# A Closed-Form Technique for the Reliability and Risk Assessment of Wind Turbine Systems

^{*}

## Abstract

**:**

## 1. Introduction

^{N}system failure events in efficient computational time O(N

^{2}) for a system with N components. They further allow the possibility to account for correlation, although through reliability bounds.

## 2. Closed-Form Technique for Wind Turbine System Reliability Assessment

#### 2.1. Combinatorial System Reliability

**S**be a vector that contains all unique possible s numbers of failed components. As an illustration, in a wind turbine system with twelve components, $\mathit{S}=\left\{1,2,\dots ,12\right\}$, the first and the last ${\mathit{k}}^{\mathit{*}}$ vectors of the ${\mathit{K}}_{12\times 4096}$ matrix are [000000000000] and [111111111111], respectively, with the remaining 4094 vectors filled with combinations of zeros and ones. This arrangement characterizes a system with undefined system events and the failure probability is estimated either with definitions from the wind turbine industry or within bounds representing a series to parallel system definition. If the failure probability of the i

^{th}component is denoted as ${P}_{i}$, then, the probability of system failure as a result of exactly $s\in \mathit{S}$ number of components failing is given by:

^{th}component and ${\mathit{K}}_{s}$ is a subset of $\mathit{K}$ containing all ${\mathit{k}}^{\mathit{*}}$ vectors whose sum equals $s$. A probability mass function of the system failure can be obtained by evaluating Equation (1) for all ${\mathit{k}}^{\mathit{*}}\in {\mathit{K}}_{\mathit{s}}$ and then for all integers $s\in \mathit{S}$.

^{N}). This makes the application of the non-recursive formulation to wind turbines with a large number of documented components infeasible. The next section presents a recursive version of the closed-form method which is applicable to a wind turbine system with any number of components N.

#### 2.2. Recursive Solution to Combinatorial System Reliability

^{th}wind turbine component in ${\mathit{k}}^{\mathit{*}}$ and only dependent on the component failure probabilities ${P}_{i}$, it can be calculated beforehand. Hence, the computation load is only on the term $\sum _{{\mathit{k}}^{\mathit{*}}\in {\mathit{K}}_{\mathit{s}}}\left(\prod _{i=1}^{N}{\alpha}_{i}^{{k}_{i}}\right)$ denoted by $Q\left(s,{K}_{s}\right)$. This $Q\left(s,{K}_{s}\right)$ term can be evaluated for each s in a recursive manner. By defining ${\mathit{K}}_{\mathit{s}}\left(r\right)=\left\{{\mathit{k}}^{\mathit{*}}\in {\mathit{K}}_{\mathit{s}}:{k}_{r}^{*}=0\right\}$ with r = 1, although it can take any value from $\left\{1,2,\dots N\right\}$ but the authors recommend the first term for tractability, this function can be expanded as:

^{N}) using the combinatorial non-recursive approach to a polynomial increasing time O(N

^{2}) as recently shown [18]. The recursive solution is an efficient technique to generate an entire probability mass density function for wind turbine systems with a large number of components. This is a prospective technique to handle risk or consequence analyses. Though such capability is already possible in the non-recursive approach, it is yet to be developed in the context of the recursive method.

## 3. Closed-Form Technique Application to Wind Turbine Risk and Sensitivity Analyses

#### 3.1. Consequence-Based Reliability Analysis

^{th}component failure and ${\mathit{q}}^{\mathit{*}}$ is a vector in the ${\mathrm{\mathbb{R}}}^{N}$ space containing ${q}_{i}$ elements. Let C be a vector containing all possible unique quantities of failure consequences c such as component downtimes in h or repair costs in dollars. Each vector in $\mathit{K}$ is multiplied by ${\mathit{q}}^{\mathit{*}}$ to form ${K}_{C}=\left\{{\mathit{k}}^{\mathit{*}}\in \mathit{K}:{\mathit{k}}^{\mathit{*}}\cdot {\mathit{q}}^{\mathit{*}}=c\right\}$ representing all possible k

^{*}configurations that yield $c\in \mathit{C}$ system failure consequence. Equation (2) is modified slightly to evaluate the wind turbine system risk, ${P}_{rsys}$ for given c units as shown below:

#### 3.2. Component Criticality Analysis

^{th}component. It is expressed as:

^{th}component which relies on the system risk in Equation (5) is expressed as:

^{th}component does not fail. In this way, researchers, designers and engineers are able to identify the components whose reliabilities must be improved to increase the system reliability or reduce risks.

## 4. Illustrative Examples

_{i}failures in time t for component i is given as:

^{th}component is the intensity function of the Poisson process. Failure rates are often determined using historical data and operational data recorded on wind farms or experimental testing of turbine components. In the absence of empirical data, expert opinions are sampled through surveys to estimate failure rates. The annual probability of at least one occurrence of the i

^{th}component failure is given by ${P}_{i}=1-{e}^{-{\lambda}_{i}}$.

#### 4.1. Example 1: 12-Subassembly Wind Turbine System

**Table 1.**Failure data for 12-subassembly wind turbine system [23].

Subassembly | I | Failure Rate (failures/yr/turbine) | Downtime (hours/failure) | Average cost (US$/failure) | |
---|---|---|---|---|---|

Lower limit | Upper limit | ||||

Electrical subsystem | 1 | 0.320 | 251 | 5,520 | 87,056 |

Rotor or blades | 2 | 0.190 | 120 | 6,581 | 52,956 |

Electrical controls | 3 | 0.239 | 60 | 440 | 6,000 |

Yaw system | 4 | 0.116 | 58 | 401 | 9,121 |

Generator | 5 | 0.139 | 161 | 332 | 53,228 |

Hydraulic subsystem | 6 | 0.131 | 70 | 158 | 1,276 |

Gear box | 7 | 0.134 | 345 | 1,476 | 153,601 |

Pitch control | 8 | 0.083 | 65 | 2,087 | 17,832 |

Air brakes | 9 | 0.040 | 105 | 3,076 | 3,076 |

Mechanical brake | 10 | 0.055 | 48 | 200 | 1,483 |

Main shaft | 11 | 0.031 | 135 | 4,318 | 15,668 |

All others | 12 | 0.367 | 60 | 94,801 | 94,801 |

Total | 1.846 | 1478 | 119,390 | 496,098 | |

Average | 0.154 | 123 | 9,949 | 41,342 | |

COV * | 0.698 | 0.741 | 2.695 | 1.176 |

#### 4.1.1. System Failure Distribution

**k***with each of the entries in $\left\{01\right\}$ from which unique combinations of numbers of failed subassemblies are realized, i.e., $\mathit{S}=\left\{0,1,\dots ,12\right\}$. As an illustration of the vectors that contribute to system failure, defined by at least one, two or three subassembly failures, the specific vector event when yaw system, the generator and the main shaft jointly fail is

**k***= [000110000010]. The distribution of the failure probability as a function of the number of failed subassemblies is shown in Figure 1. It is observed that the annual reliability of the system ${P}_{sys}={P}_{sys}\left(s=0\right)$ is 0.158, if the wind turbine is considered to fail when at least a single subassembly fails. The wind turbine has an 84% chance of failure in any given year according to this definition. The annual likelihood of the system failing owning to at least two subassembly failures is 51%. Interestingly, there is a negligible chance of wind turbine failure if at least six subassemblies have to fail before the system is considered as failed. Figure 2 compares the cumulative distribution functions (CDFs) of the system in terms of number of component failures evaluated using the closed-form technique and a Monte Carlo simulation routine consisting of 10,000 samples. The excellent agreement between the two CDFs confirms the adequate implementation of the exact closed-form combinatorial formulation.

**Figure 1.**Annual system failure probability mass function in terms of the number of subassembly failures.

**Figure 2.**Comparison of annual system failure CDFs obtained by the closed-form technique and a naïve Monte Carlo simulation (MCS) approach.

#### 4.1.3. System Repair Cost Probability

#### 4.1.4. Component Importance Metrics

Subassembly | CIM | Rank | CAM | Rank | CCM | Rank |
---|---|---|---|---|---|---|

Electrical system | 1.076 | 2 | 1.239 | 1 | 1.259 | 2 |

Rotor or blades | 1.041 | 4 | 1.122 | 2 | 1.183 | 3 |

Electrical controls | 1.053 | 3 | 1.061 | 6 | 1.087 | 9 |

Yaw system | 1.024 | 8 | 1.034 | 8 | 1.097 | 7 |

Generator | 1.029 | 5 | 1.085 | 3 | 1.153 | 4 |

Hydraulic system | 1.027 | 7 | 1.037 | 7 | 1.083 | 10 |

Gear box | 1.028 | 6 | 1.082 | 4 | 1.150 | 5 |

Pitch control | 1.016 | 9 | 1.025 | 9 | 1.122 | 6 |

Air brakes | 1.008 | 11 | 1.024 | 10 | 1.080 | 12 |

Mechanical brake | 1.011 | 10 | 1.018 | 12 | 1.080 | 11 |

Main shaft | 1.006 | 12 | 1.020 | 11 | 1.094 | 8 |

All others | 1.091 | 1 | 1.081 | 5 | 1.287 | 1 |

#### 4.2. Example 2: 45-Component Wind Turbine System

^{45}= 3.518 × 10

^{13}possible ${\mathit{k}}^{\mathit{*}}\in \mathit{K}$ vectors (or unique combinations) describing system events. This value represents at least the number of operations that would have been required by using the non-recursive combinatorial approach. However, Equation (5) reduces this number to 91,125 computations via the recursive formulation yielding the system failure probability distributions shown in Figure 5. The distribution contains all possibilities of system event. If a series system is considered, this wind turbine system has a reliability of 0.166. The annual probability of system failure due to at least 2 components is 51%. The likelihood of system failure at the instance of at least 5 components is about 2%. There is an insignificant chance of failure occurrence if system failure is defined by at least 6 out of the 45 components. The recursive approach also provides a reliability bound of 0.32 to 0.83 as a series represented system with positive event correlation. The recursive approach proves to agree perfectly with Monte Carlo simulations (MCS) as shown in Figure 6.

**Figure 6.**Comparison of annual system failure CDFs obtained for 45-component wind turbine by the recursive solution and a naïve Monte Carlo simulation (MCS) approach.

## 5. Conclusions

^{N}possible system configurations using a naïve approach in polynomial time O(N

^{2}).

## Acknowledgments

## References

- Hameed, Z.; Vatn, J.; Heggset, J. Challenges in the reliability and maintainability data collection for offshore wind turbines. Renew. Energy
**2011**, 36, 2154–2165. [Google Scholar] [CrossRef] - 2010 U.S. Wind Industry Market Update. 2011. America Wind Energy Association Web site. Available online: http://www.awea.org/learnabout/publications/ (accessed on 15 August 2011).
- 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] - Song, J.; Kang, W.-H. System reliability and sensitivity under statistical dependence by matrix-based system reliability method. Struct. Saf.
**2009**, 31, 148–156. [Google Scholar] [CrossRef] - Wangdee, W.; Billinton, R. Reliability assessment of bulk electric systems containing large wind farms. Int. J. Electr. Power Energy Syst.
**2007**, 29, 759–766. [Google Scholar] [CrossRef] - Dueñas-Osorio, L.; Craig, J.I.; Goodno, B.J. Seismic response of critical interdependent networks. Earthq. Eng. Struct. Dyn.
**2007**, 36, 285–306. [Google Scholar] [CrossRef] - Adachi, T.; Ellingwood, B.R. Serviceability assessment of a municipal water system under spatially correlated seismic intensities. Comput.-Aided Civ. Infrastruct. Eng.
**2009**, 24, 237–248. [Google Scholar] [CrossRef] - Dueñas-Osorio, L.; Rojo, J. Reliability assessment of lifeline systems with radial topology. Comput.-Aided Civ. Infrastruct. Eng.
**2011**, 26, 111–128. [Google Scholar] [CrossRef] - Duenas-Osorio, L.; Padgett, J.E. Seismic reliability assessment of bridges with user-defined system failure events. J. Eng. Mech.
**2011**, 137, 680–690. [Google Scholar] [CrossRef] - Kang, W.-H.; Song, J.; Gardoni, P. Matrix-based system reliability method and applications to bridge networks. Reliab. Eng. Syst. Saf.
**2008**, 93, 1584–1593. [Google Scholar] [CrossRef] - Karki, R.; Billinton, R. Cost-effective wind energy utilization for reliable power supply. Energy Convers. IEEE Trans.
**2004**, 19, 435–440. [Google Scholar] [CrossRef] - Arabian-Hoseynabadi, H.; Oraee, H.; Tavner, P.J. Wind turbine productivity considering electrical subassembly reliability. Renew. Energy
**2010**, 35, 190–197. [Google Scholar] [CrossRef] - Arabian-Hoseynabadi, H.; Oraee, H.; Tavner, P.J. Failure modes and effects analysis (FMEA) for wind turbines. Int. J. Electr. Power Energy Syst.
**2010**, 32, 817–824. [Google Scholar] [CrossRef] [Green Version] - Sørensen, J.D. Framework for risk-based planning of operation and maintenance for offshore wind turbines. Wind Energy
**2009**, 12, 493–506. [Google Scholar] [CrossRef] - Sørensen, J.D.; Toft, H.S. Probabilistic design of wind turbines. Energies
**2010**, 3, 241–257. [Google Scholar] [CrossRef] - Continuous Reliability Enhancement for Wind (CREW) Database. 2011; Sandia National Laboratories Web site. Available online: http://energy.sandia.gov/?page_id=6682 (accessed on 12 October 2011).
- U.S. Wind Turbine Reliability Survey Web site. Available online: http://duenas-osorio.rice.edu/survey/ (accessed on 15 December 2011).
- Rojo, J.; Dueñas-Osorio, L. Recursive reliability assessment of radial lifeline systems with correlated component failures. In Proceeding of the 11th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP11); Zurich, Switzerland, 1–4 August, 2011, Faber, M.H., Kohler, J., Nishijima, K., Eds.; CRC Press, Taylor and Francis Group: London, UK, 2011; pp. 1435–1443. [Google Scholar]
- Volkanovski, A.; Čepin, M.; Mavko, B. Application of the fault tree analysis for assessment of power system reliability. Reliab. Eng. Syst. Saf.
**2009**, 94, 1116–1127. [Google Scholar] [CrossRef] - Tavner, P.J.; Xiang, J.; Spinato, F. Reliability analysis for wind turbines. Wind Energy
**2007**, 10, 1–18. [Google Scholar] [CrossRef] - Eggersgl, W. Wind Energie; Landwirtschaftskammer Schleswig-Holstein: Rendsburg, Germany, 1993–2004; Volume V–XVI. [Google Scholar]
- Spinato, F. The Reliability of Wind Turbines; Durham University: Durham, UK, 2008. [Google Scholar]
- Poore, R.; Walford, C. Development of an Operations and Maintenance Cost Model to Identify Cost of Energy Savings for Low Wind Speed Turbines; National Renewable Energy Laboratory: Seattle, WA, USA, 2008. Available online: http://www.nrel.gov/docs/fy08osti/40581.pdf (accessed on 22 January 2012 ).
- Nielsen, J.J.; Sørensen, J.D. On risk-based operation and maintenance of offshore wind turbine components. Reliab. Eng. Syst. Saf.
**2011**, 96, 218–229. [Google Scholar] [CrossRef] - Reliawind Reliability Focused Research on Optimizing Wind Energy Systems Design, Operation and Maintenance: Tools, Proof of Concepts, Guidelines & Methodologies for a New Generation. 2011. Relia Wind Project Web site. Available online: http://www.reliawind.eu/files/file-inline/ (accessed on 2 September 2011).

© 2012 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 license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Mensah, A.F.; Dueñas-Osorio, L.
A Closed-Form Technique for the Reliability and Risk Assessment of Wind Turbine Systems. *Energies* **2012**, *5*, 1734-1750.
https://doi.org/10.3390/en5061734

**AMA Style**

Mensah AF, Dueñas-Osorio L.
A Closed-Form Technique for the Reliability and Risk Assessment of Wind Turbine Systems. *Energies*. 2012; 5(6):1734-1750.
https://doi.org/10.3390/en5061734

**Chicago/Turabian Style**

Mensah, Akwasi F., and Leonardo Dueñas-Osorio.
2012. "A Closed-Form Technique for the Reliability and Risk Assessment of Wind Turbine Systems" *Energies* 5, no. 6: 1734-1750.
https://doi.org/10.3390/en5061734