Electromagnetic-Thermal Design Optimization of the Brushless Dfig

1710 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 4, APRIL 2014

Electromagnetic-Thermal Design Optimization of theBrushless Doubly Fed Induction Generator

Hamed Gorginpour, Hashem Oraee, Senior Member, IEEE, and Richard A. McMahon

Abstract—In view of its special features, the brushless doublyfed induction generator (BDFIG) shows high potentials to beemployed as a variable-speed drive or wind generator. However,the machine suffers from low efficiency and power factor andalso high level of noise and vibration due to spatial harmonics.These harmonics arise mainly from rotor winding configuration,slotting effects, and saturation. In this paper, analytical equationsare derived for spatial harmonics and their effects on leakage flux,additional loss, noise, and vibration. Using the derived equationsand an electromagnetic-thermal model, a simple design procedureis presented, while the design variables are selected based onsensitivity analyses. A multiobjective optimization method usingan imperialist competitive algorithm as the solver is establishedto maximize efficiency, power factor, and power-to-weight ratio,as well as to reduce rotor spatial harmonic distortion and volt-age regulation simultaneously. Several constraints on dimensions,magnetic flux densities, temperatures, vibration level, and con-verter voltage and rating are imposed to ensure feasibility of thedesigned machine. The results show a significant improvementin the objective function. Finally, the analytical results of theoptimized structure are validated using finite-element method andare compared to the experimental results of the D180 frame sizeprototype BDFIG.

Index Terms—Brushless doubly fed induction generator(BDFIG), design optimization, power-to-weight ratio, spatialharmonics, wind generator.

NOMENCLATURE

| · | Denotes the absolute value.�(·) Denotes the phase angle.μ Magnetic permeability.B Magnetic flux density.f Frequency.I Current.Im{·} Denotes imaginary part.Kcs/Kcr Stator/rotor Carter factor.L Number of series loops per nest.lfe Stack length.Nc Number of conductors per coil.Nl Number of winding layers.

Manuscript received February 12, 2013; revised April 21, 2013; acceptedMay 21, 2013. Date of publication June 11, 2013; date of current versionSeptember 19, 2013.

H. Gorginpour and H. Oraee are with the Department of Electrical Engi-neering, Sharif University of Technology, Tehran 11365-8639, Iran (e-mail:[email protected]; [email protected]).

R. A. McMahon is with the Department of Engineering, University ofCambridge, CB3 0FA Cambridge, U.K., and also with Wind Technologies Ltd.,CB4 0WS Cambridge, U.K. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2013.2267705

nm Mechanical speed.Nss/Nrs Number of stator/rotor slots.P Power/pole pairs.Pr Number of rotor nests.Re{·} Denotes real part.s Slip.S Apparent power.X Reactance.Z Impedance.α Convection heat transfer constant.θ Spatial angle.θamb Ambient temperature.λ Conduction heat transfer constant.σcu/σfe Copper/iron electric conductivity.τ Pitch length.ω Angular frequency.

SUBSCRIPTS AND SUPERSCRIPTS

ins. Insulator.max/min/av Maximum/minimum/average value.nl/fl No load/full load.p(pw)/c(cw) Power/control winding.ss/rs Stator slot/rotor slot.rt/ry Rotor tooth/rotor yoke.st/sy Stator tooth/stator yoke.

I. INTRODUCTION

IN recent decades, electricity generation using renewableenergy sources, especially wind, has gained considerable

attention worldwide [1], [2]. Various generating systems havebeen proposed for wind turbines to convert wind mechanicalpower into electrical power [3]. Up to 70% of the installed windturbines incorporates doubly fed induction generator (DFIG)and a fractionally rated power converter. Other schemes stillhave technological and economic penalties such as high costof active materials, complexity of machine manufacturing andcontrol systems, and large and expensive converters, whichlimit their spread in wind power plants [3]. Among these,brushless DFIG (BDFIG) has attractive features to be thenext generation of wind generators. Having no brush and sliprings, a robust structure, and lower operating and maintenancecosts and requiring a smaller mechanical gear box besides itsadvantages in grid connection issues, such as better low-voltageride-through capability, are the benefits of BDFIG over DFIG[4]–[8]. The disadvantages of BDFIG are related to its designsince it has slightly larger dimensions and manufacturing cost

0278-0046 © 2013 IEEE

GORGINPOUR et al.: ELECTROMAGNETIC-THERMAL DESIGN OPTIMIZATION OF BDFIG 1711

as well as lower efficiency in comparison to a DFIG withthe same rating [9]. However, its promising feature warrantsfurther investigations on the design possibilities. It is thereforenecessary to optimize the machine structure in order to enhanceits performance, both technical and economic, against otheroptions.

Generally, optimization of an electrical machine is a multi-objective problem with several variables and constraints. First,the optimization variables, i.e., geometrical, electrical, andmagnetic parameters, are chosen based on the results of thesensitivity analyses. In the following, the objective functions(OFs) and constraints are formulated using these variables,and finally, an optimization solver is employed to find theoptimal values of the variables. The most crucial part of thisprocedure is the OF formulation, which can be considered as acombination of power loss, active mass weight or cost, volume,efficiency, cogging torque, noise, and vibration. Constraints areimposed to the problem due to mechanical, electrical, mag-netic, thermal, cost, volume, and manufacturing limitations.Optimization algorithms can be either single- or multiobjective,constrained or unconstrained, and nonlinear programming orrandom search-based approaches.

A limited number of investigations on BDFIG design isreported in literature. The relations of the machine rating andperformance characteristics based on dimensions are presentedin [9] using a simplified electric equivalent circuit (EEC; coremodel) and the definitions of the equivalent electric and mag-netic loadings. It is shown that the BDFIG suffers a reducedrating of about 30% in comparison to comparable DFIG arisingfrom its magnetic and electric loadings. A design optimizationis studied in [10] using the derived equations in [9] to maximizethe electromagnetic torque by optimal division of the equivalentelectric and magnetic loadings between stator windings. Therated produced torque of the optimized machine is increased to78% of the induction machine (IM) with the same frame size.

In this paper, the energy conversion mechanism in BDFIGis first discussed. Then, an analytical electromagnetic-thermalmodel based fully on design parameters is extracted by consid-ering spatial harmonics and their effects. The OF which is acombination of power-to-weight ratio, efficiency, power factor,rotor differential leakage inductance, and voltage regulation isformulated, and the design algorithm is presented in the nextstep. The results of an optimized BDFIG are verified using2-D finite-element (FE) analyses and are compared to theresults of the D180 frame size prototype machine.

II. ELECTROMAGNETIC-THERMAL MODEL OF BDFIG

A. Generating Operation

BDFIG has two stator windings with different pole pairnumbers to avoid direct magnetic coupling, known as thepower winding (PW) and the control winding (CW). PW isconnected to the grid, and CW is excited via a partially ratedbidirectional converter, which allows the machine to operatesynchronously in a limited range of shaft rotational speeds.The rotor is traditionally designed as nested loop (Fig. 1) [11],which couples the stator magnetic fields indirectly. The process,

Fig. 1. Nested-loop rotor of a D180 frame size BDFIG [11].

Fig. 2. Wind turbine system based on BDFIG.

which is called cross coupling, is studied in several referencessuch as [12]. The number of rotor nests or poles should beequal to the summation of stator winding pole pair numbersto provide indirect coupling between PW and CW magneticfields [9]. A wind turbine system based on BDFIG is shownin Fig. 2.

Turbine speed (8–15 r/min) is increased via gearbox to aspeed within the range of ±25% of the natural speed (nn =60fp/Pr). The power converter regulates the CW frequency(fc) at Prnm/60−fp. The rotor magnetic field due to CWrotational field comprises many spatial harmonics and two fun-damental components having Pp and Pc pole pairs rotating inbackward and forward directions, respectively. Rotor quantitiesalternate in slip frequency of fc − Pcnm/60. Hence, the rotorfield component of Pp pole pairs has the frequency of fp fromthe PW point of view. The PW-induced voltage due to thisrotating field is proportional to the field magnitude and thusto the CW voltage magnitude. Therefore, CW plays the roleof the field winding in the case of a conventional synchronousgenerator. A braking torque is developed because of the inter-action between PW field due to load currents and the rotor fieldof Pp pole pairs. One of the rotor harmonic fields in responseto the PW field has Pc pole pairs and forward direction. TheCW-induced voltage due to this component reduces the CWback electromotive force, and thus, the converter current isincreased. Hence, the decrement of the rotational speed and theincrement of the converter power are the effects of increasingthe electrical load. As mentioned previously, the rotor fieldcomprises many undesired spatial harmonics, which impairs theeffective performance by increasing the power loss, tempera-tures, and vibration level. The air-gap flux density distributionof D180 prototype BDFIG under specified operating conditionsis plotted in Fig. 3(a), and the spatial harmonic spectrum isshown in Fig. 3(b). It is obvious that the teeth saturation andslotting effects increase the spatial harmonics distortion. Hence,these harmonics should be considered in design studies.

Improving rotor configuration from spatial harmonics pointof view is investigated in [13], and a novel scheme comprisingseries loops is proposed. The new configuration has some


Fig. 3. (a) Air-gap magnetic field distribution of the nested-loop rotor ofFig. 1 and (b) its spatial harmonics spectrum (Pp = 2, Pc = 4, and nm =600 r/min).

Fig. 4. EEC of BDFIM [11].

interesting advantages, including lower rotor leakage flux,equal current magnitude in rotor bars, more uniform flux den-sity distribution in rotor teeth, and considerably lower rotor hot-spot temperature. Hence, the series loop configuration is usedin this study instead of nested loop.

B. EEC Model

The EEC is depicted in Fig. 4. The modified procedure ofcalculating EEC parameters when the rotor carries series loopconfiguration is presented in [13].

Spatial harmonics are considered in calculation of statorand rotor leakage inductances because of using the windingfunction approach in the calculation algorithm. Moreover, thefollowing points should be taken into account.

1) For low-power designs, a uni-stack stator with no axialduct is used, and both stator and rotor slots have asemiclosed shape. Also, parallel teeth shape is chosen inorder to have a higher fill factor in stator slots and limitedflux densities in rotor teeth. Fig. 5 indicates some of thegeometrical dimensions of the BDFIG structure.

Fig. 5. Structure of BDFIG.

TABLE IRELATIONS OF STATOR SLOT AND END-WINDING LEAKAGE

INDUCTANCES

The required relations for calculating slot and end-winding leakage inductances are stated in Table I.

2) The rotor winding often has deep bars in order to decreasethe rotor loop resistances. Hence, it is necessary to con-sider the effects of current displacement on the reductionof the bar leakage inductance and increment of the barresistance. The correction factors, as stated in (1) and(2), modify the slot leakage inductance and slot resistanceterms, respectively [14]

Φ(ζ) = ζsinh(2ζ) + sin(2ζ)

cosh(2ζ)− cos(2ζ)(1)

Ψ(ζ) =3

2ζ

sinh(2ζ)− sin(2ζ)

cosh(2ζ)− cos(2ζ)(2)

where ζ is the penetration depth and its relation is [14]

ζ = Hrs

√μ0πσcufr. (3)


3) The values of the stator winding magnetizing inductancesare evaluated by assuming infinite magnetic permeabilityfor stator and rotor cores. Hence, the magnetizing currenthas much lower magnitude than its actual value. Thisleads to underestimation of the power converter rating asthe machine is designed to have unity power factor in PW,while all reactive power including magnetizing power ishandled by CW. Therefore, a correction factor as the ratioof air-gap magnetomotive force (MMF) drop to the totalMMF drop in the flux route of the related winding shouldbe considered [14].

C. Flux Density Distributions

The MMF distribution of a three-phase and P pole pairwinding can be expressed as follows:

�s(t, θ) =3ImaxNsw

π

⎧⎨⎩

∞∑h=P (6n+1)

1

hkwh sin(ωt− hθ)

+∞∑

h=P (6n−1)

1

hkwh sin(ωt+ hθ)

⎫⎬⎭ , n = 0, 1, 2, . . . (4)

where Imax is the current amplitude, Nsw is the number ofwinding turns per phase, and kwh is the winding factor of thehth spatial harmonic order.

The MMF of the nested-loop rotor can be expressed as

�r(t, θr) =Pr

π×

L∑m=1

Im,max

×{ ∞∑

h=nPr+Pc

1

hsin

(hαm

2

)sin

× (ωrt+ hθr − nπ(Pr − 1))

+

∞∑h=nPr+Pp

1

hsin

(hαm

2

)sin

× (ωrt− hθr + nπ(Pr − 1))

}(5)

where Im,max is the magnitude of the current flowing in themth loop with αm span angle.

As can be seen in Fig. 3, in addition to Pp and Pc pole paircomponents, low-order spatial harmonics with considerablemagnitudes are present in the rotor magnetic field. Undesiredharmonics will increase when the rotor teeth are magneticallysaturated. It can be shown that saturation causes harmonics ofnPr ± 3Pp and nPr ± 3Pc orders to be superimposed on theair-gap magnetic field [12].

Examinations show that an appropriate description of mag-netic field distribution on a stator slot pitch is [15]

B(θ)=

{[1−βs−βs cos

(πDag

1.6wssoθ)]Bmax |θ|≤ 0.8wsso

Dag

Bmax0.8wsso

Dag≤|θ|≤ π

Nss

(6)

where

βs =(1− u)2

2(1 + u2), u =

wsso

2g+ 1 +

(wsso

2g

)2

. (7)

The distribution of air-gap conductance 1/g(θ) can be ob-tained using (6) as [15]

Gag(θ, θr) ≈1

gkcskcr+

a1kcr

cos(Nssθ) +a2kcr

cos(2Nssθ)

+b1kcs

cos (Nrs(θ − θr))

+b2kcs

cos (2Nrs(θ − θr)) +a1b12

g

× [cos ((Nss +Nrs)θ −Nrsθr)

+ cos ((Nss −Nrs)θ +Nrsθr)] (8)

where kcs and kcr are stator and rotor Carter’s factors, respec-tively. These factors are calculated using (9) by substitutingappropriate parameters of stator and rotor. an and bn are relatedto stator and rotor conductance distributions, respectively, andcan be obtained using Fourier series calculations of (6). Thegeneral forms of an and bn coefficients are stated in (10).Hence, the air-gap conductance consists of spatial harmonicsof c1Nrs, c2Nss, and c3Nss ± c4Nsr orders superimposed onthe large average value of 1/(gkcskcr), where c1−4 = 1, 2, . . .

kcs(\cr) =τss(\rs)

τss(\rs) − 1.6βs(\r)wsso(\rso)(9)

an =βs

gFn(wsso/τss), bn =

βr

gFn(wrso/τrs)

Fn(x) =4

nπsin(1.6nπx)

{0.5 +

(nx)2

0.78− 2(nx)2

}. (10)

The air-gap distributions of stator and rotor magnetic fields areevaluated by multiplying their MMF distributions by μ0Gag.

The flux densities predicted using the presented relationsare slightly larger than the FE results because of neglectingsaturation. Stator and rotor teeth saturation leads to flat-toppedflux density distribution in air-gap. The physical air-gap lengthis modified in order to consider teeth saturation. The correctionfactor for the hth harmonic is (11) [15]. The factor is a functionof the magnetic permeability of the stator and rotor teeth,i.e., (μfe)s and (μfe)r, respectively. Hence, an iterative processusing the μ-B curve is necessary for finding the correctionfactor

kgh = 1 +

[sin(hπ/Nss)

hπ/Nss

τssτss − wsst

hss1 + hss2

(μfe)s

+sin(hπ/Nrs)

hπ/Nrs

τrsτrs − wrst

hrs

(μfe)r

]μ0

0.92g. (11)


Fig. 6. Equivalent thermal network of a radial sector of BDFIG.

D. Lumped Parameter Thermal Model

The design of an electrical machine is a combined elec-tromagnetic and thermal problem. Optimal design is reachedby maximizing the active mass utilization while limiting thetemperature to its maximum allowed value. The maximum tem-perature rise of an industrial machine with insulation class F,in accordance to IEC 60034-1, is 80 K (thermal class B)considering ambient temperature of 40 ◦C and hot-spot tem-perature rise of 10 K. There are valuable design and manufac-turing experiences on IM (cage or wound rotor) providing thedesigners with the optimal boundaries of average air-gap fluxdensity and stator and rotor current densities. Choosing fluxand current densities in these ranges guarantees optimal designwith no excessive temperature rise. However, such informationis not available in the case of BDFIG. In addition, the rotorcore loss, in contrast to conventional IM, is considerable, andthe total iron and additional losses are higher than IM. Hence,an accurate thermal model has to be developed in conjunctionwith the electromagnetic model. The actual distributions of thepower losses should be considered in this model, while therequired data for calculating the network elements are designparameters.

The equivalent thermal network of a radial sector of onestator slot span angle (2π/Nss) is depicted in Fig. 6. Pfe,s(\r)denotes the iron loss dissipated in a stator (\rotor) radial sector,and Pcu,slot and Pcu,ew are the resistive losses occurring inslot and overhang regions, respectively. The definitions of thethermal resistances are given in Table II. Note that all of therotor thermal resistances except Rr1 should be modified byNss/Nrs factor.

TABLE IIDEFINITION OF THERMAL RESISTANCES IN FIG. 6

TABLE IIISOME PARAMETERS AND RELATIONS USED IN CALCULATION OF

THERMAL RESISTANCES

The additional iron losses due to surface losses of spatialharmonics and flux pulsation are superimposed to the losssources of stator and rotor cores.

Examinations show that the Taylor number value for conven-tional ratings of BDFIGs is much lower than the critical valueof 1740. Hence, the flow mode in the air-gap region is laminar,and the air-gap thermal resistance can be calculated as [16]

Rag =gNss

λairπDaglfe. (12)

The external-frame area in calculation of Rs0 is considered tobe three times wider because of the existence of fins. However,the total surface of the external frame should be multiplied bythe overall surface efficiency (η0) which indicates the ther-mal performance of finned surface and is a function of findimensions, thermal conductivity of frame, and convection heattransfer coefficient of cooling air [17]. η0 and some of theparameters with considerable effects and high uncertainty intheir related values are given in Table III.

The thermal resistances can be evaluated based on dimen-sions and thermal constants by utilizing simple relations pre-sented in [16] and [18] for IMs of the TEFC design.


Friction loss of bearings which increases the temperature inthe bearing region can be expressed as [14]

Pfric. = 0.0005πnmDriMrotor (13)

where Mrotor is the rotor mass.Also, windage loss due to the friction between cooling air

and surfaces is expressed as follows [14]:

Pwindage = 0.0042D3roπ

2n2mlfe. (14)

III. PERFORMANCE CALCULATION

In this section, analytical equations are derived in orderto calculate electromagnetic torque, efficiency, power factor,voltage regulation, and rotor differential leakage inductance.For higher accuracy, the effects of all loss components areconsidered in calculation of efficiency. As the rotational speedis allowed to vary in the range of ±0.25nn, there is a mainconcern on the value of rotor speed in the calculations. Toaddress this, investigations on generating performance revealthat the minimum torque is developed at the lower band of thespeed range. In addition, copper and core losses increase bydeviating from the natural speed. Hence, satisfaction of designcriteria at minimum speed warrants the desired performancethroughout the speed range.

Total copper loss is the summation of PW, CW, and rotorwinding losses

Pcu = 3RpwI2pw + 3RcwI

2cw + PrRrI

2r . (15)

The core loss is evaluated using modified Steinmetz equation(MSE) as [19]

Pfe = kkfα−1eq Bβ

maxf + kef2B2

max (16)

where kh, ke, α, and β are MSE constants determined for eachsteel type via experiments and

feq =2

(Bmax −Bmin)2π2

1/f∫o

(dB

dt

)2

dt. (17)

Using the MSE approach makes it possible to handle thecomplex situation occurring for flux density distributions, i.e.,magnetic fields of different frequencies, and also eliminates thethird term of iron loss, i.e., excess eddy current loss.

As mentioned previously, the additional losses due to spatialharmonics are more pronounced in BDFIG compared to IM.Stator teeth surface and pulsation losses are computed using(18) and (19), respectively. These components can be expressedfor rotor teeth by appropriate substitutions in (18)–(21) [14]

Psurf.,s =0.08Nss(τss − wss1)lfe

(τss4

)2(√

2

πβKcrBag,av

)2

×(Nrs

nm

60

)1.5

×√

π3σfe/μst,av (18)

Ppul.,s=1.8keΔB2st,av

(Nrsnm/60

50

)2

Mst (19)

where

β =0.5

⎛⎝1− 2/h√

1 + (2/h)2

⎞⎠ h =

wss1

g(20)

ΔBst,av =Bst,avβKcr√

2

sin (πNss/Nrs)

πNss/Nrs. (21)

The efficiency of the energy conversion process is defined asfollows:

η =Pout

Pin(22)

where

Pin =Tel(1− smax)πnn/30 + 3VcwIcw(P.F.cw) (23)

Pout =Pin − Pcu − Pfe − Padd. − Pfric. − Pwindage. (24)

The electromagnetic torque (Tel) is as follows [9] (Fig. 4):

Tel =3

2πfp|Zr||Epw||Ecw|

√P 2p + P 2

c − 2PpPc cos(2�Zr)

× sin

(�Epw − �Ecw − tan−1 (Pp − Pc) cos(�Zr)

(Pp + Pc) sin(�Zr)

).

(25)

The power factor from CW point of view is

P.F.cw = cos

(tan−1 Im {Zcw

in }Re {Zcw

in }

)(26)

where

Zcwin =

scsp

R′′c+jX ′′

c+jX ′′

mc (Zr+jXmp‖(R1+jX1+Zload))

Zr+jX ′′mc+jXmp‖(R1+jX1+Zload)

(27)

where Zload is the full-load impedance of the unity powerfactor.

Furthermore, the rotor differential leakage inductance for themth loop with τm span pitch length which measures the spatialharmonic distortion can be expressed as

Drilfeμ0P2r

πg

∑h

(sin (hτm/Dri) /h)2 . (28)

The terminal voltage is developed from PW back EMF(Epw) after some internal drops. The magnitude of Epw is in-creased by increasing the load current in order to have constantterminal voltage. Under extreme operational conditions, theCW voltage and rating should not exceed their limits, i.e., Vter

and 30%Spw, respectively. An important operational parameterwhich indicates the internal voltage drop of the generator isvoltage regulation (VR). VR is defined as (29) for full loadcondition

V R =Vter,nl − Vter

Vter,nl

∣∣∣∣∣Vcw=cst.

(29)


As previously stated, higher levels of noise and vibration arereported in BDFIGs compared to IMs. Hence, it is necessary totake into account the magnetic radial forces and the resultingmechanical vibration in the design stage.

The natural frequency of mth vibration mode (m ≥ 2) is [15]

Fm≥2 =hsy

π(Dso − hsy)2

√Es

3kfe.Δ.ρfe

m(m2 − 1)√1 +m2

[Hz] (30)

where Es, i.e., module of elasticity, is equal to 1.521×1011 N/m2 for laminated magnetic steels. Core stacking factor(kfe) and mass density (ρfe) are typically 0.95 and 7800 kg/m3,respectively. Δ is defined as the increment factor of stator massdue to windings and teeth [15].

Only the frequencies of low-order magnetic forces (m ≤ 6)are considered in the vibration analysis. The frequency of thelowest order harmonic (Nss −Nrs − Pr) can be concluded as

(Nrs + Pr)nm/60. (31)

IV. DESIGN PROCEDURE

In this paper, the optimization goals are chosen as maximiz-ing the power-to-weight ratio, efficiency, and power factor aswell as minimizing VR and rotor differential leakage induc-tance. Therefore, the OF is defined as

OF(x) =(Pout/weight)

k1 ηk2(P.F.cw)k3

(VR)k4

(Lrleak,diff

)k5(32)

where x is the optimization variable vector and ki(i = 1 . . . 5)can be set equal to 0 or 1.

Different design variables affect the machine performancein various ways. The dimensions and specifications of D180frame size prototype BDFIG are employed to investigate theeffects of design variables. The prototype machine has a naturalspeed of 500 r/min and two and four pole pairs on its PW andCW, respectively. The machine is manufactured by changingthe winding arrangements of a 22-kW four-pole industrialIM [11].

The implemented model in MATLAB software makes itpossible to investigate the effects of variation of each parameteron the performance characteristics. The results are important toselect the design variables.

In Fig. 7, the dependence of electromagnetic torque on someparameters is presented while keeping the other parametersconstant. As expected, the increments of machine diameter,axial length, PW and CW turns, and current magnitudes as wellas the decrement of air-gap length increase the produced torque.However, the rotor slot pitch angle affects the rotor leakageinductance, and the CW phase angle affects the air-gap fluxdensity distribution in more complex behaviors. Also, thereis an optimum value for the number of loops per rotor seriesgroup. The rotor impedance and turn ratios are functions of thenumber of loops.

Fig. 8 shows the effects of air-gap diameter, axial length, androtor and stator slot heights on the system efficiency. Stator

Fig. 7. Electromagnetic torque versus some design variables.

Fig. 8. Efficiency versus some design variables.

Fig. 9. Variations of PW-induced voltage, rotor leakage inductance, CWpower factor, and output power versus design variables.

winding resistances and thus copper loss are subjected to de-crease by increasing the stator slot height. However, increasingthe rotor slot height causes decrement of rotor copper loss andincrement of rotor leakage inductance simultaneously.

Fig. 9 reveals the dependences of terminal voltage on CWphase angle, rotor leakage inductance on number of rotor loops,


TABLE IVDESIGN VARIABLES

CW input impedance on stack length, and output power on rotorslot pitch angle.

The design variables are selected based on the performedsensitivity investigations and by considering all the effectiveparameters (Table IV).

The variation range of each variable is wide to ensure reach-ing the global optimum point. Before presenting the designprocedure, it is necessary to determine other parameters whichare not considered as design variables. Note that the aim ofthis paper is to address the design of low-power BDFIGs.1) The optimum pole pair numbers for PW and CW are 2 and 4,respectively [9]. The stator is assumed to have 48 slots, whichmeans 4 and 2 slots/pole/phase for PW and CW, respectively.2) The PW is considered having 2 layers with 10/12 shortpitching in order to reduce the spatial harmonics. 3) To havea higher slot fill factor, the CW is considered as one layer andfull pitch. 4) Because of the small changes in slot and windinginsulation thicknesses for below 1 kV, both stator windings areconnected in delta (Δ) arrangement. This causes the reductionof phase currents (in comparison to star (Y) connection) andthus reduces loss and temperature rises. 5) Also, the CW andPW terminal voltages are set equal to eliminate the matchingtransformer. 6) The stator slot fill factor (0.62) is dividedbetween the coils of two windings. Hence, the cross-sectionalareas of the winding conductors can be calculated based on thecoil turns and rated currents.

The flowchart of the optimization procedure is depicted inFig. 10.

The imperialist competitive algorithm (ICA) is effectivelyemployed as the numerical solver. ICA has better stabilityand convergence rate in comparison to the other optimizationalgorithms [21]. The concepts of ICA are summarized in thenext section.

V. ICA

ICA is a new optimization algorithm, which is based onsocial and political evolution of humanity. The algorithm, sim-ilar to other evolutionary optimization methods, starts with apopulation. Each population in ICA is called a country. The

Fig. 10. Flowchart of the BDFIG design procedure [∗Sy is the yield strength(190 MPa for stainless steel at 50 ◦C), n ≥ 1 is the safety factor, and ∗∗Δciis set to 0.1 times the maximum or minimum bound of cith constraints(0.1ci,max \min)].

cost of a country is found by evaluating the cost function f atthe variable positions. The initial population size is generatedto start the optimization. Specified number of this population isconsidered as imperialists and the remaining as colonies, eachof which belongs to an empire.

Imperialist countries start to improve their countries. Thetotal power of an imperialist, which is the counterpart of thefitness value in genetic algorithm, is defined as the summationof the imperialist country and a percent of dominated countrypowers, i.e.,

T.C.n = Cost(imperialistn)

+ξmean {Cost(colonies of empiren)} . (33)

All empires try to take possession of colonies of other em-pires and control them. This imperialist competition graduallybrings about a decrease in the power of weaker empires andan increase in the power of more powerful ones. The ICAflowchart is depicted in Fig. 11.


Fig. 11. ICA flowchart.

TABLE VICA PARAMETER SETTINGS

TABLE VISPECIFICATIONS OF LAMINATED CORE (M800-65A)

After a while, all the empires except the most powerful onewill collapse, and all the colonies will be under the control ofthe unique empire. At this time, ICA reaches the optimal pointand stops.

VI. OPTIMIZATION RESULTS

The ICA parameter settings used in the optimization problemare listed in Table V. The specifications of electric steel used instator and rotor core are given in Table VI.

The optimization is carried out for different OFs using (32).In the first step, K1−2 = 1, and K3−5 = 0, which means that

TABLE VIIPROTOTYPE AND OPTIMIZED MACHINE SPECIFICATIONS

only the efficiency and power-to-weight ratio are optimized,and then, all of the considered functions are optimized simul-taneously by setting K1−5 = 1. The optimal design values aswell as the prototype machine values are stated in Table VII.The calculations are carried out for a constant rotational speedof 375 r/min. The intervals of design variables are set broadin order to create a large searching domain. High convergencerate and low computational time are achieved by using ICA.The visual descriptions of the prototype and optimum designsare provided in Fig. 12.

Optimized structures have significantly better OF values thanthe prototype. In comparison, an optimal BDFIG has a higherDag/lfe ratio than conventional IMs. This is due to the factthat BDFIG is inherently a high-pole-number machine. In thisway, more spaces are available for stator and rotor windings byincreasing the air-gap diameter. Hence, lower resistances andhigher magnetizing inductances for stator windings and betterformation of rotor loops with lower impedance are achieved.The main goal of optimization, i.e., power-to-weight ratio,is improved considerably. One of the important reasons, inaddition to increasing Dag, is the use of the series loop configu-ration instead of the nested-loop scheme, which results in equalcurrent magnitude in all of the rotor bars and more uniformdistribution of rotor flux density. Furthermore, elimination oflocally saturated hot spots of the rotor teeth allows an increasein the average air-gap flux density.

The value of the power-to-weight ratio of the optimizedBDFIG is now comparable to that for standard IMs with a


Fig. 12. Two-dimensional cross section of the (a) prototype machine andoptimum designed machines by setting (b) K1−2 = 1, K3−5 = 0, and(c) K1−5 = 1 (vertical line indicated the axial length).

synchronous speed of 500 r/min (120–180 W/kg at this rangeof powers). It should be noted that the rating is increased byincreasing the rotational speed. This is due to the improvementof the heat transfer capability and the reduction of the converterabsorbing power as well as injection of power to the gridat supernatural speeds. The key limitations against increasingpower to weight are maximum boundaries of flux densities andtemperature rise. Higher temperatures are reported in BDFIGsthan IMs since BDFIG has lower rotational speeds and thuslower cooling air speeds, higher rotor power loss due to largeslip values, and higher stator power loss due to dedicated slotspace to coils of two windings and existence of two rotatingflux waveforms in the core. In addition, air-gap flux densityconsists of two rotating flux density waveforms of differentfrequencies and pole numbers. This causes a larger differencebetween maximum and average values of air-gap flux densityin comparison to IMs. Hence, there is local saturation in lowaverage air-gap flux densities in the case of BDFIGs. Thisimplies that the maximum boundaries of flux densities can bechosen slightly higher.

The machine rating can be improved by using external cool-ing system, choosing higher insulation class and electric steelsof higher grades (according to EN 10106). These increase theinitial cost and are suitable for large machines.

VII. FE ANALYSIS AND EXPERIMENTAL STUDY

The effectiveness of the design mainly depends on the va-lidity of the developed model. To validate the model and toconfirm the results of the optimization, 2-D FEM is employed.The optimal design by setting K1−5 = 1 is analyzed using acommercial software (Maxwell 12.0). Two-dimensional pat-terns of flux density and flux lines in the cross section of the

Fig. 13. Two-dimensional patterns of (a) flux density and (b) flux lines inoptimized BDFIG.

TABLE VIIICALCULATION AND FEM RESULTS

machine are shown in Fig. 13. The maximum flux densities indifferent parts of the machine are well limited to the specifiedbounds. As evident, the flux line pattern is completely differentfrom the pattern occurring in IMs.

The analytical and FE results are presented in Table VIII. Theoverall efficiency in FE analysis is greater than the calculatedvalue because of the inaccuracies in formulation of core loss inFE software. There is a good agreement between the values ofother parameters, which confirms the validity of the modelingapproach.


Fig. 14. Experimental test rig.

Fig. 15. PW-induced voltage versus rotor speed (CW is excited via a three-phase 240-Vrms 50-Hz voltage source, and PW is open circuited).

Also, a study has been conducted to evaluate the performanceof the optimized machine. The experimental setup is shown inFig. 14. A dc machine equipped with an ABB DCS800 drivesystem is mechanically coupled with the D-180 BDFIG as theprime mover. A Magtrol TMB312 torque transducer and anincremental encoder with 10 000 pulses per revolution are usedto measure torque and shaft rotational speed, respectively. Thevoltage and current of each stator phase are measured usingLEM LV 25-p and LEM LTA 100-p transducers, respectively.

The CW is excited via a three-phase 240-Vrms 50-Hz voltagesource, and the induced voltages in open-circuited PW aremeasured. Fig. 15 shows the measured voltages versus rotorspeed for the prototype machine and calculated ones for theoptimized machine of Fig. 12(c)(K1−5 = 1). As can be seen,the optimized machine has a significantly higher ability tocreate cross coupling between stator windings. It should benoted that the low values of the induced voltage magnitude arebecause of high CW frequency (fc,max = 0.3fp).

VIII. CONCLUSION

In this paper, an affective, practical, and simple design pro-cedure based on the electromagnetic-thermal models has beenpresented for BDFIG. Magnetic saturation, current displace-ment, spatial harmonics, vibration, and manufacturing limita-tions have been considered in the procedure. A multiobjectiveoptimization using ICA has been developed to maximize thepower-to-weight ratio, efficiency, and power factor and simul-taneously to minimize the rotor leakage inductance and voltageregulation. The results show that the objectives, particularlypower-to-weight ratio, can be significantly improved by appro-priate selection of the machine parameters. A 2-D FEM andexperiments have been carried out to confirm the precision of

the derived models and equations, and the effectiveness of theoptimization method.

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Hamed Gorginpour was born in Bushehr, Iran,in 1985. He received the B.Sc. degree in electricalengineering from Shiraz University, Shiraz, Iran, in2007 and the M.Sc. degree in electrical engineeringfrom Sharif University of Technology, Tehran, Iran,in 2009, where he is currently working toward thePh.D. degree.

His research interests include electrical machinedesign and modeling, finite-element analysis, andpower electronics and drives.

Hashem Oraee (SM’98) received the B.Eng. degreein electrical and electronic engineering (first classhonors) from the University of Wales, Cardiff, U.K.,in 1980 and the Ph.D. degree in electrical machinesfrom the University of Cambridge, Cambridge, U.K.,in 1984.

He is currently a Professor of electrical engineer-ing with Sharif University of Technology, Tehran,Iran. His research interests include electrical energyconversion and power quality. He is also active incommercialization of brushless doubly fed induction

generators in wind generation industry.

Richard A. McMahon received the B.A. degreein electrical sciences and the Ph.D. degree fromCambridge University, Cambridge, U.K., in 1976and 1980, respectively.

Following his postdoctoral work on semiconduc-tor device processing, he was appointed as Uni-versity Lecturer in electrical engineering with theDepartment of Engineering, Cambridge University,in 1989 and became a Senior Lecturer in 2000. Hisresearch interests include electrical machines andpower electronics, particularly for wind and wave

power systems.

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Electromagnetic-Thermal Design Optimization of the Brushless Dfig