2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Modified Genetic Algorithm for Layout Optimization of Multi-type Wind Turbines Bin Duan1 , Jun Wang1∗ and Huajie Gu1 Abstract— The wind farm micro-siting is an important strategy for reducing the cost of wind energy. In this paper, two different types of wind turbines are considered for a wind farm to take the full advantage of wind resources at different altitudes. The phenotype of the problem is described by an integer encoding method, and a repair operator of genetic algorithm is proposed to handle the position constraint. The optimal objective is to maximize the net present value of a wind farm under a certain initial budget. Simulation results demonstrate the effectiveness of the proposed algorithm and the significance of planting multi-type wind turbines.
I. INTRODUCTION Nowadays, wind energy has received increasing attention due to its availability, low cost and environment-friendly operation. Wind energy has superior advantages over conventional energy resources. For one thing the gradually diminishing fossil fuel resources have caused adverse effects on the environment, and for another wind resources are inexhaustible and emission-free. According to the data in [1], worldwide nameplate capacity of wind-powered generators was 282,482 megawatts at the end of 2012, growing by 44 gigawatts over the preceding year. Moreover, wind energy has been widely developed in Europe, Asia, and the United States. The wind farm micro-siting, which is an approach for optimizing the layout of wind turbines in a wind farm, is one of the most effective strategies for reducing the cost of wind energy and pursuing greater economic profits. A few algorithms were proposed for the wind turbine layout optimization, among which the evolutionary algorithms were frequently adopted. Mosetti et al. [2] first utilized the binary-coded genetic algorithm to optimize the wind turbine placement by extracting the maximum energy for the minimum installation costs, and their study were revised by Grady et al. [3] for better results. Mora et al. [4] introduced the net present value (NPV) to calculate the revenue of a wind farm instead of the traditional cost model. Wan et al. [5] improved the wind and turbine models, and Emami et al. [6] proposed the matrix chromosome to improve the optimization algorithm. Empirically, a cluster of turbines should be placed apart with a distance of more than three times that of a rotor diameter to reduce the wake losses [7]. The common way to deal with this problem is to divide the wind farm into finite uniform square grids, which turns the constrained problem into an unconstrained one.
It is generally known that the higher the altitude, the faster the wind speed. In the above studies, only one type of wind turbine was considered on a wind farm. In order to make full use of wind resources at different altitudes, two types of wind turbines with different hub heights and diameters are taken into account in this paper. An integer encoding approach rather than the traditional binary-coded one is utilized to solve the specific layout problem. A novel operator of the genetic algorithm named “repair operator” is also proposed to satisfy the position constraint of different turbines. The remainder of the paper is organized as follows. Section II formulates the models for the wind farm micrositing and gives a brief introduction to the two different wind turbine types. Section III describes the encoding method and the operators of genetic algorithm. Section IV demonstrates the simulation results and analyses the advantage of the proposed algorithm. Section V makes concluding remarks and points out future work. II. MICRO-SITING MODELS On a wind farm, different turbines may have interactions due to the wake effect that upstream turbines reduce the wind speed of downstream turbines. A wake model similar to the Jensen’s analysis [8] is used to simplify the calculation of the wake effect in this paper. Fig. 1 is the schematic of the wake effect, which roughly shows the expansion of the wake effect area downstream. The increasing wake diameter Dd is proportional to the distance d between the downstream turbine and the upstream one. Based on the momentum conservation and Betz theory, the wind speed of the downstream turbine can be described by [8], [10] ( )2 ) ( √ ( ) D u˜k (d) = uk 1 − 1 − 1 −CT (1) D + 2α d where u˜k (d) is the wake speed generated by the Turbine k at the downstream distance d, uk is the initial wind speed of
1 Bin Duan, Jun Wang and Huajie Gu are with the Dept. of Control Science and Engineering, Tongji University, Shanghai 201804, P. R. China. ∗ Corresponding author, Email:
[email protected].
978-1-4799-3274-0/$31.00 ©2014 AACC
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u0
D uk
Dd
D
Turbine k
uk (d )
D
x k , yk
A k ,i
x i , yi
d Turbine i
Fig. 1.
Schematic of wake effect [9]
the Turbine k, CT is the thrust coefficient, D is the diameter of the turbine, α is the wake spreading constant related to the hub height h of the turbine and the surface roughness z0 of the farm. The calculation formula of α is [3] 0.5 α= (2) ln(h/z0 ) Suppose that the downstream turbine is affected by the wake effect from only one upstream turbine. According to the theory of kinetic energy balance, the initial wind speed of the Turbine i affected by the Turbine k can be calculated by [9] ( )) ( u˜k (d) (3) ui = uk 1 − qk,i 1 − u0 where qk,i is the velocity deficit proportion of the Turbine k on the Turbine i, and it is defined by Ak,i (4) A0 where A0 is the swept area of the turbine rotor, Ak,i is the overlapped area between the wake area of the Turbine k at the downstream distance and the rotor of the Turbine i, i.e. the shadow area in Fig. 1. When the downstream turbine is affected by the wake effect from several upstream turbines, the initial wind speed of the Turbine i can be calculated by [9] v ( u N ( ( ))2 ) u u ˜ (d ) k k,i ui = u0 1 − t ∑ qk,i 1 − (5) u0 k=1,k̸=i qk,i =
where u0 is the free wind speed of the whole farm, dk,i is the distance between the kth and ith turbines along the wind direction, and u˜k (dk,i ) is the wake speed of the kth turbine where the ith turbine locates. On a real wind farm, the complicated wind regime that consists of wind speed, wind direction and the probability for each direction at each wind speed should be investigated, which is the fundamental precondition to calculate the initial wind speed of each turbine. Combined with its power curve, the power output of the wind turbine can be evaluated approximately. Let uin be the cut-in wind speed of the turbine and uout the cut-out wind speed. If the initial wind speed of the turbine lies in the range between uin and uout , the power output can be calculated based on its power curve by interpolation. Otherwise, the turbine will stop working and the power output is 0. The V 80 wind turbine from Vestas [11] and E33 wind turbine from Enercon [12] are selected for the simulation, which can be regarded as the “megawatt wind turbine” and the “kilowatt wind turbine” respectively. To illustrate the distinction of the two types of wind turbines, their power curves and characteristics are shown in Fig. 2 and Table I respectively. The differences between the two types are so obvious in both the hub height and the rotor diameter that the wake effect caused by one type hardly influence the other, that is to say, the wake effect interactions between different types of wind turbines can be ignored.
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V80 E33
Power (kW)
1500
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0 0
5
Fig. 2.
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Power curves of both types [11], [12]
In order to distinguish the initial cost of different wind turbine types, their capacity-weighted average project costs are taken into consideration in this paper. The statistics in [13] contain the capacity-weighted average project costs of the selected turbines mentioned above, which are also listed in Table I. Then their initial costs can be calculated approximately. The objective in the optimization is to maximize the NPV [14] under a certain initial budget B, and it is defined by NPV = −IW F + ∑LT k=1 AEP ·
pkW h (1+∆pkW h )k+1 , (1+r)k
IW F ≤ B (6) where IW F is the initial cost of the wind farm, LT is the life time of the wind farm, AEP is the annual energy production, pkW h is the energy cost per kilowatt-hour, ∆pkW h is the annual increment of pkW h , and r is the interest rate. The wake loss of a wind farm is defined in the following to evaluate the efficiency of a wind farm [15] Wake loss =
AEPg − AEPw × 100% AEPg
(7)
where AEPw and AEPg are the annual energy production of the wind farm with and without taking into account the wake effect under the same condition. III. MODIFIED GENETIC ALGORITHM Genetic algorithms are globally probabilistic search techniques based on the principle of natural genetics. These algorithms combine the mechanics of natural selection and survival of the fittest to find the optimum solution for complex problems. Compared with other calculus-based methods, genetic algorithms have great advantages over them because of the robustness and global searching. In an optimization problem solved by genetic algorithms, a population of candidate solutions represented by chromosomes is evolved toward better solutions. Traditionally,
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TABLE I CHARACTERISTICS OF THE TURBINES
Wind turbines Rated power (kW) Hub height (m) Rotor diameter (m) Wind turbine cost ($/kW)
[11], [12], [13]
V80 2000 100 80 2100
E33 335 37 33.4 2500
chromosomes in the population are coded in binary as strings of 0s and 1s, but other possible encoding methods can also be adopted for specific problems. A. Codification In [2], [3], the authors studied the wind farm micro-siting problem on a flat square farm divided into 10×10 cells. The width of each cell was five times turbine rotor diameter and its center was regarded as the possible position for turbines to place. Nevertheless, in the present study, the layout of two different types of wind turbines is considered on a wind farm. The scale of the wind farm is decreased and divided into 7×7 cells for simplification, and the width of each cell is four times the diameter of the kilowatt wind turbine. In general, a typical genetic algorithm requires the genotype of the solution domain by encoding and the evaluation of the solution domain by the fitness function. In the previous literatures, the square wind farm was represented by a binary bit string, in which “1” represented that a turbine was placed while “0” represented no turbine. The relevant information of the placed turbines could also be obtained by the bit string, including their numbers and positions, and then the performance of each solution could be evaluated by the fitness function. However, when it comes to two types of wind turbines, the previous method using “0” and “1” to represent whether a turbine is placed in the cell is not suitable for describing it clearly. In order to solve this problem, two encoding methods are introduced in the current study. The first one is similar to the traditional binary-coded method but the length of the string is doubled, and the turbine information of each cell is represented by two bits. More specifically, “00” and “11” represent that no turbine is placed in the cell, while “01” and “10” represent the megawatt and kilowatt turbine respectively. However, this method may bring about lower calculation and convergence rate for the doubled length of the chromosome and redundant genes. Another integer encoding approach is proposed that the chromosome is made up of “0”, “1” and “2”, representing no turbine, the megawatt turbine and the kilowatt turbine in a cell. By contrast, the latter method has the advantage over the previous one for the shorter length of the chromosome and nonexistent redundant genes, and it also has the less computation time and better fitness values. Besides, this encoding method is problem-based and extensible, that is to say, the placement of more types of wind turbines can also be optimized by this integer encoding method. In the present study, the square farm divided into 49 cells is used as the computational domain, so both the number of independent variables and the length of the chromosome are 49.
For each new solution to be produced, a pair of parent solutions should be selected for breeding from the pool. In this simulation, the type of selection operator is set as “Stochastic Uniform Selection” for selecting potentially useful solutions. The crossover operator set as “Uniform Crossover” is applied to generate offsprings that inherit the mix of the chromosomes from the two selected parents, and an improved bit-flip mutation operator is utilized that inverts the selected genome bit to the other alternative bits with the same probability. These operators help the population to avoid the trap of local minima and maintain diversity by preventing the chromosomes from becoming too similar to each other. In this paper, the width of each cell is set to be four times the diameter of the kilowatt turbine to handle its position constraint, but this distance cannot satisfy the spacing requirements with regard to the megawatt turbine because of its larger diameter. A “repair operator (RO)” is proposed to tackle the constraint problem instead of traditional penalty functions. The width of two cells is larger than three times that of the diameter of the megawatt turbine, so if no turbines are placed around the cell where a megawatt turbine locates, its position constraint can be satisfied. The process of the repair operator is explained as follows. •
•
•
After crossover and mutation, the chromosome string is arrayed into the wind farm cells. Fig. 3 illustrates the process of the array; In the matrix obtained in the process one, every cell in the matrix should be considered in a anticlockwise way outside-in because the turbine near the boundary of the wind farm is more likely to be a upstream turbine, which may capture more wind energy than that in the middle of the farm. If a megawatt wind turbine is placed in a cell of the wind farm, no turbines should be placed around the cell. That is to say, if the number in the current cell is “1”, the number in the cells around it will be set to “0” to satisfy the position constraint. A specific example of this process is demonstrated in Fig. 4, and the direction of the dashed line describes the current cell to be repaired.
Bit 1
Bit 2
B. Operators In the genetic algorithm, the initial population is set either randomly or heuristically, and genetic reproduction is performed by means of a few fundamental genetic operators to evolve into better individuals. 3635
Fig. 3.
Bit 3
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Ă
Ă
Bit 43
Bit 44
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Bit 49
Ă
Ă
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Bit 8
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Bit 14
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Bit 7
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Process one of the repair operator
Bit 49
cases. The parameters in the simulation are listed in Table II, and the currency unit has been exchanged from the euro to the dollar to unify the currency unit in [13] and [14]. Fig. 5 illustrates the wind farm configurations of the three cases with the repair operator, and the two types of turbines are differentiated by the size of the turbine motif in the figure.
Repair operator
Fig. 4.
(a) Case 1
Repair process
With the repair operator, the algorithm can overcome the problems that traditional penalty function may cause. For one thing the repair operator saves the computation time in the algorithm, and for another the repair operator preserves the diversity of the population because it just repairs the part violating the position constraint instead of adding a penalty to the whole chromosome. Theoretically, the chromosome tackled by the penalty function is less likely to survive into the next generation regardless of other parts that may have superior genes, which may bring about negative effects to the whole evolution.
(b) Case 2
IV. SIMULATION RESULTS In this paper, wind speeds at different altitudes should be considered because of the different hub heights of the turbines, which is known as the wind shear effect. The thrust coefficient CT is set to be a constant 0.88 throughout the process for simplification and the surface roughness of the site z0 is 0.3 m. According to [16], once the wind speed at a given reference height hre f is given, the corresponding wind speed at another height h can be calculated by an exponential function: ln(h/z0 ) (8) v(h) = v(hre f ) ln(hre f /z0 ) As with the previous literatures, three cases of different wind conditions are investigated in the present study. The wind conditions are the same as that in [3] at the altitude of 50m, and the wind speeds can be calculated at the hub heights of both types of turbines. A population of 100 individuals is allowed to evolve over 2000 generations in this 3636
(c) Case 3 Fig. 5. •
•
Wind farm configurations with the repair operator
Case 1: A uniform wind direction with a constant wind speed of 12 m/s at the height of 50m is considered. As is demonstrated in Fig. 5(a), eleven megawatt turbines and four kilowatt turbines are planted in four rows respectively so as to reduce the wake losses to the minimum, and the layout of the wind turbines is symmetric in the wind farm. Case 2: This case is very similar to the previous one for the same wind speed, and the only difference is that the uniform wind direction is extended to four cardinal
•
directions with the equal probability of occurrence. The optimal solution is shown in Fig. 5(b), and eleven megawatt turbines and two kilowatt turbines are planted symmetrically with regard to the center of the wind farm. Case 3: A more realistic wind condition that multidirectional wind with different wind speeds of 8, 12 and 17 m/s is taken into consideration at the height of 50m. Fig. 5(c) is the optimal configuration that eleven megawatt turbines and four kilowatt turbines are planted in the wind farm. As is shown in the figure, the wind turbines are planted dispersively to reduce the wake effect from each direction.
(a) Case 1
In order to demonstrate the effectiveness of the repair operator for this optimization problem, the same cases are optimized without the repair operator and the position constraint is tackled by the penalty function. Fig. 6 shows the simulation results of the three cases without the repair operator. The comparison of the algorithm with the repair operator (RO) and without it (NRO) is listed in Table III. It can be manifested that the algorithm with the repair operator performs better in the net present value, and its initial investment is more likely to approach the initial budget of the wind farm. When the algorithm without the repair operator is adopted to optimize the problem, the distance between two turbines should be computed in each chromosome every generation. And if the calculated spacing violates the position constraint, a penalty function is utilized in this chromosome to tackle it. In this way, megawatt turbines are less likely to be planted in the wind farm because of their higher probability of violating the constraints, which leads to less initial investment of the wind farm. Besides, the layout of the turbines is more reasonable and brings less wake losses than the algorithm without the operator. Fig. 7 compares the fitness curves of both algorithms, and the objective value is set to be 0 if the calculated NPV is a minus because of the penalty function. Since the genetic algorithm with RO is fit for solving the problem, the fitness curve outperforms in both convergence rate and convergence value during the whole optimal stage. V. CONCLUSIONS In this paper, an integer encoding approach and a repair operator of genetic algorithm are utilized to optimize the layout of two types of wind turbines. The significance of planting multi-type wind turbines in a wind farm is demonstrated to make full use of wind resources at different TABLE II VARIABLES IN THE SIMULATION [14] Parameters Life span of the wind farm LT Energy price pkW h Yearly increase of energy price ∆pkW h Interest rate r Initial budget B
Values 20 0.0945 3% 6% 50
Units years $/kWh M$
(b) Case 2
(c) Case 3 Fig. 6.
Wind farm configurations without the repair operator
altitudes. Simulation results indicate that the genetic algorithm with the repair operator always performs better under different wind conditions. It is also manifested that megawatt wind turbines always predominate over the kilowatt ones for the lower capacity-weighted average project cost, while the kilowatt turbines can be regarded as supplementaries to the megawatt ones because of the fixed budget of the wind farm. In addition, different types of wind turbines may bring about less wake losses and take full advantage of the field in a relatively small-scale wind farm. In the future, some simplified models should be perfected, including the cost model and the objective function. What’s more, it is of great importance to study more realistic situations, and complex wind conditions and terrains should be taken into consideration so as to apply this micro-siting approach into actual projects.
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ACKNOWLEDGMENT TABLE III C OMPARISON OF THREE CASES Case 1 Case 2 RO NRO RO NRO NPV (M$) 245.9 141.9 228.1 134.1 Initial cost (M$) 49.6 36.9 47.9 34.4 Number of MT 11 5 11 5 Number of KT 4 19 2 16 Wake loss (%) 0.2 11.3 3.2 10.9 MT megawatt turbine, KT kilowatt turbine Cases
Case 3 RO NRO 220.3 148.8 49.6 37.8 11 5 4 20 4.2 5.7
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This work was supported in part by the National Natural Science Foundation of China under Grant No. 61075064, and by the International Science and Technology Co-operation Program of China under Grant No. 2011DFG13020. R EFERENCES [1] Global Wind Energy Council, “Global wind report: annual market update 2012,” GWEC Report, April, 2012. [2] G. Mosetti, C. Poloni, and B. Diviacco, “Optimization of wind turbine positioning in large wind farms by means of a genetic algorithm,” Journal of Wind Engineering and Industrial Aerodynamic, vol. 51, no. 1, pp. 105–116, 1994. [3] S. A. Grady, M. Y. Hussaini, and M. M. Abdullah, “Placement of wind turbines using genetic algorithms,” Renewable Energy, vol. 30, no. 2, pp. 259–270, 2005. [4] J. C. Mora, J. M. C. Bar´on, J. M. R. Santos, and M. Pay´an, “An evolutive algorithm for wind farm optimal design,” Neurocomputing, vol. 70, no. 16-18, pp. 2651–2658, 2007. [5] C. Wan, J. Wang, G. Yang, X. Li, and X. Zhang, “Optimal micrositing of wind turbines by genetic algorithms based on improved wind and turbine models,” in proceedings of The 48th IEEE Conference on Decision and Control, Shanghai, China, 2009, pp. 5092–5096. [6] A. Emami and P. Noghreh, “New approach on optimization in placement of wind turbines within wind farm by genetic algorithms,” Renewable Energy, vol. 35, no. 7, pp. 1559–1564, 2010. [7] M. Patel, Wind and power solar systems. CRC Press, 1999. [8] N. O. Jensen, “A note on wind turbine interaction,” Denmark: Risø National Laboratory, 1983. [9] C. Wan, J. Wang, G. Yang, H. Gu, and X. Zhang, “Wind farm micro-siting by gaussian particle swarm optimization with local search strategy,” Renewable Energy, vol. 48, pp. 276–286, 2012. [10] I. Katic, J. Højstrup, and N. O. Jensen, “A simple model for cluster efficiency,” in European Wind Energy Association Conference and Exhibition, Rome, Italy, 1986, pp. 407–410. [11] Vestas wind systems, http://www.vestas.com, 2012. [12] ENERCON GmbH, “ENERCON Wind energy converters Product overview,” http://www.enercon.de, 2012. [13] M. Bolinger, “2011 wind technologies market report,” Lawrence Berkeley National Laboratory, Tech. Rep., 2013. [14] J. S. Gonz´alez, A. G. G. Rodriguez, J. C. Mora, and J. R. Santos, “Optimization of wind farm turbines layout using an evolutive algorithm,” Renewable Energy, vol. 35, no. 8, pp. 1671–1681, 2010. [15] H. Gu and J. Wang, “Irregular-shape wind farm micro-siting optimization,” Energy, 2013. [16] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind Energy Handbook. John Wiley & Sons Ltd, 2005.
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Comparisons of optimization processes
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