# Point-to-Point-Based Optimization Method of Ballast Water Allocation for Revolving Floating Cranes with Experimental Verification

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## Abstract

**:**

## 1. Introduction

## 2. PTP-Based Optimization Method of Ballast Water Allocation

#### 2.1. PTP Theory

#### 2.2. The PTP-Based Optimization Method for Ballast Water Allocation

- (i)
- By considering the given information of the lifting task, including the lifting trajectory, the lifting load, the lifting speed, etc., the ballast water allocation model of the RFC is established, and the water level changes of each ballast tank are described using the PTP theory. In this way, the optimal control problem of the ballast water allocation is transfered to a nonlinear constraint problem and the polynomial coefficients are treated as design variables;
- (ii)
- Time discretization is performed within the entire time domain of the RFC’s lifting processes. For all the time instants obtained from the discretization, one-to-one correspondences between the lifting trajectory positions and the water levels in each ballast tank are established based on the principle of torque balance. This step effectively connects the lifting trajectory with the water-level variation in each tank, laying the foundation for subsequent optimization processes;
- (iii)
- The optimization model of the PTP-based optimization method of ballast water allocation is built by considering relevant system constraints while the RFC is working, such as vessel tilt angles and the moment equilibrium;
- (iv)
- An appropriate intelligent optimization algorithm is selected to solve the PTP-based optimization problem. The convergence condition of the optimization process is set as the residual error being less than the residual error standard. If the converge condition is met, the optimal ballast water-allocation scheme is obtained; otherwise, the optimizer will update the design variables and continue with the iterative optimizing process.

## 3. Ballast Water-Allocation Optimization Problem Formulation

#### 3.1. Objective Function

#### 3.2. Design Variables

#### 3.3. Constraints

- (i)
- Hull inclination constraints

- (ii)
- Ballast water level constraint

- (iii)
- Constraint of the ballast water allocation rate

- (iv)
- Total ballast water constraint

#### 3.4. Optimization Model

## 4. Case Study and Superiority Verification

#### 4.1. Case Study of the PTP-Based Method

#### 4.2. Comparison Analysis of the PTP-Based Method

#### 4.2.1. Conventional Optimization Method for Ballast Water Allocation

#### 4.2.2. Comparison Results Analysis

#### 4.3. Superiority Discussion

## 5. Numerical Experiments for the PTP-Based Optimization Method

#### 5.1. Numerical Experiments Regarding Different Lifting Trajectories

_{1}, TR

_{2}, and TR

_{3}, are listed in Table 6, and the corresponding three-dimensional representations of the trajectories are illustrated in Figure 9. In addition, in this numerical experiment, the lifting loads are uniformly set to be 300 t.

_{1}, the optimal ballast scheme is shown in Figure 10, including the duration curves of the water level and the hull inclination. It can be seen in Figure 10a that, as the lifting operation continues, the water level in Tank 2 on the port side shows a downward trend approximately, while the water level in Tank 7 on the starboard side shows rising and then decreasing trends. The water level in Tank 3 decreases slightly, and the corresponding magnitudes of change are minimal. As seen, the actual ballast operation regarding TR

_{1}is mainly determined by three tanks, including Tanks 1, 2, and 7. In addition, it is observed from Figure 10b that the hull inclinations are completely within the safe range.

_{2}and TR

_{3}are presented in Figure 11 and Figure 12, respectively. From Figure 11a and Figure 12a, the obtained duration curves of the water levels in each ballast tank are continuous and smooth without abrupt changes, which conforms to the practical ballasting controls. From Figure 11b and Figure 12b, both the heeling and trim angles of the hull remain within the safe range throughout the lifting process, fully complying with the safety regulations.

#### 5.2. Numerical Experiments Regarding Different Lifting Loads

_{2}, as listed in Table 6, so as to facilitate the analysis of the impact of load mass on the optimization results.

#### 5.3. Resulting Discussion of Numerical Experiments

## 6. Experimental Validation of the PTP-Based Optimization of the Ballast Water Allocation

#### 6.1. RFC Physical Test Stand Establishment

#### 6.2. Experimental Scheme and Procedure

- (i)
- The RFC test stand is slowly launched in the designated water area by the overhead crane and its position angle is adjusted for experimental observation, as illustrated in Figure 19a. After that, the test stand needs to stand for 10 min in order to eliminate the influence of ripples on the accuracy of the subsequent test data;
- (ii)
- According to the per-set lifting trajectory and load weight, the optimal ballast water allocation scheme is obtained by performing the PTP-based method, and the corresponding logical instructions of the operations of the test stand are written, as shown in Figure 19b;
- (iii)
- Based on the written logic instructions, the test stand is controlled to perform the lifting operations according to the pre-set lifting trajectory and the optimal ballast water allocation scheme obtained using the PTP-based method, as illustrated in Figure 19c. During the test process, the key experimental data, including the heeling angle, the trim angle, and the time consumption are collected and recorded in real time for subsequent analysis;
- (iv)
- The data of the heeling and trim angles of the hull are processed and compared with the theoretical values to evaluate the reliability of the ballast water allocation scheme obtained from the PTP-based method. Additionally, the heeling and trim angles are also assessed to determine whether they meet the safety requirements for offshore lifting operations. If the requirements are met, the feasibility of the PTP-based method in practical applications can be confirmed.

#### 6.3. Resulting Discussions of Physical Experiments

_{1}as mentioned in Section 5, and the lifting load is determined to be 25 kg. In order to reduce the experimental error and improve the corresponding reliability, the same experiment is repeated five times for the above load and trajectory. By collecting and processing the real-time data of the lifting experiments, comparisons of the hull inclination between the theoretical and experimental values are illustrated in Figure 20. The dotted lines with markers represent the histories of the heeling and trim angles obtained from the five repeated lifting experiments. The dotted lines without markers represent the average values of the heeling and trim angles obtained by the five repeated experiments. The theoretical values of the heeling and trim angles obtained from the PTP-based optimization method of ballast water allocation are represented by solid lines.

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${A}_{\mathrm{c}}$ (m^{3}) | Transverse projected area of the ship above the waterline |

a (m) | Wave amplitude |

${b}_{n}$ | Polynomial coefficients |

${b}_{p0}\sim {b}_{p6}$ | Polynomial coefficients |

${C}_{\mathrm{mx}}$, ${C}_{\mathrm{my}}$ | Wave moment coefficients in x and y directions, respectively |

${C}_{\mathrm{wx}}$, ${C}_{\mathrm{wy}}$ | Wind moment coefficients in x and y directions, respectively |

${d}_{\mathrm{x}}$ (m) | Longitudinal positions of the load barycenter |

${d}_{\mathrm{y}}$ (m) | Transverse positions of the load barycenter |

${f}_{B}$ | The optimization objective |

$\overline{GM}$ (m) | Initial transverse metacentric height before loading |

$\overline{G{M}_{\mathrm{L}}}$ (m) | Initial longitudinal metacentric height before loading |

$\overline{{G}_{1}{M}_{1}}$ (m) | Transverse metacentric height influenced by free surface effect |

$\overline{{G}_{1}{M}_{1\mathrm{L}}}$ (m) | Longitudinal metacentric height influenced by free surface effect |

${\mathit{g}}_{\mathrm{B}}$ | The inequality constraints |

${\mathit{h}}_{\mathrm{B}}$ | The equality constraints |

${h}_{p}\left(t\right)$ (m) | Water level change of ballast tank p at time t |

${\dot{h}}_{p}$ (m/s) | Speed of water level change in ballast tank p |

${\ddot{h}}_{p}$ (m/s^{2}) | Acceleration of water level change in ballast tank p |

${h}_{p0}$ (m) | Initial water levels in ballast tank p |

${h}_{p\mathrm{f}}$ (m) | Final water levels in ballast tank p |

${I}_{\mathrm{x}p}$ (m^{4}) | Moment of inertia with respect to the x-axis generated by the free-surface effect in ballast tank p |

${I}_{\mathrm{y}p}$ (m^{4}) | Moment of inertia with respect to the y-axis generated by the free-surface effect in ballast tank p |

k | Position of the system at the final instant |

L (m) | Overall hull length |

${\mathit{M}}_{\mathrm{Ex}}$ (N·m) | Longitudinal environmental moments acting on the hull |

${\mathit{M}}_{\mathrm{Ey}}$ (N·m) | Transverse environmental moments acting on the hull |

${\mathit{M}}_{\mathrm{wavex}}$ (N·m) | Wave moments on longitudinal directions |

${\mathit{M}}_{\mathrm{wavey}}$ (N·m) | Wave moments on transverse directions |

${\mathit{M}}_{\mathrm{windx}}$ (N·m) | Wind moments on longitudinal directions |

${\mathit{M}}_{\mathrm{windy}}$ (N·m) | Wind moments on transverse directions |

m (t) | Load mass |

${m}_{2}$ (t) | Mass of the crane boom |

n | Non-negative integer |

Q (t) | Total amount of ballast water loaded in all the tanks at time t |

${Q}_{0}$ (t) | Initial loading capacity of the ballast water |

q | Total number of ballast tanks in RFC |

${S}_{p}$ (m^{2}) | Base area of ballast tank p |

$s\left(t\right)$ | Position |

$\dot{s}\left(t\right)$ | Velocity |

$\ddot{s}\left(t\right)$ | Acceleration |

T (m) | Average draft of the hull |

${t}_{\mathrm{f}}$ | Final time |

${t}_{\mathrm{i}}$ | Initial time |

V (m/s) | Wind speed |

${x}_{2\mathrm{g}}$ (m) | Longitudinal coordinates of the center of gravity of the lifting arm |

${\mathit{x}}_{\mathrm{B}}$ | The design variables |

${x}_{\mathrm{b}}$ (m) | Longitudinal positions of the barycenter of the equivalent tank |

${x}_{p}$ (m) | Longitudinal position of the barycenter of ballast tank p |

${y}_{2\mathrm{g}}$ (m) | Transverse coordinates of the center of gravity of the lifting arm |

${y}_{\mathrm{b}}$ (m) | Transverse positions of the barycenter of the equivalent tank during |

${y}_{p}$ (m) | Transverse position of the barycenter of ballast tank p |

z (m) | Vertical height of the load barycenter |

$\Delta {h}_{\mathrm{pf}}$ (m) | Water level change in ballast tank p |

$\Delta $ (t) | Displacement of the hull |

$\Delta T$ (m) | Increment in the average draft of the hull |

${\rho}_{\mathrm{a}}$ (kg/m^{3}) | Air density |

${\rho}_{\mathrm{w}}$ (kg/m^{3}) | Density of the ballast water |

$\chi $ (°, deg) | Wave angle, representing the direction from which waves are coming |

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**Figure 6.**The duration curves of the ballast water levels and hull’s inclination obtained from the PTP-based optimization method: (

**a**) the duration curves of the ballast water levels, (

**b**) the duration curves of the hull’s inclination.

**Figure 10.**The optimal ballast scheme regarding Trajectory TR

_{1}(300 t load mass): (

**a**) the duration curves of the ballast water levels (

**b**) and the duration curves of the heeling and trim angles.

**Figure 11.**The optimal ballast scheme regarding Trajectory TR

_{2}(300 t load mass): (

**a**) the duration curves of the ballast water levels (

**b**) and the duration curves of the heeling and trim angles.

**Figure 12.**The optimal ballast scheme regarding Trajectory TR

_{3}(300 t load mass): (

**a**) the duration curves of the ballast water levels (

**b**) and the duration curves of the heeling and trim angles.

**Figure 13.**The optimal ballast scheme while lifting a 200 t load mass: (

**a**) the duration curves of the ballast water levels and (

**b**) the duration curves of the heeling and trim angles.

**Figure 14.**The optimal ballast scheme while lifting 400 t load mass: (

**a**) the duration curves of the ballast water levels and (

**b**) the duration curves of the heeling and trim angles.

**Figure 16.**Layout diagram of the ballast tanks of the RFC test stand: (

**a**) top view and (

**b**) side view.

**Figure 20.**Comparisons of the trim and heeling angles between the theoretical and experimental values: (

**a**) comparison of the trim angle between the theoretical and experimental values and (

**b**) comparison of the heeling angle between the theoretical and experimental values.

Design Variable (Unit) | Lower Boundary | Upper Boundary |
---|---|---|

${b}_{p6}$ | $-\infty $ | $+\infty $ |

${h}_{p\mathrm{f}}$ (m) | 0 | ${h}_{p\mathrm{T}}$ |

Key Structure and Performance Parameters (Unit) | Value | Key Structure and Performance Parameters (Unit) | Value |
---|---|---|---|

Total length (m) | 100.00 | Main hook load (t) | 450.00 |

Base height (m) | 32.00 | Base mass (t) | 1571.46 |

Boom length (m) | 75.00 | Base radius (m) | 11.64 |

Rated drive power of the sling (kW) | 200.00 | Ballast tank height (m) | 4.00 |

Rated driving power of the boom (kW) | 400.00 | Longitudinal position of crane (m) | 35.00 |

Molded breadth (m) | 30.00 | Boom mass (t) | 314.29 |

Molded depth (m) | 8.00 | Base rated drive power (kW) | 300.00 |

Total flow rate of the ballast system (t/h) | 3000.00 | The pump head (m) | 6.00 |

The pump efficiency | 0.80 |

Wind Speed (m/s) | Wind Direction ($\xb0$, deg) | Wave Scale | Atmospheric Temperature (°C) | Wave Direction ($\xb0$, deg) |
---|---|---|---|---|

10.30 | 80.00 | 3.00 | 20.00 | 90.00 |

Design Variable (Unit) | Optimal Value | Design Variable (Unit) | Optimal Value |
---|---|---|---|

${b}_{16}$ | 8.36 | ${b}_{1f}$ (m) | 0 |

${b}_{26}$ | 1.14 | ${b}_{2f}$ (m) | 3.71 |

${b}_{36}$ | 2.12 | ${b}_{3f}$ (m) | 0 |

${b}_{46}$ | −0.38 | ${b}_{4f}$ (m) | 0 |

${b}_{56}$ | −0.52 | ${b}_{5f}$ (m) | 0 |

${b}_{66}$ | 1.45 | ${b}_{6f}$ (m) | 0 |

${b}_{76}$ | −0.90 | ${b}_{7f}$ (m) | 0.04 |

${b}_{86}$ | 0.78 | ${b}_{8f}$ (m) | 1.05 |

Metric | DPT-MO | PTP-Based | Improvement (%) |
---|---|---|---|

Ballast water allocation quantity (t) | 595.22 | 489.54 | 17.75 |

Ballast energy (${10}^{3}$ kJ) | 8.75 | 7.19 | 17.75 |

Decision-making time (min) | 6.30 | 1.76 | 73.49 |

Lifting Trajectory | Starting Position | Ending Position | Lifting Time (min) |
---|---|---|---|

TR_{1} | (25.65, 0, 62.00) | (0, 31.70, 55.00) | 13.92 |

TR_{2} | (25.65, 0, 62.00) | (0, 25.65, 62.00) | 10.05 |

TR_{3} | (25.65, 0, 62.00) | (0, 19.41, 69.00) | 12.15 |

**Table 7.**Comparing the optimal lifting schemes obtained using the PTP-based method regarding different lifting trajectories.

Lifting Trajectory | TR_{1} | TR_{2} | TR_{3} |
---|---|---|---|

Ballast water quantity (t) | 500.12 | 440.23 | 300.37 |

Maximum heeling angle (°, deg) | 1.01 | 2.82 | 2.47 |

Maximum trim angle (°, deg) | 0.72 | 0.99 | 0.99 |

Energy consumption (${10}^{3}$ kJ) | 7.35 | 6.46 | 4.41 |

**Table 8.**Optimization results of the lifting scheme obtained by the PTP method regarding different lifting load masses.

Lifting Load | 200 t | 300 t | 400 t |
---|---|---|---|

Ballast water amount (t) | 303.23 | 440.23 | 577.21 |

Maximum heeling angle (°, deg) | 0.37 | 2.82 | 3.02 |

Maximum trim angle (°, deg) | 0.68 | 0.99 | 0.99 |

Energy consumption (${10}^{3}$ kJ) | 4.45 | 6.46 | 8.49 |

Parameter (Unit) | Value | Parameter (Unit) | Value |
---|---|---|---|

Total length (m) | 4.50 | The number of ballast tanks | 8.00 |

Molded breadth (m) | 1.50 | Ballast pump flow (m^{3}/h) | 2.00 |

Molded depth (m) | 0.70 | Boom length (m) | 1.12 |

Empty ship mass (t) | 0.80 | Boom amplitude range (°, deg) | 0∼60 |

Tank length (m) | 0.80 | Lifting capacity (kg) | 50.00 |

Tank width (m) | 0.30 | Tank height (m) | 0.40 |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Yu, Y.; Li, S.; Zhang, J.; Liu, Z.
Point-to-Point-Based Optimization Method of Ballast Water Allocation for Revolving Floating Cranes with Experimental Verification. *J. Mar. Sci. Eng.* **2024**, *12*, 437.
https://doi.org/10.3390/jmse12030437

**AMA Style**

Wang X, Yu Y, Li S, Zhang J, Liu Z.
Point-to-Point-Based Optimization Method of Ballast Water Allocation for Revolving Floating Cranes with Experimental Verification. *Journal of Marine Science and Engineering*. 2024; 12(3):437.
https://doi.org/10.3390/jmse12030437

**Chicago/Turabian Style**

Wang, Xiaobang, Yang Yu, Siyu Li, Jie Zhang, and Zhijie Liu.
2024. "Point-to-Point-Based Optimization Method of Ballast Water Allocation for Revolving Floating Cranes with Experimental Verification" *Journal of Marine Science and Engineering* 12, no. 3: 437.
https://doi.org/10.3390/jmse12030437