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On the passivity-based power control of a doubly-fed induction machine

I. Lpez-Garca a, G. Espinosa-Prez a,,1, H. Siguerdidjane b, A. Dria-Cerezo c,2

a Universidad Nacional Autnoma de Mxico, FIUNAM, Apartado Postal 70-256, 04510 Mxico DF, MexicobAutomatic Control Department, Suplec, Plateau du Moulon, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, Cedex, Francec Universitat Politcnica de Catalunya, DEE and IOC, EPSEVG, Av. V. Balaguer s/n, Vilanova i la Geltr 08800, Spain

a r t i c l e i n f o

Article history:

Received 5 September 2011

Received in revised form 24 August 2012

Accepted 29 August 2012

Available online 23 October 2012

Keywords:

Double-fed induction machines

Passivity-based control

Power regulation

a b s t r a c t

In this paper the stator-side power regulation control problem of DFIM is approached. It is shown how a

passivity-based controller can deal with this problem establishing a viable solution from both a dynamic

performance perspective and a practical implementation point of view. The evaluation of the presented

scheme is carried out by considering typical operation conditions, namely: active power generation with

demanded or delivered reactive power. Special interest is given to the rigorous establishment of the sta-

bility properties of the closed-loop system, which allows for achieving remarkable dynamic responses, in

terms of convergence rates as well as the region of attraction of the equilibrium point defined by a given

operation regime. From a practical viewpoint, the usefulness of the scheme is concluded by realizing that

the requirements imposed concerning the structural features of both the generator by itself and the

power converter required for its operation, can be fulfilled by commercially available off-the-shelf

devices.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

Doubly-fed induction machines (DFIMs) have become a funda-

mental element in several applications related with efficient meth-

ods for electric energy generation. This fact is due, on the one hand,

to the high performance that can be achieved with this kind of

devices under variable-speed operation and, on the other hand,

to the possibility for feeding directly the rotor windings using high

efficient power converters, which makes feasible to carry out the

energy conversion process using only a small fraction of the energy

managed by the whole power system[13].

Motivated by these remarkable features, DFIM has been deeply

studied and several useful properties are currently well-known, for

example, that if the model is represented in a line-voltage vector-

oriented reference frame, then the controller design can be carried

out in a simpler and more robust way, since the stator active andreactive power can be controlled in a decoupled way while the

coordinate transformation for representing the model in the afore-

mentioned reference frame is parameter-independent[4].

Starting with the pioneering work of[1], currently several con-

trollers have been proposed in the literature, specially in applica-

tions related with wind generation, e.g. [5,6]. Unfortunately, in

most of these contributions both the design and/or the perfor-

mance analysis rely on the application of classical linear control

techniques[8,9], simplifying model assumptions (either consider-

ing linearized models[7], negligible some of the machine parame-

ters[10]or using reduced order models [11]) or by implementing

non-robust open-loop integration methods for estimation of

unmeasurable variables (e.g. stator fluxes[12]), while (to the best

of the authors knowledge) only a few references can be found,

where a rigorous complete stability analysis is presented, being

the results presented in[13,14]some illustrative examples.

The purpose of this paper is to contribute to the proposition of

control schemes for DFIM that can deal with the stator-side power

regulation control problem with rigorously established stabilityproperties. The motivation for contributing in this way to the solu-

tion of this problem lies in the conviction that better structured

control schemes leads to the achievement of higher performances.

To this aim, in this paper it is shown that the controller presented

by the authors in[15], where it was considered the case of having

an isolated load, can be directly implemented to solve the power

control problem, in the sense that some constant reference for

the stator-side active and reactive powers can be reached asymp-

totically. This objective is achieved by identifying, for some given

power reference, the corresponding values for stator and rotor cur-

rents and mechanical speed that must be achieved and showing

that the considered control scheme indeed states a stabilization

0142-0615/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijepes.2012.08.067

Corresponding author.

E-mail addresses: [email protected](I. Lpez-Garca), [email protected]

(G. Espinosa-Prez), [email protected] (H. Siguerdidjane), arnau.

[email protected](A. Dria-Cerezo).1 On sabbatical leave at LSS-SUPELEC, France, supported by SUPELEC, DGAPA-

UNAM (IN111211), II-FI-UNAM (Grant 1111) and Universit Paris-Sud XI.2 Arnau Dria-Cerezo was partially supported by the Spanish government research

project DPI2010-15110.

Electrical Power and Energy Systems 45 (2013) 303312

Contents lists available atSciVerse ScienceDirect

Electrical Power and Energy Systems

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j e p e s
http://dx.doi.org/10.1016/j.ijepes.2012.08.067mailto:[email protected]:[email protected]:[email protected]:arnau.%[email protected]:arnau.%[email protected]://dx.doi.org/10.1016/j.ijepes.2012.08.067http://www.sciencedirect.com/science/journal/01420615http://www.elsevier.com/locate/ijepeshttp://www.elsevier.com/locate/ijepeshttp://www.sciencedirect.com/science/journal/01420615http://dx.doi.org/10.1016/j.ijepes.2012.08.067mailto:arnau.%[email protected]:arnau.%[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.ijepes.2012.08.067


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mechanism for this operation conditions, provided the mechanical

torque delivered to the generator is constant.

The usefulness of the controller scheme is evaluated assuming

the more typical operation conditions for the DFIM, i.e. active

power is always fed into the grid (for different values of stator

power factor) while it is allowed both to deliver or demand reac-

tive power to/from the grid. This different scenarios are considered

by letting the constant mechanical torque to take different values.

The achieved stability and performance properties of the closed-

loop system are illustrated, first, under an stringent scenario,

where the generator is at standstill at the beginning of the exper-

iments, and, second, under a more practical situation, where the

operation departs from non-zero initial conditions. The attractive-

ness of the control scheme is concluded from two different per-

spectives, namely: From the dynamic point of view, due to the

stability properties and performance exhibited by the closed-loop

system, and from a practical viewpoint, since the magnitude of

all the system variables correspond to commercially available

off-the-shelf experimental setups (including both the generator

and the power converter).

The rest of the paper is organized as follows: In Section2, the

considered model for the DFIM is presented, together with the con-

trol problem formulation and its solvability analysis. The proposed

controller is developed in Section3 while its evaluation is carried

out in Section 4. Section 5 is devoted to state some concluding

remarks.

2. Problem formulation

In this section the considered model for the DFIM is presented

to later on state the approached control problem and the condi-

tions under which it is solvable.

2.1. DFIM model

Under the assumption of linear magnetic circuits and balanced

operating conditions, the equivalent two-phase model of the sym-

metrical DFIM, represented in a rotating dq reference frame fixed

to the stator voltage vector, is given by[4]

disdt

xsJis xbJkr cis abkrbLrLsr

us bur 1

_kr xs xJkr aLsris akr ur 2

J_xLsrLr

iTsJkr Bx Tm 3

where xs is the rotation speed for the reference frame, x is therotor speed, is= [is1, is2]

T are the stator currents, kr= [kr1,kr2]T are

the rotor fluxes, us and ur are the stator and rotor voltages,

respectively, while the all positive parameters are given by

aRrLr

; bLsrl

; c1

lRsL

2r RrL

2sr

Lr

!

withl LsLr L2sr and

J 0 1

1 0

JT

HereLs,Lrare stator and rotor proper inductances, Lsris the mutual

inductance, Rs,Rrare the winding resistances, Jis the inertia coeffi-

cient,B is the damping coefficient andTmis the applied mechanical

torque.

Considering the flux vector k Lei with k kTs; k

Tr

Tand

i iT

s

; iT

rh i

T

, whereksare the stator fluxes andirthe rotor currents,

while

Le LsI2 LsrI2

LsrI2 LrI2

; I2

1 0

0 1

4

model(1)(3)can be equivalently written as

_ks xsLsJis xsLsrJir Rsis us 5

_kr xs xLsrJis xs xLrJir Rrir ur 6

J_x LsriTsJir Bx Tm 7

Remark 1. Under generator operation of the DFIM, Tmis the torque

delivered to the machine by a controlled primary mover while

TgLsriTsJiris torque produced by the machine itself. In this paper

it is considered that Tm is constant, assumption that is not

restrictive since usually the primary mover is equipped with a

speed controller[16].

Remark 2. From a controller design perspective, one advantage

exhibited by the DFIM is that the complete state vector is measur-

able, i.e. mechanical speed and both stator and rotor currents can

be used to structure of the control scheme.

2.2. Power control problem

From the generated power viewpoint, model (5)(7) exhibits

the following structure if it is assumed that the stator terminals

are connected to an infinite bus with voltage magnitudeUand fre-

quency determined byxs. Active P and reactive Qpower at thestator side are given by

Pab ITs Vs; Qab I

TsJVs

where Is and Vs are the vectors of stator currents and voltages,

respectively, in the natural ab reference frame for the inductionmachine.3 In the dq reference frame considered for representing

the machine model, these expressions take the form

P Uis1; Q Uis2 8

which clearly exhibit the advantage of the representation, since

control of the active and reactive power can be carried out by con-

trolling each of the components of the stator current vector is.

Taking into account the information presented above, the con-

trol problem solved in this paper can be stated as.

Consider the DFIM model given by (5)(7). Assume that

A.1 The mechanical speed and both stator and rotor currents are

available for measurement.

A.2 The torque delivered by the prime mover is constant andknown.

A.3 The model parameters are known.

A.4 Stator voltages have fixed frequency and amplitude, i.e.

stator windings are directly connected to the line grid.

Under these conditions, design a control law for the rotor

voltagesur= ur(is, ir,x) such that

limt!1

P PI; limt!1

Q QI

with the desired power defined by PI, QI, guaranteeing internal

stability.

3 In theabreference frame the stator variables are represented in a fixed reference

frame while the rotor variables are expressed with respect to a reference frame thatrotates at an angular speed given by x [4].

304 I. Lpez-Garca et al./ Electrical Power and Energy Systems 45 (2013) 303312


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2.3. Solvability analysis

Once the control problem has been stated, this section is

devoted to analyze under which conditions it is possible to find a

viable solution. This analysis is carried out by studying the behav-

ior of the machine operating in steady state conditions with the

aim to identify the required values that must be reached by the

state variables of the system in order to achieve the desired oper-

ation. From a stability viewpoint, this procedure corresponds to

identify, among the different equilibrium points of the dq model,

the assignable equilibrium points, i.e. those that satisfy the im-

posed conditions. From a controller design perspective, identifying

these points has the additional advantage of establishing the

desired operation point that must stabilized in order to succeed

in the solution of the control problem.

The equilibria of system (5)(7) are determined by the set of

algebraic equations given by

xsJisI xsLsrJirI RsisI us 0 9

xs xILsrJisI xs xILrJirI RrirI urI 0 10

LsriT

sIJirI BxI Tm 0 11

where the subindex q stands for the value of the variables under

equilibrium operation. Notice that under the stated assumptions

us andTm remain constant.

The first point to be noticed is that, due to the existence of the

control input given by the rotor voltages ur, Eq.(10)can always be

satisfied by a proper choice of this variable. Thus, the analysis must

be concentrated in Eqs.(9) and (11). On the other hand, from Eq.

(8)it is clear that for a prescribed value of the active and reactive

powers, given byPI andQIrespectively, the corresponding stator

currents are given by

isI 1

U

1 0

0 1

" # PI

QI

" # 12

sinceus= [U, 0]T.

With the desired values for the stator currents, it is easy to

obtain, from(9), the value of the rotor current irq that must be

achieved as

irI 1

xsLsrJTxsJ RsI2isI us

1

xsLsrxsI2 RsJisI J

Tus 13

where it was used the fact that JTJ=JJT = I2.

The final step, which is related with the computation of the

required rotor speedxq, is carried out by substituting Eq.(13) into(11)to obtain that

xI 1Bxs

iTsIJxsI2 RsJTisIJTus Tm

B

1

BxsPI RskisIk

2

TmB

14

sinceJJ = I2 andkisIk2 i

TsIisI, withk k the Euclidean norm.

Even that from a mathematical perspective Eqs. (12)(14)

clearly state the conditions that must be satisfied in order to

achieve a given power in the stator side of the machine, from a

practical point of view several conditions must be taken into

account in order to solve the stated control problem, namely:

C.1 Given a desired power behavior (in terms of a given isq) it

must be considered the capacity of the machine concerning

the nominal rotor current that can be managed, i.e. careful

selection must be done in order to avoid demanding a rotor

current beyond the limits imposedby the construction of the

DFIM. This constraint is also related with the size of the

power converter required to operate the machine.

C.2 Eq.(14) exhibits the power balance that must be satisfied by

the machine. In this sense, for a given torqueTm, selection of

the desired active power (and the associated losses) will

determine the speed under which the machine must oper-

ate. In the same spirit than the mentioned in the previous

condition, the limits imposed by the physical structure of

the DFIM must be taken into consideration.

3. Stabilization mechanism

In this section it is presented the proposed controller that ren-

ders asymptotically stable the assignable equilibrium point intro-

duced in Section2.3. The design is carried out under a passivity-

based approach[17,18], i.e. by exploiting the energy dissipation

properties of the machine, due to the attractive results previously

obtained solving power systems related problems, e.g. [1921].

The section starts by briefly recalling the rational behind the design

approach to continue with the presentation of the controller.

3.1. Passivity-based control

The problem of stabilizing an equilibrium point of nonlinear

systems of the form

_x fx; t gxu

wherex 2 Rn is the state vector,u 2 Rm (m< n) is the control action

andg(x) is assumed full rank, is approached from the Interconnec-

tion and Damping Assignment-Passivity Based Control (IDA-PBC)

perspective by finding a control lawu(x) that leads to a closed-loop

system of the form

_x Fdx@Hdx

@x 15

whereFdx FTdx 6 0 andHd(x)P 0 is a scalar (energy) function,which satisfies the condition

xI arg min Hdx

with xq the equilibrium to be stabilized.4

A system with structure like the introduced in(15)is known as

a Hamiltonian system, and the aim of achieving such structure is

that it enjoys several properties [23]. Among them, one that is

quite important from the stability point of view is related with

the fact that ifHd(x) is considered as Lyapunov function of the sys-

tem, then its time derivative along the trajectories of(15)is given

by

_Hd @Hdx

@x T

Fdx FTd x

@Hdx

@x

allowing for the possibility of proving the asymptotic stability prop-

erties ofxq [22], due to the negativeness ofFdx FTdx.

A particular case, that will be exploited in this paper, is when

Hd(x) is linear, i.e. is a quadratic function of the form

Hdx 1

2xTPx

with P= PT > 0 a symmetric positive definite matrix. Under these

conditions

@Hdx

@x Px

4 Notice that the gradient of the scalar function Hd(x), given by@H

dx@x , is considered

as a column vector throughout the paper.

I. Lpez-Garca et al./ Electrical Power and Energy Systems 45 (2013) 303312 305
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and therefore

_Hd xTP Fdx F

Tdx

Px

guaranteeing automatically the asymptotic stability of xq if

Fdx FTdx < 0.

3.2. Controller design

In order to apply the IDA-PBC approach, it is convenient at this

point to recall that the total energy stored by the DFIM is given by

Hz 1

2zTL1z

where L diagfLe;Jg while z= [kT,Jx]T. Thus, a natural candidate

for the desired energy function of the closed-loop system is

Hdz 1

2zzI

TPdzzI

withPd PTd >0 andzI k

TI;JxI

T, wherekI LeiI. As required,

this function has a minimum in zzq = 0, i.e. when k = kq and

x = xq, the former implying that the minimum is achieved when

i= iq

.If the matrix Pd has the following structure

Pd

psI2 0 0

0 prI2 0

0 0 pm

264

375

withps, prandpm positive constants (to be defined latter) and the

matrix Fd(z) is partitioned in a conformal way as

Fdz

F11z F12z F13z

F21z F22z F23z

F31z F32z F33z

264

375

then the desired closed-loop system(15)takes the form

_z FdzPdzzI FdzPdLe 16with

e

es

er

em

264

375

is isI

ir irI

x xI

264

375

which (after carrying out all the required computations) can be

written as

_z

LspsF11 LsrprF12 LsrpsF11 LrprF12 JpmF13

LspsF21 LsrprF22 LsrpsF21 LrprF22 JpmF23

LspsF31 LsrprF32 LspsF31 LsrprF32 JpmF33

264

375e 17

In order to find the structure ofFd(z) and the values forps,prand

pmthat satisfy the required conditions, consider the first constraintgiven in terms of the stator flux dynamics(5)as

xsJis xsLsrJir Rsis us LspsF11 LsrprF12es LsrpsF11

LrprF12er JpmF13em

This equation, under the value obtained for usfrom the equilib-

rium Eq.(9)can be written as

xsJ RsI2es xsLsrJer LspsF11 LsrprF12es LsrpsF11

LrprF12erJpmF13em

leading directly to the solution given byF33= 0 and

F11

F12

LspsI2 LsrprI2

Lsr

prI

2 L

rp

rI

2

1 xsJ RsI2

xsL

srJ 18

Concerning the second constraint, obtained from the rotor flux

dynamics (6), the presence of the control input u r allows for the

possibility of satisfying the corresponding equation no matter the

value ofFd(z),ps,prandpm. In fact, taking advantage of this facility,

an useful choice for the corresponding components of the matrix

Fd(z) (whose usefulness will be better recognized during the stabil-

ity analysis) is given by

F21 FT12; F23 F

T32; F22

kr2pr

I2 19

withkra gain (to be defined later).

Once the second constraint has been studied, the attention

must be concentrated in the third one, imposed by (7) and given as

LsriTsJir Bx Tm LspsF31 LsrprF32es LspsF31 LsrprF32er

JpmF33em

Again, exploiting the value obtained for Tmfrom the equilibrium

Eq.(11), this last expression can be presented as

LsriTsJir Lsri

TsIJirI Bem LspsF31 LsrprF32es LspsF31

LsrprF32erJpmF33em

from which it can immediately recognized the necessity for having

F33 B

Jpmand leading to the expression

LsriTsJir Lsri

TsIJirI LspsF31 LsrprF32es LspsF31 LsrprF32er

After substitution of the errors esanderin the left-hand side of

this last expression and manipulating it in a proper way, it can be

rewritten as

eTs eTr

T psFT31prF

T32

" # LsrL

1e

JirI

Jis

( ) 0

from, where a possible solution forF31and F32could be obtained by

letting the term in the brackets equal to zero. Unfortunately, this

solution is not useful sinceF31would depend on the stateis, feature

that (as will be clear later) does not allow for achieving the desiredstability properties. Therefore it is necessary to write this condition

in an equivalent way given by

eTs eTr

T psFT31prF

T32

" # LsrL

1e

JirI

Jis

Gz

( )0

provided that

eTs eTr

TGz eTs e

Tr

T G1zG2z

0

Under these conditions, if

Gz LsL

2sr

l Jes L3srl Jer

L3

srl Jes L

rL2

srl Jer24 35

withl as in the DFIM model, then

F31 Lsrlps

kTrIJ; F32

Lsrlk

TsJ 20

are solutions of the constraint.

Until this point, putting together the results obtained in (18)

(20), it has been obtained a matrix Fd(z) that satisfies Eq. (16)for

the given structure ofPd. Indeed, this matrix is given by

Fdz

1ps

xsJ LrRsl I2

LsrRs

lprI2 0

LsrRslpr I2 kr2pr

I2Lsrl J

Tks

Lsr

lpskTrIJ

Lsr

l

kTsJ B

Jpm

26664

37775

21

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With matrix(21)at hand, then the remaining task is to find the

conditions for guaranteeing thatFdz FTd z 6 0. To this end, con-

sider that under the structure ofFd(z) given above, it is obtained

that

Fdz FTdz

2LrRslpsI2 0

Lsrlps

JTkrI

0 krpr

I2 0

Lsrlps kTrIJ 0 2BJpm

2

664

3

775where the advantage of choosing the skew-symmetric terms in the

design ofFd(z), in particular the feature of leaving the element F33depending only on the constant value krq, is now evident, since

using the standard tool given by the Schur complement it is possi-

ble to prove that the required condition Fdz FTdz < 0 is achieved

if

ps > JL2sr4BlLr

kkrIk2

!pm 22

Under the condition found for the parameters psandpm, all the

elements that are necessary for implementing the control law are

at disposition. Then from(6) and the corresponding row of (16),

it is possible to state the following

Result 1. Consider the DFIM model (5)(7) and assume thatA.1 to

A.4hold. Let the rotor control input be given as

ur xs xJLsris Lrir Rrir psF21Lses Lsrer

prF22Lsres Lrer JpmF23em 23

where

F21 LsrRslpr

I2; F22 kr2pr

I2; F23Lsrl

JTLsis Lsrir

while kr> 0, pr> 0, pm> 0 and

ps >

JL2sr4BlLr kLsrisI LrirIk

2 !pm

Define the equilibrium point of the closed-loop as

isI 1

U

1 0

0 1

PI

QI

irI 1

xsLsrxsI2 RsJ

TisI JTus

xI 1

BxsPI RskisIk

2 TmB

Under these conditions, it is achieved that

limt!1

P PI; limt!1

Q QI

guaranteeing internal stability.

Remark 3. As pointed out in[15],the structure of the presented

controller can be further simplified, from a tuning point of view,

by defining

ks psLsrRslpr

; km pmJLsrlpr

since, under these conditions, the controller takes the form

ur xs xJLsris Lrir Rrir ksLses Lsrer krLsres Lrer

kmJLsis Lsrirem

with kr> 0,km> 0 and

ks > L2sr4BlLr

kLsrisI LrirIk2

!km

4. Controller evaluation

The usefulness of the presented control scheme is analyzed inthis section. This analysis is divided into two parts, namely: First,

Table 1

DFIM parameters.

Rs= 4.92[X]

Rr= 4.42[X]

Ls= 0.725[H]

Lr= 0.715[H]

Lsr= 0.71[H]

J= 0.00512[kg m2]

B= 0.005[N m/rad/s]

0 0.1 0.2 0.31

0

1

2x 10

4

Ps[W]

Time [s]

0 0.1 0.2 0.34

3

2

1

0

1x 10

4

Qs[VAR]

Time [s]

0 0.1 0.2 0.35

0

5

10

15

20

x 104

Pr[W]

Time [s]

0 0.1 0.2 0.32

0

2

4

6

8

x 104

Qr[VAR]

Time [s]

Fig. 1. Stator and rotor powers forPI 1750:7W,QI 0VARwith zero initial

conditions.

0 1 2 3 4 5 60

50

100

150

200

250

w[rad/s]

Time [s]

Fig. 2. Mechanical Velocity for PI 1750:7W, QI 0VAR with zero initial

conditions.

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the stability properties of the control law are numerically evalu-

ated. It is illustrated how, as predicted by the theory, all the vari-

ables converge to their desired values both, under an extreme

operation regime (zero initial conditions) and under a more realis-

tic condition (non-zero initial conditions). Second, for several oper-

ation regimes the corresponding values for the rotor and speed

variables are given with the aim to evaluate if the requirements

imposed by the controller operation are feasible to achieve in a

practical setting. This evaluation procedure was carried out under

the MATLAB/Simulink environment.

Throughout all the operation conditions, there were considered

the same machine parameters, which are included inTable 1, the

same controller gainsks= 1700, kr= 1000 andkm= 0.18, while the

infinite bus was modeled in such a way that the amplitude of the

stator terminal voltages was fixed at U= 220[V].

To illustrate the stability (convergence) features exhibited by

the control law, it is first considered the case when it is demanded

some amount of active power in the stator side with zero reactive

power. With the aim to evaluate the scheme under drastic condi-

tions, it was assumed that the generator was at standstill (all the

initial conditions were set to zero). In this case, the reference for

the stator active power was PI 1750:7Wwith a desired stator

power factor PFsq= 1 while the mechanical torque was kept at

Tm= 5[Nm]. From Figs. 14 the stator and rotor powers, the

mechanical speed, the stator and rotor currents and a phase

voltage of the stator and rotor are shown, respectively. In all these

pictures it can be shown how the desired behavior is achieved in a

very short time. It is also clear that a transient response of consid-

erable magnitude is exhibited, but this behavior is due to the

stringent initial conditions imposed to the generator, as will be

clear in the next experiments when a more practical situation is

considered.

A second experiment, as well as the first one more oriented to

stress the stability properties of the control scheme, is presented

from Figs. 58. In this case the reference for the stator active power

was PI 1400:6W while for the reactive power was

QI 1050:4VAR, with a desired reactive power of PFsq= 0.8

0 0.1 0.2 0.320

10

0

10

20

30

isd[A]

Time [s]

0 0.1 0.2 0.320

0

20

40

60

isq[A]

Time [s]

0 0.1 0.2 0.330

20

10

0

10

20

ird[A]

Time [s]

0 0.1 0.2 0.3

60

40

20

0

20

irq[A]

Time [s]

Fig. 3. Statorand rotor currents for PI 1750:7W, QI 0VAR withzero initial

conditions.

0 2 4 6

215

220

225

230

Vsd[V]

Time [s]

0 2 4 62

1

0

1

2

Vsq[V]

Time [s]

0 2 4 60

100

200

300

400

500

Vrd[V]

Time [s]

0 2 4 62000

1500

1000

500

0

500

Vrq[V]

Time [s]

Fig. 4. Statorand rotor voltages for PI 1750:7W, QI 0VARwith zero initialconditions.

0 0.1 0.2 0.31

0

1

2x 10

4

Ps[W]

Time [s]

0 0.1 0.2 0.34

3

2

1

0

1x 10

4

Qs[VAR]

Time [s]

0 0.1 0.2 0.35

0

5

10

15

20x 10

4

Pr[W]

Time [s]

0 0.1 0.2 0.3

0

2

4

6

8x 10

4

Qr[VAR]

Fig. 5. Stator and rotor powers forPI 1400:6W,QI 1050:4VAR with zero

initial conditions.

0 1 2 3 4 5 60

50

100

150

200

250

300

350

400

w[rad/s]

Time [s]

Fig. 6. Mechanical Velocity for PI 1400:6W, QI 1050:4VAR with zeroinitial conditions.



7/10

and considering a mechanical torque ofTm= 5[Nm]. In the same

way than in the first experiment, the convergence properties of the

controller are well illustrated. Again, the same remark than in the

previous experiment concerning the considerable transient

response applies.With the aim to further evaluate the controller scheme consid-

ering not only its stability but also its performance properties, the

two additional experiments were carried out now considering a

more realistic operation. To this end, instead of considering the

limit case of having zero initial conditions, it was considered that

the system operates under some given conditions and a step

change in the required power is imposed, i.e. different from zero

initial conditions were considered. In the first case, it was required

that the stator active power passes from an initial value of

PI 2100W to a final one ofPI 1750Wwhile the reactive

power changes from an initial value ofQI 300VARto a desired

value of QI 0VAR. The desired behavior considered a unitary

power factor and the mechanical torque was set to Tm= 5[Nm].

Fig. 9presents the stator and rotor powers evolution, where it canbe noticed how the required values are achieved in around 3 s,

while inFig. 10the corresponding mechanical speed is exhibited.

InFig. 11both the stator and rotor currents are included. In these

last three pictures, it can be noticed that although the currents

exhibit some transient response, its magnitude is now reasonable

in a practical context, as well as the required speed. This remarkis also applicable to the demanded rotor voltages which are

included inFig. 12together with the stator voltages.

The second case under this non-zero initial conditions scenario,

was devoted to the case when the final operation conditions

involve a non-zero reactive power value. In this case the initial

conditions were set in such a way that generator was operating

atPI 1250W withQI 1150VAR while the final desired

value for the power was PI 1400W with QI 1050VAR.

The considered power factor was PFsq = 0.8 and the mechanical

torque wasTm= 5[Nm]. The dynamic behavior of the generator

variables are exhibited in Fig. 13for the stator and rotor powers,

Fig. 14for the mechanical speed, Fig. 15 for the stator and rotor

currents andFig. 16for the stator and rotor voltages. In all of them

it can be noticed how the desired behavior is achieved underreasonable transient values for all the variables.

0 0.1 0.2 0.3

20

10

0

10

20

30

isd[A]

Time [s]0 0.1 0.2 0.3

20

0

20

40

60

isq[A]

Time [s]

0 0.1 0.2 0.330

20

10

0

10

20

ird[A]

Time [s]

0 0.1 0.2 0.360

40

20

0

20

irq[A]

Time [s]

Fig. 7. Stator and rotor currents forPI 1400:6W,QI 1050:4VARwith zero

initial conditions.

0 2 4 6

215

220

225

230

Vsd[V]

Time [s]0 2 4 6

2

1

0

1

2

Vsq[V]

Time [s]

0 2 4 6200

0

200

400

Vrd[V]

Time [s]

0 2 4 62000

1500

1000

500

0

500

Vrq[V]

Time [s]

Fig. 8. Stator and rotor voltages forPI 1400:6W,QI 1050:4VARwith zero

initial conditions.

0 2 4 6

2500

2000

1500

1000

Ps[W]

Time [s]0 2 4 6

1000

500

0

500

1000

Qs[VAR]

Time [s]

0 2 4 6500

1000

1500

2000

2500

Pr[W]

Time [s]

0 2 4 60

500

1000

1500

Qr[VAR]

Time [s]

Fig. 9. Stator and rotor powers for PI 1750W, QI 0VAR with non-zero

initial conditions.

0 1 2 3 4 5 640

60

80

100

120

140

160

180

200

220

w[rad/s]

Time [s]

Fig. 10. Mechanical Velocity for PI 1750W,QI 0VARwith non-zero initial

conditions.



8/10

The second part in the evaluation of the presented controller

includes several operation conditions that are presented in a

compact way in Table 2. Concerning the procedure followed for

carrying out this part of the evaluation, it was assumed that thegenerator always fed active power into the grid while there were

considered two conditions for the reactive power: when the DFIM

operates as a capacitor (delivering reactive power to the grid) and

when it operates as an inductive load (demanding reactive power

from the grid). In all these cases the system was simulated for dif-

ferent values of the power factor including both lagging and lead-

ing values. To achieve these operation conditions, as in the

previous experiments, once the desired value for power (PI and

QI) was defined, the corresponding components of the desired sta-

tor currents were computed to later on obtain the corresponding

desired rotor current and speed, i rq andxq respectively. Lookingfor an uniform evaluation context, the components of the desired

stator current vector were defined in such a way that the magni-

tude of this vector was kept at the same value in all the experi-ments, i.e. the condition that k isqk

2 = 2.6[A] always was satisfied.

In addition, variation of the delivered mechanical torque was also

considered.

In the first six columns ofTable 2 are presented the required

behavior in terms of the applied mechanical torque, the desiredpower factor, () Lagging, (+) Leading, the desired power (active

and reactive) and the fixed conditions considered for the infinite

bus. The second part is devoted to present the conditions that

are necessary to match in the rotor side of the machine in order

to achieve the desired conditions. Here negative values of power

represent active or reactive power supplied by the stator or the

rotor and positive values represent the amount of power that they

demand. The voltages and currents correspond to therms magni-

tude of one of the phases while the last column stands for the slip

frequency, i.e. the difference between the actual and the synchro-

nous speeds in steady-state. It must be noticed that after conver-

gence of the actual variables to the desired ones, the values

included for the stator and rotor currents and the mechanical

speed, correspond to the computed from Eqs.(18)(20), neverthe-less, they are included for completeness purposes.

0 2 4 6

3.5

3

2.5

2

1.5

isd[A]

Time [s]0 2 4 6

1

0.5

0

0.5

1

isq[A]

Time [s]

0 2 4 61.5

2

2.5

3

3.5

ird[A]

Time [s]

0 2 4 62

1.5

1

0.5

0

irq[A]

Time [s]

Fig. 11. Stator and rotor currents for PI 1750W, QI 0VAR with non-zero

initial conditions.

0 2 4 6

215

220

225

230

Vsd[V]

Time [s]

0 2 4 62

1

0

1

2

Vsq[V]

Time [s]

0 2 4 650

100

150

200

250

Vrd[V]

Time [s]

0 2 4 610

0

10

20

30

40

Vrq[V]

Time [s]

Fig. 12. Stator and rotor voltages for PI 1750W, QI 0VAR with non-zero

initial conditions.

0 2 4 61700

1600

1500

1400

1300

1200

Ps[W]

Time [s]

0 2 4 61300

1200

1100

1000

900

800

Qs[VAR]

Time [s]

0 2 4 6500

400

300

200

100

0

Pr[W]

Time [s]

0 2 4 6800

600

400

200

Qr[VAR]

Time [s]

Fig. 13. Stator and rotor powers for PI 1400W, QI 1050VAR with non-

zero initial conditions.

0 1 2 3 4 5 6360

370

380

390

400

410

420

430

w[rad/s]

Time [s]

Fig. 14. Mechanical Velocity for PI 1400W, QI 1050VAR with non-zero

initial conditions.



9/10

The usefulness of the information included inTable 2is mainly

concentrated in the information concerning the rotor side of the

machine. It can be viewed from two different perspectives. The

first one is related with conditions C.1 andC.2 introduced in Sec-

tion2.3. In this sense, all the values obtained for the motor vari-

ables could be used to identify safe operation regimes in terms of

the nominal characteristics of the rotor side when an specific DFIM

is analyzed. The information can be used to conclude if it is possi-

ble or not to achieve a given amount of desired generated power.

The second point of view refers to the possibility of using this

information from a design perspective. If the analysis is carried

out without considering any specific machine, then the informa-

tion could be used to carry out a proper selection of both, the gen-

erator by itself and the size of the converter that must be

implemented for feeding the rotor windings.

In the particular numeric example considered inTable 2, from

the information provided by the column that displays the mechan-

ical speed, it can be concluded that for a proper operation of the

scheme it is not necessary to consider something else but a stan-

dard prime mover, since the speed values required under the sev-

eral operation conditions can be reached with a standard device for

the corresponding power requirements.

Concerning the power converter that must be considered for

equipping the generator, its structure is mainly defined by the

power that it must handle and the frequency at which it must

operate, i.e. the information contained in the last three columns

ofTable 2. In this context it is important to recognize that having

positive and negative values of the slip frequency (which means

the necessity for operating at both sub and super-synchronous

speeds) implies that the converter must be able to change the rotor

phase sequence. Moreover, since there are also positive and nega-

tive (active and reactive) rotor powers, then it will be also required

a bidirectional operation for this device. Fortunately, these charac-

teristics are easily fulfilled with standard back to back converters

and the only point that is necessary to study is the size that is

related with the switching frequency operation and the power

capacity. However, even from this perspective, the required valuesfor all the operation regimes can be easily managed with a

standard converter of reasonable size.

The only point that must be treated with more detail is related

with the required rotor current. As can be seen in the correspond-

ing column of the table, the required value of this variable in-

creases when both active and reactive power is delivered to the

grid. This fact generates the necessity for considering a DFIM capa-

ble of working under this demanded current. However, even that

this is the major limitation of the scheme, achieving values of

0 2 4 6

2.6

2.4

2.2

2

1.8

isd[A]

Time [s]0 2 4 6

1.4

1.6

1.8

2

isq[A]

Time [s]

0 2 4 61.8

2

2.2

2.4

2.6

ird[A]

Time [s]

0 2 4 63.2

3

2.8

2.6

2.4

2.2

irq[A]

Time [s]

Fig. 15. Stator and rotor currents for PI 1400W, QI 1050VAR with non-


0 2 4 6

215

220

225

230

Vsd[V]

Time [s]

0 2 4 62

1

0

1

2

Vsq[V]

Time [s]

0 2 4 680

60

40

20

Vrd[V]

Time [s]

0 2 4 625

20

15

10

Vrq[V]

Time [s]

Fig. 16. Stator and rotor voltages for PI 1400W, QI 1050VAR with non-


Table 2

DFIM response to fixed stator power exchange.

Tm [N-m] PFsq Psq [W] Qsq [VAR] isa [A] usa [V] ira [A] ura [V] x [rad/s] Pr[W] Qr[VAR] fr[H]

4.5 0.8 1400.6 1050.4 2.6 220 2.28 47.48 261.2 322.9 43.2 8.42

5 0.8 1400.6 1050.4 2.6 220 2.28 23.43 361.2 156.1 38.4 7.49

5.5 0.8 1400.6 1050.4 2.6 220 2.28 94.17 461 635.2 120.1 23.41

4.5 0.9 1575.6 763.1 2.6 220 2.47 102.63 186.9 761.6 22.2 20.25

5 0.9 1575.6 763.1 2.6 220 2.47 30.55 286.9 226.8 4.7 4.33

5.5 0.9 1575.6 763.1 2.6 220 2.47 41.52 386.9 308 12.7 11.48

4.5 1.0 1750.7 0 2.6 220 2.90 162.28 112.6 1302 557.4 32.07

5 1.0 1750.7 0 2.6 220 2.90 87.66 212.6 711.8 280.8 16.16

5.5 1.0 1750.7 0 2.6 220 2.90 13.89 312.6 121.2 4.2 0.24

4.5 0.9 1575.6 763.1 2.6 220 3.28 108.08 186.9 823.3 675.3 20.25

5 0.9 1575.6 763.1 2.6 220 3.28 32.76 286.9 288.5 144.5 4.33

5.5 0.9 1575.6 763.1 2.6 220 3.28 46.49 386.9 246.2 386.3 11.58

4.5 0.8 1400.6 1050.4 2.6 220 3.41 51.30 261.2 407.9 330.7 8.42

5 0.8 1400.6 1050.4 2.6 220 3.41 29.55 361.2 71.1 294.3 7.49

5.5 0.8 1400.6 1050.4 2.6 220 3.41 104.62 461.2 550.2 919.2 23.41



10/10

around 3.5[Arms] can be considered yet under a reasonable limit for

a DFIM as the considered for the numerical evaluation.

5. Concluding remarks

It was shown in this paper that the controller proposed in[15]

can be directly applied in order to solve the stator-side power reg-

ulation control problemof DFIM. This conclusion is valid for both, adynamic performance scenario and a practical implementation

perspective. The procedure followed for evaluating the control

scheme involved typical operation conditions for a generator of

the approached kind and the obtained results allowed for the illus-

trating the stability (convergence) properties of the control law,

including stringent (zero initial conditions) and more practical

conditions (non-zero initial conditions). In addition, it was shown

that for all the operation regimes the structural requirements for

the generator as well as for the power converter required to oper-

ate the machine, correspond to standard devices that can be easily

obtained in a practical context. It remains to develop a deeper

research work with the aim to relax the assumption of having a

constant mechanical torque delivered by the prime mover in order

to enlarge the possible applications of the proposed controller.

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