Modelling and parameter identification of synchronous ... - IEEE Xplore

SPEEDAM 2010 International Symposium on Power Electronics, Electrical Drives, Automation and Motion

Modelling and Parameter Identification of Synchronous Machine by PWM Excitation Signals Hasni Mourad*, Touhami Omar**, Ibtiouen Rachid**, M. Fadel*** and Stephane Caux*** * Electrical & Industrial Systems Laboratory-LSEI, Faculty of Electrical Engineering & Computing, University of Sciences & Technology Houari Boumediene of Algiers, BP.32 El-Alia, Bab Ezzouar 16111 Algiers, Algeria. Email : [email protected] ** Laboratoire de Recherche en Electrotechnique – Ecole Nationale Polytechnique10, Av. Pasteur, El Harrach, Algiers, ALGERIA, Bp.182, 16200 *** Laboratoire Plasma et Conversion d’Energie -Unité mixte CNRS-INP Toulouse 2, rue Camichel - BP 7122- 31071 Toulouse Cedex 7 FRANCE

II.

Abstract-This paper presents the results of a time-domain identification procedure to estimate the linear parameters of a salient-pole synchronous machine at standstill. The objective of this study is to use PWM input signal to identify the model structure and parameters of a salient-pole synchronous machine from standstill test data. The procedure consists to define, to conduct the standstill tests and also to identify the model structure. The signals used for identification are the excitation voltages at standstill and the flowing current in different windings. We estimate the parameters of operational impedances, or in other words the reactance and the time constants. The tests were carried out on synchronous machine of 1.5 kVA 380V 1500 rpm. Index Terms--PWM signal excitation, Standstill tests, Synchronous machine.

I.

NOMENCLATURE

p ω, ω0 d, ϕq Vd, Vq id, iq Vf, if r a, r f xd, xq xd, xd xq, xq xmd xa xf Yd,q(p) Tdo, Td

Laplace’s operator angular speeds d and q-axis flux linkage d- and q-axis stator voltages d- and q-axis stator currents d-axis field voltage and current Armature and field resistances d and q-axis synchronous reactance. d-axis transient and subtransient reactances. q-axis transient and subtransient reactances. d-axis magnetizing reactance Armature leakage reactance Field leakage reactance d- and q-axis operational admittances d-axis transient open circuit and short-circuit time constant. Tdo, Td d-axis subtransient open circuit and shortcircuit time constant. Tqo, Tq q-axis transient open circuit and short-circuit time constant. Tqo, Tq q-axis subtransient open circuit and shortcircuit time constant.

978-1-4244-4987-3/10/$25.00 ©2010 IEEE

INTRODUCTION

Synchronous machine parameter identification is a problem that has attracted the attention of many researchers. Knowledge of the operational parameters of generators is necessary for performing accurate stability studies and post mortem analysis of power systems. Additionally, anomalous machine operation might be identified through parameter estimation. Traditionally, synchronous machine parameters are obtained by offline tests as described in the IEEE Standards [1]. However standards methods may not be practical and parameters obtained by these methods may not be accurate. Currently, there are many suggested methods for the determination of synchronous machine parameters. Among the modern techniques used are neural networks, finite elements, on-line and off-line statistical methods, frequency response, load rejection and others [2-5]. Different measurement techniques, identification procedures and models structures are developed to obtain models as accurate predictors for the transient behaviour of generators. The standstill modelling approach has received great emphasis due to its relatively simple testing method where the d and q axis are decoupled. This paper deals with the experimental determination of synchronous machine parameters and models. The purpose of these studies is to determine the static experimental conditions to estimate the parameters of a salient pole synchronous machine when its armature is fed at standstill by PWM excitation signals The tests are performed and studied on a synchronous machine (Sn=3 kVA, Un=220 V, In=8 A, Nn=1500 rpm). III.

SYNCHRONOUS GENERATOR MODELING

The aim of this section is to set up a theoretical model of a synchronous generator suited for both time-domain simulation, and model based control design. In essence, there are two aspects of a synchronous generator that need to be modeled. The (electro) mechanical and the

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electromagnetic part. In the present paper we will restrict ourselves to the dynamic modeling of the electromagnetic part. From a modeling point of view all synchronous generators have similar representations. They differ only with respect to some model parameters. Because the round-rotor synchronous generator is a special case of the salient-pole rotor synchronous generator, we will treat only the latter for an arbitrary number of pole-pairs. Fig. 1 depicts a salient-rotor synchronous generator with only one pole-pair (p=1). The machine has the usual three stator windings, each 120 (electrical) degrees apart. The stator windings are star connected. The rotor has one accessible circuit, the eld or excitation winding, and two sets of inaccessible circuits, called damper windings. Damper windings are real or ctitious windings that can be used to represent for example the damping effects of eddy currents in the machine. In Figure 1, one damper winding is located along the direct-axis, and one along the quadrature-axis.

IV. GENERALIZED MATHEMATICAL MODELS OF THE SYNCHRONOUS MACHINE The Park's transform permits the treatment of the generator as two electrical networks representing the direct (d) and quadrature (q) axes as viewed from the stator. Research some years ago led various authors to suggest structures for these networks ranging from the simple to the extremely complex [69]. A general consensus seems to be emerging that favours one particular structure for equivalent circuits (Figs. 2). This is aimed at providing a compromise between older models and present requirements for increasing accuracy and realism in describing complex rotor machines [9]. Ra

Xa

Xkf1

XD

XD2

XDnd

RD

RD2

RD

ed

ib vb

(a)

q

Ra

Xa

iq

iD if v f

Phase a

Vq

a

XQ1

XQ2

XQnq

RQ1

RQ2

RQnq

Xmq

ia va

Rotor

Xf

Vf

b

iQ

Rf

Xm

Vd

Phase b

Xkfnd

if

id

d

Xkf2

ed vc

Stator

ic

(b)

Phase c

Fig; 2. Generalised equivalent circuits of the synchronous machine (a) d-axis circuit (b) q-axis circuit

c

From these equivalent circuits, a convenient state model may be obtained by expressing their loop equations as a set of coupled, first order differential equations. The effects of magnetic saturation can be approximated by adding a nonlinear correction term to the total magnetizing. Saturation is represented by determining the excursion of the open circuit saturation curve from the linear, no saturated, relationships (air gap line), and using this difference to correct the magnetizing flux for a given current. The effects of saturation on leakage inductance are usually neglected. The structure of the synchronous machine model used in this study is a standard second order model with one damper in the d-axis and two dampers in the q-axis given in Figure 3.

Fig. 1. Three-phase Synchronous machine with dampers

The Six windings representing the figure 1 are described by the following equations

dϕ a ½ dt ° ° dϕ b ° V b = R a ib + ¾ dt ° dϕ c ° V c = R a ic + dt °¿

V a = R a ia +

dϕ f ½ dt °° dϕ D ° 0 = R D iD + ¾ dt ° dϕ Q ° 0 = R Q iQ + ° dt ¿

armature ( Stator )

(1.a)

Vf = Rfif +

(1.b)

A. Reduced Models Order In theory we can represent a synchronous machine by an unlimited stator and rotor circuits. However, the experience show, in modelling and identification there is seven models structures which can be used. The complex model is the 3x3 model which have a field winding, two

field ( Rotor )

The voltage applied to the D and Q circuits are null, since they are in short-circuit

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damper windings on the direct axis, and three on the quadrature axis. The more common representation is the one deduced from the second order characteristic equation which describes the 2x2 model [3, 5, 9-11] Damper circuits, especially those in the quadrature axis provide much of the damper torque. This particularity is important in studies of small signal stability, where conditions are examined about some operating point [10]. The second order direct axis models includes a differential leakage reactance. In certain situations for second order models, the identity of the transients field winding. Alternatively, the field circuit topology can alter by the presence of an excitation system, with its associated non-linear features. For synchronous machine studies, two-axis equivalent circuits with two or three damping windings are usually assumed at the proper structures [10]. Using the Park’s d and q-axis reference frame, the synchronous machine is supposed to be modelled with one damper winding for the d-axis and two windings for the q-axis (2x2 model) as shown in figs. 3.

xkf

xa

ra id

rf

ϕ d ( p ) = X d ( p )id ( p ) + G ( p )V f ( p )

(3.c)

ϕq ( p ) = X q ( p )iq ( p )

(3.e)

and G ( p ) , which depend on the number of dampers. For machine at standstill, the rotor speed is zero ( ω = 0 ) and the voltage equations are: - For the d-axis

P X ( p )]id + pG ( p )v f Vd = [ ra + ω0 d (4.a)

xf

xmd

Ud

(3.b)

This writing form of the equations has the advantage of being independent of the number of damper windings considered on each axis. In fact, it is the order of the functions X d ( p ) , X ( p ) q

V f = [r f +

if

xD

Vq ( p ) = ra iq ( p ) + pϕ q ( p ) + ωrϕ d ( p )

Uf

eq

Vq = [ ra + xa

(a)

iq

xQ1

xQ2

rQ1

rQ2

p X f ( p )]i f + X i ω0 ω 0 md d

p

ω0

X q ( p )]iq

(4.c)

With the operational reactances are: ' " (1 + pTd ,q, f )(1 + pTd ,q, f )

xmq

Uq

(4.b)

- For the q-axis

rD

ra

p

X d ,q, f ( p) = X d ,q, f

ed (b) Figs. 3. 2x2 model circuit (a) d-axis circuit (b) q-axis circuit

' " (1 + pTd 0,q0, f 0 )(1 + pTd 0,q0, f 0 )

(4.a)

And the operational function G( p ) is:

G( p) =

Voltage Equations

Vd ( p ) = ra id ( p ) + pϕ d ( p ) − ω rϕ q ( p )

(2.a)

Vq ( p ) = raiq ( p ) + pϕ q ( p ) + ωr ϕ d ( p )

(2.b)

V f ( p ) = r f i f ( p ) + pϕ f ( p )

(2.c)

0 = rDiD ( p ) + pϕ D ( p )

(2.e)

0 = rQ iQ ( p ) + pϕQ ( p )

(2.f)

X md

1

rf

' 1 + pTd 0

(4.b)

d, q, f denote the d-axis, q-axis and field respectively. From these equations it follows that only the three functions X d ( p ) , X ( p ) and G ( p ) are necessary to q identify a synchronous machine. In the original theory, the quadrature axis has no transient quantities. However, the DC measurements at standstill recommended by IEC, and also the short-circuit tests in the quadrature axis show ' that the real machine also has transient values X q ( p ) , ' " " Tq in addition to the sub transient values X q ( p ) , Tq In this study, voltage is the input signal and current is the output signal. Therefore, the transfer functions are in the form of admittances. The reduced operational admittances of d-axis and q-axis are reduced from the input-output signals

While eliminating ϕ f , i f , ϕ D , iD , ϕ , iQ we obtains Q the following equations: (3.a) Vd ( p ) = ra id ( p ) + pϕ d ( p ) − ωrϕ q ( p )

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V.

id , q ( p ) Yd , q ( p ) = Vd ,q ( p )

(4.c)

In this section, the practical aspects of measurements are described and machine conditions for standstill tests are also given. The tests are described in IEC34-4A and ANSI-IEEE Std.115 A publications [12, 1]. It is very easy to perform in practice. PWM voltages are applied to two terminals of the three-phase stator windings at standstill. The alignment of the rotor can be accomplished with shorted excitation winding. A sinewave voltage is applied between two phases of the stator. The duration of the voltage application should be limited to avoid serious overheating of solid parts. The rotor is slowly rotated to find the angular positions corresponding to the maximum value of the excitation current that gives the direct axis and zero value of the excitation winding current that corresponds to the quadrature axis. Figure 4 shows the experimental procedure of identification.

Or in other terms: Yd,q( p) =

' " 2 ' " 1+ p(Td0,q0 +Td0,q0) + p Td0,q0Td0,q0 xd,q xd,q ' " ' " 2 ' " (T +T ) ) + p (raTd0Td0 + ra + p(ra(Td0,q0 +Td0) + ω0 ω0 d,q d,q

The reduced operational admittances take the following forms

2 b0 + b1. p + b2 . p H d ,q ( p ) = 2 3 1 + a1. p + a2 . p + a3. p

(5.a)

The synchronous machine parameters are identified by:

1 b0 = ra b1 =

b2 =

Identification of Synchronous Machine Parameters

(5.b)

Stator winding Field winding

' " Td 0,q 0 + Td 0,q 0

(5.c)

ra

Excitation signals

' " Td 0, q 0Td 0, q 0

(5.d)

d-axis Test

ra xd , q

Input/output board

' " a1 = Td 0,q 0 + Td 0, q 0 + ω0ra

(5.e)

x d, q ' ' " " )+T (T T a = +T 2 ω r d,q d,q d 0, q 0 d 0, q 0 0a

(5.f)

xd , q

' " a3 = T T ω0ra d , q d , q

q-axis test

PC

Field winding

Stator winding

Excitation signals

(5.g) d-axis

Equations (5.a-5.g) show that to determine the various parameters and time-constants of the machine, we have to calculate the constants values a1, a2, a3, b0, b1 and b2 by using the non-linear programming method. For this reason we have used a program, which makes it possible to compute from the input-output signals for each axis (see Figs (3-6)), the parameters quoted above. The model structure selected, which means that the form of the transfer function is known. The numerator order and the denominator one are respectively: Num=2 and Den=3. The objective of our identification task is to compare the simulated model response and the actual response by minimizing the error between them. For minimizing this error, a good optimisation technique is needed. For that we used a quadratic criterion to quantify the difference between the process and the model. Our approach is based on Levenberg-Marquardt algorithm.

Input/output board

PC

Fig. 4. Experimental procedure of identification

The machine is not saturated during standstill tests; in fact, the flux densities are below those on the more linear part of the permeability characteristic that is commonly referred to as “unsaturated”. The determination of quantities referred to the unsaturated state of the machine must be done from tests, which supply voltages (1 to 2%) of the nominal values. A. Experimental Procedure. The two principal characteristic parameters, which relate to the listed definitions, are:

445

0 .2 2Vq(V) 0 Iq(A)

•

Zd(p): the direct-axis operational impedance equal to ra +pld(p), where ra is the DC armature resistance per phase • Zq(p): the quadrature-axis operational impedance equal to ra + plq(p). The above two quantities are the stator driving point impedances. While the above quantities are consistent with the definitions, an alternative method of measuring theses parameters is given as follow:

G( p) =

0 .1 5 1 5

Vq(V), iq(A)

0 .1 1 0

C urrent Voltage

0 .0 5 5 0 0 - 0 .0 5 - 5 - 0 .1 - 1 0 - 0 .1 5 - 1 5 - 0 .2 - 2 0 0

I fd ( p )

for E fd = 0

pI d ( p )

0 .0 0 5

0 .0 1 t(ms)

0 .0 1 5

0 .0 2

c) q-axis stator voltage and current for field winding open

(6)

0 .2 2Vq(V) 0 Iq(A) 0 .1 5 1 5

Short-circuited field winding: with a shorted field winding (V f = 0) , the d- and q- axis operational

Vq(V), iq(A)

0 .1 1 0

admittances are given:

id , q ( p ) Yd , q ( p ) = vd , q ( p )

for

Vf = 0

For

v f ( p)

Vd = 0

(8)

Parameters Ra(p.u) Rf (p.u) T’d (s) T’’d (s) T’d0 (s) T’’d0 (s) T’q (s) T’’q (s) T’q0 (s) T’’q0 (s) Xd (p.u) X’d (p.u) Xq (p.u) X’q (p.u) X’’d (p.u) X’’q (p.u)

Vd(V), Id(A)

C urre nt

00 -0.05 -5 -0.1 -10 -0.15 -15 t(ms) 0.02

a) d-axis stator voltage and current for field winding shorted g

0 .0 2

TABLE I. SYNCHRONOUS MACHINE PARAMETER VALUES IDENTIFIED

0.05 5

0.015

0 .0 1 5

B. Experimental results Table I presents the parameter values of a synchronous machine from tests using in this study.

Voltage

0.01

0 .0 1 t(ms)

Figs. 5. Windings excited by PWM voltages

0.15 15

0.005

0 .0 0 5

d) . q-axis stator voltage and current for field winding shorted

0.2 20Vd(V) Id(A)

-0.2 -20 0

0 0 - 0 .0 5 - 5

- 0 .2 - 2 0 0

Some experimental input-output data, stator voltages and currents, for open and short-circuited field winding, and field winding voltages and currents, are presented in Figures 5. It should be noted that the figures presented correspond to the d-axis; similar curves were measured for the q-axis.

0.1 10

Voltage

- 0 .1 5 - 1 5

can be obtained by:

Y f ( p) =

C urrent

- 0 .1 - 1 0

(7)

Short-circuited d-axis stator winding: with a d-axis armature shorted ( Vd = 0 ), the field winding parameters

i f ( p)

0 .0 5 5

g p

Id(A) 0.2 20Vd(V)

PWM voltages 0.149 4.95 0.2045 0.0471 1.1943 0.5060 0.1782 0.0432 1.0360 0.4348 1.9314 0.3857 1.6328 0.2158 0.0343 0.0209

0.15 15 Vo ltag e

Vd (V) id(A)

0.1 10

The time domain approach offers a method, which can yield useful models, particularly if the data are correctly treated. Taking into account the difficulties associated to the classical test analysis, the identification of synchronous machine is more and more oriented to the static tests. Nevertheless, the choice of supply signals is very important as the choice of model.

Current

0.05 5 0 0 -0.05 -5 -0.1 -10 -0.15 -15 -0.2 -20 0

t(ms) 0.005

0.01

0.015

0.02

b) d-axis stator voltage and current for field winding open

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MODEL VALIDATION

VI.

VII.

The identified d-q axis models are verified by comparing their simulated d-q axis stator currents responses against the measured standstill response, for that we present in Figs. 6 simulation results for chosen signals among those presented previously. The measured and simulated responses to off-line excitation disturbances comparison show that the machine linear parameters are accurately estimated to represent the machine at standstill.

This paper presents a step-by-step procedure to identify the parameter values of the d-q axis synchronous machine models using the standstill time-domain data analysis. A three-phase salient-pole laboratory machine rated 1.5 kVA and 380 V is tested at standstill and its parameters are estimated. Both the transfer function model and the equivalent circuit model parameters are identified using the Levenberg-Marquardt algorithm. The standstill test concept is preferred because there is no interaction between the direct and the quadrature axis, and it can be concluded that the parameter identification for both axis may be carried out separately. This tests method had been used successfully on our synchronous machine at standstill and gave all the Parks model parameters. Among the advantages claimed for the timedomain approach at standstill is that the tests are safe and relatively inexpensive.

0.15 Id me a sure d

0.1

0.05 Id S imula te d

Id(A)

CONCLUSION

0

-0.05

-0.1

-0.15

-0.2

REFERENCES

t(ms) 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

a) d-axis stator voltage and current for field winding shorted

IEEE Guide: Test procedures for synchronous machines, IEEE Std 115-1995 [2] E C Bortoni, J A Jardini: A standstill frequency response method for large salient pole Synchronous machines, IEEE Trans on E.C, Vol 19, N°4, December 2004, pp. 687-691 [3] Guesbaoui H., C. Iung and O Touhami : Towards a real time Parameter identification methodology for electrical machine, IEEE-PES Stockholm Tech. Conf, 1995, pp 91-96. [4] S.Horning, A.Keyhani , I.Kamwa,: On-line Evaluation of round rotor synchronous machine Parameter set estimation from standstill time-domain data, IEEE Trans on E.C, Vol 12, N°4, December 1997, pp. 192-198. [5] M. Hasni, O Touhami, R Ibtiouen, M. Fadel: Modelling and Identification of A Synchronous Machine by Using Singular Perturbations. IREE, Aug 2006, pp 418-425. [6] P. Krause, : Analysis of electric machinery. Mc Graw-hill Book Company, 1986. [7] C. Concordia,: Synchronous machines theory and performances. John Willey and sons, 1951. [8] Nele Dedene, Rik Pintelon and Philipe Lataire: Estimation of a global synchronous machine model using a multiple input multiple output estimators, IEEE Trans. On E.C, vol.18, n°1, Feb.2003, pp11-16. [9] Karnwa I., Viarouge P., J. Dickinson: Direct estimation of the generalized equivalent circuits of synchronous machines from short-circuits osciIlographs, IEE PROC, V. 137, No. 6, NOV 1990, pp. 445-452. [10] Touhami O., Guesbaoui H. and C. Iung: Synchronous machine parameter identification by multitime scale Technique. IEEE Trans. On Ind. Ap. , vol. 30, n°5, 1994, pp 1600- 1608. [11] V.Graza, M.Biriescu, Gh.Liuba, V.Cretu: Experimental determination of synchronous machines reactances from DC decay at standstill: IEEE Instrumentation and measurement technology conference, Budapest, Hungary, May 2001, pp19541957. [12] IEC Recommendations for rotating electrical machinery, Part.4 : methods for determining synchronous machine quantities, IEC, 34-4A. 1985. [1]

0 .1 5 S imula te d curre nt

0 .1

M e asure d curre nt

Id(A)

0 .0 5

0

-0 .0 5

t(ms) -0 .1

0

0 .0 0 5

0 .0 1

0.0 1 5

0.02

b) d-axis stator voltage and current for field winding open 0.12 0.1 Iq me a sure d

0.08 Iq simula te d

0.06

Iq(A)

0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 0

t(ms) 0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

c) q-axis stator voltage and current for field winding open q

0.1 2

g p

0.1 0.0 8 m e a sure d c urre nt

0.0 6

S imula te d c urre nt

Iq(A)

0.0 4 0.0 2 0 -0.0 2 -0.0 4 -0.0 6 -0.0 8 0

t(m s) 0.002

0 .004

0 .006

0 .008

0 .01

0 .012

0 .014

0 .016

0 .018

0.02

d) . q-axis stator voltage and current for field winding open Figs. 6. Comparison between real data output and simulated data output, armature current for field winding open and short circuited (PWM voltages) respectively

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