- Email: [email protected]
Irregular-shape wind farm micro-siting optimization
Energy 57 (2013) 535e544
Contents lists available at SciVerse ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Irregular-shape wind farm micro-siting optimization Huajie Gu, Jun Wang* Department of Control Sci. & Engn., Tongji University, Shanghai 201804, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 November 2012 Received in revised form 4 May 2013 Accepted 29 May 2013 Available online 10 July 2013
Landscape constraints inevitably cause the irregularity of the shape or boundary of a wind farm, which was not fully considered in previous literature. In this paper, a single-boundary constraint model and a novel multi-boundary constraint model incorporated with ray intersection method are developed to quantify the irregular boundary constraint for wind farm micro-siting optimization. In order to obtain high-fidelity wind farm shape information, an edge detection algorithm is employed to extract wind farm contour data from digital maps, and an optimal polygonal approximation algorithm is applied to compress the contour data so as to make the computation of boundary constraints less time-consuming. Simulations of four commercial wind farms comprehensively demonstrate the effectiveness of the proposed boundary constraint models and the significance of irregular-shape wind farm micro-siting optimization. 2013 Elsevier Ltd. All rights reserved.
Keywords: Wind farm Wind farm shape Wind farm boundary constraint Wind farm micro-siting Polygonal approximation Point-in-polygon
1. Introduction Wind energy takes the leading place among the other renewable energy, due to its abundant distribution and well developed exploitation technology. With the growing demand of wind energy, more and more wind farms have been built onshore or offshore [1]. The object of wind farm micro-siting is to find the optimal layout of wind turbines so that the wind farm can reach the maximal electricity output or economic benefit. In some wind farms, wind turbines are empirically placed apart with a distance of several times of a rotor diameter in order to reduce wake losses [2]. This empirical layout is easy to achieve, but is usually not the optimal solution to the wind farm micro-siting problem. A genetic algorithm was first utilized by Mosetti et al. [3] to optimize wind turbine placement in a rectangular-shape wind farm. The proposed method first discretized the wind farm into finite uniform square grids, and laid down a solid foundation for different recent approaches [4e10]. A mixed-integer discrete optimization model was proposed in Ref. [11], and nonlinear programming algorithms were applied to simultaneously achieve optimal selection of the type and number of wind turbines and optimal placement of all the turbines. A PSO (particle swarm optimization) algorithm was utilized by Wan et al. [12e15] and
* Corresponding author. Tel./fax: þ86 (0)21 69580069. E-mail addresses: [email protected], [email protected] (J. Wang). 0360-5442/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.05.066
Chowdhury et al. [16] to optically place turbines in any locations of a wind farm rather than in the cell centers of a given grid. The “continuous” search space provided the opportunity to explore every potential location on a wind farm so as to further reduce wake effects. It happens that some parts of a chosen wind farm can be unsuitable for placing wind turbines due to a number of landscape constraints, for example, lakes, parks as well as ecologically sensitive areas. Therefore, the shape of a wind farm is usually irregular. However, most previous optimization approaches in the literature only consider rectangular-shape wind farms [3e9,11e20]. Saavedra-Moreno et al. [10] used a binary template matrix to store wind farm shape information, and a much smaller grid size should be used in order to get an accurate shape model. Besides, the “discrete” gird-like locating scheme is not suitable for irregularshape wind farms as the optimal size and direction of the grid is difficult to choose especially when the farm is narrow and long. Using real location variables can avoid choosing optimal grid size and direction, and furthermore it can achieve better results. However, for the “continuous” micro-siting problem, the irregular shape of a wind farm introduces complicated boundary constraints, i.e. how to determine whether turbine locations yielded by optimization algorithms are inside permitted areas. In this paper, an edge detection algorithm used in image processing is employed to extract the contour data from the map, and an optimal polygonal approximation algorithm used in pattern recognition is then introduced to compress the contour data. A
536
H. Gu, J. Wang / Energy 57 (2013) 535e544
single-boundary constraint model and a novel multi-boundary model incorporated with the ray intersection method are proposed to calculate the boundary constraint violation value of turbine locations. An improved PSO algorithm proposed by Wan et al. [15] is adopted to tackle the optimal micro-siting problem. The remainder of this paper is organized as follows. Sections 2 and 3 describe the wind farm shape model and the boundary constraint model respectively. Section 4 presents the simulation results and the discussions. Section 5 summaries the contributions of the paper. 2. Wind farm shape model 2.1. Wind farm contour extraction Edge detection is widely used in image processing to extract useful structural information about object boundaries and many successful computational approaches have been developed for it. The Canny edge detection algorithm [21] is used in this paper. The contour of a wind farm in a digital map should be represented by closed curves in distinguishing color so that it can be extracted easily and accurately. The coordinates of pixels in the curves are then used to locate the contour in a Cartesian coordinate system in order to facilitate the calculation of following boundary constraint. This paper aims at developing a general approach to optimizing the irregular-shape wind farm micro-siting problem, so the practical contour can be achieved if the actual map is available. Shepham wind farm map shown in Fig. 1 is used as an example to illustrate the whole extraction process. It consists of two sub wind farms and a lake (the infeasible zone), which are marked with numbers as shown in Fig. 1. Each time only the contour of one of above three areas is extracted, and the extraction of a single contour is executed by the following steps. Step 1 Apply edge detection operation by using the Canny algorithm. The contour of the wind farm represented by a closed
blue curve is a distinguishing object from the background of the map. In this case, two contours can be achieved (the external one and the internal one), as shown in Fig. 2, and the width of each achieved contour is one pixel unit. After the edge detection, we transform the image coordinates to the Cartesian coordinates, and store the coordinates of the contour pixels in a vector. Step 2 Apply contour tracing operation. pffiffiffiSince the distance between two adjacent pixels is less than 2 pixel unit, we sort the coordinates in the vector in a clockwise or counterclockwise order. In this paper, the external contour V ¼ [v1, v2, /, vNv] is traced and the result is shown in Fig. 3, where Nv is the number of pixels in the traced contour. Note that, the traced contour has an aliasing effect. Step 3 Apply anti-aliasing operation. Downsampling is used to smooth the traced contour. Based on comprehensive simulations, the recommended downsampling rate in this paper is 4. The result is shown in Fig. 3. Thus, the final adjacently linked contour vector is S ¼ [s1, s2, /, sN] and N ¼ Nv/4. S represents a closed digital curve with N number of vertices. As wind turbines are of large dimensions, the contour location is not necessarily accurate and a more precise imaging processing method is out the scope of this paper. 2.2. Optimal polygonal approximation The shape or contour information of a wind farm is represented by a closed N-vertex polygonal curve S, which is stored as the ordered set of vertices S ¼ [s1, s2, /, sN]. The actual wind farm contour P should be
P ¼ lS ¼ ½p1 ; p2 ; /; pN
(1)
where l is the map scale. In this paper, the closed curve of a wind farm is further compressed by an approximate polygon to reduce
Fig. 1. The map of Shepham wind farm [22].
H. Gu, J. Wang / Energy 57 (2013) 535e544
537
between qj and qj þ 1 to the edge qj qjþ1 . The total approximation error E (P, Q) between P and Q is defined as
EðP; Q Þ ¼
M X e qj ; qjþ1 ε0
(3)
j¼1
The threshold ε0 used in this paper is [23]
ε0 ¼ Ns20
Fig. 2. Detected edge of sub wind farm marked No. 1 in Shepham wind farm.
computing burden of wind farm micro-siting while preserving the curve local properties well. It aims at finding minimal M vertices among P so that the total approximation error of the polygon Q constructed by directly connecting these vertices does not exceed a given threshold ε0. The minimal M vertices to be found among P is denoted by
i h Q ¼ q1 ; q2 ; /; qj ; /; qM ; qMþ1
(2)
where the last vertex coincides with the first one: qM þ 1 ¼ q1. The approximation error of the jth edge, denoted by e(qj, qj þ 1), is defined as the sum of squared Euclidean distance from each point
600 Traced contour Anti−aliased contour 500
(4)
where s0 is the error threshold parameter and the recommended value in this paper is s0 ˛ [4, 5]. A small s0 forces the approximation polygon to be close to the original curve, yet at the cost of decreasing the graphic compression ratio. Optimal polygonal approximation can be solved by dynamic programming [23e25]. The SPPA (shortest-path polygonal approximation) algorithm [23] is used in this paper. The algorithm searches for the shortest path in the feasible graph constructed by the vertices of the input polygon P. The shortest path found in the graph from the first vertex to the last one with minimal vertices is the optimal approximation polygon Q. The SPPA algorithm significantly improves the solution of optimal polygonal approximation, even the input polygon has a biased distribution. In this paper, the average absolute deviation b d between the vertices of P and the edges of Q, the area ratio A of the polygon P to Q and the perimeter ratio C of P to Q are used as three evaluation indices for a polygonal approximation. The optimal polygonal approximation of wind farms Shepham [22], Ironstone [26] (Fig. 4) and Grasmere and Albany [27] (Fig. 5) are shown in Figs. 6e8 respectively. The simulation results and their analyses are shown in Table 1 in detail. The average absolute deviation b d of all the polygonal approximations is less than 0.6 m, and the area ratio A and the perimeter ratio C larger than 98%, which clearly demonstrate that approximation polygons have a high fidelity to the original wind farm shapes. If a wind farm design needs high accuracy, a much smaller s0 could be used. The approximation compression ratio of each wind farm contour is larger than 5, hence the proposed error threshold s0 is appropriate. Optimal polygonal approximation of a wind farm shape not only removes noise sides of the original contour in an acceptable degree of accuracy so as to make the placement of wind turbines easier, but also significantly compresses the contour data so that the computation of wind farm boundary constraint can be more efficient. 3. Boundary constraint model
Length (pixel)
400
3.1. Single-boundary constraint model The shape of a wind farm adds a boundary constraint to the micro-siting problem, that is how to judge whether the locations of wind turbines yielded by optimization algorithms are inside the wind farm. When a wind turbine is regarded as a point and the wind farm as a polygon, the boundary constraint is simplified as determining whether the point lies in the interior of the polygon. This operation is called the point-in-polygon query in computational geometry. In this paper, the ray intersection method [28] applicable for both convex and concave polygons is used to deal with the point-in-polygon query. The basic idea is to determine the spatial relationship between a point (e.g. T1) and a polygon (e.g. ABCDE) as illustrated in Fig. 9:
300
200
100
0
0
50
100
150
200
250
300
350
Width (pixel) Fig. 3. Contour tracing and anti-aliasing.
400
450
500
Step 1 Draw a vertical half line upward from the point T1 (if the half line crosses a vertex of the polygon ABCDE, remove the half line and draw a new horizontal half line from left to right from
H. Gu, J. Wang / Energy 57 (2013) 535e544
Fig. 4. The map of Ironstone wind farm [26].
T1) and count the number of intersections made by the half line and the edges of the polygon ABCDE. Rarely, if both the vertical and horizontal half lines pass a vertex of the polygon, an oblique half line is desirable. Step 2 If the number of intersections is odd, T1 is inside the polygon, otherwise it is not. (The number of intersections of T2 is 2, thus T2 is outside the polygon ABCDE.)
3000 Original contour Approximate polygon 2500
2000
Length (m)
538
1
1500 2
1000
3
500
0 0
500
1000
1500
2000
Width (m) Fig. 5. The map of Grasmere and Albany wind farm [27].
Fig. 6. Optimal polygonal approximation of Shepham wind farm.
2500
H. Gu, J. Wang / Energy 57 (2013) 535e544
2000
4j ðxÞ ¼
yjþ1 yj x xj þ y j ; xjþ1 xj
x˛ xj ; xjþ1
(8)
If xj > xj þ 1, then x ˛ (xj þ 1,xj). We assume xj < xj þ 1 hereafter. Hence, the number xj of intersections made by li and qj qjþ1 is
1500
Length (m)
539
xj ¼ 1000
1; 0;
bi < 4j ðai Þ and ai ˛ xj ; xjþ1 otherwise
(9)
Original contour
where xj ¼ 1 means the intersection exists and xj ¼ 0 means not. Therefore, the number of intersections made by the half line li from the ith wind turbine zi and the polygon Q is
Approximate polygon 500
Cðzi ; Q Þ ¼
M X
xj
(10)
j¼1
0 0
500
1000
1500
Width (m)
Define
qi ¼ modðCðzi ; Q Þ; 2Þ
Fig. 7. Optimal polygonal approximation of Ironstone wind farm.
Define a location vector of Nt wind turbines
z ¼ z1 ; z2 ; /; zi ; /; zNt ¼ ða1 ; b1 Þ; ða2 ; b2 Þ; /; ðai ; bi Þ; /; aNt ; bNt
(5)
It is clear that qi represents the location feasibility of the ith wind turbine zi in the polygon Q. The parameter qi ¼ 1 means that zi is inside Q, and qi ¼ 0 means not. Therefore, the boundary constraint violation value of z can be calculated by
and a single polygon with M vertices
BðzÞ ¼ Nt
i h Q ¼ q1 ; q2 ; /; qj ; /; qM ; qMþ1 i h ¼ ðx1 ; y1 Þ; ðx2 ; y2 Þ; /; xj ; yj ; /; ðxM ; yM Þ; ðxMþ1 ; yMþ1 Þ
where the last vertex coincides with the first one qM þ 1 ¼ q1. The ray intersection method starts with searching the intersection made by the half line from the ith wind turbine zi and the jth edge qj qjþ1 . Let li represents the half line from zi. It may be vertical, horizontal or oblique. For simplicity, we assume li is vertical hereafter, and the equation of li is
y˛½bi ; þNÞ
(7)
and the equation of the jth edge qj qjþ1 of Q is
2500 Grasmere Wind Farm
Length (m)
qi
(12)
The pseudo-code of the ray intersection method is given in Algorithm 1.
3.2. Multi-boundary constraint model A large wind farm may have many sub wind farms and infeasible areas, namely the wind farm has a multi-boundary constraint. There exist the following three cases when applying optimal polygonal approximation to the extracted wind farm contours. Case 1: a single convex or concave polygon, e.g. the wind farm in Fig. 4. Case 2: two or more adjacent polygons, e.g. the wind farm in Fig. 5. Case 3: polygons with holes (infeasible areas), e.g. the wind farm in Fig. 1. The above three cases can be regarded as the combination of multiple independent polygons. Generally, we assume that a given wind farm has Mw independent-area sub wind farms and the kth sub wind farm has nk infeasible areas inside itself. Thus the number of all approximation polygons of the wind farm is
2000
1500
Nt X i¼1
(6)
x ¼ ai ;
(11)
Albany Wind Farm
1000
Nw ¼ Mw þ
Mw X
nk
Mw 1; nk 0
(13)
k¼1
500 Original contour Approximate polygon 0 0
500
1000
1500
2000
2500 3000 Width (m)
3500
4000
4500
5000
Fig. 8. Optimal polygonal approximation of Grasmere and Albany wind farm.
Clearly, a wind turbine must be installed inside one of the Mw sub wind farms but outside all Nw Mw infeasible areas. In order to find the global optimal location of the wind turbine, all feasible areas of Mw sub wind farms should be explored simultaneously. Therefore, there exists Mw candidate feasible areas for the wind turbine in an exploration. Let Dk be the area feasible operator of the kth sub wind farm
540
H. Gu, J. Wang / Energy 57 (2013) 535e544
where dk ˛ {0, 1} and dk ¼ 1 means the kth sub wind farm is chosen as a candidate feasible area for the wind turbine, dk ¼ 0 means not. The entries of the last Nw Mw columns of Dk represent all infeasible areas of the wind farm, thus they should be 0. Therefore, the set of area feasible operators is
D ¼ D1 ; D2 ; /; Dk ; /; DMw
(15)
The Nw approximation polygons of the wind farm is denoted by
Q ¼ Q1 ; Q2 ; /; Qh ; /; QNw
(16)
where h ¼ {1, 2, /, Nw}. By repeatedly using the single-boundary constraint model for each approximation polygon Qh, the spatial relationship between the ith wind turbine zi and Q can be achieved as
Qi ¼ qi1 ; qi2 ; /; qih ; /; qiNw
(17)
where qih ˛ {0, 1} and qih ¼ 1 means the ith wind turbine zi is inside the hth polygon Qh, qih ¼ 0 means not. Finally, according to the area feasible operators set D, the location feasibility gi of the ith wind turbine zi on the wind farm can be calculated by
gi ¼
1; 0;
Qi ˛D
(18)
otherwise
where gi ¼ 1 means zi is inside the feasible areas of the wind farm and gi ¼ 0 means not. Therefore, for the multi-boundary constraint, Equation (12) should be reformed as
BðzÞ ¼ Nt
Nt X
gi
(19)
i¼1
Let us take Shepham wind farm as an example and assume five wind turbines placed on the farm as shown in Fig. 10, where z ¼ [T1, T2, T3, T4, T5]. It has two sub wind farms, and one infeasible area i.e., Mw ¼ 2 and n2 ¼ 1. Thus there are Nw ¼ 3 approximation polygons and the area feasible operators set of Shepham wind farm is
Algorithm 1. Ray intersection method.
2
Nw Mw
3
zfflfflffl}|fflfflffl{7 6 7 Dk ¼ 6 4d1 ; d2 ; /; dk ; /; dMw ; 0; /; 0 5
D ¼ f½1; 0; 0; ½0; 1; 0g
1Nw
The spatial relationship set of z and Q by Equation (17) is
3
2 M
N M
w w w 6zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ zfflfflffl}|fflfflffl{7 7 6 ¼ 60; /; 0 ; 1; 0; /; 0; 0; /; 07 5 4|fflfflfflffl{zfflfflfflffl}
k1
(14) 1Nw
Q ¼ fQ1 ; Q2 ; Q3 ; Q4 ; Q5 g ¼ f½1; 0; 0; ½0; 0; 0; ½0; 1; 1; ½0; 1; 0; ½0; 0; 0g Therefore, by using Equations (18) and (19), the multi-boundary constraint violation value B(z) is 3. Clearly, the multi-boundary
Table 1 Optimal polygonal approximation results. Variables
Wind farms Shepham 1
Shepham 2
Shepham 3
Ironstone
Grasmere
Albany
Map scale l Error threshold parameter s0 Number of original pixels Nv Number of vertices after anti-aliasing N Number of vertices after approximation M Original area after anti-aliasing (km2) Original perimeter after anti-aliasing (km) Area ratio A (%) Perimeter ratio C (%) Average absolute deviation b d (m) Approximation compression ratio N/M Total compression ratio Nv/M
1:500 5 910 227 25 1.29 5.29 99.87 98.52 0.33 9.08 36.40
1:500 5 1042 260 20 2.06 6.13 99.96 99.12 0.33 13.00 52.10
1:500 4 316 79 16 0.146 1.73 99.64 98.20 0.52 4.94 19.75
1:500 4 991 247 31 1.66 5.41 99.91 98.00 0.28 7.97 31.96
1:800 4 590 147 24 1.03 5.12 99.53 99.26 0.38 6.13 24.58
1:800 4 655 163 31 1.32 5.99 99.63 99.38 0.38 5.26 21.13
H. Gu, J. Wang / Energy 57 (2013) 535e544
541
A
330°
E
300 T1
N 0° 20%
°
°
15%
60
10% 5%
W 270°
B
30°
° 90 E
D
240°
120° 210°
T2 C
180° S
150°
Fig. 9. Illustration of the ray intersection method.
Fig. 11. The wind rose map [15].
constraint model is suitable to wind farms of any shape and any optimization algorithms using real or integer location variables. Since the Jensen wake model of wind turbines is only valid when the turbine distance is larger than four times its diameter D [29,30], the minimal distance dmin among wind turbines in this paper is set to 4D. Define
solutions inside the wind farm feasible areas firstly. Clearly, j(z) ¼ 0 if and only if the turbine layout z is feasible, otherwise j(z) > 0.
gij ¼
dmin dij ; 0;
dij < dmin otherwise
(20)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where dij ¼ ðai aj Þ2 þ ðbi bj Þ2 is the distance between the ith and jth wind turbines. Then the distance constraint violation value of Nt wind turbines z can be calculated by
GðzÞ ¼
NX t 1
Nt X
gij
(21)
i ¼ 1 j ¼ iþ1
Therefore, the constraint function of irregular-shape wind farm micro-siting optimization is
jðzÞ ¼ aBðzÞ þ bGðzÞ
(22)
where a and b are the weights of the boundary constraint and the distance constraint respectively. In this paper, a is set to 104 and b is set to 1 in order to heavily force optimization algorithms to find
4. Simulation The rose map is used to describe the characteristic of wind direction variations on a wind farm [15]. In the map, the compass is equally divided into several sectors. The length of each sector represents the annual frequency of wind in the respective range of direction. The Weibull probability distribution is used to represent the statistical behavior of wind speed [2,15,31,32]. It is a function of the shape parameter and the scale parameter describing the frequency for a given wind speed. The shape parameter relates to the range of wind speed variations, and the scale parameter relates to the mean wind speed. The actual wind direction and speed distributions of the wind farms Shepham, Ironstone, Grasmere or Albany are not available in public. To demonstrate the effectiveness of the proposed boundary constraint model, the same wind distributions are used for the above four wind farms, as shown in Fig. 11 and Table 2. The wake loss of a wind farm is defined in the following to evaluate the performance of a layout z [18]
WakeðzÞ ¼
AAPg ðzÞ AAPn ðzÞ 100% AAPg ðzÞ
(23)
where AAPg(z) is the annual average power of the wind farm without taking into account the wake losses and AAPn(z) is the annual average power of the wind farm considering wake losses.
Table 2 Wind distribution parameters [15].
T1
T2
T3 T4 T5
Fig. 10. An example for multi-boundary constraint model.
Direction
Frequency
Scale parameter
Shape parameter
0 30 60 90 120 150 180 210 240 270 300 330
0.020 0.044 0.056 0.076 0.061 0.053 0.078 0.083 0.123 0.158 0.167 0.079
6.4 7.4 8.8 7.0 7.3 7.5 8.0 8.9 9.8 9.9 9.3 7.8
1.79 1.81 1.65 1.96 1.83 1.81 1.89 1.85 1.96 1.92 1.93 1.75
542
H. Gu, J. Wang / Energy 57 (2013) 535e544 Table 3 Simulation parameters and results.
1.2
Variables
Wind farms
1 Population size Simulation generations Step length of local search Feasible installation area (km2) Number of wind turbines Nt Total installed capacity (MW) AAPg(z) (MW) AAPn(z) (MW) Wake(z) (%)
Thrust coefficient
0.8
0.6
0.4
0.2
Ironstone
Grasmere
Albany
Shepham
14 2000 1.2 1.66 14 21.00 10.68 10.37 2.93
8 2000 1.2 1.02 8 12.00 6.11 5.97 2.14
11 2000 1.2 1.32 11 16.50 8.39 8.07 3.92
26 2000 1.2 3.20 26 39.00 19.84 18.95 4.51
farm in this paper. Therefore, the irregular-shape wind farm micrositing optimization problem is to maximize the annual average power AAPn(z) generated by Nt turbines
0 3
5
7
9
11
13
15
17
19
21
23
25
Wind speed (m/s)
max AAPn ðzÞ
Fig. 12. Thrust coefficient of UP1500-86 [33].
subject to the boundary and the distance constraints on wind turbine locations In this paper, the models of wind direction and speed distributions, wake effects and wind turbine power are the same as those in Ref. [15]. Guodian UP (United Power) [33] is chosen as the wind turbine supplier, and the turbine type used is UP1500-86. The thrust coefficient and power curve of the wind turbine are shown in Figs. 12 and 13 respectively. The blade diameter of the wind turbine is 86.086 m and the hub height is 80 m. The Gaussian PSO with local search strategy algorithm [15] is used to optimize the layout of Nt wind turbines z in an irregular-shape wind farm. It incorporated a local search strategy into Gaussian PSO algorithm to search around the population’s historical best layout so as to improve both the precision and robustness of the optimization algorithm. Furthermore, to reduce the computation of objective functions and improve the computational efficiency it also taken a feasibilitybased method to compare the performance of different layouts. Only three parameters need to be tuned in simulations, i.e., the population size, the number of simulation generations and the step length of local search, which are given in Table 3. Since the Gaussian PSO with local search strategy cannot optimize the number of wind turbines, a fixed number of wind turbines is used for a given wind
jðzÞ ¼ 0 The Ironstone wind farm is first considered, which only has a single-boundary constraint. The optimal layout of 14 wind turbines on the farm is shown in Fig. 14. All turbines are inside the feasible area with a small wake loss of 2.93% (shown in Table 3). It is worth noting that the multi-boundary constraint model makes the simultaneous layout optimization of adjacent wind farms possible, and this can further reduce the wake losses among wind turbines. The simultaneous layout optimization result of the two adjacent wind farms Grasmere and Albany is shown in Fig. 15. The wake loss of Grasmere is 2.14%, and Albany is 3.92% which has more wind turbines. More generally, Shepham wind farm, which has two sub wind farms and an infeasible area (a lake), is used to further test the proposed multi-boundary constraint model, and the optimal layout and result of 26 wind turbines are shown in Fig. 16 and Table 3 respectively. Clearly, all turbines are inside two sub wind farms but outside the lake shaded by blue lines in Fig. 16.
2000
2000
1750
1500
1250
Length (m)
Power (kW)
1500
1000 750 500
1000
500
250 0 3
5
7
9
11
13
15
17
19
Wind speed (m/s) Fig. 13. Power curve of UP1500-86 [33].
21
23
25
0 0
500
1000
Width (m) Fig. 14. Optimal layout of Ironstone wind farm.
1500
H. Gu, J. Wang / Energy 57 (2013) 535e544
Based on the high-fidelity shape information provided by these two map processing steps, a single-boundary constraint model and a novel multi-boundary constraint model incorporated with the ray intersection method are proposed to quantify the wind farm boundary constraint for optimization algorithms. The simulations of typical cases have been carried out, where the effectiveness of the proposed boundary constraint models and the significance of irregular-shape wind farm micro-siting optimization have been clearly demonstrated.
2500 Grasmere Wind Farm
Length (m)
2000
1500
543
Albany Wind Farm
1000
Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant No. 61075064, and by the International Science and Technology Co-operation Program of China under Grant No. 2011DFG13020.
500
0 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
References
Width (m) Fig. 15. Optimal layout of Grasmere and Albany wind farm.
3000
2500
Length (m)
2000
1500
1000
500
0 0
500
1000
1500
2000
2500
Width (m) Fig. 16. Optimal layout of Shepham wind farm.
A more scattered layout of wind turbines guarantees a less wake loss of a wind farm, but forces the micro-siting of wind turbines to be closer to the boundary of the wind farm, as shown in Fig. 14 w16. The proposed boundary constraint model ensures that all feasible areas especially corners of the wind farm can be fully used. Therefore, simply considering commercial wind farms as rectangular-shape ones can not obtain an optimal layout of wind turbines, the significance of irregular-shape wind farm optimization has been clearly demonstrated.
5. Conclusions In this paper, a systematic approach to dealing with the irregular shape of a wind farm is originally developed for the optimization of wind farm micro-siting. Edge detection is used to automatically extract wind farm shape information from digital maps and optimal polygonal approximation is applied to compress wind farm contour data with an appropriate degree of accuracy so as to simplify the calculation of the wind farm boundary constraint.
[1] Sun X, Huang D, Wu G. The current state of offshore wind energy technology development. Energy 2012;41(1):298e312. [2] Abbes M, Belhadj J. Wind resource estimation and wind park design in El-Kef region, Tunisia. Energy 2012;40(1):348e57. [3] Mosetti G, Poloni C, Diviacco B. Optimization of wind turbine positioning in large wind farms by means of a genetic algorithm. Journal of Wind Engineering and Industrial Aerodynamic 1994;51(1):105e16. [4] Grady SA, Hussaini MY, Abdullah MM. Placement of wind turbines using genetic algorithms. Renewable Energy 2005;30(2):259e70. [5] Castro Mora J, Calero Barón JM, Riquelme Santos JM, Burgos Payán M. An evolutive algorithm for wind farm optimal design. Neurocomputing 2007;70(16e18):2651e8. [6] Marmidis G, Lazarou S, Pyrgioti E. Optimal placement of wind turbines in a wind park using Monte Carlo simulation. Renewable Energy 2008;33(7): 1455e60. [7] Wan C, Wang J, Yang G, Li X, Zhang X. Optimal micro-siting wind turbines by genetic algorithms based on improved wind and turbine models. In: Proceedings of the 48th IEEE Conference on Decision and Control. Shanghai, P. R. China 2009. p. 5092e6. [8] Emami A, Noghreh P. New approach on optimization in placement of wind turbines within wind farm by genetic algorithms. Renewable Energy 2010;35(7):1559e64. [9] Serrano González J, González Rodriguez AG, Castro Mora J, Riquelme Santos J, Burgos Payan M. Optimization of wind farm turbines layout using an evolutive algorithm. Renewable Energy 2010;35(8):1671e81. [10] Saavedra-Moreno B, Salcedo-Sanzand S, Paniagua-Tineo A, Prieto L, PortillaFigueras A. Seeding evolutionary algorithms with heuristics for optimal wind turbines positioning in wind farms. Renewable Energy 2011;36(11):2838e44. [11] Mustakerov I, Borissova D. Wind turbines type and number choice using combinatorial optimization. Renewable Energy 2010;35(9):1887e94. [12] Wan C, Wang J, Yang G, Zhang X. Optimal micro-siting of wind farms by particle swarm optimization. In: . Proceedings of the 1st International Conference on Advances in Swarm Intelligence 2010a;vol. 6145. p. 198e205. Beijing, China. [13] Wan C, Wang J, Yang G, Zhang X. Particle swarm optimization based on gaussian mutation and its application to wind farm micro-siting. In: Proceedings of the 49th IEEE Conference on Decision and Control 2010. p. 2227e32. [14] Wan C, Wang J, Yang G, Zhang X. Gaussian particle swarm optimization with differential evolution mutation. In: . Proceedings of the 2nd International Conference on Advances in Swarm Intelligence 2011;vol. 6728. p. 439e46. Chongqing, China. [15] Wan C, Wang J, Yang G, Gu H, Zhang X. Wind farm micro-siting by Gaussian particle swarm optimization with local search strategy. Renewable Energy 2012;48(0):276e86. [16] Chowdhury S, Zhang J, Messac A, Castillo L. Unrestricted wind farm layout optimization (UWFLO): investigating key factors influencing the maximum power generation. Renewable Energy 2012;38(1):16e30. [17] Ozturk UA, Norman BA. Heuristic methods for wind energy conversion system positioning. Electric Power Systems Research 2004;70(3):179e85. [18] Rivas RA, Clausen J, Hansen K, Jensen L. Solving the turbine positioning problem for large offshore wind farms by simulated annealing. Wind Engineering 2009;33(3):287e97. [19] Wan C, Wang J, Yang G, Zhang X. Optimal siting of wind turbines using realcoded genetic algorithms. In: . European Wind Energy Association Conference and Exhibition 2009;vol. 6. p. 3710e5. Marseille, France. [20] Serrano González J, González Rodríguez AG, Castro Mora J, Burgos Payán M, Riquelme Santos J. Overall design optimization of wind farms. Renewable Energy 2011;36(7):1973e82. [21] Canny J. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence 1986;8(6):679e98.
544
H. Gu, J. Wang / Energy 57 (2013) 535e544
[22] Shepham wind farm. http://www.shephamwindfarm.co.uk/site-location. aspx; 2013. [23] Kolesnikov A. ISE-bounded polygonal approximation of digital curves. Pattern Recognition Letters 2012;33(10):1329e37. [24] Horng J, Li JT. An automatic and efficient dynamic programming algorithm for polygonal approximation of digital curves. Pattern Recognition Letters 2002;23(1e3):171e82. [25] Kolesnikov A, Fränti P. Polygonal approximation of closed discrete curves. Pattern Recognition 2007;40(4):1282e93. [26] Ironstone wind farm. http://server13.web-mania.com/users/mollingt/regenco. pdf; 2013. [27] Grasmere and Albany wind farm. http://www.grasmerewindfarm.com.au/ project/site-location; 2013.
[28] Huang C, Shih T. On the complexity of point-in-polygon algorithms. Computers & Geosciences 1997;23(1):109e18. [29] Jensen NO. A note on wind generator interaction. Roskilde, Denmark: Tech. Rep.; Risø National Laboratory; 1984. [30] Katic I, Højstrup J, Jensen NO. A simple model for cluster efficiency. In: European Wind Energy Association Conference and Exhibition. Rome, Italy 1986. [31] Ouammi A, Sacile R, Zejli D, Mimet A, Benchrifa R. Sustainability of a wind power plant: application to different Moroccan sites. Energy 2010;35(10): 4226e36. [32] Albadi MH, El-Saadany EF. Optimum turbine-site matching. Energy 2010;35(9): 3593e602. [33] Guodian United Power Technology Co., Ltd. http://www.gdupc.com.cn/ english/tabid/282/language/zh-CN/Default.aspx; 2013.
Contents lists available at SciVerse ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Irregular-shape wind farm micro-siting optimization Huajie Gu, Jun Wang* Department of Control Sci. & Engn., Tongji University, Shanghai 201804, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 November 2012 Received in revised form 4 May 2013 Accepted 29 May 2013 Available online 10 July 2013
Landscape constraints inevitably cause the irregularity of the shape or boundary of a wind farm, which was not fully considered in previous literature. In this paper, a single-boundary constraint model and a novel multi-boundary constraint model incorporated with ray intersection method are developed to quantify the irregular boundary constraint for wind farm micro-siting optimization. In order to obtain high-fidelity wind farm shape information, an edge detection algorithm is employed to extract wind farm contour data from digital maps, and an optimal polygonal approximation algorithm is applied to compress the contour data so as to make the computation of boundary constraints less time-consuming. Simulations of four commercial wind farms comprehensively demonstrate the effectiveness of the proposed boundary constraint models and the significance of irregular-shape wind farm micro-siting optimization. 2013 Elsevier Ltd. All rights reserved.
Keywords: Wind farm Wind farm shape Wind farm boundary constraint Wind farm micro-siting Polygonal approximation Point-in-polygon
1. Introduction Wind energy takes the leading place among the other renewable energy, due to its abundant distribution and well developed exploitation technology. With the growing demand of wind energy, more and more wind farms have been built onshore or offshore [1]. The object of wind farm micro-siting is to find the optimal layout of wind turbines so that the wind farm can reach the maximal electricity output or economic benefit. In some wind farms, wind turbines are empirically placed apart with a distance of several times of a rotor diameter in order to reduce wake losses [2]. This empirical layout is easy to achieve, but is usually not the optimal solution to the wind farm micro-siting problem. A genetic algorithm was first utilized by Mosetti et al. [3] to optimize wind turbine placement in a rectangular-shape wind farm. The proposed method first discretized the wind farm into finite uniform square grids, and laid down a solid foundation for different recent approaches [4e10]. A mixed-integer discrete optimization model was proposed in Ref. [11], and nonlinear programming algorithms were applied to simultaneously achieve optimal selection of the type and number of wind turbines and optimal placement of all the turbines. A PSO (particle swarm optimization) algorithm was utilized by Wan et al. [12e15] and
* Corresponding author. Tel./fax: þ86 (0)21 69580069. E-mail addresses: [email protected], [email protected] (J. Wang). 0360-5442/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.05.066
Chowdhury et al. [16] to optically place turbines in any locations of a wind farm rather than in the cell centers of a given grid. The “continuous” search space provided the opportunity to explore every potential location on a wind farm so as to further reduce wake effects. It happens that some parts of a chosen wind farm can be unsuitable for placing wind turbines due to a number of landscape constraints, for example, lakes, parks as well as ecologically sensitive areas. Therefore, the shape of a wind farm is usually irregular. However, most previous optimization approaches in the literature only consider rectangular-shape wind farms [3e9,11e20]. Saavedra-Moreno et al. [10] used a binary template matrix to store wind farm shape information, and a much smaller grid size should be used in order to get an accurate shape model. Besides, the “discrete” gird-like locating scheme is not suitable for irregularshape wind farms as the optimal size and direction of the grid is difficult to choose especially when the farm is narrow and long. Using real location variables can avoid choosing optimal grid size and direction, and furthermore it can achieve better results. However, for the “continuous” micro-siting problem, the irregular shape of a wind farm introduces complicated boundary constraints, i.e. how to determine whether turbine locations yielded by optimization algorithms are inside permitted areas. In this paper, an edge detection algorithm used in image processing is employed to extract the contour data from the map, and an optimal polygonal approximation algorithm used in pattern recognition is then introduced to compress the contour data. A
536
H. Gu, J. Wang / Energy 57 (2013) 535e544
single-boundary constraint model and a novel multi-boundary model incorporated with the ray intersection method are proposed to calculate the boundary constraint violation value of turbine locations. An improved PSO algorithm proposed by Wan et al. [15] is adopted to tackle the optimal micro-siting problem. The remainder of this paper is organized as follows. Sections 2 and 3 describe the wind farm shape model and the boundary constraint model respectively. Section 4 presents the simulation results and the discussions. Section 5 summaries the contributions of the paper. 2. Wind farm shape model 2.1. Wind farm contour extraction Edge detection is widely used in image processing to extract useful structural information about object boundaries and many successful computational approaches have been developed for it. The Canny edge detection algorithm [21] is used in this paper. The contour of a wind farm in a digital map should be represented by closed curves in distinguishing color so that it can be extracted easily and accurately. The coordinates of pixels in the curves are then used to locate the contour in a Cartesian coordinate system in order to facilitate the calculation of following boundary constraint. This paper aims at developing a general approach to optimizing the irregular-shape wind farm micro-siting problem, so the practical contour can be achieved if the actual map is available. Shepham wind farm map shown in Fig. 1 is used as an example to illustrate the whole extraction process. It consists of two sub wind farms and a lake (the infeasible zone), which are marked with numbers as shown in Fig. 1. Each time only the contour of one of above three areas is extracted, and the extraction of a single contour is executed by the following steps. Step 1 Apply edge detection operation by using the Canny algorithm. The contour of the wind farm represented by a closed
blue curve is a distinguishing object from the background of the map. In this case, two contours can be achieved (the external one and the internal one), as shown in Fig. 2, and the width of each achieved contour is one pixel unit. After the edge detection, we transform the image coordinates to the Cartesian coordinates, and store the coordinates of the contour pixels in a vector. Step 2 Apply contour tracing operation. pffiffiffiSince the distance between two adjacent pixels is less than 2 pixel unit, we sort the coordinates in the vector in a clockwise or counterclockwise order. In this paper, the external contour V ¼ [v1, v2, /, vNv] is traced and the result is shown in Fig. 3, where Nv is the number of pixels in the traced contour. Note that, the traced contour has an aliasing effect. Step 3 Apply anti-aliasing operation. Downsampling is used to smooth the traced contour. Based on comprehensive simulations, the recommended downsampling rate in this paper is 4. The result is shown in Fig. 3. Thus, the final adjacently linked contour vector is S ¼ [s1, s2, /, sN] and N ¼ Nv/4. S represents a closed digital curve with N number of vertices. As wind turbines are of large dimensions, the contour location is not necessarily accurate and a more precise imaging processing method is out the scope of this paper. 2.2. Optimal polygonal approximation The shape or contour information of a wind farm is represented by a closed N-vertex polygonal curve S, which is stored as the ordered set of vertices S ¼ [s1, s2, /, sN]. The actual wind farm contour P should be
P ¼ lS ¼ ½p1 ; p2 ; /; pN
(1)
where l is the map scale. In this paper, the closed curve of a wind farm is further compressed by an approximate polygon to reduce
Fig. 1. The map of Shepham wind farm [22].
H. Gu, J. Wang / Energy 57 (2013) 535e544
537
between qj and qj þ 1 to the edge qj qjþ1 . The total approximation error E (P, Q) between P and Q is defined as
EðP; Q Þ ¼
M X e qj ; qjþ1 ε0
(3)
j¼1
The threshold ε0 used in this paper is [23]
ε0 ¼ Ns20
Fig. 2. Detected edge of sub wind farm marked No. 1 in Shepham wind farm.
computing burden of wind farm micro-siting while preserving the curve local properties well. It aims at finding minimal M vertices among P so that the total approximation error of the polygon Q constructed by directly connecting these vertices does not exceed a given threshold ε0. The minimal M vertices to be found among P is denoted by
i h Q ¼ q1 ; q2 ; /; qj ; /; qM ; qMþ1
(2)
where the last vertex coincides with the first one: qM þ 1 ¼ q1. The approximation error of the jth edge, denoted by e(qj, qj þ 1), is defined as the sum of squared Euclidean distance from each point
600 Traced contour Anti−aliased contour 500
(4)
where s0 is the error threshold parameter and the recommended value in this paper is s0 ˛ [4, 5]. A small s0 forces the approximation polygon to be close to the original curve, yet at the cost of decreasing the graphic compression ratio. Optimal polygonal approximation can be solved by dynamic programming [23e25]. The SPPA (shortest-path polygonal approximation) algorithm [23] is used in this paper. The algorithm searches for the shortest path in the feasible graph constructed by the vertices of the input polygon P. The shortest path found in the graph from the first vertex to the last one with minimal vertices is the optimal approximation polygon Q. The SPPA algorithm significantly improves the solution of optimal polygonal approximation, even the input polygon has a biased distribution. In this paper, the average absolute deviation b d between the vertices of P and the edges of Q, the area ratio A of the polygon P to Q and the perimeter ratio C of P to Q are used as three evaluation indices for a polygonal approximation. The optimal polygonal approximation of wind farms Shepham [22], Ironstone [26] (Fig. 4) and Grasmere and Albany [27] (Fig. 5) are shown in Figs. 6e8 respectively. The simulation results and their analyses are shown in Table 1 in detail. The average absolute deviation b d of all the polygonal approximations is less than 0.6 m, and the area ratio A and the perimeter ratio C larger than 98%, which clearly demonstrate that approximation polygons have a high fidelity to the original wind farm shapes. If a wind farm design needs high accuracy, a much smaller s0 could be used. The approximation compression ratio of each wind farm contour is larger than 5, hence the proposed error threshold s0 is appropriate. Optimal polygonal approximation of a wind farm shape not only removes noise sides of the original contour in an acceptable degree of accuracy so as to make the placement of wind turbines easier, but also significantly compresses the contour data so that the computation of wind farm boundary constraint can be more efficient. 3. Boundary constraint model
Length (pixel)
400
3.1. Single-boundary constraint model The shape of a wind farm adds a boundary constraint to the micro-siting problem, that is how to judge whether the locations of wind turbines yielded by optimization algorithms are inside the wind farm. When a wind turbine is regarded as a point and the wind farm as a polygon, the boundary constraint is simplified as determining whether the point lies in the interior of the polygon. This operation is called the point-in-polygon query in computational geometry. In this paper, the ray intersection method [28] applicable for both convex and concave polygons is used to deal with the point-in-polygon query. The basic idea is to determine the spatial relationship between a point (e.g. T1) and a polygon (e.g. ABCDE) as illustrated in Fig. 9:
300
200
100
0
0
50
100
150
200
250
300
350
Width (pixel) Fig. 3. Contour tracing and anti-aliasing.
400
450
500
Step 1 Draw a vertical half line upward from the point T1 (if the half line crosses a vertex of the polygon ABCDE, remove the half line and draw a new horizontal half line from left to right from
H. Gu, J. Wang / Energy 57 (2013) 535e544
Fig. 4. The map of Ironstone wind farm [26].
T1) and count the number of intersections made by the half line and the edges of the polygon ABCDE. Rarely, if both the vertical and horizontal half lines pass a vertex of the polygon, an oblique half line is desirable. Step 2 If the number of intersections is odd, T1 is inside the polygon, otherwise it is not. (The number of intersections of T2 is 2, thus T2 is outside the polygon ABCDE.)
3000 Original contour Approximate polygon 2500
2000
Length (m)
538
1
1500 2
1000
3
500
0 0
500
1000
1500
2000
Width (m) Fig. 5. The map of Grasmere and Albany wind farm [27].
Fig. 6. Optimal polygonal approximation of Shepham wind farm.
2500
H. Gu, J. Wang / Energy 57 (2013) 535e544
2000
4j ðxÞ ¼
yjþ1 yj x xj þ y j ; xjþ1 xj
x˛ xj ; xjþ1
(8)
If xj > xj þ 1, then x ˛ (xj þ 1,xj). We assume xj < xj þ 1 hereafter. Hence, the number xj of intersections made by li and qj qjþ1 is
1500
Length (m)
539
xj ¼ 1000
1; 0;
bi < 4j ðai Þ and ai ˛ xj ; xjþ1 otherwise
(9)
Original contour
where xj ¼ 1 means the intersection exists and xj ¼ 0 means not. Therefore, the number of intersections made by the half line li from the ith wind turbine zi and the polygon Q is
Approximate polygon 500
Cðzi ; Q Þ ¼
M X
xj
(10)
j¼1
0 0
500
1000
1500
Width (m)
Define
qi ¼ modðCðzi ; Q Þ; 2Þ
Fig. 7. Optimal polygonal approximation of Ironstone wind farm.
Define a location vector of Nt wind turbines
z ¼ z1 ; z2 ; /; zi ; /; zNt ¼ ða1 ; b1 Þ; ða2 ; b2 Þ; /; ðai ; bi Þ; /; aNt ; bNt
(5)
It is clear that qi represents the location feasibility of the ith wind turbine zi in the polygon Q. The parameter qi ¼ 1 means that zi is inside Q, and qi ¼ 0 means not. Therefore, the boundary constraint violation value of z can be calculated by
and a single polygon with M vertices
BðzÞ ¼ Nt
i h Q ¼ q1 ; q2 ; /; qj ; /; qM ; qMþ1 i h ¼ ðx1 ; y1 Þ; ðx2 ; y2 Þ; /; xj ; yj ; /; ðxM ; yM Þ; ðxMþ1 ; yMþ1 Þ
where the last vertex coincides with the first one qM þ 1 ¼ q1. The ray intersection method starts with searching the intersection made by the half line from the ith wind turbine zi and the jth edge qj qjþ1 . Let li represents the half line from zi. It may be vertical, horizontal or oblique. For simplicity, we assume li is vertical hereafter, and the equation of li is
y˛½bi ; þNÞ
(7)
and the equation of the jth edge qj qjþ1 of Q is
2500 Grasmere Wind Farm
Length (m)
qi
(12)
The pseudo-code of the ray intersection method is given in Algorithm 1.
3.2. Multi-boundary constraint model A large wind farm may have many sub wind farms and infeasible areas, namely the wind farm has a multi-boundary constraint. There exist the following three cases when applying optimal polygonal approximation to the extracted wind farm contours. Case 1: a single convex or concave polygon, e.g. the wind farm in Fig. 4. Case 2: two or more adjacent polygons, e.g. the wind farm in Fig. 5. Case 3: polygons with holes (infeasible areas), e.g. the wind farm in Fig. 1. The above three cases can be regarded as the combination of multiple independent polygons. Generally, we assume that a given wind farm has Mw independent-area sub wind farms and the kth sub wind farm has nk infeasible areas inside itself. Thus the number of all approximation polygons of the wind farm is
2000
1500
Nt X i¼1
(6)
x ¼ ai ;
(11)
Albany Wind Farm
1000
Nw ¼ Mw þ
Mw X
nk
Mw 1; nk 0
(13)
k¼1
500 Original contour Approximate polygon 0 0
500
1000
1500
2000
2500 3000 Width (m)
3500
4000
4500
5000
Fig. 8. Optimal polygonal approximation of Grasmere and Albany wind farm.
Clearly, a wind turbine must be installed inside one of the Mw sub wind farms but outside all Nw Mw infeasible areas. In order to find the global optimal location of the wind turbine, all feasible areas of Mw sub wind farms should be explored simultaneously. Therefore, there exists Mw candidate feasible areas for the wind turbine in an exploration. Let Dk be the area feasible operator of the kth sub wind farm
540
H. Gu, J. Wang / Energy 57 (2013) 535e544
where dk ˛ {0, 1} and dk ¼ 1 means the kth sub wind farm is chosen as a candidate feasible area for the wind turbine, dk ¼ 0 means not. The entries of the last Nw Mw columns of Dk represent all infeasible areas of the wind farm, thus they should be 0. Therefore, the set of area feasible operators is
D ¼ D1 ; D2 ; /; Dk ; /; DMw
(15)
The Nw approximation polygons of the wind farm is denoted by
Q ¼ Q1 ; Q2 ; /; Qh ; /; QNw
(16)
where h ¼ {1, 2, /, Nw}. By repeatedly using the single-boundary constraint model for each approximation polygon Qh, the spatial relationship between the ith wind turbine zi and Q can be achieved as
Qi ¼ qi1 ; qi2 ; /; qih ; /; qiNw
(17)
where qih ˛ {0, 1} and qih ¼ 1 means the ith wind turbine zi is inside the hth polygon Qh, qih ¼ 0 means not. Finally, according to the area feasible operators set D, the location feasibility gi of the ith wind turbine zi on the wind farm can be calculated by
gi ¼
1; 0;
Qi ˛D
(18)
otherwise
where gi ¼ 1 means zi is inside the feasible areas of the wind farm and gi ¼ 0 means not. Therefore, for the multi-boundary constraint, Equation (12) should be reformed as
BðzÞ ¼ Nt
Nt X
gi
(19)
i¼1
Let us take Shepham wind farm as an example and assume five wind turbines placed on the farm as shown in Fig. 10, where z ¼ [T1, T2, T3, T4, T5]. It has two sub wind farms, and one infeasible area i.e., Mw ¼ 2 and n2 ¼ 1. Thus there are Nw ¼ 3 approximation polygons and the area feasible operators set of Shepham wind farm is
Algorithm 1. Ray intersection method.
2
Nw Mw
3
zfflfflffl}|fflfflffl{7 6 7 Dk ¼ 6 4d1 ; d2 ; /; dk ; /; dMw ; 0; /; 0 5
D ¼ f½1; 0; 0; ½0; 1; 0g
1Nw
The spatial relationship set of z and Q by Equation (17) is
3
2 M
N M
w w w 6zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ zfflfflffl}|fflfflffl{7 7 6 ¼ 60; /; 0 ; 1; 0; /; 0; 0; /; 07 5 4|fflfflfflffl{zfflfflfflffl}
k1
(14) 1Nw
Q ¼ fQ1 ; Q2 ; Q3 ; Q4 ; Q5 g ¼ f½1; 0; 0; ½0; 0; 0; ½0; 1; 1; ½0; 1; 0; ½0; 0; 0g Therefore, by using Equations (18) and (19), the multi-boundary constraint violation value B(z) is 3. Clearly, the multi-boundary
Table 1 Optimal polygonal approximation results. Variables
Wind farms Shepham 1
Shepham 2
Shepham 3
Ironstone
Grasmere
Albany
Map scale l Error threshold parameter s0 Number of original pixels Nv Number of vertices after anti-aliasing N Number of vertices after approximation M Original area after anti-aliasing (km2) Original perimeter after anti-aliasing (km) Area ratio A (%) Perimeter ratio C (%) Average absolute deviation b d (m) Approximation compression ratio N/M Total compression ratio Nv/M
1:500 5 910 227 25 1.29 5.29 99.87 98.52 0.33 9.08 36.40
1:500 5 1042 260 20 2.06 6.13 99.96 99.12 0.33 13.00 52.10
1:500 4 316 79 16 0.146 1.73 99.64 98.20 0.52 4.94 19.75
1:500 4 991 247 31 1.66 5.41 99.91 98.00 0.28 7.97 31.96
1:800 4 590 147 24 1.03 5.12 99.53 99.26 0.38 6.13 24.58
1:800 4 655 163 31 1.32 5.99 99.63 99.38 0.38 5.26 21.13
H. Gu, J. Wang / Energy 57 (2013) 535e544
541
A
330°
E
300 T1
N 0° 20%
°
°
15%
60
10% 5%
W 270°
B
30°
° 90 E
D
240°
120° 210°
T2 C
180° S
150°
Fig. 9. Illustration of the ray intersection method.
Fig. 11. The wind rose map [15].
constraint model is suitable to wind farms of any shape and any optimization algorithms using real or integer location variables. Since the Jensen wake model of wind turbines is only valid when the turbine distance is larger than four times its diameter D [29,30], the minimal distance dmin among wind turbines in this paper is set to 4D. Define
solutions inside the wind farm feasible areas firstly. Clearly, j(z) ¼ 0 if and only if the turbine layout z is feasible, otherwise j(z) > 0.
gij ¼
dmin dij ; 0;
dij < dmin otherwise
(20)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where dij ¼ ðai aj Þ2 þ ðbi bj Þ2 is the distance between the ith and jth wind turbines. Then the distance constraint violation value of Nt wind turbines z can be calculated by
GðzÞ ¼
NX t 1
Nt X
gij
(21)
i ¼ 1 j ¼ iþ1
Therefore, the constraint function of irregular-shape wind farm micro-siting optimization is
jðzÞ ¼ aBðzÞ þ bGðzÞ
(22)
where a and b are the weights of the boundary constraint and the distance constraint respectively. In this paper, a is set to 104 and b is set to 1 in order to heavily force optimization algorithms to find
4. Simulation The rose map is used to describe the characteristic of wind direction variations on a wind farm [15]. In the map, the compass is equally divided into several sectors. The length of each sector represents the annual frequency of wind in the respective range of direction. The Weibull probability distribution is used to represent the statistical behavior of wind speed [2,15,31,32]. It is a function of the shape parameter and the scale parameter describing the frequency for a given wind speed. The shape parameter relates to the range of wind speed variations, and the scale parameter relates to the mean wind speed. The actual wind direction and speed distributions of the wind farms Shepham, Ironstone, Grasmere or Albany are not available in public. To demonstrate the effectiveness of the proposed boundary constraint model, the same wind distributions are used for the above four wind farms, as shown in Fig. 11 and Table 2. The wake loss of a wind farm is defined in the following to evaluate the performance of a layout z [18]
WakeðzÞ ¼
AAPg ðzÞ AAPn ðzÞ 100% AAPg ðzÞ
(23)
where AAPg(z) is the annual average power of the wind farm without taking into account the wake losses and AAPn(z) is the annual average power of the wind farm considering wake losses.
Table 2 Wind distribution parameters [15].
T1
T2
T3 T4 T5
Fig. 10. An example for multi-boundary constraint model.
Direction
Frequency
Scale parameter
Shape parameter
0 30 60 90 120 150 180 210 240 270 300 330
0.020 0.044 0.056 0.076 0.061 0.053 0.078 0.083 0.123 0.158 0.167 0.079
6.4 7.4 8.8 7.0 7.3 7.5 8.0 8.9 9.8 9.9 9.3 7.8
1.79 1.81 1.65 1.96 1.83 1.81 1.89 1.85 1.96 1.92 1.93 1.75
542
H. Gu, J. Wang / Energy 57 (2013) 535e544 Table 3 Simulation parameters and results.
1.2
Variables
Wind farms
1 Population size Simulation generations Step length of local search Feasible installation area (km2) Number of wind turbines Nt Total installed capacity (MW) AAPg(z) (MW) AAPn(z) (MW) Wake(z) (%)
Thrust coefficient
0.8
0.6
0.4
0.2
Ironstone
Grasmere
Albany
Shepham
14 2000 1.2 1.66 14 21.00 10.68 10.37 2.93
8 2000 1.2 1.02 8 12.00 6.11 5.97 2.14
11 2000 1.2 1.32 11 16.50 8.39 8.07 3.92
26 2000 1.2 3.20 26 39.00 19.84 18.95 4.51
farm in this paper. Therefore, the irregular-shape wind farm micrositing optimization problem is to maximize the annual average power AAPn(z) generated by Nt turbines
0 3
5
7
9
11
13
15
17
19
21
23
25
Wind speed (m/s)
max AAPn ðzÞ
Fig. 12. Thrust coefficient of UP1500-86 [33].
subject to the boundary and the distance constraints on wind turbine locations In this paper, the models of wind direction and speed distributions, wake effects and wind turbine power are the same as those in Ref. [15]. Guodian UP (United Power) [33] is chosen as the wind turbine supplier, and the turbine type used is UP1500-86. The thrust coefficient and power curve of the wind turbine are shown in Figs. 12 and 13 respectively. The blade diameter of the wind turbine is 86.086 m and the hub height is 80 m. The Gaussian PSO with local search strategy algorithm [15] is used to optimize the layout of Nt wind turbines z in an irregular-shape wind farm. It incorporated a local search strategy into Gaussian PSO algorithm to search around the population’s historical best layout so as to improve both the precision and robustness of the optimization algorithm. Furthermore, to reduce the computation of objective functions and improve the computational efficiency it also taken a feasibilitybased method to compare the performance of different layouts. Only three parameters need to be tuned in simulations, i.e., the population size, the number of simulation generations and the step length of local search, which are given in Table 3. Since the Gaussian PSO with local search strategy cannot optimize the number of wind turbines, a fixed number of wind turbines is used for a given wind
jðzÞ ¼ 0 The Ironstone wind farm is first considered, which only has a single-boundary constraint. The optimal layout of 14 wind turbines on the farm is shown in Fig. 14. All turbines are inside the feasible area with a small wake loss of 2.93% (shown in Table 3). It is worth noting that the multi-boundary constraint model makes the simultaneous layout optimization of adjacent wind farms possible, and this can further reduce the wake losses among wind turbines. The simultaneous layout optimization result of the two adjacent wind farms Grasmere and Albany is shown in Fig. 15. The wake loss of Grasmere is 2.14%, and Albany is 3.92% which has more wind turbines. More generally, Shepham wind farm, which has two sub wind farms and an infeasible area (a lake), is used to further test the proposed multi-boundary constraint model, and the optimal layout and result of 26 wind turbines are shown in Fig. 16 and Table 3 respectively. Clearly, all turbines are inside two sub wind farms but outside the lake shaded by blue lines in Fig. 16.
2000
2000
1750
1500
1250
Length (m)
Power (kW)
1500
1000 750 500
1000
500
250 0 3
5
7
9
11
13
15
17
19
Wind speed (m/s) Fig. 13. Power curve of UP1500-86 [33].
21
23
25
0 0
500
1000
Width (m) Fig. 14. Optimal layout of Ironstone wind farm.
1500
H. Gu, J. Wang / Energy 57 (2013) 535e544
Based on the high-fidelity shape information provided by these two map processing steps, a single-boundary constraint model and a novel multi-boundary constraint model incorporated with the ray intersection method are proposed to quantify the wind farm boundary constraint for optimization algorithms. The simulations of typical cases have been carried out, where the effectiveness of the proposed boundary constraint models and the significance of irregular-shape wind farm micro-siting optimization have been clearly demonstrated.
2500 Grasmere Wind Farm
Length (m)
2000
1500
543
Albany Wind Farm
1000
Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant No. 61075064, and by the International Science and Technology Co-operation Program of China under Grant No. 2011DFG13020.
500
0 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
References
Width (m) Fig. 15. Optimal layout of Grasmere and Albany wind farm.
3000
2500
Length (m)
2000
1500
1000
500
0 0
500
1000
1500
2000
2500
Width (m) Fig. 16. Optimal layout of Shepham wind farm.
A more scattered layout of wind turbines guarantees a less wake loss of a wind farm, but forces the micro-siting of wind turbines to be closer to the boundary of the wind farm, as shown in Fig. 14 w16. The proposed boundary constraint model ensures that all feasible areas especially corners of the wind farm can be fully used. Therefore, simply considering commercial wind farms as rectangular-shape ones can not obtain an optimal layout of wind turbines, the significance of irregular-shape wind farm optimization has been clearly demonstrated.
5. Conclusions In this paper, a systematic approach to dealing with the irregular shape of a wind farm is originally developed for the optimization of wind farm micro-siting. Edge detection is used to automatically extract wind farm shape information from digital maps and optimal polygonal approximation is applied to compress wind farm contour data with an appropriate degree of accuracy so as to simplify the calculation of the wind farm boundary constraint.
[1] Sun X, Huang D, Wu G. The current state of offshore wind energy technology development. Energy 2012;41(1):298e312. [2] Abbes M, Belhadj J. Wind resource estimation and wind park design in El-Kef region, Tunisia. Energy 2012;40(1):348e57. [3] Mosetti G, Poloni C, Diviacco B. Optimization of wind turbine positioning in large wind farms by means of a genetic algorithm. Journal of Wind Engineering and Industrial Aerodynamic 1994;51(1):105e16. [4] Grady SA, Hussaini MY, Abdullah MM. Placement of wind turbines using genetic algorithms. Renewable Energy 2005;30(2):259e70. [5] Castro Mora J, Calero Barón JM, Riquelme Santos JM, Burgos Payán M. An evolutive algorithm for wind farm optimal design. Neurocomputing 2007;70(16e18):2651e8. [6] Marmidis G, Lazarou S, Pyrgioti E. Optimal placement of wind turbines in a wind park using Monte Carlo simulation. Renewable Energy 2008;33(7): 1455e60. [7] Wan C, Wang J, Yang G, Li X, Zhang X. Optimal micro-siting wind turbines by genetic algorithms based on improved wind and turbine models. In: Proceedings of the 48th IEEE Conference on Decision and Control. Shanghai, P. R. China 2009. p. 5092e6. [8] Emami A, Noghreh P. New approach on optimization in placement of wind turbines within wind farm by genetic algorithms. Renewable Energy 2010;35(7):1559e64. [9] Serrano González J, González Rodriguez AG, Castro Mora J, Riquelme Santos J, Burgos Payan M. Optimization of wind farm turbines layout using an evolutive algorithm. Renewable Energy 2010;35(8):1671e81. [10] Saavedra-Moreno B, Salcedo-Sanzand S, Paniagua-Tineo A, Prieto L, PortillaFigueras A. Seeding evolutionary algorithms with heuristics for optimal wind turbines positioning in wind farms. Renewable Energy 2011;36(11):2838e44. [11] Mustakerov I, Borissova D. Wind turbines type and number choice using combinatorial optimization. Renewable Energy 2010;35(9):1887e94. [12] Wan C, Wang J, Yang G, Zhang X. Optimal micro-siting of wind farms by particle swarm optimization. In: . Proceedings of the 1st International Conference on Advances in Swarm Intelligence 2010a;vol. 6145. p. 198e205. Beijing, China. [13] Wan C, Wang J, Yang G, Zhang X. Particle swarm optimization based on gaussian mutation and its application to wind farm micro-siting. In: Proceedings of the 49th IEEE Conference on Decision and Control 2010. p. 2227e32. [14] Wan C, Wang J, Yang G, Zhang X. Gaussian particle swarm optimization with differential evolution mutation. In: . Proceedings of the 2nd International Conference on Advances in Swarm Intelligence 2011;vol. 6728. p. 439e46. Chongqing, China. [15] Wan C, Wang J, Yang G, Gu H, Zhang X. Wind farm micro-siting by Gaussian particle swarm optimization with local search strategy. Renewable Energy 2012;48(0):276e86. [16] Chowdhury S, Zhang J, Messac A, Castillo L. Unrestricted wind farm layout optimization (UWFLO): investigating key factors influencing the maximum power generation. Renewable Energy 2012;38(1):16e30. [17] Ozturk UA, Norman BA. Heuristic methods for wind energy conversion system positioning. Electric Power Systems Research 2004;70(3):179e85. [18] Rivas RA, Clausen J, Hansen K, Jensen L. Solving the turbine positioning problem for large offshore wind farms by simulated annealing. Wind Engineering 2009;33(3):287e97. [19] Wan C, Wang J, Yang G, Zhang X. Optimal siting of wind turbines using realcoded genetic algorithms. In: . European Wind Energy Association Conference and Exhibition 2009;vol. 6. p. 3710e5. Marseille, France. [20] Serrano González J, González Rodríguez AG, Castro Mora J, Burgos Payán M, Riquelme Santos J. Overall design optimization of wind farms. Renewable Energy 2011;36(7):1973e82. [21] Canny J. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence 1986;8(6):679e98.
544
H. Gu, J. Wang / Energy 57 (2013) 535e544
[22] Shepham wind farm. http://www.shephamwindfarm.co.uk/site-location. aspx; 2013. [23] Kolesnikov A. ISE-bounded polygonal approximation of digital curves. Pattern Recognition Letters 2012;33(10):1329e37. [24] Horng J, Li JT. An automatic and efficient dynamic programming algorithm for polygonal approximation of digital curves. Pattern Recognition Letters 2002;23(1e3):171e82. [25] Kolesnikov A, Fränti P. Polygonal approximation of closed discrete curves. Pattern Recognition 2007;40(4):1282e93. [26] Ironstone wind farm. http://server13.web-mania.com/users/mollingt/regenco. pdf; 2013. [27] Grasmere and Albany wind farm. http://www.grasmerewindfarm.com.au/ project/site-location; 2013.
[28] Huang C, Shih T. On the complexity of point-in-polygon algorithms. Computers & Geosciences 1997;23(1):109e18. [29] Jensen NO. A note on wind generator interaction. Roskilde, Denmark: Tech. Rep.; Risø National Laboratory; 1984. [30] Katic I, Højstrup J, Jensen NO. A simple model for cluster efficiency. In: European Wind Energy Association Conference and Exhibition. Rome, Italy 1986. [31] Ouammi A, Sacile R, Zejli D, Mimet A, Benchrifa R. Sustainability of a wind power plant: application to different Moroccan sites. Energy 2010;35(10): 4226e36. [32] Albadi MH, El-Saadany EF. Optimum turbine-site matching. Energy 2010;35(9): 3593e602. [33] Guodian United Power Technology Co., Ltd. http://www.gdupc.com.cn/ english/tabid/282/language/zh-CN/Default.aspx; 2013.