Multi-objective design optimization of a large-scale direct-drive permanent magnet generator for wind energy conversion systems

Abstract

This paper presents a simultaneous multi-objective optimization of a direct-drive permanent magnet synchronous generator and a three-blade horizontal-axis wind turbine for a large scale wind energy conversion system. Analytical models of the generator and the turbine are used along with the cost model for optimization. Three important characteristics of the system i.e.,the total cost of the generator and blades, the annual energy output and the total mass of generator and blades are chosen as objective functions for a multi-objective optimization. Genetic algorithm (GA) is then employed to optimize the value of eight design parameters including seven generator parameters and a turbine parameter resulting in a set of Pareto optimal solutions. Four optimal solutions are then selected by applying some practical restrictions on the Pareto front. One of these optimal designs is chosen for finite element verification. A circuit-fed coupled time stepping finite element method is then performed to evaluate the no-load and the full load performance analysis of the system including the generator, a rectifier and a resistive load. The results obtained by the finite element analysis (FEA) verify the accuracy of the analytical model and the proposed method.

Graphical abstract

Keywords

permanent magnet synchronous generator / wind turbine / direct-drive / multi-objective optimization / cost / mass / annual energy output / finite element analysis (FEA)

1 Introduction

The world-wide increasing demands for energy causes societies to face two major crises i.e., the future energy provision challenge and the climate changes due to environmental pollutions caused by greenhouse emissions [1]. Therefore, renewable energy resources have received increasing attention during the last few decades. Wind energy is one of the most attractive forms of renewable energy which is widely installed in many countries. The worldwide installed power capacity has undergone a significant increase in the last decade [2].

Wind turbine can be connected to a generator directly by the so-called direct-drive or through a gear box. In the direct-drive connection the gearbox is omitted which reduces the cost and the maintenance of wind turbines and increases the reliability [3,4]. Removing the gearbox form the system causes the generator to run at lower rotational speeds. Therefore, generators with high number of magnetic poles are required to maintain the efficiency and the power factor at desirable values. The power factor and the efficiency of induction generators (IGs) considerably decline as the number of poles increases because of a larger air gap length in multi-pole machines [5]. This issue limits their utilization for direct-drive applications although some efforts have been made to overcome this limitation [6]. In contrast with IGs, permanent magnet synchronous generators (PMSGs) are appropriate candidates for direct-drive applications because of keeping high efficiency and high power factor performance in multi-pole configurations with low rotational speeds [7,8]. PMSGs have been proposed from small-scale to large-scale direct-drive wind turbines in several researches so far [9−11]. The proper performance of these generators needs optimal design of their dimensions and characteristics. Different objectives have been considered for the optimization of PMSGs. Loss minimization and efficiency improvement have been the concern of some researches [12,13]. Since the efficiency is optimized for a specific operation speed, it does not guarantee to reduce the total losses in a wide range of generator operation speeds. Therefore, maximization of the annual energy output (AEO) has been considered as an objective function in optimal design of a direct-drive PMSG involving the generator efficiency in all operation speeds [14,15]. Reduction of the payback period has been also presented for both surface-mounted and interior type PMSGs [16,17]. Minimization of the cogging torque in surface-mounted PMSGs and improvement of the value and quality of output torque are objectives of some design optimizations [18−20]. Multi-phase PMSGs have been designed and their performance have been assessed and compared with conventional three-phase PMSGs [21,22]

Weight of a nacelle and turbine blades of a wind turbine installed on the top of the tower is an important issue affecting the mechanical design, the installation labor, and the installation cost. Although some researches consider structural solutions for weight reduction [23−25], this issue has not received enough attention in the optimization of the wind energy conversion system (WECS). Besides, in previous researches, the generator is usually optimized for a given turbine dimensions and afterwards the site matching technique tries to optimize a generator for a specific region [26]. However, this cannot ensure the optimal performance of the system, and a simultaneous optimization of turbine and generator specifications regarding the site wind characteristics is required to achieve a superior performance of the WECS.

This paper presents a multi-objective optimization of a direct-drive PMSG and a horizontal-axis three-blade turbine for a 1-MW WECS designed to be installed in a region with a low average wind speed. Seven generator parameters as well as turbine nominal speed are selected as design variables. The genetic algorithm (GA) method along with analytical and cost models of the generator and the turbine are then used to find a set of Pareto optimal solutions. Some restrictions are then applied to the Pareto optimal solutions omitting non-practical designs. Four optimal designs finally remain where one of them is selected for finite element verification. Both no-load and full load performances of the generator are investigated using a field-circuit coupled finite element analysis (FEA) including the generator, a rectified resistive load. The results obtained from the FEA confirm the validity of the analytical model and the proposed method.

2 Modeling of the system

The schematic view of a WECS is depicted in Fig. 1, which consists of a turbine, a PMSG, and grid connection apparatuses including an AC/DC/DC converter, a DC link capacitor, a DC/AC converter and possibly a transformer. The turbine is directly coupled to the generator. The output of the generator is rectified. The level of the DC voltage is regulated by the DC/DC converter, which is then used to make sinusoidal voltage by the inverter. The inverter output is then connected to the grid through the transformer or directly.

2.1 Turbine model

The output power of the wind turbine directly transmitted to the generator shaft is given by [27]

(1) P_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{t}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathrm{\pi }{\rho }_{\mathrm{a}\mathrm{i}\mathrm{r}}{R}^{\mathrm{2}}{U}^{\mathrm{3}}{C}_{\rho }(\lambda ).

The characteristics of C_{\rho } as a function of \lambda is approximately parabolic and there is an optimal value for tip-speed ratio ( {\lambda }_{opt}) maximizing the torque coefficient C_{\rho \max }. The characteristics of the wind turbine used in this paper are listed in Table 1. Determination of the rated speed is an important issue which depends on several factors and should be done through an optimization procedure.

Recent investigations show that the average wind speed in most regions of Iran is not high confirming by the wind atlas of Iran for elevation of 80 m above the ground [28]. The wind speed is between 6.8 m/s and 8.2 m/s in most windy sites in this elevation and therefore, the average wind speed of 8 m/s is selected for the generator design.

The Rayleigh distribution is selected as a probability density function for the wind speed. The annual energy output is then given by [29]

(2) AEO={\displaystyle \sum _{i=\mathrm{1}}^{n}P(U_i)}H(U_i),

where P(U_i) is the output power of the wind generator at the speed of U_i; H(U_i), the total number of hours per year in which the wind speed U_i is given by

(3) H(U_i)=\mathrm{365}\times \mathrm{24}\times f(U_i)\mathrm{\Delta }U.

2.2 PMSG model

The per-phase electrical equivalent circuit of a PMSG is depicted in Fig. 2.

Different parameters used in the circuit can be determined using the dimensions and material characteristics of the generator. The main dimensions of the generator are determined by [27]

(4) D^{\mathrm{2}}L={\displaystyle \frac{\mathrm{2}\epsilon P_{gen}}{k_{w\mathrm{1}}{n}_s{\pi }^{\mathrm{2}}{SEM}_{pk}{SEL}_{pk}\cos {\phi }_{est}}},

where {SEL}_{pk} stands for the specific electrical loading which is the linear current density in the machine. {SEM}_{pk} indicates the specific magnetic loading that is the root mean square (rms) value of magnetic flux density in the middle of air-gap.

By selecting the air gap diameter to machine length ratio, the air gap diameter and the machine length can be calculated. In this paper, the aforementioned ratio is chosen as an optimization variable. The rms value of the fundamental component of phase excitation voltage in the PMSG is calculated as [30]

(5) E_f=\mathrm{4.44}f{N}_{ph}{k}_{w\mathrm{1}}{\varphi }_{PM},

where the flux per pole is given by [30]

(6) {\varphi }_{PM}={\displaystyle \frac{\mathrm{8}}{{\pi }^{\mathrm{2}}}}\tau L{B}_{mg}\sin \left({\displaystyle \frac{\alpha \pi }{\mathrm{2}}}\right).

The air gap magnet flux density is obtained using the magnetic equivalent circuit (MEC) of the machine. The synchronous inductance and the stator resistance of the machine are also given by [31]

(7) X_s={\displaystyle \frac{\mathrm{6}{\mu }_{\mathrm{0}}LDf{k}_{\mathrm{w}\mathrm{1}}^{}{N}_{\mathrm{p}\mathrm{h}}^{\mathrm{2}}}{p^{\mathrm{2}}(k_C{l}_g+l_m/{\mu }_r)}}+X_l,

(8) R_a={\displaystyle \frac{l_{ph}}{a{\sigma }_{cu}{A}_{cond}}}.

The leakage inductance depends on the shape and dimensions of stator slots [30]. To have a constant width throughout a stator tooth, the shape of the stator slot is selected as trapezoidal and its dimensions are determined by using a design program shown in Fig. 6, where the slot height is given by [27]

(9) h_s={\displaystyle \frac{{SEL}_{pk}}{\sqrt{\mathrm{2}}J{K}_{sf}(\mathrm{1}-w_{\mathrm{t}}/{\tau }_{\mathrm{s}})}}.

Generator losses are divided into Joules loss, core loss, mechanical loss and extra loss. Joules loss is calculated as

(10) P_{copper}={\sigma }_{cu}(V_{cu}+V_{end})J^{\mathrm{2}}.

The core loss density in each part of the machine, composed of hysteresis loss, eddy current loss and extra loss, is calculated as

(11) p_c=k_h{f}_B{(B_{\max })}^{\beta }+{\displaystyle \frac{\mathrm{2}}{T}}{\displaystyle {\int }_{\mathrm{0}}^{T_B}\left(k_e{\left|{\displaystyle \frac{dB}{dt}}\right|}^{\mathrm{2}}+k_{ex}{\left|{\displaystyle \frac{dB}{dt}}\right|}^{\mathrm{3}/\mathrm{2}}\right)dt},

where loss coefficients are provided by lamination manufacturer. The mechanical loss and extra losses can also be calculated by the equations presented in Ref. [30].

2.3 Mass calculation

The weight of the nacelle and blades installed on the top of the tower is an important issue in the mechanical design of wind turbines and directly affects the installation labor and the installation cost. Since the gearbox is omitted in the direct-drive system, the mass of nacelle mainly stems from the generator, blades, brake and drive train. Furthermore, as only generator dimensions and blades diameter are variable during the optimization, the mass of these parts is considered as the total mass. The mass of the generator is considered as the mass of its active consumed materials obtained by

(12) M_{gen}=(V_{PM}{\rho }_{PM}+V_{cu}{\rho }_{cu}+V_{lam}{\rho }_{lam}).

The volume of blades approximately varies as a cubic function of their length. Therefore, the mass of blades is expressed as [32]

(13) M_b(D)\cong M_b(\mathrm{70}){\left({\displaystyle \frac{D}{\mathrm{70}}}\right)}^{\mathrm{3}},

where M_b(\mathrm{70}) is the mass of blades of a baseline turbine with a diameter of 70 m. Therefore, the estimated mass of nacelle is calculated as

(14) M_t=M_{gen}+M_b\mathrm{.}

2.4 Cost model

The total cost of a direct-drive WECS includes the generator cost, the turbine and tower cost, the grid connection and control system cost, and the installation costs. However, taking into consideration the fact that the total cost of the system can hardly be calculated since the installation and labor costs are different from one place to the other, and the fact that the tower cost and its foundations depend on soul properties and environmental issues, this paper only focuses on the raw material cost of the generator and a specific cost function for the turbine including blades and tower. Since the output power of the generator does not change, the converter size should not be changed. Besides, the configuration of converter is not the aim of this work, the cost of the converter is assumed to be constant and is considered in optimization.

(15) C_t=C_{gen}+C_{tur}+C_{grid}+C_{inst}.

The cost of the generator depends on the cost of active materials consumed. Active materials used in the generator are permanent magnet, ferromagnetic laminations and copper. Therefore the cost of the generator can be calculated as

(16) C_{gen}=(C_{PM}+C_{cu}+C_{lam})k_{manu}.

The base costs of raw materials used in this paper are listed in Table 2.

Since the rated output power of the generator is kept constant through the optimization procedure, the cost of grid connection apparatus and the control system are assumed to be constant as shown in Table 2. The installation cost is also assumed to be constant but is not included in the optimization procedure. The turbine and tower cost depends on blades diameter and can be modeled by a cost function as [32]

(17) C_{tur}(D)=C_T(\mathrm{70})\left[\mu {\left({\displaystyle \frac{D}{\mathrm{70}}}\right)}^{\mathrm{3}}+(\mathrm{1}-\mu )\right],

where C_T(\mathrm{70}) is the total cost of a baseline turbine with a diameter of 70 m which is presented in Table 2, ρ is a weight coefficient which depends on the size of the turbine and is assumed to be 0.3 in this paper [32]. It is worthy of note that all prices are related to the time of project, early 2011, and may not be valid at the time of publication.

3 Optimization problem

The AEO, the cost of the system and total mass are selected as objective functions for a multi-objective design optimization of PMSG and turbine dimensions and characteristics. GA is employed to find Pareto optimal designs in a search space. Eight design variables are selected for the optimization, as listed in Table 3 along with their boundaries. A design algorithm is also used to find other generator parameters and dimensions. The flowchart of the design algorithm as well as its connection to GA program is demonstrated in Fig. 3. First, all design parameters are initialized by GA to form the initial population and are used as inputs for design algorithm. At the end of design algorithm, calculated objective functions i.e., mass, cost and AEO are transferred to GA program and are used to calculate fitness functions for the current population. Then this population goes through genetic operators to form the new population. Fitness functions of the new population is then calculated using design algorithm. After that the Pareto optimal solutions are updated based on the definition of a Pareto optimal solution. It is worth noticing that this optimization does not combine fitness functions as a single objective function. This procedure continues until the termination condition is satisfied. This condition can be a number of total populations or variation of Pareto optimal solutions. GA toolbox of Matlab software is used in this paper and probability of cross over and mutation operators are chosen as 0.7 and 0.05 respectively.

The final sets of Pareto optimal solutions, the so-called as Pareto front, is exhibited in a three dimensional space of objective functions in Fig. 4. The set of Pareto optimal solutions including 16 optimal designs are listed in Table 4. In this step, some practical restrictions are applied to these solutions to remove impractical designs. A limit of 1 million USD is set for the total cost. Therefore, some designs which are indicated by a star in front of their cost are omitted from the final choices. Also, a maximum total mass equal to 70 ton and a minimum AEO equal to 4.5 GWh are imposed to optimal designs. Some other designs which do not satisfy these conditions indicated by stars in front of their mass or AEO are removed. The two dimensional Pareto fronts are illustrated in Fig. 5 in terms of each pair of objectives where regions containing acceptable solutions are indicated. The final solutions are common solutions located in these regions. Therefore, four final designs remained, one of which can be selected based on the designer’s point of view. In this paper, as an example, the generator number 7 is selected for FE validation. The dimension and parameters of this generator are listed in Table 5.

The value of AEO is higher than that of available commercial wind generators in this scale for the selected site. The reason for this is that the optimal value obtained for the turbine rated speed, around 9.5 m/s, is lower than usual rated speed values of conventional wind turbines which are between 12 m/s and 14 m/s. This is a special feature of presented WECS which makes it capable of working with a full potential of energy production in regions with a relatively lower average wind speed. Slot opening is chosen for easily placement of conductors into slots. It is also seen that the frequency, the current density and the voltage optimal values are obtained near their lower boundaries whereas the optimal value of tooth maximum flux density is close to its upper boundary. The machine length is roughly one fourth of the air gap diameter which is mainly caused by a high number of pole pairs.

4 Finite element verification

The FEA is conducted to verify the results obtained by analytical model and to analyze the performance of the selected optimal PMSG. A 2-D FEA is therefore utilized to solve the electromagnetic equation in the generator. Thanks to the symmetry, only 1/50 of the generator is modeled by FEA resulting in a significant reduction in the solution time. The meshed model includes 5018 triangular second-order element with 10153 nodes. The flux lines and no-load condition is depicted in Fig. 6.

The no-load excitation voltage due to permanent magnets and its harmonic contents are also depicted in Fig. 7. It is observed that the rms value of its first harmonic is approximately 637 V. The difference between the analytical and FE results is caused by the approximations of correction factors used in the analytical model for saturation and leakage effects and the approximation considered in Carter factor estimation. Since the voltage drop on stator resistance in full load condition is approximately 32 V, the terminal voltage of the optimal generator in full load operation will approximate 605 V which is in good agreement with the value obtained by the analytical calculations.

Although only the no-load excitation voltage of PMSG is evaluated by FEA in some researches, the performance of the generator under full load condition is also of great importance to investigating the armature effect and ensuring the generator is capable of providing its rated power. Therefore, in this section a field-circuit coupled time stepping FEA is presented to study the full load performance of the generator. Instead of the grid side convertor and the grid, a resistive load is used just to evaluate the performance under load, as shown in Fig. 8. The generator stator coils are connected to the load using a rectifier and a capacitor as a DC filter. The load experiences a voltage which is almost DC as illustrated in Fig. 9. The load power also shown in Fig. 9 demonstrates the capability of the generator to produce nominal power. Since the current density of the generator at nominal condition is selected in a safe limit and also due to the high efficiency of the generator, the temperature rise in the machine would be in the safe range.

Cogging torque is an important issue in a low-speed direct-drive PMSG where it may cause start-up failure if the cogging torque is not much less than turbine torque at cut-in speed [17,18]. In this paper, the magnet skewing method is employed for cogging torque reduction [33]. Each pole is divided into two segments which have been installed with a skewing angle equal to half of a slot pitch. The cogging torque before and after magnet skewing is computed by the multi-slice two-dimensional finite element method, as illustrated in Fig. 10 [34]. It is observed that the cogging torque is reduced from 2180 N·m to less than 600 N·m.

The transmitted torque from the turbine to the generator shaft declines as quadratic function of wind speed is [27]:

(18) T_{shaft}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\rho }_{air}\pi R^{\mathrm{3}}{U}^{\mathrm{2}}{\displaystyle \frac{C_{\rho }(\lambda )}{\lambda }}.

The ratio of cut-in speed to the rated speed of the turbine is 0.37 m/s. Therefore, the ratio of the turbine torque at the cut-in speed to the turbine torque at the rated speed is 0.137 m/s. The turbine torque at the rated speed is given by

(19) T_{rated}={\displaystyle \frac{P_n}{\eta {\omega }_{rn}}}={\displaystyle \frac{{\mathrm{10}}^{\mathrm{6}}}{\mathrm{0.94}\times \mathrm{13.3}\pi /\mathrm{30}}}\approx \mathrm{764}(kN·m).

Consequently, the turbine torque at the cut-in speed is determined as

(20) T_{cut-in}=\mathrm{0.137}{T}_{rated}\approx \mathrm{104.7}(kN\cdot m).

Thus, the cogging torque is less than 0.6% of the turbine torque at the starting which considerably reduces the possibility of start-up failure.

5 Conclusions and future works

A multi-objective optimization of a PMSG and a horizontal-axis wind turbine for a large-scale wind energy conversion system is presented. Analytical models of the generator and the turbine as well as the cost model for different parts of the system are used. GA is then employed for a multi-objective optimization aimed at simultaneous optimization of the total cost of the system, mass of nacelle and blades, and AEC. Seven generator parameters and a turbine parameter are selected as optimization variables. A set of 16 Pareto optimal solutions is achieved by the optimization method. Three practical restrictions on the AEO, mass and cost are then applied to the optimal designs. Four eligible designs are remained, one of which is selected for verification. The selected optimal design produces 4.57 GWh of energy per year. The FEA is employed to evaluate the performance of the final design at no-load condition and full load condition by coupling to a rectified resistive load. The results confirm the validity of the analytical model and the optimization method. Afterwards, the magnet skewing method is used to reduce the cogging torque of the machine. A multi-slice FEA shows that by using two magnet layers with a 2.4 degree of magnet rotation, the cogging torque is decreased up to 72.5%. In future works the thermal analysis of the PMSG should be considered and the cooling method should be designed. The mechanical analysis of the generator including modal analysis and stress analysis is necessary before prototyping the machine. Besides, by considering the installation cost of the system, the labor cost due to the nacelle installation can be modeled and used in the optimization. Considering the installation and maintenance costs in addition to the energy production cost and inflation and interest rates provides the possibility of calculating the accurate payback period which can also be used in the optimization.

References

[1] Bevrani H, Ghosh A, Ledwich G. Renewable energy sources and frequency regulation: survey and new perspectives. IET Renewable Power Generation, 2010, 4(5): 438−457

[2] REN 21. Renewables 2011 Global Status Report. 2013-05-06

[3] Arabian-Hoseynabadi H, Tavner P J, Oraee H. Reliability comparison of direct- drive and geared-drive wind turbine concepts. Wind Energy, 2009, 13(1): 62−73

[4] Polinder H, Van der Pigl F F A, Tavner P J. Comparison of direct-drive and geared generator concepts for wind turbines. IEEE Transactions on Energy Conversion, 2006, 21(3): 725−733

[5] Lampola P. Directly driven, low-speed permanent-magnet generators for wind power applications. Dissertation for the Doctoral Degree. Helsinki: Helsinki University of Technology, 2000

[6] Colli D, Marignetti F, Attaianese C. Analytical and multiphysics approach to the optimal design of a 10-MW DFIG for direct-drive wind turbines. IEEE Transactions on Industrial Electronics, 2012, 59(7): 2791−2799

[7] Abbasian M, Isfahani A H, Shahghasemi S, Sheikholeslam F. Effects of permanent magnet synchronous generator and wind turbine parameters on the performance of a small-scale wind power generation system. Przegląd Elektrotechniczny (Electrical Review), 2011, 87: 360−363

[8] Dubis M R. Optimized permanent magnet generator topologies for direct-drive wind turbines. Dissertation for the Doctoral Degree. Delft: Delft University, 2004

[9] Chen Y, Pillay P, Khan M A. PM wind generator topologies. IEEE Transactions on Industry Applications, 2005, 41(6): 1619−1626

[10] Muljadi E, Butterfield C P, Wan Y. Axial-flux modular permanent-magnet generator with a toroidal winding for wind-turbine applications. IEEE Transactions on Industry Applications, 1999, 35(4): 831−836

[11] Spooner E, Williamson A C. Direct coupled, permanent-magnet generators for wind turbine applications. IEE Proceedings-Electric Power Applications, 1996, 143(1): 1−8

[12] Eriksson S, Bernhoff H. Loss evaluation and design optimisation for direct driven permanent magnet synchronous generators for wind power. Applied Energy, 2011, 88(1): 265−271

[13] Pinilla M, Martinez S. Selection of main design variables for low-speed permanent magnet machines devoted to renewable energy conversion. IEEE Transactions on Energy Conversion, 2011, 26(3): 940−945

[14] Jung H, Lee C G, Hahn S C, Jung S Y. Optimal design of a direct-driven PM wind generator aimed at maximum AEP using coupled FEA and parallel computing GA. Journal of Electrical Engineering & Technology, 2006, 3(4): 552−558

[15] Jung S Y, Jung H, Hahn S C, Jung H K, Lee C G. Optimal design of direct-driven PM wind generator for maximum annual energy production. IEEE Transactions on Magnetics, 2008, 44(6): 1062−1065

[16] Abbasian M, Isfahani A H. Optimal design of a direct-drive permanent magnet synchronous generator for small-scale wind energy conversion systems. Journal of Magnetism, 2011, 16(4): 379−385

[17] Zhang J, Cheng M, Chen Z. Optimal design of stator interior permanent magnet machine with minimized cogging torque for wind power application. Energy Conversion and Management, 2008, 49(8): 2100−2105

[18] Potgieter J H J, Kamper M J. Torque and Voltage quality in design optimisation of low cost non-overlap single layer winding permanent magnet wind generator. IEEE Transactions on Industrial Electronics, 2012, 59(5): 2147−2156

[19] Sun X, Cheng M, Hua W, Xu L. Optimal design of double-layer permanent magnet dual mechanical port machine for wind power application. IEEE Transactions on Magnetics, 2009, 45(5): 4613−4619

[20] Hassanain N E A M, Fletcher J E. Steady-state performance assessment of three- and five-phase permanent magnet generators connected to a diode bridge rectifier under open-circuit faults. IET Renewable Power Generation, 2010, 4(5): 420−427

[21] Brisset S, Vizireanu D, Brochet P. Design and optimization of a nine-phase axial-flux pm synchronous generator with concentrated winding for direct-drive wind turbine. IEEE Transactions on Industry Applications, 2008, 44(3): 707−715

[22] Morandin M, Fornasiero E, Bolognani S, Bianchi N. Torque/power rating design of an IPM machine for maximum profit-to-cost ratio in wind power generation. In: IEEE International Electric Machines and Drives Conference. Niagara Falls, Canada, 2011, 1113−1118

[23] McDonald A S, Mueller M A, Polinder H. Structural mass in direct-drive permanent magnet electrical generators. IET Renewable Power Generation, 2008, 2(1): 3−15

[24] Shrestha G, Polinder H, Bang D, Ferreira J A. Structural flexibility: a solution for weight reductionof large direct-drive wind-turbine generators. IEEE Transactions on Energy Conversion, 2010, 25(3): 732−740

[25] Mueller M A, McDonald A S. A lightweight low-speed permanent magnet electrical generator for direct-drive wind turbines. Wind Energy, 2009, 12(8): 768−780

[26] Li H, Chen Z. Design optimization and site matching of direct-drive permanent magnet wind power generator systems. Renewable Energy, 2009, 34(3): 1175−1184

[27] Khan M A. Contributions to permanent magnet wind generator design including the application of soft magnetic composites, Dissertation for the Doctoral Degree, New York: Clarkson University, 2006

[28] Renewable Energy Organization of Iran. Iran wind atlas. 2014-03-21

[29] Zinger D S, Muljadi E. Annualized wind energy improvement using variable-speeds. IEEE Transactions on Industry Applications, 1997, 33(6): 1444−1447

[30] Gieras J F, Wing M. Permanent magnet motor technology: design and application. 2nd ed. Marcel Dekker Inc., 2002

[31] Khan M A, Pillay P, Visser K D. On adapting a small PM wind generator for a multiblade, high solidity wind turbine. IEEE Transactions on Energy Conversion, 2005, 20(3): 685−692

[32] Burton T, Sharpe D, Jenkins N, Bossanyi E. Wind Energy Handbook. England: John Wiley&Sons, Ltd, 2001

[33] Dosiek L, Pillay P. Cogging torque reduction in permanent magnet machines. IEEE Transactions on Industry Applications, 2007, 43(6): 1565−1571

[34] Gyselinck J J C, Vandevelde L, Melkebeek J A A. Multi-slice FE modeling of electrical machines with skewed slots- the skew discretization error. IEEE Transactions on Magnetics, 2011, 37(5): 3233−3237

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