# A Minimalistic Prediction Model to Determine Energy Production and Costs of Offshore Wind Farms

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Wind Turbine Modeling

_{out}. In the subsequent model applications, it is assumed that ${U}_{in}=$ 3 m/s and ${U}_{out}=$ 25 m/s. The corresponding thrust coefficient ${C}_{T}$ is approximated as

#### 2.2. Wind Farm Modeling

_{T}and the wind farm area A. Denoting the assumed uniform distance between the wind turbines as ${L}_{T}$, and assuming that each wind turbine occupies a square of area ${L}_{T}^{2}$, the required wind farm area relates to the number of wind turbines and the spacing distance as

#### 2.3. Wake Modeling

- The wind farm is large enough for the vertical wind profile to be horizontally homogeneous.
- The thrust on the wind turbine rotors is assumed concentrated at hub height.
- The horizontally homogeneous vertical wind profile is logarithmic both below and above hub height.
- The vertical wind profile is continuous at hub height.
- The height of the planetary boundary layer is considerably larger than the wind turbine hub height.
- Turbulent wind speed fluctuations are horizontally homogeneous.

#### 2.4. Wind Resource Modeling

_{y}may be formulated as a convolution of the wind turbine production characteristics, Equation (1), with the mean wind speed probability density function expressed in Equation (14). Thus

#### 2.5. Finite-Size Wind Farm Correction

#### 2.6. Cost Models

#### 2.6.1. Cost of Wind Turbine

_{WT}, may according to Lundberg [6] be expressed as ${C}_{WT}=-0.15+0.92{P}_{G}$, where ${P}_{G}$ is the installed generator power in MW. However, this pricing refers to the year 2003, when the report was compiled. The inflationary development in (Danish) consumer prices in general from the year 2003 up to 2015 was 23% [5,24]. In this study, we will assume wind turbine prices follow the inflation in general consumer prices during this period, and we will further add 2% to approximately include the wind turbine price development up to today (i.e., 2019). With these assumptions, we finally arrive at the following expression for wind turbine prices in MEUR:

#### 2.6.2. Cost of Support Structure

_{FM}, may in a first order approximation be simplified as (Buhl and Natarajan [25])

_{R}denotes the wind turbine rated power in MW, and H is the water depth in meters.

_{FJ}, may in a first order approximation be simplified as (Buhl and Natarajan [25])

#### 2.6.3. Cost of Wind Farm Electrical Grid

_{C}is given by

#### 2.6.4. Cost of Operation and Maintenance

_{WT},

_{Ref}= 106 EUR/kW (Chaviaropoulos and Natarajan [29]). This number, which forms the basis for the OM model in Equation (26), is determined as an average value based on analyses of different offshore located wind farms. However, the distance to the shore is an additional important parameter, as the OM expenses depend on the transport time, and hence the distance from the shore to the location of the wind farm. Therefore, the cost model formulated in Equations (26)–(29) forms the basis for OM expenses at a certain average location in terms of distance to service harbor. To determine the actual OM expenses, Equation (26) needs to be corrected by an additional term, which depends on the distance to the shore:

_{ref}= 20 km. The gradient $\frac{\Delta {C}_{O\&M}}{\Delta Y}$ needs to be determined from experience. One way to do this is to consider the difference in agreed cost price between two nearby located wind farms. Assuming that the invested installation expenses per produced energy unit is the same for the two wind farms, then this difference approximately corresponds to the difference in OM expenses. Using the Horns Rev I and Horns Rev II wind farms, which are located 14 km from each other, and which operate at agreed cost prices of 0.432 DKK/kWh and 0.518 DKK/kWh ([30,31]), respectively, we obtain the following estimate:

#### 2.6.5. Levelized Cost of Energy

_{Y}is the life time of the wind farm in years, P

_{E}is the yearly average power production for the wind farm (Equation (20)), and the denominator is the total electricity production in kWh. For the present study, we assume a wind farm life time of 20 years, i.e., N

_{Y}= 20.

## 3. Results

#### 3.1. Wind Farm Cases

- Lillgrund Wind Farm (LG)

- Rødsand 1 (RS1)

- Rødsand 2 (RS2)

- Horns Rev 1 (HR1)

- Horns Rev 2 (HR2)

- Horns Rev 3 (HR3)

#### 3.2. Computed Results

## 4. Discussion

^{2}(or equivalently W/m

^{2}). Comparing the power intensity, Lillgrund is, due its compactness, noticeably different from the rest, with 7.85 MW/km

^{2}, whereas the remaining wind farms have a power intensity in the range from 2.2 to 3.3 MW/km

^{2}. The relative difference between actual and predicted values are obviously identical to the values in Figure 3. Although the Lillgrund wind farm has very high power intensity, the costs are also high, with a LCOE that is more than double that for the other wind farms. The reason is that the wake loses are quite high, and that OPEX increases inversely with the distance between the turbines (Equations (26) and (27)). In fact, there will always be a tradeoff between erecting wind turbines close to each in order to maximize the power harvest per square unit, and to minimize wake losses by erecting the wind turbines far from each other. An example of the latter is the Horns Rev 3 wind farm, where the wind turbines in average are located about 9.5 diameters from each other, and the expected and predicted power intensity is 2.2 MW/km

^{2}and 2.4 MW/km

^{2}, respectively, but the predicted energy costs only amounts to 54.9 EUR/MWh.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Calibration constant |

c_{t} | Dimensionless auxiliary parameter |

f | Coriolis parameter |

f (*,*,*) | Weibull probability density function |

f_{C} | Wind turbine capacity factor |

f_{S}(*) | Wind turbine load factor |

f_{WF} | Wind farm capacity factor |

f_{WT}(*|*) | Wind turbine size factor |

h | Hub height |

k | Weibull shape parameter |

u*_{lo} | Friction velocity for the lower part of the boundary layer |

u*_{hi} | Friction velocity for the upper part of the boundary layer |

x | Realization of a stochastic variable X |

z | Height above sea surface in m |

z_{ref} | Reference height above sea surface in m |

z_{0,lo} | Roughness length characteristic for the lower part of the boundary layer |

z_{0,hi} | Roughness length characteristic for the upper part of the boundary layer |

A | Wind farm area |

A_{R} | Rotor area |

C_{add} | Additional costs in EUR |

C_{C} | Wind farm grid financial costs pr. running meter in EUR |

C_{G} | Aggregated internal wind farm grid costs in EUR |

C_{FJ} | Cost of a jacket support structure in MEUR |

C_{FM} | Cost of a monopile support structure in MEUR |

C_{O&M}(*,*,*) | Cost of operation and maintenance (OM) in EUR |

C_{O&M,base}(*,*) | Cost of operation and maintenance (OM) excluding transportation to site in EUR |

C_{O&M,L}(*) | Cost of transportation associated with operation and maintenance (OM) in EUR |

C_{p} | Planning costs in MEUR |

C_{P} | Power coefficient |

C_{P,rated} | Power coefficient at rated wind speed |

C_{s} | Cost of substation in MEUR |

C_{T} | Thrust coefficient |

C_{total} | Total cost of an offshore wind farm installation in MEUR |

C_{T,rated} | Thrust coefficient at rated wind speed |

C_{WT} | Cost of a wind turbine in MEUR |

C_{WTref} | Yearly cost of OM for a reference wind turbine in EUR |

C_{yref} | Cost of a 20 km export cable in MEUR |

CAPEX | Total cost of an offshore wind farm installation (i.e., capital expenditures) |

D | Rotor diameter |

Dw | Water depth in m |

E | Annual energy production in MWh |

G | Geostrophic wind speed |

Ht | Wind turbine tower height |

L_{C} | Aggregated length of internal wind farm grid cables |

Ls | Average distance from wind farm to the shore |

L_{T} | Wind turbine inter spacing |

LCOE | Levelized cost of energy |

N_{T} | Number of wind farm wind turbines |

N_{Y} | Life time of the wind farm in years |

OPEX | Operational expenditures |

P(U) | Wind turbine power production at mean wind speed U |

P_{E} | Average annual wind farm power production |

P_{g} | Name plate generator capacity |

P_{G} | Generator power |

P_{R,ref} | Rated power of a reference wind turbine in MW |

P_{S,y} | Average annual power yield of a solitary turbine in MWh |

P_{WF,y} | Average annual power yield of a wind farm turbine in MWh |

P_{y} | Yearly average production of a wind turbine in MWh |

PoA | Power density |

PPA | Power purchase agreement |

S | Normalized wind turbine inter spacing |

U | Mean wind speed |

U_{h} | Mean wind speed at wind turbine hub height |

U_{h,0} | Ambient mean wind speed at wind turbine hub height |

U_{lo} | Mean wind speed at lower part of the boundary layer |

U_{hi} | Mean wind speed at upper part of the boundary layer |

U_{in} | Cut-in mean wind speed |

U_{out} | Cut-out mean wind speed |

U_{r} | Rated mean wind speed |

X | Stochastic variable |

Y | Distance from site to service harbor in km |

Y_{ref} | Reference distance from site to service harbor in km |

α | Auxiliary coefficient |

β | Auxiliary coefficient |

δ | Auxiliary parameter |

ε_{1} | Auxiliary parameter |

ε_{2} | Auxiliary parameter |

φ | Latitude |

$\gamma $ | Auxiliary parameter |

${\gamma}_{F}$ | Fraction of wind turbines erected on monopole foundations |

κ | von Kármán constant |

λ | Weibull scale parameter |

ρ | Air density |

τ_{w} | Surface friction stress |

τ_{w,hi} | Surface friction stress |

τ_{w,lo} | Surface friction stress |

Γ (*,*) | Incomplete Gamma function |

Ω | Rotational speed of the earth |

## Appendix A

#### Appendix A.1. Average Production under Ambient Flow Conditions

_{y}may be formulated as a convolution of the wind turbine production characteristics with the mean wind speed probability density function expressed in Equation (A1). Thus

_{C}, expresses the ratio of the actual yearly output to its potential output, if it were possible to operate at full nameplate capacity continuously over the year. For a solitary turbine it is accordingly defined as

_{y}obtained from Equation (A6).

#### Appendix A.2. Average Production under Wind Farm Flow Conditions

_{h}inside the wind farm is given by Equation (13). For ${c}_{t}=0$, we obtain the ambient mean wind speed at hub height as

_{H}in the below rated regime can now be formulated in closed form by combining Equations (A14) and (A15):

_{h}and U

_{h,}

_{0}. A precondition for obtaining the simple degenerated expressions resulting from this transformation, given by Equations (A15) and (A16), is that ${U}_{h}={U}_{h}({U}_{h,0})$ is a monotonic function. For the below rated wind speed case this is easily shown as ${\epsilon}_{1}$ in Equation (A16) is a constant. For the above rated wind speed case a formal proof is given in Appendix B.

## Appendix B

_{r}, and U

_{H}being positive, $d{U}_{h,0}/d{U}_{h}$ is positive, and thereby $d{U}_{h}/d{U}_{h,0}$ is positive for any (positive) value of ${U}_{h,0}$, which in turn means that ${U}_{h}({U}_{h,0})$ is strictly monotonic. As seen, this qualitative result has been obtained without knowing the explicit form of the function ${U}_{h}({U}_{h,0})$.

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**Figure 1.**Sketch of a wind farm (Horns Rev 1) with wind vector from northwest, where the wind turbines along the first northern and the first western line (red dots) are seen to be directly exposed to the free wind.

**Figure 2.**Sketch of the three Horns Rev wind farms (published by permission of Vattenfall Wind Power). To the bottom left, the scale indicates the size of the wind farms and the map at the bottom right shows the position on a map of Denmark. It is seen that the layout of the oldest wind farm (Horns Rev 1) has a nearly quadratic shape, whereas Horns Rev 2 is curved, and the wind turbines forming Horns Rev 3 are distributed according to an optimization algorithm, which locates them in a seemingly less systematic manner.

**Figure 4.**Comparison between computed and actual power per area unit for the investigated wind farms.

**Figure 6.**Comparison between agreed power purchase costs (PPA) and computed and actual levelized costs of energy for the analyzed wind farms. Since actual OM costs are not publically available, ”actual” refers to actual CAPEX and actual production data, but with computed OPEX values.

**Figure 7.**Comparison between computed and actual CAPEX to production ratio for the different wind farms.

Pg [MW] | D [m] | Ht [m] | Dw [m] | Ls [km] | A [km ^{2}] | Nt | $\mathit{\lambda}$ [m/s] | k | CAPEX [MEUR] | E [GWh] | PPA [EUR/MWh] | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

LG | 2.3 | 93 | 65 | 4–8 | 10 | 4.8 | 48 | 9.7 | 2.4 | 214 | 330 | N/A |

RS1 | 2.3 | 82 | 69 | 6–10 | 13 | 22 | 72 | 10.5 | 2.4 | 322 | 540 | 83.9 |

RS2 | 2.3 | 93 | 68 | 6–10 | 16 | 35 | 90 | 10.5 | 2.4 | 460 | 790 | 83.9 |

HR1 | 2.0 | 80 | 70 | 6–14 | 16 | 20 | 80 | 11.0 | 2.4 | 354 | 580 | 57.6 |

HR2 | 2.3 | 93 | 68 | 9–17 | 30 | 33 | 91 | 11.2 | 2.4 | 524 | 880 | 69.0 |

HR3 | 8.0 | 164 | 105 | 11–19 | 35 | 88 | 49 | 11.5 | 2.4 | 1000 | 1700 | 78.7 |

S [-] | CF [%] | PoA [MW/km ^{2}] | OPEX [EUR/MWh] | OPEX [MEUR] | CAPEX [MEUR] | E [GWh] | LCOE [EUR/MWh] | |
---|---|---|---|---|---|---|---|---|

LG | 3.98 | 30.9 | 7.13 | 97.6 | 580 | 206 | 299 | 132 |

R1 | 7.64 | 39.0 | 2.94 | 32.7 | 371 | 336 | 566 | 62.4 |

R2 | 7.50 | 43.6 | 2.58 | 32.5 | 514 | 431 | 791 | 59.7 |

HR1 | 7.04 | 42.7 | 3.41 | 36.9 | 441 | 334 | 598 | 64.8 |

HR2 | 7.23 | 47.6 | 3.02 | 40.7 | 710 | 482 | 872 | 68.4 |

HR3 | 9.53 | 54.0 | 2.41 | 29.7 | 1100 | 937 | 1855 | 54.9 |

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## Share and Cite

**MDPI and ACS Style**

Sørensen, J.N.; Larsen, G.C.
A Minimalistic Prediction Model to Determine Energy Production and Costs of Offshore Wind Farms. *Energies* **2021**, *14*, 448.
https://doi.org/10.3390/en14020448

**AMA Style**

Sørensen JN, Larsen GC.
A Minimalistic Prediction Model to Determine Energy Production and Costs of Offshore Wind Farms. *Energies*. 2021; 14(2):448.
https://doi.org/10.3390/en14020448

**Chicago/Turabian Style**

Sørensen, Jens Nørkær, and Gunner Christian Larsen.
2021. "A Minimalistic Prediction Model to Determine Energy Production and Costs of Offshore Wind Farms" *Energies* 14, no. 2: 448.
https://doi.org/10.3390/en14020448