Renewable Energy xxx (2011) 1e8
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Renewable Energy
journal homepage: www.elsevier.com/locate/renene
A fast algorithm based on the submodular property for optimization of wind
turbine positioning
Zhang Changshui a, Hou Guangdong a, *, Wang Jun b
a
b
Department of Automation, Tsinghua University, FIT Building 3-120, 100084 Beijing, China
Department of Control Science and Engineering, Tongji University, 201804 Shanghai, China
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 29 December 2010
Accepted 31 March 2011
Available online xxx
In the design of a wind farm, the placement of turbines is an important factor that affects the efficiency
and profit, but automatic placement of turbines is still a challenging problem.
This study reveals the “submodular” property of the wind turbine positioning problem based on
Jensen wake model. Based on this property, a “lazy greedy” algorithm is used to optimize the placement.
This method can obtain solutions with theoretical guarantee of quality. It can also estimate the lower
bound of the optimal value of the objective function. This method is tested on three types of wind
scenarios. Compared to previous research, this algorithm takes much less time, and always gains a better
solution.
To enlarge the application scope, the wake model is extended to the large scale complex terrain in this
study. The present algorithm and some other algorithms are tested in the simulation of the complex
terrain. The experimental results demonstrate the present method’s superior performance.
2011 Elsevier Ltd. All rights reserved.
Keywords:
Wind turbine positioning
Wake model
Submodular
Optimization
Genetic algorithm
1. Introduction
Due to the depletion of fossil fuel resources and the impact of
environmental concerns, attention has increasingly focussed on
renewable energies, among which wind is one of the fastest
developing energies. In order to extract more power while using
wind energy, many factors should be taken into account, such as
the type of wind energy conversion systems, the location of the
farm, the local wind condition. When these factors are determined,
the individual position of turbines in the wind farm becomes
significant, mainly because the turbines will change wind condition
and influence each other. That also makes wind turbine positioning
a challenging task in wind farm design.
There have been several studies on the optimization of wind
turbine positioning [1e4]. Mosetti et al. designed a genetic algorithm
for optimizing placement of turbines [1]. The objective function
contained the investment cost and the total power produced by the
wind farm. And their solutions were compared with configurations
of randomly placed turbines. Though the solutions given by Mosetti
et al. were proved better than random placement, they did not
outperform the empirical placement. Grady et al. also used a genetic
algorithm to demonstrate its effectiveness in turbine positioning [2].
* Corresponding author. Tel.: þ86 10 62796872.
E-mail addresses: zcs@mail.tsinghua.edu.cn (Z. Changshui), hougd05@gmail.
com (H. Guangdong), junwang@tongji.edu.cn (W. Jun).
They run the algorithm with more individuals for sufficient generations and obtained improved solutions.
Those works mentioned above are based on the genetic algorithm. Generic algorithms always take a lot of time to get a solution,
and this will be an obstacle when the wind condition is complicated. Besides, there is no guarantee about the quality of the
solution. That means there is no knowledge about how good the
solution is. And in those works, only the flat terrain is considered.
While in many real-world instances, wind farms are built in
mountains or some other types of terrain, not always in flat area.
In this work, a completely different method based on the
conception named “submodular” is used. It takes much less time and
always gains a better solution. And both the wake model and this
turbine positioning approach are extended to the complex terrain.
2. Wake and cost model
In both Mosetti and S.A. Grady’s work [1,2], the wake model used
for wind farm calculation is similar to the model developed by Jensen
et al. [5e7]. The model will be introduced in this section. It is based
on the assumption that the momentum is conserved in the wake.
For one single turbine, the downstream wake area will be
regarded as a trapezoid. It means that the radius of the area
increases at a rate proportional to the distance from the turbine. As
shown in Fig. 1:
0960-1481/$ e see front matter 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.renene.2011.03.045
Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
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3. The algorithm for flat terrain
The mean wind speed obeys the formulation as follows:
2
3
6
u ¼ u0 6
41 7
2a
7
x2 2 5
1þa
3.1. Introduction of submodular property
(1)
r
where a is the axial induction factor, x is the distance from the
turbine, a is the entrainment constant, r is the downstream rotor
radius. The relationship between r and the turbine radius rr is
shown by equation (2):
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1a
r ¼ rr
1 2a
(2)
The axial induction factor can be calculated by the turbine thrust
coefficient CT using the relationship CT ¼ 4a(1 a). The entrainment constant a, is empirically given as:
a ¼
0:5
z
ln
z0
(3)
where z is the hub height of the wind turbine, and z0 is the surface
roughness. Assuming that in multiple wakes, the kinetic energy loss
is equal to the sum of the energy deficits. So for the downstream of
N turbines, the velocity downstream can be expressed by the
following equation:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
u N uX
u 25
1
ui ¼ u0 41 t
u0
Submodular is a useful property of some set functions. And it
can be proved that when using Jensen’s model, the function which
describes the extracted power P from certain placement of turbines
is submodular. So the algorithm to find the maximum of a submodular function can be designed for turbine positioning.
Definition. Assume that V ¼ {1,2,.,N} is a finite set, F: 2V/R is
a function defined on the set V. If F meets the following condition:
for all A; B4V, FðAÞ þ FðBÞ FðAWBÞ þ FðAXBÞ, then F is called
submodular.
And there is an equal definition: A set function F on V is called
submodular if for all A4B4V; s;B, it holds that FðAWfsgÞ FðAÞ FðBWfsgÞ FðBÞ.
The following is a simple example of a submodular function: V is
the set of positions where circles can be placed. A, B are two subsets of
V, F(A) is the total area covered by all the circles in A, as shown by Fig. 2.
A, B are subsets of V, and A4B, it is obvious that for all s;B, it
holds that FðAWfsgÞ FðAÞ FðBWfsgÞ FðBÞ, so F is submodular.
In next subsection, the proof is presented that the turbine
positioning problem based on current wake model is in nature
a problem of maximizing a submodular function.
2
(4)
3.2. The submodular property of the turbine positioning problem
i¼1
The total power P extracted from the wind is a function of local
wind speed, as shown in the following expression:
P ¼
N
X
0:3 u3i
(5)
i
The investment cost of a wind farm is assumed to be determined
only by the number of turbines. The total cost per year for the entire
wind park can be expressed as follows [1]:
cost ¼ N
2 1 0:00174 N2
þ e
3 3
(6)
PðAWfsgÞ PðAÞ ¼
The following objective function will be optimized:
Objective ¼
cost
Ptotal
In the turbine positioning problem, the available terrain can be
subdivided into cells. To keep necessary distance between two
adjacent turbines, the size of a cell is suitably chosen and every
turbine is installed only on the center of a cell. So the available
positions are finite, V denotes the set of all these positions, thus one
possible solution A is a subset of V. F(A) is the total power of A
extracted for a period of time.
There is the proof that F(A) is submodular. Assume A and B are
two solutions, B contains all the positions in A. s is an available
position which neither in A nor B. Adding s to A, the total power
increment is:
jAWfsgj
X
i¼1
(7)
Minimizing this objective function leads to a solution with
lowest cost of per unit of wind energy production.
¼ Ps0
in
Pi0 A
þ
jAj
X
Pi
i¼1
jAj
X
i¼1
Pi0
in
A
Pi
in
(8)
A
Adding s to solution B, the total power increment is:
PðBWfsgÞ PðBÞ ¼
jBWfsgj
X
i¼1
þ
jAj
X
i¼1
Ps0 in A
Pi0 Pi0
jBj
X
i¼1
in
B
Pi ¼ Ps0
in
Pi
B
in
B
þ
jBj
X
i ¼ jAjþ1
Pi0 Pi
ð9Þ
Ps0 in B
where
and
are respectively the power extracted by s
PjAj
after added in A and B. It can be proved that i ¼ 1 ðPi0 in A Pi in B Þ PjAj
0
ðP
Pi in B Þ, so the following inequality is obtained:
i ¼ 1 i in A
PðBWfsgÞ PðBÞ ½PðAWfsgÞ PðAÞ
jBj
X
i ¼ jAjþ1
Fig. 1. The wake model.
Pi0 Pi þ Ps0
in
B
Ps0
in
A
(10)
After a new turbine s placed, the total power of an original
turbine decreases, if this turbine is in an area influenced by the
Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
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Z. Changshui et al. / Renewable Energy xxx (2011) 1e8
3
While using this algorithm to solve turbine positioning problem
in present study, it always leads to much better solutions than 63%
P (Aoptimal), and even better than genetic algorithm solutions.
Fig. 2. A simple example of a submodular function.
wake of s, otherwise its extracted power does not change. So Pi0 Pi ,
PjBj
and i ¼ jAjþ1 ðPi0 Pi Þ 0
Let uA,uB be the wind speeds of s in A,B solutions, u0 is the wind
speed of the same position before s added. According to the
multiple wakes model, for A:
uA
u0
1
2
¼
jAj X
1
i¼1
ui
u0
2
IðiÞ ¼ SA
(11)
for B:
1
uB
u0
and
2
IðiÞ ¼
¼
jBj X
j¼1
1
uj
u0
2
IðjÞ ¼ SA þ
jBj X
j ¼ jAjþ1
1
uj
u0
2
IðjÞ
(12)
1 ; s is in the wake area of i
0 ; s is out of the wake area of i
PjBj
For
ð1 uj =u0 Þ2 IðjÞ 0 and P ¼ 0.3u3, so uA uB
j ¼ jAjþ1
and Ps0 in A Ps0 in B , PðAWfsgÞ PðAÞ PðBWfsgÞ PðBÞ, it means
that the total extracted power P is a submodular function defined
on the set of turbine positions V.
3.3. Algorithms for the turbine positioning
3.3.1. A basic greedy algorithm for turbine positioning
In last subsection, it is proved that P(A) is submodular, so the
turbine positioning problem can be considered as the maximization
of a submodular function. The maximization of a submodular function is NP-hard unfortunately, however there are efficient algorithms
to get an approximate solution which has a guarantee of quality.
A very simple greedy algorithm (Algorithm 1) is designed to
solve this optimization problem. This algorithm can give a placement for turbines with specified number k to (locally) maximize
the total extracted power.
Algorithm 1. Greedy Algorithm For Turbine Positioning
1:
2:
3:
4:
5:
6:
3.3.3. The bound of the best solution
Using submodular property, the bound of the optimal placement’s total power can be got. A rough bound is contained in
equation 13, while a better bound can be gained by a data dependent method developed by Minoux [9]. This method starts with
a random solution A, for cs˛V=A, gets ds ¼ PðAWfsgÞ PðAÞ. The
results are sorted, in the order d1 d2 . dn, then Minoux
P
proved that PðAÞ þ ki¼ 1 di is a limit of the P (Aoptimal).
4. Experiment and results
Initialize: A0 ¼ B;
for all i such that 1 i k do
si ¼ arg maxfPðAi1 WsÞ PðAi1 Þg
s
Ai ¼ Ai1 Wsi
end for
return Ak
4.1. Numerical procedure
First, a square area is used as the farm and subdivided into 100
cells. In the center of one cell a turbine would be placed. The width of
each cell is 200 m, which is five times the length of the rotor diameter
For such a simple algorithm, Nemhauser et al. proved that it can
provide a solution with a guarantee of quality [8]. They proved that
for a monotone submodular function PðAÞ and PðBÞ ¼ 0, the
greedy solution Agreedy with k elements has the property [8]:
1
maxjAjk PðAÞz63% P Aoptimal
P Agreedy 1 e
3.3.2. The lazy greedy algorithm for turbine positioning
In the above algorithm, i < j0Ai 4Aj . And according to submodular property, Ai 4Aj 0ds ðAi Þ ds ðAj Þ. This property can be
used to speed up the basic greedy algorithm. Interestingly, Minoux
developed an accelerated greedy algorithm which speeds up the
greedy algorithm substantially [9], when the submodular property
holds. Compared to the basic greedy algorithm, Minoux’s algorithm
can find a solution with a guarantee of quality in much less time.
The algorithm used in present study with name “lazy greedy” is
derived from Minoux’s method with some slight modifications.
Algorithm 2 shows its whole procedure. This “lazy greedy” algorithm starts with solution A0 ¼ B, adds elements in this solution
one by one, like Algorithm 1 does. The difference is: Algorithm 2
sorts the increments ds ¼ PðAWsÞ PðAÞ of all elements s in
descending order to create a list at first, then updates this list in
every iteration and according to this list, selects elements to add to
the solution.
This procedure helps to save a lot of running time. Using submodular property, the values in this list do not increase during the
whole procedure, so it is easy to update this list. In each iteration,
the values of ds are tested from the top of the list, if it decreases, its
position will be updated in the list, otherwise the corresponding
element will be removed to the solution set then one iteration is
completed. Most times, only several values need to be updated in
each iteration, while the basic greedy algorithm actually calculates
all the values in this list in one iteration. That makes the “lazy
greedy” algorithm much faster than basic greedy algorithm, especially when the turbines number is large.
It is noted that in “lazy greedy” algorithm, the power increment
brought by a newly added turbine decreases as the turbines number
increases. When the increment in one iteration is less than 60% of
that the fist added one provided, the algorithm stops. Then it gets
a series of solutions: Ai, i ¼ 1,.,j. Among these solutions, the one with
the max value of cost ðiÞ=PðiÞ will be chosen as the final solution.
(13)
Table 1
Wind turbine properties.
Wind turbine properties
Value
Hub height (z)
Rotor radius (rr)
Thrust coefficient (CT)
60m
40m
0.88
Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
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Case c): Multiple directions and variable speeds of 8, 12, and
17 m/s. The distribution is shown in Fig. 3 [2].
Wind Distribution in Case c)
7
17 m/s
12 m/s
8 m/s
6
Algorithm 2. “lazy greedy” Algorithm For Turbine Positioning
Wind Frequency (%)
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Angle(o)
Fig. 3. The wind distribution in Case c) [2].
D in this numerical experiment. Other parameters of the turbines are
shown in Table 1. The values are the same as those in [1] and [2].
Three types of wind scenarios will be considered.
Case a): Uniform wind direction and a constant wind speed of
12 m/s;
Case b): A constant mean wind speed of 12 m/s and 36 directions (every 10 );
Initialize: A0 ¼ B;
2: for all i such that 1 i N do
di ¼ PðfigÞ
4: end for
Sort fdi ; 1 i Ng in descending order to create a list L
6: Poriginal ¼ d1
j ¼ 0;
*
8: while ds > 60% Poriginal do
*
Pick the first s in list L, calculate ds ¼ arg maxfPðAj WsÞ PðAj Þg
*
10: if ds < ds in the list then
ds ¼ d*s ;
12: Update the position of ds to keep the descending order of the
list
else
14: Ai ¼ Ai1 Ws
j¼jþ1
16: Remove ds from the list.
end if
18: end while
return Agreedy ¼ arg maxfPðAk Þjk ¼ 1; .; jg
Ak
4.2. Results and analysis
Case a). Since “lazy greedy” algorithm picks the positions one by
one, the total power and the objective (cost/power) value of every
stage will be gained. The less objective value means the less cost
Fig. 4. Optimal configurations and objective curve in Case a).
Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
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Table 2
Comparison of solutions in Case a).
5
Table 3
Comparison of solutions in Case b).
Placement
Turbine number
Total power
Efficiency
Objective
Placement
Turbine number
Total power
Efficiency
Objective
Mosetti
Grady
Present Study
26
30
30
12352
14310
14310
91.645
92.015
92.015
16197
15436
15436
Mosetti
Grady
Present study
Present study
Present study
19
39
19
39
40
9245
17220
9354
17611
17991
93.86
85.17
94.97
87.11
86.76
17371
15666
17169
15318
15280
and the more extracted power. The “lazy greedy” algorithm takes
the minimum point on the curve as the result. It is shown in Fig. 4d.
Fig. 4a and b illustrate Mosetti et al. and Grady et al. solutions. Both
of them use genetic algorithms. Fig. 4c is the optimal solution given
by present study. Table 2 gives the comparison of solutions
numerically.
In this simple wind scenario, the turbines in one column cannot
influence the ones in other columns. Grady pointed out that each
10-cell column can be viewed as a single independent optimized
unit in Case a). The optimization for such a column can be extended
to the whole considered domain. So the solution for the whole
domain can be simply verified on a column by testing all the
210 ¼ 1024 placements. That shows Grady’s method and the present
method both give the optimal solution. But the present method
uses much less time.
In this scenario, the turbines could be influenced only by the one
in the same column. As the number of turbines in one column
increases, the average power and the average cost of each turbine
decrease. From Fig. 4d, it can be seen that when each column has 3
turbines, the objective function is optimal. Mosetti et al. did not find
the best solution. Compared to the work by Grady et al., that may be
caused by not sufficient individuals or generations.
Case b). In this case, any two turbines influence each other. It
makes hard to place the turbines by experience. Fig. 5 shows the
solutions given by Mosetti et al., Grady et al. and present study.
Table 3 is the comparison of the numerical results of these
methods. When using the same number of turbines, the present
work increases the total power by 1.19% than Mosetti et al. and
2.27% than Grady et al. Using 40 turbines, the objective can be
further improved.
Case c). In this case, the present study also gets good solutions.
Fig. 6 shows the solutions given by Mosetti et al., Grady et al. and
the present study. Table 4 is the comparison of the numerical
results of these methods. The present method increases the total
power by 6.39%, when using the same number of turbines with
Mosetti et al. It also increases the total power by 4.73%, when using
the same number 39 with Mosetti et al. The method chooses 40
turbines at final to get a better objective value.
In these solutions, the density of placed turbines is high in the
edge and low in the inner. This character is more obvious than
Case b).
Fig. 6f also illustrates the comparison of time cost between basic
greedy and “lazy greedy” algorithm in Case c). It can be found that
the latter performs much better in time cost. And generally speaking,
even the basic greedy algorithm always costs less time than genetic
algorithm in the same case. This is shown in next section.
5. Complex terrain
5.1. Modified model
For many real-world instances, wind farms are not always built
in flat area. Mountains or some other types of terrain are often used
Fig. 5. Optimal configurations and objective curve in Case b).
Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
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Z. Changshui et al. / Renewable Energy xxx (2011) 1e8
Fig. 6. Optimal configurations and objective curve in Case b).
for the rich wind resource. Some differences must be taken into
account when placing turbines in these complex terrains.
Firstly, even before the turbines placed, the wind conditions in
different parts of a complex terrain are usually different, unlike in
flat areas. In a complex terrain area, wind conditions may have
opposite polarities in different parts. Some parts, like a ridge or
a valley between two mountains, may have strong wind, while
some may be opposite, like the lee side of a hill. Thus, the wind
distribution for the whole terrain needs to be simulated before the
evaluation of the algorithms.
Secondly, in flat terrain, the same type of turbines has the same
hub heights. So all the hubs of the turbines are in one plane. The
influence of the turbine wake can be calculated simply in this plane,
and the wake area of one single turbine is a trapezoidal in this plane
as shown in Fig. 1. In complex terrain, the different positions make
the hub heights different, and the shape of the influenced area is
like a cone behind the turbine shown in Fig. 7. Besides this, the axis
of the wake area (L0 in Figs. 1 and 7) is horizontal in flat area, while
this property does not always hold in complex terrain.
These differences require us to extend the Jensen wake model to
the stereoscopic space. For this task, an assumption needs to be
made at first. In large scale complex terrain, when the local slope
does not change drastically, its tangent plane can be taken as the
local terrain with very little loss of accuracy. Based on this
assumption, the modified wake model shown in Fig. 7 is used. The
wake area of one turbine will be regarded as a cone. The sectional
radius increases at a rate proportional to the distance from the
turbine. The wind speed decreases according to equation (3). The
axis of this area has the same direction with the average of the wind
vectors in the windward of the turbine.
In Fig. 7, V0, V1 are the average vector of the wind in the windward
of the turbine. L2 is the distance vector of these two turbines and L1 is
the length of the projection perpendicularity to L0. It can be determined that whether the influence exits by comparing L1 with r1.
5.2. Result and analysis
The modified wake model and the “lazy greedy” positioning
algorithm are tested in simulated complex terrain, and compared
with the genetic algorithm under the same situation. The terrain is
shown in Fig. 8a and b. The wind distribution is illustrated by
Fig. 8c. The simulation of the wind distribution is not very precise.
That does not influence the research much, since the purpose here
is evaluating the optimal algorithms but not getting accurate
simulation result.
Table 4
Comparison of solutions in Case c).
Placement
Turbine number
Total power
Objective
Mosetti
Grady
Present study
Present study
Present study
15
39
15
39
40
13460
32038
14320
33553
34271
994.05
803.14
934.38
802.36
802.15
Fig. 7. The modified wake model.
Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
positioning, Renewable Energy (2011), doi:10.1016/j.renene.2011.03.045
Z. Changshui et al. / Renewable Energy xxx (2011) 1e8
7
Fig. 8. Solutions of “lazy greedy” and genetic algorithms.
As shown in Fig. 8, two normal functions are used to simulate
two mountains, with heights about 45 m and 15 m respectively. The
area is 4 km 4 km, and is subdivided into 400 cells. The top is
north and the input wind is 15 m/s Fig. 8b shows the direction of
the input wind.
Although the distribution of the wind power is more uneven in
complex terrain, in this scenario, the total power is also submodular.
Fig. 8d shows the result gained by the “lazy greedy” algorithm,
the total power is 79584.8 kW, objective value is 3382. In this
solution, turbines mainly distribute on the windward side of the
ridge and the valley between two hills. Fig. 8e is the result of the
genetic algorithm in the same situation. The two solutions are
similar, while Table 5 gives the details of them which prove that the
“lazy greedy”’s result is better.
In Table 5, the genetic algorithm uses 40 individuals. When it
stops, the total generation is about 2100. It is noted that the lazy
Table 5
Comparison of characters of algorithms on complex terrain.
Algorithm
Greedy
Lazy Greedy
Genetic Algorithm
Time Cost
Total Power
Cost/Total Power (Objective)
2 m 32 s
79584.8
3383
About 10 s
79584.8
3383
1 h 40 m
78850
3414
greedy algorithm has outstanding performance in time cost.
Besides this the total power and objective are both better than
traditional genetic algorithm.
6. Conclusion
The present study reveals the submodular property of the
turbine positioning problem based on Jenson model. Taking
advantage of this property, a “lazy greedy” algorithm is used to
handle the turbine positioning problem in much less time than
some other popular methods. The solution gained by this algorithm
has a theoretical guarantee. In fact, the experiments in present
work on three scenarios show that this algorithm always finds
a better solution than genetic algorithms.
An effort is made to extend the wake model to complex terrain,
which makes it possible to handle the more general cases. And for
complex terrains, the more uneven wind distribution makes the
greedy based algorithms more effective.
Acknowledgments
This work was supported by 973 Program (2009CB320602) and
NSFC (Grant No. 61021063 and No. 61075064).
Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
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Please cite this article in press as: Changshui Z, et al., A fast algorithm based on the submodular property for optimization of wind turbine
positioning, Renewable Energy (2011), doi:10.1016/j.renene.2011.03.045