Design optimization and site matching of direct-drive permanent magnet wind power generator systems

Renewable Energy 34 (2009) 1175–1184

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Technical Note

Design optimization and site matching of direct-drive permanent magnet wind power generator systems H. Li a, b, *, Z. Chen a a b

Institute of Energy Technology, Aalborg University, Aalborg East DK-9220, Denmark State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing 400044, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 July 2007 Accepted 2 April 2008 Available online 22 October 2008

This paper investigates the possible site matching of the direct-drive wind turbine concepts based on the electromagnetic design optimization of permanent magnet (PM) generator systems. Firstly, the analytical models of a three-phase radial-flux PM generator with a back-to-back power converter are presented. The optimum design models of direct-drive PM wind generation system are developed with an improved genetic algorithm, and a 500-kW direct-drive PM generator for the minimal generator active material cost is compared to demonstrate the effectiveness of the design optimization. Forty-five PM generator systems, the combinations of five rated rotor speeds in the range of 10–30 rpm and nine power ratings from 100 kW to 10 MW, are optimally designed, respectively. The optimum results are compared graphically in terms of the generator design indexes. Next, according to the design principle of the maximum wind energy capture, the rotor diameter and the rated wind speed of a direct-drive wind turbine with the optimum PM generator are determined. The annual energy output (AEO) is also presented using the Weibull density function. Finally, the maximum AEO per cost (AEOPC) of the optimized wind generator systems is evaluated at eight potential sites with annual mean wind speeds in the range of 3–10 m/s, respectively. These results have shown the suitable designs for the optimum site matching of the investigated PM generator systems. Ó 2008 Published by Elsevier Ltd.

Keywords: Wind turbines Direct-drive Permanent magnet generator Design optimization Site matching

1. Introduction Most cost-effective and reliable large wind energy conversion systems are becoming the main focus of wind energy technology development in order to make wind energy to be competitive with the other more traditional sources of electrical energy like coal, gas and nuclear generation. Various wind turbine concepts and wind generators have been developed during last two decades. According to the construction of drive trains, these wind turbine concepts may be classified into the geared drives and the direct-drive concepts. Compared to the geared drive concepts, direct-drive concepts may be more attractive due to the advantages of simplified drive train and higher overall efficiency, reliability and availability by omitting the gearbox [1–19]. Especially, most of larger machines are intended for offshore use where the trend has been toward very low maintenance designs. On the other hand, in various wind power generators, permanent magnet (PM) machines have higher ratio of power to weight, reliability, and efficiency than electrically

* Corresponding author. Institute of Energy Technology, Aalborg University, Aalborg East DK-9220, Denmark. E-mail addresses: [email protected] (H. Li), [email protected] (Z. Chen). 0960-1481/$ – see front matter Ó 2008 Published by Elsevier Ltd. doi:10.1016/j.renene.2008.04.041

excited machines. In addition, the use of PM machines becomes attractive than before, because the performance of PM materials is improving and the cost of PM materials is decreasing in recent years. Therefore, these advantages and trends may make directdrive PM wind generator systems more attractive for wind turbine concepts, especially for offshore applications [6–10]. Currently, Zephyros (currently Harakosan) and Mitsubishi are using this concept, and the largest capacity is up to 2 MW on the market [11]. Fig. 1(a) is a scheme of grid connection of the direct-drive concept, and Fig. 1(b) illustrates a sketch of a wind turbine nacelle of a direct-drive PM synchronous generator (PMSG) configuration. With rapid development of technology and with significant growth of wind power capacity installed worldwide, one of the trends of wind turbines is to increase the scale of direct-drive wind turbine concepts with the maximum energy yield and the minimum cost, especially for offshore wind power applications. However, the energy production of wind turbines depends upon many factors, for example wind climates of a potential site, the hub height, the rated wind speed, the cut-in and cut-out wind speed of wind turbines and the generator design [12–18]. In addition, direct-drive PM generator costs are mainly dependent on the chosen generator diameter. Larger generator diameters decrease the necessary generator length and active magnetic material costs, but increase the generator structural costs, technical difficulties of

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a

b

PMSG Grid Converter

Fig. 1. Configuration of a direct-drive PM wind generator system. (a) Scheme of a grid-connected PM generator system and (b) sketch of a direct-drive PM wind turbines of Zephyros [11].

transport and assembly. Furthermore, according to Ref. [13], experience with the existing wind farms has shown that some of the wind power plants have failed completely or performed poorly especially in some developing countries because the installed wind turbine system do not match the wind site. Consequently, the problem of optimum site matching of a suitable wind generator system for higher energy yield and lower cost needs to be investigated. Though some comparisons and choices of suitable wind turbines at a specific site have been researched, comparisons are done usually upon collecting the commercial wind turbines’ data so that it is difficult to understand the interrelationship between design and economical performances [12,13]. In addition, the often used capacity factor (CF) for assessing the optimum site matching of wind turbine generators may be not reasonable without considering the system costs. Moreover, some optimum designs of PM wind generators at different power levels are mainly based on the comparison of the generator design indexes [5–10], the characteristics of wind turbines and wind distribution have been seldom considered. Therefore, in order to assess the site matching of the direct-drive wind generator under various combinations of rated rotor speeds and rated power levels, the electromagnetic optimization of direct-drive PM generators is necessary, and the index of annual energy output per cost (AEOPC) can be taken into consideration to select the most cost-effective PM synchronous generator (PMSG) system for a potential site. The contribution of this paper investigates the optimum potential site matching of large directdrive PM wind power generator systems by considering both the electromagnetic optimization and the maximizing wind energy capture characteristics. In this paper, firstly, the electromagnetic design models of a three-phase radial-flux surface mounted PM generator with a back-to-back power electronic converter are presented. The electromagnetic design optimization is developed with an improved genetic algorithm and demonstrated by the electromagnetic design of a 500-kW direct-drive PM generator system. Next, design optimization of 45 PM generator systems with various rated power levels and rated rotational speeds is obtained and compared graphically based on the generator design indexes, including the generator system cost, the generator outer diameter, the torque density and the torque per cost. Furthermore, according to the design principle of the maximum wind energy capture, the rotor diameter and the rated wind speed of the corresponding direct-drive wind turbine with the optimum PM generator are determined. Finally, by using the performance index of the maximum AEOPC, the optimum potential site matching is analyzed for each annual mean wind speeds ranging 3–10 m/s, respectively.

2. Design optimization of PM generator systems Though various topologies of PM machines are possible to be used for direct-drive wind turbines, the radial-flux PM (RFPM) machine with surface mounted magnets seems to be a better choice for low-speed, direct-drive large wind turbines due to its simple structure and reliability [5–9]. In this paper, a three-phase RFPM machine with a back-to-back power electronic converter is investigated. Considering the rotor speed range of a medium or large wind turbine is typical in the range of 10–30 rpm [14,15], 45 direct-drive PMSG systems with the combinations of five rated rotor speeds from 10 rpm to 30 rpm and nine rated power levels from 100 kW to 10 MW are designed. At each given rated rotational speed and rated power, the direct-drive PMSG with a back-to-back power converter is optimally designed for the minimum generator system cost by using an improved genetic algorithm. 2.1. Design modeling of RFPM machines The machine parameters are calculated based on well-known analytical methods using equivalent circuit models. This subsection describes the key analytical models used to determine the dimensions of the active parts and the important equations used to determine the parameters of the equivalent circuit. Fig. 2 illustrates one pole of an RFPM machine. The slots and teeth are characterized by the slot height (hs), the slot pitch (s) and the tooth width (bd). The stator yoke heights hys is determined by the allowed flux densities in the stator

hys ¼

b bm Le B g0 b ys Lu 2B

(1)

b ys is the peak stator yoke flux where bm is the magnet width, B density, Le is the equivalent core length, and Lu is the available iron stack length. In this case, the stator and rotor yoke heights are assumed as the same value. The magnet thickness hm as a function of air gap flux density is given by [2]

hm ¼

b m g B g0 rm eff Brm

(2)

where Brm is the remanent flux density of the magnets (1.2 T), mrm is the relative permeability of the permanent magnet material, and geff is the effective air gap, which can be given as

H. Li, Z. Chen / Renewable Energy 34 (2009) 1175–1184

τ

bd

hys

pFe

1177

  b !2  2 b !2 fe fe B Fe B Fe ¼ 2pFe0h þ2pFe0e b b f0 f 0 B B 0 0

(9)

where fe is the frequency of the field in the iron, pFe0h and pFe0e are the specific hysteresis loss and the specific eddy-current loss (in W/kg) in the laminated stator core for a given frequency f0 (50 Hz) and a flux b (1.5 T). density B 0 Table 1 gives the specific parameters of the investigated PM generators.

hs

g hm

2.2. Power electronic converter modeling

hyr bm τp

rs

Fig. 2. Basic dimension of one radial-flux PMSG pole.

  hm geff ¼ kc g þ

mrm

(3)

where kc is the Carter factor for the stator slots [16] and g is the mechanical air gap. As the machine has PM poles on the rotor, the magnetizing inductance can be given by [2,16]

Lsm ¼

6m0 Le rs ðkw Ns Þ2 p p2 geff

(4)

where m0 is the constant of the permeability of vacuum, rs is the stator radius, Ns is the number of turns of the phase winding, kw is the winding factor, and p is the number of pole pairs. The leakage inductance Lsl has to be added to complete the synchronous inductance expression. Slot, air gap, and end-winding leakage inductances are calculated as given in Refs. [9,16]. The fundamental harmonic of the air gap flux density can be calculated as

Bg1 ¼ Brm

  pbm hm 4 sin mrm geff p 2sp

(5)

where sp is the pole pitch. The no-load voltage induced by this flux density in a stator winding can be calculated as [2]

Ep ¼

pffiffiffi 2kw Ns um rs Le Bg1

(6)

where um is the mechanical angular speed of the rotor. The copper losses are calculated from the currents and the resistance. The phase resistance is calculated as

Rs ¼

rcu Lcu Acu

(7)

where rcu is the resistivity of copper, Lcu is the length of the conductor of the phase winding and Acu is the cross-sectional area of the conductor, which can be given as [2]

Acu ¼ kfills

pqbs ðhs  hw Þ Ns

In direct-drive wind turbine concepts, a back-to-back PWM full power converter can be used as the interface between the stator of PM generator and grids in order to ensure that the generator currents and the grid currents are sinusoidal [5,17]. Fig. 3 shows a main circuit topology of a back-to-back PWM power converter, which is composed of a generator side converter, a grid side converter and a dc-link capacitor. A cost estimate is given in Table 1. By using the full power converter, variable speed operation of PM generator systems can be controlled so that the wind turbine can operate at its maximum efficiency. In order to reduce saturation and to get a compromise between the converter rating and generator rating, the controller can adjust the phase angle between generator angle and current, so that the amplitude of the stator terminal voltage equals the internal induced voltage. This mode of operation introduces the lowest power rating requirements on both generator and rectifier, so that it can utilize the PMSG and converter best. According to the investigation by Ref. [5], the type of rectifier to be selected is strongly dependent on the machine synchronous reactance Xs. For values of Xs between 0.5 pu and 1.5 pu, the IGBT PWM rectifier can be used, and the DC voltage can be kept constant for all generator rotational speeds. However, for values of Xs above 1.5 pu, even with a PWM rectifier, the power factor is very low, and the generator and rectifier rating must be increased strongly. Therefore, the synchronous reactance is limited to 0.5–1.5 pu in the design optimization as a constraint There are various ways of modeling converter losses [2,9,16,17]. In this case, the losses in the power electronic converter pconv are modeled as [2]

(8)

where q is the number of slots per pole per phase, kfills is the slot fill factor, and hw is the slot wedge thickness. The iron losses are approximated with Steinmetz formula [2,10,16]

pconv

Ig2 Ig p Is I2 ¼ convN 1 þ 10 þ 5 2s þ 10 þ5 2 31 IsN IgN IsN IgN

! (10)

where pconvN is the dissipation in the converter at the rated power (3% of the rated power of the converter), Is is the generator side converter current, IsN is the generator side converter rated current, Ig is the grid side converter current, and IgN is the grid side converter rated current. Table 1 Loss modeling and cost modeling parameters Main performance constants Peak teeth flux density Bt0 (T) Remanent flux density of PM Br (T)

1.6 1.2

Loss modeling Hysteresis losses at 1.5 T and 50 Hz pFe0h (w/kg) Eddy-current losses at 1.5 T and 50 Hz pFe0e (w/kg)

2 0.5

Cost modeling Specific cost of Specific cost of Specific cost of Specific cost of Specific cost of Specific cost of

laminated iron cFe [Euro/kg] copper ccu [Euro/kg] NdFeB magnets cm [Euro/kg] reference generator structure cstr [Euro] power electronics ccon [Euro/kW] sub electrical system csubsystem [Euro/kW]

3 15 30 15,000 40 38

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Fig. 3. Main circuit topology of a back-to-back PWM full power converter for PMSG.

2.3. Optimization models of PM generator systems In order to investigate the most cost-effective PM wind generator system, the generator system cost is used as the optimized objective function.

Cw ¼ Cg

act

þ Cg

str

þ Ccon þ Csubsystem

(11)

where Cg_act ¼ ccuGcu þ cFeGFe þ cmGm is the active generator material cost; ccu, cFe, cm are the unit costs of the copper, the active iron and the permanent magnets; Gcu, GFe, Gm are the weight of the copper, the active iron and the permanent magnets, respectively. Cg_str is the generator structure cost, which is approximated as Cg str ¼ cstr ð1=2Þ½ðD1 =2Þ3 þ ðLtot Þ3  and cstr is the cost of a reference structure of 2 m diameter and 1 m length [9]. Ccon is the cost of power electronic converter. Csubsystem is the other electrical subsystem cost, which includes transformer, cable, switchgear and so on. As the basis for this criterion, the different specific component cost is given in Table 1, respectively. In order to optimize the machines to the criterion (11), six variables are chosen to vary within a certain range, including the air gap radius (rs), the stator length (L), the slot height (hs), the pole b go Þ and the peak stator pitch (sp), the peak air gap flux density ð B b ys Þ. The following assumptions are used in the yoke flux density ð B optimization program. (1) The number of slots per pole per phase is q ¼ 1. A two-layer winding with two conductors per slot (Nslot ¼ 2) is used to make the end-windings simple due to an integer slot winding. The stator slots are open and a non-magnetic wedge thickness is hw ¼ 5 mm. The slot filling factor is set to 0.65 for the outer diameter larger than 2 m, and is set to 0.45 below 2 m outer diameter. The slot width bs is 45% of the slot pitch and the stator slots are skewed by one slot pitch, so that the torque ripple can be reduced [8,16]. (2) For mechanical reasons, the ratio of slot depth to slot width is limited over the range of 4–10, which prevents excessive tooth mechanical vibrations from occurring. The air gap is equal to 0.001 of the air gap diameter. (3) The magnet width bm is kept at 70% of the pole pitch, but the minimum value is limited to be larger 3 times the air gap thickness to reduce the tangential fringing flux of the PMs in the air gap [16]. The magnet material is NdFeB with a remanent magnet flux density of 1.2 T. To avoid demagnetization of the magnets, the peak flux density generated by the stator b (0.7–1.0 T). Durb s is limited to be smaller than B winding B g0 b ing normal operation, B s depends on the peak value of the

b s and the effective air gap stator magnet-motive force (mmf) V bs ¼ m ðV b s =g Þ. In addition, length geff, which is given as B 0 eff since the PM cover a pole arc of 126 (bm ¼ 7.0sp), the ratio of leakage inductance to magnetizing inductance is also limited to be larger than 1.27 to avoid the risk of demagnetization at a short circuit at the generator terminals [9]. (4) The maximum flux density in the stator and rotor yoke is set to 1.2 T, in order to reduce the drop in mmf in those parts. This also reduces iron losses in the stator yoke. (5) The current density in the stator windings is limited to 3–6 A/ mm2, and the current loading is limited to 40–60 kA/m to prevent excessive and avoid critical cooling requirements. 2.4. Optimization approach The genetic algorithm (GA) belongs to the group of probabilistic searching methods, which have high probability of locating the global optimum in the multidimensional searching space discarding all existing local optimal solutions. The idea of genetic optimization is to imitate evolution in nature. A standard GA is described by the following steps [20]: Step (1) Initialize a population of solutions. Step (2) Evaluate each solution in the population. Step (3) Create new solutions by mating current solutions: apply mutations and recombination as the parents mate. Step (4) Delete members of the population to make room for the new solutions. Step (5) Evaluate the new solutions and insert them into the population. Step (6) If the available generations have expired, halt and return to the best solution; otherwise go to step [20]. The aim of the algorithm is to find the right genes for a population member thrive in the environment described by the objective functions and the constraints. The feasibility of the design is guaranteed by adding a penalty to the objective function f(X) (e.g. cost) due to constraint violations.

FðXÞ ¼ f ðXÞ þ

X

ai ½maxð0; gi ðXÞÞ2

(12)

i

where ai is the scaling parameter, gi(X) is the constraint function and X is the vector of the optimal design variables. In this study, an improved genetic algorithm (IGA) is used to optimize the direct-drive PM generator system, which has been developed and applied to the design optimization of induction machines and power transformers in our previous works [21,22]. In the IGA model, each string (chromosome) is expressed by the real number code for the chosen optimal design variable; the stochastic crossover method incorporating the arithmetic crossover technique and the uniform crossover scheme is developed to increase the solution space and speed up the convergence of optimization;

H. Li, Z. Chen / Renewable Energy 34 (2009) 1175–1184

the crossover rate and mutation rate can be dynamically regulated with the processing of optimization. Forty-five PM generators with the combinations of five rated rotational speeds from 10 rpm to 30 rpm and nine rated power levels from 100 kW to 10 MW are designed optimally by using the above mentioned IGA. Fig. 4 depicts the flow chart of the main computation procedure for a PM generator system at a given rotor speed. Firstly, for a given rated rotational speed and a rated power, the initial population is randomly generated by the six variables within a specific range. Next, according to the described procedures of the IGA and the analytical models of PM generator, the radial-flux PM generator with a back-to-back power converter is optimized to obtain the minimal generator system cost. The above mentioned assumptions of mechanical limitations and performance constraints are applied in the optimization. Once the best design is obtained, the program will update the rated power and repeat the optimization in the specific range of rated power levels. After the optimal design in a given range of rated powers is finished, the program will update the rated rotational speed to repeat the optimization. Finally, when all of the combinations in the ranges of rated rotational speeds and rated power levels are implemented, the optimal design procedure will stop. The optimal results with the combinations of rated rotor speeds and rated power levels can be obtained.

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2.5. Test of design optimization In order to demonstrate the electromagnetic designs and the optimization models, the 500-kW direct-drive PM generator system with a rated speed 32 rpm has been chosen. Table 2 shows the detailed comparison of the IGA and the numerical optimization by Grauers [9]. From the optimum results presented in Table 2, it can be seen that the good agreement exists with respect to the generator dimensions and performance. Also a lower total active material weight can be obtained by using the IGA. 2.6. Comparisons of generator design indexes The comparisons of the optimized results of 45 PM generators with different rated rotational speeds and rated power levels are given in three-dimensional graphical format. Including the generator system cost, the criteria used for comparisons are the stator outer diameter, the torque density and the torque per cost, which are the key design indexes of PM machines. Figs. 5 and 6 depict three-dimensional representations in terms of the generator system cost (criterion) and the stator outer diameter for the optimized PM generator system. The results show the cost of the generator system becomes higher as the rated power

Fig. 4. The flow chart of the optimal design procedure by using IGA.

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H. Li, Z. Chen / Renewable Energy 34 (2009) 1175–1184

Table 2 Comparison of the optimized 500 kW PM direct-drive generator system IGA

Optimization in Ref. [9]

Design specification Air gap radius rs [m] Stator length L [m] Stator slot height hs [mm] Pole pitch Tp [mm] Peak air gap flux density Bg0 [T] Peak stator yoke flux density Bysm [T]

1.13 0.54 60 61.1 0.78 1.13

1.08 0.55 64 68.3 0.77 1.2

Main dimensions Air gap [mm] Stator slot width bs [mm] Stator tooth width bd [mm] Stator yoke height hys [mm] Rotor yoke height hyr [mm] Magnet height hm [mm] Magnet width bm [mm]

2.26 9.2 11.2 14.2 14.2 5.8 42.4

2.15 11.7 11.1 15.9 15.4 6.3 47.8

Performance parameters and material weight Generator output frequency [Hz] Current density [A/mm2] Stator yoke losses [W] Stator teeth losses [W] Full load efficiency [%] Magnet weight [kg] Copper weight [kg] Core weight [kg] Total active material weight [ton]

30.9 4.02 784 2510 94.15 121 630 1860 2611

26.5 3.6 760 1950 94.2 124 779 1786 2690

Fig. 6. Three-dimensional representation of the stator outer diameter.

further consider wind climates to choose a suitable direct-drive PM generator systems.

3. Site matching of the optimized PM wind generator systems increases and the required rotational speed decreases. In addition, when the power rating is larger than 3 MW and the rated speed is in the range of 10–25 rpm, the stator outer diameters of the optimized generator systems may exceed 5 m, which may lead to have more difficulties of transport and assembly. Figs. 7 and 8 depict three-dimensional representations in terms of the torque density (torque per active weight) and the torque per cost for the optimized PM generator system. For the same rated power level, it can be seen that the torque density is almost similar at the different rated rotation speed, because it is independent of the choice of rotation speed in a certain speed range. Furthermore, both the torque per weight and the torque per cost become larger as the rated power increases. This is also a reason that the modern direct-drive wind turbines will become larger and larger in term of rating powers. From the viewpoint of the torque per cost, the larger PM wind generators with lower rated rotational speed may be more suitable, however, the cost of the generator system is inevitable to be more expensive, and the outer diameter also become larger. So it is necessary to

Fig. 5. Three-dimensional representation of the optimized system cost.

3.1. Wind turbine modeling Most large three-bladed horizontal axial wind turbines are designed to operate at the maximum aerodynamic efficiency, Cp max at the optimal tip speed ratio ranges between lopt ¼ 6–8 [15]. The maximum aerodynamic efficiencies typically vary between 0.4 and 0.5 [15]. The available rated shaft power, PT can be usually calculated as a function of the rated wind speed as

PT ¼

1 rCp max pD2 v3r 8

(13)

where r (kg/m3) is the air density, vr (m/s) is the rated wind speed and D (m) is the rotor diameter of a wind turbine. In this study, PM wind generator systems for different rated power and rated wind speed are assumed to have the same maximum aerodynamic efficiencies and the optimum tip speed ratio at rated loads. The system efficiency is also assumed to be 90% at the rated load for each design. The rotor diameter D and the rated wind

Fig. 7. Three-dimensional representation of the torque density.

H. Li, Z. Chen / Renewable Energy 34 (2009) 1175–1184

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Fig. 10. Rated wind speeds versus rated powers and rated rotor speeds.

Fig. 8. Three-dimensional representation of the torque per cost.

3.2. Calculation of annual energy output speed vr of wind turbines can be rewritten as a function of the rated rotor speed and the rated power

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3 5 8ðP =0:9Þ 60lopt N D ¼ 4 nr p rCp max

vr ¼

n1 pD 60lopt

(14)

(15)

where PN (W) is the rated output power of wind generator systems and nr (rpm) is the rated operating shaft speed. According to the various combinations of the rated power levels and the rated rotational speeds, Figs. 9 and 10 show the rotor diameter of wind turbine blades and the rated wind speed with Cp max ¼ 0.45 and lopt ¼ 7, respectively. For a given rated rotational speed, it can be seen that both the rotor diameter and the rated wind speed increase as the required output power of the generation system increases. Additionally at the same rated power level, the rated wind speed increases as the rated rotor speed increases in order to keep a constant of the optimum tip speed ratio, however, the corresponding rotor diameter of wind turbines has to decrease to keep the same rated power.

The production of electricity by a wind turbine at a specific site depends on many factors. These factors include the annual mean wind speed of a potential site and the wind speed characteristics of wind turbines, namely, the cut-in wind speed (vi), the rated wind speed (vr) and the cut-out wind speed (vc) and the hub height (H). Fig. 11 shows a typical power characteristic of a wind turbine. In this case, the typical cut-in wind speed (vi) is chosen to be 25% of its rated wind speed. The cut-out wind speed (vc) is set to be a constant (25 m/s) independent of the rated wind speed. The relationship of the output power and wind speed can be described as

PðvÞ ¼

8 0 > > > < ðvvi Þ3

3

ðvr vi Þ

0  v < vi PN

> > P > : N 0

vi  v < vr

(16)

vr  v < vc vc  v

The annual energy output (AEO) can be approximately estimated by integrating the product of the output power and the probability density function of wind speed for a specific site [13,14], which can be described as

AEO ¼ 8760

Z

vc

PðvÞf ðvÞdv vi

2

¼ 8760PN 4

Z

1 ðvr  vi Þ3

vr

vi

3

ðv  vi Þ f ðvÞdv þ

Z

vc

vr

3 f ðvÞdv5 (17)

where f(v) is Weibull density distribution, which is given as

Output power

PN

vi Fig. 9. Rotor diameters versus rated powers and rated rotor speeds.

vr

vc Wind speed

Fig. 11. Power characteristic of a wind turbine.

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H. Li, Z. Chen / Renewable Energy 34 (2009) 1175–1184

Table 3 Wind distribution parameters of potential sites at 10 M height Classifications of Sites no. Annual mean wind Shape parameter k Scale parameter c wind speed speed v0 [m/s] Low

1 2

3 4

1.52 1.95

3.33 4.52

Medium

3 4 5 6

5 6 7 8

1.96 2 2.1 2.15

5.64 6.78 7.91 9.05

High

7 8

9 10

2.3 2.4

11.35 12.57

kvk1 ððv=cÞk Þ e c c

ðk > 0; c > 1Þ

f ðvÞ ¼

(18)

where k is the shape parameter and c is the scale parameter. The wind speed of candidate sites is usually measured at 10 m anemometer height [14]. If these heights do not match the hub height of a wind turbine, it is necessary to extrapolate the wind speeds to the hub height of the turbine. The extrapolated wind speed, vH corresponding to the hub height is given by [13,14]

vH ¼ v0 ðH=H0 Þa

(19)

where v0 is the wind speed at height H0 ¼ 10 m above the ground level and a is the power index constant. In this case, a is assumed to be 1/7. The hub height of wind turbine is approximately calculated with 1.2 times of turbine blades diameter.

3.3. Site matching of the optimized PM generators Since the minimum required wind speed for a typical wind farm is about 2.54 m/s at 10 m anemometer height [14], eight different sites with annual mean wind speeds in the range of 3–10 m/s at 10 m height are used to determine the suitable PM generator systems. The majority of wind farm sites around the world have the annual mean wind speeds in the range of 5–8 m/s [15], sites with the wind speed ranges of 3–4 m/s and 9–10 m/s are taken into consideration as the low and high speed regions, respectively. According to the main wind data provided in Refs. [12–14], the wind distribution parameters of the investigated eight potential sites can be estimated by Weibull statistical model, which are presented in Table 3, respectively. For each potential site, the AEO is calculated for the 45 optimized PM generator systems, then the performance indexes of AEO per cost (AEOPC) are determined and compared, respectively, in which the cost is referred to the investigated generator system cost. At a given rated power, the maximum AEOPC (kWh/Euro) for each potential wind site is obtained, as shown in Table 4. Fig. 12 depicts

Fig. 12. Three-dimensional representation of the AEOPC.

a three-dimensional representation according to the results shown in Table 4. In Table 4, the bold face represents the maximum AEOPC at a give mean wind speed. As it can be seen that the performance of AEOPC increases for the same rated power machine as the annual mean wind speed of a potential site increases, it is beyond doubt that the wind turbine is more cost-effective when it is installed at a site with higher mean wind speed. In addition, at any site, the AEOPC is different for the wind turbines with different rated power levels. When the rated power is larger than 3 MW, the optimized direct-drive PM generator systems have a decreasing in AEOPC as the power ratings increase, this may be a reason that the cost of the direct-drive generator structure could rise more rapidly than the energy production for the system, because these generators have an optimum outer diameter that is larger than 5.0 m. Furthermore, though the AEOPC decreases as the MW wind turbines sizes increase, the decrease trend may be rather small as the annual wind speed increase of the installed sites. Fig. 13 depicts the AEOPC of the optimized PM generator systems in typical wind sites with annual mean wind speeds of 5–8 m/s at 10 m height. The results show that the PM generator systems with rated powers of 500 kW, 1 MW, and 1.5 MW have highest AEOPC in the typical wind sites. Moreover it could be concluded that the rated powers ranging from 500 kW up

Table 4 The maximum AEOPC (kWh/Euro) of each potential site for a given rated power Mean wind 100 kW 500 kW 1 MW 1.5 MW 3 MW 5 MW 7.5 MW 8.5 speed [m/s] MW

10 MW

3 4

9.89 15.55

11.21 18.37

11.09 10.97 17.76 17.94

10.07 16.99

8.55 7.65 13.341 13.34

7.17 6.74 12.56 11.86

5 6 7 8

19.84 24.30 27.91 30.43

25.07 30.54 36.54 42.34

24.53 30.82 36.90 42.37

24.362 30.09 34.83 39.36

21.97 28.14 33.14 37.63

20.70 27.24 32.63 36.97

19.72 26.17 31.53 36.66

9 10

32.16 33.44

46.47 49.62

46.48 46.69 50.19 50.26

43.26 46.33

42.07 45.58

40.66 43.22

40.47 39.38 43.13 42.12

24.36 30.18 36.78 42.52

18.91 25.36 30.78 35.51

Fig. 13. The AEOPC variation in typical wind regions.

H. Li, Z. Chen / Renewable Energy 34 (2009) 1175–1184

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Table 5 Rated rotational speed and rated wind speed of the optimum siting of wind turbine Mean wind 100 kW rpm 500 kW rpm (m/s) 1 MW rpm (m/s) 1.5 MW rpm (m/s) 3 MW rpm (m/s) 5 MW rpm (m/s) 7.5 MW rpm (m/s) 8.5 MW rpm (m/s) 10 MW speed [m/s] (m/s) rpm (m/s) 3 4

20(6.48) 20(6.48)

15(7.98) 15(7.98)

10(7.79) 10(7.79)

10(8.45) 10(8.45)

10(9.71) 10(9.71)

10(10.75) 10(10.75)

10(11.66) 10(11.66)

10(11.95) 10(11.95)

10(12.35) 10(12.35)

5 6 7 8

30(7.63) 30(7.63) 30(7.63) 30(7.63)

20(8.95) 20(8.95) 25(9.79) 30(10.53)

15(9.16) 15(9.16) 20(10.28) 30(12.09)

15(9.94) 15(9.94) 15(9.94) 20(11.15)

10(9.71) 10(9.71) 15(11.42) 15(11.42)

10(10.75) 10(10.75) 15(12.64) 15(12.64)

10(11.66) 10(11.66) 10(11.66) 15(13.71)

10(11.95) 10(11.95) 10(11.95) 15(14.06)

10(12.35) 10(12.35) 10(12.35) 15(14.52)

9 10

30(7.63) 30(7.63)

30(10.53) 30(10.53)

30(12.09) 30(12.09)

25(12.19) 25(12.19)

20(12.81) 20(12.81)

20(14.19) 20(14.19)

15(13.71) 15(13.71)

15(14.06) 15(14.06)

15(14.52) 15(14.52)

to 1.5 MW may be more suitable to match some potential sites of low, medium and high mean wind speed in term of AEOPC. Table 5 shows the corresponding rated rotor speeds and rated wind speeds of the optimum wind turbines with the maximum AEOPC in Table 4. The bold face represents the corresponding values with the maximum AEOPC at a give mean wind speed. As it can be seen in Table 5, the rated rotational speed or designed wind speed of a wind turbine may be different at the different sites in order to obtain the better AEOPC for a given rated power. That is, it is not possible to keep the most cost-effectiveness for all wind climates for any one optimum wind turbines. Moreover, when the rated power is larger than 3 MW, the range of rated rotational speed with the better AEOPC is about 10–15 rpm in the typical annual mean wind speed range of 5–8 m/s. It should be mentioned that the presented design in this study is mainly limited to the generator part. In addition, the generator system cost is roughly estimated, and the specific costs of power electronics converter and generator active materials may vary depending on the market. These factors may have a significant effect on the optimization results, so that the obtained indexes may not be necessarily to reflect exactly practical performances. In addition, the most cost-effective wind turbine size may be affected with sitespecific wind resources, transportation logistics, and construction and erection issues. However, the developed optimal design procedures and the optimized results may still be useful as a guide for design and choice of the suitable direct-drive PM generator systems. 4. Conclusions In order to investigate the performances of large direct-drive wind turbines with PM generators and the site matching at some potential sites in terms of the AEOPC, the optimal design models of the radial-flux PM generator have been developed with an improved genetic algorithm, and have been demonstrated with a 500-kW direct-drive PM wind generator. Forty-five PM generator systems with combinations of rated power ranges of 100 kW to 10 MW and rated rotational speed ranges of 10–30 rpm have been designed for the minimum generator system cost. The optimum results have been presented and compared by using the generator design indexes including the generator system cost, the stator outer diameter, the torque density and the torque per cost. The results have shown the PM machines with the lower rated rotational speed and the higher rated power level have much better performance of the torque density and the torque per cost, however, these generator also become more expensive and larger outer diameters. In addition, based on the design principle of the maximum wind energy capture of wind turbines and the optimized PM generators, the site matching of direct-drive wind turbines have been investigated for eight potential wind sites. According to the performance of AEOPC, the direct-drive PM generator systems with the rated power of 500 kW, 1 MW and 1.5 MW may be more suitable to respectively match some potential sites of low, medium and high

annual mean wind speed. From the cost-effective viewpoint of energy capture and costs, the wind generator system that has the highest AEOPC may be the best matched turbine by considering the information of the potential sites, wind turbine characteristics and generator design, though the decision of choosing the best wind generator system depends on the various factors, including utility loads, installation costs and grid requirements. Furthermore, the developed procedure could be used to choose the cost-effective direct-drive PM generator system for the wind power developer or the power utilities carrying out the planning of wind power station installation. Acknowledgements The research was supported by a grant from the EU sixth framework program UP-WIND project. The authors are grateful for the support. References [1] Bianchi N, Lorenzoni A. Performance magnet generators for wind power industry: an overall comparison with traditional generators. In: Opportunities and advances in international power generation; 18–20th March 1996. p. 49–54. [2] Polinder H, van der Pijl FFA, de Vilder GJ, Tavner P. Comparison of direct-drive and geared generator concepts for wind turbines. IEEE Transactions on Energy Conversion September 2006;21:725–33. [3] Poore R, Lettenmaier T. Alternative design study report: windPACT advanced wind turbine drive train designs study. Golden, Colorado: NREL; August 2003. Report no. NREL/SR-500-33196. [4] Cotrell JR. A preliminary evaluation of a multiple-generator drive train configuration for wind turbines. In: 2002 ASME wind energy symposium, AIAA aerospace sciences meeting and exhibit, 40th, collection of technical papers, Reno, NV; January 14–17, 2002. [5] Dubois MR. Review of electromechanical conversion in wind turbines. Report EPP00.R03; April 2000. [6] Chen Y, Pillay P, Khan A. PM wind generator topologies. IEEE Transactions on Industry Applications November 2005;41(6):1619–26. [7] Polinder H, Slootweg JG. Design optimization of a synchronous generator for a direct-drive wind turbine. In: Proceedings of the European wind energy conference and exhibition. Copenhagen; July 2–6, 2001. p. 1067–70. [8] Dubois MR. Optimized permanent magnet generator topologies for directdrive wind turbines. Ph.D. dissertation, Delft Univ. Technol., Delft, The Netherlands; 2004. [9] Grauers A. Design of direct-driven permanent-magnet generators for wind turbines. Ph.D. dissertation, Chalmers University of Technology, Goteborg; 1996. [10] Parviainen A. Design of axial-flux permanent-magnet low-speed machines and performance comparison between radial-flux and axial-flux machines. Ph. D. dissertation, Acta universitatis Lappeenrantaensis; 2005. [11] Versteegh CJA. Design of the Zephyros Z72 wind turbine with emphasis on the direct drive PM generator. In: NORPIE 2004. Trondheim, Norway: NTNU; 14–16 June 2004. [12] Jangamshetti SH, Rau VG. Site matching of wind turbine generators: a case study. IEEE Transactions on Energy Conversion December 1999; 14(4):1537–43. [13] Jangamshetti SH, Rau VG. Optimum siting of wind turbine generators. IEEE Transactions on Energy Conversion March 2001;16(1):8–13. [14] Marafia AH, Ashour HA. Economics of off-shore/on-shore wind energy systems in Qatar. Renewable Energy 2003;28:1953–63. [15] Harrison R, Hau E, Snel H. Large wind turbines design and economics. John Wiley & Sons Ltd, ISBN 0471-494569; 2000.

1184

H. Li, Z. Chen / Renewable Energy 34 (2009) 1175–1184

[16] Boldea I. The electric generators handbook – variable speed generators. Taylor & Francis; 2006. [17] Lundberg S. Performance comparison of wind park configurations. Ph.D. dissertation, Chalmers University of Technology, Goteborg, Sweden; 2003. [18] Li H, Chen Z. Overview of generator topologies for wind turbines. IET Renew. Power Gener 2008;2(2):123–38. [19] Jianzhong Zhang, Chen Z, Cheng M. Design and comparison of a novel stator interior permanent magnet generator for direct-drive wind turbines. IET Renew. Power Gener 2007;1(4):203–10.

[20] Back T, Fogel D, Michalewicz Z. Handbook of evolutionary computation. Oxford, England: Oxford University Press; 1997. [21] Hui L, Li H, Bei H, Chang YS. Application research based on improved genetic algorithm for optimum design of power transformers. In: Proceedings of the fifth international conference on electrical machines and systems, ICEMS 2001, vol. 1; 18–20 August 2001. p. 242–5. [22] Li H, Hui L, Can LJ, Guo ZJ. Optimization for induction motor design by improved genetic algorithm. In: Australasian universities power engineering conference (AUPEC 2004), Brisbane, Australia; 26–29 September 2004.