# Benchmarking Wind Farm Projects by Means of Series Two-Stage DEA

## Abstract

**:**

## 1. Introduction

_{j}= (x

_{1}, …, x

_{m}) Equations (1) and (2), respectively are assumed to provide frontier and observed production function [8]:

^{f}≥ y for all observations, then the frontier is deterministic and DEA can be used. (ii) If $\epsilon $ is a two-error term of noise and inefficiency then the frontier is stochastic and SFA should be used.

## 2. Problem Statement

## 3. Methods—Data Set

#### 3.1. Series Two-Stage DEA Structure

#### 3.2. Model Building

^{m}

_{+}and non-discretionary inputs Z ∈ $\mathcal{R}$

^{p}

_{+}to generate outputs Y ∈ $\mathcal{R}$

^{k}

_{+}, the following BCC (envelopment form) or variable returns to scale (VRS) input-oriented Model (3) [22,23] and output-oriented Model (4) [29] are selected to assess the investment and operational efficiency of projects, respectively:

_{ij}, and z

_{lj}are the ith discretionary and lth non-discretionary input, respectively, used by the jth project; y

_{rj}is rth output produced by the jth project; $\theta $ and $1/\phi $ denote the efficiency score of WF

_{0}derived by the model (3) and model (4), respectively; WF

_{0}denotes the farm under evaluation; and ${\lambda}_{j}$ is intensity factor that shows the contribution of WFj in the derivation of efficiency of WF

_{0}.

_{0.}The solving process is repeated for each WF. The efficiency of WF

_{0}deals with the WF which has inputs ${x}_{ij0}$, ${z}_{ij0}$ and outputs ${y}_{rj0}$, respectively. The model searches for a group of WFs created by weighting each WFj by a coefficient ${\lambda}_{j}$ so that they do not generate more outputs than WF

_{0}and minimize inputs in comparison to those of WF

_{0}. WFs for which ${\theta}^{\ast}=1$ and ${\theta}^{\ast}<1$ are deemed efficient and inefficient, respectively.

_{0}is given by $q=1/{\phi}^{\ast}$, where ${\phi}^{\ast}$ is the optimal solution of Model (4). The model searches for a group of WFs created by weighting each WFj by a coefficient ${\lambda}_{j}$ so that they do not use more inputs than WF

_{0}and maximize outputs in comparison to those of WF

_{0}. WFs for which $q=1$ and $q<1$ are deemed efficient and inefficient, respectively.

#### 3.3. Data Set

^{3}is the air density, A is the swept area (A = πr

^{2}), i.e., the area through which the rotor blades of a WT spin, $\nu $ is the average wind speed, and ${C}_{p}$ = 0.593 is a power coefficient that reflects the maximum power which can be extracted. The diameter of WT is taken from equipment technical data. It is worth noting that the average wind speed at each WF site was used to estimate the wind power density in line with previous studies [7]. A more precise estimation will require taking into account the distribution of wind speed, which is typically a Weibull distribution [7]. This was difficult to obtain in our case, because of the lack of data. The purpose of the investment phase is to arrange WTs reasonably and thus, the number of WTs is selected as the output variable [21].

## 4. Results

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Model | Input-Oriented Model—Stage 1 | Output-Oriented Model—Stage 2 | Efficiency Definition |
---|---|---|---|

BCC input oriented efficiency (PTEin) | $\theta $ | Scalar derived from Model (3) | |

CCR output oriented efficiency (GTEout) | ${\theta}^{\prime}$ | Scalar derived from a modified Model (4) without the restriction $\sum _{j=1}^{n}{\lambda}_{j}=1$ | |

BCC output oriented efficiency (PTEout) | $q=1/\phi $ | Reciprocal of scalar $\phi $ derived from Model (4) | |

Scale output oriented efficiency (SEout) | ${\theta}^{\prime}/q$ | SEout = GTEout/PTEout |

Variable | Definition | Units | Source |
---|---|---|---|

Project cost | Estimated cost to complete the project | 10^{6} Euros | [13] |

Wind speed | Speed (velocity) of air | m/s | [13] |

Wind power density | Wind power available per square meter of swept area of a turbine | W/m^{2} | This study |

Number of wind turbines | The number of wind turbines | 1 | This study |

Installed capacity | Facility capacity | MW | [13] |

Operating and maintenance cost | Estimated cost of annual operating and maintenance cost | 10^{6} Euros | [13] |

Power generation | Estimated annual generated electricity | MWh | This study |

Features | WT1 | WT2 |
---|---|---|

Rotor diameter, m | 90 | 52 |

Area swept, m^{2} | 6362 | 2124 |

Number of blades | 3 | 3 |

Wind speed cut-in, m/s | 4 | 4 |

Wind speed rated, m/s | 15 | 14 |

Wind speed cut-out, m/s | 25 | 25 |

Nominal output, kW | 3000 | 850 |

Installation | Onshore | Onshore |

Project No. | Wind Turbine Type | Autonomous System | Down-Rated Capacity | Voltage Rise Substation Construction |
---|---|---|---|---|

WF1 | WT1 | ✓ | ||

WF2 | WT2 | ✓ | ||

WF3 | WT1 | ✓ | ||

WF4 | WT1 | ✓ | ||

WF5 | WT1 | ✓ | ||

WF6 | WT1 | ✓ | ||

WF7 | WT1 | ✓ | ||

WF8 | WT1 | ✓ | ||

WF9 | WT1 | ✓ | ✓ | |

WF10 | WT1 | |||

WF11 | WT2 | |||

WF12 | WT1 | |||

WF13 | WT2 | ✓ |

Descriptive Statistics | Project Cost (10^{6} Euros) | Wind Speed (m/s) | Wind Power Density (W/m^{2}) | Number of Wind Turbines | Installed Capacity (MW) | Annual Operating and Maintenance Cost (10^{6} Euros) | Power Generation (10^{3} MWh) |
---|---|---|---|---|---|---|---|

Mean | 20.78 | 7.57 | 3.25 | 7.15 | 17.88 | 0.55 | 38.10 |

Standard deviation | 12.07 | 1.11 | 1.29 | 3.29 | 10.97 | 0.32 | 20.57 |

Median | 25.15 | 7.47 | 3.16 | 8.00 | 18.00 | 0.49 | 38.39 |

Min | 2.86 | 5.63 | 1.65 | 3.00 | 2.55 | 0.09 | 5.77 |

Max | 34.66 | 9.37 | 5.83 | 12.00 | 36.00 | 1.12 | 67.20 |

**Table 6.**WF project efficiency measures for investment and operating phase and descriptive statistics.

Project No. | Investment Phase | Operating Phase | ||
---|---|---|---|---|

Investment Performance (%) | Operating Efficiency (%) | Scale Efficiency (%) | RTS | |

WF1 | 100.00 | 100.00 | 67.42 | DRS |

WF2 | 100.00 | 100.00 | 100.00 | CRS |

WF3 | 75.07 | 100.00 | 77.08 | DRS |

WF4 | 100.00 | 75.99 | 98.91 | IRS |

WF5 | 85.01 | 77.66 | 84.10 | DRS |

WF6 | 78.04 | 78.15 | 99.46 | IRS |

WF7 | 67.52 | 100.00 | 74.50 | DRS |

WF8 | 35.19 | 100.00 | 99.30 | DRS |

WF9 | 100.00 | 66.64 | 81.56 | DRS |

WF10 | 46.03 | 87.31 | 85.96 | DRS |

WF11 | 100.00 | 100.00 | 94.37 | IRS |

WF12 | 31.42 | 100.00 | 100.00 | CRS |

WF13 | 66.43 | 74.59 | 92.36 | IRS |

Mean | 75.75 | 89.26 | 88.85 | |

Standard deviation | 25.30 | 12.82 | 11.19 | |

Median | 78.04 | 100.00 | 92.36 | |

Min | 31.42 | 66.64 | 67.42 | |

Max | 100.00 | 100.00 | 100.00 |

Variables | Investment Performance | Operating Efficiency | ||||
---|---|---|---|---|---|---|

Original Model (Model (3)) | Modified Model (3′) | Modified Model (3″) | Original Model (Model (4)) | Modified Model (4′) | Modified Model (4″) | |

Project cost | ✓ | ✓ | ✓ | |||

Wind speed | ✓ | ✓ | ||||

Wind power density | ✓ | ✓ | ||||

Number of turbines | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

Installed capacity | ✓ | ✓ | ||||

Annual O&M cost | ✓ | ✓ | ||||

Annual generated electricity | ✓ | ✓ | ✓ | |||

Average efficiency score (%) | 75.75 | 53.90 | 75.75 | 89.26 | 86.11 | 87.29 |

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**MDPI and ACS Style**

Tsolas, I.E.
Benchmarking Wind Farm Projects by Means of Series Two-Stage DEA. *Clean Technol.* **2020**, *2*, 365-376.
https://doi.org/10.3390/cleantechnol2030022

**AMA Style**

Tsolas IE.
Benchmarking Wind Farm Projects by Means of Series Two-Stage DEA. *Clean Technologies*. 2020; 2(3):365-376.
https://doi.org/10.3390/cleantechnol2030022

**Chicago/Turabian Style**

Tsolas, Ioannis E.
2020. "Benchmarking Wind Farm Projects by Means of Series Two-Stage DEA" *Clean Technologies* 2, no. 3: 365-376.
https://doi.org/10.3390/cleantechnol2030022