=Paper= {{Paper |id=Vol-1458/E16_CRC20_Lueckehe |storemode=property |title=Simulated Annealing With Parameter Tuning for Wind Turbine Placement Optimization |pdfUrl=https://ceur-ws.org/Vol-1458/E16_CRC20_Lueckehe.pdf |volume=Vol-1458 |dblpUrl=https://dblp.org/rec/conf/lwa/LuckeheKW15 }} ==Simulated Annealing With Parameter Tuning for Wind Turbine Placement Optimization== https://ceur-ws.org/Vol-1458/E16_CRC20_Lueckehe.pdf

Simulated Annealing with Parameter Tuning for
Wind Turbine Placement Optimization

Daniel Lückehe1 , Oliver Kramer2 , and Manfred Weisensee3
1
Department of Geoinformation, Jade University of Applied Sciences, Oldenburg,
Germany daniel.lueckehe@uni-oldenburg.de
2
Department of Computing Science, University of Oldenburg, Oldenburg, Germany
oliver.kramer@uni-oldenburg.de
3
Department of Geoinformation, Jade University of Applied Sciences, Oldenburg,
Germany weisensee@jade-hs.de

Abstract. Because of wake effects and geographical constraints, the
search for optimal positions of wind turbines has an important part
to play for their efficiency. The determination of their positions can be
treated as optimization problem and can be solved by various methods.
In this paper, we propose optimization approaches based on Simulated
Annealing (SA) to improve solutions for the wind turbines placement
problem. We define neighborhoods of solutions and analyze the influence
of specific parameters, e.g., neighborhood distance and SA temperature
in experimental studies. The experiments are based on a real-world sce-
nario with a wind model, wind data from a meteorological service, and
geographical constraints. Inspired by adaptive step size control applied in
Evolutionary Strategies (ES), we propose an approach using an adaptive
neighborhood distance and compare the results to optimization runs with
a constant neighborhood distance. Also the best and worst optimization
run and the corresponding placement results are shown and compared.

1 Introduction
Planning and optimization of renewable energy resources is an important part of
today’s activities towards an ecologically friendly smart grid. As the environment
of wind turbines is significant for their efficiency, the determination of their
locations should be handled carefully and with consideration of various aspects.
In this paper, we use a wind model based on wind distributions using data from
the German Weather Service and take geographical constraints into account.
We apply different optimization approaches using SA to the turbine placement
problem with the objective to maximize the power output. SA is often used for
combinatorial problems, but can also be applied to continuous solution spaces.
For this, we define neighborhoods in the continuous solution space of turbine
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academic purposes. In: R. Bergmann, S. Görg, G. Müller (Eds.): Proceedings of
the LWA 2015 Workshops: KDML, FGWM, IR, and FGDB. Trier, Germany, 7.-9.
October 2015, published at http://ceur-ws.org

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positions. We also propose an adaptive variant of neighborhoods for SA inspired
by an adaptive method from the field of ES, as the definition of neighborhood
corresponds to the step size of ES.
This paper is structured as follows. In Section 2, we give an overview to
related work. The wind model is explained in Section 3 and includes the definition
of the employed scenario. In Section 4, we introduce the optimzation approaches,
followed by the experimental results in Section 5. In Section 6, conclusions are
drawn.

2 Related Work
The wind turbine placement problem is widely known and there are a lot of
different models and approaches to solve it [5]. We observe a trend towards more
realistic representations of the optimization problem, e.g., Kusiak and Song [8]
are using Weibull [18] distribution and the Jensen [13] wake model to describe
the behavior of the wind. Their model is able to compute the power output of
turbines on a continuous map considering wake effects. They solved the optimiza-
tion problem with a simple ES. Further works exploit more complex approaches
like the covariance matrix adaptation evolution strategy (CMA-ES) [4] to solve
the optimization problem [17]. There are also approaches that include geograph-
ical information from map services to the turbine placement problem [9].
To solve the turbine placement problem, we employ stochastic search al-
gorithms [12]. In particular, we aim for an approach that exploits SA. For a
comprehensive overview of SA, we refer to [6]. In this work, the basic idea of
SA is explained, critically analyzed, and different variants of SA are experimen-
tally considered. Nourani and Andresen [14] put a focus on the cooling schedules
for SA. In their work, constant thermodynamic speed, exponential, logarithmic,
and linear cooling schedules are analyzed. Rivas et al. [15] published an ap-
proach to solve the turbine placement problem for large offshore wind farms by
SA. Their algorithm employs three types of local search operations: add, move,
and remove. The operations are performed recursively and each has its own tem-
perature. Based on the experimental results, in the conclusion of this work SA is
called a suitable method for the wind turbine placement optimization problem.

3 Wind Setting
In this section, we introduce the wind turbine model that we use in the experi-
mental part of this work. This model computes the produced energy of a wind
farm. The description is followed by the specification of the real-world scenario
used in this paper.

3.1 Wind Turbine Model
As the objective in this work is to maximize the power output of a wind farm,
we apply a wind turbine model to calculate the produced energy of the turbines.

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The wind turbine model f exploits a scenario that consists of wind turbines
and their power curves based on the Enercon E101, wind data from the German
Weather Service, wake effects computation using the Jensen wake model [13], and
geographical constraints based on data from OpenStreetMap [3]. The COSMO-
DE [2] wind data from the German Weather Service are used to calculate Weibull
distributions [18] for every position and wind direction. With the model from
Kusiak and Song [8] the power output E is calculated. For one wind turbine
with position ti , it applies:
Z 360
E(ti ) = pθ (ti , θ) · Eθ (ti , θ)dθ (1)
0

with the power output Eθ (ti , θ) for one wind direction:
Z ∞
Eθ (ti , θ) = βi (v) · pv (v, k(ti , θ), c(ti , θ))dv. (2)
0

The distribution of wind angles is described in pθ (ti , θ), the function βi (v) spec-
ifies the power curve of the used wind turbine, and the Weibull distribution
pv (v, k(ti , θ), c(ti , θ)) represents the wind speed distribution. In the evolutionary
optimization process, the geographical constraints are modeled by a variant of
death penalty that is similar to the approach used by Morales and Quezada [11].
For a detailed description of the wind model, we refer to our depiction in [10].
The model f computes the power output of a solution x that describes
the positions of multiple wind turbines for a defined scenario, i.e., a scenario
specifies the map section and the wind distributions. Thereby, the optimiza-
tion objective is to maximize the sum of the power output E of all turbines t:
PN/2
f (x) = i=1 E(ti ). The solution x is a vector of elements x = (x1 , x2 , . . . , xN )T
with the length N coding the x- and y-coordinate of every turbine xt and y t ,
i.e.:
x = (xt1 , y1t , xt2 , y2t , . . . , xtN/2 , yN/2
t
).
With ti = (xti , yit ) the solution vector can be written as:

x = (t1 , t2 , . . . , tN/2 ).
x
We denote the position ti of a specific turbines of solution xj as ti j . This notation
is required for the definition of neighborhoods in Section 4.1.

3.2 Scenario
To specify a scenario, we keep in mind the objective to model realistic settings
for Lower Saxony, Germany. There are more than 5, 000 wind turbines in Lower
Saxony. Most of them are grouped in wind farms smaller than 30 turbines [16].
We define an onshore scenario with 22 turbines in an area of 5 km x 5 km, which
leads to a 44-dimensional solution space. To take into account the constraints
outside the feasible area, we also consider the geographical information within

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668.0

665.5

663.0

660.5

658.0

Fig. 1. Visualization of the scenario.

a distance of 1 km beyond each border. The coordinates of the scenario are
53.3925◦ - 53.45648◦ , 7.7395◦ - 7.84304◦ in decimal degrees. Figure 1 shows the
scenario with red constrains and yellow to blue potential map. It consists of 311
buildings and 355 streets consisting of 1987 parts modeled in OpenStreetMap.
The potential for a turbine without wake effect by other turbines is about 660 kW
becoming lower in the middle of the map. Figure 2 shows an exemplary wind
rose in our scenario. Most wind is coming from south-west in this scenario.

> 18 m/s
15 - 18 m/s
12 - 15 m/s
9 - 12 m/s
6 - 9 m/s
3 - 6 m/s
0 - 3 m/s

Fig. 2. Wind rose at location of the scenario.

4 Simulated Annealing
In the following, we introduce the SA variants that are compared in the ex-
perimental study. First, the concept of SA is introduced. Then, we define the
neighborhood of two different solutions xi and xj using a neighborhood dis-
tance dn , followed by the introduction of two optimization algorithms. The first
one is an SA approach with a deterministic cooling schedule. In the second ap-
proach, we propose an adaptive control of the neighborhood distance dn , which
is inspired by the step size control used by ES.

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SA is based on the cooling process of metallic elements that reduce defects in
crystals. The algorithm starts with an initial solution x0 . In our work, we start
with a feasible randomly created solution. Then, the optimization algorithm
creates a neighbor x0 of the actual solution x. The neighboring solution x0 is
analyzed w. r. t. fitness function f . If f (x0 ) > f (x) the solution x0 replaces x
in the following iteration. For a better chance to leave local optima, SA can
also accept worse solutions x0 with a fitness function value f (x0 ) < f (x). The
probability M that describes the chance to accept a worse solution is depending
on the difference of the fitness function values f (x0 ) and f (x) and on a parameter
T that is called temperature, i.e.:

f (x)−f (x0 )
M (x, x0 , T ) = e− T (3)

The temperature T is variable during an optimization run with SA. Equation 3
shows how the temperature T affects the probability M to accept a worse solu-
tion: The higher the temperature T is, the higher is the probability M .

4.1 Turbine-Oriented Neighborhood

To define the neighborhood between two solutions xi and xj , we first specify a
neighborhood distance dn . Solution xj is in the neighborhood of solution xi , if
one turbine tk with k ∈ {1, . . . , N/2} is different in both solutions but within
the distance dn :
x
max txk i − tk j ≤ dn (4)

and all other turbine positions from both solutions are equal:
x
∀tl : txl i = tl j (5)

with l ∈ {1, . . . , N/2} and l 6= k.

4.2 Deterministic Cooling Schedule

In SA, the cooling process starts with a high probability M at the beginning
of the optimization process, while M is reduced in the course of the optimiza-
tion. Various options to reduce M have been introduced in the past, mainly
by decreasing temperature T . A common rule to cool down is a deterministic
temperature control with T 0 = α · T . Defining i as iteration number and T0 as
starting temperature, the cooling process can be described by:

Ti = αi · T0 (6)

with 0 < α < 1.

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4.3 Approach with Constant Neighborhood Distance
To implement an optimization algorithm using SA with a deterministic cooling
schedule, we have to determine the neighborhood distance dn , the initial tem-
perature T0 , and the cooling factor α. In this work, we use a small and a large
neighborhood distance dn . The small distance d− n is set to 50 m, which means
a turbine t can be shifted up to 50 m per iteration in both dimensions. With
this distance the solution can reach the next local optimum, but is not able to
shift a turbine t over a constraint like a street. Therefore, we define the large
distance d+n = 500 m which makes it possible to shift a turbine over a street but
the fine-tuning is more difficult.
As we can see in Equation 3, the temperature T must be chosen depending
on the scale of the difference ∆f = f (x) − f (x0 ). Preliminary experiments show
that the difference ∆f depends on the neighborhood distance dn . Using d− n,
in the first 100 iterations with a new feasible solution x0 , it applies for the
difference ∆f that its mean value and standard deviation are ∆f ≈ 0 ± 6 with
max(∆f ) ≈ 20. It means better and worse solutions are equally distributed, in
about 70% of the iterations the difference is smaller than 6, and the maximum
value is approximately 20. Using d+ n , it applies for the first 100 iterations with
a new feasible solution x0 : ∆f ≈ 10 ± 30 with max(∆f ) ≈ 120. We define two
different initial temperatures using this information. A high temperature T0h has
the objective that at the beginning of the optimization process in 10% of the
cases a worse solution is accepted. As we are not focusing on extreme values, we
use the standard deviation. It applies for d− n:

− 6h 6
M = 0.1/0.7 = e T0
⇒ T0h = − ≈ 3.08 (7)
ln(0.1/0.7)

And a low temperature T0l with the objective to accept 1% of the worse solutions
at the beginning of the optimization process:
6
T0l = − ≈ 0.71 (8)
ln(0.01/0.7)

For d+ h l
n , we are using T0 = 15.4 and T0 = 3.55. To determine α, we define
that the temperature T should be decreased by 10% every 100 iterations. So it
applies: √
100
α= 0.9 ≈ 0.99895 (9)

4.4 Approach with Adaptive Neighborhood Distance
In this section, we propose an approach based on the algorithm using the de-
terministic cooling schedule from the last section extending it with an adaptive
technique. In the field of ES, an adaptive step size control is a common tool
to improve optimization results. The idea is that the solution space conditions
change during an optimization run and therefore the optimal step size is not
the same during the whole run. At the beginning, a large step size allows the

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exploration of the solution space, while at the end, a small step size allows the
fine-tuning of solutions. A well-known example is Rechenberg’s step size con-
trol [1]. In our approach using SA, the neighborhood distance dn play a similar
role like the step size, i.e., controlling the exploration characteristics of the al-
gorithm. According to Rechenberg’s step size control, we count the number of
improved solutions x0 with f (x0 ) > f (x) and compute ratio of improved so-
lutions w.r.t. all solutions. If more than 1/5th of the new solutions have been
improved, the neighborhood distance dn is increased. Otherwise, it is decreased.
To become independent from short-term fluctuations, we evaluate the ratio af-
ter 100 new solutions. To increase or decrease, the neighborhood distance dn is
modified with factor τ = 1.1 representing a change of 10%.

Algorithm 1 Adaptive Neighborhood Distance
Require: dn , T , α
x ← x0 , i ← 1, o ← 0
while i ≤ I do
Create neighbor x0 from x
if f (x0 ) > f (x) then
x ← x0
o←o+1
f (x)−f (x0 )
else if U ∼ U [0, 1] > e− T then
x ← x0
dn ← dn · 2.0
end if
if i mod 100 = 0 then
if o ≥ 20 then
dn ← dn · 1.1
else
dn ← dn /1.1
end if
o←0
end if
T ← α · T, i ← i + 1
end while
return x

Preliminary experiments show that the acceptance of a worse solution can
change the optimization process. The adaptive control may result in an inappro-
priate neighborhood distance dn . To prevent this, we implement a neighborhood
distance boost bdn in case of accepting a worse solution. Hence, the optimization
process can explore a larger area after taking a worse solution and thus it is less
sensitive to changes in the solution space. We set this neighborhood distance
boost to bdn = 2.0, which turned out to be reasonable in our experimental stud-
ies. The neighborhood distance dn is doubled after accepting a worse solution.

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Algorithm 1 shows the pseudocode of the optimization approach with the total
number of iterations I and a random value U ∼ U[0, 1].

5 Experimental Results
Every optimization is run for 10, 000 iterations. As we use SA which is a heuristic
optimization approach and also apply random initializations, we repeat every
experiment 100 times and interpret the mean value and standard deviation.
Additionally, we test the significance of the experimental results with a Wilcoxon
signed rank-sum test [7].

5.1 Comparison of the Configurations
We use eight different configurations to test the capability of SA for the wind
turbine placement problem with geographical constraints. The optimization runs
are separated into two categories. First, runs that use the deterministic cooling
schedule with a constant neighborhood distance and second, runs that use the
adaptive neighborhood distance. In each category, we test the initial neighbor-
hood distances d− + h
n and dn and the starting temperatures T0 and T0 .
l

Table 1. Experimental comparison between constant and adaptive neighborhood dis-
tance control.

Algorithm Mean ± Std Max
P in kW ± P in kW P in kW
Without Optimization 12 923.81 ± 186.64 13 397.48
Constant Neighborhood Distance 13 233.57 ± 183.49 13 598.25
Adaptive Neighborhood Control 13 384.33 ± 121.81 13 654.61

Table 1 shows the comparison of the experimental results for the two differ-
ent categories including various configurations. The values specify the average
power production from the wind turbines in kilowatts. Both approaches are able
to clearly improve the initial solution. The approach with adaptive neighbor-
hood control performs significantly better than the approach with a constant
neighborhood distance, confirmed by a Wilcoxon signed rank-sum test with a
p-value = 6.39 · 10−30 . It should also be noted, that the approach with the adap-
tive neighborhood control is able to reduce the standard deviation considerably,
which means that the approach is more reliable. Also the best solution is created
by the approach with adaptive neighborhood control.
In Table 2 concentrates on a detailed comparison between different config-
urations for start temperatures and neighborhood sizes. First, we can observe
that the random initialization leads to slightly different initial values. But clearly
confirmed by a Wilcoxon signed rank-sum test with a p-value = 0.761, there is
no significant difference between the initial solutions. With a constant neighbor-
hood distance, the approaches using T0l perform better than the approaches using

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Table 2. Experimental comparison with various start temperatures and neighborhood
sizes.

Constant Adaptive
Neighborhood Distance Neighborhood Distance
Algorithm Mean ± Std Max Mean ± Std Max
P in kW ± P in kW P in kW P in kW ± P in kW P in kW
Without optimization 12 925.07 ± 182.41 13 309.67 12 922.55 ± 190.78 13 397.48
SA with T0l & d−
n 13 340.32 ± 123.44 13 598.25 13 401.58 ± 132.44 13 624.78
SA with T0h & d−
n 13 173.90 ± 140.71 13 418.70 13 402.24 ± 101.37 13 633.63
SA with T0l & d+
n 13 370.38 ± 111.74 13 565.51 13 395.63 ± 123.92 13 643.72
SA with T0h & d+
n 13 049.68 ± 140.22 13 306.36 13 337.86 ± 115.20 13 654.61

T0h . This is probably due to the fact, that a higher temperature increases the
number of accepted worse solutions. As we have a highly complex 44-dimensional
solution space, too many accepted worse solutions can reduce the quality of the
optimization. This effect is increased by the use of a larger neighborhood dis-
tance, as we can see comparing the results of SA with T0h & d− n and SA with T0
h
+
& dn using a constant neighborhood distance.
The adaptive neighborhood distance adapts itself to the solution space and
can counteract this effect. But also here, we can observe that the worst mean
value is achieved by SA with T0h & d+ n . Interestingly, this configuration created
the best overall solution. Although the higher acceptance rate of worse solutions
decreases the optimization results in mean, there is a small chance that the
algorithm chooses exactly the worst solutions which helps to leave local optima.
A run with good choices is able to create the best solution with this configuration.
This indicates that it might be promising future research to further analyze the
highly complex solution space. This may allow better predictions, if accepting
worse solutions may help to leave local optima. Between the approaches with the
best mean values SA with T0l & d− h − l +
n , SA with T0 & dn , and SA with T0 & dn using
the adaptive neighborhood distance is not significant difference. With carefully
chosen parameters, the adaptive neighborhood control is able to operate well.

13800

13400
best run
13000 worst run

12600

0 4000 8000

Fig. 3. Comparison of the best and worst optimization run.

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5.2 Optimization Runs

Figure 3 shows the dynamic of the best with and the worst run. Both runs are
achieved by SA with T0h & d+ n but the best run uses the adaptive neighborhood
distance while the worst run uses the constant neighborhood distance. The im-
portant parts of the fitness development are enlarged and visualized in Figure 4.
The y-scale in Figure 4(a) and 4(b) is different by factor 5 because the changes
in the worst run are larger than the changes in the best run. In the best run,
we can see the acceptance of worse solutions, but only with few deteriorations
of the fitness function. This observation is different in case of the worst solution.
Clearly worse solutions are accepted and the optimization process is unable to
improve the fitness function to the starting level. This also explains, why it is
possible that the best solution optimized with SA with T0h & d+ n is worse than
the best initial solution, see Table 2.

12900
13540 worst run

12700
13500

12500
13460 best run
0 1000 2000 0 1000 2000
(a) Best run (b) Worst run

Fig. 4. Zoomed visualization of best and worst optimization run.

5.3 Placement Result

In the last experimental section, we show the placement results of the best solu-
tion, see Figure 5(a), and the worst solution, see Figure 5(b). The best solution
maximizes the distances between the turbines w. r. t. the wind rose, see Figure 2.
We can observe four lines of turbines. The largest line has been placed on the
right, because most of the wind comes from direction south-west. This line of
turbines does no cause wake effects for other turbines. The second line on the
right is curved, also reducing the wake effects, e.g., for Turbine T 9 and T 18,
as the wind comes from direction south-west. The geographical constraints are
considered, e.g., on the left of the area with Turbines T 10, T 16, T 19 placed in
the free areas between the constraints. The placement of the worst solution is
interesting. The turbines are pulled to the upper right corner. The distances
between the turbines are small, so major wake effects reduce the power output.
We can also see this behavior in other solutions created with a high tempera-
ture. Accepting to many worse solutions can lead to deadlock situations in the

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626 636
T9 T9
T2
T5 T18 T7T1 T21 T7T22 T1
T8
T10T5T14T18
T16 T20 T4 T17 536
538 T16 T20
T19 T17 T2 T13
T21 T19
T10 T6 436
T3 450 T11
T22 T12
T3
T4 T15
T14 T15 362 T6 336

T12 T13
T8
T11 274 236

(a) Best placement (b) Worst placement

Fig. 5. Placement results.

optimization process, where multiple turbines blockade each other, i.e., increas-
ing the distance between two turbines will decrease the distance to a different
turbine. An aggravation of this effect can be the geographical constraints. SA
has serious difficulties to resolve deadlock situations. Again, this confirms the
need to analyze in detail, if accepting worse solutions in highly complex solution
spaces may help to leave local optima and avoiding deadlock situations.

6 Conclusions
Finding optimal turbine locations is important for the power output of wind
turbines. The optimization process to improve the positions of turbines induces
a highly complex solution space. SA is able to optimize the solutions, but the
choice of appropriate parameters for neighborhood distance and temperature is
important. Our approach using an adaptive neighborhood distance is more reli-
able than the variant with constant neighborhood distance and can significantly
improve the results.
Our experiments show that an intelligent choice of accepted worse solutions
can be a promising field for further research. Further, the integration of addi-
tional techniques from ES to SA may be an interesting research direction, e.g.,
the employment of a population like in parallel SA. The success of ES in turbine
placement is an indicator that a smart combination of both fields could improve
the results.

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