CFD Modeling of Wind CFD Modeling of Wind Farms in Flat and Farms in Flat and Complex TerrainComplex Terrain

J. M. Prospathopoulos, E. S. Politis, P. K. Chaviaropoulos

K. G. Rados, G. Schepers, D. Cabezon, K. S. Hansenand R. J. Barthelmie

Numerical issues in modelingNumerical issues in modelingCorrection of the velocity deficit

underestimation in the near wake◦Modification of the turbulence model ◦Realizability constraint

Definition of the reference wind speed for thrust estimation◦Independent of the distance from the W/T rotor◦Induction factor concept

Test cases examinedTest cases examined5 W/T in a row for stable conditions◦ECN test wind farm, flat terrain◦WT distance = 3.8 D◦Wind speed: 6-8 m/s◦Wind directions: ±30 degs

Test cases examinedTest cases examinedReal wind farm in complex terrain with 43

W/Ts◦Complex terrain in Spain, neutral conditions◦Distance between rows ≈ 11D◦Distance between WTs at the same row ≈ 1.8D◦Wind speed 8 m/s

Wind direction 327 degs

Navier–Stokes modeling Navier–Stokes modeling RANS solver based on the pressure correction

scheme◦ Body fitted coordinate transformation◦ Numerically integration of equations with an implicit multi-

block scheme◦ A matrix-free, conjugate gradient type, solver handles the

pressure correction◦ Developed, used and verified in European research projects

(UpWind)

Turbulence model k-ω modified for atmospheric flows ◦ Constants:

Rotor modeling◦ Momentum sink through actuator force

α 0.3706, β 0.0275, β 0.033,

σ 0.5, σ 0.5

2t refDF 0.5 ρC U dS

Boundary conditionsBoundary conditionsWind speed profile at inlet

k & ω profiles at inlet

x 0

0, neutraluU ln z / z c z , c(z )

5z / L, stableK

0.52

0.5ωω m

m

fu uk , ω f f

fβ β Κ z

m ω

1, neutral 1, neutralf , f

1 5z / L, stable 1 4z / L, stable

Modeling of stable conditions Modeling of stable conditions Additional buoyancy term is added for

turbulence

Add buoyancy term to k and ω equations: ◦k-equation:◦ω-equation:◦Dirichlet inflow conditions (common approach):◦Neumann inflow conditions (calculate coefficients

to satisfy N-S equations):◦Similar results

2

t m2m

U Ri 0.74 4.7ζG μ , Ri ζ , f 1 5ζ , ζ z / L

z f 1 4.7ζ

k ωf 1, f 0

k ωf 1 4.9ζ , f 14 (1 1.28ζ )

Computational GridsComputational GridsHorizontal grid spacing 0.05 D close to the W/Ts

◦ Grid refinement in vertical direction close to the ground◦ 1st grid line 0.01 D above ground◦ Grid refinement in W/T rotor disk◦ 21 grid points along rotor diameter

Computational GridsComputational GridsMinimum grid spacing at xy-plane: 0.08 D / 0.1 D

close to the W/Ts First vertical grid-line at 0.5 m above ground100 grid points over the rotor disk area7 million grid-points for the total simulation

Turbulence model correctionTurbulence model correctionVelocity deficit underestimation ↔ Turbulence

overestimation◦Concept from stagnation point flows where

turbulence overestimation is also observed◦Realizibility constraint for turbulent velocities ◦Apply the constraint on the eddy viscosity formula in

the principal axes of the strain tensor◦Relationship for turbulent time scale:

◦Substitution of the turbulent time scale T in: Calculation of turbulent viscosity ω-transport equation

22k u 0

jiij ji , ij2

j i

UU1 2 3 1T min , , S S S S

ω 3 2 x x8S

Definition of the reference wind Definition of the reference wind speed speed

Typical definition: 1 D upstream of the W/T◦Mean value over the rotor disk area◦Hub height value (centre of the rotor disk)

This stems from isolated W/Ts in flat terrain considerations

Issues that arise:◦Is this valid in complex terrain?◦Is this valid in wake simulations?

Induction factor concept Induction factor concept Definition of induction factor:

Relationship between CT and induction factor

Iterative procedure starting from an initial guess of Uref

ref NS disk refa U U / U

2T

4a(1 a), a 0.4C

0.889 – 0.0203 – a 0.143 / 0.6427, a 0.4

T T ref T ref NS diskC C (U ) a a(C ) U U / (1 a)

5 W/Ts in flat terrain5 W/Ts in flat terrain

Induction factor method: Overestimation of power is in accordance to the single W/T predictions

Under-performance of the 2nd W/T is not reflected in the predictions

1D Upstream Induction factor

5 W/Ts in flat terrain5 W/Ts in flat terrain

Predictions performed using induction factor methodOverestimation of W/Ts performance is partially

correctedUnder-performance of the 2nd W/T is reproduced by

the calculation

Baseline model Turbulence model correction

43 W/Ts in complex terrain43 W/Ts in complex terrain“No wakes”: Predictions without W/Ts (terrain

effect)“Flat terrain”: Predictions in flat terrain (1D

upstream)“Terrain+wakes”: Complete simulation (1D

upstream)

43 W/Ts in complex terrain43 W/Ts in complex terrain Uncertainty of operational data is related to the lack of

calibration for the power converter and yaw position signals. So, the estimation of the reference WT’s yaw position was not better than ±5 degs.

43 W/Ts in complex terrain43 W/Ts in complex terrain “Fine grid”: dx=0.05D, dy=0.07D, dz=0.25m (5 million

nodes) 1D upstream reference wind speed and induction factor

taken at hub height

43 W/Ts in complex terrain43 W/Ts in complex terrain 1D upstream reference velocity gives better predictions Finer discretization improves the results Fine grids are necessary to simulate complex terrain

SummarySummary

Baseline predictions underestimate near wake deficit◦Modeling approaches decrease the turbulence

production in the near wake & correct the deficit◦Adjustment of additional parameters is needed

Durbin’s correction bounds the turbulent time scale◦Based on a general constraint for the turbulent velocities◦No adjustment of additional parameters is needed

Reference wind speed is defined through induction factor concept◦Applicable on W/Ts located in wakes of neighboring W/Ts

SummarySummary

Durbin’s correction improves the power prediction in a 5 W/Ts wind farm

Induction factor method does not produce satisfactory predictions in complex terrain◦Its use should be further investigated

AcknowledgementsAcknowledgementsThis work has been partially financed by the EC

within the FP6 UpWind project (# SES6 079945) and by the Greek Secretariat for Research and Technology

The wind farm owners for supplying the data for the model evaluation

Thank you for your attention!Thank you for your attention!

Correspondence to:

Evangelos S. Politis

Wind Energy Department19th km Marathonos Avenue, GR19009, Pikermi, GreeceEmail : vpolitis@cres.gr