# Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components

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## Abstract

**:**

## 1. Introduction

## 2. A Single-Component Model

**Proposition**

**1.**

## 3. Optimal Maintenance Algorithm in the One-Component Case

**Definition**

**1.**

- If $\{u\le t\}$, then the breakdown happens before the planned PM time, and the expected total maintenance cost is estimated to be $g+(T-u)c$, with c given by (1),
- If $\{u\ge t+1\}$, so that there is no breakdown before the planned PM time, then the expected total maintenance cost is estimated to be$$h+(t-s+a)m+(T-t)c.$$

**Proposition**

**2.**

## 4. Multiple-Component Model

- ${g}_{0}=$ the shared logistic and downtime costs associated with a CM activity,
- ${g}^{j}=$ the component-specific CM cost,
- ${h}_{0}=$ the fixed logistic cost plus the downtime cost during a PM activity,
- ${h}^{j}+t{m}^{j}=$ the component-specific PM replacement cost for j-th component at age t.

## 5. Virtual Replacement Cost

**Proposition**

**3.**

## 6. Optimal Maintenance Algorithm for Multiple Components

## 7. Case Studies

#### 7.1. Sensitivity Analysis 1

#### 7.2. Sensitivity Analysis 2

#### 7.3. Sensitivity Analysis 3

#### 7.4. Case Study 1

#### 7.5. Case Study 2

## 8. Conclusions and Future Work

## 9. Proofs of Propositions

**Proof of Proposition**

**1.**

**Proof of Proposition**

**2.**

**Proof of Proposition**

**3.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CM | Corrective maintenance |

PM | Preventive maintenance |

## References

- Márquez, F.P.G.; Karyotakis, A.; Papaelias, M. Renewable Energies: Business Outlook 2050; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
- Röckmann, C.; Lagerveld, S.; Stavenuiter, J. Operation and maintenance costs of offshore wind farms and potential multi-use platforms in the Dutch North Sea. In Aquaculture Perspective of Multi-Use Sites in the Open Ocean: The Untapped Potential for Marine Resources in the Anthropocene; Springer: Berlin/Heidelberg, Germany, 2017; pp. 97–113. [Google Scholar]
- Lee, H.; Cha, J.H. New stochastic models for preventive maintenance and maintenance optimization. Eur. J. Oper. Res.
**2016**, 255, 80–90. [Google Scholar] [CrossRef] - Sarker, B.R.; Faiz, T.I. Minimizing maintenance cost for offshore wind turbines following multi-level opportunistic preventive strategy. Renew. Energy
**2016**, 85, 104–113. [Google Scholar] [CrossRef] - Moghaddam, K.S.; Usher, J.S. Sensitivity analysis and comparison of algorithms in preventive maintenance and replacement scheduling optimization models. Comput. Ind. Eng.
**2011**, 61, 64–75. [Google Scholar] [CrossRef] - Guo, H.; Watson, S.; Tavner, P.; Xiang, J. Reliability analysis for wind turbines with incomplete failure data collected from after the date of initial installation. Reliab. Eng. Syst. Saf.
**2009**, 94, 1057–1063. [Google Scholar] [CrossRef] [Green Version] - Yu, Q.; Patriksson, M.; Sagitov, S. Optimal scheduling of the next preventive maintenance activity for a wind farm. Wind. Energy Sci.
**2021**, 6, 949–959. [Google Scholar] [CrossRef] - Santos, F.P.; Teixeira, Â.P.; Soares, C.G. Modeling, simulation and optimization of maintenance cost aspects on multi-unit systems by stochastic Petri nets with predicates. Simulation
**2019**, 95, 461–478. [Google Scholar] [CrossRef] - Chen, Y.L. A bivariate optimal imperfect preventive maintenance policy for a used system with two-type shocks. Comput. Ind. Eng.
**2012**, 63, 1227–1234. [Google Scholar] [CrossRef] - Liu, B.; Xu, Z.; Xie, M.; Kuo, W. A value-based preventive maintenance policy for multi-component system with continuously degrading components. Reliab. Eng. Syst. Saf.
**2014**, 132, 83–89. [Google Scholar] [CrossRef] - Grimmett, G.S.; Stirzaker, D.R. Probability and Random Processes; Oxford University Press: Singapore, 2020. [Google Scholar]
- Tian, Z.; Jin, T.; Wu, B.; Ding, F. Condition based maintenance optimization for wind power generation systems under continuous monitoring. Renew. Energy
**2011**, 36, 1502–1509. [Google Scholar] [CrossRef] - Ziegler, L.; Gonzalez, E.; Rubert, T.; Smolka, U.; Melero, J.J. Lifetime extension of onshore wind turbines: A review covering Germany, Spain, Denmark, and the UK. Renew. Sustain. Energy Rev.
**2018**, 82, 1261–1271. [Google Scholar] [CrossRef] [Green Version] - Yu, Q.; Strömberg, A.B. Mathematical optimization models for long-term maintenance scheduling of wind power systems. arXiv
**2021**, arXiv:2105.06666. [Google Scholar]

**Figure 3.**Plots of ${B}_{a}^{4}$ for different combinations of parameters $(g,{h}_{0},{h}^{4},{m}^{4})$.

**Figure 4.**

**Left panel**: the optimal time to perform the next PM as a function of the parameter m.

**Right panel**: different average cost based on different PM plans and different monthly value loss m.

Component (j) | ${\mathit{g}}^{\mathit{j}}$, CM Replacement Cost (USD 1000) | ${\mathit{m}}^{\mathit{j}}$, Value Loss per Month (USD 1000) | Weibull Shape ${\mathit{\beta}}^{\mathit{j}}$ | Weibull Scale ${\mathit{\theta}}^{\mathit{j}}$ |
---|---|---|---|---|

Rotor $(j=1)$ | 162 | 0.5 | 3 | 1 × ${10}^{-6}$ |

Main Bearing $(j=2)$ | 110 | 0.25 | 2 | 6.4 × ${10}^{-5}$ |

Gearbox $(j=3)$ | 202 | 1 | 3 | 1.95 × ${10}^{-6}$ |

Generator $(j=4)$ | 150 | 0.45 | 2 | 8.26 × ${10}^{-5}$ |

Initial Ages | $\mathit{j}=1$ | $\mathit{j}=2$ | $\mathit{j}=3$ | $\mathit{j}=4$ | PM | CM |
---|---|---|---|---|---|---|

$(0,0,0,0)$ | 62 | x | 62 | x | 9.937 | 13.756 |

$(30,30,30,30)$ | 32 | x | 32 | x | 10.815 | 15.064 |

$(30,30,0,30)$ | 46 | x | 46 | 46 | 10.469 | 14.868 |

$(20,60,0,30)$ | 47 | 47 | 47 | 47 | 10.458 | 14.668 |

$(0,0,40,0)$ | x | x | 12 | x | 10.364 | 14.229 |

$\mathit{d}=1$ | 1 | 2 | 3 | 4 | Monthly Maintenance Cost | CPU Time |
---|---|---|---|---|---|---|

NextPM | x | x | 43 | x | 4.731 | 49 s |

New algorithm | x | x | 43 | x | 4.703 | 2 s |

$\mathit{d}=\mathbf{5}$ | 1 | 2 | 3 | 4 | Monthly Maintenance Cost | CPU Time |

NextPM | 50 | 50 | 50 | 50 | 4.964 | 54 s |

New algorithm | 51 | 51 | 51 | 51 | 4.881 | 2 s |

$\mathit{d}=\mathbf{10}$ | 1 | 2 | 3 | 4 | Monthly Maintenance Cost | CPU Time |

NextPM | 52 | 52 | 52 | 52 | 5.061 | 55 s |

New algorithm | 52 | 52 | 52 | 52 | 5.040 | 2 s |

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**MDPI and ACS Style**

Yu, Q.; Carlson, O.; Sagitov, S.
Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components. *Algorithms* **2023**, *16*, 334.
https://doi.org/10.3390/a16070334

**AMA Style**

Yu Q, Carlson O, Sagitov S.
Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components. *Algorithms*. 2023; 16(7):334.
https://doi.org/10.3390/a16070334

**Chicago/Turabian Style**

Yu, Quanjiang, Ola Carlson, and Serik Sagitov.
2023. "Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components" *Algorithms* 16, no. 7: 334.
https://doi.org/10.3390/a16070334