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Optimization of wind farm layout with complex land divisions
Renewable Energy 105 (2017) 30e40
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimization of wind farm layout with complex land divisions Longyan Wang a, Andy C.C. Tan a, b, Michael E. Cholette a, *, Yuantong Gu a, ** a b
School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, Brisbane 4001, Australia LKC Faculty of Engineering & Science, Universiti Tungku Abdul Rahman, Bandar Sungai Long, Cheras, 43000 Kajang, Selangor, Malaysia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 March 2016 Received in revised form 7 November 2016 Accepted 12 December 2016 Available online 13 December 2016
The study of wind farm layout optimization considering the decisions of land owners has rarely been reported in literature. In this paper, the common situation of complex land divisions (e.g. unequallyspaced plots) is addressed for the first time. A new constraint handling and fitness evaluation technique is developed to address the more complex wind farm boundaries and integrated into two common wind farm optimization approaches: the grid based method and the unrestricted coordinate method. Enable by the new technique, a numerical optimization study is conducted with the goal of evaluating the impact of the participation of land owners on the economic performance of the wind farm. In particular, two scenarios are considered: 1) the varying land plot scenario, where the land plot availability is included in the decision variables of the optimization, and 2) the sequential land plot scenario, where the land plot availability is fixed prior to optimization. The study reveals that the unrestricted coordinate method under the sequential land plot scenario yields the best optimization results, with the smallest cost of energy and the largest wind farm efficiency. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Layout optimization Wind farm Complex land divisions Constraint handling technique
1. Introduction Due to the depletion of tradition fossil fuels like coal, oil and natural gas, renewable energy sources have recently attracted great attention. Among the different forms of renewable energy including solar power, tidal power, biomass, etc., wind power serves as an attractive alternative energy source, owing to its abundance and cost-competitiveness [1]. Wind power is primarily used for the electricity generation by wind turbines that are typically clustered in a wind farm [2]. After a wind turbine extracts the kinetic energy from the wind, the air speed behind the wind turbine decreases drastically. This reduced velocity zone forms a highly turbulent flow field called the wake region and its recovery to the free stream wind can take dozens of kilometers [3]. Inevitably, some turbines are located in the wakes of others leading to power losses. This is a common issue for wind farm development.
* Corresponding author. School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, GP Box 2434, Brisbane, Queensland 4001, Australia. ** Corresponding author. School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, GP Box 2434, Brisbane, Queensland 4001, Australia. E-mail addresses: [email protected] (M.E. Cholette), yuantong.gu@ qut.edu.au (Y. Gu). http://dx.doi.org/10.1016/j.renene.2016.12.025 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
Through the optimization of wind farm layout, the power losses due to the wake interaction between turbines can be mitigated and the wind farm power output can be maximized for improved profitability [4,5]. Currently, wind farm layout optimization methods can be categorized into two groups: (i) grid based method, which discretize the wind farm area into uniform grids where each wind turbine can only be placed at the center point of the grid; and (ii) unrestricted coordinate method, which directly employ the Cartesian coordinates to represent the wind turbine locations in the wind farm. Mosetti [6] first studied the optimization problem for a 2 km 2 km dimension square wind farm by dividing it into one hundred 0.2 km 0.2 km even cells. Subsequently, many papers were published targeting the same topic with most of them employing the same 10 10 grids regardless of the wind farm dimensions studied [7e9]. Wang [10] applied three types of grids to optimize the wind farm layout. He found that 20 20 grids yield the best optimization results. The unrestricted coordinate method was first used by Beyer [11] to investigate the optimization of three wind farm layouts with different sizes. He found that better results are achieved for all wind farms by using the optimized wind farm layout compared with the empirical design. Nevertheless, almost all the wind farm layout optimization studies have been conducted based on the assumption that a continuous piece of land is readily available before developing a
L. Wang et al. / Renewable Energy 105 (2017) 30e40
wind farm. In reality, the willingness of landowners to participate in the wind farm project plays a crucial role in the success of developing the wind farm. Therefore, it is extremely important to study the impact of land plots’ availability on the wind farm design when doing the wind farm layout optimization. By assessing this impact, the wind farm developer can make a rational decision on the acceptable compensation for different landowners (e.g. the most important land plots can be compensated with more money). Chen [12e14] studied the wind farm layout optimization by considering the factor of landowners’ decision. Since then, no studies on the wind farm layout optimization with different landowners have been reported. The aforementioned studies typically consider only simple land plot divisions, while the complex land plot divisions have been reported in only one study [15]. The difference between simple and complex land divisions is illustrated in Fig. 1. The bold solid lines indicate the wind farm boundaries and the bold dash lines indicate the land plot divisions. The distinction between simple and complex land plot divisions are made as in Ref. [16]: simple divisions refer to equally-spaced square plots while the complex divisions allow for unequally-spaced rectangular plots, as illustrated in Fig. 1. Clearly, the wind farm complying with the above strict rules of simple divisions is quite restrictive and likely overly-simplistic. It is obvious that the wind farm with complex land divisions is more common in the real situation, and hence it is of great significance to develop the method for its study. In light of the above discussion, the study of wind farm optimization with complex land divisions is extremely crucial to facilitate the application of the research for a real commercial wind farm design. To the authors’ best knowledge, the sole study to address the complex land divisions is reference [15], in which only the single grid based wind farm design method is applied. However, this study utilized a standard penalty method to penalize infeasible solutions which the authors of the current paper have shown is typically outperformed by repair infeasible solution (RIS) method [16]. In this paper, different optimization approaches, including three optimization methods and two kinds of land plot scenarios, are considered and compared to investigate the effectiveness of different approaches. A new RIS constraint handling technique developed to handle the optimization constraints of complex wind farm divisions, together with the means of fitness evaluation. The remainder of the paper is organized as follows. Section 2 establishes the fundamental models for wind farm layout optimization study applied in this paper. It contains the Weibull wind scenario model, wind farm wake model, wind turbine power and cost model, and the objective function calculation. Section 3 describes
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the optimization solution encodings, optimization constraints, feasible solutions generation techniques, fitness evaluation process, and finally the optimization algorithm. Section 4 compares and discusses the wind farm layout optimization results with different approaches. Conclusions are drawn in Section 5. 2. Modelling 2.1. Wind characteristic Realistic wind descriptions are based on wind data measured on site with continuous wind speed and wind direction. It is widely accepted that wind scenarios in most areas of the world can be represented by the two-parameter Weibull distribution, whose probability density function is given by
pðvÞ ¼
k k vk1 v exp c c c
(1)
where p(v) is the probability density of occurrence of wind speed v, c is the scale parameter and k is the shape parameter. The cumulative Weibull distribution, P(v), giving the probability of the wind speed less than or equal to a certain value v, is given by
k v PðvÞ ¼ 1 exp c
(2)
In this paper, the wind condition of Weibull distribution used in Ref. [5] is applied and the detail of the wind scenario can be referred in the literature. 2.2. Wake model To describe the properties of wind after passing through the wind turbine rotor quantitatively, a simple wake model called the PARK model is utilized in this paper, which was based on N.O. Jensen theory [17]. It was then tuned for applying the improved model to optimize the wind farm configuration as discussed in Ref. [18]. Its effectiveness on predicting the wind farm power output considering the wake effect has been validated by comparison to the real wind farm observational data and computational simulation results [19]. As shown in Fig. 2, the PARK model assumes a linear expansion of the wake. Based on the theory of momentum conservation, the velocity in the wake of an upstream wind turbine at a distance of x along the wind direction can be obtained, it is given by:
Fig. 1. Illustration of the wind farm model with: (a) simple land plot division and (b) complex land plot division.
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
8 < Pi ¼ 0 P ¼ 0:3v3i : i Pi ¼ 630
if vi < 2:3 m=s or vi > 18 m=s if 2:3 m=s < vi < 12:8 m=s if 12:8 m=s < vi < 18 m=s
(7)
As can be seen from Eq. (7), the power is zero when the velocity is less than the cut-in speed or more than the cut-out speed; it is proportional to the cubic of velocity magnitude between cut-in and rated speeds; and it maintains constant rated power between rated speed and cut-out speeds. For the cost model, most of the wind farm layout optimization studies have applied the simplified model proposed by the Mosetti et al. [6] which is given by:
cost ¼ N
v1 ¼ v0
1 2a
r0 r0 þ ax
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1a r0 ¼ R 1 2a
(3)
(4)
in which a is axial induction factor which is the percentage of wind speed reduction from free stream wind to the wind at the rotor place. In Eq. (3), a is the wake spreading coefficient (or entrainment constant), which is related to the relative surface roughness length (z0) and the WT hub height (h) by empirical equation [21]:
2.4. Objective function Based on the different models introduced above, the objective function (CoE) for the optimization study in this paper is formulated and described below. For the Weibull distribution wind condition, only scaling parameter c will be affected by the wake loss [24], and it is given by:
ci ðqÞ ¼ cðqÞ 1 Vel defi
Z360 Pi ¼
(5)
According to reference [22], for a single wake between two wind turbines the velocity deficit (Vel_def) is given by:
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aoverlap r0 r0 þ ax Arotor
(6)
where Aoverlap is the overlap area between the upstream wind turbine wake and downstream wind turbine rotor, and its value is between zero (no wake pattern) and rotor area Arotor (full wake pattern). In the case of partial wake pattern, the overlap area can be calculated through arithmetical operations based on the relationship between wind turbine positions [22].
2.3. Power and cost model In this paper, it is assumed that the wind turbine applied is identical with the wind turbine from Refs. [6,23] with the same power characteristics. The magnitudes of the minimum (cut-in) operating wind speed, rated wind speed and maximum (cut-out) operating wind speed are 2.3 m/s, 12.8 m/s and 18 m/s, respectively. The power Pi of turbine i in kW is given by:
(9)
The produced power of a single WT Pi based on the Weibull distribution can be represented by:
pq ðqÞ
Z∞
0
0:5 ln zh0
Vel def ¼ 2a
(8)
2 #
where v0 is the free stream wind speed; v1 is the wind speed in the wake; r0 is the downstream rotor radius which is calculated based on the WT rotor radius R as:
a¼
The wind farm cost is thus a function of the number of wind turbines N. It is normalized to the single wind turbine cost (and is therefore non-dimensional). Each wind turbine cost consists of two cost components: constant cost portion of 2/3 and variable cost portion of 1/3.
Fig. 2. Diagram of PARK wake model [20].
"
2 1 0:00174N2 þ e 3 3
exp
PðvÞ 0
v ci ðqÞ
kðqÞ1 kðqÞ v ci ðqÞ ci ðqÞ ! kðqÞ
(10)
dqdv
Through discretizing both wind speed and wind direction, the analytical expression of a single wind turbine power output can be calculated. For the wind turbine model P(v) working under three different regions, the wind turbine power Pi is obtained by the separate power calculation for the region. When the wind turbine produces constant rated power, the power P1 under Weibull distribution is given by Ref. [25]:
(
P1 ¼ Prated
NX q þ1 l¼1
ul1 exp
k !) 2vrated ci ðl 1Þ þ ci ðlÞ
(11)
When the wind turbine power increases along with wind speed, the power P2 under Weibull distribution is given by Ref. [25]:
( 24 ( k ! vj1 þ vj 3 X 2vj1 0:3 ul1 exp 2 ci ðl 1Þ þ ci ðlÞ j¼1 l¼1 k !)) 2vj exp ci ðl 1Þ þ ci ðlÞ
P2 ¼
NX v þ1
(12) where Nq is the number of bins for the wind direction discretization while Nv is the number of bins for the wind speed discretization.
L. Wang et al. / Renewable Energy 105 (2017) 30e40
And ul-1 is the relative frequency of occurrence for the wind direction within the interval of [ql-1, ql]. The final wind turbine power is calculated by summing up the separate power production and it is given by:
Pi ¼ P1 þ P2
(13)
The total wind farm power output can be calculated as the aggregation of every individual wind turbine. The final objective function is given by:
CoE ¼ N
2 1 0:00174N2 þ e 3 3
, X N
(14)
i¼1
3.1. Optimization solution encodings Two different land plot scenarios are explored in this paper. The first is a varying land plot scenario, for which the unavailable land plot index is variable and determined by the optimization solution X. Hence, both wind farm layout and selected land plots are optimized. For the grid based method, X is represented by:
100000… … …000001 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
1…1 |fflffl{zfflffl}
(15)
100 or 400 bits for wind turbines 4 bits land plotðc* Þ
where the first 100 or 400 bits applied for 10 10 or 20 20 grids, are used to represent the existence of wind turbine in the specific grid. The land plot string, c*, is used to represent the land plot that is excluded. In this paper, only the scenario of one unavailable land plot is studied and hence 4 bit land plot string is sufficient. For the situation where more unavailable land plots are possible, this encoding can be easily adapted by adding additional bits to c* to indicate additional unavailable plots. For unrestricted coordinate method, X is represented by:
X¼
xi yi xj yj … … …xp yp xq yq |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
method, the distance between two adjacent cells is less than the minimum value of 5D. Hence, the proximity constraint should be added to the objective function to ensure that the wind turbines are placed far enough from each other to avoid greater power losses and potential damage caused by closer distance. For the grid density of 20 20 applied in this paper, the constraint can be represented by:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½2:5DðmðjÞ mðiÞÞ2 þ ½2:5DðnðjÞ nðiÞÞ2 < 5Dðfor c i and j; isjÞ
Pi
3. Methodology
X¼
33
1…1 |fflffl{zfflffl}
(16)
where m() and n() are integer arrays storing the row and column indices of wind turbines locations respectively, and 2.5D is the spacing between adjacent grid points. For the wind farm layout optimization with unrestricted coordinate method, the proximity constraint must be imposed on the decision variables due to the free-moving property of wind turbines in this method. The constraint is given by:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxðjÞ xðiÞÞ2 þ ðyðjÞ yðiÞÞ2 < 5D ðfor c i and j; isjÞ
3.2. Optimization constraints 3.2.1. Proximity constraint When 10 10 grids are utilized, a proximity constraint is not required since coarse discretization will ensure that the wind turbines are well-separated. For the wind farm studied in this paper, with 50D 50D (D is the wind turbine diameter) dimensions, the distance between two adjacent cells is 5D which is exactly the criterion of proximity constraint applied in this paper. When the grids are denser than 10 10 for the grid based
(18)
where x() and y() are real number arrays storing the X and Y coordinates for all wind turbines.
3.2.2. Land plot constraint To conduct the wind farm layout optimization by considering landowner decisions, the constraint of wind turbine placement should be imposed on the objective function so that the wind turbine can only be located in the available land plots. To facilitate the explanation of the constraints for different optimization methods, we assume that land plot 1 is unavailable because of nonparticipation of the landowner. For the grid based method with 10 10 grids as shown in Fig. 3(a), the constraint is represented by: N P
4ðX; cÞ ¼
2$N real numbers for N wind turbines 4 bits land plotðc* Þ
where the first 2N number of real numbers are used to represent the X and Y coordinates of N number of wind turbines. For both methods, the unavailable land plot index is determined with the expression value of (bin2dec(c*)%9 þ 1), where bin2dec() is the operation of converting the binary string in the bracket into decimal number and % is the modulo operator [15]. The second scenario is a sequential land plot scenario, where the unavailable land plot index is specified prior to the optimization. The encoding of solutions is identical to the varying land plot scenario except for the exclusion of the bits associated with the nonparticipating plots, c*.
(17)
4ðX; cÞ ¼ 0
c¼1
1 0
if 1 mðX; cÞ 3&1 nðX; cÞ 5 otherwise
(19)
Note that the variable c represents cell index of the wind turbine. For 20 20 grids as shown in Fig. 3(b), the constraint is represented by: N P
4ðX; cÞ ¼
4ðX; cÞ ¼ 0
c¼1
1 0
if 1 mðX; cÞ 6&1 nðX; cÞ 10 otherwise
(20)
where f(X, c) indicates the situation of wind turbine cell c for the binary solution X, and it is equal to 1 if the cell is located in the unavailable land plot, otherwise it is 0. For the unrestricted coordinate method as shown in Fig. 3(c), the constraint is represented by (with the dimension of 2 km 2 km for wind farm): N P
4ðX; iÞ ¼
i¼1
1 0
4ðX; iÞ ¼ 0
if 0 xðX; iÞ 1000&0 yðX; iÞ 600 otherwise
(21)
where f(X, i) indicates the situation of i-th wind turbine, and it is equal to 1 when the turbine is located in the unavailable land plot, otherwise it is 0.
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
Fig. 3. Schematic of wind farm layout optimization characterized by complex divisions of nine land plots through different methods: grid based method with 10 10 grid density (left) and with 20 20 grid density (middle), and unrestricted coordinate method (right).
3.3. Feasible solution generation Among all the different techniques that have been employed to handle the constraints for wind farm optimization study, the penalty method is most widely applied due to its flexibility in handling various types of constraints. However, there are limitations of applying the penalty method for the wind farm layout optimization study with complex land divisions, e.g., the determination of the penalty parameter and representation of land plot index [12,13,26]. Therefore, a new constraint handling technique is developed in this paper to cope with the proximity and land plot constraint without requiring the determination of additional penalty parameters. The detailed new constraint handling procedures for the two different land plot scenarios are described below. Let us consider the condition of grid based method with 20 20 grid density. The pseudo codes of varying land plot scenario are shown in Fig. 4. Let LP be the unavailable land plot index. The range ðiÞ
of row and column for land plot i is defined by nrow (lower bound) ðiÞ Nrow
(upper bound) and
ðiÞ ncolumn
(lower bound)
ðiÞ Ncolumn
(upper
bound), respectively. Take the first land plot (i ¼ 1) in Fig. 3(b) for example, where lower and upper bound of row index are 1 and 6 while the lower and upper bound of column index are 1 and 10, respectively. Based on the unavailable land index, the row and column arrays are processed by eliminating the misplaced wind turbine indices that violate the land plot constraint. Based on the proximity constraint in order to meet the minimum distance requirement, the arrays are further processed. In this way, all wind turbines in arrays will satisfy both of the two constraints. For the sequential land plot scenario, the code of handling land plot constraint part is indicated in Fig. 5 with land plot 1 assumed to
Algorithm 2: Land plot constraint handling
(assuming unavailable land plot to be 1 for example ) for i from N (X ) to 1, do if 1 ≤ m(i ) ≤ 6 and 1 ≤ n(i ) ≤ 10, do for j from i to N(X ), do m(i ) = m(i + 1);
Algorithm 1a: Land plot constraint handling
n(i ) = n(i + 1);
LP = bin 2dec(c )%9 + 1 ∗
switch ( LP ) case 1, do for i from N (X ) to 1, do
Record the number of times N LP ; Fig. 5. New procedures to obtain feasible solutions for 20 20 grid based method under sequential land plot scenario (only the land plot constraint handling part is shown while the proximity constraint handling part is the same as Fig. 4).
(1) (1) (1) (1) if nrow and ncolumn , do ≤ m(i ) ≤ N row ≤ n(i ) ≤ N column
Delete i-th wind turbine from m and n arrays
Algorithm 3a: Grid based method fitness evaluation
Record the number of times N LP ;
1. for i from 1 to (N (X ) − N LP (X ) − N PC (X )), do
case 2, do likewise; # case 9, do likewise; Algorithm 1b: Proximity constraint handling for ii from (N (X ) - N LP ) to 1, do for jj from ii to 1, do if
[2.5D ((m( jj ) − m(ii) )]2 + [2.5D ((n( jj ) − n(ii) )]2 < 5D, do
calculate the accumulated wind turbine power output Ptotal ; 2. caculate the wind farm cost Ccost with N (X ) number of turbines; 3. calculate the fitness Ffitness = Ccost / Ptotal + multiplier ⋅ abs ( N (X ) - N); 4. end; Algorithm 3b: Unrestricted coordinate method fitness evaluation 1. for i from 1 to (N − N LP − N PC ), do calculate the accumulated wind turbine power output Ptotal ;
Delete jj-th wind turbine from m and n arrays
2. caculate the wind farm cost Ccost with N number of turbines.
Record the number of times N PC ;
3. calculate the fitness Ffitness = Ccost / Ptotal .
Fig. 4. New procedures to obtain feasible solutions for 20 20 grid based method under varying land plot scenario (N(X) represents the number of wind turbines in solution X).
4. end; Fig. 6. Fitness evaluation procedures for the three different optimization methods.
L. Wang et al. / Renewable Energy 105 (2017) 30e40
35
Fig. 7. Schematic of the fitness evaluation process for the two wind farm layout optimization methods.
Table 1 Settings of GA optimization under different situations (N is the fixed number of turbines for unrestricted coordinate method) [28]. Parameter indices
Population size Population type Generations Crossover rate Mutation rate
Varying land plot scenario
Sequential land plot scenario
10 10 grids
20 20 grids
Coordinates
10 10 grids
20 20grids
Coordinates
104 Binary 1 million 0.8 0.1
404 Binary 100,000 0.8 0.1
2$Nþ4 Mixed 1 million 0.8 0.1
100 Binary 1 million 0.8 0.1
400 Binary 100,000 0.8 0.1
2$N Real 1 million 0.8 0.1
be unavailable, and the situation of other available land plots are considered in the same way. Based on the codes of 20 20 grid based method, the new constraint handling procedures for the other two methods can be obtained with small changes. For 10 10 grid based method, the land plot constraint part is basically the same with the critical row and column numbers changed according to the grid density, while the proximity constraint part is discarded since it is unnecessary. For the unrestricted coordinate method, first the row and column arrays are replaced with the arrays of Cartesian X and Y coordinates. Then the critical row and column numbers of unavailable land plot is replaced with the real coordinate value. Finally, the expression of any two wind turbine distance is reformulated by using coordinates instead of row and column numbers.
this paper, genetic algorithm (GA) is utilized and the parameter settings of different optimization method and land plot scenario are described in Table 1. The number of generations for both 10 10 grid based method and unrestricted coordinate method are set to be 1 million, while it is set to be 100,000 for 20 20 grid based method because the computational domain is much larger compared with other methods. All GAs share the same parameters settings for crossover and mutation rates, which describe the ratio of next generation population born by crossover and mutation operation [28,29]. Both the two operators performed using
3.4. Fitness evaluation In light of the objective function and techniques applied for handling the optimization constraints, the final fitness to evaluate the quality of a potential solution can be obtained and following the procedures shown in Fig. 6. For all methods, the total power output is calculated by excluding the misplaced wind turbines while the wind farm cost is calculated considering all wind turbines. In this manner, the fitness values of infeasible solutions will be penalized and will be automatically eliminated during the optimization due to its inferiority to the feasible solutions. The detailed schematic of the fitness evaluation process for the optimization is described in Fig. 7. It begins with the representation of the optimization solution X. For the two different land plot scenarios, different expressions of the solutions are employed. In order to obtain the feasible optimization solutions, the techniques to handle the wind turbine constraint and land plot constraint are performed which are shown in Algorithm 1 and Algorithm 2. After obtaining the feasible optimization solutions, the fitness of the objective function (cost of energy production) is evaluated according to Algorithm 3. 3.5. Optimization algorithm Due to the huge computational demand with manifold convexity for the wind farm layout optimization problem, traditional exact optimization approaches (e.g. linear programming, branchand-bound, and dynamic programming) are inapplicable [27]. In
Fig. 8. Variation of fitness (CoE) value results as a function of optimized number of turbines for the three methods.
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
Fig. 9. Variation of wind farm efficiency and total power output results as a function of optimized number of turbines for the three methods: (a) Wind farm efficiency and (b) Total wind farm power output.
standard GA techniques [30]. For all grid based methods, the binary scattered crossover and uniform mutation operators are applied. For unrestricted coordinate method under sequential land plot scenario, the decimal scattered crossover and uniform mutation operators are applied and due to the mixed population type property of the method under varying land plot scenario, both binary and decimal operators are incorporated. 4. Results and discussion In this paper, the wind farm layout optimization with complex land divisions is conducted with one land plot excluded, and the wind turbines can only be placed in the remaining eight land plots. The aim of this study is to examine the capability of our new constraint handling methods for both the grid based and unrestricted coordinate methods. We also consider two approaches to deciding which land plot(s) to exclude: the varying land plot scenario in which the land plot availability is included in the optimization, and the sequential land plot scenario in which the land plot availability is fixed prior to the optimization and a grid search on the land plot is used to determine the optimally-excluded plot.
of turbines optimized. When the number is small, e.g., less than 20, the discrepancy of fitness values is trivial and hard to distinguish. As the number increases, the unrestricted coordinate method clearly yields better solutions. As for the variation of fitness value as a function of the number of turbines optimized, they all share the same trend of variation for different methods, which decreases monotonically until reaching the minimal value around 40 to 50 turbines and increasing thereafter. The trend between 40 and 50 turbines is displayed in Fig. 8. In this range, but the fitness values for both grid methods display small increases while the finesses for the unrestricted coordinate method are nearly flat, indicating that the cost of energy is insensitive to the number of turbines near the optimum. Note that this is not the conclusion if one uses the gird based methods, which both indicate an increase in the optimal CoE. Then the wind farm efficiency and total wind farm power output results are shown in Fig. 9. The wind farm efficiency is defined as the actual wind farm power output divided by the theoretical
4.1. Varying land plot optimization with different methods The presentation of wind farm layout optimization results begins with varying land plot scenario. The fitness value results are shown in Fig. 8 for different optimized number of turbines. In order to consider the situation of the optimization under a wide range of turbine numbers, the top figure shows the optimized results for 10 to 80 turbines. We do not consider any more turbines since there are only 100 available positions for 10 10 grid based method. Due to the presence of randomness in the GA optimization, all the optimizations were repeatedly performed at 5 times with the same GA parameters and solutions with the best fitness are reported. For different optimization methods, it is found that the unrestricted coordinate method achieves the best results (lowest fitness values) in all cases. Meanwhile, the differences of optimization results between different methods are dependent on the targeted number
Fig. 10. Computational time with different number of wind turbines for the three methods.
L. Wang et al. / Renewable Energy 105 (2017) 30e40
Fig. 11. Fitness value (CoE) results of the sequential land plot scenario for different number of turbines optimized with the three methods, i.e., (a) 15 wind turbines, (b) 45 wind turbines and (c) 75 wind turbines.
power output excluding wake losses. Unsurprisingly, this efficiency decreases as the number of turbines increases according to the figure. Furthermore, the optimization results using unrestricted coordinate method has higher efficiency with less wake power losses compared to the grid based methods. Together with the fitness results in Fig. 8, it is clear that the unrestricted coordinate method achieves smaller fitness values by using the extra freedom to assign turbines to positions. Furthermore, this capability leads to significant power increases and CoE reduction for more than 20 turbines. Considering that the optimal number of turbines is around 40 (see Fig. 8), this performance difference is important. The computational time with different methods as a function of the number of turbines optimized is plotted in Fig. 10 to compare the effectiveness of the optimization methods more rationally. Obviously, the time increases as the optimized number of turbines increases. When the number is small (less than 40), the optimization time for different optimization methods are close to each other. When the number increases, the 10 10 grid based method is the most computationally costly, followed by the unrestricted
37
Fig. 12. Percentage of total power increase of 20 20 grid based and unrestricted coordinate methods compared to 10 10 grid based method for different number of turbines optimization: (a) 15 wind turbines, (b) 45 wind turbines and (c) 75 wind turbines.
coordinate method and then the 20 20 grid based method. The higher computational cost for the 10 10 grid is due to the low resolution of the grid yielding a high number of infeasible solutions which require further processing by the constraint handling method. 4.2. Sequential land plot optimization with different methods The results of sequential land plot optimization are shown in Fig. 11 for the fitness values with different nonparticipating land plots (from 1 to 9). Since the study of sequential land plot optimization demands the wind farm to be optimized considering every single land plot to be unavailable in series, the optimization with the wide range of turbines like the varying land plot optimization situation would be very time-consuming. Three different number of turbine situations are considered that are with small number (15), medium number (45) and large number (75). With 15 turbines, the fitness values are close to each other for the wind farm
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
layout optimization with the different methods, indicating that the selection of optimization methods has little influence on the optimization results. When the number of turbines increases, differences in the optimized fitness values become more pronounced for the different methods (note the different scale of y-axes). Like the above varying land plot scenario, the unrestricted coordinate method also achieves the best optimization results for this sequential land plot scenario, followed by the 20 20 grid based method, and the results using the 10 10 grid based method is still the worst. The increase of power output by utilizing 20 20 grid method and unrestricted coordinate method with respect to 10 10 grid method are shown in Fig. 12. Note that the wind farm efficiency has the same value percentage increase since the theoretical power output are identical for different nonparticipating land plots. As can be seen, the power output increases quite strongly as the optimized number of turbines increases. When the number is small, the power output increase with the unrestricted coordinate method is always positive even though the value is relatively small. For the 20 20 grid based method, however, the power output increase can be even negative at certain points (land plot 5 and land plot 8), meaning that the result is inferior to that optimized by the 10 10 grid based method. In order to assess the importance of considering complex land divisions in evaluating the importance of different land plots, it is instructive to compare the results to a simple optimization-andexclusion approach (which currently-available methods can handle). Firstly, the whole wind farm layout is optimized including all land plots. When a land plot is made unavailable, a naive evaluation of the new wind farm cost can be obtained by simply excluding the misplaced wind turbines and re-calculating the CoE. This method is hereafter referred to as the naive exclusion method. The comparison between the sequential land plot optimization method and the naive exclusion method is shown in Fig. 13, Since the unrestricted coordinate wind farm design method is superior according to the previous discussion, all the results are obtained using the unrestricted coordinate method. For all three different numbers of wind turbines studied, it is evident that the proposed sequential land plot optimization method achieves a much smaller cost of energy production than the naive exclusion method. The fitness of different nonparticipating land plots using the sequential land plot method have small variations, while the change is much bigger for the naive exclusion method. Such a difference in performance is practically important; this discrepancy would cause wind farm planners to over-estimate the importance of any given land plot. In contrast, the complex land division optimization enables a more accurate assessment of the impact of excluding a nonparticipating land plot on the economic performance of the whole wind farm. 4.3. Comparison between varying and sequential land plot scenario Fig. 13. Effectiveness of the proposed sequential land plot method by comparison to the naive infeasible wind turbine exclusion method with (a) 15 wind turbines; (b) 45 wind turbines and (c) 75 wind turbines.
Based on the above discussion on the results of varying and sequential land plot scenario, the comparison between the two different optimization studies is made in this section. The fitness
Table 2 Best fitness values (103) of different number of turbines optimization with the three methods for varying and sequential land plot scenario. Number of turbines
15 45 75
10 10 grids
20 20 grids
Coordinates
Varying
Sequential
Varying
Sequential
Varying
Sequential
4.18 3.483 4.187
4.18 3.48 4.143
4.172 3.444 3.821
4.175 3.429 3.767
4.171 3.377 3.63
4.169 3.363 3.601
L. Wang et al. / Renewable Energy 105 (2017) 30e40
39
Table 3 Results of the best nonparticipating land plot index for the optimization with varying and sequential land plot scenario. Number of turbines
15 45 75
10 10 grids
20 20 grids
Coordinates
Varying
Sequential
Varying
Sequential
Varying
Sequential
3 or 4 or 5 3 7
3 or 4 or 5 4 5
3 or 4 or 5 7 6
3 or 4 or 5 5 4
3 or 5 7 7
3 or 5 or 8 4 7
results with different optimization methods are tabulated and shown in Table 2. When applying 10 10 grid based method, the fitness values with varying and sequential land plot scenarios are equal. As the number of turbines increases, however, the sequential land plot scenario becomes increasingly superior with approximately 0.1% and 1.1% improvement of the fitness values for medium (45) and large number (75) of turbines, respectively. For the 20 20 grid based method, the fitness of sequential land plot scenario is even worse with small number of turbines (15), which could be possibly due to the fact that it is stuck in the local optima. When the number of turbines increases (45 and 75), the fitness of sequential land plot scenario decreases by 0.4% and 1.4%, respectively. As for the unrestricted coordinate method, the decrease in fitness values for all number of turbines has been detected although they are relatively small (0.05%, 0.4% and 0.8% for the three tested numbers, respectively). Thus, the sequential optimization procedure produces superior solutions, but must be run for each combination of excluded land plot and number of turbines. In contrast, the varying land plot scenario optimization only needs to be conducted once for each number of wind turbines, since it determines the optimal land plot(s) to exclude. For the study of wind farm optimization with lands owned by different owners, one of the primary objectives is to determine the importance of different land plots so that the least important land plot to the wind farm project can be obtained. Hence, indices of best nonparticipating land plots for different optimized number of
Fig. 14. Optimal wind farm layouts with 15 turbines, through varying land plot scenario (left figure) and sequential land plot scenario (right figure) and different optimization methods (the same for the figure below): (a) 10 10 grid method, (b) 20 20 grid method and (c) unrestricted coordinate method.
turbines with the two different optimization studies are indicated in Table 3. For the two grid methods, the land plot indices with the two studies are exactly the same when the number of turbines is small. Since the optimized wind turbines are less, three land plots are equally unimportant to the project (plots 3, 4, and 5). When the number of turbines increases, the least important land plot becomes unique, though it varies with the number of turbines studied. For the unrestricted coordinate method, the least important land plot indices are also multiple while they are different for the two different land plot scenarios and differ from the grid method results as well. Of course, since the sequential land plot scenario using the unrestricted coordinate method yields the best CoE, it’s identification of the least important plots are the most trustworthy. We also see that despite the small differences in CoE, the identified least important land plot is apparently quite sensitive to these small differences and it is thus important to utilise the most accurate method available to conclude which plots are the least important. 4.4. Optimal wind farm layouts In this section, results of optimal wind farm layouts for different number of turbine optimization through the varying and sequential land plot scenarios are presented using the three different optimization methods. First, the optimal layouts with less turbines are shown in Fig. 14, in which the wind turbines are located in the center of grids marked with black color for grid based method, while they are represented
Fig. 15. Optimal wind farm layouts with 75 turbines through varying land plot scenario (left figure) and sequential land plot scenario (right figure) and different optimization methods (the same as figure above).
40
L. Wang et al. / Renewable Energy 105 (2017) 30e40
by black circles for the unrestricted coordinate method. The nonparticipating land plots are shaded with grey color. For 10 10 grid based method, the optimal layouts with the two land plot scenarios are exactly the same. The wind turbines are distributed evenly with 10 rotor diameters away from each other along row 1, row 3 and row 10. For 20 20 grid based method, the optimal layout with varying land plot scenario are organized with 4, 5 and 6 turbines in rows 1, 5 and 20, respectively, but the layout is abnormal for sequential land plot as the result is suboptimal. For the unrestricted coordinate method, even though the layouts are irregular in both scenarios, they are similar to each other and spreading along the diagonal of wind farm which is perpendicular to the dominant wind direction. Next, the optimal wind farm layouts with more wind turbines are presented in Fig. 15, in which the wind turbines are represented in the same way as the above figure for the two methods. Unlike the regular wind farm layout pattern for a small number of turbines, the wind turbines occupy most of the potential locations for the 10 10 grids, while they are similarly distributed for both varying and sequential studies. The optimal layouts for 20 20 grid based method also show similar patterns: the wind turbines are staggered and spread along every other row. The wind turbine distributions by the grid based method are quite different for the two different land plot scenarios since the nonparticipating land plot indices differ. However, they have the same nonparticipating land plots for the optimization with the two land plot scenarios by the unrestricted coordinate method. They share very similar optimal layouts which are generally aligned with the diagonal normal to the dominant wind direction.
5. Conclusions In this work, the problem of the wind farm layout optimization with complex land plot divisions is addressed. Due to the inapplicability of traditional penalty technique for the wind farm optimization with complex land divisions, a new constraint handling technique is proposed to be applied to cope with both proximity constraint and land plot constraint. The means of fitness evaluation for different optimization methods incorporating the new constraint handling technique have been presented. In addition, different optimization approaches including three wind farm design methods in conjunction with two land plot scenarios have been developed in terms of the representation of optimization solutions and different types of constraints imposed on the objective function. The results show that unrestricted coordinate method yields the best optimization results with lower fitness values and higher wind farm efficiency for both varying and sequential scenarios. The discrepancy between different optimization methods increases with the number of turbines optimized. When the number is relatively big, the sequential scenario is always able to produce better optimization results than the varying scenario for all methods. When the number is small, however, it varies depending on the optimization methods applied. For small number of turbines, the results indicate that the designer has significant freedom in exclusion of different land plots since there can be a number of optimal unavailable land plots. For a larger number of turbines, the best unavailable land plot tends to become unique and is sensitive to the optimization accuracy. Based on the above discussion, it is concluded that the sequential land plot optimization study with unrestricted coordinate method is the best selection which yields the most reliable results for the complex land divisions.
Acknowledgements The High Performance Computer resources provided by Queensland University of Technology (QUT) are gratefully acknowledged. Also the financial support of China Scholarship Council (CSC) from Chinese government and Top-up scholarship from QUT are greatly appreciated.
References [1] P. Gipe, Wind Power: Renewable Energy for Home, Farm, and Business, Chelsea Green Publishing, 2004. [2] T.B. Johansson, Renewable Energy: Sources for Fuels and Electricity, Island Press, 1993. [3] P.M. McKay, Group Dynamics of Commercial Scale Wind Turbines, University of Windsor, (Canada): Ann Arbor, 2011, p. 98. [4] P.Y. Zhang, Topics in Wind Farm Layout Optimization: Analytical Wake Models, Noise Propagation, and Energy Production, University of Toronto, 2013. [5] L. Wang, A.C. Tan, Y. Gu, Comparative study on optimizing the wind farm layout using different design methods and cost models, J. Wind Eng. Ind. Aerod. 146 (2015) 1e10. [6] G. Mosetti, C. Poloni, B. Diviacco, Optimization of wind turbine positioning in large wind farms by means of a genetic algorithm, J. Wind Eng. Ind. Aerod. 51 (1) (1994) 105e116. [7] C. Wan, et al., Wind farm micro-siting by Gaussian particle swarm optimization with local search strategy, Renew. Energy 48 (2012) 276e286. €kçeada using multi[8] S. S¸is¸bot, et al., Optimal positioning of wind turbines on Go objective genetic algorithm, Wind Energy 13 (4) (2010) 297e306. [9] S.D.O. Turner, et al., A new mathematical programming approach to optimize wind farm layouts, Renew. Energy 63 (0) (2014) 674e680. [10] F. Wang, D. Liu, L. Zeng, Study on computational grids in placement of wind turbines using genetic algorithm, in: World Non-Grid-Connected Wind Power and Energy Conference, 2009. WNWEC 2009, IEEE, 2009. [11] H.G. Beyer, et al., Optimization of wind farm configurations with variable number of turbines, in: Proceedings European Union Wind Energy Confer€teborg, 1996. ence, Go [12] L. Chen, E. MacDonald, Considering landowner participation in wind farm layout optimization, J. Mech. Des. 134 (8) (2012) 084506. [13] L. Chen and E. MacDonald, Effects of uncertain land availability, wind shear, and cost on wind farm layout, in: Under Review, ASME 2013 IDETC Conference, Portland, Oregon, August. [14] L. Chen, E. MacDonald, A new model for wind farm layout optimization with landowner decisions, in: ASME International Design Engineering Technical Conference/Design Automation Conference, Washington DC, August, 2011. [15] L. Chen, E. MacDonald, A system-level cost-of-energy wind farm layout optimization with landowner modeling, Energy Convers. Manag. 77 (2014) 484e494. [16] L. Wang, et al., A new constraint handling method for wind farm layout optimization with lands owned by different owners, Renew. Energy 83 (2015) 151e161. [17] N.O. Jensen, A Note on Wind Generator Interaction, 1983. [18] I. Katic, J. Højstrup, N. Jensen, A Simple Model for Cluster Efficiency, 1986. [19] L. Wang, et al., Comparison of the effectiveness of analytical wake models for wind farm with constant and variable hub heights, Energy Convers. Manag. 124 (2016) 189e202. [20] 2014; Available from: http://offwind.eu/Help/EngWindSim. [21] S. Frandsen, On the wind speed reduction in the center of large clusters of wind turbines, J. Wind Eng. Ind. Aerod. 39 (1) (1992) 251e265. [22] S. Pookpunt, W. Ongsakul, Optimal placement of wind turbines within wind farm using binary particle swarm optimization with time-varying acceleration coefficients, Renew. Energy 55 (2013) 266e276. [23] S. Grady, M. Hussaini, M.M. Abdullah, Placement of wind turbines using genetic algorithms, Renew. Energy 30 (2) (2005) 259e270. [24] M.A. Lackner, C.N. Elkinton, An analytical framework for offshore wind farm layout optimization, Wind Eng. 31 (1) (2007) 17e31. [25] A. Kusiak, Z. Song, Design of wind farm layout for maximum wind energy capture, Renew. Energy 35 (3) (2010) 685e694. [26] G.C. Larsen, et al., Wake meandering: a pragmatic approach, Wind Energy 11 (4) (2008) 377e395. [27] E. Horowitz, S. Sahni, Fundamentals of Computer Algorithms, Computer Science Press, 1978. [28] L. Wang, et al., Different Approaches of Applying Single-Objective Binary Genetic Algorithm on the Wind Farm Design, 2014. [29] M. Zhao, Z. Chen, J. Hjerrild, Analysis of the behaviour of genetic algorithm applied in optimization of electrical system design for offshore wind farms, in: IEEE Industrial Electronics, IECON 2006-32nd Annual Conference on, IEEE, 2006. [30] V. Kayvanfar, et al., Minimizing total tardiness and earliness on unrelated parallel machines with controllable processing times, Comput. Oper. Res. 41 (2014) 31e43.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimization of wind farm layout with complex land divisions Longyan Wang a, Andy C.C. Tan a, b, Michael E. Cholette a, *, Yuantong Gu a, ** a b
School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, Brisbane 4001, Australia LKC Faculty of Engineering & Science, Universiti Tungku Abdul Rahman, Bandar Sungai Long, Cheras, 43000 Kajang, Selangor, Malaysia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 March 2016 Received in revised form 7 November 2016 Accepted 12 December 2016 Available online 13 December 2016
The study of wind farm layout optimization considering the decisions of land owners has rarely been reported in literature. In this paper, the common situation of complex land divisions (e.g. unequallyspaced plots) is addressed for the first time. A new constraint handling and fitness evaluation technique is developed to address the more complex wind farm boundaries and integrated into two common wind farm optimization approaches: the grid based method and the unrestricted coordinate method. Enable by the new technique, a numerical optimization study is conducted with the goal of evaluating the impact of the participation of land owners on the economic performance of the wind farm. In particular, two scenarios are considered: 1) the varying land plot scenario, where the land plot availability is included in the decision variables of the optimization, and 2) the sequential land plot scenario, where the land plot availability is fixed prior to optimization. The study reveals that the unrestricted coordinate method under the sequential land plot scenario yields the best optimization results, with the smallest cost of energy and the largest wind farm efficiency. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Layout optimization Wind farm Complex land divisions Constraint handling technique
1. Introduction Due to the depletion of tradition fossil fuels like coal, oil and natural gas, renewable energy sources have recently attracted great attention. Among the different forms of renewable energy including solar power, tidal power, biomass, etc., wind power serves as an attractive alternative energy source, owing to its abundance and cost-competitiveness [1]. Wind power is primarily used for the electricity generation by wind turbines that are typically clustered in a wind farm [2]. After a wind turbine extracts the kinetic energy from the wind, the air speed behind the wind turbine decreases drastically. This reduced velocity zone forms a highly turbulent flow field called the wake region and its recovery to the free stream wind can take dozens of kilometers [3]. Inevitably, some turbines are located in the wakes of others leading to power losses. This is a common issue for wind farm development.
* Corresponding author. School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, GP Box 2434, Brisbane, Queensland 4001, Australia. ** Corresponding author. School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, GP Box 2434, Brisbane, Queensland 4001, Australia. E-mail addresses: [email protected] (M.E. Cholette), yuantong.gu@ qut.edu.au (Y. Gu). http://dx.doi.org/10.1016/j.renene.2016.12.025 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
Through the optimization of wind farm layout, the power losses due to the wake interaction between turbines can be mitigated and the wind farm power output can be maximized for improved profitability [4,5]. Currently, wind farm layout optimization methods can be categorized into two groups: (i) grid based method, which discretize the wind farm area into uniform grids where each wind turbine can only be placed at the center point of the grid; and (ii) unrestricted coordinate method, which directly employ the Cartesian coordinates to represent the wind turbine locations in the wind farm. Mosetti [6] first studied the optimization problem for a 2 km 2 km dimension square wind farm by dividing it into one hundred 0.2 km 0.2 km even cells. Subsequently, many papers were published targeting the same topic with most of them employing the same 10 10 grids regardless of the wind farm dimensions studied [7e9]. Wang [10] applied three types of grids to optimize the wind farm layout. He found that 20 20 grids yield the best optimization results. The unrestricted coordinate method was first used by Beyer [11] to investigate the optimization of three wind farm layouts with different sizes. He found that better results are achieved for all wind farms by using the optimized wind farm layout compared with the empirical design. Nevertheless, almost all the wind farm layout optimization studies have been conducted based on the assumption that a continuous piece of land is readily available before developing a
L. Wang et al. / Renewable Energy 105 (2017) 30e40
wind farm. In reality, the willingness of landowners to participate in the wind farm project plays a crucial role in the success of developing the wind farm. Therefore, it is extremely important to study the impact of land plots’ availability on the wind farm design when doing the wind farm layout optimization. By assessing this impact, the wind farm developer can make a rational decision on the acceptable compensation for different landowners (e.g. the most important land plots can be compensated with more money). Chen [12e14] studied the wind farm layout optimization by considering the factor of landowners’ decision. Since then, no studies on the wind farm layout optimization with different landowners have been reported. The aforementioned studies typically consider only simple land plot divisions, while the complex land plot divisions have been reported in only one study [15]. The difference between simple and complex land divisions is illustrated in Fig. 1. The bold solid lines indicate the wind farm boundaries and the bold dash lines indicate the land plot divisions. The distinction between simple and complex land plot divisions are made as in Ref. [16]: simple divisions refer to equally-spaced square plots while the complex divisions allow for unequally-spaced rectangular plots, as illustrated in Fig. 1. Clearly, the wind farm complying with the above strict rules of simple divisions is quite restrictive and likely overly-simplistic. It is obvious that the wind farm with complex land divisions is more common in the real situation, and hence it is of great significance to develop the method for its study. In light of the above discussion, the study of wind farm optimization with complex land divisions is extremely crucial to facilitate the application of the research for a real commercial wind farm design. To the authors’ best knowledge, the sole study to address the complex land divisions is reference [15], in which only the single grid based wind farm design method is applied. However, this study utilized a standard penalty method to penalize infeasible solutions which the authors of the current paper have shown is typically outperformed by repair infeasible solution (RIS) method [16]. In this paper, different optimization approaches, including three optimization methods and two kinds of land plot scenarios, are considered and compared to investigate the effectiveness of different approaches. A new RIS constraint handling technique developed to handle the optimization constraints of complex wind farm divisions, together with the means of fitness evaluation. The remainder of the paper is organized as follows. Section 2 establishes the fundamental models for wind farm layout optimization study applied in this paper. It contains the Weibull wind scenario model, wind farm wake model, wind turbine power and cost model, and the objective function calculation. Section 3 describes
31
the optimization solution encodings, optimization constraints, feasible solutions generation techniques, fitness evaluation process, and finally the optimization algorithm. Section 4 compares and discusses the wind farm layout optimization results with different approaches. Conclusions are drawn in Section 5. 2. Modelling 2.1. Wind characteristic Realistic wind descriptions are based on wind data measured on site with continuous wind speed and wind direction. It is widely accepted that wind scenarios in most areas of the world can be represented by the two-parameter Weibull distribution, whose probability density function is given by
pðvÞ ¼
k k vk1 v exp c c c
(1)
where p(v) is the probability density of occurrence of wind speed v, c is the scale parameter and k is the shape parameter. The cumulative Weibull distribution, P(v), giving the probability of the wind speed less than or equal to a certain value v, is given by
k v PðvÞ ¼ 1 exp c
(2)
In this paper, the wind condition of Weibull distribution used in Ref. [5] is applied and the detail of the wind scenario can be referred in the literature. 2.2. Wake model To describe the properties of wind after passing through the wind turbine rotor quantitatively, a simple wake model called the PARK model is utilized in this paper, which was based on N.O. Jensen theory [17]. It was then tuned for applying the improved model to optimize the wind farm configuration as discussed in Ref. [18]. Its effectiveness on predicting the wind farm power output considering the wake effect has been validated by comparison to the real wind farm observational data and computational simulation results [19]. As shown in Fig. 2, the PARK model assumes a linear expansion of the wake. Based on the theory of momentum conservation, the velocity in the wake of an upstream wind turbine at a distance of x along the wind direction can be obtained, it is given by:
Fig. 1. Illustration of the wind farm model with: (a) simple land plot division and (b) complex land plot division.
32
L. Wang et al. / Renewable Energy 105 (2017) 30e40
8 < Pi ¼ 0 P ¼ 0:3v3i : i Pi ¼ 630
if vi < 2:3 m=s or vi > 18 m=s if 2:3 m=s < vi < 12:8 m=s if 12:8 m=s < vi < 18 m=s
(7)
As can be seen from Eq. (7), the power is zero when the velocity is less than the cut-in speed or more than the cut-out speed; it is proportional to the cubic of velocity magnitude between cut-in and rated speeds; and it maintains constant rated power between rated speed and cut-out speeds. For the cost model, most of the wind farm layout optimization studies have applied the simplified model proposed by the Mosetti et al. [6] which is given by:
cost ¼ N
v1 ¼ v0
1 2a
r0 r0 þ ax
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1a r0 ¼ R 1 2a
(3)
(4)
in which a is axial induction factor which is the percentage of wind speed reduction from free stream wind to the wind at the rotor place. In Eq. (3), a is the wake spreading coefficient (or entrainment constant), which is related to the relative surface roughness length (z0) and the WT hub height (h) by empirical equation [21]:
2.4. Objective function Based on the different models introduced above, the objective function (CoE) for the optimization study in this paper is formulated and described below. For the Weibull distribution wind condition, only scaling parameter c will be affected by the wake loss [24], and it is given by:
ci ðqÞ ¼ cðqÞ 1 Vel defi
Z360 Pi ¼
(5)
According to reference [22], for a single wake between two wind turbines the velocity deficit (Vel_def) is given by:
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aoverlap r0 r0 þ ax Arotor
(6)
where Aoverlap is the overlap area between the upstream wind turbine wake and downstream wind turbine rotor, and its value is between zero (no wake pattern) and rotor area Arotor (full wake pattern). In the case of partial wake pattern, the overlap area can be calculated through arithmetical operations based on the relationship between wind turbine positions [22].
2.3. Power and cost model In this paper, it is assumed that the wind turbine applied is identical with the wind turbine from Refs. [6,23] with the same power characteristics. The magnitudes of the minimum (cut-in) operating wind speed, rated wind speed and maximum (cut-out) operating wind speed are 2.3 m/s, 12.8 m/s and 18 m/s, respectively. The power Pi of turbine i in kW is given by:
(9)
The produced power of a single WT Pi based on the Weibull distribution can be represented by:
pq ðqÞ
Z∞
0
0:5 ln zh0
Vel def ¼ 2a
(8)
2 #
where v0 is the free stream wind speed; v1 is the wind speed in the wake; r0 is the downstream rotor radius which is calculated based on the WT rotor radius R as:
a¼
The wind farm cost is thus a function of the number of wind turbines N. It is normalized to the single wind turbine cost (and is therefore non-dimensional). Each wind turbine cost consists of two cost components: constant cost portion of 2/3 and variable cost portion of 1/3.
Fig. 2. Diagram of PARK wake model [20].
"
2 1 0:00174N2 þ e 3 3
exp
PðvÞ 0
v ci ðqÞ
kðqÞ1 kðqÞ v ci ðqÞ ci ðqÞ ! kðqÞ
(10)
dqdv
Through discretizing both wind speed and wind direction, the analytical expression of a single wind turbine power output can be calculated. For the wind turbine model P(v) working under three different regions, the wind turbine power Pi is obtained by the separate power calculation for the region. When the wind turbine produces constant rated power, the power P1 under Weibull distribution is given by Ref. [25]:
(
P1 ¼ Prated
NX q þ1 l¼1
ul1 exp
k !) 2vrated ci ðl 1Þ þ ci ðlÞ
(11)
When the wind turbine power increases along with wind speed, the power P2 under Weibull distribution is given by Ref. [25]:
( 24 ( k ! vj1 þ vj 3 X 2vj1 0:3 ul1 exp 2 ci ðl 1Þ þ ci ðlÞ j¼1 l¼1 k !)) 2vj exp ci ðl 1Þ þ ci ðlÞ
P2 ¼
NX v þ1
(12) where Nq is the number of bins for the wind direction discretization while Nv is the number of bins for the wind speed discretization.
L. Wang et al. / Renewable Energy 105 (2017) 30e40
And ul-1 is the relative frequency of occurrence for the wind direction within the interval of [ql-1, ql]. The final wind turbine power is calculated by summing up the separate power production and it is given by:
Pi ¼ P1 þ P2
(13)
The total wind farm power output can be calculated as the aggregation of every individual wind turbine. The final objective function is given by:
CoE ¼ N
2 1 0:00174N2 þ e 3 3
, X N
(14)
i¼1
3.1. Optimization solution encodings Two different land plot scenarios are explored in this paper. The first is a varying land plot scenario, for which the unavailable land plot index is variable and determined by the optimization solution X. Hence, both wind farm layout and selected land plots are optimized. For the grid based method, X is represented by:
100000… … …000001 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
1…1 |fflffl{zfflffl}
(15)
100 or 400 bits for wind turbines 4 bits land plotðc* Þ
where the first 100 or 400 bits applied for 10 10 or 20 20 grids, are used to represent the existence of wind turbine in the specific grid. The land plot string, c*, is used to represent the land plot that is excluded. In this paper, only the scenario of one unavailable land plot is studied and hence 4 bit land plot string is sufficient. For the situation where more unavailable land plots are possible, this encoding can be easily adapted by adding additional bits to c* to indicate additional unavailable plots. For unrestricted coordinate method, X is represented by:
X¼
xi yi xj yj … … …xp yp xq yq |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
method, the distance between two adjacent cells is less than the minimum value of 5D. Hence, the proximity constraint should be added to the objective function to ensure that the wind turbines are placed far enough from each other to avoid greater power losses and potential damage caused by closer distance. For the grid density of 20 20 applied in this paper, the constraint can be represented by:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½2:5DðmðjÞ mðiÞÞ2 þ ½2:5DðnðjÞ nðiÞÞ2 < 5Dðfor c i and j; isjÞ
Pi
3. Methodology
X¼
33
1…1 |fflffl{zfflffl}
(16)
where m() and n() are integer arrays storing the row and column indices of wind turbines locations respectively, and 2.5D is the spacing between adjacent grid points. For the wind farm layout optimization with unrestricted coordinate method, the proximity constraint must be imposed on the decision variables due to the free-moving property of wind turbines in this method. The constraint is given by:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxðjÞ xðiÞÞ2 þ ðyðjÞ yðiÞÞ2 < 5D ðfor c i and j; isjÞ
3.2. Optimization constraints 3.2.1. Proximity constraint When 10 10 grids are utilized, a proximity constraint is not required since coarse discretization will ensure that the wind turbines are well-separated. For the wind farm studied in this paper, with 50D 50D (D is the wind turbine diameter) dimensions, the distance between two adjacent cells is 5D which is exactly the criterion of proximity constraint applied in this paper. When the grids are denser than 10 10 for the grid based
(18)
where x() and y() are real number arrays storing the X and Y coordinates for all wind turbines.
3.2.2. Land plot constraint To conduct the wind farm layout optimization by considering landowner decisions, the constraint of wind turbine placement should be imposed on the objective function so that the wind turbine can only be located in the available land plots. To facilitate the explanation of the constraints for different optimization methods, we assume that land plot 1 is unavailable because of nonparticipation of the landowner. For the grid based method with 10 10 grids as shown in Fig. 3(a), the constraint is represented by: N P
4ðX; cÞ ¼
2$N real numbers for N wind turbines 4 bits land plotðc* Þ
where the first 2N number of real numbers are used to represent the X and Y coordinates of N number of wind turbines. For both methods, the unavailable land plot index is determined with the expression value of (bin2dec(c*)%9 þ 1), where bin2dec() is the operation of converting the binary string in the bracket into decimal number and % is the modulo operator [15]. The second scenario is a sequential land plot scenario, where the unavailable land plot index is specified prior to the optimization. The encoding of solutions is identical to the varying land plot scenario except for the exclusion of the bits associated with the nonparticipating plots, c*.
(17)
4ðX; cÞ ¼ 0
c¼1
1 0
if 1 mðX; cÞ 3&1 nðX; cÞ 5 otherwise
(19)
Note that the variable c represents cell index of the wind turbine. For 20 20 grids as shown in Fig. 3(b), the constraint is represented by: N P
4ðX; cÞ ¼
4ðX; cÞ ¼ 0
c¼1
1 0
if 1 mðX; cÞ 6&1 nðX; cÞ 10 otherwise
(20)
where f(X, c) indicates the situation of wind turbine cell c for the binary solution X, and it is equal to 1 if the cell is located in the unavailable land plot, otherwise it is 0. For the unrestricted coordinate method as shown in Fig. 3(c), the constraint is represented by (with the dimension of 2 km 2 km for wind farm): N P
4ðX; iÞ ¼
i¼1
1 0
4ðX; iÞ ¼ 0
if 0 xðX; iÞ 1000&0 yðX; iÞ 600 otherwise
(21)
where f(X, i) indicates the situation of i-th wind turbine, and it is equal to 1 when the turbine is located in the unavailable land plot, otherwise it is 0.
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
Fig. 3. Schematic of wind farm layout optimization characterized by complex divisions of nine land plots through different methods: grid based method with 10 10 grid density (left) and with 20 20 grid density (middle), and unrestricted coordinate method (right).
3.3. Feasible solution generation Among all the different techniques that have been employed to handle the constraints for wind farm optimization study, the penalty method is most widely applied due to its flexibility in handling various types of constraints. However, there are limitations of applying the penalty method for the wind farm layout optimization study with complex land divisions, e.g., the determination of the penalty parameter and representation of land plot index [12,13,26]. Therefore, a new constraint handling technique is developed in this paper to cope with the proximity and land plot constraint without requiring the determination of additional penalty parameters. The detailed new constraint handling procedures for the two different land plot scenarios are described below. Let us consider the condition of grid based method with 20 20 grid density. The pseudo codes of varying land plot scenario are shown in Fig. 4. Let LP be the unavailable land plot index. The range ðiÞ
of row and column for land plot i is defined by nrow (lower bound) ðiÞ Nrow
(upper bound) and
ðiÞ ncolumn
(lower bound)
ðiÞ Ncolumn
(upper
bound), respectively. Take the first land plot (i ¼ 1) in Fig. 3(b) for example, where lower and upper bound of row index are 1 and 6 while the lower and upper bound of column index are 1 and 10, respectively. Based on the unavailable land index, the row and column arrays are processed by eliminating the misplaced wind turbine indices that violate the land plot constraint. Based on the proximity constraint in order to meet the minimum distance requirement, the arrays are further processed. In this way, all wind turbines in arrays will satisfy both of the two constraints. For the sequential land plot scenario, the code of handling land plot constraint part is indicated in Fig. 5 with land plot 1 assumed to
Algorithm 2: Land plot constraint handling
(assuming unavailable land plot to be 1 for example ) for i from N (X ) to 1, do if 1 ≤ m(i ) ≤ 6 and 1 ≤ n(i ) ≤ 10, do for j from i to N(X ), do m(i ) = m(i + 1);
Algorithm 1a: Land plot constraint handling
n(i ) = n(i + 1);
LP = bin 2dec(c )%9 + 1 ∗
switch ( LP ) case 1, do for i from N (X ) to 1, do
Record the number of times N LP ; Fig. 5. New procedures to obtain feasible solutions for 20 20 grid based method under sequential land plot scenario (only the land plot constraint handling part is shown while the proximity constraint handling part is the same as Fig. 4).
(1) (1) (1) (1) if nrow and ncolumn , do ≤ m(i ) ≤ N row ≤ n(i ) ≤ N column
Delete i-th wind turbine from m and n arrays
Algorithm 3a: Grid based method fitness evaluation
Record the number of times N LP ;
1. for i from 1 to (N (X ) − N LP (X ) − N PC (X )), do
case 2, do likewise; # case 9, do likewise; Algorithm 1b: Proximity constraint handling for ii from (N (X ) - N LP ) to 1, do for jj from ii to 1, do if
[2.5D ((m( jj ) − m(ii) )]2 + [2.5D ((n( jj ) − n(ii) )]2 < 5D, do
calculate the accumulated wind turbine power output Ptotal ; 2. caculate the wind farm cost Ccost with N (X ) number of turbines; 3. calculate the fitness Ffitness = Ccost / Ptotal + multiplier ⋅ abs ( N (X ) - N); 4. end; Algorithm 3b: Unrestricted coordinate method fitness evaluation 1. for i from 1 to (N − N LP − N PC ), do calculate the accumulated wind turbine power output Ptotal ;
Delete jj-th wind turbine from m and n arrays
2. caculate the wind farm cost Ccost with N number of turbines.
Record the number of times N PC ;
3. calculate the fitness Ffitness = Ccost / Ptotal .
Fig. 4. New procedures to obtain feasible solutions for 20 20 grid based method under varying land plot scenario (N(X) represents the number of wind turbines in solution X).
4. end; Fig. 6. Fitness evaluation procedures for the three different optimization methods.
L. Wang et al. / Renewable Energy 105 (2017) 30e40
35
Fig. 7. Schematic of the fitness evaluation process for the two wind farm layout optimization methods.
Table 1 Settings of GA optimization under different situations (N is the fixed number of turbines for unrestricted coordinate method) [28]. Parameter indices
Population size Population type Generations Crossover rate Mutation rate
Varying land plot scenario
Sequential land plot scenario
10 10 grids
20 20 grids
Coordinates
10 10 grids
20 20grids
Coordinates
104 Binary 1 million 0.8 0.1
404 Binary 100,000 0.8 0.1
2$Nþ4 Mixed 1 million 0.8 0.1
100 Binary 1 million 0.8 0.1
400 Binary 100,000 0.8 0.1
2$N Real 1 million 0.8 0.1
be unavailable, and the situation of other available land plots are considered in the same way. Based on the codes of 20 20 grid based method, the new constraint handling procedures for the other two methods can be obtained with small changes. For 10 10 grid based method, the land plot constraint part is basically the same with the critical row and column numbers changed according to the grid density, while the proximity constraint part is discarded since it is unnecessary. For the unrestricted coordinate method, first the row and column arrays are replaced with the arrays of Cartesian X and Y coordinates. Then the critical row and column numbers of unavailable land plot is replaced with the real coordinate value. Finally, the expression of any two wind turbine distance is reformulated by using coordinates instead of row and column numbers.
this paper, genetic algorithm (GA) is utilized and the parameter settings of different optimization method and land plot scenario are described in Table 1. The number of generations for both 10 10 grid based method and unrestricted coordinate method are set to be 1 million, while it is set to be 100,000 for 20 20 grid based method because the computational domain is much larger compared with other methods. All GAs share the same parameters settings for crossover and mutation rates, which describe the ratio of next generation population born by crossover and mutation operation [28,29]. Both the two operators performed using
3.4. Fitness evaluation In light of the objective function and techniques applied for handling the optimization constraints, the final fitness to evaluate the quality of a potential solution can be obtained and following the procedures shown in Fig. 6. For all methods, the total power output is calculated by excluding the misplaced wind turbines while the wind farm cost is calculated considering all wind turbines. In this manner, the fitness values of infeasible solutions will be penalized and will be automatically eliminated during the optimization due to its inferiority to the feasible solutions. The detailed schematic of the fitness evaluation process for the optimization is described in Fig. 7. It begins with the representation of the optimization solution X. For the two different land plot scenarios, different expressions of the solutions are employed. In order to obtain the feasible optimization solutions, the techniques to handle the wind turbine constraint and land plot constraint are performed which are shown in Algorithm 1 and Algorithm 2. After obtaining the feasible optimization solutions, the fitness of the objective function (cost of energy production) is evaluated according to Algorithm 3. 3.5. Optimization algorithm Due to the huge computational demand with manifold convexity for the wind farm layout optimization problem, traditional exact optimization approaches (e.g. linear programming, branchand-bound, and dynamic programming) are inapplicable [27]. In
Fig. 8. Variation of fitness (CoE) value results as a function of optimized number of turbines for the three methods.
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
Fig. 9. Variation of wind farm efficiency and total power output results as a function of optimized number of turbines for the three methods: (a) Wind farm efficiency and (b) Total wind farm power output.
standard GA techniques [30]. For all grid based methods, the binary scattered crossover and uniform mutation operators are applied. For unrestricted coordinate method under sequential land plot scenario, the decimal scattered crossover and uniform mutation operators are applied and due to the mixed population type property of the method under varying land plot scenario, both binary and decimal operators are incorporated. 4. Results and discussion In this paper, the wind farm layout optimization with complex land divisions is conducted with one land plot excluded, and the wind turbines can only be placed in the remaining eight land plots. The aim of this study is to examine the capability of our new constraint handling methods for both the grid based and unrestricted coordinate methods. We also consider two approaches to deciding which land plot(s) to exclude: the varying land plot scenario in which the land plot availability is included in the optimization, and the sequential land plot scenario in which the land plot availability is fixed prior to the optimization and a grid search on the land plot is used to determine the optimally-excluded plot.
of turbines optimized. When the number is small, e.g., less than 20, the discrepancy of fitness values is trivial and hard to distinguish. As the number increases, the unrestricted coordinate method clearly yields better solutions. As for the variation of fitness value as a function of the number of turbines optimized, they all share the same trend of variation for different methods, which decreases monotonically until reaching the minimal value around 40 to 50 turbines and increasing thereafter. The trend between 40 and 50 turbines is displayed in Fig. 8. In this range, but the fitness values for both grid methods display small increases while the finesses for the unrestricted coordinate method are nearly flat, indicating that the cost of energy is insensitive to the number of turbines near the optimum. Note that this is not the conclusion if one uses the gird based methods, which both indicate an increase in the optimal CoE. Then the wind farm efficiency and total wind farm power output results are shown in Fig. 9. The wind farm efficiency is defined as the actual wind farm power output divided by the theoretical
4.1. Varying land plot optimization with different methods The presentation of wind farm layout optimization results begins with varying land plot scenario. The fitness value results are shown in Fig. 8 for different optimized number of turbines. In order to consider the situation of the optimization under a wide range of turbine numbers, the top figure shows the optimized results for 10 to 80 turbines. We do not consider any more turbines since there are only 100 available positions for 10 10 grid based method. Due to the presence of randomness in the GA optimization, all the optimizations were repeatedly performed at 5 times with the same GA parameters and solutions with the best fitness are reported. For different optimization methods, it is found that the unrestricted coordinate method achieves the best results (lowest fitness values) in all cases. Meanwhile, the differences of optimization results between different methods are dependent on the targeted number
Fig. 10. Computational time with different number of wind turbines for the three methods.
L. Wang et al. / Renewable Energy 105 (2017) 30e40
Fig. 11. Fitness value (CoE) results of the sequential land plot scenario for different number of turbines optimized with the three methods, i.e., (a) 15 wind turbines, (b) 45 wind turbines and (c) 75 wind turbines.
power output excluding wake losses. Unsurprisingly, this efficiency decreases as the number of turbines increases according to the figure. Furthermore, the optimization results using unrestricted coordinate method has higher efficiency with less wake power losses compared to the grid based methods. Together with the fitness results in Fig. 8, it is clear that the unrestricted coordinate method achieves smaller fitness values by using the extra freedom to assign turbines to positions. Furthermore, this capability leads to significant power increases and CoE reduction for more than 20 turbines. Considering that the optimal number of turbines is around 40 (see Fig. 8), this performance difference is important. The computational time with different methods as a function of the number of turbines optimized is plotted in Fig. 10 to compare the effectiveness of the optimization methods more rationally. Obviously, the time increases as the optimized number of turbines increases. When the number is small (less than 40), the optimization time for different optimization methods are close to each other. When the number increases, the 10 10 grid based method is the most computationally costly, followed by the unrestricted
37
Fig. 12. Percentage of total power increase of 20 20 grid based and unrestricted coordinate methods compared to 10 10 grid based method for different number of turbines optimization: (a) 15 wind turbines, (b) 45 wind turbines and (c) 75 wind turbines.
coordinate method and then the 20 20 grid based method. The higher computational cost for the 10 10 grid is due to the low resolution of the grid yielding a high number of infeasible solutions which require further processing by the constraint handling method. 4.2. Sequential land plot optimization with different methods The results of sequential land plot optimization are shown in Fig. 11 for the fitness values with different nonparticipating land plots (from 1 to 9). Since the study of sequential land plot optimization demands the wind farm to be optimized considering every single land plot to be unavailable in series, the optimization with the wide range of turbines like the varying land plot optimization situation would be very time-consuming. Three different number of turbine situations are considered that are with small number (15), medium number (45) and large number (75). With 15 turbines, the fitness values are close to each other for the wind farm
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
layout optimization with the different methods, indicating that the selection of optimization methods has little influence on the optimization results. When the number of turbines increases, differences in the optimized fitness values become more pronounced for the different methods (note the different scale of y-axes). Like the above varying land plot scenario, the unrestricted coordinate method also achieves the best optimization results for this sequential land plot scenario, followed by the 20 20 grid based method, and the results using the 10 10 grid based method is still the worst. The increase of power output by utilizing 20 20 grid method and unrestricted coordinate method with respect to 10 10 grid method are shown in Fig. 12. Note that the wind farm efficiency has the same value percentage increase since the theoretical power output are identical for different nonparticipating land plots. As can be seen, the power output increases quite strongly as the optimized number of turbines increases. When the number is small, the power output increase with the unrestricted coordinate method is always positive even though the value is relatively small. For the 20 20 grid based method, however, the power output increase can be even negative at certain points (land plot 5 and land plot 8), meaning that the result is inferior to that optimized by the 10 10 grid based method. In order to assess the importance of considering complex land divisions in evaluating the importance of different land plots, it is instructive to compare the results to a simple optimization-andexclusion approach (which currently-available methods can handle). Firstly, the whole wind farm layout is optimized including all land plots. When a land plot is made unavailable, a naive evaluation of the new wind farm cost can be obtained by simply excluding the misplaced wind turbines and re-calculating the CoE. This method is hereafter referred to as the naive exclusion method. The comparison between the sequential land plot optimization method and the naive exclusion method is shown in Fig. 13, Since the unrestricted coordinate wind farm design method is superior according to the previous discussion, all the results are obtained using the unrestricted coordinate method. For all three different numbers of wind turbines studied, it is evident that the proposed sequential land plot optimization method achieves a much smaller cost of energy production than the naive exclusion method. The fitness of different nonparticipating land plots using the sequential land plot method have small variations, while the change is much bigger for the naive exclusion method. Such a difference in performance is practically important; this discrepancy would cause wind farm planners to over-estimate the importance of any given land plot. In contrast, the complex land division optimization enables a more accurate assessment of the impact of excluding a nonparticipating land plot on the economic performance of the whole wind farm. 4.3. Comparison between varying and sequential land plot scenario Fig. 13. Effectiveness of the proposed sequential land plot method by comparison to the naive infeasible wind turbine exclusion method with (a) 15 wind turbines; (b) 45 wind turbines and (c) 75 wind turbines.
Based on the above discussion on the results of varying and sequential land plot scenario, the comparison between the two different optimization studies is made in this section. The fitness
Table 2 Best fitness values (103) of different number of turbines optimization with the three methods for varying and sequential land plot scenario. Number of turbines
15 45 75
10 10 grids
20 20 grids
Coordinates
Varying
Sequential
Varying
Sequential
Varying
Sequential
4.18 3.483 4.187
4.18 3.48 4.143
4.172 3.444 3.821
4.175 3.429 3.767
4.171 3.377 3.63
4.169 3.363 3.601
L. Wang et al. / Renewable Energy 105 (2017) 30e40
39
Table 3 Results of the best nonparticipating land plot index for the optimization with varying and sequential land plot scenario. Number of turbines
15 45 75
10 10 grids
20 20 grids
Coordinates
Varying
Sequential
Varying
Sequential
Varying
Sequential
3 or 4 or 5 3 7
3 or 4 or 5 4 5
3 or 4 or 5 7 6
3 or 4 or 5 5 4
3 or 5 7 7
3 or 5 or 8 4 7
results with different optimization methods are tabulated and shown in Table 2. When applying 10 10 grid based method, the fitness values with varying and sequential land plot scenarios are equal. As the number of turbines increases, however, the sequential land plot scenario becomes increasingly superior with approximately 0.1% and 1.1% improvement of the fitness values for medium (45) and large number (75) of turbines, respectively. For the 20 20 grid based method, the fitness of sequential land plot scenario is even worse with small number of turbines (15), which could be possibly due to the fact that it is stuck in the local optima. When the number of turbines increases (45 and 75), the fitness of sequential land plot scenario decreases by 0.4% and 1.4%, respectively. As for the unrestricted coordinate method, the decrease in fitness values for all number of turbines has been detected although they are relatively small (0.05%, 0.4% and 0.8% for the three tested numbers, respectively). Thus, the sequential optimization procedure produces superior solutions, but must be run for each combination of excluded land plot and number of turbines. In contrast, the varying land plot scenario optimization only needs to be conducted once for each number of wind turbines, since it determines the optimal land plot(s) to exclude. For the study of wind farm optimization with lands owned by different owners, one of the primary objectives is to determine the importance of different land plots so that the least important land plot to the wind farm project can be obtained. Hence, indices of best nonparticipating land plots for different optimized number of
Fig. 14. Optimal wind farm layouts with 15 turbines, through varying land plot scenario (left figure) and sequential land plot scenario (right figure) and different optimization methods (the same for the figure below): (a) 10 10 grid method, (b) 20 20 grid method and (c) unrestricted coordinate method.
turbines with the two different optimization studies are indicated in Table 3. For the two grid methods, the land plot indices with the two studies are exactly the same when the number of turbines is small. Since the optimized wind turbines are less, three land plots are equally unimportant to the project (plots 3, 4, and 5). When the number of turbines increases, the least important land plot becomes unique, though it varies with the number of turbines studied. For the unrestricted coordinate method, the least important land plot indices are also multiple while they are different for the two different land plot scenarios and differ from the grid method results as well. Of course, since the sequential land plot scenario using the unrestricted coordinate method yields the best CoE, it’s identification of the least important plots are the most trustworthy. We also see that despite the small differences in CoE, the identified least important land plot is apparently quite sensitive to these small differences and it is thus important to utilise the most accurate method available to conclude which plots are the least important. 4.4. Optimal wind farm layouts In this section, results of optimal wind farm layouts for different number of turbine optimization through the varying and sequential land plot scenarios are presented using the three different optimization methods. First, the optimal layouts with less turbines are shown in Fig. 14, in which the wind turbines are located in the center of grids marked with black color for grid based method, while they are represented
Fig. 15. Optimal wind farm layouts with 75 turbines through varying land plot scenario (left figure) and sequential land plot scenario (right figure) and different optimization methods (the same as figure above).
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L. Wang et al. / Renewable Energy 105 (2017) 30e40
by black circles for the unrestricted coordinate method. The nonparticipating land plots are shaded with grey color. For 10 10 grid based method, the optimal layouts with the two land plot scenarios are exactly the same. The wind turbines are distributed evenly with 10 rotor diameters away from each other along row 1, row 3 and row 10. For 20 20 grid based method, the optimal layout with varying land plot scenario are organized with 4, 5 and 6 turbines in rows 1, 5 and 20, respectively, but the layout is abnormal for sequential land plot as the result is suboptimal. For the unrestricted coordinate method, even though the layouts are irregular in both scenarios, they are similar to each other and spreading along the diagonal of wind farm which is perpendicular to the dominant wind direction. Next, the optimal wind farm layouts with more wind turbines are presented in Fig. 15, in which the wind turbines are represented in the same way as the above figure for the two methods. Unlike the regular wind farm layout pattern for a small number of turbines, the wind turbines occupy most of the potential locations for the 10 10 grids, while they are similarly distributed for both varying and sequential studies. The optimal layouts for 20 20 grid based method also show similar patterns: the wind turbines are staggered and spread along every other row. The wind turbine distributions by the grid based method are quite different for the two different land plot scenarios since the nonparticipating land plot indices differ. However, they have the same nonparticipating land plots for the optimization with the two land plot scenarios by the unrestricted coordinate method. They share very similar optimal layouts which are generally aligned with the diagonal normal to the dominant wind direction.
5. Conclusions In this work, the problem of the wind farm layout optimization with complex land plot divisions is addressed. Due to the inapplicability of traditional penalty technique for the wind farm optimization with complex land divisions, a new constraint handling technique is proposed to be applied to cope with both proximity constraint and land plot constraint. The means of fitness evaluation for different optimization methods incorporating the new constraint handling technique have been presented. In addition, different optimization approaches including three wind farm design methods in conjunction with two land plot scenarios have been developed in terms of the representation of optimization solutions and different types of constraints imposed on the objective function. The results show that unrestricted coordinate method yields the best optimization results with lower fitness values and higher wind farm efficiency for both varying and sequential scenarios. The discrepancy between different optimization methods increases with the number of turbines optimized. When the number is relatively big, the sequential scenario is always able to produce better optimization results than the varying scenario for all methods. When the number is small, however, it varies depending on the optimization methods applied. For small number of turbines, the results indicate that the designer has significant freedom in exclusion of different land plots since there can be a number of optimal unavailable land plots. For a larger number of turbines, the best unavailable land plot tends to become unique and is sensitive to the optimization accuracy. Based on the above discussion, it is concluded that the sequential land plot optimization study with unrestricted coordinate method is the best selection which yields the most reliable results for the complex land divisions.
Acknowledgements The High Performance Computer resources provided by Queensland University of Technology (QUT) are gratefully acknowledged. Also the financial support of China Scholarship Council (CSC) from Chinese government and Top-up scholarship from QUT are greatly appreciated.
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