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A Control Strategy to Maximise Energy Extraction in

Windfarms

Vahid S. Bokharaie∗and Johan Meyers†

Departmebt of Mechanical Engineering, KU Leuven,

Celestijnenlaan 300A, Heverlee, Belgium

Abstract

The common practice in existing commercial wind-

farms is to set the control parameters of each indi-

vidual wind turbine such that it extracts as much

energy as possible from the wind, a setting that

we refer to as greedy setting. But since each wind

turbine reduces wind velocity in its wake, down-

stream wind turbines will generate less power. In

this study, using a Large Eddie Simulation (LES)

environment, we show that if the operating points

of some wind turbines are changed to non-greedy

setting, the total energy extracted by windfarm can

be increased. This holds when the control parame-

ters are kept constant for relatively large time peri-

ods. In other words, it is possible to set the control

parameters of some wind turbines in a windfarm

such that the loss in energy extraction caused by

deviating from greedy case is less than the gain in

power generation in the wind turbines which are

located downstream. We also present an optimisa-

tion scheme which is a step towards extending these

results into a more dynamic approach to windfarm

control.

1 Introduction

There are many studies which have considered op-

timisation of windfarm performance, some of them

focussing on optimisation of farm layout both in

small farms [14, 15, 3] and in large arrays [17, 16].

Also diﬀerent works have been done on windfarm

∗Email address: vahid.bokharaie@kuleuven.be; Corre-

sponding author

†Email address: johan.meyers@kuleuven.be

control, focussing on various aspects of windfarm

operation such as reduction of structural loads,

power regulation and grid support [23, 10, 13, 5].

Our focus in this study is on increasing energy

extraction in an existing windfarm. Some at-

tempts have been made in this area, for example

in [18, 6, 21]. However, as far as windfarm-ﬂow in-

teractions are included in these studies, they are all

based on fast heuristic models, for example mod-

els for wake interaction and merging such as the

ones presented in [20, 12, 19]. One of the excep-

tions is [9], in which the authors have considered

the optimal control of windfarms using large-eddy

simulations of the windfarm boundary layer as the

state model, allowing for a detailed optimisation

of the dynamic interaction of the windfarm’s tur-

bines with the boundary layer and its large-scale

three-dimensional turbulent structures. Their re-

sults accumulated over one hour of operation shows

16% of increase in energy extraction compared to

greedy case. But the downside of this approach is

that it is computationally intensive and by order of

magnitude too slow for online optimisation.

In this study, we have provided evidence that

shows by an appropriate choice of control param-

eters for the wind turbines (which are the disk-

average thrust coeﬃcients in our study), even when

they are kept constant for a relatively long time

period, an increase in energy extraction can be

achieved. As a step towards adapting these results

to more dynamical windfarm control algorithm, we

present an optimisation scheme which is performed

on the data collected from various simulations. The

simulations are performed using SP-Wind, which

is an in-house Large Eddy Simulation environment

1

developed in TFSO group in KU Leuven.

The paper is organised as follows: in Section 2,

we present the physical model governing the sys-

tem. In Section 3, we state the mathematical for-

mulation of the optimisation problem. Section 4 in-

cludes description on simulation environment and

simulation setup. Section 5 provides evidence to

demonstrate that the greedy setting is not an opti-

mum setting for a windfarm, even when the control

parameters are set to ﬁxed values. In Section 6, we

presents the results of an optimisation scheme for

the windfarm control problem. In Section 7 we ex-

plain the conclusions and future works.

2 System Model

The governing equations are the ﬁltered incom-

pressible Navier-Stokes equations for neutral ﬂows

and the continuity equation [9]:

5 · ˜u= 0

∂˜u

∂t + ˜u· 5˜u=−1

ρ˜p+f∞e1+5 · τM+f(1)

where ˜u= [˜u1,˜u2,˜u3] is the resolved velocity ﬁeld, ˜p

the remaining pressure ﬁeld (after subtracting p∞),

τMis the subgrid-scale model, and the density ρis

assumed to remain constant. Furthermore, frep-

resents the forces (per unit mass) introduced by

turbines on the ﬂow.

Modern wind turbines are controlled by the gen-

erator torque, and by a blade-pitch controller,

which controls the aerodynamic torque by changing

the angle of attack of the turbine blades. In simu-

lations, two models are widely used for interactions

between wind turbine and the environment: Actua-

tor Line Models and Actuator Disk Models. In the

simulations whose results are reported in the fol-

lowing sections, Actuator Disk Model (ADM) has

been used to calculate the force f. In Actuator Disc

Models, tangential forces are usually neglected. It

is demonstrated that standard ADMs are an accu-

rate representation of the overall wake structure be-

hind turbine except for the near wake, i.e., x/D < 3

with Dbeing the diameter of the turbine [25].

In ADM, the axial force of a turbine on the ﬂow

ﬁeld is expressed as [9]:

F=−1

2C0

Tρb

V2A

where C0

Tis the disk-based thrust coeﬃcient, b

V

is the average axial ﬂow velocity at turbine rotor

disk, and A=πD2/4 is the rotor-disk surface.

The disk-based thrust coeﬃcient C0

Tresults from

integrated lift and drag coeﬃcient over the turbine

blade, taking design geometry and ﬂow angles into

account and is considered as the control parameter

in our optimisation scheme. C0

T= 2.0 corresponds

to the case where the wind turbine is set to extract

maximum amount of energy from the wind, and

C0

T= 0.0 represents the case where the wind tur-

bine is shut down and extracts no energy. Hence in

our search to ﬁnd the optimum setting for a wind-

farm, we limit our feasible region to [0.0,2.0] for all

wind turbines.

3 Problem Description

In existing commercial windfarms, the eﬀects of

turbine wakes and the accumulated local energy ex-

traction from the atmospheric boundary layer leads

to a reduction in farm eﬃciency, with power gen-

erated by turbines in a farm being lower than that

of a lone-standing turbine by up to 50% [1]. This

eﬀect is shown in [11] with detailed measurements

in the Horn’s Rev windfarm. We will also provide

evidence to this end in the following section.

Since our aim is to maximise the power gener-

ated by a windfarm, the natural choice of objective

function in the optimisation problem is:

F:= ZT

0

Ptotal(t) (2)

where Trepresents the optimisation horizon and

Ptotal is the total power produce by windfarm. In

the results presented in Section 6, we have assumed

Tto be the whole simulation time period, which is

500 seconds. Therefore, the general form of optimi-

sation problem for maximising energy extraction in

a windfarm can be stated as a PDE-constrained op-

timisation problem whose aim is to maximise Fas

deﬁned in (2), subject to the Navier-Stokes equa-

tions as deﬁned in (1). The following constraint

should also be added to the optimisation setup:

0.0≤C0

Ti≤2.0∀i∈ {1,· · · , n}

where nis the total number of wind turbines in

the windfarm. This is to ensure that the values of

control parameter are physically meaningful.

2

This is the general setup considered in [9] and

as stated earlier, leads to a computationally inten-

sive optimisation problem. In this study, we have

applied a number of simpliﬁcations and approxi-

mations. Firstly, since the simulation results are

sampled every 1 seconds, we have used a discrete-

time formulation for the optimisation problem:

maximise F:=

Ns

X

t=1

n

X

i=1

Pi(t) (3)

where Nsrepresents the total number of available

samples and Pirepresent the electrical power gener-

ated by wind turbine i. Apart from that, instead of

considering velocities in the whole domain, we have

only considered the hub-height velocities and disc-

averaged thrust coeﬃcient of each wind turbine as

state variables. We then use the Extremely Ran-

domised Trees regression method [8] to estimate the

total power produced by the windfarm, instead of

solving Navier-Stokes equations in each iteration.

More on that will be discussed in Section 6.

4 Simulation Environment

and Simulation Setup

Simulations are performed in SP-Wind, which is

developed in TFSO group in KU Leuven and is

based on previous studies on Large Eddie Simula-

tions. SP-Wind uses a pseudo-spectral discretisa-

tion in the horizontal direction. Message Passing

Interface (MPI) is used to run simulations in par-

allel. FFTW library is employed for Fourier Trans-

forms. In the vertical direction, a fourth-order

energy-conservative ﬁnite diﬀerence discretisation

is used, while time-integration is performed using a

four-stage Runge-Kutta scheme.

The focus has been on aligned windfarms, which

are windfarms whose wind turbines are arranged

in rows aligned with the wind direction. We take

a boundary layer height at H= 1 km, and use a

domain size of Ly×Lx×H= 6.28 ×3.14 ×1 km.

It is assumed that stream-wise direction aligns with

the x-axis. The computational grid corresponds to

Ny×Nx×NZ= 192 ×128 ×160. In simulations,

25 wind turbines are arranged in a 5 ×5 grid with

a diameter of D= 100 meach and hub height

of zh= 100 m. Stream-wise spacing is Sy≈5D

and span-wise spacing is Sx≈4D. The simulation

time-steps are set dynamically based on a convec-

tive CFL number of 0.4 in Runge-Kutta method

and is approximately 0.47 seconds in the reported

cases. These parameters are summarised in Table

1.

Domain Size 6.28 ×3.14 ×1 km3

Turbine Diameter 100m

Turbine Height 100m

Turbine Arrangements 5 ×5

Surface Roughness 0.1m

Grid Size 192 ×128 ×160

Cell Size 32.7×24.5×6.25 m3

Time-step ≈0.47 seconds

Table 1: Simulation Parameters

Simulations are performed using concurrent pre-

cursor method [24], which means an auxiliary do-

main is deﬁned to supply the main domain with a

turbulent inﬂow. Also, a fringe forcing technique is

used to generate the desired inﬂow proﬁle [22].

The Thinking node in Flemish Super-Computer

(VSC) is being utilised to run the SP-Wind code.

The simulation time depends on various settings,

but for a typical simulation with the above-

mentioned parameters and in precursor mode, it

takes around 75 minutes to simulate 1000 seconds

of implementation time using 160 processors in pen-

cil decomposition parallel processing mode. Figure

1 shows a snapshot of the hub-height plane in pre-

cursor mode simulation.

Figure 1: Windfarm layout and instantaneous ve-

locity proﬁle at hub-height plane

3

5 Simulation Results

The results presented in this section provide evi-

dence that even without any dynamical windfarm

control algorithm and by only proper choice of con-

trol parameters of wind turbines, around 1% in-

crease in total power generated in the windfarm

can be achieved. In order to make sure this gain

in generated power is not because of the diﬀer-

ences in inﬂow velocity proﬁle, all simulations are

started from the exact same initial boundary layer

conditions. The inﬂow into to the domain which

subsequently speciﬁes the energy transferred into

the domain, depends on the atmospheric boundary

layer conditions. Hence the inﬂows for two diﬀerent

simulations are the same if started from the same

atmospheric boundary layer condition. This guar-

antees that the observed diﬀerences in generated

powers are caused by changes in control parame-

ters only, not the diﬀerence in wind energy injected

into the windfarm. It should be noted that because

of the minute diﬀerences caused by round-oﬀ errors

for diﬀerent cases, the inﬂow proﬁle will have slight

diﬀerences at some points. But statistical analysis

showed that these slight changes are insigniﬁcant

compared to the reported gains in power.

Table 2 shows some of the cases for which the

total power has been more than the greedy case.

As a reminder, the greedy case corresponds to all

control parameters set at 2.0.

Control Parameters Increase in PT OT AL

2.0−2.0−2.0−2.0−2.0 BASE CASE

2.0−1.8−2.0−2.0−2.0 0.98%

2.0−2.0−1.85 −2.0−2.0 0.34%

2.0−1.75 −2.0−1.8−2.0 0.13%

2.0−2.0−2.0−1.9−2.0 0.46%

2.0−1.8−2.0−1.9−2.0 0.37%

2.0−1.8−2.0−1.7−2.0 0.12%

Table 2: Comparing Total Power generated by

windfarm for diﬀerent control settings.

It should be noted that in these simulation, con-

trol parameters for diﬀerent rows are considered to

be the same. This means that as an example, for

the case 2.0−1.8−2.0−2.0−2.0, C0

Tfor the second

wind turbine in each row is set to 1.8, and C0

Tfor

all other wind turbines are set to 2.0. Simulation

results have showed that the wake eﬀects between

rows is negligible compared to wake eﬀects within

each row. Therefore, we have focused our study to

the eﬀects of changing C0

Tin each row to make them

easier to compare the results with other windfarms

with diﬀerent layouts.

Figure 2 shows the evolution of total power in the

windfarm over 500 seconds, for the greedy case and

for those cases which have performed better than

the greedy case. The changes of control parameters

are applied at time t= 0. It should come as no sur-

prise that greedy case performs better than other

cases at the earliest stages of simulation. This can

be explained considering the fact that it will take

some time for the ﬂow to travel between diﬀerent

wind turbines in each row.

0 100 200 300 400 500

220

240

260

280

300

320

Figure 2: Instantaneous Power for some of the cases

which have performed better than the greedy case.

The greedy case represented by colour red.

Figure 3 shows the total power produced by each

wind turbine, average over 5 rows, normalised to

corresponding wind turbine in greedy mode. These

diagrams show that in these cases, the loss in power

generation for some wind turbines is compensated

by a larger gain in subsequent wind turbines.

A minute point in reading values in Figure 3 is

that they are normalised to values of the corre-

sponding wind turbines in the greedy case, which

are not the same themselves. Figure 4 shows the

total power produced by wind turbines (averaged

over 5 rows), in greedy setting.

These results are valid for cases where the wind

direction is aligned with the grid’s x-axis. Chang-

ing the wind direction will make the problem qual-

itatively diﬀerent which is out of the scope of this

study and the subject of future research.

4

2.00 - 2.00 - 2.00 - 2.00 - 2.00 -

12345

0.95

1

1.05

2.00 - 2.00 - 2.00 - 1.90 - 2.00 -

12345

0.95

1

1.05

2.00 - 2.00 - 1.90 - 2.00 - 2.00 -

12345

0.95

1

1.05

2.00 - 1.80 - 2.00 - 2.00 - 2.00 -

12345

0.95

1

1.05

Figure 3: Total Power generated by each wind tur-

bine averaged over 5 rows, normalised to corre-

sponding wind turbines in greedy setting.

1 2 3 4 5

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4: The Total Power generated in each wind

turbine in greedy case, averaged over 5 rows, nor-

malised to the ﬁrst wind turbine.

6 Optimisation based on

Cross-Entropy method

As we showed in the previous section, by chang-

ing the control setting of certain wind turbines in

a windfarm, we can have an increase in total en-

ergy extraction in the windfarm. In this section, as

a ﬁrst step towards a more dynamic approach, we

have utilised the Cross Entropy (CE) optimisation

method in order to ﬁnd the optimum setting based

on the available data obtained from various simu-

lations. Cross Entropy method was originally de-

veloped to estimate probability of rare events and

later on, it was realised that it could be used to

solve diﬃcult optimisation problems [4]. Cross En-

tropy involves two general steps; to generate ran-

dom samples under certain conditions, and to up-

date samples based on the most optimum samples

in each iteration.

One of the advantages of Cross Entropy method

is the limited number of parameters that needs to

be set. They include total number of samples in

each generation, which is set to 1000 in our case,

number of best samples chosen in each generation

to generate the next generation which is set at 50,

and a weight given to previous best samples which

is set at 0.3. Also the mean and standard devi-

ation of the initial population should be speciﬁed.

We have considered diﬀerent values for these initial

values to ensure the optimisation method does not

converge to local minima.

As already explained in the previous section, sim-

ulation results have shown that with the considered

span-wise spacing between the rows, inter-row in-

teractions can be ignored. The objective function is

considered to be the same as deﬁned in (3) with the

diﬀerence that the optimisation is performed over a

row of 5 wind turbines, instead of the whole wind-

farm of 5×5. This has become possible by splitting

the simulation results for each 5 ×5 windfarm into

5 sets of one-row windfarms with 5 wind turbines

each. Also, since the last turbine in each row does

not have any other turbine in its wake, its control

value is set at a ﬁxed value of 2.0. Therefore, the

optimisation parameters are C0

Tifor i= 1,· · · ,4.

To evaluate the value of objective function, we

have used the Extremely Randomised Trees regres-

sion method, which is an ensemble of decision trees

[8]. Decision Trees are a non-parametric super-

vised learning method used for classiﬁcation and

regression. The goal is to create a model that pre-

dicts the value of a target variable by learning sim-

ple decision rules inferred from the data features

[2]. In Extremely Randomised Trees we establish a

collection of decision trees. And while calculating

the splits in the trees, instead of looking for most

discriminative thresholds, thresholds are drawn at

random for each tree and the best of these ran-

domly generated thresholds is picked as the split-

ting rule. In this study, we have used the imple-

mentation of Extremely Randomised Trees as in

[7]. Figure 5 shows the evolution of an optimisa-

tion run, with initial population built around the

5

mean initial value of [1.0,1.0,1.0,1.0] with a stan-

dard deviation of [0.5,0.5,0.5,0.5]. Figure 6 shows

the estimated calculated power in each iteration.

As can be seen, the algorithm converges to a set

of control parameters with values of approximately

[2.0,1.8,2.0,2.0], which is the optimum solution ob-

served in simulations.

Figure 5: Changes in control parameter in the CE

optimisation scheme.

5 10 15 20 25 30

52

52.5

53

53.5

54

best

greedy

Figure 6: Changes in power for best case in each

iteration of CE optimisation scheme.

7 Conclusions

In this paper, we showed that the standard practice

in commercial windfarm, in which each wind tur-

bine is set to extract maximum amount of energy

from the wind, is not the optimum setting when the

objective function is the total power generated by

the windfarm. We showed that by suitable changes

in settings, even when these settings are considered

to be ﬁxed for a relatively long period of time, total

energy extracted by the windfarm can be increased.

We then presented an optimisation scheme that

can be considered as a ﬁrst step towards designing

a more dynamic control methodology to increase

the total gain in power generation. This work can

be extended in a number of ways. It can be ap-

plied to windfarms with diﬀerent layout, both in

terms of number of wind turbines and in alignment.

More interestingly, the optimisation results can be

adapted to be applied to shorter time-horizons in

order to achieve a more dynamical control scheme

and higher gains in total energy extraction in wind-

farms.

8 Acknowledgements

The authors would like to thank Dr. Bert Claessens

from Vito NV, Belgium and Prof. Damien Ernst

from University of Liege, Belgium, for their contri-

butions.

References

[1] R. J. Barthelmie, S. Pryor, S. T. Frand-

sen, K. S. Hansen, J. Schepers, K. Rados,

W. Schlez, A. Neubert, L. Jensen, and S. Neck-

elmann. Quantifying the impact of wind tur-

bine wakes on power output at oﬀshore wind

farms. Journal of Atmospheric and Oceanic

Technology, 27(8):1302–1317, 2010.

[2] L. Breiman, J. Friedman, R. Olshen, and

C. Stone. Classiﬁcation and regression trees

(wadsworth, belmont, ca, 1984). In Proceed-

ings of the Thirteenth International Confer-

ence, Bari, Italy, page 148, 1996.

[3] S. Chowdhury, J. Zhang, A. Messac, and

L. Castillo. Unrestricted wind farm layout op-

timization (uwﬂo): Investigating key factors

inﬂuencing the maximum power generation.

Renewable Energy, 38(1):16–30, 2012.

[4] P.-T. De Boer, D. P. Kroese, S. Mannor, and

R. Y. Rubinstein. A tutorial on the cross-

entropy method. Annals of operations re-

search, 134(1):19–67, 2005.

[5] P. Fleming, P. Gebraad, J.-W. van Winger-

den, S. Lee, M. Churchﬁeld, A. Scholbrock,

J. Michalakes, K. Johnson, and P. Moriarty.

6

The sowfa super-controller: A high-ﬁdelity

tool for evaluating wind plant control ap-

proaches. In Proceedings of the EWEA Annual

Meeting, Vienna, Austria, 2013.

[6] P. Gebraad and J. Wingerden. Maximum

power-point tracking control for wind farms.

Wind Energy, 2014.

[7] P. Geurts. Regression tree package.

http://www.montefiore.ulg.ac.be/

~geurts/Software.html, 2015. Last Re-

trieved: 01 March 2015.

[8] P. Geurts, D. Ernst, and L. Wehenkel. Ex-

tremely randomized trees. Machine learning,

63(1):3–42, 2006.

[9] J. Goit and J. Meyers. Optimal control of en-

ergy extraction in wind-farm boundary layers.

Journal of Fluid Mechanics, 768:5–50, 2015.

[10] A. D. Hansen, P. Sørensen, F. Iov, and

F. Blaabjerg. Centralised power control of

wind farm with doubly fed induction genera-

tors. Renewable Energy, 31(7):935–951, 2006.

[11] K. S. Hansen, R. J. Barthelmie, L. E. Jensen,

and A. Sommer. The impact of turbulence

intensity and atmospheric stability on power

deﬁcits due to wind turbine wakes at horns

rev wind farm. Wind Energy, 15(1):183–196,

2012.

[12] N. O. Jensen. A note on wind generator inter-

action. 1983.

[13] K. E. Johnson and N. Thomas. Wind farm

control: addressing the aerodynamic interac-

tion among wind turbines. In American Con-

trol Conference, 2009. ACC’09., pages 2104–

2109. IEEE, 2009.

[14] F. Kaminsky, R. Kirchhoﬀ, and L.-J. Sheu.

Optimal spacing of wind turbines in a wind

energy power plant. Solar energy, 39(6):467–

471, 1987.

[15] A. Kusiak and Z. Song. Design of wind farm

layout for maximum wind energy capture. Re-

newable Energy, 35(3):685–694, 2010.

[16] J. Meyers and C. Meneveau. Optimal turbine

spacing in fully developed wind farm boundary

layers. Wind Energy, 15(2):305–317, 2012.

[17] B. Newman. The spacing of wind turbines in

large arrays. Energy Conversion, 16(4):169–

171, 1977.

[18] J. Park and K. Law. A bayesian optimiza-

tion approach for wind farm monitoring and

power maximization. In The 6th International

Conference on Structural Health Monitoring of

Intelligent Infrastructure, 2013.

[19] O. Rathmann, S. T. Frandsen, and R. J.

Barthelmie. Wake modelling for intermediate

and large wind farms. In 2007 European wind

energy conference and exhibition, 2007.

[20] P. S. Lissaman. Energy eﬀectiveness of arbi-

trary arrays of wind turbines. Journal of En-

ergy, 3(6):323–328, 1979.

[21] M. Soleimanzadeh, R. Wisniewski, and

S. Kanev. An optimization framework for load

and power distribution in wind farms. Journal

of Wind Engineering and Industrial Aerody-

namics, 107:256–262, 2012.

[22] P. R. Spalart and J. H. Watmuﬀ. Experimental

and numerical study of a turbulent boundary

layer with pressure gradients. Journal of Fluid

Mechanics, 249:337–371, 1993.

[23] C. J. Spruce. Simulation and control of wind-

farms. PhD thesis, University of Oxford, 1993.

[24] R. J. Stevens, J. Graham, and C. Meneveau. A

concurrent precursor inﬂow method for large

eddy simulations and applications to ﬁnite

length wind farms. Renewable energy, 68:46–

50, 2014.

[25] Y.-T. Wu and F. Port´e-Agel. Large-eddy sim-

ulation of wind-turbine wakes: evaluation of

turbine parametrisations. Boundary-layer me-

teorology, 138(3):345–366, 2011.

7