# Improved Fractional-Order PID Controller of a PMSM-Based Wave Compensation System for Offshore Ship Cranes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Models

- Assuming the vessel is operating in the level 4 sea state;
- Assuming that the lifting vessel is a square box and that the crane boom is not deformed during operation;
- The mass of the rope is negligible compared to the mass of the load being lifted;
- The influence of air resistance on the load being lifted is neglected.

#### 2.1. Motion Coordinate System

_{N}Y

_{N}Z

_{N}), the translation coordinate system frame (O–X

_{O}Y

_{O}Z

_{O}), and the Body-axis coordinate system (BODY) frame (O–X

_{b}Y

_{b}Z

_{b}); the positions of these coordinate systems are shown in Figure 2:

**x**is the displacement vector of the vessel in all DOF (m); ω

_{1}is the oscillation frequency (Hz); M is the mass of the vessel (Kg); A(ω

_{1}) is the additional quality (Kg); B(ω

_{1}) is the damping ratio; C is the resilience factor; F(ω

_{1}) is the harmonic excitation force (N).

**F**is the wave linear excitation force (N); ζ

_{0}_{a}is the amplitude of the harmonic vector (m); using the MSS toolbox module in MATLAB, the response of the vessel in the different DOF can be modeled using the RAO principle.

**r**; the vector from the bottom to the top lifting point of the crane is

^{b}_{base}**r**.

^{b}_{basetip}**S(a)**is defined as:

#### 2.2. Wave Model

_{a}(i) in equation (15) and wave frequency spectrum (S

_{ζ}) is:

_{n}is a causeless constant; g is the acceleration of gravity (m/s

^{2}); U is the average wind speed (m/s); γ is the spectral peak factor and takes the value 3.3; ω

_{p}is the spectral peak frequency (Hz).

#### 2.3. Model of the Vessel Crane Load Motion System

**G**and

^{n},**G**is the position of the load in the coordinate system with the lifting point (P

^{p}_{tip}) as the original point; α, β is the inner angle of the plane and the outer angle of the plane. The vector relationship gives:

**G**is expressed as:

^{p}#### 2.4. Hydrodynamic Calculations of Splash Zones

_{t}is the elasticity of the rope (N); F

_{h}is the hydrodynamic force on the rope (N).

_{B}, F

_{A}, and F

_{D}are the forces caused by buoyancy, added mass, and drag effects, respectively (N). In this paper it is assumed that the load is a sphere with a diameter of d = 2 m. The hydrodynamic force can therefore be simplified as [16]:

_{p}> 0:

_{p}≤ 0:

_{p}is the distance below the surface of the sea (m); m

_{a}is the additional mass of the spherical load (kg); V

_{p}(d

_{p}) is the volume of the object submerged in water (m

^{3}); d is the diameter of the object to be lifted (m); ρ is the density of seawater, ρ ≈ 1024 (kg/m

^{3}). d

_{p}can be found from the distance between the load and the water surface (z

_{w}):

_{w}= 0, the load just touches the water surface. The remaining terms in Equation (33) can be derived from the following equation:

#### 2.5. Prediction of Heave Displacement Data

_{1}, ϕ

_{2}, …, ϕ

_{p}denote the autoregressive coefficient, θ

_{1}, θ

_{2}, …, θ

_{q}denote the sliding average coefficient, ε

_{t}denotes the white noise sequence. The expected value of the ε

_{t}is 0 and the variance value of the ε

_{t}is σ

_{ε}. The following are statistical characteristics:

#### 2.6. Mathematical Model of PMSM

_{r}is the mechanical angular velocity of the motor (rad/s); ω is the electrical angular of the motor (rad/s); u

_{d}, u

_{q}; ψ

_{d},ψ

_{q}; i

_{d}, i

_{q}; L

_{d}, L

_{q}are the respectively the voltages (V), flux linkage component (Wb), current (A) and inductance (H) components expressed in the d-q frame; R

_{s}is the stator equivalent resistance (Ω); ψ

_{f}is the permanent magnet flux of the rotor (Wb); T

_{L}is the load torque (N·m); J

_{m}is the moment of inertia (kg/m²); T

_{e}is the electromagnetic torque (N·m); P

_{n}is the number of pole pairs.

_{d}= L

_{q}= L). So, the corresponding electromagnetic torque equation can be simplified as:

_{d}is controlled to be 0 and i

_{q}current is used for torque control in current loop which will be described in the next section.

## 3. Fractional-Order PID Controller

#### 3.1. Basic Principle of FOPID Controller

^{λ}can be approached by connecting an integer-order integrator and a fractional-order differentiator approximated by an Oustaloup filter. As shown in Figure 7, and Equation (50) 5 parameters have to be determined to tune a FOPID. If the five parameters of the controller are set appropriately, a better control than conventional PID controllers can be achieved [25].

_{m}is the phase margin (rad), ω

_{c}is the gain crossover frequency (rad/s).

#### 3.2. Improved FOPID Based on GAPSO Algorithm

## 4. Presentation and Discussion of Test Results

_{tip}in Figure 2) is Z = −25 m. The simulation parameters are shown in Table 3.

#### 4.1. Simulation When the Initial Rope Length L = 5 m and Fixed Position for the Load (V = 0 m/s)

_{1}(t) is the compensated heave error; e

_{2}(t) is the heave error without the compensation.

#### 4.2. Simulation When the Initial Rope Length L = 5 m and the Lowering Speed of the Rope V = 0.2 m/s

#### 4.3. Simulation When the Initial Rope Length L = 27 m and the Rising Speed of the Rope of 0.2 m/s

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Yang, J. The development and prospect of the offshore oil and gas industry. Adv. Resour. Res.
**2021**, 1, 1–3. [Google Scholar] - Liu, Y.; Lu, H.; Li, Y.; Xu, H.; Pan, Z.; Dai, P.; Wang, H.; Yang, Q. A review of treatment technologies for produced water in offshore oil and gas fields. Sci. Total Environ.
**2021**, 775, 145–485. [Google Scholar] [CrossRef] [PubMed] - Woodacre, J.K.; Bauer, R.J.; Irani, R.A. A review of Vertical Motion Heave Compensation Systems. Ocean Eng.
**2015**, 104, 140–154. [Google Scholar] [CrossRef] - Wu, W.; Liu, X.; Guo, Z.; Wang, H. Real time-delay control of active heave compensation system for marine crane. Chin. Hydraul. Pneumatics.
**2021**, 45, 167. [Google Scholar] - Shrenik, Z.; Abhilash, S. A comparative study of different active heave compensation approaches. Ocean. Syst. Eng.
**2020**, 10, 373–399. [Google Scholar] - Southerland, A. Mechanical systems for ocean engineering. Nav. Eng. J.
**1970**, 82, 63–74. [Google Scholar] [CrossRef] - Do, K.; Pan, J. Nonlinear control of an active heave compensation system. Ocean Eng.
**2008**, 35, 558–571. [Google Scholar] [CrossRef] - Johansen, T.A.; Fossen, T.I.; Sagatun, S.I.; Nielsen, F.G. Wave synchronizing crane control during water entry in offshore moonpool operations. IEEE J. Ocean. Eng.
**2003**, 28, 720–728. [Google Scholar] [CrossRef] - Yan, F.; Fan, K.; Yan, X.; Li, S. Constant tension control of hybrid active-passive heave compensator based on adaptive integral sliding mode method. IEEE Access
**2020**, 8, 103782–103791. [Google Scholar] [CrossRef] - Sandre-Hernandez, O.; Morales-Caporal, R.; Rangel-Magdaleno, J.; Peregrina-Barreto, H.; Hernandez-Perez, J.N. Parameter identification of PMSMs using experimental measurements and a PSO algorithm. IEEE Trans. Instrum. Meas.
**2015**, 64, 2146–2154. [Google Scholar] [CrossRef] - Sun, X.; Wu, M.; Lei, G.; Guo, Y.; Zhu, J. An Improved Model Predictive Current Control for PMSM Drives Based on Current Track Circle. IEEE Trans. Ind. Electron.
**2021**, 68, 3782–3793. [Google Scholar] [CrossRef] - Gao, J.; Liu, J.; Gong, C. A High-efficiency PMSM Sensorless Control Approach Based on MPC Controller. In Proceedings of the ICEON 2018-44th Annual Conference of the IEEE Industrial Electronics Society, Washington, DC, USA, 21–23 October 2018. [Google Scholar]
- Zhang, H.; Liu, W.; Chen, Z.; Luo, G.; Liu, J.; Zhao, D. Asymmetric space vector modulation for PMSM sensorless drives based on square-wave voltage-injection method. IEEE Trans. Ind. Appl.
**2018**, 54, 1425–1436. [Google Scholar] [CrossRef] - Liu, G.; Chen, B.; Wang, K.; Song, X. Selective current harmonic suppression for high-speed PMSM based on high-precision harmonic detection method. IEEE Trans. Ind. Inform.
**2018**, 15, 3457–3468. [Google Scholar] [CrossRef] - Yasukawa, H.; Yoshimura, Y. Introduction of MMG standard method for ship maneuvering predictions. J. Mar. Sci. Technol.
**2015**, 20, 37–52. [Google Scholar] [CrossRef] - Johansen, T.A.; Fossen, T.I. Modeling and Identification of Offshore Crane-Rig System; Tech. rep; Department of Engineering Cybernetics, NTNU: Trondheim, Norway, 2001. [Google Scholar]
- Chu, Y.; Li, G.; Zhang, H. Incorporation of ship motion prediction into active heave compensation for offshore crane operation. In Proceedings of the 2020—15th IEEE Conference on Industrial Electronics and Applications (ICIEA), Kristiansand, Norway, 9–13 November 2020. [Google Scholar]
- Cai, B.; Wang, Z.; Zhu, H.; Liu, Y.; Hao, K.; Yang, Z.; Ren, Y.; Feng, Q.; Liu, Z. Artificial Intelligence Enhanced Two-Stage Hybrid Fault Prognosis Methodology of PMSM. IEEE Trans. Ind. Inform.
**2021**, 18, 7262–7273. [Google Scholar] [CrossRef] - Yu, L.; Wang, C.; Shi, H.; Xin, R.; Wang, L. Simulation of PMSM field-oriented control based on SVPWM. In Proceedings of the 2017—29th Chinese Control And Decision Conference (CCDC), Chongqing, China, 28–30 May 2017. [Google Scholar]
- Zhang, T.; He, L. Secondary development of nonlinear creep model for element with fractional derivative. J. East China Jiaotong Univ.
**2017**, 34, 21–28. [Google Scholar] [CrossRef] - Karthikeyan, A.; Prabhakaran, K.; Nagamani, C. FPGA based direct torque control with speed loop Pseudo derivative controller for PMSM drive. Clust. Comput.
**2018**, 22, 13511–13519. [Google Scholar] - Wang, D.; Song, B. Design of Fractional-order Sliding Mode Controller for Permanent Magnet Synchronous Motor Based on PSO. Electr. Drive.
**2017**, 50, 8–12. [Google Scholar] - Shah, P.; Agashe, S. Review of fractional PID controller. Mechatronics
**2016**, 38, 29–41. [Google Scholar] [CrossRef] - Zheng, W.; Luo, Y.; Chen, Y.; Pi, Y. Fractional-order modeling of permanent magnet synchronous motor speed servo system. J. Vib. Control
**2016**, 22, 2255–2280. [Google Scholar] [CrossRef] - Zheng, W.; Luo, Y.; Chen, Y.; Wang, X. A simplified fractional order PID controller’s optimal tuning: A case study on a PMSM speed servo. Entropy
**2021**, 23, 130. [Google Scholar] [CrossRef] [PubMed] - Yeroglu, C.; Tan, N. Note on fractional-order proportional-integral-differential controller design. IET Control Theory Appl.
**2011**, 5, 1978–1989. [Google Scholar] [CrossRef] - Chaoui, H.; Khayamy, M.; Okoye, O.; Gualous, H. Simplified Speed Control of Permanent Magnet Synchronous Motors Using Genetic Algorithms. IEEE Trans. Power Electron.
**2019**, 34, 3563–3574. [Google Scholar] [CrossRef] - Mesloub, H.; Benchouia, M.T.; Goléa, A.; Goléa, N.; Benbouzid, M.E.H. Predictive DTC schemes with PI regulator and particle swarm optimization for PMSM drive: Comparative simulation and experimental study. Int. J. Adv. Manuf. Technol.
**2016**, 86, 3123–3134. [Google Scholar] [CrossRef] - Wu, X.; Wang, Y.; Zhang, T. An improved GAPSO hybrid programming algorithm. In Proceedings of the 2009 International Conference on Information Engineering and Computer Science (ICIECS), Wuhan, China, 19–20 December 2009. [Google Scholar]
- Skaare, B.; Egeland, O. Parallel Force/Position Crane Control in Marine Operations. IEEE J. Ocean. Eng.
**2006**, 31, 599–613. [Google Scholar] [CrossRef]

**Figure 5.**The heave displacement autocorrelation test chart and autocorrelation test chart. (

**a**) Heave displacement autocorrelation test chart; (

**b**) Heave displacement partial autocorrelation test chart.

**Figure 17.**L = 5 m, V = 0.2 m/s rope tension and hydrodynamic forces on the load without wave synchronization.

**Figure 18.**L = 5 m, V = 0.2 m/s rope tension and hydrodynamic forces on the load with wave synchronization.

**Figure 21.**L = 27 m, V = −0.2 m/s Position of the load with compensation with the 3 studied methods.

**Figure 23.**L = 27 m, V = −0.2 m/s, rope tension and hydrodynamic forces without using adding wave synchronization strategy.

**Figure 24.**L = 27 m, V = −0.2 m/s, rope tension and hydrodynamic forces on the load with wave synchronization.

6 DOF | Linear Velocity and Angular Velocity | Displacement and Offset Angle |
---|---|---|

surge | u | x |

sway | v | y |

heave | w | z |

roll | p | ϕ |

pitch | q | θ |

yaw | r | Ψ |

Model | ACF | PACF |
---|---|---|

AR(p) | trailing | p-order truncation |

MA(q) | q-order truncation | trailing |

ARMA(p,q) | trailing | trailing |

Parameters | Value | Parameters | Value |
---|---|---|---|

Vessel draft | 6 m | Load diameter | 2 m |

Vessel width | 19.2 m | Wave spreading factor | 1 |

Vessel quality | 63,622 kg | Significant wave height | 3 m |

Elasticity of the rope | 8.2 × 106 N/m | Mean wave direction | Pi/4 rad |

Load mass | 21,000 kg | The density of water | 1025 kg/m^{3} |

**Table 4.**ITAE metrics for each algorithm for the results of Figure 12.

Algorithms | ITAE |
---|---|

PID | 5.4437 |

FOPID | 2.3731 |

GAPSO | 0.3024 |

**Table 5.**Compensation efficiency of different algorithms using results of Figure 13.

Algorithms | η |
---|---|

PID | 96.432% |

FOPID | 96.949% |

GAPSO | 97.369% |

**Table 6.**Compensation efficiency for different algorithms using results of Figure 19.

Algorithms | η |
---|---|

PID | 96.773% |

FOPID | 95.268% |

GAPSO | 96.135% |

**Table 7.**Compensation efficiency for different algorithms and the results of Figure 22.

Algorithms | η |
---|---|

PID | 92.426% |

FOPID | 93.607% |

GAPSO | 94.068% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, H.; Wang, X.; Benbouzid, M.; Charpentier, J.-F.; Aϊt-Ahmed, N.; Han, J.
Improved Fractional-Order PID Controller of a PMSM-Based Wave Compensation System for Offshore Ship Cranes. *J. Mar. Sci. Eng.* **2022**, *10*, 1238.
https://doi.org/10.3390/jmse10091238

**AMA Style**

Chen H, Wang X, Benbouzid M, Charpentier J-F, Aϊt-Ahmed N, Han J.
Improved Fractional-Order PID Controller of a PMSM-Based Wave Compensation System for Offshore Ship Cranes. *Journal of Marine Science and Engineering*. 2022; 10(9):1238.
https://doi.org/10.3390/jmse10091238

**Chicago/Turabian Style**

Chen, Hao, Xin Wang, Mohamed Benbouzid, Jean-Frédéric Charpentier, Nadia Aϊt-Ahmed, and Jingang Han.
2022. "Improved Fractional-Order PID Controller of a PMSM-Based Wave Compensation System for Offshore Ship Cranes" *Journal of Marine Science and Engineering* 10, no. 9: 1238.
https://doi.org/10.3390/jmse10091238