SYNCHRONOUS MACHINE PARAMETER ESTIMATION BY

Journal of ELECTRICAL ENGINEERING, VOL. 59, NO. 2, 2008, 75–80

SYNCHRONOUS MACHINE PARAMETER ESTIMATIONBY STANDSTILL FREQUENCY RESPONSE TESTS

Mourad Hasni∗— Omar Touhami

∗— Rachid Ibtiouen

∗

Maurice Fadel∗∗

— Stephane Caux∗∗

This paper presents the steps of the approach to identify the linear parameters of a salient-pole synchronous machine atstandstill and by frequency response tests data (SSFR). It is currently established and demonstrated that stability parametersfor synchronous machines can be obtained by performing frequency response tests with the machine at standstill. Theobjective of this study is to use several input-output signals, according to a setup test recommended by the standard IEEEStd 115, to identify the model structure and parameters of a salient-pole synchronous machine from frequency responsetests data. This procedure consists of defining and conducting the frequency response tests, identifying the model structure,estimating the corresponding parameters, and making valid the resulting model. We estimate the parameters of operationalimpedances, or in other terms, the reactances and the time constants. The results are presented from tests on the synchronousmachine of 1.5 kVA/380V/1500 rpm.

K e y w o r d s: synchronous machine, parameter estimation, modeling, standstill tests, frequency response

1 INTRODUCTION

Accurate identification of the field circuit is a desir-

able feature for present day stability analyses where ex-

citation controls play an important role. This is not pos-

sible with the standard tests described in IEEE Std 115

(1995). Another difficulty with these tests lies in defin-

ing adequate tests for quadrature-axis parameters, there

are not practical or acceptable procedures for obtaining

quadrature-axis transient or subtransient reactances or

time constants. Present day studies require quadrature-

axis as well as direct-axis values for an accurate and ad-

equate synchronous machine stability simulation [1].

A new approach has demonstrated that stability pa-

rameters for synchronous machines can be obtained by

performing frequency response tests with the machine at

standstill.

Many papers have been published on the modelling

and parameter estimation of synchronous machines [2–

8] using the standstill frequency response test data and

the time-domain response data. In reference [4–10], the

standstill frequency response test data method is used to

estimate machine parameters. The results given in [10–

14] indicate that when two-rotor winding or three rotor

winding models are used, a good estimate of machine

parameters can be obtained

In reference [11] methods are described for obtain-

ing synchronous machine parameters in the form of re-

actances and time constants.

Frequency response data describe the response of ma-chine fluxes to stator current and field voltage changesin both the direct and quadrature axes of a synchronousmachine. Some advantages of the method can be done ei-ther in the factory or on site, it poses a low of risk to themachine being tested, and it provides complete data inboth direct and quadrature axes. Resistances and reac-tances for the associated models can be calculated usingthe methods in the Paragraph 3.

2 SYNCHRONOUS MACHINE MODELLING

Today a synchronous machine is normally modelled inthe two axes with transient and subtransient quantities(2nd order) [11]. For synchronous machine studies, thetwo-axis equivalent circuits with two or three dampingwindings are usually assumed at the proper structures[6]. In this work, and using the Park’s d and q -axis ref-erence frame, the synchronous machine is supposed to bemodelled with one damper winding for the d-axis andtwo windings for the q -axis (2 × 2 model) as shown inFig. 1 [2–6].

Voltage equations

Vd(p) = raid(p) + pϕd(p) − ωrϕq(p) , (1a)

Vq(p) = raiq(p) + pϕq(p) + ωrϕd(p) , (1b)

Vf (p) = rf if (p) + pϕf (p) , (1c)

0 = rDiD(p) + pϕD(p) , (1d)

0 = rQiQ(p) + pϕQ(p) . (1e)

∗Laboratoire de Recherche en Electrotechnique — Ecole Nationale Polytechnique 10, Av. Pasteur, El Harrach, Algiers, Algeria,

Bp.182, 16200; [email protected], [email protected];∗∗

Laboratoire Plasma et Conversion d’Energie — Unit mixteCNRS-INP Toulouse 2, rue Camichelle – BP 7122-31071 Toulouse, Cedex 7 France

ISSN 1335-3632 c© 2008 FEI STU

76 M. Hasni — O. Touhami — R. Ibtiouen — M. Fadel — S. Caux: SYNCHRONOUS MACHINE PARAMETER ESTIMATION . . .

While eliminating ϕf , if , ϕD , iD , ϕQ , iQ we obtainthe following equations:

Vd(p) = raid(p) + pϕd(p) − ωrϕq(p) , (2a)

Vq(p) = raiq(p) + pϕq(p) − ωrϕd(p) , (2b)

ϕd(p) = Xd(p)id(p) + G(p)Vf (p) , (2c)

ϕq(p) = Xq(p)iq(p) . (2d)

This form of writing of the equations of the machinehas the advantage of being independent of the numberof dampers considered on each axis.

Fig. 1. Standard d -q axis circuit models.

In fact, it is the order of the functions Xd(p), Xq(p)and G(p), which depend on the number of dampers. Intheory we can represent a synchronous machine by an un-limited stator and rotor circuits. However, the experienceshow, in modelling and identification there is seven mod-els structures which can be used. The complex model isthe 3× 3 model which have a field winding, two damperwindings on the direct axis, and three on the quadratureaxis.

The more common representation is the one deducedfrom the second order characteristic equation which de-scribes the 2 × 2 model [8–10]

Damper circuits, especially those in the quadratureaxis provide much of the damper torque. This particular-ity is important in studies of small signal stability, whereconditions are examined about some operating point [10].The second order direct axis models includes a differen-tial leakage reactance. In certain situations for second or-der models, the identity of the transients field winding.Alternatively, the field circuit topology can alter by thepresence of an excitation system, with its associated non-linear features.

For machine at standstill, the rotor speed is zero(ω = 0) and using the p Laplace’s operator, the volt-age equations are:

– For the d-axis

Vd =[

ra +p

ω0Xd(p)

]

id + pG(p)vf , (3a)

Vf =[

rf +p

ω0Xf (p)

]

if +p

ω0Xmdid . (3b)

– For the q -axis

Vq =[

ra +p

ω0Xq(p)

]

iq . (3c)

With the operational reactances:

Xd,q,f(p) = Xd,q,f

(

1 + pT ′

d,q,f

)(

1 + pT ′′

d,q,f

)

(

1 + pT ′

d0,q0,f0

)(

1 + pT ′′

d0,q0,f0

) (4a)

and the operational function G(p):

G(p) =Xmd

rf

1

1 + pT ′

d0

. (4b)

d , q , f denote the d-axis, q -axis and field respectively.From these equations it follows that only the three func-tions Xd(p), Xq(p) and G(p) are necessary to identify asynchronous machine. In the original theory, the quadra-ture axis has no transient quantities. However, the DCmeasurements at standstill recommended by IEC, andalso the short-circuit tests in the quadrature axis showthat the real machine also has transient values X ′

q(p), T ′

q

in addition to the sub transient values X ′′

q (p), T ′′

q .

3 TEST PROCEDURE

The procedure for determining the values of syn-chronous machine parameters, using the frequency re-sponse tests data, follows several steps:

• Step 1 is the standstill frequency response process; itdetermines the data of operational impedances andtransfer functions, which punitively describe the in-teractions of voltages and currents as functions of fre-quency.

• Step 2 is the determination of transfer functions toquantize the current-flux-voltage relations in simple,standard forms, such as Xd(P ). Step 2 is a conven-tional curve fitting process.

• Step 3 is the determination of equivalent circuit data(ri, Xi, Ti, . . . ) that are used in simulations from thetransfer functions of step 2.

These are various steps that we follow to identify ourmachine.

The tests are described in ANSI-IEEE Std.115A pub-lications [1]. It is very easy to perform in practice.. Therotor is placed so that the magnetic field axis due toDC stator current, be along the direct axis (d-axis test)and then along the quadrature axis (q -axis test).

Journal of ELECTRICAL ENGINEERING 59, NO. 2, 2008 77

Fig. 2. Test Setup for Direct-Axis Measurements, using the current shunts according to the IEEE standard 115 A

It should be recognized that during standstill fre-quency response tests, the capability of the machine willbe reduced with respect to its capability at normal op-erating conditions. Therefore, test levels of currents andvoltages shall be maintained at sufficiently low levels toavoid any possible damage to either stator or rotor com-ponents. This can be achieved by limiting the maximumoutput of the power source to levels equal to or less thanthe standstill capability of the machine.

The tests cannot be performed at the rated stator volt-age or current; the determination of quantities referred tothe unsaturated state of the machine must be done fromtests with supply voltages (1 to 2%) of the nominal val-ues.

The acquisition of the experimental data is made bya data acquisition system connected to the computer forprocessing by the Software Matlab. The remainder con-sists of classical devices such as, ampermeter, voltmeter,wattmeter, frequency response analyzer, power amplifier,etcthat are necessary in controlling the physical parame-ters of the system to be identified.

A. Positioning the rotor for d- and q - axis tests

In this section, the practical aspects of measurementsare described and machine conditions for standstill testsare also given [1–2].

The tests are described in ANSI-IEEE Std.115 A pub-lications [1]. It is very easy to perform in practice. Thealignment of the rotor can be accomplished with shortedexcitation winding. A sine wave voltage is applied be-tween two phases of the stator. The duration of the volt-age application should be limited to avoid serious over-

heating of solid parts. The rotor is slowly rotated tofind the angular positions corresponding to the maxi-mum value of the excitation current that gives the directaxis and zero value of the excitation winding current, andthat corresponds to the quadrature axis. This procedureis used by the authors [2, 5 and 6].

B. Typical test setup

The power amplifier should create readily measurablesignal levels for the armature and field winding volt-ages and currents. Tests currents should be small enoughto avoid temperature changes in the armature, field, ordamper circuits during the test. Voltages at the armatureor field winding terminals shall not exceed rated voltagelevels

The magnitude and phase of the desired quantities

Zd(p), Zq(p) and∆ifd(p)∆iq(p) are measured over a range of

frequencies (Fig. 3 to Fig. 6).

The minimum frequency (fmin ) should be at least oneorder of magnitude less than that corresponding to thetransient open circuit time constant of the generator, thatis, fmin = 0.016

T ′

d0

. The maximum frequency for the test

should be greater than twice the rated frequency of themachine being tested.

It is, therefore, suggested that despite the high mea-surement cost, it is preferable for the sake of accuracy touse at least 40 points/decade in the very low frequencyrange, such as, 1 to 100 MHz. The experience suggeststhat the users of the successful tests and model devel-opment, (1) careful rotor positioning for d-q axes and(2) accurate data acquisition in low frequency range.


Fig. 3. d -axis Impedance (Field Shorted) Fig. 4. Standstill Armature to Field Transfer Impedance

Fig. 5. Standstill Armature Field Transfer Function pG(p) Fig. 6. q -axis Impedance

Fig. 7. d -axis Operational Impedance (Field Shorted) mm Fig. 8. q -axis Operational Impedance (Field Shorted)

C. Measurable parameters at standstill

According to the various test setups, the tests that we

have realized correspond to the following equations:

Zd = −∆ed(p)

∆id(p)

∣

∣

∣

∆efd=0, (5a)

Zq = −∆eq(p)

∆iq(p), (5b)

G(p) =∆ed(p)

p∆efd(p)

∣

∣

∣

∆id=0. (5c)

An alternative method of measuring this parameter issuggested as follows:

pG(p) =∆efd(p)

∆id(p)

∣

∣

∣

∆efd=0. (5d)

The advantage of the latter form is that it can be mea-sured at the same time as Zd(p). A fourth measurableparameter at standstill is the armature to field transferimpedance:

Zaf0(p) = −∆efd(p)

∆id(p)

∣

∣

∣

∆ifd=0. (5e)

Journal of ELECTRICAL ENGINEERING 59, NO. 2, 2008 79

Table 1. Synchronous Machine Parameter Values Identified

Parameters ValuesRa (pu) 0.150

Rf (pu) 4.942

T ′

d (s) 0.1842

T ′′

d (s) 0.0475

T ′

d0 (s) 1.0706

T ′′

d0 (s) 0.4290

T ′

q (s) 0.1450

T ′′

q (s) 0.0390

T ′

q0 (s) 0.8995

T ′′

q0 (s) 0.4520

Xd (pu) 1.8850X ′

d (pu) 0.3450

Xq (pu) 1.3825

X ′

q (pu) 0.2130

X ′′

d (pu) 0.380

X ′′

q (pu) 0.203

The relationships between the measured quantitiesand desired variables are given by: – d-axis parameters

Ld(p) =Zd(p) − Ra

p, (6a)

where

Zd(p) =1

2Zarmd(p) , (6b)

Rz =1

2

[

lims→0

|Zarmd(p)|]

(6c)

and p = jω .

To obtain ra , we plot the real component of thisimpedance as a function of frequency, and we extrapo-late it to zero frequency to get the dc resistance of thetwo phases of the armature winding in series. Then wecalculate pG(p) by:

∆ifd(p)

∆id(p)=

√3∆ifd(p)

2∆i(p). (6d)

Finally, we measure the ratio∆efd

∆i, and we calculate

Zaf0 =∆efd(p)

∆id(p)=

√3

2

(∆efd(p)

∆i(p)

)

. (6e)

– q -axis parameters

Lq(p) =Zq(p) − Ra

p, (7a)

where

Zq(p) =1

2Zarmq(p) (7b)

and

Rz =1

2

[

lims→0

|Zarmq(p)|]

. (7c)

Figure 7 represents the direct axis operational induc-tances for each frequency at which Zd(p) was measured.

The quadrature-axis operational reactance, Xq(p),plotted in Fig. 8, is obtained in the same way from Zd(p).

Table 1 presents the parameter vaues of a synchronousmachine from tests using in this study.

4 CONCLUSION

A step-by-step procedure to identify the parametervalues of the d-q axis synchronous machine models usingthe standstill Frequency response testing is presented inthis paper.

A three-phase salient-pole laboratory machine rated1.5 kVA and 380 V is tested at standstill and its parame-ters are estimated. Both the transfer function model andthe equivalent circuit model parameters are identified.

The standstill test concept is preferred because there isno interaction between the direct and the quadrature axis,and it can be concluded that the parameter identificationfor both axis may be carried out separately. This testsmethod had been used successfully on our synchronousmachine at standstill and gave all the Parks model pa-rameters. Among the advantages claimed for the SSFR isthat the tests are safe and relatively inexpensive.

Furthermore, the information about the quadratureaxis, as well as the direct axis of the machine is obtained.We notice that, it can be possible to realize a parameterestimation of large power synchronous machine or turbogenerators.

The results show that the machine linear parame-ters are accurately estimated to represent the machineat standstill condition.

Acknowledgments

The authors of Laboratories LRE and LAPLACErespectively of the ENP Algiers and the ENSEEIHTToulouse make a point of announcing that this workenters within the framework of an international projectof co-operation, Agreement — CMEP Tassili, under thecode 05 MDU 662.

List of symbols

ed, eq direct and quadrature-axis armature voltageefd field voltageifd field currentG(p) armature to field transfer functioni instantaneous value of armature current dur-

ing testid, iq direct and quadrature-axis armature currentZarmd Operational impedance measured between

two armature terminals during direct-axistests


Zarmq Operational impedance measured betweentwo armature terminals during quadrature-axis tests

P Laplace’s operatorω, ω0 angular and rated speedϕd, ϕq flux leakage in the direct and quadrature axisXf field leakage reactanceXd, Xq d- and q -axis synchronous reactancesXmd, Xmq d- and q -axis magnetizing reactancesXQ1, XQ2 q -axis damper leakage reactancerQ1, rQ2 q -axis damper resistancesYd,q(p) d- and q -axis operational admittancesT ′

d, T′

d0 d-axis transient open circuit and short-circuittime constant

T ′

q, T′

q0 q -axis transient open circuit and short-circuittime constant

T ′′

d , T ′′

d0 d-axis subtransient open circuit and short-circuit time constant

T ′′

q , T ′′

q0 q -axis subtransient open circuit and short-circuit time constant

ra, rf armature and field resistancesVd, Vq d- and q -axis stator voltagesid, iq d- and q -axis stator currentsVf , if d-axis field voltage and current

References

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Received 17 April 2007

Mourad Hasni received the Engineer, Master and Doc-torate degrees in Electrical Engineering in 1991, 1996 and 2007respectively from National Polytechnic School of Algiers. Cur-rently he is an assistant professor of Electrical Engineeringat Houari Boumediene University of sciences and technology(Algiers). His research interests are in modeling, parameterestimation, and fault detection of electrical machines.

Omar Touhami received the Engineer, Master and Doc-torate degrees in Electrical Engineering in 1981, 1986 and1994 respectively from National Polytechnic School of Al-giers. He is currently a professor in Electrical EngineeringDepartment at National Polytechnic School of Algiers. Hisresearch interests have including Electric Machines, VariableSpeed Drives, and Power Systems. From 1989 to 1994, hewas associate researcher in the Research Center in Automaticof Nancy (CRAN-ENSEM-INPL). He is actually reviewer inIEEE Transaction On Energy Conversion. He is also Direc-tor of Research Laboratory in National Polytechnic School ofAlgiers since 2000 on 2005.

R. Ibtiouen received the PhD in Electrical Engineeringfrom Ecole Nationale Polytechnique of Algiers (ENP) andINPLorraine France, in 1993. His fields of interest includepower electronics, power quality and electric machines. He iscurrently professor of electrical engineering at ENP Algiers.From 1988 to 1993, He was associate Researcher in the Groupede Recherche en Electronique et Electrotechnique (GREEN -Nancy - ENSEM-INPL) of Nancy.

Maurice Fadel received the Doctorate degree from the In-stitut National Polytechnique de Toulouse in 1988. He is cur-rently a Professor at the Ecole Nationale dElectrotechniquedElectronique dInformatique et dHydraulique de Toulouse(ENSEEIHT). In 1985 he joined the Laboratoire Plasma etConversion dEnergie (LAPLACE). Its work concerns the mod-elling and the control of the electric systems. Currently itshead of the Laplace Laboratiry

Stephane Caux received the Doctorate degree from theInstitut National Polytechnique de Toulouse. He is currentlyan assistant professor at the (ENSEEIHT Toulouse),He ismember of the Laboratoire Plasma et Conversion dEnergie(LAPLACE). Its work concerns the control of the electric sys-tems.

SYNCHRONOUS MACHINE PARAMETER ESTIMATION BY

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