# Optimal Life Extension Management of Offshore Wind Farms Based on the Modern Portfolio Theory

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## Abstract

**:**

## 1. Introduction

## 2. Modern Portfolio Theory (MPT) and Portfolio Optimisation

_{p}is the mean value of returns of the portfolio, w

_{i}is the weighting factor related to each asset and µ

_{i}is the mean value of returns of each asset. The variance of returns is expressed as follows:

_{p}is the standard deviation of return of the portfolio, which indicates the level of risk of the overall portfolio and σ

_{ij}is the covariance between two assets which is calculated as:

_{ij}is the correlation between the two assets. For a large number of assets, the standard deviation of return of the portfolio is as follows:

_{f}tangent to the efficient frontier is called the optimised risk/return relationship (Sharpe ratio) along the efficient frontier. The most efficient portfolio is identified as the highest Sharpe ratio. An investor who does not define a preference function (utility function), which is a function of risk-averseness, should accept the portfolio with the highest Sharpe ratio.

## 3. Life Extension Assessment and Offshore Wind Asset Classification

_{OWT}is the number of offshore wind turbines in a wind farm, t is the duration of life extension, r is the discount rate and FCF

_{i}is the free cash flow for a given year. The free cash flow is calculated by subtracting the revenue from the operating income as:

_{p}is the capacity factor, X

_{OI}is the operational intensity factor, X

_{ME}is the management efficiency factor, X

_{WH}is the working hours per year and FiT is the feed-in tariff. The annual maintenance cost is estimated together with the fixed operational cost C

_{O}, which is assumed to be 40 €/MWh for an expected wind speed of 8.5 m/s. The corrective maintenance cost C

_{M}of a fixed support structure is considered to be 50 K€/MW.

_{f}is the risk-free interest rate, β is the volatility measure of the equity, r

_{m}is the mean value of the market return and α is the sector return relative to the market. The volatility measure, β, represents the likelihood that the offshore wind asset cannot produce the predicted free cash flow, covering the revenues from electricity sales and the operational costs to keep the offshore wind structure above the permissible safety limits.

## 4. Case Studies and Discussion

_{1}), which demonstrates the power of diversification on the offshore wind farm consisting of multiple offshore wind turbines with varying features.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Classification of life extension projects in four groups (A–D) with the centroids of the classified groups (Yellow star).

**Figure 6.**Correlation matrix for 50 simulated offshore wind assets (

**a**) and four representative assets (

**b**).

Wind turbine | 5 MW NREL |

Rated wind speed | 11.8 m/s |

The expected value of wind speed | 9 m/s |

Hub height (from the MSL) | 86 m |

Water depth (from the MSL) | 40 m |

Turbulence intensity | 0.12 |

Integral length scale | 340 m |

Thrust coefficient (N/(m/s)) | 0.73 |

Structural and aerodynamic damping | 4% and 1% |

Natural frequency | 0.281 Hz |

Diameter of support structure | 6 m |

Thickness of support structure | 50 mm |

Material constant | 5.21 × 10^{−13} |

Material exponent | 3 |

Threshold stress intensity factor | 2 MPa·m^{l/2} |

Critical stress intensity factor | 69 MPa·m^{l/2} |

Yield stress | 355 MPa |

Plate breadth | 0.1 m |

Sigmoid slope | 0.2 |

Shift parameter | 0.01 |

Variables | E [] | COV (%) | Distribution |

Expected wind speed (m/s) | 10 | 10 | Weibull |

Operation intensity | 0.75 | 10 | Normal |

Wind farm size (unit) | 30 | 20 | Uniform |

Life extension (year) | 20 | 40 | Uniform |

Management efficiency | 0.85 | 10 | Normal |

Availability and capacity factor | 0.95 and 0.44 | 10 | Normal |

Measured crack size (m) | 0.010 | 20 | Lognormal |

Feed-in tariff (€/MWh) | 120 | 20 | Normal |

Safety class | 5 | 50 | Uniform |

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**MDPI and ACS Style**

Yeter, B.; Garbatov, Y.
Optimal Life Extension Management of Offshore Wind Farms Based on the Modern Portfolio Theory. *Oceans* **2021**, *2*, 566-582.
https://doi.org/10.3390/oceans2030032

**AMA Style**

Yeter B, Garbatov Y.
Optimal Life Extension Management of Offshore Wind Farms Based on the Modern Portfolio Theory. *Oceans*. 2021; 2(3):566-582.
https://doi.org/10.3390/oceans2030032

**Chicago/Turabian Style**

Yeter, Baran, and Yordan Garbatov.
2021. "Optimal Life Extension Management of Offshore Wind Farms Based on the Modern Portfolio Theory" *Oceans* 2, no. 3: 566-582.
https://doi.org/10.3390/oceans2030032