determination of steady-state and dynamic control

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 49, NO. 3, MAY/JUNE 2013 1343

Determination of Steady-State and Dynamic ControlLaws of Doubly Fed Induction Generator Using

Natural and Power VariablesAdeola Balogun, Member, IEEE, Olorunfemi Ojo, Fellow, IEEE,

Frank Okafor, Member, IEEE, and Sosthenes Karugaba, Member, IEEE

AbstractA doubly fed induction generator model is presentedwhereby the natural and power variables are the state variables.The natural variables are the electromagnetic torque (Te), thereactive torque (Tr), the magnitude of the rotor flux linkage(r), the magnitude of the stator flux linkage (s), and therotor speed (r). The power variables are the real power (Pf )and reactive power (Qf ) generated/absorbed by the grid-sideconverter into/from the grid. Simulation of the dynamic naturalvariable model of the induction machine is compared with a vectorvariable simulation. Steady-state operating regions are establishedfor various power factor operations. The optimal stator powerfactor operation is estimated. A direct control of torque and powervariables is developed. The robustness of the developed controlagainst rotor parameter variation is investigated using small signalanalysis and is compared with vector control. Results are shownfor a 5-hp machine.

Index TermsDoubly fed induction generator (DFIG), naturalvariables, power variables, robustness, stator power factor, windenergy conversion.

I. INTRODUCTION

THE CONVENTIONAL ways of modeling induction ma-chines entail obtaining the machine model in terms ofits flux linkages or rotor and stator currents as state variables,as given in [1] and [2]. These variables in qd, however, donot remain the same in all reference frames of transformation,which implies that they differ from one reference frame tothe other. The reasons for this can be derived from the factthat the reference frequency , which determines the angleof transformation , is inherent in such models. Hence, theqd state variables are dependent on choice of reference frame

Manuscript received September 12, 2011; revised August 12, 2012; acceptedAugust 30, 2012. Date of publication March 20, 2013; date of current versionMay 15, 2013. Paper 2011-EMC-494.R1, presented at the 2010 IEEE EnergyConversion Congress and Exposition, Atlanta, GA, USA, September 1216,and approved for publication in the IEEE TRANSACTIONS ON INDUSTRYAPPLICATIONS by the Electric Machines Committee of the IEEE IndustryApplications Society.

A. Balogun and F. Okafor are with the Department of Electrical andElectronics Engineering, University of Lagos, Lagos 101017, Nigeria (e-mail:[email protected]; [email protected]).

O. Ojo is with the Center for Energy Systems Research, Department ofElectrical and Computer Engineering, Tennessee Technological University,Cookeville, TN 38505 USA (e-mail: [email protected]).

S. Karugaba is with the Department of Electrical Engineering, Dar esSalaam Institute of Technology (DIT), Dar es Salaam, Tanzania (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/TIA.2013.2253532

of transformation. However, these state variables are invariantafter inverse transformation back to the abc reference frame.Such models usually form the basis of various vector controlschemes for induction machine applications, which are evidentin [3][8]. In vector control schemes, the torque and powerquantities of the doubly fed induction generator (DFIG) areregulated indirectly by some inner loop current control usedto generate the corresponding rotor voltage vectors. Becausevector control schemes are generally known to be dependenton system parameters [8], alternative control schemes are con-sidered in literature to avoid such dependences. Direct scalarcontrol schemes such as direct torque control (DTC) and directpower control (DPC) have been alternative choices of interestwhich are independent on system parameters. ConventionalDTC and DPC use nonlinear controllers such as hysteresisregulators to control the torque and power directly [9][15].Although some efforts have been made in literature to make theswitching frequency of the DTC and DPC techniques constant[9], [12], tables of optimal switching are still usually required.

Models of squirrel-cage induction machines developed in[16][18] had the torque quantities as state variables. In [17]and [18], the model termed the natural variable was used toachieve controller design for efficiency optimization. In [19],a wound rotor induction machine of a DFIG was presentedin terms of torque variables, but details were not given aboutmodeling the grid-side converter (GSC) having power as statevariables. The multivariable dynamic model of the voltagesource converter presented in [20] had the power quantity asthe state variables. However, the converter was not connectedback to back to another converter.

In this paper, therefore, the methods of [16][21] are adoptedsuch that inputoutput linearization is used to linearly relatethe torque and power outputs of the DFIG to the rotor voltagevectors. The resultant model is referred to as the natural/powervariable model because the state variables are the torque vari-ables of the induction machine, the power variables of the GSC,and the rotor speed r. Since the natural and power variablesare scalar quantities, they do not change with respect to changein angle of reference frame transformation . Hence, they areinsensitive to the reference frame angular velocity . Therefore,such a model can be effectively utilized in developing directcontrol of torque and power schemes, which do not require anyform of inner loop current regulation. Another advantage is that,since a linear relationship exists between the output variable

0093-9994/$31.00 2013 IEEE

1344 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 49, NO. 3, MAY/JUNE 2013

and the respective rotor voltage vector, then the switching ofthe rotor voltage quantities becomes symmetrical.

II. INDUCTION MACHINE MODEL

Fig. 1 shows a schematic representation of a DFIG driven bya mechanical turbine via a gearbox. The stator and rotor voltageequations of the induction machine in qd arbitrary referenceframe [1] are given in (1)(8). The model accounts for the coreloss with a core loss resistance rc, as shown in Appendix A,such that kc = 1 + (rs/rc)

vqs = rsiqs + kcpqs + kcds (1)

vds = rsids + kcpds kcqs (2)

vqr =mqrvdc/2 = rriqr + pqr + ( r)dr (3)vdr =mdrvdc/2 = rridr + pdr ( r)qr (4)

where

qs =Lsiqs + Lmiqr (5)

ds =Lsids + Lmidr (6)

qr =Lriqr + Lmiqs (7)

dr =Lridr + Lmids. (8)

The qd components of the stator voltage (vqs and vds) and therotor voltage (vqr and vdr) in (1)(4) are the transformed valuesof the abc reference frame stator voltage and rotor voltage, re-spectively. The stator voltage in (1) and (2) is expressed in termsof the stator current (iqs and ids), stator flux linkage (qs andds), and the stator reference speed (). Similarly, in (3) and(4), the rotor voltage is likewise expressed in terms of the rotorcurrent (iqr and idr), rotor flux linkage (qr and dr), statorreference speed (), and r, which is the rotor speed. Ls andLr represent the stator and rotor self-inductances, respectively,and rs and rr represent the stator and rotor resistances, respec-tively, while Lm is a value 3/2 the magnetizing inductance [1].Observe in (1)(4) that the operator p = d/dt. A change ofvariable from the abc reference frame to the arbitrary qd ref-erence frame is achieved using the transformation matrix givenin (13) and (15) for the stator and rotor circuit variables, re-spectively. In (9)(12), fqd and fabc represent the voltage, cur-rent, and flux variables in the qd and abc reference frames,respectively. The transformation angle given in (14) is in itsdefinite integral form and is defined in terms of an arbitraryelectrical speed . The angular displacement of the rotor r isdefined in terms of the rotor electrical speed r by (17)

fqds =Ts()fabcs (9)fqdr =Tr()fabcr (10)

where

(fqds)T = [fqs fds] (11)

(fqdr)T = [fqr fdr]. (12)

Fig. 1. DFIG for wind power applications.

All of the variables defined in (1)(4) change from one refer-ence frame to the other. However, the electromagnetic torqueTe, reactive torque Tr, stator flux linkage magnitude s, rotorflux linkage magnitude r, and rotor speed r remain the samein every reference frame, and that is the reason why they arereferred to as natural variables

Ts() =2

3

[cos cos

( 23

)cos

( + 23

)sin sin

( 23

)sin

( + 23

) ] (13) =

t0

()d + (0) (14)

Tr() =2

3

[cos cos

( 23

)cos

( + 23

)sin sin

( 23

)sin

( + 23

) ] (15)where

= r (16)

r =drdt

. (17)

When (1)(4) are rearranged in terms of the flux linkages asstate variable equations, a conventional simulation model instatespace form is obtained as given in

pqs = vqs rsiqs ds (18)pds = vds rsids + qs (19)pqr = vqr rriqr ( r)dr (20)pdr = vdr rridr + ( r)qr. (21)

III. NATURAL/POWER VARIABLE MODEL OF DFIG

The outputs of the DFIG that are generally of interest tocontrol are the following: the electromagnetic torque (Te), thereactive torque (Tr), the stator real power (Ps), the statorreactive power (Qs), the GSC real power (Pf ), and the GSC

BALOGUN et al.: DETERMINATION OF STEADY-STATE AND DYNAMIC CONTROL LAWS OF DFIG 1345

reactive power (Qf ). These outputs vary nonlinearly with themagnitude of the rotor voltage of the DFIG. However, sinceDFIG can be regarded as a multi-inputmulti-output (MIMO)system, then a feedback (inputoutput) linearization theorydescribed in [22] for a MIMO system can be adopted. Thetheory summarily entails differentiating the outputs until thecontrol inputs appear.

A. Natural Variable Model of Induction Machine and MSCThe stator flux linkage referred equations for the electromag-

netic torque, reactive torque, and square of the magnitude ofstator flux linkage magnitude (ss) are given in

Te = k(dsiqs qsids) (22)Tr = k(dsids + qsiqs) (23)ss =

2qs +

2ds. (24)

Similarly, the rotor flux linkage referred equations for Te, Tr,and rr (square of the magnitude of rotor flux linkage) aregiven in

Te = ko(driqs qrids) (25)Tr = ko(drids + qriqs) (26)rr =

2qr +

2dr (27)

where k = 3P/4 and ko = 3PLm/4Lr.However, the stator flux linkage referred model is considered

throughout this paper. Both stator flux linkage referred and rotorflux linkage referred models were given in [21].

The method of obtaining the natural variable model in astatespace form in the stator reference frame stems fromdifferentiating (22)(24) until the rotor voltage vectors appear.

If the rotor flux linkage and the rotor current are eliminated in(20) and (21) using (5)(8) and simplification is made with (18)and (19), then (28) and (29) are obtained such that the statorcurrents are the state variable. Hence, combining (18), (19),(28) and (29) into the derivative of (22)(24) enables the qdrotor voltage vectors to appear. Hence, (30)(32) are obtained.The q-axis and d-axis components of the rotor voltage in (30)and (31) represent half the product of the respective modulationindex (mqr and mdr, respectively) of the machine-side con-verter (MSC) and the dc-link voltage (vdc). This incorporatesthe switching actions of the MSC into the machine dynamics.The rotor speed dynamics is given in (33) to form a completemodel in natural reference frame. In (33), Tm represents themechanical load torque

L1piqs = vqr rrLm

qs + rliqs LrLmkc

vqs

r LrLm

ds ( r)L1ids (28)

L1pids = vdr rrLm

ds + rliqs LrLmkc

vds

r LrLm

qs + ( r)L1iqs (29)

pTe = k

{(ds Lids

L

)vqskc

(qs Liqs

L

)vdskc

}

rTL

Te rTr krL

ss

LmkdsLLr

vqr +LmkqsLLr

vdr (30)

pTr = k

((qs + Li

qs

L

)vqskc

(ds + Li

ds

L

)vdskc

rrLLr

ss

)

rTL

Tr + rTe +rs

kss

(T 2e + T

2r

)+

LmkqsLLr

vqr +LmkdsLLr

vdr (31)

pss =2

(qs

vqskc

+ dsvdskc

)+

2rskkc

Tr (32)

pr =P

2J(Te Tm) (33)

where

rl =rrLsLm

+LrrsLmkc

rT =

(rskc

+rrLsLr

)

L1 =Lm LrLsLm

L = Ls L2m

Lr

vqr =mqrvdc

2vdr =

mdrvdc2

.

B. Power Variable Model of GSCThe voltage equation at the output of the GSC across the filter

network toward the point of shunt injection of current into thegrid is given as

vqf =mqfvdc/2

= rf iqf + Lfpiqf + Lf idf + nvqL (34)vdf =mdfvdc/2

= rf idf + Lfpidf Lf iqf + nvdL (35)

Cdpvdc =3

2(io id)

=3

4(mqriqr +mdridr mqf iqf mdf idf ). (36)

In (34) and (35), the inverter voltages vqf and vdf are ex-pressed in terms of the filter network (rf , Lf ), the current (iqfand idf ) injected through the shunt transformer into the grid,and the voltage at the grids point of common coupling (vqL andvdL). The q-axis and d-axis components of the GSC ac voltagemodulation index are mqf and mdf , respectively. Equation (36)establishes Kirchhoffs current law at the dc-link between thetwo converters in terms of the dc capacitor current, the MSCdc output current (io), and the GSC dc input current (id). Cd is


the capacitance of the dc-link capacitor. In (34)(36), the realpower and reactive power injected in shunt into the grid by theGSC are as follows:

Pf =3

2n(vqLiqf + vdLidf ) (37)

Qf =3

2n(vdLiqf vqLidf ). (38)

The power model of the GSC is obtained by differentiating (37)and (38) and making appropriate substitutions and simplifica-tions, where n in (34)(38) represents the turns ratio of theshunt injection transformer. If the stator terminal connectionto the grid is , for example, then vqs = vqL and vds =vdL in (37) and (38). Rearranging (37) and (38) results in(39) and (40)

iqf =2

3nvss(vqsPf + vdsQf ) (39)

idf =2

3nvssvdsPf VqsQf ) (40)

where vss = v2qs + v2ds.Substitute (39) and (40) in (34) and (35), and then, rearrange

yield (41) and (42). If (39)(42) are used to make appropriatesubstitutions in the time derivatives of (37) and (38)

Lfpiqf =mqfvdc

2 2rf

3nvss(vqsPf + vdsQf )

2eLf3nvss

(vdsPf vqsQf ) nvqs (41)

Lfpidf =mdfvdc

2 2rf

3nvss(vdsPf vqsQf )

+2eLf3nvss

(vqsPf + vdsQf ) nvds (42)

then (43) and (44) evolve as the dynamic power model ofthe GSC, with the real power and reactive power as the statevariables

pPf = rfLf

Pf + eQf 3n2vss

2Lf+

3mqfvdcnvqs4Lf

+3mdfvdcnvds

4Lf(43)

pQf = rfLf

Qf ePf + 3mqfvdcnvds4Lf

3mdfvdcnvqs4Lf

. (44)

C. DC-Link Dynamics

The MSC and the GSC are linked by (36). The MSC dcoutput current (io) and the GSC dc input current (id) areexpressed in terms of rotor current with MSC modulation indexand filter current with GSC modulation index, respectively. Forease of analysis, the stator flux linkage is aligned such thatthe q-axis of the stator flux linkage corresponds to the stator

flux linkage magnitude so that the d-axis component becomeszero, given in (45). Such orientation is similar to the methodof alignment done in vector control. Hence, with appropriatesubstitution and simplification of (36), (46) evolves

qs =s ds = 0 (45)

pvdc =3mqr4Cd

(qsLm

+LsTr

Lmqsk

)+

3mdr4Cd

(LsTe

Lmqsk

)

+1

2Cdnvss[Pf (mqsvqs +mdsvds)

+Qf (mqsvds mdsvqs)] . (46)

IV. OPEN-LOOP SIMULATION OF INDUCTION MACHINES

For the purpose of simulation, the rotor of the wound rotorinduction machine is electrically shorted so that it acts like asquirrel-cage induction machine. Hence, the rotor voltage iszero. The machine parameters are given in Appendix A.

A. Vector Variable (Conventional) SimulationThe vector variable model of (18)(21) along with (32) is

simulated such that the stator flux linkage and rotor flux linkageare the state variables. Observe that the reference velocity isevident in this model, which implies that the qd componentsof the state variables are dependent on the choice of the ref-erence frame of transformation. The implication of this is thatthey are not bound to the same orientation or angular directionin the different reference frames.

B. Natural Variable Simulation

In order to ease the simulation of the natural variable modelof (30)(33), it is essential, therefore, to align the stator fluxlinkage along the q-axis component flux linkage such that thed-axis component is equal to zero (45). An angular velocitygiven in (47), which is obtained by substituting (45) into (19),gives such an orientation. Integrating (47) gives the precise an-gle of transformation. Recall that the transformation angulardisplacement in definite integral is given in (14). Consequently,the qd stator currents are given in

= 1s

(vds rsids) (47)

iqs = Trks

(48)

ids = Teks

. (49)

C. Open-Loop Simulation Results

MATLAB/Simulink is employed for simulation. Observethat the electrical speed of 376.9911 rad/s is thrice the mechan-ical speed at 60 Hz because the induction machine is a six-polemachine.


Fig. 2. No-load transient of the electromagnetic torque.

Fig. 3. No-load transient of the reactive torque.

The results obtained from simulating the vector variablemodel and the natural variable model are presented in Figs. 29.Observe that the results from the two models overlap oneanother, which validates that the natural variable model canalso be effectively used to predict the electrical dynamics of themachine. No-load transients of the electromagnetic torque forboth simulations are shown in Fig. 2 to decay to zero. Figs. 35show the transients of the reactive torque, the square of themagnitude of the stator flux linkage, and the stator angularvelocity.

At steady state, some step changes are introduced by themechanical torque. At 2 and 4 s, 16 and 16N m are applied,respectively, to compare the responses of both models. InFigs. 69, the corresponding changes in the rotor speed, elec-tromagnetic torque, reactive torque, and stator flux linkage areobserved to be the same for both models. In Fig. 7, theelectromagnetic torque was positive for 16-N m changein load torque and negative for 16N m change in loadtorquerespectively, motoring and generating modes asexpected.

The reactive torque, like the reactive power as regards topower, is the inactive torque component. In fact, it is related tothe reactive power by a factor of 2/P , where P is the numberof magnetic poles of the machine. The sign convention adopted

Fig. 4. No-load transient of the square of the magnitude of the stator fluxlinkage.

Fig. 5. No-load transient of the stator angular velocity in radians per second.

Fig. 6. Stepped response of the rotor electrical speed.

for this paper is such that a negative reactive torque implies thatreactive torque is supplied to the machine. Hence, in Figs. 3 and8, the reactive torque remains negative irrespective of changein load (mechanical) torque because the machine operates atlagging power factor. Fig. 5 shows the stator frequency whichis computed to align the stator flux. Such alignment ensures thatthe d-axis stator flux remains zero at all time. Fig. 9 shows the


Fig. 7. Stepped response of the electromagnetic torque.

Fig. 8. Stepped response of the reactive torque.

Fig. 9. Stepped response of the stator flux linkage.

stepped response of the stator flux linkage which is the squareroot of the magnitude of ss.

V. STEADY-STATE OPERATING LIMITS

At steady state, all derivatives become zero. Hence, whenthe reference frame is the synchronous speed, i.e., at sta-

tor frequency, (30)(33) and (43)(46) translate to (50)(56),respectively,

0 = k

{TeVqskqskc

(qsL

+Trkqs

)Vdskc

}

rT TeL

rTr krL

ss +

(LmkqsLLr

MdrVdc2

)(50)

0 = k

{(qsL

Trkqs

)Vqskc

+TeVdskqskc

rrLLr

ss

}

rT TrL

+ rTe +rs

kss

(T 2e + T

2r

)+

(LmkqsLLr

MqrVdc2

)(51)

0 =2

(qs

Vqskc

+rskkc

Tr

)(52)

0 =P

2J(Te Tm) (53)

0 = rfLf

Pf + eQf 3n2Vss2Lf

+3MqfVdcnVqs

4Lf+

3MdfVdcnVds4Lf

(54)

0 = rfLf

Qf ePf + 3MqfVdcnVds4Lf

3MdfVdcnVqs4Lf

(55)

0 =3Mqr4

(qsLm

+LsTr

Lmqsk

)+

3Mdr4

(LsTe

Lmqsk

)

+1

2nVss[Pf (MqsVqs +MdsVds)

+Qf (MqsVds MdsVqs)] . (56)

A. Power BalanceEquation (56) establishes power balance between the two

converters in terms of the modulation indexes and the statevariables. In general, it is desired to operate the GSC at unitypower factor to improve efficiency [4]. For such a situation,Qf = 0, which means that the GSC does not consume orgenerate reactive power from or into the grid. Hence, (54)(56)translate to (57)(59)

0 = rfLf

Pf 3n2Vss2Lf

+3MqfVdcnVqs

4Lf

+3MdfVdcnVds

4Lf(57)

0 = ePf + 3MqfVdcnVds4Lf

3MdfVdcnVqs4Lf

(58)

0 =3Mqr4

(qsLm

+LsTr

Lmqsk

)+

3MdrLsTe4Lmqsk

+Pf

2nVss(MqsVqs +MdsVds) (59)

Ps =3

2

[(Trqsk

)2rs +

(Teqsk

)2rs +

sTek

](60)

Qs =3

2

[Trks

]=

2sTrP

(61)

Tr =2kQG3s

(62)


Vqs =2rsQG3qss

(63)

Vds =Vss V 2qs =

Vss

(2rsQG3qss

)2(64)

2s2ss +(Vss 2rsTes

k

)ss

(rsTek

)2(2rsQG

3qss

)2= 0 (65)

ss = b b2 4ac2a

(66)

where

a = 2sb =

(Vss 2rsTes

k

)

c = (rsTek

)2(2rsQG

3s

)2.

When the stator real power (Ps) and reactive power (Qs) areexpressed in terms of the natural variables, then (60) and (61)evolve. If the stator reactive power is equated to the grid reactivepower (QG) because the GSC is set not to exchange reactivepower with the grid, then (62) is obtained. Substituting (62)into (52) yields the q-axis of the stator voltage, which is givenin (63). The d-axis of the stator voltage is given in (64), fromwhich (65) is obtained by using (49) to substitute for the d-axisstator current. Therefore, the stator flux linkage is determinedfrom the square root of (66) while specifying the grid reactivepower. Hence, appropriate substitution and simplification in(50) and (51) and (57)(59) yield (67)(71). The qd vectorsof the MSC modulation indexes are given in (67) and (68),respectively. Similarly, the qd vectors of the GSC modulationindexes are given in (69) and (70), respectively. Equation (71)gives the real power of the GSC. Observe that if (59) ismultiplied by Vdc, then the first two expressions after theequality sign in terms of Mqr and Mdr represent the rotorreal power, which is given in (72). The rotor reactive poweris given in (73). Equation (74) gives the mechanical torque asa quadratic function of the shafts mechanical speed m (inradians per second) [13], where kopt is a constant which isobtained from the mechanical torque versus shaft speed curveof associated wind turbine. Hence, the turbines mechanicalshaft power Pm of (75) in terms of Tm and m is furtherexpressed in terms of Te, damping coefficient Bm, and rotorelectrical speed r (in radians per second), where m = 2r/Pand P = number of poles of the machine. The copper and corepower losses of the generator are given in (76)

Mdr = (

2LLrLmkqsVdc

)

[k

{TeVqskqskc

(qsL

+Trkqs

)Vdskc

}

rT TeL

rTr krL

ss

](67)

Mqr = (

2LLrLmkqsVdc

)

[k

{(qsL

Trkqs

)Vqskc

+TeVdskqskc

rrLLr

ss

}

rT TrL

+ rTe +rs

kss

(T 2e + T

2r

)]. (68)

B. Unity Power Factor Operation

The DFIG can operate at a general unity power factor withthe stator and GSC set at unity power factor. Such unity powerfactor operation will require that the MSC supplies the machinewith the reactive power needed for excitation. The machineparameters for this study are given in Appendix A. Fig. 10shows the turbine power, the stator power, the rotor power,and the sum of the stator power and rotor power at generalunity power factor operation of stator and GSC. The differencebetween the turbine power and the sum of the stator powerand rotor power is accounted by the copper and core lossesof the machine, assuming that all other losses were neglectedfor the purpose of analysis. In Figs. 1117, the unity powerfactor is compared to leading and lagging stator power factors.It is clearly shown, therefore, that a general unity power factoroperation requires a higher power rating of MSC than laggingstator power factor but a lower MSC power rating for leadingstator power factor

Mds =bmds

b2mds 4amdscmds2amds

(69)

where

amds =3rfVdcV

2ss

42eL2f

bmds =3V 2ss2eLf

(rfVdseLf

+ Vqs

)and

cmds =3V 2ssVqs2eLf

+

(rfVdseLf

+ Vqs

)2

[3VssMqr

2

(qsLm

+LsTr

Lmqsk

)+

3VssMdrLsTe2Lmqsk

]

Mqs =

(rfVqseLf

+ Vds

)Mds +

2VssVdc(rfVds

eLf+ Vqs

) (70)

Pf =

3VssVdc4eLf

Mds +3VssVds2eLf(rfVds

eLf+ Vqs

) (71)

Pr =3MqrVdc

4

(qsLm

+LsTr

Lmqsk

)+

3MdrVdcLsTe4Lmqsk

(72)

Qr =3MdrVdc

4

(qsLm

+LsTr

Lmqsk

) 3MqrVdcLsTe

4Lmqsk(73)

Tm = kopt2m (74)


Fig. 10. Turbine power, stator power, rotor power, and sum of stator powerand rotor power at unity power factor operation of stator and GSC.

Pm =Tmm =

(Te +Bm

2rP

)2rP

(75)

PL =APl

(Trkqs

)2+APl

(Tekqs

)2

+BPlTrkqs

+ CPlTekqs

+DPl (76)

where

APl =1.5

[rs + rr

(LsLm

)2]

BPl =3rrLsqs

L2m

CPl =3ersqs

rc

DPl =1.52qs

(rrL2m

+ kc

(erc

)2).

C. Leading and Lagging Stator Power Factor Operation

In the case of the motoring convention in which the modelis based upon, which is evident in (1)(4), a negative statorcurrent should normally imply that power flows to the grid, andwhen positive, it should mean that power flows back into themachines stator. This is true for the stator reactive power, butfor the stator, it is vice versa. The reasons for this are derivedfrom the fact that the stator flux orientation which aligns theq-axis of the stator flux linkage with the stator flux magnitudeimposes that the Ids controls electromagnetic torque Te and thatIqs controls the reactive torque Tr, as evident in (22) and (23).Notice in (60) and (61) that the real power is mainly regulatedby Te while Tr controls the reactive power. It is intuitive that anegative Te implies that the machine is generating and a positiveTe means motoring.

Consequently, a negative Te will imply that Ids is positivefor generating. Therefore, a negative stator real power means

Fig. 11. Rotor voltage versus rotor electrical speed for unity, leading, andlagging stator power factors.

Fig. 12. Rotor current versus rotor electrical speed for unity, leading, andlagging stator power factors.

generating operation and vice versa. However, from (62), basedon the orientation of the stator current, the stator reactive poweris negative for lagging power factor and positive for leadingpower factor. Take notice that the power sign convention of theGSC of Fig. 16 differs from the rotor power of Fig. 15 becausethe GSC was modeled based on (34) and (35) such that positivepower flow means power flow toward the grid and vice versa.

For a situation whereby the DFIG is required to supply thegrid with reactive power, i.e., a leading stator power factor, therating of the MSC must be raised so as to account for the extrareactive power consumption by the grid. This definitely impliesmore losses because of the increase in the rotor voltage andcurrent, which is revealed in Figs. 11, 12, and 14. However,Figs. 1116 show that the lagging stator power factor gives thebest operating regimes since it accounts for the least losses andcertainly the least size of MSC power rating. Furthermore, it isevident in Fig. 18 that there exists an optimal lagging reactivepower for the various power profiles. This is dealt with in thenext section.


Fig. 13. Stator current versus rotor electrical speed for unity, leading, andlagging stator power factors.

Fig. 14. Core and copper losses versus rotor electrical speed for unity, leading,and lagging stator power factors.

Fig. 15. Rotor real power versus rotor electrical speed for unity, leading, andlagging stator power factors.

D. Estimation of Optimal Stator Power FactorThe losses in (76) account for the copper and core losses

of the generator. The windage and frictional losses can beconsidered to be constant over the entire operating region [4],

Fig. 16. Rotor reactive power versus rotor electrical speed for unity, leading,and lagging stator power factors.

Fig. 17. GSC real power versus rotor electrical speed for unity, leading, andlagging stator power factors.

Fig. 18. Copper and core losses versus stator reactive power for 380, 392, and410 rad/s rotor electrical speeds.

[23], while the stray load loss is also assumed to be constant[4]. Therefore, it is intuitive from the results obtained so farthat the minimal copper and core losses correspond to a laggingstator power factor, as evident in Fig. 18. A lagging stator powerfactor will guarantee the least power rating of the MSC [24].


Fig. 19. Rotor apparent power versus rotor speed for unity, leading, andlagging stator power factors.

Fig. 20. Optimal copper and core losses versus rotor electrical speed.

The stator reactive power that corresponds to the minimal lossescan be obtained analytically from the optimal Tr. Therefore, aJacobi matrix obtained from (76) and (75) with respect to Teand Tr is given in (77), which yields the optimal Tr given in(78) and consequently the optimal stator reactive power givenin (79). Hence, the optimal value is about Qs = 1250 V ARwith a qs = 0.499. Take notice that this optimal stator reactivepower agrees with the common minimum value of the reactivepower for the three rotor electrical speed conditions of Fig. 18.The negative sign implies a lagging power factor, which isintuitive from the results of Fig. 18. Therefore, this value ofstator reactive power gives an optimal lagging stator powerfactor of 0.9389. The apparent power (V A) profile of theMSC for unity stator power factor and stator reactive powerabsorption/generation is given in Fig. 19. Observe in Fig. 20that the copper and core losses are lower than those obtained inFig. 14. Hence, Fig. 20 gives the optimal copper and core lossesfor the generator. Fig. 21 gives the rotor real power and statorreal power at optimal stator reactive power.

Although the optimal stator reactive power corresponds to alagging stator power factor, for most applications, the DFIG isoperated at either unity stator power factor or even at leading

Fig. 21. Optimal steady-state results of stator real power and rotor real powerversus rotor electrical speed.

stator power factor to supply reactive power when required forsystem stability [6], [7] PmTe PmTrPL

TePLTr

= 2APlkqs Tr +BPl = 0 (77)Tr =

kqsBPl2APl

(78)

Qs =3eqsBPl

4APl. (79)

VI. DYNAMIC CONTROL

A. MSC Torque Control

An independent control of the torque variables is achievedby field orientation of qs = s and ds = 0 so that vdr regu-lates the electromagnetic torque while vqr controls the reactivetorque. Hence, (30) and (31) yield (80) and (81)

LpTe + rTTe

= Lk

{ids

vqskc

(qs Liqs

L

)vdskc

}

LrTr krss +(Lmkqs

Lr

mdrvdc2

)(80)

LpTr + rTTr

= Lk

{(qs + Li

qs

L

)vqskc

idsvdskc

rrLLr

ss

}

+ LrTe +Lrskss

(T 2e + T

2r

)+

(Lmkqs

Lr

mqrvdc2

).

(81)

Therefore, the dynamic controllers are given in

LpTe + rTTe =KTe (Te Te) = Te (82)

LpTr + rTTr =KTr (Tr Tr) = Tr. (83)

In Laplace domain, (84) and (85) are obtained

(pL + rT )Te =KTe(p)eTe (84)(pL + rT )Tr =KTr(p)eTr (85)


where KTe and KTr are PI controllers and eTe and eTr arethe errors between the reference current and the actual value.If it is assumed that KTe = KTr = KT , then, in general, theopen-loop transfer function for (84) and (85) is given

HT (p) =ITeeTe

=ITreTr

=KT (p)

(pL + rT ). (86)

The PI controller is defined as

KT (p) = Kpt +Kit/p. (87)

Dead-time, transport, and sampling delays introduced by theconverter and the analog-to-digital conversion process aremodeled by introducing into (86) a delay factor Hrd(s) =esTrd[6], [25], which can be simplified as given in

HTd(p) =1

(pTTd + 1). (88)

Hence, (89) is obtained as the new closed-loop transfer function

HTo(p) =KT (p)

(pL + rT )

1

(pTTd + 1). (89)

For a condition of the controllers zero cancelling undesiredpole of the plant, the gains of the PI controller are selected suchthat Kpt/Kit = L/rT [26]. Therefore, (90) is obtained

HcT (p) =Kpt

(p2LTrd + pL +Kpt). (90)

When the denominator of (90) is compared with the second-order Butterworth polynomial of p2 + 2np+ 2n at optimaldamping design, such that = 1/

2, then n

2 = 1/Trd,

and 2n = Kpt/(LTTd). Hence, Kpt = L/(2TTd), and con-sequently, Kit = rT /(2TTd).

B. GSC Power Control

An independent power control is achieved with voltage ori-entation of vds = vs and vqs = 0. Therefore, vdf controls thereal power generation/absorption by the GSC into/from thegrid, while the GSC reactive power generation/absorption iscontrolled by vqf

LfpPf + rfPf =LfeQf 3n2vss2

+3mdfvdcnvds

4(91)

LfpQf + rfQf = LfePf + 3mqfvdcnvds4

. (92)

Similarly, the closed-loop transfer function of (91) and (92) isgiven in

HcPQ(p) =KpPQ

(p2LfTPQd + pLf +KpPQ)(93)

where KpPQ = Lf/(2TPQd) and KiPQ = rf/(2TPQd) forpole-zero cancellation.

C. Rotor Speed Control

Speed control of the wind generator is essential to enablean optimal power extraction from the wind turbine system.Therefore, the electromagnetic torque reference command T ecan be generated from an outer loop speed control

2J

Ppr = (T

e Tm) . (94)

The speed controller and the reference T e are given in (95) and(96), respectively

r =Kr (r r) = (T e Tm) (95)

T e = r + Tm. (96)

D. DC-Link Voltage Control

Control of the dc-link voltage is achieved by the d-axis outerloop control of the GSC. It is assumed that the GSC does notgenerate/absorb reactive power into/from the grid. Therefore,(97) is obtained from (46)

pvdc =3mqr4Cd

(qsLm

+LsTr

Lmqsk

)

+3mdr4Cd

(LsTe

Lmqsk

)+

Pfmdsvds2Cdnvss

. (97)

A multiplication of (63) by vdc results in a situation such thatthe first two terms of the RHS of (97) represent the real powerof the MSC while the last term represents the real power of theGSC. Hence, further simplification of the last term eventuallyresults to Pf

Cdpvdc = (Pr + Pf )/Vdc. (98)

Therefore, the controller is developed as follows:

Vdc =KVdc (vdc vdc) =

(Pr + P

f

)/Vdc (99)

P f = VdcVdc Pr (100)

where P f is the reference command of the GSC real power.The overall power variable control schemes for the GSC is

illustrated in Fig. 22.

E. Robustness Against Rotor Parameter Variation

The impacts of parameter variation on the PI controllers forthe direct control of torque and power variables and on thePI controllers for vector control are investigated using smallsignal model of the DFIG. If a small perturbation is performedon (80) and (81) by setting the state variables x = xo +xabout an equilibrium state xo and by neglecting higher orderterms, then a small signal linear dynamic model of (101) and(102) is obtained. Hence, the transfer functions that relate small


Fig. 22. Decoupled natural and power variable controls of DFIG. (a) MSCcontrol. (b) GSC control.

change in the output to small change in the input variables arederived

LpTe + rTTe

= kLo(idsvqs

kc+

iqsvdskc

) kqsvds

kc rLoTr krss+

Lmk

Lr(qsvdr + qsvdr) (101)

LpTr + rTTr

= kL(iqsvqs

kc+

idsvdskc

)

k(qsvqs

kc+

rrLr

ss

)+ rLTe

+rsLkss

(2TeTe + 2TrTr)

+Lmk

Lr(qsvqr + qsvqr). (102)

Let us assume that the estimated parameters used in selectingthe gains of the PI controllers do not match the actual parame-ters of the plant. In that case, (103) and (104) can be assumedto be the basis for the controller design, where indicates anestimated value

KTe = kLo(idsvqs

kc+

iqsvdskc

) kqsvds

kc

rLoTr krss

+Lmk

Lr(qsvdr + qsvdr) (103)

KTr = kLo(iqsvqs

kc+

idsvdskc

)

k(qsvqs

kc+

rr

Lrss

)

+ rLoTe +rsLokss

(2TeTe + 2TrTr)

+Lmk

Lr(qsvqr + qsvqr) . (104)

If the rotor voltage vectors of (103) and (104) are substituted byusing (101) and (102), then the closed-loop transfer functionsof (105) and (106) are obtained. The coefficients of equations(105) and (106) are given in Appendix BTeT e

=A3p

3 +A2p2 +A1p+A0

B5p5 +B4p4 +B3p3 +B2p2 +B1p+B0(105)

TrT r

=C3p

3 + C2p2 + C1p+ C0

D5p5 +D4p4 +D3p3 +D2p2 +D1p+D0. (106)

Similarly, for vector control, the closed-loop transfer functionof (107) is obtained

iqriqr

=idridr

=a3p

3 + a2p2 + a1p+ a0

b4p4 + b3p3 + b2p2 + b1p+ b0(107)

where

a3 = kprL3

a2 = k2prL

2 + 2kprrrL2

a1 =2k2prrrL+ kprrrL

2

a0 = k2prr

2r

b5 =L4Tdr

b4 =L4Tdr + rrL

3Tdr + L4 + L3rr

b3 =2kprL3 + kprL

2rr + kprTdrL2rr + 2rrL

3 + L4

+ L2r3r +((e r)(L L)

)2L2Tdr

b2 =4kprL2rr + r

2rLkpr + r

2rL

2

+((e r)(L L)

)2L2 + r2rk

2pr

b1 =2kprLr2r + 2k

2prrrL

b0 = k2prr

2r .

The root loci of (105)(107) are obtained when variationsare introduced to the actual rotor resistance and rotor leakageinductance. The rotor parameters are of interest because theconverters are connected to the rotor circuit. The root loci ob-tained from (105) and (106) for the direct control of the torquevariables in Figs. 23 and 24 reveal that the poles are restrainedto the left hand plane even if the actual rotor resistance and rotorleakage inductance are each 0.5 times the estimated values. Thisinfers that, unlike the vector control, the controllers developedfor the decoupled direct control of the torque variables are


Fig. 23. Loci of poles of (105) when Llr = 0.5Llr and rr = 0.5rr .

Fig. 24. Loci of poles of (106) when Llr = 0.5Llr and rr = 0.5rr .

robust against parameter variations. However, Fig. 25 showsthat, for a vector control scheme, two poles crossed over to theinstability plane for such a rotor parameter variation.

VII. EXPERIMENTAL RESULTS

The results given in Fig. 26 were obtained from an ex-perimental set-up that was achieved by a 5-hp wound rotorinduction generator, which serves as the DFIG. A 5-hp dcmachine served as the prime mover to emulate the wind turbine.Two Semikron converters were connected back to back, oneto the rotor electrical circuit and the other to the grid witha common dc-link. A ds1104 dSPACE board was used forsignal conditioning and controller implementation for the ma-chine. The rotor speed was set at 350 rad/s, and the controlledvariables were observed to track their respective commands.Fig. 26(a) shows the electromagnetic torque which tends towardabout 22 N m, while Fig. 26(b) shows the reactive torque thatwas set to operate at 9.94, which corresponds to 1250 VARstator reactive power.

Fig. 25. Loci of poles of (107) when Llr = 0.5Llr , rr = 0.5rr .

Fig. 26. Experimental results showing the actual torque variables. (a) Electro-magnetic torque. (b) Reactive torque.

VIII. CONCLUSIONA DFIG model with the torque and power variables as state

variables has been presented. The results obtained from thenatural variable simulation were shown to conform to vectorvariable simulation. Steady-state operating regions were estab-lished for various stator power factor operations. The highestapparent power rating of the MSC was noticed for leading statorpower factor. Optimal stator reactive power was estimated.Direct control of torque and power variables was obtained usingfeedback linearization. It was revealed by the small signal anal-ysis that, unlike the vector control, the developed controllerswere robust to variations in actual and estimated parameters ofthe machine.

APPENDIX APARAMETERS

5-hp, 60-Hz, 220-V lineline (rms), and six-pole wound-rotor induction motorStator resistance (rs) 0.65 ;Rotor referred resistance 0.41 ;


Fig. 27. qd equivalent circuit model of the induction machine includingcore-loss resistance. (a) q-axis. (b) d-axis.

Stator leakage inductance (Lls) 2.6 mH;Rotor referred leakage inductance (Llr) 2.6 mH;Magnetizing Inductance (Lm) 0.0441 H;Core loss resistance (rc) 800 (Fig. 27).

APPENDIX BDEFINITION OF COEFFICIENTS

A3 = kptL3o,

A2 = kptL2o

(kpt + rT + (rsLo rsLo) 2Tr

kss+ rT

)

A1 = kptLo

(2kptrT + r

2T + (rsLo rsLo)

2Trkss

+ kptr2T

)

A0 = k2ptr

2T ,

B5 =L4oTd

B4 =L3oTdkpt + (rsLo rsLo)

2TrL4oTd

kss+ L4o(Lo + TdrT )

B3 = kptL3o + kptL

2oTdrT + L

2okpt(Lo + TdrT )

+ (Lo + TdrT )(rsLo rsLo)2TrL2o

kss

+ rtL3o L2oTd2r(Lo Lo)2

(Lo Lo)(rsLo rsLo)2TeL2oTdr

kss

B2 = k2ptL

2o + (rsLo rsLo)

2TrL2okpt

kss

+ 2kptL2orT + kptLort(Lo + TdrT )

+ (rsLo rsLo)2TrrTL2o

kss(Lor(Lo Lo)

)2

(Lo Lo)(rsLo rsLo)2TeL2or

kss

B1 =2k2ptrTLo + 2kptr

2TLo + (rsLo rsLo)

2TrLokptrTkss

B0 = k2ptr

2T

C3 = kptL3o,

C2 = kptL2o

(kpt + 2rT + (rsLo rsLo)

)C1 = kptLo

(2kptrT + r

2T

),

C0 = k2ptr

2T ,

D5 =L4oTd

D4 =L3o(Tdrt + Lo) + L

3oTd(kpt + rT )

+ (rsLo rsLo)2TrL3oTd

kss

D3 =L2orT (Lo + kptTd)

+(TdkptLo + TdLorT + L

2o

)(rsLo rsLo)2TrLo

kss

+ kptL3o + L

2oTd

2r(Lo Lo)2

+ (Lo Lo)(rsLo rsLo)2TeL2oTdr

kss

D2 = rTL2o(kpt + rT ) + Lo

(TdrTLo + L

2o

)(kpt + rT )

+ kptrT (TdrT + Lo)

+ (TdkptrT + Lokpt + LorT )(rsLo rsLo)2TrLokss

+ (rsLo rsLo)2TrrTL2o

kss L2o2r(Lo Lo)2

+ (Lo Lo)(rsLo rsLo)2TeL2or

kss

+ Lo2r(Lo Lo)2

D1 =2k2ptrTLo + 2kptr

2TLo + (rsLo rsLo)

2TrLokptrTkss

,

D0 = k2ptr

2T .

REFERENCES[1] P. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machin-

ery and Drive Systems. Piscataway, NJ, USA: IEEE Press, 2002.[2] B. Ozpineci and L. M. Tolbert, Simulink implementation of induction

machine modelA modular approach, in Proc. Int. Elect. Mach. DrivesConf., 2003, pp. 728734.

[3] M. Shin, D. Hyun, S. Cho, and S. Choe, An improved stator flux es-timation for speed sensorless stator flux orientation control of inductionmotors, IEEE Trans. Power Electron., vol. 15, no. 2, pp. 312318,Mar. 2000.

[4] W. Qiao, W. Zhou, J. M. Aller, and R. G. Harley, Wind speed estimationbased sensorless output maximization control for a wind turbine drivinga DFIG, IEEE Trans. Power Electron., vol. 23, no. 23, pp. 11561169,May 2008.

[5] R. Pena, J. C. Clare, and G. M. Asher, Doubly fed induction generatorusing back-to-back PWM converters and its application to variable-speedwind-energy generation, Proc. Inst. Elect. Eng.Elect. Power Appl.,vol. 143, no. 3, pp. 231241, May 1996.


[6] B. C. Rabelo, W. Hofmann, J. Lucas da Silva, R. Gaiba de Oliveira, andS. Rocha Silva, Reactive power control design in doubly fed inductiongenerators for wind turbines, IEEE Trans. Ind. Electron., vol. 56, no. 10,pp. 41544162, Oct. 2009.

[7] M. Kayk and J. V. Milanovic, Reactive power control strategiesfor DFIG-based plants, IEEE Trans. Energy Convers., vol. 22, no. 2,pp. 389396, Jun. 2007.

[8] M. P. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control inPower Electronics: Selected Problems. London, U.K.: Academic, 2002,pp. 190207.

[9] N. R. Nik Idris and A. H. Mohamed Yatim, Direct torque controlof induction machines with constant switching frequency and reducedtorque ripple, IEEE Trans. Power Electron., vol. 51, no. 4, pp. 758767,Aug. 2004.

[10] J. Rodrguez, J. Pontt, S. Kouro, and P. Correa, Direct torque controlwith imposed switching frequency in an 11-level cascaded inverter, IEEETrans. Ind. Electron., vol. 51, no. 4, pp. 827833, Aug. 2004.

[11] G. Abad, M. A. Rodrguez, and J. Poza, Two-level VSC based predictivedirect torque control of the doubly fed induction machine with reducedtorque and flux ripples at low constant switching frequency, IEEE Trans.Power Electron., vol. 23, no. 3, pp. 10501061, May 2008.

[12] D. Zhi and L. Xu, Direct power control of DFIG with constant switch-ing frequency and improved transient performance, IEEE Trans. EnergyConvers., vol. 22, no. 1, pp. 110118, Mar. 2007.

[13] G. S. Buja and M. P. Kazmierkowski, Direct torque control of PWMinverter-fed ac motorsA survey, IEEE Trans. Ind. Electron., vol. 51,no. 4, pp. 744757, Aug. 2004.

[14] G. Poddar and V. T. Ranganathan, Direct torque and frequency controlof double-inverter-fed slip-ring induction motor drive, IEEE Trans. Ind.Electron., vol. 51, no. 6, pp. 13291337, Dec. 2004.

[15] J. Maes and J. A. Melkebeek, Speed-sensorless direct torque control ofinduction motors using an adaptive flux observer, IEEE Trans. Ind. Appl.,vol. 36, no. 3, pp. 778785, May/Jun. 2000.

[16] E. Bogalecka and Z. Krzeminski, Control system of doubly-fed inductionmachine supplied by current controlled voltage source inverter, in Proc.6th Int. Conf. Elect. Mach. Drives, 1993, pp. 168172.

[17] G. Dong and O. Ojo, Efficiency optimization control of induction mo-tor using natural variables, IEEE Trans. Ind. Electron., vol. 53, no. 6,pp. 17911798, Dec. 2006.

[18] G. Dong, Sensorless and efficiency optimized induction machine controlwith associated converter PWM modulation schemes, Ph.D. dissertation,Tennessee Technol. Univ., Cookeville, TN, USA, 2005.

[19] Z. Krzeminski, Sensorless multiscalar control of double fed machine forwind power generators, in Proc. PCC, Osaka, Japan, 2002, pp. 334339.

[20] A. Tabesh and R. Iravani, Multivariable dynamic model and robust con-trol of a voltage-source converter for power system applications, IEEETrans. Power Del., vol. 24, no. 1, pp. 462471, Jan. 2009.

[21] A. Balogun and O. Ojo, Natural variable simulation of induction ma-chines, in Proc. IEEE AFRICON, Sep. 2325, 2009, pp. 16, [CD-ROM].

[22] J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ,USA: Prentice-Hall, 1991, pp. 207275.

[23] IEEE Standard Test Procedure for Polyphase Induction Motors and Gen-erators, IEEE Std. 112-2004, Nov. 2004.

[24] D. Aguglia, P. Viarouge, R. Wamkeue, and J. Cros, Analytical determi-nation of steady-state converter control laws for wind turbines equippedwith doubly fed induction generators, IET Renew. Power Gener., vol. 2,no. 1, pp. 1625, Mar. 2008.

[25] D. G. Holmes, T. A. Lipo, W. Y. Kong, and B. P. McGrath, Optimizeddesign of stationary frame three phase ac stationary ac current regulators,IEEE Trans. Power Electron., vol. 24, no. 11, pp. 24172426, Nov. 2009.

[26] C. Lascu, L. Asiminoaei, I. Boldea, and F. Blaabjerg, High performancecurrent controller for selective harmonic compensation in active powerfilters, IEEE Trans. Power Electron., vol. 22, no. 5, pp. 18261835,Sep. 2007.

Adeola Balogun (M10) was born in Kano, Nigeria.He received the B.Sc., M.Sc., and Ph.D. degreesin electrical and electronics engineering from theUniversity of Lagos, Lagos, Nigeria, in 1998, 2002,and 2011, respectively.

He was a Visiting/Research Scholar with the Cen-ter for Energy Systems Research, Tennessee Tech-nological University, Cookeville, TN, USA, in 2008and 2009. Presently, he is a Lecturer in the De-partment of Electrical and Electronics Engineering,University of Lagos, Akoka-Lagos, Nigeria. His cur-

rent research interests include electric machines, power electronics, control ofelectric drives, and renewable energy conversion systems.

Olorunfemi Ojo (M87SM95F10) was born inKabba, Nigeria. He received the Bachelors andMasters degrees in electrical engineering from Ah-madu Bello University, Zaria, Nigeria, and the Ph.D.degree from the University of Wisconsin, Madison,WI, USA.

He is currently a TVA Chair Professor of electricaland computer engineering with Tennessee Techno-logical University, Cookeville, TN, USA. His currentresearch interests include the areas of electric ma-chine analysis and drive control, switching converter

technology, and modern control applications in converter-enhanced power anddistributed energy generation systems.

Dr. Ojo is a Fellow of the Institution of Electrical Engineers, U.K. He iscurrently the Chair of the Industrial Power Conversion Systems Department ofthe IEEE Industry Applications Society. He is an Associate Editor of the IEEETRANSACTIONS ON POWER ELECTRONICS and a member of the EditorialBoard of IET Power Electronics.

Frank Okafor (M01) received the B.Sc., M.Phil.,and Ph.D. degrees in electrical engineering from theUniversity of Lagos, Lagos, Nigeria, in 1984, 1987,and 1993, respectively.

He was a Research Scholar with the Techni-cal University of Chemnitz, Chemnitz, Germany, in2000. He is currently a Professor of electrical andelectronics engineering with the University of Lagos.His research interests include the areas of renewableenergy systems, control engineering, power systems,electric drives, and electromagnetic compatibility.

Dr. Okafor is a Fellow of the Nigerian Society of Engineers.

Sosthenes Karugaba (S07M12) was born inMuleba, Kagera, Tanzania. He received the B.Sc.degree in electrical engineering from the Universityof Dar es Salaam, Dar es Salaam, Tanzania, in 1999and the M.S. degree in electrical engineering and thePh.D. degree in engineering from Tennessee Tech-nological University, Cookeville, TN, USA, in 2008and 2012, respectively. Presently, he is a Lecturerin the Department of Electrical Engineering, Dares Salaam Institute of Technology, Dar es Salaam,Tanzania.

His research interests include power electronics, electrical machines, andmotor drives.

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determination of steady-state and dynamic control

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