1.Introduction
With the development of information technology, ship modelling technology is required to satisfy all ships and navigational conditions. Various loading conditions of a vessel, which also affects draught and trim conditions, is one of the major constraints to determine manoeuvrability. For example, fully loaded vessel requires greater turning circle than the one in ballast condition and trimmed by stern condition has larger circle than even keel condition (Kijima et al., 1990; Oltmann, 2003).
The most reliable way to examine ship’s manoeuvrability considering with loading conditions is to conduct the model tests, such as planar motion mechanism, rotating arm and towing tank, or real ship trial for every loading condition. However, it requires expensive time and cost for the experiment (Yoon et al., 2016). IMO standards for ship manoeuvrability consider that and require only for full load condition of the vessel, also it is hard to get such data for all ships (Im et al., 2005).
International Towing Tank Conference (ITTC) summarized multiple ways to estimate and to approximate hydrodynamic coefficients for the ship manoeuvrability (ITTC, 2008). Computational methods, such as Computational Fluid Dynamics (CFD) method and system identification method are also an alternative for the model test.
As a preliminary study for estimation of all loading conditions, this paper estimates hydrodynamic coefficients with sets of sea measurement data by mathematical optimization, which is a kind of the system identification method. Based on the authors’ previous researches, coefficients are optimized through the Interior point algorithm and these are validated through comparison with the measurement data from sea trial (Kim et al., 2016; Kim et al., 2017).
2.Modelling ship and benchmark data
2.1.Mathematical model
This study applied the 3DegreesofFreedom (DOF) shipfixed and earthfixed coordinate systems. Fig. 1 illustrates basic information of the coordinate systems. The Shipfixedcoordinate $Oxy$ plane and the Earthfixedcoordinate ${O}_{0}{x}_{0}{y}_{0}$ plane are placed on the undisturbed free surface, with the ${x}_{0}$ axis pointing in the direction of the original heading of the ship. The ${z}_{0}$ axis and the z axis point downwards vertically. The angle between the directions of the ${x}_{0}$ axis and the x axis is defined as the heading angle, ψ.
where,

G : Center of gravity

Ψ : Heading

β : Drift angle

δ : Rudder angle

$\overrightarrow{V}$ : Ship’s speed

r : yaw rate
A fast time simulation tool SIMOPT with a mathematical model of a Ship Handling Simulator (SHS) systems ANS5000, developed by Rheinmetall Defence Electronics is used for the optimization process (ISSIMS GmbH, 2013). In the mathematical model of the tool, a ship is considered as a massive and rigid body and forces and moment are acting on the hull can be described as equation (1), according to the Newtonian law of motion (Rheinmetall Defence Electronic, 2008).
The model applies modular structure to each force and moment as equation (2): hull, propeller, rudder and other external forces and moments.
The hydrodynamic forces and moment acting on the hull are composed as Equation (3) and the empirical regression formulas of Norrbin and Clarke are applied to calculate the initial hydrodynamic coefficients (Norrbin, 1971; Clarke et al., 1983). Each hydrodynamic coefficients can be expressed the function of ship’s main dimension as equation (4): length, beam, draught and displacement of the ship. Y_{non} and N_{non} are nonlinear components of sway force and yaw moment. These nonlinear components vary according to the position of the ship’s turning point.
2.2.Benchmark data
A set of benchmark data is acquired from sea trials using a 4,500 TEU class container carrier. Details of the vessel are given in Table 1.
Five zigzag manoeuvres under three different loading conditions were carried out for this study. Table 2 shows detailed conditions for each manoeuvre.
3.Estimation of hydrodynamic coefficients
3.1.Mathematical optimization
The mathematical optimization is a process to minimize or maximize an objective function value, subject to several constraints on its variables (Nocedal and Wright, 2006). This can be written as equation (5):
where,

 x is the variable, which has to be optimized ;

 f is the objective function, a function which returns scalar and it contains the information of minimization or maximization;

 c_{i} are constraints, which sets equations and inequality condition those the variable x must satisfy during the whole optimization process. The constraints are optional in the optimization process.
Fig. 2 shows the whole process of the mathematical optimization to get tuned hydrodynamic coefficients. The optimization process in the process is carried out by the Optimization Toolbox of MATLAB.
The solver requires an objective function, which calculates a minimum or a maximum value of the function. Constraints also can increase the reliability of the optimization results. In this study, lower and upper bound are applied as constraints of the optimization process.
3.2.Sensitivity analysis
A lot of target variables for the optimization process requires expensive resources for the calculation. Therefore this study conducted a sensitivity analysis of each coefficient with the corresponding manoeuvre, prior to the main optimization process. The procedures of the sensitivity analysis are as follows:

1) Split coefficients into two groups according to the manoeuvre tests: the constant speed with straight motion and zigzag manoeuvre.

2) Conduct manoeuvring simulations with regard to certain changes of a specific coefficient.

3) Get derivative of each data set and conduct minmax normalization for all hydrodynamic coefficients to figure out their own sensitivity in the group.
Fig. 3 and Fig. 4 show the results of sensitivity analysis and Table 3shows the list of coefficients for the optimization of this study. Stepwise optimization is carried out based on the results of the sensitivity analysis. Two coefficients for the force acting on Xaxis is optimized with straight motion with constant speed. Also four linear coefficients for the force acting on Y and Zaxis is optimized with various zigzag manoeuvres.
3.3.Optimization conditions
Table 4 shows an example of optimization conditions, a condition for measurement data 3. The initial values of the optimization are calculated by the Clarke estimation and lower and upper bounds are set by values close to 0 for each sign or 10 times the initial values. The object function calculates differences of X and Y coordinates between the benchmark data and the optimized coefficients at each iteration.
Optimizations are carried out only for three data sets, measurement data 2 to 4, which have different trim and draught conditions. Data 1 and data 5 are used for validation of optimization results with corresponding trim and draught.
4.Validation of optimization results
4.1.Validation with corresponding benchmarks
Table 5 and Table 6 presents optimization results, coefficients and corresponding manoeuvre characteristics, respectively. Simulation results using Clarke estimation coefficients have a relatively big difference to the benchmark data compared with the simulations results using the optimized coefficients.
As seen in Fig. 937 to Fig. 10, Simulation with optimized coefficients made similar heading values to the reference data and this also enables similar trajectory compared to the simulation using Clarke estimation coefficients.Fig. 5Fig. 6Fig. 7Fig. 8Fig. 9
4.2.Validation with other manoeuvre data
An additional validation is carried out with rest manoeuvre measurements. Trajectory and heading records for Data 1 and Data 5 are compared simulation results using coefficients from optimization of Data 2 and Data 4, respectively. Table 7 and Fig. 11 to Fig. 14 presents a comparison between benchmark and simulation results. For Data 1, the second overshoot angle and Way/Lpp are still differed from the benchmark values and these are related to the difference of trajectory between them. Whereas the simulation result for Data 5 is almost similar with the benchmark data even it uses the coefficients optimized from Data 4, which are based on same zigzag manoeuvre, but different rudder angles.Fig. 12Fig. 13
4.3.General review
Two kinds of validations are carried out in this paper. Firstly a comparison among a measurement data, a simulation result using existing coefficient estimation formulas and a simulation result using optimized hydrodynamic coefficients shows that the result using optimized coefficients are relatively similar with the benchmark data than the result using original coefficients. As a second validation, an another measurement data with the same trim draught conditions for the first validation is chosen as a benchmark data. Same as the first validation, the result using optimized hydrodynamic coefficients is similar with the benchmark data. However, additional validations using other manoeuvres, such as turning manoeuvre and emergency manoeuvres, are still required for higher reliabilities of the optimization results.
5.Conclusion
This paper estimated hydrodynamic coefficients for modelling ship under various loading conditions. The mathematical optimization, which is a kind of the system identification method, is applied to calculate the coefficients. Three different loading conditions and in five sea trial measurement data are used for the benchmark data. Also, two kinds of test manoeuvres, straight motion with constant speed and zigzag manoeuvre, are applied in the optimization process in consideration of measurement data and coefficients to be optimized. The study can be summarized as follows:

1) Simulation results using optimized hydrodynamic coefficients are relatively close to the benchmark data, comparing with the one using the coefficients calculated by the Clarke estimation formulas.

2) For the additional validation, the optimized coefficients agree well with the benchmark data, which is the same loading condition with the original benchmark data.

3) However, due to limitation of data measuring, validations using other manoeuvres except a manoeuvre which is used for the optimization process are not carried out in this paper.
Based on the results of this study, it could be possible to get a new estimation formulas to complement the existing Clarke’s formulas in the future studies. In addition, more manoeuvres with various loading conditions and types are still required for higher reliability of the new suggestion.