Improved non-dominated sorting genetic algorithm (NSGA)-II in multi-objective optimization studies of wind turbine blades

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1 Appl. Math. Mech. -Engl. Ed., 32(6), (2011) DOI /s x c Shanghai University and Springer-Verlag Berlin Heidelberg 2011 Applied Mathematics and Mechanics (English Edition) Improved non-dominated sorting genetic algorithm (NSGA)-II in multi-objective optimization studies of wind turbine blades Long WANG ( ), Tong-guang WANG ( ), Yuan LUO ( ) (Jiangsu Key Laboratory of Hi-Tech Research for Wind Turbine Design, Nanjing University of Aeronautics and Astronautics, Nanjing , P. R. China) (Communicated by Wen-rui HU) Abstract The non-dominated sorting genetic algorithm (NSGA) is improved with the controlled elitism and dynamic crowding distance. A novel multi-objective optimization algorithm is obtained for wind turbine blades. As an example, a 5 MW wind turbine blade design is presented by taking the maximum power coefficient and the minimum blade mass as the optimization objectives. The optimal results show that this algorithm has good performance in handling the multi-objective optimization of wind turbines, and it gives a Pareto-optimal solution set rather than the optimum solutions to the conventional multiobjective optimization problems. The wind turbine blade optimization method presented in this paper provides a new and general algorithm for the multi-objective optimization of wind turbines. Key words wind turbine, multi-objective optimization, Pareto-optimal solution, nondominated sorting genetic algorithm (NSGA)-II Chinese Library Classification TM Mathematics Subject Classification 76N15 1 Introduction The wind turbine blade design in the field of multi-objective optimization involves many disciplines, such as aerodynamics, aeroelastics, structural mechanics, and aeroacoustics. It is also related to many influencing factors, such as constraints, optimization variables, and objectives, among which there may exist conflicts. To reduce the design complexity, singleobjective optimization algorithms are usually implemented for the wind turbine blade design with the maximum power coefficient or the annual power generation taken as the optimal objective. Over the years, a number of methods have been suggested for the wind turbine blade design. Typically, Wilson and Lissaman [1] suggested a classic design method applying the blade element momentum (BEM) theory. In the 1990s, a method based on genetic algorithms was brought into the field of wind turbine blade design by Selig and Coverstone-Carroll [2], which has been drawn more attention since then. In the aspects of the multi-objective optimization design of wind turbine blades, Wood [3] and Sale et al. [4] simplified the multi-objectives into the singleobjective by the objective weighted method. Then, the single-objective genetic algorithms were Received Jan. 15, 2011 / Revised Apr. 14, 2011 Project supported by the National Basic Research Program of China (973 Program) (No. 2007CB714600) Corresponding author Long WANG, Ph. D., wl2007@nuaa.edu.cn

2 740 Long WANG, Tong-guang WANG, and Yuan LUO used, and good results were obtained. However, fundamentally, this nominal two-objective optimization approach is a single-objective concept and unable to appropriately solve multiobjective optimization design problems in practice. Therefore, it is necessary to introduce new methodology and optimization algorithms to the multi-disciplinary design of wind turbine blades. There is no single solution that is optimal (global minimum or maximum) with respect to all the optimization objectives under multi-constraints for the multi-objective optimization design contrasted to the single-objective one, and only acceptable non-dominated solutions exist, which are the so-called Pareto optimal solutions. The approaches for solving conventional multiobjective design problems include the objective weighted method, the hierarchical optimization method, the ε constraint method, the goal programming method, etc. [5]. These algorithms deal with the problems converting the multi-objective problem to the single-objective problem with the emphasis of one particular Pareto-optimal solution at a time. Actually, they are not real multi-objective optimization algorithms and cannot be widely used in solving the complex problems. Over the past decades, to solve the complicated multi-objective optimization problems and achieve the Pareto-optimal solution set, a number of multi-objective evolutionary algorithms have been suggested, such as the niched Pareto genetic algorithm (NPGA) [6], the Pareto archived evolution strategy (PAES) [7], the strength Pareto evolutionary algorithm (SPEA) [8], the neighborhood cultivation genetic algorithm (NCGA) [9], and the non-dominated sorting genetic algorithm (NSGA)-II [10]. Among them, the recently well-known NSGA-II (i.e., the fast and elitist non-dominated sorting genetic algorithm) has drawn more attention due to its feasibility for any number of objectives although it has not been used for the wind turbine blade optimization. In this paper, the NSGA-II is introduced and improved with the controlled elitism and dynamic crowding distance (DCD). This method is then applied to the wind turbine blade design taking the maximum power coefficient and the minimum blade mass as the optimization objectives. As an example, the designs of the 5 MW wind turbine blades are given, and the optimization results are analyzed for completeness. 2 Modification of NSGA-II The NSGA [11] proposed by Srinivas and Deb is an efficient sorting algorithm and developed from the simple genetic algorithm only in the way of the selection operator. The NSGA can simplify multi-objective optimization problems into one fitness function problem and also be used for any objective of optimization problems. The main ideas of the NSGA are as follows: First, the individuals of the population must be classified into different non-dominated sets according to the non-domination relationship. Second, to maintain the diversity in the population, sharing methods are adopted to control the individual distribution of every non-dominated set. Third, the selection operator is performed on the basis of an individual s domination set and niche count. Finally, combined with other operators of the simple genetic algorithm, the Pareto-optimal solutions can be obtained. Deb et al. carried out a substantial improvement of the NSGA and proposed a fast and elitist NSGA (i.e., NSGA-II) [10] with the following three aspects of improvement: (i) a fast non-dominated sorting approach taken to reduce the computational complexity from O(MN 3 )too(mn 2 ), where M is the number of the objectives, and N is the population size, (ii) the elitism strategy introduced to improve the robustness and convergence, (iii) the sharing methods replaced by the crowding-distance methods to avoid specifying the sharing parameter δ share. Combining with the tournament selection method, the simulated binary crossover (SBX) operator, the polynomial mutation operator [12], and the efficient constraint-handling approach,

3 Improved NSGA-II in multi-objective optimization studies of wind turbine blades 741 the NSGA-II displays good convergence and robustness and becomes one of the benchmarks of multi-objective optimization algorithms. To improve the lateral diversity and the uniform diversity of non-dominated solutions, both the controlled elitism strategy [13] and the DCD approach [14] areemployedinthensga-iiin this paper. 2.1 Controlled elitism Ensuring that the diversity of the population is crucial to the success of evolutionary algorithms. Although the NSGA-II uses an efficient crowding-distance approach to control the distributions of the individuals, the local convergence can occur in dealing with multi-modal problems. This is because the crowding distance approach of the NSGA-II just works within each non-dominated set and only keeps the horizontal diversity. Meanwhile, the elitist preservation strategy and the tournament selection method cause rapid reproduction of the Paretooptimal solutions. Therefore, the numbers of the non-domination sets and the individuals of non-elitist sets decrease significantly. Then, the lateral diversity is lost, which leads to the local convergence shown in Fig. 1. Fig. 1 Controlled elitism procedure To laterally improve the diversity of the NSGA-II, a pre-defined controlled elitism strategy [13] is used. The combined population R t of the parent Pop t and the offspring Off t is sorted for nondomination, where t is the generation count. Then, the non-dominated set number K of the combined population is obtained. Subsequently, the allowable number of the individuals of every non-dominated set is determined according to Eq. (1). This method can maintain the lateral diversity effectively. N j = N 1 r 1 r K rj 1, (1) where N j is the maximum allowable number of the jth non-dominated set, N is the population size, and r [0,1] is the reduction rate. 2.2 Dynamic crowding distance Maintaining the diversity of the population not only requires a certain space distance between individuals but also needs good uniformity for the non-dominated sets. To evaluate the distribution of the individuals, the crowding distance D Ci is calculated according to Eq. (2) in the NSGA-II. D Ci = 1 N obj fi+1 m f i 1 m N obj f m m=1 max fmin m, (2) where fi+1 m and f i 1 m are the (i+1)th and (i 1)th fitness values in the mth objective, respectively; fmax m and fmin m are the maximum and minimum fitness values of the mth objective in the same

4 742 Long WANG, Tong-guang WANG, and Yuan LUO non-dominated set, respectively. Moreover, the definition of the crowding distance does not consider the uniformity of the individual distributions in each non-dominated front, and, in some conditions, it may destroy the uniformity. The individual distribution of the two objectives is shown in Fig. 2. It can be seen that, for the points B and F in the Pareto-optimal solution set, the rectangular used to calculate the crowding distance D Ci of the point B has a larger aspect ratio than the point F. Therefore, it seems that the point B rather than the point F should be eliminated. However, because the individual B helps to maintain uniform spread, the point B should be maintained actually to obtain the good horizontal diversity. Fig. 2 Crowding distance of individuals To improve the uniformity of the individual distributions, a DCD approach proposed by Luo et al. [14] is used here. The DCD D DCi is calculated as follows: where V i = 1 N obj D DCi = N obj m=1 D Ci log 1 V i, (3) ( f m i+1 f m i 1 DCi ) 2 (4) is the variance of crowding distances of the individuals that are the neighbors of the ith individual. 2.3 Implementation of algorithm The controlled elitism strategy and the DCD approach have been incorporated into the NSGA-II, obtaining an improved optimization algorithm. It is described in the following steps: (i) The variable numbers, control variable limits, crossover probability and mutation probability, crossover and mutation indices, population size N, maximum number of generation and so on are chosen, and a random initial population Pop 0 is generated. (ii) For the parent population in Pop, t the tournament selection method, the SBX operator, and the polynomial mutation operator are used to generate a new offspring population O t+1 ff. (iii) Combining the offspring population O t+1 ff with the parent population Pop, t anewpopulation R t is generated, whose size is 2N. With the objective functions and constraints, the non-dominated sorting is performed to R t. (iv) For each set of R t, the maximum number N j of the individuals is determined using the controlled elitism strategy. The reduction rate is set to be (v) If the size of the jth non-dominated set M j is greater than N j,(m j N j ) individuals are eliminated according to the DCD approach. (vi) The calculation goes back to step (ii) and repeats steps (ii) (v) until the global convergence is achieved.

5 Improved NSGA-II in multi-objective optimization studies of wind turbine blades Objective functions and constraints 3.1 Mathematical models of optimization objectives The common optimization objectives of the wind turbine blade design include the maximum annual power generation under given wind speed distributions, maximum power coefficient C P at a design wind speed, minimum blade mass, minimum cost, noise control, resonance avoidance, etc. Sometimes, based on different design requirements, the thrust and power output should also be restricted [15 17]. These objectives can simultaneously be dealt with by the improved NSGA- II as long as they can be expressed properly. In this paper, the maximum power coefficient C P at the design wind speed and the minimum blade mass, between which distinct conflicts exist, are chosen as two optimization objectives to validate the developed optimization algorithm, while the computational overhead is kept minimal. (i) Maximum power coefficient at design wind speed The modified BEM theory is used in the calculation of the maximum power coefficient C P. The BEM theory is a classic model for wind turbine aerodynamics. The modifications include the tip loss, hub loss, and correction of the axial induced velocity factor at the large thrust state [18]. The objective function for the maximum power coefficient at the design wind speed of 9 m/s is as follows: f 1 =max{c P V =9 m/s }. (5) (ii) Minimum blade mass The blade structural design requires the load input. The aerodynamic loads acting on the blade are calculated using the modified BEM theory. In this case, during the preliminary design of the blade structure, the quasi-steady calculation of the blade is performed with the velocity ranging from the cut-in wind speed to the cut-out wind speed, obtaining limit loads on each cross-section of the blade. The design loads of the blade structure are usually acquired by multiplying the limit loads by a safety factor of 2.5 in engineering. The blade structure [19] is shown in Fig. 3, which is primarily made up of a double web I- beam. The material used for the bulk of the blade is the glass fiber with reinforced polyester and foam. Fig. 3 Schematic of blade section structure The thickness of the spar is set as the variables in design. To obtain the mass distribution and the total mass of the blade, normal stress equations in the condition of free bending of thin-walled beams are applied as

6 744 Long WANG, Tong-guang WANG, and Yuan LUO ε 0 E i A i + 1 ρ x (E i S xi ) 1 (E i S yi )=Fz, ρ y ε 0 E i S xi + 1 ρ x (E i I xi ) 1 (E i I xyi )=Mx, ρ y (6) ε 0 E i S yi 1 ρ x (E i I xyi )+ 1 (E i I yi )=My. ρ y In the above formulas, k is the number of the airfoil discrete points; ε 0 is the strain, and E i is Young s modulus; S xi and S yi are the static moments at the ith point in the x- and y-directions, respectively; I xyi, I xi,andi yi are the moments of inertia, respectively; ρ x and ρ y are the radii of the curvature in the yz- andxz-coordinate planes, respectively. The stress at any point of the section can be obtained as follows: ( σ i = E i ε 0 + y x ). (7) ρ x ρ y The thickness of the spar of the section can be obtained in an iterative process according to the safety factor F S = [σ], (8) K σ i i=0 where [σ] is the maximum safety stress, and K is the number of the discretized points in the section. In this design, F S =1.75. Subsequently, the linear mass M j of the jth section can be obtained. The objective function for the minimum blade mass is as follows: R f 2 =min M j dr. (9) R hub 3.2 Constraints To cover blade shapes in given samples as many as possible, 15 design variables are applied in this paper, which are corresponded with the chord c, thetwistθ, and the thickness t of 5 key sections. Meanwhile, the variables are smoothed by the fourth-order Bezier curves, and then the design variables of 55 sections are obtained by interpolation. The chord, twist, and thickness of the 5 key sections meet the following constraints: c min <c i <c max, θ min <θ i <θ max, (10) t min <t i <t max, where i = 1, 2,, 5, and the boundaries of these three types of parameters with the subscripts min and max are determined empirically according to the BEM theory. Besides the geometry constraints, the power output is restricted to the rated power output by pitch regulation when the wind speed is higher than the rated speed until the cut-out speed is achieved.

7 Improved NSGA-II in multi-objective optimization studies of wind turbine blades Optimization of 5 MW wind turbine blade 4.1 Studing conditions The maximum power coefficient C P and the minimum blade mass are taken as the optimization objectives in this paper. The improved NSGA-II is used for the optimization design of 5 MW variable-speed constant-frequency wind turbine blades, of which the aerodynamic performance is predicted and compared with the result of the NREL-5 MW offshore wind turbine given in Ref. [20]. The material property parameters used in the blade structural design are given in Table 1. The wind rotor and algorithm parameters are shown in Tables 2 and 3. Table 1 Material performance data Parameter GFRP 45 PVC foam GFRP 0 Young s modulus/pa 1.18E E E+10 Density/(kg m 3 ) 1.90E E E+03 Stretch-allowable strength/pa 4.65E E E+08 Compress-allowable strength/pa 8.08E E E+08 Table 2 Wind rotor parameters Parameter Value Number of blades 3 Radius of rotor 63 Maximum speed/(r min 1 ) 12.1 Design wind speed/(m s 1 ) 9 Rated power/mw 5 Cut-in wind speed/(m s 1 ) 3 Cut-out wind speed/(m s 1 ) 25 Airfoil family DU/NACA 63 Parameter Table 3 Parameters used in NSGA-II Value Population size 100 Crossover probability 0.8 Mutation probability 0.05 Crossover index 2 Mutation index 20 Maximum number of generation 500 Reduction rate Optimization design results Figure 4 shows the Pareto-optimal solutions taking the maximum power coefficient C P and the minimum blade mass as the optimization objectives at the design wind speed. The curve of the Pareto-optimal solution set denoted by circles in the figure illustrates a monotone increasing trend and divides the optimal region into two parts, i.e., parts I and II. Part I is an ideal solution region that cannot be reached under the design conditions, while part II is a feasible solution region. The Pareto-optimal solutions are the only optimal values that can be reached in practice. The monotonicity of the Pareto-optimal solution set clearly shows that the blade mass increases

8 746 Long WANG, Tong-guang WANG, and Yuan LUO with the power generation, indicating that the two design goals have a certain degree of conflict. It cannot be said which point in the Pareto-optimal solution set is much better than others in theory, because they are all optimal with different combinations of the power output and the blade mass. This illustrates that there is no single point, at which all objectives are optimal in the multi-objective optimization problems of the wind turbine design. It is worthy emphasizing again that the discrete points of the Pareto optimal solution curve are all the optimal solutions under the given conditions. The choice of the solution in the practical design should be made according to the designer s favor with the less mass or more power output. To explain the formation of the Pareto optimal solution set better, three blades marked in Fig. 4 are analyzed. Figures 5 and 6 show the blade chord, twist, and thickness distributions of the blades 1, 2, and 3 and the comparison with the NREL-5 MW blade. It can be seen that the blade chords of the three blades increase with C P, while the twist angle and the relative thickness decrease. This indicates that the power generation can be improved effectively with the increase of the wind turbine solidity. The optimization of the twist distribution and the appropriate reduction of the relative thickness of the blade can also increase the power generation. Fig. 4 Pareto-optimal solution set of two objectives Fig. 5 Blade chord and twist distributions Fig. 6 Blade thickness distributions The comparison of the power coefficient curves with the blades 1, 2, and 3 as well as the NREL-5 MW blade is shown in Fig. 7. It can be seen that compared with the NREL-5 MW blade, the power coefficients of the blades 1, 2, and 3 at the design wind speed of 9 m/s increase by 4.6%, 1.7%, and 7.1%, respectively, while the values of the mass increase by 10.02%, 1.73%, and 9.67%. As mentioned above, the power increase results in the mass increase

9 Improved NSGA-II in multi-objective optimization studies of wind turbine blades 747 according to the Pareto-optimal solution shown in Fig. 4. This is identical with the result given in Fig. 8, where the spar thickness (t c ) variation with the local radius is shown for the three optimal blades, and the blade with a higher power coefficient has the thicker spar. The high performance of blades usually is accompanied by the high lift of the blade, resulting in high loads on the blade, which requires the high solidity and the thick spar to meet the requirement of the structural strength. As for the designs in this paper, since the blade 2 has both the greater power coefficient and the less mass than the NREL-5 MW blade, it seems to be a more desirable result than the other two blades by the consideration of the present two optimization objectives. Fig. 7 Power coefficient comparison Fig. 8 Blade spar thickness comparison 5 Conclusions Both the controlled elitism and the DCD have been incorporated into the NSGA-II, obtaining an improved optimization algorithm. This algorithm has good performance in the convergence and robustness when the multi-objective, multi-variable, and multi-constraint optimization problems are handled, providing a general approach for the multi-objective optimization design of the wind turbines. Meanwhile, it is illustrated from the optimization results that the Pareto-optimal solution set should be acquired rather than a single optimum solution, which are obtained usually from the conventional multi-objective optimizations. This provides a new idea for the multi-objective optimization design of wind turbines. Finally, the improved NSGA-II is used to design a 5 MW wind turbine blade. Compared with the NREL-5 MW blade, a higher-performance and less-mass blade can be obtained through this optimization without changing any design input. References [1] Wilson, R. E. and Lissaman, P. B. S. Applied Aerodynamics of Wind Power Machines, Report NSF/RA/N 7413, Oregon State University (1974) [2] Selig, M. S. and Coverstone-Carroll, V. L. Application of a genetic algorithm to wind turbine design. Journal of Solar Energy Engineering, 118(1), (1996) [3] Wood, D. H. Dual purpose design of small wind turbine blades. Wind Engineering, 28(5), (2004) [4] Sale, D., Jonkman, J., and Musial, W. Development of a hydrodynamic optimization tool for stall-regulated hydrokinetic turbine rotors. ASME th International Conference on Ocean, Offshore and Arctic Engineering, Honolulu, Hawaii, USA, (2009) [5] Roy, R., Hinduja, S., and Teti, R. Recent advances in engineering design optimisation: challenges and future trends. Manufacturing Technology, 57(2), (2008)

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