Energy Conversion and Management 82 (2014) 11–26

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Modeling, control and fault diagnosis of an isolated wind energyconversion system with a self-excited induction generator subjectto electrical faults

http://dx.doi.org/10.1016/j.enconman.2014.02.0680196-8904/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: i.atoui@csc.dz, atoui_article@yahoo.fr (I. Attoui), omeiri.

amar@univ-annaba.org (A. Omeiri).

Issam Attoui a,⇑, Amar Omeiri b

a Welding and NDT Research Centre, BP 64, Cheraga, Algeriab Electrical Laboratory LEA, Badji Mokhtar-Annaba University, BP 12, Annaba 23000, Algeria

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 November 2013Accepted 27 February 2014Available online 19 March 2014

Keywords:Wind energy conversion systemsFault detection and diagnosisSEIGFractional-Order Controllers

In this paper, a contribution to modeling and fault diagnosis of rotor and stator faults of a Self-ExcitedInduction Generator (SEIG) in an Isolated Wind Energy Conversion System (IWECS) is proposed. In orderto control the speed of the wind turbine, while basing on the linear model of wind turbine system about aspecified operating point, a new Fractional-Order Controller (FOC) with a simple and practical designmethod is proposed. The FOC ensures the stability of the nonlinear system in both healthy and faulty con-ditions. Furthermore, in order to detect the stator and rotor faults in the squirrel-cage self-excited induc-tion generator, an on-line fault diagnostic technique based on the spectral analysis of stator currents ofthe squirrel-cage SEIG by a Fast Fourier Transform (FFT) algorithm is used. Additionally, a generalizedmodel of the squirrel-cage SEIG is developed to simulate both the rotor and stator faults taking iron loss,main flux and cross flux saturation into account. The efficiencies of generalized model, control strategyand diagnostic procedure are illustrated with simulation results.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The recent technological developments of the wind turbinesystems focus on maintenance cost reduction [1–3], productioncapacity improvement [4,5], control stability and operationalreliability [6–8]. The use of an induction machine as a generatorin almost all types of these systems [9,10] as in isolated windenergy conversion system justifies the importance of thesupervision of their normal operations. A variety of faults can oc-cur within the induction machine [11], such as the rotor cagemalfunctions and the stator phase unbalance, that may resultin a complete breakdown if the progress of the fault is notdetected.

Generally, the fault diagnosis method consists of creating thereal fault into the physical system, and evaluating its effect on dif-ferent measured variables. Such approach can be dangerous for thegenerator and may lead to the destruction of the wind turbine.Therefore, adequate models of the induction machine for studyingthe behavior of the wind energy conversion system are needed and

remain an effective tool to predict the performance of the IWECSunder fault conditions.

The classical dynamic models incorporating the effect of satura-tion in the magnetic circuit of the SEIG which are based on station-ary reference frame d–q axes theory as in [12–16] suppose thatboth stator and rotor windings are symmetric. In these models,the equivalent resistance matrix is diagonal and the equivalentinductance matrix is symmetric. These models are simple for sim-ulation but they cannot reflect any asymmetries due to stator orrotor faults.

In the literature, other analytical models of induction ma-chines are still the most common choices for the emulation ofstator and rotor faults [17–19]. The simplest method to simulatea stator fault is to insert an additional resistance in series to onephase stator winding in order to cause a stator phase unbalance.While emulating a rotor fault, the classical model takes into ac-count the individual conductors in the rotor cage using R–L ser-ies circuits, with current loops defined by two adjacent rotorbars connected by portions to the end ring. When a broken rotorbar fault is considered, the corresponding bar resistance value isassumed to be high value to force the current in the bar to zero[20]. Therefore, for squirrel-cage self-excited induction generator,a model with Nr rotor bars takes into account the nonlinearity ofthe magnetic circuit characteristic of the machine. This approach

i j+1i j Re

Lb

Lb

Rb

LeRe

LeRe

12 I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26

leads to a very complicated transient model. Such a model isquite complex and its computer simulation becomes very long.So, it will not be used in this work.

In this paper, a new model based on stationary frame d–q of theSEIG incorporating the effect of stator and rotor asymmetries (dueto faults) is developed. The effects of iron losses, main flux andcross flux saturation are also incorporated in the model. The modelsupposes that stator resistances are not equal. This situation isequivalent to a dissymmetry winding due to faults in the SEIG sta-tor such as the inter-turn short circuit. The rotor is decomposed toNr loops. For a healthy rotor, the rotor loops are identical and havethe same parameters that make this model similar to the classicald–q model, but when a rotor fault occurs, some loops are affected.In this condition, the equivalent resistance of the rotor in the twoaxes d–q model is not diagonal, making this model more general-ized than the classical d–q model.

Fault diagnosis allows the detection and the identification ofany deviation of the operating parameters from their normal or ex-pected values [21,22]. Most popular methods of induction machinecondition monitoring use the steady-state spectral components ofthe current variables [23,24]. In this paper, in order to diagnosisthe stator and rotor faults in the SEIG, we use a diagnostic proce-dure based on the stator current analysis using FFT algorithm thathas the ability to identify and isolate certain frequency compo-nents of interest [25].

Recently, application of fractional calculus in the control areais increasingly used [26–30]. In comparison with the classical or-der controllers, the fractional order controllers have a potentialto improve the control performance [27], and increase thesystem robustness because of extra real parameters involved[31]. In this paper, according to the imposed three tuning con-straints, to guarantee control performance and the robustnessto the loop gain variations, a new fractional order controller(Fractional Order Proportional, Integral, Derivative and Integratororder Derivative PIaDlD) is proposed and designed for the fixed-speed operation of wind turbines by adjusting the blade-pitchangle.

The paper is organized as follows. The model of the systemrepresented by Fig. 1 (SEIG, actuator and turbine) is developedin Section 2. The pitch control design, based on proposedfractional-order PIaDlD controller, is represented in Section 3.The diagnosis procedure is represented in Section 4. The simula-tion results are given in Section 5. The conclusion is given inSection 6.

2. Modeling of the system

2.1. Modeling of the squirrel-cage SEIG

2.1.1. Stator modeling taking into account stator faultThe dynamic model for the stator windings of the three-phase

squirrel cage induction generator is developed and the relevantvolt–ampere equations are:

Fixed capacitor bank

Wind direction

RL load

CC

C

Gear box

Induction generator

Fig. 1. Basic schema: SEIG with a capacitor excitation system driven by a windturbine.

�Vs ¼ RsIs þd/s

dtð1Þ

Vs ¼ ½Vsa;Vsb;Vsc�T is the stator voltage vector; Is ¼ ½isa; isb; isc�T is thestator current vector; Rs ¼ diag½Rsa;Rsb;Rsc� is an 3 by 3 stator resis-tance matrix; /s ¼ ½/sa;/sb;/sc�

T is a stator flux vector.With the Park transformation (Ps), the voltage equations for the

stator windings can be written as:

�Vsdq¼ PsRsP�1s IsdqþPs

d P�1s /sdq

� �dt

¼RSDQ Isdqþd/sdq

dtþxs

0 �11 0

� �/sdq

ð2Þ

where RSDQ is the equivalent resistance matrix and is given by:

RSDQ ¼ PsRsP�1s ¼

Rds Rsdq

Rsdq Rqs

� �;

Ps ¼ffiffiffi23

rsinðhÞ sin h� 2p

3

� �sin hþ 2p

3

� �cosðhÞ cos h� 2p

3

� �cos hþ 2p

3

� �" #ð3Þ

To emulate a stator phase unbalance, we insert an additionalresistance Rd in series to one phase stator winding (Rsa = Rs + Rd).

2.1.2. Rotor modeling taking into account rotor faultA squirrel-cage rotor is often modeled by NR meshes, as shown

in Fig. 2. Each mesh is substituted by an equivalent circuit repre-senting the resistive and inductive nature of the cage [32,33].

From the rotor cage equivalent circuit represented by Fig. 2, theelectric equation of the j-th loop can be defined as:

0 ¼ ðRbðjÞ þ Rbðj�1Þ þ 2ReÞij � Rbðj�1Þij�1 � RbðjÞijþ1 þd/rj

dtð4Þ

Expression of /rj is given by:

/rj ¼ ðLrp þ 2ðLb þ LeÞÞij þXNr

k ¼ 1k–j

Mrrirk þX3

k¼1

Mrjskisk � Lbiðj�1Þ � Lbi jþ1ð Þ

ð5Þ

Thus, the rotor electric equations can be written as:

0 ¼ RIr þddt

/r

/r ¼ ½/r1;/r2; . . . ;/rNr�

Ir ¼ ½ir1; ir2; . . . ; irNr �

ð6Þ

where R is the equivalent rotor loop resistance matrix, its expres-sion is given by:

Le

Le

Re

Le

Re

LbRb

i j-1

Rb

Le

Re

LeRe Le

Re

Le

Re

Lb

Rb

Le

Re

Le

Re

i 1

Lb

RbLeRe

Lb

Rb

LeRe

Le

Re

Fig. 2. Rotor cage equivalent circuit.

R ¼

Rb1 þ 2Re þ RbðNrÞ �Rb1 0 0 . . . �RbðNrÞ

�Rb1 Rb2 þ 2Re þ Rb1 �Rb2 0 . . . 00 �Rbðk�1Þ Rbðk�1Þ þ 2Re þ RbðkÞ �RbðkÞ . . . 0

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

�RbðNrÞ 0 0 . . . �RbðNr�1Þ RbðNr�1Þ þ 2Re þ RbðNr Þ

2666666664

3777777775ð7Þ

I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26 13

Rb and Re are: the bar resistance and the end-ring segment resis-tance, respectively.

A change of variables which formulates transformation of the Nr

rotor variables to the arbitrary two axes reference may be done bythe use of Kr transformation matrix given by:

Kr ¼

ffiffiffiffiffiffi2Nr

scos hs � hr � Ppar

2

� �. . . cos hs � hr � Pp 2j�1ð Þar

2

� �sin hs � hr � Ppar

2

� �. . . sin hs � hr � Ppð2j�1Þar

2

� �264

375ð8Þ

with Pp is the number of pole-pairs, ar = 2p/Nr and hs, hr are the twoaxes reference angular velocity.

By the use of Kr, the relation between (d,q) components of rotorcurrents and rotor current loops can be written as:

idqr ¼ Krir ð9Þ

Consequently, the rotor electric Eq. (4) becomes:

0 ¼ mf KrRK�1r Irdq þ

d/rdq

dtþ ðxs �xrÞ

0 �11 0

� �/rdq ð10Þ

While comparing with the traditional Park model of the rotor’svoltage equations in (11),

Vrdq ¼ RRDQ Irdq þd/rdq

dtþ xs �xrð Þ

0 �11 0

� �/rdq ð11Þ

we obtain the expression of rotor resistance:

RRDQ ¼ mf KrRK�1r

� �¼

Rdr Rrdq

Rrdq Rqr

� �ð12Þ

The values of RRDQ must be considered with a multiplicativeequivalence factor (mf) given by:

mf ¼Rr

ffiffiffiffi2

Nr

q2 Re

Nrþ 2Rbð1� cosðPparÞÞ

ð13Þ

The rotor resistance in Park transformation is calculated accord-ing to the bar and the end-ring segment resistances. A sharp vari-ation of rotor resistance is directly related to a defect in the rotorbars.

2.1.3. Stator and rotor flux modelingThe stator and rotor fluxes in the dq reference frame are given,

respectively, by:

/sd ¼ lsisd þ /md

/sq ¼ lsisq þ /mq;

/rd ¼ lr ird þ k/md

/rq ¼ lr irq þ k/mqð14Þ

where /md and /mq are, respectively, the direct and quadrate mag-netizing air–gap fluxes and are defined as:

/md ¼ Mdimd þMdqimq þ /d0

/mq ¼ Mqimq þMdqimd þ /q0ð15Þ

where Md, Mq and Mdq are new inductive parameters.With [34] one gets:

Md ¼@/md

@imd; Mq ¼

@/mq

@imq; Mdq ¼

@/md

@imq¼@/mq

@imdð16Þ

The analytical expressions of main and cross flux saturation de-pended coefficients are expressed as follows [34]:

Md ¼ Lm þ@Lm

@imdimd ¼ Lm þ

dLm

djImji2md

Im

Mq ¼ Lm þdLm

dIm

i2mq

Im; Mdq ¼

dLm

djImjimdimq

Im

ð17Þ

Therefore, the (d–q) components of the [im]123 system takingiron losses into account satisfy:

imd ¼Rm

Rm þ Lmpðisd þ kirdÞ

imq ¼Rm

Rm þ Lmpðisq þ kirqÞ

ð18Þ

and

i2m ¼ i2

md þ i2mq

� �=2 ð19Þ

where k (k = 1 in this work) is the transformation report of the SEIG.The iron losses are represented by means of an equivalent iron

loss resistance Rm connected in parallel with the magnetizinginductance Lm [35]. The variation in Rm is modeled by the followingcurve fit

Rm ¼ Vph þ 1200 ð20Þ

where Vph is the phase RMS voltage.

2.1.4. The generalized two axes SEIG modelThe generalized two axes model of a squirrel-cage self-excited

induction generator to simulate both the rotor and stator faults,taking into account iron loss, main flux and cross flux saturation,is presented by:

½v� ¼ ½RT �½i� þ ½LT �ddt½i� þ ½GT �½i� ð21Þ

The matrices of Eq. (21) are defined as:

½v� ¼ ½�Vds � VqsVdrVqr�T ; ½i� ¼ ½idsiqsidriqr �T ;

½RT � ¼ ½RsdqRrdq�T ; Rsdq ¼ PsRsP�1s ; Rrdq ¼ mf KrRK�1

r :

½LT � ¼

ls þMdNEW MdqNEW MdNEW MdqNEW

MdqNEW ls þMqNEW MdqNEW MqNEW

MdNEW MdqNEW lr þMdNEW MdqNEW

MdqNEW MqNEW MdqNEW lr þMqNEW

2666437775

½GT � ¼

0 �xsðls þ LmNEWÞ 0 �xsLmNEW

xsðls þ LmNEWÞ 0 xsLmNEW 00 �ðxs �xrÞLmNEW 0 �ðxs �xrÞðls þ LmNEW Þ

ðxs �xrÞLmNEW 0 ðxs �xrÞðls þ LmNEWÞ 0

2666437775

14 I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26

where

MdqNEW ¼ MdqRm

Rm þ Lmp; MdNEW ¼ Md

Rm

Rm þ Lmp;

MqNEW ¼ MqRm

Rm þ Lmp; LmNEW ¼ Lm

Rm

Rm þ Lmp

The definition of electromagnetic torque is

TSEIG ¼32

PpRm

Zðiqsidr � idsiqrÞdt �

ZRm

LmTSEIG

dt

¼ 32

PpRmLm

Rm þ Lmpðiqsidr � idsiqrÞ ð22Þ

Fig. 3. Block diagram of the linear wind turbine model.

2.2. Load (RL) and Fixed Capacitor Bank model

Assume that the load is an RL (per phase value) series circuitconnected to the stator winding. The voltages and currents equa-tions in this case will be given by

ddt Vds ¼ 1

C ðids � idLÞ þxsVqs

ddt idL ¼ 1

LchðVds � RLidLÞ þxsiqL

ddt Vqs ¼ 1

C ðiqs � iqLÞ �xsVds

ddt iqL ¼ 1

LchðVqs � RLiqLÞ �xsidch

8>>>>><>>>>>:ð23Þ

where iqL and idL are the q- and d-axis load currents.Under no-load conditions; the minimum capacitance value

needed for the onset of self-excitation is approximated as [36]:

Cmin �1

x2r Ln

m

ð24Þ

Any change in load or rotor speed may result in a loss of excita-tion. Hence, it is recommended to use about 25% overestimatedcapacitance values C = 1.25Cmin [13].

2.3. Model of wind turbine

The mathematic expressions of wind turbine are expressed asfollows [37,38]:

Taer ¼ GTg ¼ Pw=Xt ¼ GPw=Xmec ¼12

Cpðk; bÞqARt3w=Xt ð25Þ

Cpðk; bÞ ¼ 0:5176116ki� 0:4b� 5

e�21ki þ 0:0068k ð26Þ

1ki¼ 1

kþ 0:08b� 0:035

b3 þ 1ð27Þ

k ¼ XtRtw

ð28Þ

where Pw is the aerodynamic power captured from the wind; q isthe density of air; AR is the surface covered by the wind wheel; Cp

is the power coefficient; tw is the wind speed; k is the tip speed ra-tio (TSR); b is the pitch angle; Xt is the wind turbine mechanicalangular velocity; Taer is the wind turbine output torque; Tg is thedriving torque of the generator, G is the gear ratio and Xmec is thegenerator mechanical angular velocity.

Because the torque coefficient is related to the power coeffi-cient, CP, through the following relation:

Cpðk;bÞ ¼ kCqðk; bÞ ð29Þ

The manipulation of the torque coefficient using k and b will re-sult in the manipulation of the power produced by the turbine.

The fundamental dynamics of the wind turbine are captured bythe following simple mathematical model:

Tg � TSEIG ¼ JtdXt

dtð30Þ

2.4. Wind turbine model linearization

In order to design a robust linear system controller such as PIa-

DlD, it is first necessary to consider an appropriate linear dynamicmodel of the system. This requires that the non-linear turbinedynamics are to be linearized about a specified operating point.But, because the real turbine model is nonlinear, the deviation ofthe turbine operating point provokes variations in the linearizedwind turbine model parameters. This requires that the system con-troller is to be robust to parameter variation.

The linearization of the turbine Eq. (30) would yield [39,40]:

JtD _Xt ¼ cDXt þ nDtw þ dDb ð31Þ

Where the linearization coefficients are given by:

c ¼ @Tm@Xt

���op¼ @

@XtðJt

_XtÞ���op¼ 0:5qARt3

wop@@Xt

Cpðk;bÞXt

h i���op

n ¼ @Tm@tx

���op¼ @

@twJt

_Xt

� ����op¼ 0:5qAR

1Xtop

@@tx

Cpðk;bÞ � t3w

� ��op

d ¼ @Tm@b

���op¼ @

@b Jt_Xt

� ����op¼ 0:5qAR

t3wop

Xtop

@@b ½Cpðk;bÞ�

��op

8>>>>><>>>>>:ð32Þ

where kop = RXtop/txop

Here, DX, Dtx, and Db represent deviations from the chosenoperating point, Xtop, txop, and bop. The rotational speed operatingpoint, Xtop, is selected to be the desired constant speed of the tur-bine, 450 rpm (47.1 rad/s). The blade-pitch and wind speed operat-ing points are selected as (bop = 16� and txop = 11 m/s).

After Laplace transformation, the Eq. (31) becomes:

JtpDXt ¼ cDXt þ nDtx þ dDbðpÞ ð33Þ

Let

D ¼ cJt

ð34Þ

The turbine rotor shaft speed can be represented as

Reference Speed (Ωt-ref)

Actuator+ Rate limiter

+ MDZ

Pitch Angle Limits0°: 90°

tNon-Linear

Turbine +

SEIG

Wind Speed

-

+ΔΩt Δβ

++PIαDµDController

Reference Pitch (βref)

Ω

Fig. 4. Simulation of the block diagram of controlled system.

I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26 15

DXt ¼nJt

DtxðpÞ þdJt

DbðpÞ� �

1p� D

ð35Þ

Eq. (35) describes the linearized model of the wind turbine.Such model is represented by the block diagram shown in Fig. 3.

For the control of the wind turbine speed, the factor Dtx(p)�(n/Jt) is seen as a constant perturbation and the parameters (�1/D),(�d/JtD) are the time constant and static gain respectively. Whenthe turbine operating point deviates with a constant wind turbinespeed, the static gain parameter gets the most significant variation(Appendix C). For this reason, a robust control strategy to gain vari-ations of the system should be considered. In this work, the frac-tional order PIaDlD controller is proposed.

2.5. Model of actuator

The actuator turbine blade adjustment, which may be repre-sented by the transfer function given in (36) where ba(p) andbo(p) are the Laplace transform of the pitch angle input and outputrespectively, tm is the time constant.

b0ðpÞbaðpÞ

¼ 1tmpþ 1

ð36Þ

2.6. Model of wind speed

The wind speed can be modeled as a deterministic sum of sev-eral harmonics, as in [41]:

tx ðtÞ ¼ 11þ 0:2 sinð0:1047 tÞ þ 2 sinð0:2665 tÞþ sinð1:2930 tÞ þ 0:2 sinð3:6645 tÞ ð37Þ

3. Control system

The system of the studied device including speed control isshown in Fig. 4.

The wind turbine control algorithm has focused on maintainingconstant rotor speed. In this case, the input is generally the blade-pitch angle (b) and the output is the rotor speed (Xr).

The PIaDlD controller is used for controlling the rotor speed,where DXt(p) represents the input rotor speed (error signal), andDb(p) represents the output pitch angle change.

A dead zone (MDZ) is added when the command pitch rate isless than a certain value (0.1 deg/s in this study) to eliminate thenoise in the command signal and reduce actuator motion for alonger lifetime [42]. The pitch rate commanded by the actuator isphysically limited to 10� degrees per second according to manufac-turer recommendations.

3.1. Fractional order controllers

Fractional calculus gives a generalization of ordinary differenti-ation and integration to arbitrary (non-integer) order [28].

The fractional-order differentiator can be denoted by a generaloperator aDa

t [43], given by:

aDat ¼

da

dta; RðaÞ > 0

1; RðaÞ ¼ 0R ta ðdsÞ

�a; RðaÞ < 0

8>><>>: ð38Þ

where a is the order of derivative or integrals, R(a) is the real part ofthe a.

The called Riemann–Liouville definition of fractional derivativesand integrals is

aD�at f ðtÞ ¼ 1

CðaÞ

Z t

aðt � sÞa�1f ðsÞds ð39Þ

aDat f ðtÞ ¼ 1

Cðn� aÞdn

dtn

Z t

a

f ðsÞdsðt � sÞk�nþ1 ds

" #ð40Þ

where:

CðxÞ ¼Z 1

0yx�1e�ydy ð41Þ

is the Euler’s Gamma function, a and t are the limits of the opera-tion, and a is the number identifying the fractional order. In this pa-per, a is assumed as a real number that satisfies the restrictions0 < a 6 1. Also, it is assumed that a = 0. The following conventionis used: 0D�a

t � D�at .

In this work, the Oustaloup’s approximation method [44] isused. Assuming the frequency range to fit is selected as (xb, xh),the Oustaloup algorithm is based on the approximation of a func-tion of the form:

HðpÞ ¼ pa; a 2 Rþ ð42Þ

by a rational function:

bHðpÞ ¼ CYN

k¼�N

pþx0kpþxk

ð43Þ

where the zeros, poles and the gain can be evaluated from

x0k ¼ xbxh

xb

kþNþ12ð1�aÞ

2Nþ1

; xk ¼ xbxh

xb

kþNþ12ð1þaÞ

2Nþ1

; C ¼ xh

xb

�a2 YN

k¼�N

xk

x0k

The differential equation of the proposed fractional-order PIaDl-

D controller, 0 < l and k < 2, in time domain, is given by:

uðtÞ ¼ Kp eðtÞ þ KiD�at eðtÞ þ Kd1Dl

t eðtÞ þ Kd2DteðtÞ� �

ð44Þ

where Kp is a proportional constant, Ki is an integration constant.Kd1 and Kd2 are the derivation constants. Taking l = 1, a = 1 andKd2 = 0 in (44), a classical PID controller is obtained. Hence, using La-place transforms the transfer functions of the proposed fractionalorder PIaDlD controller and classical PID controller are respectivelygiven by:

CðpÞ ¼ Kp 1þ Ki

pa þ Kd1pl þ Kd2p

ð45Þ

GðpÞ ¼ Kp 1þ Ki

pþ Kdp

ð46Þ

16 I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26

The fractional-order PIaDlD controller is more flexible than theclassical PID controller, because it has one more adjustableparameter.

3.2. Design specifications

Assuming that the gain crossover frequency is xc and the phasemargin is /m, for the system stability and robustness, three speci-fications concerned with the phase and the gain of the open-looptransfer function are proposed as follows [45,46],

(i) The phase margin specification

Arg½GðjxcÞ� ¼ Arg½CðjxcÞPðjxcÞ� ¼ �pþ /m

(ii) The gain specification at the crossover frequency

jGðjxcÞjdB ¼ jCðjxcÞPðjxcÞjdB ¼ 0

(iii) Robustness to variation in the gain of the plant demands thatthe phase derivative w.r.t. the frequency is zero, i.e., thephase Bode plot is flat, at the gain crossover frequency. Itmeans that the system is more robust to gain changes andthe overshoots of the response are almost the same.

dðArgjGðjxÞjÞdx

x¼xc

¼ 0

3.3. FO-PIDD controller design procedure to wind turbine speedcontrol

The block diagram of the wind turbine speed Xt control is givenby Fig. 5:

with

FðpÞ ¼ d=Jt

p� D

1

tmpþ 1

¼ K

1tmpþ 1

1

� 1D pþ 1

!and K ¼ ð�d=DJtÞ ð47Þ

Note: D < 0 and d < 0.The open-loop transfer function G(p) of the fractional order PIa-

DlD controller for the wind turbine speed control is that,

GðpÞ ¼ CðpÞFðpÞ

¼ Kp 1þ Ki

paþ Kd1pl þ Kd2p

:

d=Jtð Þðtmpþ 1Þðp� DÞ

ð48Þ

The proposed procedure of the FO-PIDD controller design issummarized in two steps as:

3.3.1. First step: poles compensationThe first step of the proposed FO-PIDD controller design proce-

dure consists in poles compensating of the actuator system by thezeros of the regulator.

The compensation of the time-constant tm leads to choose

Kd1 ¼ Kitm

Kd2 ¼ tm

l ¼ 1� a

8><>: ð49Þ

We obtain in open loop, the transfer function G(p)

Fig. 5. Block diagram of the wind turbine speed control.

GðpÞ ¼ CðpÞFðpÞ

¼ Kp 1þ Ki

pa

� ðtmpþ 1Þ � ðd=JtÞ

ðtmpþ 1Þðp� DÞ

ð50Þ

3.3.2. Seconde step: determination of Kp, Ki and a according toimposed three tuning constraints

According to the open loop transfer function G(p) from (50), wecan get its frequency response as follows,

GðjxÞ ¼ Kp 1þ Ki

ðjxÞa

ðtmðjxÞ þ 1Þ

� d=Jt

ðtmðjxÞ þ 1ÞððjxÞ � DÞ

¼ KpK 1þ Ki

ðjxÞa

� 1� jx

D þ 1

!ð51Þ

with

K ¼ � dJt

1D¼ � d

c

According to Specification (i), the phase of G(jx) can be ex-pressed as,

ArgjGðjxcÞj ¼ � arctanKix�a

c sinðap=2Þ1þ Kix�a

c sinðap=2Þ þ arctanðxc=DÞ

¼ �pþ /m ð52Þ

From (52), the relationship between Ki and a can be established asfollows,

Ki ¼� tan½� arctanðxc=DÞ þ /m�

x�ac sinðkp=2Þ þx�a

c cosðkp=2Þ tan½� arctanðxc=DÞ þ /m�ð53Þ

According to specification (iii) about the robustness to gain vari-ations in the plant,

d Arg G jxð Þj jð Þdx

x¼xc

¼ Kikxa�1c sin kp=2ð Þ

x2ac þ 2Kixa

c cosðkp=2Þ þ K2i

þ 1=D

1þ ðxc=DÞ2

¼ 0 ð54Þ

From (54), we can establish an another equation about Ki in the fol-lowing form,

Cx�2ac K2

i þ rKi þ C ¼ 0 ð55Þ

that is

Ki ¼�r�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � 4C2x�2a

c

q2Cx�2a

cð56Þ

where

C ¼ � 1=D

1þ ðxc=DÞ2and r ¼ 2Cx�a

c cosðap=2Þ � kx�a�1c sinðap=2Þ

ð57Þ

According to Specification (ii), we can establish an equation aboutKp,

KpKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Kix�a

c cosðap=2Þ� 2 þ Kix�a

c sinðap=2Þ� 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ ðxcÞ2

q0B@

1CA ¼ 1 ð58Þ

Clearly, we can solve Eqs. 53, 56 and 58 to get a, Ki and Kp.

NOEnd

Activate Alarms/Warnings

YES Operator ControlShutdown Equipment

NOYES

Fault component detection

Fault componentsdetection at

frequencies equal to 3fs (stator fault)and/or (1±2s) for

(rotor fault). Faulty (Stator or rotor) Healthy

Stator current data ias (t).

Begin

Data vector x(t-0.5T:t+0.5T)

t = t + T

FFT computation

FFT is computed on the stator

current data x(t)

s, r

Move the temporal window

Ω Ω

Fig. 7. Flowchart of the fault diagnosis procedure.

I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26 17

Using a graphical method [33], the used procedure for thedetermination of the a, Ki and Kp is summarized as.

(1) Given xc = 30 (rad/s), the gain crossover frequency.(2) Given um = 70�, the desired phase margin.(3) Plot curve 1, Ki w.r.t a, according to (50), and plot curve 2, Ki

w.r.t a, according to (53).Fig. 6a shows the two curves.

(4) Obtain the values of a and Ki from the intersection point onthe above two curves.a = 0.2947, Ki = 10.77, (Kd1 = tm = 0.25, Kd2 = Kitm = 2.6925and l = 1 � a = 0.7053).

(5) Calculate the Kp from (55), Kp = �3.6750.

Then, we can fix the open-loop frequency response by means ofFO-PIDD controller. The proposed FO-PIDD controller is,

CðpÞ ¼ �3:5420 1þ 10:77p0:2947 þ 2:6925p0:7053 þ 0:25p

ð59Þ

So, the Bode diagrams of designed system can be plotted as inFig. 7b and c, we can see that the phase Bode plot is flat, at the gaincrossover frequency, and the all three specifications are satisfied. Itmeans that the system is more robust to gain changes. But the threeSpecifications cannot be satisfied simultaneously for classical PID orFO-PI (a = 0.1089, Ki = �1.341, Kp = �659.94, Kd1 = Kd2 = 0) control-lers with the same design method (2nd step).

4. Implemented diagnosis procedure

During the past years, the use of induction machines in theindustry has shown that broken rotor bars and stator phase unbal-ance can cause serious mechanical damage to the insulation and aconsequential winding failure may follow, resulting in a costly re-pair and a considerable loss in production [47]. This section focuseson the application of stator current signature analysis to diagnose

Fig. 6. (a) Ki versus a; (b) bode plot of th

rotor and stator faults in three-phase squirrel-cage SEIG for isolatedwind energy conversion system. The procedure is an on-line mon-itoring technique that can reduce unexpected failures, downtime,and both maintenance and operational costs.

The implementation procedure can be illustrated with the flow-chart in Fig. 7. This procedure is based on stator current analysis byFFT algorithm that gives a frequency representation of the tempo-ral signal. TheFFT results from the healthy and faulty machine arefirst compared to identify the harmonics caused by the fault. These

e open loop transfer function G(p).

18 I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26

harmonics are then compared to the theoretical fault harmonics(Eqs. (57) and (58)). Recognition of fault type is based on differ-ences in spectral signatures on the line current of the faulty/healthy machine and theoretical fault spectra signatures.

4.1. Faulty rotor operating condition

As far as the rotor fault is concerned, a broken rotor bar is pro-duced by the assumed linear change of the value of the resistanceRbF named fault bar resistance. This asymmetry produces a chain offault components in the stator windings. The firsts are the wellknown harmonic components (fsa) at frequencies:

fsa ¼ ð1� 2sÞfs ð57Þ

where s is the slip and fs is the stator voltage frequency.

4.2. Faulty stator operating condition

As far as the stator fault is concerned, a simulation of the SEIGhas been performed with an additional resistance of the same va-lue of one stator phase resistance connected in series with one sta-tor phase winding. An unbalance in the stator produces in thestator current a component at frequency:

fsb ¼ kfs ð58Þ

with k = 3, 5, 7 . . ..

Fig. 8. (a) Wind speed profile; (b) rotational speed of the SEIG in c

5. Results and discussion

To verify the efficiencies of the generalized model, the controlstrategy and the diagnostic procedure, three simulation cases ofthe Isolated Wind Energy Conversion System (WECS) are pre-sented in Fig. 1 with the specified parameters given in theappendix A. The first case is the simulation of the SEIG modeland the regulation of the rotational speed Xt of the wind turbinein healthy conditions with wind speed fluctuations. The secondcase presents the same problems but the wind turbine is infaulty conditions with broken rotor bar. The third case presentsalso the same problems but the wind turbine is in faulty condi-tions with stator and rotor faults.

Case 1: The IWECS Simulation results in healthy conditions.The time-series traces of wind speed profile, rotor speed,

blade-pitch angle are represented in Fig. 8. The pitch rate doesnot exceed ±10 deg/s and does not less than a 0.1 deg/s. How-ever, the goal of maintaining constant rotational speed is metsatisfactorily as the wind speed fluctuates between 7.6 m/sand 14.4 m/s. Fig. 9 shows the dynamic characteristics of differ-ent parameters of the induction generator when driven by thewind turbine. At t = 0 s a capacitor with capacitance of 55 lFis connected at the stator terminals of the induction generatorwithout any load, and voltage is generated because of the avail-able rotor speed at t = 0 s. The generated voltage Vas, shown inFig. 9a, is expressed as an instantaneous value and rms value.At t = 3 s a RL load (R = 250 X, L = 200e�3H) is connected. Thegenerated voltage (Vas), magnetizing current (Im) and stator cur-

ase 1; (c) blade-pitch angle (b) and its reference (ba) in case 1.

Fig. 9. Case 1: (a) instantaneous value and rms value of the generated voltage Vas, (b) stator current ias; (c) load current ila; (d) magnetizing inductance Lm; (e) magnetizingcurrent Im; (f) electromagnetic torque TSEIG.

I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26 19

rent (ias) decrease. However, the load current (iLa), magnetizinginductance (Lm) and induced electromagnetic torque (TSEIG)increase.

At t = 7 s the load resistor is decreased to 200 X. The generatedvoltage, magnetizing current and stator current decrease further.

The load current and magnetizing inductance increase. However,the electromagnetic torque decreases.

At t = 11 s and t = 22 s the RL load is increased to 300 X, 150e�3and 300 X, 150e�3, respectively. The generated voltage, magnetiz-ing current (Im), stator current (ias) and induced electromagnetic

Fig. 10. Case 2: (a) wind speed profile; (b) rotational speed of the SEIG; (c) blade-pitch angle (b) and its reference (ba); instantaneous value and rms value of the generatedvoltage Vas.

20 I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26

torque increase. However, the load current (iLa) and magnetizinginductance (Lm) decrease.

Case 2: The IWECS simulation results in faulty conditions withbroken rotor bar.

The versus time curves of rotor speed, blade-pitch angle anddynamic characteristics of different parameters of the inductiongenerator when driven by the wind turbine are plotted in Figs.10 and 11, respectively with applied constant RL load(R = 250 X, L = 200e�3H) at t0 = 3 s. It is considered that the fail-ure of a bar starts at the moment t1 = 7 s. The oscillations of thedynamic characteristics of different parameters of the inductiongenerator justify the presence of a break rotor bar defect inthe machine.

Case 3: The IWECS simulation results in faulty conditions withstator and rotor faults.

The versus time curves of rotor speed, blade-pitch angle and dy-namic characteristics of different parameters of the induction gen-erator when driven by the wind turbine are plotted in Figs. 12 and13, respectively with applied constant RL load (R = 250 X,L = 200e�3H) at t = 3 s. It is considered that the stator unbalanceand break rotor bar starts at the moment t0 = 7 s and t1 = 15 s,respectively. At t0 = 7 s all currents, torque and generated voltageincrease. The oscillations of the dynamic characteristics of the dif-ferent parameters of the induction generator shown in Figs. 12 and13, respectively justify the presence of a stator unbalance at t = 7 sand the broken rotor bar at t = 15 s in the machine.

Fig. 11. Case 2: (a) stator current; (b) load current; (c) magnetizing inductance; (d) magnetizing current; (e) electromagnetic torque of the SEIG.

I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26 21

Fig. 14a shows the FFT of the stator current for the SEIG inhealthy condition. The spectrum is completely free of any currentcomponents around the main supply frequency, fs, and conse-quently, the expected faulty current frequency components, dueto broken rotor bars or stator unbalance, are empty. The SEIG thusshows no signs of broken rotor bars or stator unbalance.

Fig. 14b shows the FFT of the stator current for the SEIG case ofstator unbalance. The presence of the fault component at fre-quency 3fs in search band is evident. The component is a sign ofstator asymmetry.

While as Fig. 14c shows the FFT of the stator current in caseof rotor unbalance. The fault components are distributed sym-metrically around fs in search band. The components are a signof rotor asymmetry. While as Fig. 14d shows the FFT of the sta-tor current in case of stator and rotor asymmetry. The presenceof the fault components at frequencies 3fs, (1 ± 2 s) fs and3(1 ± 2 s) fs is evident. The rotor fault components are distributedsymmetrically around fs and 3fs. The components are a sign ofrotor and stator asymmetry.

6. Conclusion

In this paper, a new model of the squirrel-cage SEIG is devel-oped to simulate both the rotor and stator faults taking iron losses,main flux and cross flux saturation into account. This proposed

model can be utilized to study the behavior of the wind energyconversion systems with SEIG in healthy and faulty conditions,and provides an interesting tool to test the fault diagnosis proce-dures’ effectiveness.

The spectrum analysis of the stator currents shows an indica-tion of the SEIG health condition. It aims at detecting and diagnos-ing fault type (stator unbalance or/and bar broken) of the SEIG inwind turbine system.

A fractional-order PIDD controller with a practical and simpledesign method is proposed to control the wind turbine speed inhealthy and faulty conditions. Simulation results show that theclosed-loop wind turbine system can achieve favorable dynamicperformance and robustness. This control strategy can be also usedfor different wind turbine systems that use the blade-pitch anglecontrol.

The simulation results presented in this paper validate the pro-posed model, the chosen diagnostic method and the proposed con-trol scheme.

Appendix A

A.1. Simulation of dynamic model of SEIG

Rearranging Eq. (21) of the SEIG gives a 2nd order differentialequation represented by

Fig. 12. Case 3: (a) rotational speed of the SEIG; (b) zoom of the rotational speed of the SEIG; (c) blade-pitch angle (b) and its reference (ba); (d) zoom of the blade-pitch angle(b) and its reference (ba); (d) instantaneous value and rms value of the generated voltage Vas.

22 I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26

Fig. 13. Case 3: (a) stator current ias; (b) load current ila; (c) magnetizing inductance Lm; (d) magnetizing current Im; (e) electromagnetic torque of the SEIG TSEIG.

I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26 23

p2½i� ¼ ½A0�p½i� þ ½A1�½i� þ ½B0�p½v � þ ½B1�½v �

where

½A0� ¼

� RmLmþ MdRm

lsLmþ Rds

ls

� �� Rdqs

lsþ MdqRm

lsLm�xs

� ��MdRm

lsLm

� Rdqs

lsþ MdqRm

lsLmþxs

� �� Rm

Lmþ MqRm

lsLmþ Rqs

ls

� ��MdqRm

lsLm

�MdRmlr Lm

�MdqRm

lr Lm� Rm

Lmþ MdRm

lr Lmþ R

l

��MdqRm

lr Lm�MqRm

lr Lm� Rdqr

lrþ MdqRm

lr Lmþxs

�

2666666664

½v� ¼ ½VdsVqsVdrVqr�T ; ½i� ¼ ½idsiqsidriqr�T

�MdqRm

lsLm

�MqRm

lsLm

drr

�� Rdqr

lrþ MdqRm

lr Lm�xs þxr

� ��xr

�� Rm

Lmþ MqRm

lr Lmþ Rqr

lr

� �

3777777775

Fig. 14. (a) Stator current FFT in healthy condition; (b) stator current FFT with stator fault; (c) stator current FFT with rotor fault; (d) stator current FFT with stator and rotorfaults.

24 I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26

Fig. 15. Variations in the static gain (DG) and in the time constant (DT).

A1½ � ¼

� RmRdslsLm

� �� RmRdqs

lsLm� Rm

Lmxs � Rm

lsxs

� �0 Rmxs

ls

� RmRdqs

lsLmþ Rm

Lmxs þ Rm

lsxs

� �� RmRqs

lsLm

� �� Rmxs

ls0

0 Rmðxs�xrÞlr

� RrdRmlr Lm

� �� RdqrRm

lr Lm� xs �xrð Þ Rm

Lmþ Rm

lr

� �� �� Rmðxs�xrÞ

lr0 � RdqrRm

lr Lmþ ðxs �xrÞ Rm

Lmþ Rm

lr

� �� �� RrqRm

lr Lm

� �

2666666664

3777777775

I. Attoui, A. Omeiri / Energy Conversion and Management 82 (2014) 11–26 25

½B0� ¼

� 1ls

0 0 0

0 � 1ls

0 0

0 0 1lr

0

0 0 0 1lr

2666664

3777775; ½B1� ¼

� RmlsLm

0 0 0

0 � RmlsLm

0 0

0 0 Rmlr Lm

0

0 0 0 Rmlr Lm

2666664

3777775

A.2. Results of linearization of the wind turbine equation (MatLabCode)

% MatLab Code.

ru = 1.225.R = 1.6.Ar = pi � R � R.syms wt vw beta Ar R ru.lambda = wt � R/vw.lambdai = 1/((1/(lambda + (0.08 � beta))) � (0.035/

(1 + (beta3)))).Cp = (0.5176 � ((116/ lambdai) � (0.4 � beta) � 5) � (exp(�21/

lambdai))) + (0.0068 � lambda).CpY = 0.5 � ru � Ar � (vw3) � (Cp/wt).

gamma_op = diff (CpY,wt).xi_op = diff (CpY,vw).delta_op = diff (CpY,beta).

Appendix B

B.1. SEIG parameters

2.2 kW, 3-phase, 4-pole, 50 Hz, 415 V, 4.5 A, star connected,1440 rpm, Rs = 3.735 X, Rr = 2.91 X, Xls = 4.727 X, Xlr = 4.727 X;Re = 72e�6 X; Rb = 150e�6 X; Number of rotor bar Nr = 16. Baseimpedance = 53.24 X. Base speed = 1500 rpm, J = 0.0842 kg m2.

The magnetizing inductance Lm is related to the magnetizingcurrent in the following manner:

Lm ¼ 0:3177 for Im 6 0:75¼ 0:3502� 0:0349Im � 0:0017Im for Im < Im 6 4:25¼ 0:17667 for Im > 4:25

B.2. Wind turbine

Nominal power = 2.2 kW, Base wind speed = 11 m/s,J = 1.25 kg m2; R = 1.6 m; G = 3.18, tm = 0.25 s, nominal time con-stant = 1.334, nominal static gain = �2.5453.

Appendix C

By assuming that the wind turbine speed is kept constant in thedesigned control law, the variations in the static gain (DG = j(�d/Jt-

D) � (�dopt/JtDopt)j) and in the time constant (DT = j � 1/D + 1/Doptj)of the linear wind turbine model about their nominal operatingpoint according to the blade-pitch angle for various values of thewind speed are represented in Fig. 15a and b, respectively. The sta-tic gain parameter has the most significant variation.

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