural network model with the navigation data, and proposed a two-phase optimization framework, which involved optimizing the ship speed in the onshore planning phase and conducting the trim optimization on board based on the actual weather and sea conditions at sea. Numerical experiments with two container ships indicated that the proposed methods could achieve remarkable bunker fuel savings. Gao et al. [18] studied the trim optimization problem with the CFD numerical simulations of an oil tanker, and proved that the hull resistance as well as the fuel consumption of the tanker could be reduced significantly by trim optimization. Le et al. [19] investigated the influence of trim on hull resistance with the unsteady RANSE method, proved that trim has a significant impact on hull resistance, and further explained the phenomenon of the trim-induced variation in hull resistance with numerical simulations. Li et al. [20] calculated the wind and wave added resistance of a Very Large Crude oil Carrier (VLCC) by the Taylor expansion boundary element method using the ITTC spectrum with two parameters, and then performed trim optimization based on the wind and wave added resistance. With the VLCC example, the researchers proved that the optimal trim can improve ship energy efficiency by reducing fuel consumption or by increasing sailing speed. Xie et al. [21] developed a speed and trim joint optimization method by using sensor data and a fuel consumption prediction model to minimize the fuel consumption, and proved that joint optimization can overcome the limitations of single-factor optimization such as trim optimization and speed optimization by considering multiple factors simultaneously. Korkmaz et al. [22] investigated the trim trends by using Experimental Fluid Dynamics (EFD) compared to CFD with the example of a RoPax vessel, and came to some valuable conclusions on the comparison of the accuracy of EFD and CFD, and some suggestions were given on how to improve the precision of trim trends prediction.

As stated above, the existing literature primarily focuses on the methods to obtain optimal trim, such as CFD numerical stimulation [16,18,19,20,21], model tests [22], and analysis of real ship measured data [12,13,14,15,17], etc. The relevant research was mostly aimed at finding the optimal trim which corresponds to the minimum energy consumption at a given draft and speed, or at performing trim and speed joint optimization with a fixed draft. When a ship sails on the water, the fuel oil and other consumables are continuously consumed, which leads to constant changing in draft and trim. As a result, it is impossible to keep sailing at the optimal trim in a voyage for most ships. There are two methods to change the trim of a ship in navigation: one is increasing or decreasing the ballast water; and the other is modifying the Longitudinal Center of Gravity (LCG) by load redistribution. However, the applications of both of the two methods are limited. The first method requires that the ship has ballast tanks with enough void volume, and that the added ballast water does not cause the draft to exceed the scantling draft. Even so, the increment in draft by added ballast water will lead to an increment in hull resistance. The second method requires that there is enough space, along with enough available devices, for load redistribution in navigation. For a ship with a large enough void tank and with a draft not close to the scantling draft, such as a container ship or passenger ship, these two methods are both applicable. However, as to a traditional oil tanker or bulk carrier, these conditions are not easily met, which means it is not possible to change the ballast water or redistribute the liquid frequently during a voyage for these ships. In these cases, the selection of the initial trim is of great importance to fuel consumption over a whole voyage. It has been proved that the optimal trim of a ship varies significantly with draft and speed, and that the variation does not appear to follow any particular trend [16], which means there is no simple but accurate way to identify the optimal initial trim which could minimize the fuel consumption over a whole voyage. Thus, the first problem to be solved in this research is to calculate the optimal initial trim that corresponds to the minimum total fuel consumption, under the condition that no ballast changing or load redistribution is allowed during the voyage.

Most of the existing literature that studies single-parameter trim optimization assumes that the speed of the ship is fixed. When a ship sails on the water at a stable Main Engine (ME) power, the trim and draft are continuously changing, and in addition, the load of the wind, waves, and current is unstable. As a result, the sailing speed is constantly fluctuating, and maintaining a fixed speed requires frequent adjustment of the ME power. On the one hand, this will cause the abrasion of the ME as well as of the relevant devices; on the other hand, it is not necessary. In actual ship maneuvering, the ME power is usually fixed by locking the rotation rate, and the sailing speed is allowed to fluctuate in a small range. Based on the above characteristics of ship navigation and maneuvering, trim optimization with fixed ME power is studied in this research. There are two situations in which ship maneuvering practice is of concern. The first situation is when the voyage length is fixed, while the travel time is not strictly restricted. In this case, the ME runs at an economic rotation rate, usually at the Normal Continuous Rating (NCR), and the optimal trim should make the total fuel consumption minimum. The second situation is when the voyage length as well as the travel time is specified, which means the ship should travel a specified distance during a specified time. This is the most common situation in practice. On the one hand, a late arrival will interrupt the schedule of the port and fleet and should be avoided; on the other hand, an early arrival means more fuel oil could have been saved if a slower sailing speed had been adopted. There are several ways to solve this problem, but the most economical one is to find a ME power and an initial trim, with which the ship could travel the specified distance during the specified time at a minimum fuel consumption. That is the second main problem to be solved in this research.

As stated above, this paper focuses on the trim optimization of a ship over a whole voyage, in which no ballast changing or load redistribution is allowed. There are two main problems to be solved in this research: the first one is finding the optimal trim with a fixed ME power for a fixed voyage length; the second one is trim and ME power joint optimization on the premise of a fixed voyage length and a fixed travel time. The structure of this paper is as follows. Section 2 introduces the principles and algorithms to solve the above two problems. Section 3 details the calculation of the basic data for hydrodynamic and hydrostatic characteristics. In Section 4, a VLCC is taken as an example to demonstrate and validate the proposed methods. Section 5 is a brief conclusion and discussion.

2. Optimal Trim over a Whole Voyage with Fixed ME Power

2.1. Functions between Travel Distance and Draft

When a ship navigates with a constant ME power, the fuel consumption rate of the ME can be regarded as a fixed value C_O. It is reasonable to assume that the other consumables such as fresh water are also consumed at a constant speed C_W. The weight of consumed fuel oil and other consumables equals the variation in displacement, and it is proportional to time t. That is to say, the relationship between the differential of displacement dD and the differential of time dt can be written as follows:

$$dD=-{(C}_o+C_W)·dt$$

(1)

Displacement D can be expressed by a function of draft d as follows:

$$D=f_D\left(d\right)$$

(2)

Due to the complexity of the hull surface, the relationship between displacement and draft cannot be represented by a simple analytical expression, and hydrostatic curves are the most commonly used solution to this problem. For cargo ships such as oil tankers and bulk carriers, the consumables such as fuel oil and fresh water occupy a small proportion of the displacement, and the variation in the draft during a voyage is relatively small. As a result, the relationship between displacement and draft can be simplified into a linear relationship with acceptable accuracy for those situations. The purpose of this research is to develop a general method suitable for all kinds of ships, and the linearization assumption of the relationship between displacement and draft will cause significant errors, especially for the non-freight vessels. Considering the above factors, Equation (2) is expressed with Taylor expansion around the initial draft d₀, and the quadratic terms are kept, and then D is written as follows:

$$D=f_D\left(d_0\right)+{f_D}^{\prime }\left(d_0\right)\left(d-d_0\right)+\frac{1}{2}{f_D}^{″}\left(d_0\right){\left(d-d_0\right)}^2$$

(3)

where f_D(d₀) is the initial displacement for the initial draft d₀. The differential of displacement dD and the differential of draft dd has the following relationship:

$$dD=\left({f_D}^{\prime }\left(d_0\right)+{f_D}^{″}\left(d_0\right)\left(d-d_0\right)\right)dd$$

(4)

Substitute Equation (4) into Equation (1), and the differential of time is written as follows:

$$dt=\frac{-\left({f_D}^{\prime }\left(d_0\right)+{f_D}^{″}\left(d_0\right)\left(d-d_0\right)\right)}{{(C}_o+C_w)}dd$$

(5)

Assume that Xc is the longitudinal center of the consumed fuel oil and other consumables, and assume Xc to be fixed during the voyage. For most ships, Xc is different from the Longitudinal Center of Buoyancy (LCB) of the ship, and most commonly Xc is near the position of the engine room. As a result, the consumption of fuel and other consumables will cause a steady change in the Longitudinal Center of Gravity (LCG), which will further lead to a variation in the trim continuously. In this situation, the trim tr is a function of the draft d, and tr can be expressed by the following:

$$tr=f_T\left(d\right)$$

(6)

where tr is determined by the LCG, LCB, and Moment of per Centimeter trim (MCM) as follows:

$$tr=\frac{\left(xb-xg\right)·D}{100M}$$

(7)

where xb, xg, and M are the LCB, LCG, and MCM of the ship, respectively. Then the first derivative of tr is calculated by the following:

$${tr}^{\prime }=\frac{M\left({xb}^{\prime }D-{xg}^{\prime }D+xb{D}^{\prime }-xg{D}^{\prime }\right)-M^{\prime }(xb-xg)D}{100M^2}$$

(8)

and the second derivative of tr is written as follows:

$${tr}^{″}=\frac{A-B}{100M^4}$$

(9)

where the following apply:

$$\left\{\begin{array}{cc}A=& M^2\left(MDx{b}^{″}-MDx{g}^{″}+2M{D}^{\prime }{xb}^{\prime }-2M{D^{\prime }xg}^{\prime }+M{D}^{″}xb-M{D}^{″}xg-M^{″}Dxb+M^{″}Dxg\right)\\ B=& 2M{M}^{\prime }\left(MDx{b}^{\prime }-{MDxg}^{\prime }+M{D}^{\prime }xb-M{D}^{\prime }xg-M^{\prime }Dxb+M\prime Dxg\right)\end{array}\right.$$

(10)

Both M and xb are functions of the draft written as f_M(d) and f_B(d), respectively. Like the displacement, f_M(d) and f_B(d) do not have simple analytical expressions and they are expressed by hydrostatic curves. Similarly, Taylor expansion is used and second-order terms are kept for xb and M as follows:

$$\left\{\begin{array}{c}xb=\mathrm{x}{b}_0+f_B^{\prime }\left(d_0\right)\left(d-d_0\right)+\frac{1}{2}{f_B}^{″}\left(d_0\right){\left(d-d_0\right)}^2\\ M=M_0+f_M^{\prime }\left(d_0\right)\left(d-d_0\right)+\frac{1}{2}{f_M}^{″}\left(d_0\right){\left(d-d_0\right)}^2\end{array}\right.$$

(11)

where xb₀ and M₀ are the LCB and the MCM for the initial draft d₀. The first and second derivatives of D, xb, and M are calculated from Equations (3) and (11). The value xg is related to the variation in displacement and is calculated by the following:

$$xg=\frac{D_0·x{g}_0+\left(D-D_0\right)·X_C}{D}$$

(12)

where D₀ is the initial displacement, and xg₀ is the initial LCG. In Equation (12), xg₀, d₀, and X_C are known quantities, and D is the only variable of xg. The first and second derivatives of xg are calculated by the following:

$$\left\{\begin{array}{cc}{xg}^{\prime }=& \frac{D_0\left(X_C-{xg}_0\right)D^{\prime }}{D^2}\\ {xg}^{″}=& \frac{D_0\left(X_C-{xg}_0\right)\left(D^2{D}^{″}-2D{D^{\prime }}^2\right)}{D^4}\end{array}\right.$$

(13)

The first and second derivatives of tr are calculated via Equations (8) and (9). Then tr is written as a function of d by Taylor expansion as follows:

$$tr={tr}_0+t{r}^{\prime }\left(d_0\right)\left(d-d_0\right)+\frac{1}{2}t{r}^{″}\left(d_0\right){\left(d-d_0\right)}^2$$

(14)

where tr₀ is the initial trim, and $t{r}^{\prime }$ and $t{r}^{″}$ are indirect functions of d. With Equation (14), tr is written as a function of the draft d and the initial trim tr₀.

When a ship sails on the water, the hull resistance is related to both the draft and trim. In other words, different drafts and trims correspond to different resistance values. Therefore, if the delivered power of the propeller is fixed, the speed of a ship is a function of the draft d and the trim tr as follows:

$$v=v_{dt}\left(d,tr\right)$$

(15)

According to Equation (14), tr is a function of d, so during a voyage with fixed delivered power, the speed is only related to the draft d, and it is denoted by the following:

$$v=v_d\left(d\right)$$

(16)

It is reasonable to assume that the shaft efficiency is independent of the ME power, so fixed delivered power means fixed ME power. When the delivered power or ME power remains unchanged, the resistance is continuously changing along with the draft and trim, which will further lead to variations in sailing speed. For time T, the travel distance S can be written as follows:

$$S={\int }_0^T{v}_t\left(t\right)dt$$

(17)

where v(t) is the function of speed vs. time. Substitute Equations (5) and (16) into Equation (17), and modify the lower and upper limit of integration to the initial draft d₀ and the final draft d₁, respectively. Then the travel distance S becomes the following:

$$S=-{\int }_{d0}^{d1}{v}_d\left(d\right)\frac{f{D}^{\prime }\left(d_0\right)+f{D}^{″}\left(d_0\right)\left(d-d_0\right)}{{(C}_o+C_w)}dd$$

(18)

In Equation (18), v_d(d) is related to d₀ and tr₀, so d₀, d₁, and tr₀ are three parameters of S. When these three parameters are given, the function of speed vs. draft v_d(d), which is determined by the function of speed vs. draft and trim v_dt(d,tr), and the function of trim vs. draft f_T(d), should be calculated first. f_T(d) is expressed by Equation (14). v_dt(d,tr) is a complex function determined by the hydrodynamic characteristics of the hull, and it is related to the principal dimensions, ship form, surface roughness, ME power, propulsive efficiency, etc. So, v_dt(d,tr) does not have exact analytical expressions. In this research, hull resistance vs. different draft and trim combinations, called the trim table, is calculated by CFD numerical simulations, and a 3D surface denoted by SUR_V is created using the trim table via spline and surface modeling, interpolation, projection transformation, and other algorithms. The creation of SUR_V will be discussed subsequently. Then the relationship of S vs. d can be expressed by a 2D spline with Algorithm 1 according to SUR_V.

Algorithm 1 Calculation of travel distance vs. draft curve

1: Calculate the trim table by CFD numerical simulations, and construct the hull resistance vs. draft and trim surface SURV. 2: Establish a 3D coordinate system with the draft d as the longitudinal (X) axis, the trim tr as the transverse (Y) axis, and the speed v or travel distance S as the vertical (Z) axis. At the coordinate origin, the values of d, tr, v, and S are all zero, as shown in Figure 1. 3: As the trim tr is only related to the initial trim tr0 and draft d, calculate the trim vs. draft relationship according to Equation (14), and create a spline with the trim and draft as the X and Y coordinates, respectively. Denote the calculated trim vs. draft spline by SPd-tr. The X and Y coordinates for the start and end points of SPd-tr are the initial draft d0, initial trim tr0, final draft d1, and final trim tr1, respectively. 4: Calculate the projection spline curve of SPd-tr onto SURV, and denote the projection spline curve by SP3d. The X, Y, and Z coordinates of each point on SP3d are the draft, trim, and speed, respectively. The X and Y coordinates of the start and end points of SP3d are the same as the corresponding items of SPd-tr. The Z coordinates of the start and end points of SP3d are the initial speed and final speed. 5: Project SP3d onto plane OXZ, and obtain a planar spline curve SPd-v. The X and Z coordinates of each point on SPd-v are the draft and speed. This spline curve indicates how the speed changes with the draft during the voyage. 6: SPd-v is a 2D spline curve, and it can be used to represent the function of speed vs. draft vd(d) in Equation (16). Substitute SPd-v into Equation (18), and the travel distance according to the initial draft d0, final draft d1, and initial trim tr0 can be calculated. In this research, the integral algorithm for spline curves is used to obtain the integral in Equation (18), and the integral result is another spline curve, denoted by SPd-S, which is the travel distance vs. draft spline curve. The X and Z coordinates of each point on SPd-S are the draft and travel distance, respectively. The travel distance for a given draft can be calculated by the spline interpolation algorithm of SPd-S.

Figure 1. Calculation of travel distance vs. draft spline curve SP_d-S.

Figure 1. Calculation of travel distance vs. draft spline curve SP_d-S.

2.2. Calculation of Optimal Trim over a Whole Voyage with Fixed ME Power

For a voyage with a fixed ME power and no changes in ballast water or load redistribution operations, if the travel distance is specified, then different initial trims will result in different fuel oil consumptions. The initial trim that corresponds to the minimum total fuel consumption of the whole voyage is called the Optimal Trim over the Whole Voyage (OTWV).

During the voyage, the decrease in displacement equals the sum weight of consumed fuel oil and other consumables. The decrease in displacement is the difference between the initial displacement and final displacement, written as $f_D\left(d_0\right)-f_D\left(d_1\right)$, where d₀ and d₁ are the initial draft and the final draft, respectively. The fuel consumption during the voyage can be calculated by the following:

$$W_o=\frac{C_o}{C_o+C_w}(f_D(d_0)-f_D(d_1))$$

(19)

where Wo is the weight of consumed fuel oil, and C_W is the consumption rate of other consumables mainly consisting of fresh water.

Assume that the given travel distance is S_R, and the ship travels with fixed ME power. Substitute this condition into Equation (18), and an equation is obtained as follows:

$$S_R=-{\int }_{d0}^{d1}{v}_d\left(d\right)\frac{f{D}^{\prime }\left(d_0\right)+f{D}^{″}\left(d_0\right)\left(d-d_0\right)}{{(C}_o+C_w)}dd$$

(20)

As the purpose of this section is to determine the optimal initial trim that corresponds to the minimum fuel consumption over a whole voyage, the initial draft d₀ is a known quantity, and d₁ and tr₀ are regarded as optimization variables. Then this problem can be described by the following optimization model:

$$\begin{array}{c}\min W_o=\frac{C_o}{C_o+C_w}(f_D(d_0)-f_D(d_1))\hfill \\ \hspace{0.17em}s.t. S\left(d_1,{tr}_0\right)=S_R\hfill \\ {tr}_{min}\le {tr}_0\le {tr}_{max}\hfill \end{array}$$

(21)

where tr_min and tr_max are the permissible minimum and maximum trims determined by other requirements of the ship such as bridge visibility, propeller and rudder immersion, and fore bottom wave slap, etc. [23]. The optimization model (21) involves the calculation of travel distance, which makes it hard to obtain the global optimal solution by traditional optimization methods. In this research, this optimization model is solved with an algorithm based on spline curve and surface. The procedure is outlined in Algorithm 2.

Algorithm 2 Calculation of optimal initial trim with minimum fuel consumption for a fixed ME power and fixed voyage

1:  Create a set of initial trims within trmin and trmax with equal spacing in ascending order, and denote the initial trims by tr0,1, tr0,2, … tr0,nt, where nt is the number of initial trims, tr0,1 = trmin, and tr0,nt = trmax.   2:  for i = 1 to nt step 1.   3:  Set the initial trim of the ship equal to tr0,i, and calculate the travel distance vs. draft curve, SPd-S,i, with Algorithm 1.   4:  With the interpolation algorithm of the spline curve, calculate the X coordinate of SPd-S,i corresponding to the required travel distance SR, and mark the calculated X coordinate as d1i where d1i is the final draft which makes the travel distance equal to SR with a trim of tr0,i.   5:  Take d1i as the final draft, and calculate the fuel consumption WOi by substituting d1i into Equation (19).   6:  end for   7:  Construct a spline curve, Str-FC, by Taking tr0,i (1 ≤ i ≤ nt) as X coordinates and WOi as Y coordinates, sequentially. Str-FC is the initial trim vs. fuel consumption curve, which describes the relationship between the initial trim and the consumed fuel oil of the whole voyage with fixed ME power.   8:  Find all the points on the curve Str-FC which has a zero first derivative, and add the zero first derivative points into the point set PA. Append the start and end points of Str-FC into PA. The optimal solution is among PA.   9:  Traverse PA and find the point which has a minimum Y coordinate, and denote the point by Popt. Then Popt is the optimal solution. The X coordinate of Popt is the optimal initial trim corresponding to the minimum fuel consumption over the whole voyage.

2.3. Trim and ME Power Joint Optimization with Fixed Voyage Length and Time

As mentioned above, it is a common situation for a ship to travel a specified distance S_R using a specified time T_R. In this case, both the initial trim tr₀ and the ME power are variables, and different tr₀ values require different ME power values to meet the requirements of S_R and T_R. There are several ways to solve this problem; for example, the ship travels with a relatively high ME power for a period, and then with a relatively low ME power for the rest time. By adjusting the length of the two periods and the ME power in the two phases, both S_R and T_R could meet the requirement. As the hull resistance is highly sensitive to speed, it is obvious that the best solution is to find the exact ME power with which the ship will travel distance S_R using time T_R. For an arbitrary reasonable tr₀, there is an exact such ME power that makes S_R and T_R meet the requirement, and the one that has the lowest fuel consumption is the optimal trim for this issue, and the corresponding ME power is the required optimal ME power.

Equation (5) indicates the relationship of the differential of time dt and the differential of draft dd. The total travel time T could be written as the integral of the draft by the following:

$$T=-{\int }_{d_0}^{d_1}\frac{{f_D}^{\prime }\left(d_0\right)+f{D}^{″}\left(d_0\right)\left(d-d_0\right)}{{(C}_o+C_w)}dd$$

(22)

In Equation (22), T is a function of the initial trim tr₀ and the final draft d₁. Similarly, in Equation (18), the travel distance S is also a function of d₁ and tr₀. If the ME power, which is denoted by P_B, is regarded as a variable, then the travel time and travel distance can be written in the form of T(P_B, d₁, tr₀) and S(P_B, d₁, tr₀), respectively. The problem of trim and ME power joint optimization with fixed travel distance and time can be expressed by the following optimization model:

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\min P_B \\ s.t. T\left(P_B,d_1,{tr}_0\right)=T_R \end{array}\\ S\left({P_B,d}_1,t{r}_0\right)=S_R\end{array}\\ {tr}_{min}\le {tr}_0\le {tr}_{max}\end{array}$$

(23)

Compared to the optimization model (21), model (23) has one more variable, P_B. As a result, the solution of the optimization mode (23) is more complex. In this research, a practical method is proposed, and the optimization mode (23) is solved by Algorithm 3.

Algorithm 3 Calculation of optimal initial trim with minimum fuel consumption on condition of fixed travel distance and time

1: Assume that PBmin and PBmax are the minimum and maximum permissible ME power values. Create a set of ME power values within the range of PBmin and PBmax with equal spacing in ascending order, denoted by PB1, PB2,…, PBnb, where nb is the number of ME power values, PB1 = PBmin, and PBnb = PBmax. 2: for i=1 to nb step 1. 3: Construct two point sets, one for travel time denoted by PTA and the other for optimal trim denoted by PTRA. 4: Calculate the speed vs. draft and trim surface, SurVi, according to ME power PBi. 5: According to SurVi, obtain the optimal initial trim tr0i and the corresponding final draft d1i for travel distance SR by Algorithm 2. 6: Calculate the travel time Ti according to tr0i and d1i with Equation (22). Add point (PBi, tr0i) into point set PTRA, and add point (PBi, Ti) into point set PTA. 7: end for 8: Create a spline curve SPPB-T with point set PTA, and create a spline with point set PTRA denoted by SPPB-tr. 9: With the interpolation algorithm of the spline curve, obtain the X coordinate of SPPB-T where the Y coordinate is TR, and denote the calculated X coordinate by PBopt. 10: Calculate the Y coordinate of SPPB-tr where the X coordinate is PBopt, and denote it by tr0opt. 11: PBopt is the optimal ME power, and tr0opt is the corresponding optimal initial trim. The combination of PBopt and tr0opt is the optimal solution which has the minimum fuel consumption over the whole voyage and meets the requirement of travel distance SR using time TR.

It should be noted that the fuel consumption rate of the ME is not constant; instead, it changes along with the ME power. Generally speaking, the fuel consumption rate is minimum when the ME power is near NCR. When applying trim and ship power joint optimization, it is necessary to calculate the accurate fuel consumption rate C_O according to the corresponding ME power.

3. Hydrodynamic Data and Hydrostatic Functions

3.1. Calculation of Hull Resistance as Well as the Speed vs. Draft and Trim Surface

In this paper, the two proposed optimization methods are both based on the speed vs. draft and trim surface, the foundation of which is hull resistance under different combinations of speed, draft, and trim. In this research, the hull resistance is calculated by CFD numerical simulations and saved in the form of trim tables. Assume the ME power is P_BC, and the speed vs. draft and trim surfaces are created based on the trim tables by Algorithm 4.

Algorithm 4 Calculation of speed vs. draft and trim surfaces

1: Create a set of speeds within the range of minimum speed vmin and maximum speed vmax with equal spacing in ascending order, denoted by v1, v2, …, vnv, where nv is the number of speeds, v1 = vmin, and vnv = vmax. 2: Create a set of initial trims within the range of trmin and trmax with equal spacing in ascending order, denoted by tr1, tr2, … trnt, where nt is the number of trims, tr1 = trmin, and trnt = trmax. 3: Create a set of final drafts within the range of minimum draft dmin and maximum draft dmax with equal spacing in ascending order, denoted by d11, d12, … d1nd, where nd is the number of drafts, d11 = dmin, and d1nd = dmax. 4: Based on CFD numerical simulations, calculate the resistance for each combination of di (1 ≤ i ≤ nd), trj (1 ≤ j ≤ nt), and vk (1 ≤ k ≤ nv), and construct the trim table. 5: Based on the trim table, create nd×nt curves of ME power, which indicate the function of ME power vs. speed for each draft and trim combination. The ME power curve for draft di and trim trj is denoted by SPBij. 6: Based on the interpolation algorithm of the spline curve, calculate the speed corresponding to PBC with draft di (1 ≤ I ≤ nd) and trim trj (1 ≤ j ≤ nt) by the ME power curve SPBij, and denote the speed by vij. 7: After all the nd × nt speeds are obtained, construct a surface, SURPBC, with points (di,trj,vij) as fit points. SURPBC is the speed vs. draft and trim surface for ME power PBC, as shown in Figure 2. 8: Calculate the speed vs. draft and trim surfaces for all required ME powers using the above procedure.

Figure 2. Creation of speed vs. draft and trim surface according to ME power curve.

Figure 2. Creation of speed vs. draft and trim surface according to ME power curve.

3.2. Calculation of Functions Related to Hydrostatic Curves

In this research, three hydrostatic related functions, which are the displacement vs. draft function f_D(d), the LCB vs. draft function f_B(d), and the MCM vs. draft function f_M(d), are used when calculating the relationship of the draft with the trim of a ship during a voyage. Moreover, the first and second derivatives of f_D(d), f_B(d), and f_M(d) are also required. These three functions are all represented by cubic NURBS curves in this study, denoted by S_D, S_B, and S_M, respectively. A key problem is how to obtain the first and second derivatives of S_D, S_B, and S_M at a given X coordinate.

The expression of the NURBS curve is written as follows:

$$P\left(u\right)=\frac{\sum _{i=0}^{n-1}{B}_{i,k}\left(u\right)W_i{V}_i}{\sum _{i=0}^{n-1}{B}_{i,k}\left(u\right)V_i}$$

(24)

where u is the parameter, k is the degree, B(i,k) is the basic function, W_i is the weight, and V_i is the control point. Assume the following:

$$\left\{\begin{array}{c}U={\displaystyle \sum _{i=0}^{n-1}}{B}_{i,k}\left(u\right)W_i{V}_i\\ L={\displaystyle \sum _{i=0}^{n-1}}{B}_{i,k}\left(u\right)V_i\end{array} \right.$$

(25)

Then the first derivative of Equation (24) is written as follows:

$$P^{\prime }\left(u\right)=\frac{{(U}^{\prime }L-{UL}^{\prime })}{L^2}$$

(26)

The second derivative of Equation (24) is obtained by calculating the derivative of Equation (26) as follows:

$$P^{″}\left(u\right)=\frac{(L^3{U}^{″}-U{L}^2{L}^{″}-{2L^{2}U}^{\prime }{L}^{\prime }+2UL{L^{\prime }}^2)}{L^4}$$

(27)

S_D, S_B, and S_M are 2D splines, which means $P^{\prime }\left(u\right)$ and $P^{″}\left(u\right)$ are vectors with a length of two, and consist of the derivative of X and Y to the parameter u as follows:

$$\left\{\begin{array}{c}{P}^{\prime }\left(u\right)={\left\{x^{\prime }\left(u\right),y^{\prime }\left(u\right)\right\}}^T\\ P^{″}\left(u\right)={\left\{x^{″}\left(u\right),y^{″}\left(u\right)\right\}}^T\end{array}\right.$$

(28)

As the horizontal coordinates of S_D, S_B, and S_M are monotonically increasing, these spline curves could be written in the following form:

$$y=f\left(x\right)$$

(29)

The first derivative of f(x) is written as follows:

$${ f}^{\prime }\left(x\right)=\frac{y^{\prime }\left(u\right)}{x^{\prime }\left(u\right)}$$

(30)

Similarly, the second derivative of f(x) is obtained by the following:

$$f^{″}\left(x\right)=\frac{y^{″}\left(u\right)x^{\prime }\left(u\right)-y^{\prime }\left(u\right)x^{″}\left(u\right)}{{x^{\prime }}^3\left(u\right)}$$

(31)

where $x^{\prime }\left(u\right), y^{\prime }\left(u\right), x^{″}\left(u\right)$, and $y^{″}\left(u\right)$ are calculated by Equation (28), and they are substituted into Equation (30) and Equation (31), and then the first and second derivatives of f_D(d), f_B(d), and f_M(d) are obtained.

4. Numerical Examples

In this section, a 307000 DWT VLCC is taken as an example to demonstrate the application of the proposed optimization methods, and also to validate the efficiency and practicability of the methods.

4.1. Principal Dimensions of the Ship

The target ship is a typical VLCC, and the principal dimensions and other related parameters are shown in

Table 1. Principal dimensions of the ship.

Item	Value
Mold length	320m
Mold Breadth	60 m
Mold Depth	29.5 m
Design Draft	20.5 m
Scantling Draft	21.5 m
Service Speed	15 kn
Endurance	25,600 n miles
Deadweight (Design)	286,000 t
Deadweight (Scantling)	307,000 t
Fuel Oil Capacity	6998 t
Other consumables Capacity	1901 t

.

4.2. Hydrodynamic Analysis with CFD Numerical Simulations

The computational domain and boundary conditions are shown in Figure 3. In order to balance the computational resources and simulation precision, the unstructured mesh is used for local mesh refinement near the hull surface, and different mesh configurations are adopted for the areas near the bow, stern, and water plane. The size of the mesh is determined by the independence verification for each set of parameters, and the mesh number is about 3 million. The governing equation is the RANS equation with the standard k-ε model. The free surface was captured by the Volume of Fluid (VOF) method, and the implicit unsteady solver was used to solve the non-uniform flow field around the ship.

The commercial software StarCCM+ (Version 15.02.007) is used to perform the hydrodynamic analysis and obtain the hull resistance for different conditions. The range and spacing of the draft, trim, and speed are selected according to the characteristics of the ship. Specific parameters include four drafts (20.5 m, 18.5 m, 16.5 m, 14.5 m), four trims (−1 m, 0 m, 1.5 m, 3 m), and three speeds (12 kn, 14 kn, 16 kn). There are 48 combinations, and the typical wave pattern from the results for draft 20.5 m is shown in Figure 4.

Based on the hull resistance results by CFD simulations, Effective Horse Power (EHP) under different draft, trim, and speed combinations is calculated. Subsequently, the delivered power as well as the corresponding ME power are calculated. Then with Algorithm 4, the speed vs. draft and trim surfaces for each assumed ME power are created, and the surfaces for the ME power at 100% NCR, 80% NCR, 60% NCR, and 40% NCR are as shown in Figure 5.

4.3. Calculation of Optimal Initial Trim with Fixed ME Power

A typical route of the VLCC is taken as an example to perform the trim optimization and calculate the optimal initial trim with fixed ME power. The length of the route is 17,000 n mile, and the required travel time is 1152 h (48 days). The fuel oil and other consumables at departure are 6998 t and 1901 t, respectively. The optimization is carried out for different displacements, which are 340,000 t, 320,000 t, 300,000 t, 280,000 t, 260,000 t, and 240,000 t, and those displacements cover the range of the typical fully loaded and partly loaded conditions of this ship. Two ME powers, which are 100% NCR and 60% NCR, are taken as the fixed ME power. With the proposed trim optimization method, the optimal initial trims over the whole voyage at the above two ME powers for each assumed displacement are calculated.

4.3.1. Results for ME Power at 100% NCR

For the case of ME power at 100% NCR, the speed vs. draft curves for each trim and displacement combination are calculated with Algorithm 1, as shown in Figure 6. The horizontal coordinate of the curve is the draft, the vertical coordinate is the speed, and each curve stands for one initial trim. Being different from the speed vs. draft and trim surface, the meaning of the trim in Figure 6 is the initial trim at departure, and the real trim during the voyage is calculated by Equation (14), and it is not shown in these figures.

The trim optimization results with ME power at 100% NCR are shown in

Table 2. Results of trim optimization with ME power at 100% NCR.

D0,t	OTWV Trim, m	OTID Trim, m	OTWV Fuel Oil, t	OTID Fuel Oil, t	Saved Fuel Oil, t
220,000	0.990	0.011	3706.4	3735.2	28.8
240,000	1.195	0.075	3774.1	3820.4	46.2
260,000	1.220	0.058	3800.6	3828.5	27.9
280,000	0.585	0.000	3821.6	3822.6	1.0
300,000	0.000	0.000	3840.6	3840.6	0.0
320,000	0.000	0.000	3899.0	3899.0	0.0
340,000	1.410	0.555	3968.1	3972.9	4.7

, in which the first column is the displacement, the second column is the OTWV, the third column is the Optimal Trim at Initial Draft (OTID), the fourth column is the fuel consumption with the OTWV, the fifth column is the fuel consumption with the OTID, and the sixth column is the difference between column five and four, which is the fuel oil saved when using the OTWV in comparison to using the OTID. The table indicates that the OTWV is significantly different from the OTID for five out of the seven displacements, and that the OTWV has a lower fuel consumption than the OTID. For the case of the 240,000 t initial displacement, the OTWV saves up to 46.2 t fuel oil, which accounts for 1.2% of the total fuel consumption compared to the OTID. The results confirm that it is necessary to use the OTWV rather than the OTID for a voyage without ballast changing and load redistribution operations.

As the travel distance is a variable of the trim optimization over a whole voyage, further study for travel distances in the range of 15,000 to 20,000 n miles with a spacing of 1000 n miles is presented, as shown in Figure 7, where only the energy saving effect is shown due to limited space. In this figure, the horizontal coordinate is the displacement, the vertical coordinate is the fuel oil saved by the OTWV compared to the OTID, and each bar stands for a specified travel distance. This histogram indicates that the OTWV has a significant fuel oil saving effect for different voyage lengths, and the fuel oil saved has a positive correlation to the length of the voyage. For the case of the 240,000 t displacement and 20,000 n mile voyage length, the OTWV could save 67.6 t more fuel oil than the OTID.

4.3.2. Results for ME Power at 60% NCR

The trim optimization based on the OTWV with ME power at 60% NCR is carried out. For the case with a voyage length of 17,000 n miles, the speed vs. draft curves for different displacements are shown in Figure 8, and the optimal trim as well as the fuel oil saving effect of the OTWV compared to the OTID are listed in

Table 3. Results of trim optimization with ME power at 60% NCR.

D0, t	OTWV Trim, m	OTID Trim, m	OTWV Fuel Oil, t	OTID Fuel Oil, t	Saved Fuel Oil, t
220,000	0.810	0.000	2568.7	2577.8	9.1
240,000	1.010	0.059	2622.3	2642.5	20.2
260,000	0.985	0.056	2656.1	2669.9	13.8
280,000	1.005	0.000	2686.6	2687.8	1.3
300,000	0.000	0.000	2719.3	2719.3	0.0
320,000	0.000	0.000	2769.0	2769.0	0.0
340,000	1.440	0.535	2820.6	2823.8	3.2

.

The results of ME power at 60% NCR are similar to those at 100% NCR. When the voyage length is 17,000 n miles, and the displacement is 240,000 t, 260,000 t, 280,000 t, 300,000 t, 320,000 t, and 340,000 t, respectively, the optimal trim by the OTWV is different from the OTID for five out of the seven specified displacements, and for those cases, the OTWV could save more fuel oil than the OTID. For the case with a displacement of 240,000 t, the OTWV consumes 20.2 t less fuel oil than the OTID, which occupies about 0.8% of the total fuel consumption.

Besides 17,000 n miles, the cases of other specified voyage lengths, as described in Section 4.3.1, are also analyzed. The results of the fuel oil consumption difference between the OTWV and the OTID are shown in Figure 9. When the ME power is at 60% NCR, using the trim by the OTWV allows for saving more fuel oil than that by the OTID for different voyage lengths. The maximum fuel oil savings occurs when the displacement is 240,000 t and the voyage length is 20,000 n miles, and the OTWV could save 29.2 t, about 0.9% of the total fuel consumption compared to the OTID in this case.

4.4. Trim and ME Power Joint Optimization with Fixed Voyage Length and Fixed Travel Time

The trim and ME power joint optimization with fixed voyage length and fixed travel time is carried by taking a typical route as an example. The configurations of the voyage, travel time, and fuel oil and other consumables at departure are the same as those described in Section 4.3.

The optimal ME powers for different displacements and travel times are shown in Figure 10, where the horizontal coordinate is the required travel time, the vertical coordinate is the optimal ME power in percentage of NCR, and each curve corresponds to a specified displacement. With the proposed method, the optimal trim and ME power combination can be found for different voyage lengths and travel times.

The optimal initial trim corresponding to the optimal ME power in Figure 10 is shown in

Table 4. Results of OTWV and OTID for the combination of different specified required travel times and displacements.

D0, t	TR = 1100 h	TR = 1150 h	TR = 1200 h	TR = 1250 h	TR = 1300 h	TR = 1350 h	TR = 1400 h	TR = 1450 h
220,000	0.89/0.00	1.18/0.00	1.20/0.00	0.59/0.00	0.50/0.00	0.47/0.00	0.49/0.00	0.53/0.00
240,000	1.48/0.07	0.68/0.07	0.76/0.06	0.81/0.06	0.81/0.07	0.80/0.06	0.78/0.06	0.79/0.06
260,000	1.20/0.06	1.22/0.06	1.18/0.06	0.97/0.05	1.24/0.05	1.14/0.05	0.86/0.05	0.64/0.05
280,000	0.58/0.00	0.48/0.00	0.06/0.00	0.40/0.00	0.59/0.00	0.60/0.00	0.59/0.00	0.60/0.00
300,000	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.00
320,000	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.00	0.00/0.04	0.00/0.10	0.00/0.01
340,000	0.96/0.00	0.96/0.20	0.98/0.25	0.99/0.18	1.39/0.53	1.40/0.54	1.39/0.54	1.38/0.58

, in which the first column is the displacement and the first row is the specified travel time. Within each data grid, two trims are presented, separated by a diagonal line; the left value represents the OTWV, and the right one represents the OPID. Although sharing the same initial displacement, the trim data within the same line may vary slightly due to each travel time corresponding to an optimum ME power. From this table, it can be seen that there are significant differences between the OTWV and the OPID for the fixed voyage length and fixed travel time problem at the initial displacement of 220,000 t, 240,000 t, 260,000 t, 280,000 t, and 340,000 t. It proves once again the necessity of using the OTWV as the initial trim.

The fuel oil saving effect results corresponding to Figure 10 and Table 4 are shown in Figure 11. The horizontal coordinate of Figure 11 is the initial displacement, and the vertical coordinate is the decrease in fuel oil consumption by the OTWV compared to the OTID. Each column represents a combination of a specified displacement and travel time. In the cases of the 300,000 t and 320,000 t displacements, the initial trims by the OTWV and those by the OTID are the same, which means the fuel oil decrease is zero, so these two cases are not shown in the figure. Significant energy-saving benefits are observed within the displacement range of 220,000 t to 260,000 t, with a maximum fuel oil savings of 33 t, equivalent to approximately 1.0% of the total oil consumption.

5. Discussion

This paper develops a trim optimization method to find the optimum initial trim for minimizing fuel consumption over a whole voyage with fixed ME power. Based on this trim optimization method, a trim and ME power joint optimization method is presented on the premise that the voyage length and travel time are all specified. The proposed two optimization methods are applied to a typical VLCC. In the trim optimization with fixed ME power and voyage length, the proposed method can obtain the optimal trim for minimizing the fuel oil consumption over a whole voyage. The results demonstrate that utilizing the OTWV can result in savings of up to 1.2% of the total fuel consumption compared to the OTID. In the application of trim and ME power joint optimization with fixed voyage length and travel time, the proposed method can find the optimal combination of trim and ME power. As the ship sails at a relatively steady speed over the whole voyage, it will save a larger amount of energy than the variable ME power sailing plan. Furthermore, using the OTWV as the initial trim leads to an additional reduction in fuel consumption, up to approximately 1.0% compared to using the OTID as the initial trim.

It should be noted that the aforementioned energy-saving results are calculated based on the hull resistance by CFD simulations in calm water. Due to the fact that CFD simulations are usually not accurate enough for the hull resistance estimation, and, in addition, the random variables in a real cruise are not taken into account, the calculated fuel oil consumption quantity in this paper may have significant errors compared to the real ship oil consumption. However, that does not have much adverse influence on the conclusion of this study. This study focuses on the comparison of hull resistance with different trims. Although the absolute values obtained by CFD simulations may lack precision, they are accurate enough for the tangency analysis of hull resistance with different parameters, including trim. A good example is that CFD is widely used in hull form optimization as well as trim and speed optimization in engineering practice.

The proposed methods take consideration of the characteristics of ship navigation and maneuvering, so the optimization results can be easily adopted in engineering practice. The concept of the OTWV is developed to obtain the optimal initial trim which makes the total fuel oil consumption minimum if no ballast changing or load redistribution operations are allowed during the voyage, and it has been proved that the OTWV could save more fuel oil than the OTID. In the cases when both voyage length and travel time are specified, the proposed trim and ME power joint optimization method could find the optimal combination of trim and ME power. The proposed methods are practical in both the design and operation phases of a ship. In the design stage, these methods could be used in the optimization of loading conditions to deduce the EEDI of a cargo ship. In the operation stage, these methods could be applied in the loading computer software, with which the loading plan could be efficiently optimized, and this would be an efficient way to reduce the EEOI for a cargo ship.

This study indicates that, when applying trim optimization for a voyage with no intermediate ballast changing or load redistribution, it is necessary to take the OTWV rather than the OTID as the initial trim, for the OTWV could save more fuel oil than the OTID. It also reveals that if the ship speed is monotonically increasing or monotonically decreasing with the trim when the ME power and draft are fixed, the values of the OTWV and the OTID are identical. Otherwise, if the speed is not monotonous to the trim, which means the maximum or minimum speed occurs in the middle of the trim range, the OTWV is different from the OTID, and taking the OTWV as the initial trim could save more energy than the OTID.

In this paper, the ship resistance, based on which the relationship of speed vs. draft and trim surface is created, is obtained by CFD numerical simulations under different combinations of draft, trim, and speed. As CFD is low in accuracy if the hydrodynamic model or parameters are not properly set [22], model tests or full-scale measurements of real ships could be used instead of CFD simulations to gain more accurate resistance results for the proposed optimization methods. The hull resistance is the input data only for the proposed methods, and can be obtained using various methods. Model tests or real ship measurements have a high cost, but they could obtain more accurate and reliable optimization results.

The hull resistance in calm water is used as the foundation of the two optimization methods in this research. When a ship sails on the water, the hull resistance is affected by wind, waves, and current, especially in a rough sea state. The added resistance from wind, waves, and current will influence the accuracy of optimization. For example, in the proposed trim and ME power joint optimization with fixed voyage length and fixed travel time, the ship will not be able to travel the specified distance with the optimized ME power due to the decrease in speed by wind, waves, and current. Therefore, in practice, it is necessary to leave some margin based on the sea state and condition of the ship for this case. In further studies, trim optimization along with trim and ME power joint optimization considering the added resistance of wind, waves, and current will be studied, and that would be a more efficient way to achieve better energy saving and emission reduction for various types of ships.