# Modeling of the German Wind Power Production with High Spatiotemporal Resolution

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

#### 2.1. Plant Dataset

^{1}, rated power, hub height, turbine type, and the date of (de-)commissioning, as shown in Table 1. It should be stated, that the rotor diameter of a wind turbine is not needed for the model approach introduced in this study, and therefore all uncertainties regarding this parameter do not influence the simulation results.

#### 2.2. Calibration Data

^{3}, are available in the technical datasheets of the turbine manufacturer or can be found on internet platforms, e.g., at The Wind Power (www.thewindpower.net) [15].

_{cut-in}) there is insufficient torque exerted by the wind stream on the turbine blades to make them rotate. However, when the wind speed increases, the turbine begins to rotate and generates electricity. The wind speed when the turbine starts to deliver energy into the power grid is called cut-in speed. For most wind turbines it is between 2 and 4 m/s. When the wind speed rises above this cut-in speed, the level of the generated electricity increases rapidly. Typically, somewhere in the range of 12 and 14 m/s, the electrical output reaches the nominal value that the wind turbine is designed for. This value is called rated power (P

_{R}) and the concerning wind speed is the rated speed (v

_{rated}). At higher wind speeds, the turbine is designed to limit the output to this level. How this behavior is done depends on the technical design, but for most onshore turbines, it is achieved by adjusting the blade angles depending on the wind speed to keep the electrical output constant. This kind of power regulation is called pitch control. When the wind speed increases well above the rated speed, the forces on the turbine structure continue to rise and, at some point, there is a risk of damage. As a result, a switch-off and braking system is employed to bring the rotor to a standstill. The wind speed required for a switch-off is called cut-out speed (v

_{cut-out}) which is about of 25 m/s for many onshore turbines.

- Power losses due to mutual shading of adjacent turbines (wake effect).
- Switch-offs due to wind turbine revisions or bird and bat protection.
- Feed-in interruptions due to energy surpluses in the power grids.

#### 2.3. Weather Database

#### 2.4. Validation Data

## 3. Model

- Extrapolation of the weather data provided for the specified plant location to the hub height.
- Wind-to-power conversion with the help of the specific power curve of the wind turbine.
- Correction of the output power using the air temperature and pressure at hub height.
- Calculation of the produced electricity considering additional losses and the date of (de-)commissioning.
- Temporal aggregation of the simulated time series and data storage.

_{0}stands for the known wind speed at the height H

_{0}, which is 10 m above the ground and obtained from the weather data. The exponent α is called the friction coefficient or Hellmann exponent. This coefficient is a function of the topography at a specific location and frequently assumed as a value of 1/7 for open land [29,30], which corresponds well with the sites of most onshore turbines in Germany. Hence, this value is used for the simulations, but, if necessary and reasonable, other averaged values, e.g., 1/5 or 1/6, can be easily applied in the wind power model. The next step converts the calculated wind speed at hub height to the corresponding output power of the wind turbine. This operation is performed with the help of the power curve, which returns the output power at a standardized air density of 1.225 kg/m

^{3}. The main advantage of using the power curve is that no further information about the turbine technology, e.g., the specific rotor diameter or electrical losses of the power generator and converter electronics, has to be known to calculate the output power. Thus, in the presented simulation model, each wind turbine can be treated like a black box with the power curve as its transfer function. The nonlinear section of a typical power curve, as depicted in Figure 6, is typically given by the manufacturer as a table of values. In order to integrate such a nonlinear and discrete relationship into the simulation model with high accuracy and without any additional interpolation routines for intermediate values, the specific power curve is divided into separate sections and these sections are developed to analytical functions. Moreover, to be able to assign the same power curve to similar wind turbines with a different rated power, it is beneficial to use normalized power curves for the wind power model. The specific power curve of most wind turbines, which are often pitch-controlled like the example given in Figure 2, is implemented according to Equation (2). In this relationship, N(v) stands for the normalized output power as a function of the wind speed, i.e., N(v) = P(v)/P

_{R}∈ [0, 1]:

_{i}of Equation (3) are determined by polynomial regression using the Excel software. It turns out, as shown in Figure 6 for an Enercon E-40, that the development of the nonlinear section of the power curve, i.e., the range between the specific cut-in and rated speed, to a sixth-order polynomial results in precise approximations with coefficients of determination (R

^{2}) better than 0.99 and root-mean-square errors (RMSE) lower than 0.01.

_{N}at a fixed air density of 1.225 kg/m

^{3}, which corresponds to an ambient temperature of 288.15 K (15 °C) at normal atmospheric pressure of 1.013 bar. Hence, the obtained power value has to be corrected with the air temperature and pressure at hub height. For this, the ambient temperature provided by the weather data is extrapolated according to the general rule given in [31]:

_{0}stands for the ambient temperature at height H

_{0}, which is 2 m above ground level. After estimating the air temperature at hub height, assuming an average temperature gradient over all weather conditions of 0.0065 K/m [31], the output power can be corrected by the following expression derived from the barometric height formula and the power curve correction according to [32]:

_{T}is given by the sum of the hub height and the ground elevation above sea level, and the so-called scale height has a value of about 8430 m at an isothermal temperature of 288.15 K. Since changes in atmospheric pressure caused by the weather are not considered in these calculations, due to the lack of highly resolved air pressure data in PVGIS, this correction of the output power takes only into account the air temperature at hub height and the air pressure as a function of the total height.

## 4. Results

#### 4.1. Simulation of a Single Wind Turbine

#### 4.2. Simulation of the Plant Ensemble

- Deviations through the Hellmann’s exponential law with wind speeds at 10 m.
- The uncertainties of the weather data and the fact of hourly averaged values.
- Weather-related changes in air pressure are not considered in the model.
- The assignment of wind turbines to the corresponding power classes.

## 5. Discussion

_{st}can be calculated according to Equation (6):

_{wt}is the wind turbine capacity installed in the considered region, and E

_{wt}is the produced electricity from wind turbines in this region and period. Figure 11 shows the monthly capacity factors at the spatial resolution of LAU2 in Germany for the year 2016.

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Installed wind turbine capacity and (

**b**) intra-annual capacity increase at LAU2 level in Germany for 2016. In the grey areas no wind turbines are (

**a**) installed or (

**b**) have been added during this year.

**Figure 2.**Typical power curve (blue line) of a wind turbine using the example of an Enercon E-40 with 0.5 MW rated power. The black dots on the power curve mark the discrete values as given in the manufacturer’s datasheet [16]. The depicted characteristic parameters of the power curve are explained in the following text.

**Figure 3.**Measured output power (black line) and wind speed (blue line) time series of a wind turbine General Electric GE 1.5sl located in Zodel near the town of Görlitz. The depicted measurements are performed every 10 min over a period of 60 h.

**Figure 6.**Approximation of the nonlinear section of the normalized power curve of an Enercon E-40 to a sixth-order polynomial (blue line) with the corresponding coefficient of determination (R

^{2}) and root-mean-square error (RMSE). The black dots on the blue line mark the normalized values determined from the discrete values shown in Figure 2 and given in the manufacturer’s datasheet [16].

**Figure 7.**Required section of the guaranteed power curve (blue line) of the General Electric GE 1.5sl, which is given by the discrete values (black dots on the blue line) taken from the turbine database of The Wind Power [15].

**Figure 8.**Simulated (black line) and measured (blue line) output power of the wind turbine General Electric GE 1.5sl over a period of 60 h with a temporal resolution of 10 min.

**Figure 9.**Simulated electricity production (black line) and measured feed-in data (blue line) from wind turbines in Germany for 2016 with a daily resolution.

**Figure 10.**Aggregated monthly electricity production from onshore turbines at LAU2 level in Germany for 2016. In the grey areas no wind turbines are installed.

**Figure 11.**Determined monthly capacity factors of wind turbines at LAU2 level in Germany for 2016. In the grey areas no wind turbines are installed.

Parameter | Usage |
---|---|

Latitude | required |

Longitude | required |

LAU-Id ^{1} | optional |

Rated power | required |

Hub height | required |

Turbine type | optional |

Commission date | required |

Decommission date | optional |

^{1}The LAU-Id is an eight-digit identifier which determines a municipal area or local administrative unit (LAU).

**Table 2.**Power classes with the corresponding ranges of rated power (P

_{R}) and the applied turbine types.

Power Class (MW) | Power Range (MW) | Turbine Type |
---|---|---|

0.1 | P_{R} ≤ 0.15 | Fuhrländer FL100 |

0.2 | 0.15 < P_{R} ≤ 0.25 | Enercon E-30 |

0.5 | 0.25 < P_{R} ≤ 0.75 | Enercon E-40 |

1 | 0.75 < P_{R} ≤ 1.50 | Vestas V52 |

2 | 1.50 < P_{R} ≤ 2.50 | Enercon E-82 |

3 | 2.50 < P_{R} ≤ 3.50 | Vestas V112 |

5 | P_{R} > 3.50 | Enercon E-126 |

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**MDPI and ACS Style**

Lehneis, R.; Manske, D.; Thrän, D. Modeling of the German Wind Power Production with High Spatiotemporal Resolution. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 104.
https://doi.org/10.3390/ijgi10020104

**AMA Style**

Lehneis R, Manske D, Thrän D. Modeling of the German Wind Power Production with High Spatiotemporal Resolution. *ISPRS International Journal of Geo-Information*. 2021; 10(2):104.
https://doi.org/10.3390/ijgi10020104

**Chicago/Turabian Style**

Lehneis, Reinhold, David Manske, and Daniela Thrän. 2021. "Modeling of the German Wind Power Production with High Spatiotemporal Resolution" *ISPRS International Journal of Geo-Information* 10, no. 2: 104.
https://doi.org/10.3390/ijgi10020104