a “three transfer functionsn approach for the standstill

7/28/2019 A “THREE TRANSFER FUNCTIONSn APPROACH FOR THE STANDSTILL

http://slidepdf.com/reader/full/a-three-transfer-functionsn-approach-for-the-standstill 1/10

740 IEEE TransactionsonEnergyConversion,Vol. 5, No. , December 1990

A “THREE TRANSFER FUNCTIONSn APPROACH FOR THE STANDSTILL

FREQUENCY RESPONSE TEST OF SYNCHRONOUS MACHINES

Y. Jin A. M. El-Serafi, Senior Member

Power Systems Research Group

Department of Electrical Engineering

University of Saskatchewan, Saskatoon, Canada

ABSTRACT

One of the factors which affect the accuracy of th e

determination of synchronous machine parameters from the

Standstill Frequency Response (SSFR) tests is the number

of transfer functions involved in the d-axis model fitting.

This paper proposes to use three transfer functions instead

of the commonly used two transfer functions for the

fitting. Simulation studies and experimental investigations

on a 3KVA microalternator have indicated that this L‘three

transfer functions” approach 111 can enhance the accuracy

of the synchronous machine parameter determination from

the SSFR test.

KEYWORDS

Synchronous machine parameters, standstill frequency

response test, model fitting using two and three transfer

functions approaches.

Introduction

In recent years, the Standstill Frequency Response

(SSFR) test [2-5] has been accepted as a powerful tool for

the determination of synchronous machine parameters,

particularly those of the improved synchronous machine

models [2] in which unequal mutual inductances and more

damping effect are included. However, the existing

techniques of the SSFR tes t still have a number of

shortcomings IS] which affect the accuracy of t he tes t

measurement and the derived machine parameters. In this

respect, the number of transfer functions involved in the d-

axis model fitting has a nonnegligible effect. In the initial

stage of the development of the SSFR test, only one d-axis

transfer function ( L d ( s ) ) was measured and used in the

fitting. Later, the field-tearmature transfer function (G(u))

was introduced into the fitting. The use of the twofunctions ( L d ( u ) and G(s) ) has really improved the rotor

representation of the derived models. so it has been

recommended in the IEEE standard (IEEE Std 115A) [2].

However, the adequacy of this “two-function’’ approach is

still in question 151.

This paper suggests to use three transfer functions in

the d-axis model determina tion instead of the two transfer

functions. It is expected that such a “three transfer

functions” approach would provide better results than the

‘‘two transfer functions” approach. In this paper, the

89 SM 755-0 EC

by the IEEE Rotating Machinery Committee of the IEEEPower Engineering S ociet y for pres entat ion at the IEEEj

PES 1989 Summer Meeting, Long Beach, California,

July 9 - 14, 1989. Manuscript submi tted January 25 , 1989;

made ava ila ble f or pr inti ng June 7 , 1989.

A paper recommended and approved

network theory is employed for the explanation of thethree-function approach. The verification of the approach is

obtained through simulations and an application to an

SSFR test on a 3 KVA laboratory microalternator.

D-Axis Synchronous Machine Models

The d-axis synchronous machine model can be

expressed by two operational equations 171:

u d ( s )= Zd ( s ) i d ( s )+woLq(s ) iq ( s ) sG ( s )u f ( s ) 0)

i f ( s )=s G ( 8 ) i d ( s )+L~8 )Zf0(4

where ‘ s is the Laplace operator, Zd(s) the d-axis

operational impedance, G ( s ) the field-tearmature transfer

function, Z f 0 ( s ) the field operational impedance, and L q( u)

the q-axis operational inductance. The quantities u d ( u ) ,

u f ( s ) , i d (8 ) , i q ( s ) and i f ( s ) are the voltages and currentsof the d-axis equivalent circuit.

Generally, Zd ( s ) ,G ( s ) and Zfo(s)f Eqs. 1 and 2 are

called transfer functions. They are all functions of complex

frequency s, resistances R’ s and inductances L’s. The

expressions of these functions are model dependent. Ifthese functions are measured through frequency response

tests, all of the synchronous machine parameters (R’s and

L’s) can be determined.

At standstill ( w o = O ) , Eq. 1 becomes:

u ~ ( s )=-Zd(s ) id(s )+s G ( u ) u ~ ( s ) (3 )

Equations 2 and 3 can be written in matrix notation as:

(5 )

which includes only three independent matrix elements:

Z d ( s ) , G(8) and l / Z f o ( s ) .

It is obvious that t he synchronous machine d-axis

operational equation (Eq. 4) is a typical two-port network

equation. According to the network theory 181, a passive

two-port network can be uniquely characterized by and

only by three independent functions in the characteristic

matrix. This means that three functions are essential to

characterize the d-axis two-port network of synchronous

machines. Th is situation is similar to th e case of thetransmission line network with a characteristic matrix

1 il, n which three of the four elements are independent.

All are functions of frequency.

0885-8%9/90/1200-0740$01.00 0 1990 IEEE



741

Because of the independence of the three functions, th e

third one can not be derived from or expressed by the

other two. Thus, it is doubted that the model fitting using

only two functions could correctly determine all machine

parameters. In other words, the machine parameters

derived from two functions could not adequately represent

the property of the third independent function and, thus,

could not adequately represent the whole properties of the

character istic matrix of the network. Therefore, it is

logical to consider the use of three transfer functions in

the d-axis machine model determination.

All of the three transfer functions in the matr ix of Eq.

4 can be defined and determined by the external

measurement of the network. The operational impedance

z d ( 8 ) is defined [2] by

u d ( 8 )

a d ( 8 ) fd ( 8 ) =-7-‘-v =O (6 )

which is the rat io of the Laplace transform of the d-axis

voltage to the Laplace transform of the d-axis current

when all rotor windings are short-circuited. The other

transfer functions can be similarly defined.

i f ( 4

8G ( 8 ) = - l u --o (7)

Uf ( 4

Z f o M =- i =o (8)

add(8) f

I f ( # ) d

To determine the parameters of a tested synchronous

machine, these transfer functions are measured by an

SSFR test and used in the model fitting (or often called

“curve fitting”). In the standard procedure of the SSFR

test [Z], two” functions ( Z d ( s ) and s G ( s ) ) are measured

and involved in the model fitting. In practice, the

measurements of Z d ( s ) are used to obtain the values of the

operational inductance L d ( s ) , and then, L d ( 8 ) is used in

the fitting. In contrast t o this approach, a &‘three ransfer

functions” approach is suggested in this paper.

The Third Function

In considering the involvement of a third function in

the model fitting, a straightforward choice is to use the

field operational impedance Z f o ( s ) , which is identified byEq. 8. However, another function, the armature-to-field

transfer inductance L a f o ( s ) , s selected in this paper as the

third function for the d-axis model fitting. This function is

related to the measurable armature-to-field transfer

impedance Z a f o ( s ) . The measurements of Z a f o ( 8 ) can be

obtained easily from an SSFR test with the field winding

open-circuited, which requires a few changes from the set-

up of the SSFR test with the field winding short-circuited.

The armature-to-field impedance Za,,,(8) has been

defined in the IEEE Std 115A [2] as

V f ( 4

zof 0 8 ) = - - l i d ( 8 ) f 0 (9)

The data of L a f o ( s ) can be easily derived from the

measurements of Z , o ( s ) by the equation:

The selection of L p f o ( 8 ) as the third function can be

validated, since there is an analytic relationship between

L a f o ( 8 ) and l /Zfo(s). As shown in Appendix A, the

function 1/Zfo(8) can be replaced by -G ( u ) / L a f o ( 8 ) .

Thus, Eq. 4 can be modified into a new form as

which contains again three transfer functions z d ( 8 ) , G ( 8 )

and L a f o ( u ) in its characteristic matrix. It is obvious that

the machine model can be fully and uniquely determined ifthese three transfer functions are measured and used for

the model fitting.

;==+rad

*d

Lk d ’I

II I

I I0

I I

I I0

Figure 1: Equivalent Circuit of Model-D7

Physical explanation of the function Z

The function Za fo (u) for any of the d-axis equivalent

circuits is the armature-to-field transfer impedance when

the field winding is open-circuited. For example, the

function Z O f o ( 8 ) for Model-D7 (a d-axis model with 7

parameters) of Fig. 1 is given by

where La d is the d-axis magnetizing inductance, Rk d and

Lk d are the resistance and inductance of the d-axis

equivalent damper circuit, and Lk f is the rotor local mutual

inductance*. All of the quantities are in per-unit. The

function Zafo(s) is exactly the transfer impedance of thenetwork included in the dashed box of Fig. 1. It is obvious

that the function Z a f o ( s ) is only related to the remaining

part of the d-axis equivalent circuit when excluding the

armature branch (L , ) and the field branch ( R f and L f ) . In

other words, the function zafo(8)epends only on the

mutual inductance Lad, the damper branch parameters Rkd

and Lkd, and the rotor local mutual inductance L k f This

implies that the inclusion of Z a f o ( s ) in the d-axis model

fitting would help the identification of th e parameters

within the dashed box of Fig. 1, particularly the damper

branch parameters . Therefore, utilizing the third function

L,+,(s) together with L d ( 8 ) and G(B ) in the d-axis model

fitting would improve the overall solution of the fit ting.

In the classical synchronous machine theory, the rotor

local mutual inductance L k f is omitted in the d-axis

*I n this paper, the parameter Lk, is simply referred to as a

“rotor local mutual inductance” since it is related only with thelocal flux linkage between rotor circuits (the damper and thefield). This name describes the physical situation more clearly.



142

model. Equation 12 can then be rewritten into the

following form:

which is identical to th e equivalent impedance of the two

is deleted.

The above discussion can also be extended to the

models with more parameters, e.g. 10-parameter model.The transfer inductance L. f0 (8 ) would be still related to

the remaining part of circuits when excluding th e armat ure

and field branches, i.e. related mainly to the damper

circuits. The expression of L a 0(.9) becomes, however, more

complicated than Eq. 12.

Simulations

parallel branches Lad and Zk d =Rk d +8 Lk d Of Fig. 1 if Lkf

To examine the effect of the number of transfer

functions used in the d-axis model fitting and to verify the

three-function approach, a number of simulations have been

carried out. In the following discussion, only the

simulations for the d-axis 7-parameter model (Model-D7 of

Fig. 1) are presented. The three-function and two-function

approaches, as well as the one-function approach, are

compared.

1. Simulation data generation

The simulated model parameters, which are obtained

by an SSFR test conducted by Ontario Hydro [4] on a

Monticello turbogenerator, are listed in Table 1. In the

simulation, these parameters are used to compute the data

of the frequency response characteristics of the three

transfer functions L d ( 8 ) , G ( 8 ) and L a f o ( 8 ) . The

expressions of these three transfer functions for Model-D7

Table 1: Parameters of the Monticello Maehine

I Parameters I Values (pu) IR f

Ll

Lf

Rkd

La d

Lkd

‘k f

0.000811

0.00727

0.20901.6910

0.0171

0.0051

0.1258

are given in Appendix B. Each function has 61 data points

with frequencies from 1 mHz to 1 KHz. Random noises

are added to these produced simulation data in a way that

is expressed by the following equations.

IT(8i) l* =IY8i)I*(1+ li ) (14)

(15)

where IT(a ) I * and lT(8)I are the magnitudes of the transfer

functions with and without noise respectively; B* and B are

the phase angles with and without noise respectively; nl

and n Z are random noises produced by a GAUSS

sub rou tine of the Scientific Subroutine Package (SSP) (91.The noise ral added to L d ( 8 ) , G ( 8 ) and L a f o ( 8 ) has a

standard deviation of 0.015, 0.025 and 0.015 respectively.

In fact, noise nl appears as a relative error. The random

noise n 2 has a standard deviation of 0.007, which is

equivalen t to a deviation of 0.4 degree. Such added noises

make the simulation closer to a real test. These generated

data are treated as “measurementsn in the simulations. Inthe following discussion, the noise with the above

deviations is called “Noise-1” .

2. Simulation results

The model fitting program with the Marquardt

algorithm [l,lO] has been applied to identify theparameters of Model-D7 by fitting to the simulation data

(as “measurements” with Noiee-l) of three cases: (1) one

set of data, L d ( 8 ) only; (2) two sets of da ta, Ld (8 ) and

G(8) ; and (3) three sets of dat a, L d ( 8 ) , G(8) and L a f 0 ( 8 ) .

The initial settings of the parameters are calculated [ l , l O ]

by using inputted values of L,/L,d from 0.06 to 0.15. This

program applies an iteration process to yield the find

estimates. The estimated parameters are listed in Tables 2,

3 and 4, corresponding to these three cases of fitting

approaches.

In order to obtain an intuitive idea of the model

fitting, the simulated data with Noise1 (as

“measurements”) are drawn in Figs. 2 and 3, with symbols

“+” for their magnitudes and “ A ” for their phase angles.

The estimated parameters listed in Tables 2 to 4 are used

to produce corresponding frequency response characteristic

curves, that are drawn in these figures as well.

Based on these results, the following analyses and

conclusions can be drawn:

(1) One-functio n app roa ch: The fitting in this

case, where only Ld (8 ) is involved, can not determine the

model parameters correctly. As shown in Table 2,

different initial settings of &,/Lad result in different sets of

parameters, neither of which is close to the Monticello

model parameters of Table 1. Even using the exact

Monticello parameters as the initial setting (Setting (A))for the fitting program still can not result in an accurate

estimated model. This indicates obviously th at one-

function approach is inadequate in the model determination

of synchronous machines.

All of the models determined by this approach do not

represent the frequency response characteristics of the two-

port d-axis network of the Monticello generator. Figure 2

displays the frequency response characteristics of the

estimated model listed in the second column of Tabh 2.

This model is obtained from initial setting (A). Although

Table 2: Model Parameters by Fitting Model-D7 to

the Simulated Data of Ld (8 ) With Noise-l

(Initial Settings: (A) exact parameters of

Table 1; (B) Ll/Lad=o.07-o.15, L k F 0 . 1 2 )

R f

Ll

-L f

LY

Rk d

La d

Lkd

0.0008161

0.0067738

0.208 1299

1.7048966

0.0216999

0.0192925L.1243133

Setting (B) I0.0013177

0.0017832

0.1740746

1,7389518

0.0183916

-0.0016422

0.1764928

Ll/L,d =0.15

0.0011103

0.0019198

0.2375254

1.6755010

0.0167364

1.1014148

-0.0003679



143

Estimated Values (pu)

Parameters L , / L , d =0.07 &, / t o ,0.15'

setting (A) is the "best" setting in this case of simulation

and its estimated model is expected to be the "best" one

among the models determined by the one-function

approach, this model does not fit well to the data of d ( 8 )

and Zafo(8) , as shown in Fig. 2. Moreover, it has been

found that the models obtained from initial setting (B) can

produce much poorer fitting to the data of 8G ( 8 ) and

ZafO(8) than that of the curves in Fig. 2, though all of

them can fit well to the data of L d ( 8 ) .

Errors

(%)

(2) Tw d" t io n approach: In this case, where

G(8) as well as L d ( 8 ) are involved, the fitting

implementation is greatly improved. Table 3 demonstrates

Estimated Values (pu)

Parameters I L1/Lad=0.07 L , / L ~ ~0.15 '

0

--IO ;

. - 2 0 =YA

--30 4

w)

I0.6- YuLWREIIENlS - - 4 0

"""'4d' """'4d -404 401 ' lb.2 -

FREQUENCY lHZ1

Errors

(%)

I . " , . .Y

3 Wc

W Yv)

0 - 2

S 0 .6 - - - s o

I '

0 . 2 -- . -70 -. + UWME

A R U S E W s L E-0.2- , ,,,,, , ,,,, -9 0

io-' '"""48"" ' r id' ""'"98 '?FREQUENCY (HZ)

Figure 2: "Measurements" with Noise-l and Frequency

Response Characteristics of the Estimated

Model by Fitting Model-D7 to the Dat a of

L d ( 8 ) for the Case of Initial Setting (A )

R f

LIRk d

La d

L fLkd

Lk f

0.0007845

0.0053785

0.2607326

1.6515949

0.0130064

0.0038237

0.0715908

0.0007845

0.0053785

0.2607328

1.6515946

0.0130064

0.0038237

0.0715907

-3.268

-26.017

24.752

-2.330

-23.939

-25.025

-43.091

R f

LI

Rkd

La d

L fLk d

'k f

0.0008069

0.0072718

0.2109984

1.6981270

0.0169154

0.0051504

0.1246777 I .0008069

0.0072718

0.2 109986

1.6981270

0.0169154

0.0051504

0.1246775

-0.506

0.025

0.956

0.421

-1.080

0.988

-0.893

(3) Three-function app roa ch: Involving the three

transfer functions, L d ( 8 ) , G ( 8 ) and L a f 0 ( 8 ) ,can yield best

results among these three approaches of fitting. This is

clearly demonstrated by Table 4, where all the parameters

have much less errors than those of Table 3. The model

obtained from this three-function approach can duplicate

precisely the original Monticello model. Besides, the modeldetermination is also independent of the initial settings.

As expected, the model derived by the three-function

approach has frequency response characteristics which can

fit very well to the "measurements". The solid l ies in

Fig. 3 correspond to this model. Thie verifmi the

theoretical analysis given previously that three t r a d e r



744

functions are necessary in determining the characteristic

parameters of the d-axis two-port network of synchronous

machines. The involvement of the third function, L a f o ( # ) ,

in turn, helps to identify all other parameters of the d-axis

equivalent circuit accurately.

3. Effect of the accuracy of “measurements”

In order to explore the effect of the accuracy of the

“measurements”, random noise with larger deviations are

Can help to identify L a , R k d , Lk d and L kj accurately. This,

OPERATIONAL INDUCTANCE L d (9 )

1 ISIWULATION OATA FROM WONTICELLO MACHINE1 f

(4TRANSFER FUNGTION sG (SI

FREQUENCY (HZ)

(4Figure 3: “Measurements” with Noise l and Frequency


Models by Fitting Model-D7 to the Dataof L d ( 8 ) and G(s) and by Fitting to the

(solid lines: three-function approach;

dashed lines: two-function approach)

(Note: The dashed lines merge into the solid lines

if there are no dashed lines shown in the plots.)

Data of L d ( d ) , G(8) and L a f 0 ( 8 )

adde d to the simulated frequency response dat a of the

Monticello model of Table 1. It is obvious that the larger

the deviations of the added noise, the less the accuracy of

the generated “measurement” data. As a result, it is

expected that the models determined by any of the fitting

approaches would be less accura te. In the following

discussion, only one case of noise is reported. In this case,

the noise nl (Eq. 14) added to L d ( 8 ) , G(8) and L q f 0 ( 8 )

has a standard deviation 0.03, 0.05 and 0.03 respectively.

These are twice the deviations of the noise in the

simulations previously discussed. The noise 9 (Eq. 15)

has the same standard deviation (0.007) s in the previous

simulations. This means that the magnitudes of the noise

in the data of L d ( 8 ) , G(8) and L a f 0 ( 8 ) are doubled, while

the angles of the data remain unchanged. This higher noise

is simply referred to as “Noise-2” in this paper.

Table 5: Model Parameters by Fitting Model-D7

to the Simulated Data of L d ( 8 ) and G ( 8 )

with Noise-2 (Two-Function Approach)

Estimated

Rf

Li

Lf

Lkf

Rk d

La d

Lk d

0.0007427

0.0034914

0.3492703

1.5739334

0.0089515

0.0024853

-0.0168483

0.0007427

0.0034914

0.3492702

1.5739335

0.0089515

0.0024853

-0.0168482

Errors

(% I

-8.422

-51.975

67.115

-6.923

-47.652

-51.269

-1 13.393

The models derived by the two-function approach from

the data with Noi-2 are listed in Table 5. Comparing

this table with Table 3, it can be seen that due to the

higher noise all the parameters of the models are identified

poorly. The errors in this case are twice the errors in the

case of Table 3. All rotor inductances and resistances

become smaller. The rotor local inductance L k f becomes

even negative. This shows that the two-function approach

is very sensitive to the level of the noise added to the

magnitude data. In other words, this approach is very

sensitive to the accuracy of the “measurements”.

On the other hand, the results of the three-function

approach have indicated, as shown in Table 6, that thia

approach is practically insensitive to the level of the noise

added to the magnitude data. Although the added noise is

higher, the estimated models in Table 6 are still very

accurate, and the errors between the estimated models and

the original Monticello model (Table 1) are still

considerably small.

Table 6: Model Parameters by F itting Model-D7 to

Simulated Data of L d ( 8 ) , G(8) and L a f o ( s )

with Noise-2 (Three-Function Approach)

Parameters

R f

Rk d

Ll

La d

Lf

Lk d

LY

Estimated

L l / L a d=0.07

0.0008052

0.00727760.2103343

1.7048321

0.0167640

0.0051946

0.1262187

0.0008052

0.00727760.2103343

1.7048322

0.0167640

0.0051946

0.1262186

Errors

(% I

-0.715

0.1050.639

0.818

-1.965

1.855

0.333



745

As shown in Fig. 4, the fitness of the frequency

response characteristics of the estimated models from the

two approaches confirms the advantage of the three-

function approach. By comparing Fig. 4 with Fig. 3, it can

be seen that the models derived by the two-function

approach in the case of Noise2 have more deviations on

their curves of Zaf0(8) (dashed lines) than those in the

OPERATIONAL INDUCTANCE Ld ( 5 )1 I S I M I UT I O H M TA FRon Y W I I CE UO M A CHINE1 r

OJ

- 8. -7 0Iw -

-16. -40-03 --

-24- -1 0U

Yv)

-32- . - 2 0YEASuRD(ENT3

, t LU8NfTUY

1w

WY

a

-W-1

U

A W M L E-40

, 1 , , , , ,

o*40* ' """40" ""'46 '"""4d' ' '""'4 d '

FREQUENCY (HZ)

(b)TRANSFER INDUCTANCE L a f o (SI

1

-' -10 3

0. - 3 0 =aW- . Ir*

WW 20 - WI-

-

.-so

wIW2 0.6-

4 . U

.-70 -.2 ' m-TS

t YABNITWE

A PHASE AN6LE- 9 04c' '"""4f """4 d' '"""4f " " T0 . 2 - ,,,,,, * ,,,,,,,

FREQUENCY IHz)

(4

Figure 4: "Measurements" with Noise2 and Frequency


Models by Fitting Model-D7 to the Data

of L d ( 8 ) and G ( 8 ) and by Fitting to theData of L d ( 8 ) , G(8) and L a f o ( 8 )

(solid lines: Three-function approach;

dashed lines: Two-function approach)



case of Noise-l, while the models derived by the three-

function approach still keep a good curve fitness. This

simulation study increases confidence in the use of the

three-function approach for the determination of the d-axis

parameters of synchronous machines.

An SSFR Test on a Microalternator

An SSFR test on a 3 KVA microalternator has been

carried out. It provides a further confirmation for the

three-function approach proposed in this paper.

The nominal ratings of the microalternator are:

3 KVA, 220/127 V, 7.9 A, 60 Hz, 1800 rpm, and 0.8 pf.

The stator has a tphase, 4-wire, star-connected winding,

while the salient-pole rotor has a field winding on the d-

axis and two wound damper windings, one on each axis.

The used rotor is completely laminated.

Table 7: Microalternator Parameters of Model-D7

Obtained from an SSFR Test

0.0043110.048480

0.106265

0.889739

0.321319

0.389799

-0.092149

0.0045450.062007

0.074126

0.926512

0.354234

0.468788

-0.080757

The test on the microaltemator was carried out for

both th e d- and q-axis. In the case of the d-axis test, the

field winding was short-circuited an d open-circuited for

measuring the different transfer functions. Sinusoidal

signals of 61 discrete frequencies were applied. The

recorded signals were processed by a VA X 11/780

computer to obtain the estimates of the transfer functions.

These estimated data were transformed into their

appr opri ate per unit forms. Finally, the model fittingprograms were applied to determine the microalternator's

parameters.

The determined microalternator parameters of Model-

D7 are listed in Table 7, where Model (A ) is obtained by

fitting to three functions ( L d ( 8 ) , G ( 8 ) and L a f y ( 8 ) ) , while

Model (B) is obtained by fitting to two functions ( L d ( 8 )

and G(8) ) . It is clear that the two sets of parameters of

these two models are different from each other, particularly

the armature leakage inductance L, which is unreasonably

small in Model (B). However, it is difficult to judge these

two models directly from the parameter values themselves.

A way to evaluate these two models is to produce

their frequency response characteristics and compare with

the test measurements. To do this, the frequency response

characteristics of these two models are computed and

drawn together with the measurements in Fig. 5. Themeasurements are drawn with the symbols "+" and "A"for the magnitude and phase angle respectively. The solid

lines correspond to Model (A), while the dashed lines to

Model (B). These plots demonstrate clearly that both

Models (A) and (B) can fit well to the measurements of

the functions L d ( 8 ) and G ( 8 ) , but Model (B) can not fit to



746

the measurements of the function Lafo(8) as well as Model

(A) does. Thus, Model (A) is considered to be more

accurate than Model (B) in producing the frequency

characteristics of the microalternator. This confirms further

that three transfer functions should be involved in the d-

axis model fitting process for obtaining an accurate

parameter determination.

OPERATIONAL INDUCTANCE Ld (s )I YICRO-4LTERIAT(R DATA, LE3 METHOD11

1.1

0.7 c -20 w!O.Pv. 5 -3 0tIt MADNITWE

A PUhSE L W L E

( 4TRANSFER FUNCTION SG ( 9 )

t MADllITWf

Yo-b arm-’ ‘“-‘ rYu- ‘””‘ Y55 -.

FREQUENCY (Hzl

(b)

TRANSFER INDUCTANCE Lafo (9 )

1 I MICRO-4LTERN4TOR D4 TL LES nut” r

h2 2 4 1 2

[ =‘ h21 h , , ][z]o f 31 MEASUREMENTS Tseo*/%;

16 M I I N I T W E L- 5 0

A PW3E AN=€

0-b ’ ”””$o+ ””’\0 . 1 6‘ I ““‘4ct 1 ““‘k d’ ‘“““48 ‘“, d

FREQUENCY IHzI

(4Figure 5: D-axis Measurements and Frequency

Characteristics of Model-D7 of Microalternator

(Solid lines -- Model (A), Three-function approach;

Dashed lines -- Model (B), Two-function approach)



Conclusion

This paper has proposed a “three transfer functions”

appro ach for the SSFR test of synchronous machines. The

simulations and the test on a microalternator have verified

that the accuracy of the determined d-axis model

parameters, particularly of the rotor circuits, can be

improved by the use of the suggested third transfer

function (Lafo(8)) together with the two transfer functions( L d ( 8 ) and G(8)). This requires that the measurements of

the three transfer functions be taken in the SSFR test and

be involved in the d-axis model fit ting - instead of the

common practice that only two transfer functions (Ld ( 8 )

an d G(8)) are involved.

APPENDIX A

Relationship of the Function

l /Z fo ( s ) with L,(s) and G ( s )

In order to determine the machine’s characteristic

matrix of Eq. 4, three transfer functions (Ld(8), 8G(8) and

l/Zfo(8)) are needed. Among them, the function l/Zfo(b)

is not usually measured in the SSFR test. However, this

function can be replaced by the measured functions Lafo(8)

an d G(a). The relationship of the function l/Zfo(8) with

LOf0(8) and G(a) can be derived as follows.

Equation 4 has the same form as one of the hybrid

parameter equations used in the general reciprocal two-port

network analysis (81. This is given by

[>]=[::: 1 ][ (16)

where

E1

h l l = F I E 2 = 0 = -zd(8)1

12 E1

h1 2 = h2 1 = T 1 E 2 = 0 = =o = 8G(8)2 1

1

--

2

h22 =el1 =o - z (*)2 1 fo

The inverse form of Eq. 16 is given by

D = 1 hl l h12 1 = h,, h 2 2 - h I 2h 2 ,

I h21 h22 IBy using the definition of ZaIo(8) given by Eq. (Q),

Lafo(8)he relationship between the functions l/Zfo(8),

and G(8) can be found:



747

G(8)

1 /Z j0 (S )= --L o o b )

APPENDIX B

Expressions of L,(s), G(s)

and L,(s) of Model-D7

The transfer functions L d ( s ) , G(s) and Z j o ( s ) of

Model-D7 shown in Fig. 1 are expressed as follows:

1+Cs +D s 2

d ( = L d 1 + A a + B a 2L e

Lk d

R j Rk d

1 + A +B s 2

- ( l + s - )

G(8) =

1 1 Lk d L j

kd j Rk d R jwhere A = ( L a d + L k )(R+R)+- -

1 +R t d

REFERENCES

1. Y. Jin, A Study of the Standstill Frequency ResponseTest for Synchronous Machines, M.Sc. Thesis,

University of Saskatchewan, 1988.

2. IEEE Std 115A-1987, Standard Procedures forObtaining Synchronous Machine Parameters byStandstill Frequency Reuponse Testing (Supplement to

3. IEEE Jo int Working Group on Determination ofSynchronous Machine Stability Constants,“Supplementary Definition and Associated TestMethods for Obtaining Parameters for SynchronousMachine Stability Study Simulations” IEEE Trans.on Power Apparatus and Systems, Vol. PAS-99, 1980,

4. Ontar io Hydro, Determination of SynchronousMachine Stability Study Constants, EPRI ReportEL-1424, Vol. 2, Dec. 1980.

5. S. D. Umans, J. A. Mallick, and G. L. Wilson,

“Modeling of Solid Rotor Turbogenerators, Pa rt 1:

Theory and Techniques”, IEEE Trans. on Power

ANSI/IEEE Std 115-1085), 1987.

pp. 1625-30.

Apparatus and Systems, Vol. PAS-97, 1978,

pp. 269-77.

6. A. M. El-Serdi and Y. Jin, “Sources of Errors in theStandstill Frequency Response Testing of Synchronous

Machines”, Proceeding of International Conference onElectrical Machines (ICEM’88), vo1.3, Pisa, Italy,

September 12-14, 1988, pp.417-22.

7. B. Adkins and R. Harley, The General Theory ofAlternating Current Machines, Application to PracticalProblems, Chapman & Hall, 1975.

8. J. B. Murdoch, Network Theory, McGraw-Hill BookCo., 1970.

9. DEC, Scientific Subroutines Programmer’s Reference. Manual, AA-1101C-TC, Digital Equipment Co., June,1980.

10. Y. Jin, and A. M. El-Serafi. ”Application ofMarquardt Algorithm to the Determination ofSynchronous Machine Parameters from theirFrequency Response Data”, Paper No. B12,Proceedings of Beijing International Conference on

Electrical Machines, (BI CEM W) , Beijing, August

10-14, 1987, pp.218-21.

Yusun Jin was born in Shanghai,

China in February 1945. He

graduated from the Department of

Electrical Engineering, Qinghua

University, Beijing, China in 1968,

and received the M.Sc. degree fromthe University of Saskatchewan,

Canada in 1988.

From 1968 to 1978, he worked as an

engineer in the First Refinery, China.

From 1978 to 1983, he WBS an instructor in the North-

China College of Water Conservancy and Hydro-power.

From 1983 to 1985, he joined the Power Systems Research

Group of the University of Saskatchewan as a visiting

scholar. His research interest is signal processing and

application of digital techniques in power systems.

Ahmed M. El-Serafi (AM’54-

M’56-SM’70) was born in Cairo,

Egypt in March 1929. He received

the B.Sc. in Electrical Engineering

from Cairo University, Egypt in

1950, the Ph.D degree from theManchester College of Science and

Technology, England in 1955 and

the Dr.-Ing. from the Technical

University (T.H.) Darmstadt,

W. Germany in 1964.

He was with Cairo University from 1950 to 1953 and

from 1957 to 1961, with the Manchester College of Science

and Technology, England from 1953 to 1957, with the

Technical University (T.H.) Darmstadt, W. Germany from

1961 to 1965 and with the University of Libya from 1965

to 1968. From 1957 to 1961, he was also a Consultant for

the Egyptian Commission of Electricity. In 1968, he joined

the University of Saskatchewan in Saskatoon, Canada

where he is presently Professor of Electrical Engineering

and member of the Power Systems Research Group. From

1986 to 1988, he was the Director of Graduate Studies and

the Chairman of the Research Committee of theDepartment of Electrical Engineering.

Dr. El-Serafi is a member of the Canadian Electrical

Association, the Engineering Inst itute of Canada, the

Insti tutio n of Electrical Engineers in England, the VDE in

W. Germany and of CIGRE; and is a Registered

Professional Engineer in the Province of Saskatchewan.



748

DISCUSSION

ROBERT M. SAUNDERS, (Unive rs ity of Ca li fo rn ia ,Irv ine ): The frequency response t e s t has been used i nmany insta nces t o obtai n a linear model which w i l l

yield useful re su lt s over a limited range. Thistechniq ue of modeling has been employed i n such div er seareas as th e dynamics of a ir cr af t and chemicalprocesses a l l of which ar e als o inheren tly nonliear.I t i s always helpful i f one can pinpoint t he physical

reasons for the re sult ing poles, zeros, and gainfacto rs. However, i n the case of a synchronousmachine, (and the proc ess es mentioned above ), we have aprocess which is inherently nonlinear without anymanufacturing or operating tolerances or defects sucha s el li pt ic al roto rs, out-of-alignment sha fts, o r , i nth e case of salie nt- pol e machines, uneven airg apsaround the circumference taken into consideration.These mitigate against finding tran sfe r functions tha tw i l l reconci le clos ely with theory. Thus it is best tose t the theory into the background and “take w h a t you

get” and f i t a l inear model to that set of tes t data.

The authors’ rea ctio n t o the se comments might be usef ulfo r tho se who must model synchronous machines fo r usei n system analysis.

Manuscript received J u l y 31, 1989.

A. Keyhani (The Ohio Stat e University, Columbus, Ohio): Th e

authors are to be commended for the s tudy of noise effects

and the use of three-transfer functions in estimating machine

paramete rs. At the Ohio State University, we have studied

t,he same problem, and our result,s were reported in references

(1,2]. Some of ou r findings report,ed in reference [ l ] , re contrary

to the results reported in this paper. I would appreciate the

authors’ comments on the following points:

1. The d-axis operational impedance &(s) is highly noisy

at low frequencies, and the ar matu re resistance cannot

be accurately estimated from Zd ( s ) .

2. The d-axis operational inductance is calculated from

&(s) using an estimated R,. Namely, & ( S ) = (& (s ) -R,)/s. he results reported in [ l] ndicate that even a

half percent change in the value of R, results in wide

variations of L d ( 8 ) . The estimated d -ax i s parameter s will

vary widely depending on the value of R, used in the

study [l].

3. The equations which relate the d-axis parameters to the

time constants of Ld(s) and s G ( s ) can be obtained from

Eq. 11of the paper. These relationships are complex and

nonlinear (see Ref. (11)and can be written as

f i ( E ) =gi+g;(z,g) +ti =0

where i = 1 ...,8. The g is a known vector, and it is

given in terms of the estimated time constants and the

gain of & ( 8 ) and s G (s ) . TheZ

s an unknown vectorwhich represents the seven parameters of the d-axis cir-

cuit model (see Fig. 1 of the paper). These equations

are nonlinear in nature and are not consistent with each

other. This is due to the noise imbedded in vector 3.Naturally, these equations would be consistent if simu-

lated noise-free data are used in the analysis. A unique

solut ion will be obtained regardless of the equation which

is discarded in the solution process. However, when the

measured dat,a are used, the rquations would be inconsis-

tent because of the inherent noise in the d ata, and multi-

ple solutions are obtained depending on which equation is

ignored in the solution process [l].Adding a third func-

tion, will not solve the problem of inherent noise in the

da ta and will increase the number of nonlinear equations

wit.h only seven unknown parameters.

The authors generated their noise-corrupted d ata by adding

noise to the magnitude and phase of operational inductances

(see Eq. 14 and 15 of the paper). The operational inductances

cannot be measured directly. For study purposes, the noise

should be added to the operational d-axis impedance Zd(s).

If the authors add their noise to the Z d ( 3 ) rather than L d ( $ ) ,

then they will obtain a set of overdet,ermined nonlinear and

inconsistent equations (eight equations and seven parameters)

which will result in multiple solutions for the parameters. The

authors ’ comments concerning t,he above points will be appre-

ciated.

[ l] A. Keyhani, S. Hao, G. Dayal, “The Effects of Noise on

Frequency-Domain Parameter Estimation of SynchronousMachine Models,” IEEE Paper 89WM228-SEC. Presented

at the IEEE/PES 1989 Winter Meeting, New York, NY.

[2] A. Keyhani, S.Hao, G. Dayal, “Maximum Likelihood Es-

timation of Solid-Rotor Synchronous Machine Parameters

from SSFR Test Data, IEEE paper 89WM224-7EC, pre-

sented at the IEEE/PES 1989 Winter Meeting, New York,

NY.

Manuscript received July 2 4 , 1989.

Y. J I N AND A.M. EL-SERAFI: The au th ors would l i k e t othank the discussers for the interest they have showni n this paper and for the cormnents and the relevant

questions they have asked.

Concerning the points raised by Prof. A.Keyhani, i t would be approp riate here t o emphasizethat the main objective of our paper is t odemonstrate that three transfer functions, instead ofthe comonly used t w o functions, are needed i n thed-axis model fi tt in g of the SSFR test to representcompletely and accurately the nature of the d-axistwo-port network. With t h i s approach, a se t ofmachine parameters i n terms of res istances andinductances, including the armature leakageinductance, can be uniquely determined. In fa ct , theiden tifi cati on of these parameters i n the SSE’R t e s tdepends on many fact ors , e.g. th e order of t he modelused , the choice of the number of the transferfunctions used in the i dent ifica tion and the accuracyof t he measured frequency response dat a.

In our paper and in reference 1 of Prof.

Keyhani’s discussion, i t has been clearly shown thatthe iden tifi cati on of the parameters of the d-axismodel of synchronous machines by using two transferfunctions i s strongly affected by t he no i se in t hefrequency response dat a. Having more noi se in the seda ta re su l t s i n that the used f it t i n g program w i l lconverge a t another se t of parameters. All the setsof parameters derived from data with different noise



749

levels can fit well to the frequency response data ofthe two transfer functions used in the fitting andcould be accepted as a solution from this point ofview. However, among these sets, only one set ofparameters can fit also to the data of the thirdfunction. This set of parameters is truly the mostaccurate one. In order to determine this set, the"Three Transfer Functions" approach is proposed inour paper. By applying this three-function approach,the identification becomes less sensitive to themeasured noise level in the simulated data or to theaccuracy of the measured data.

It is well-known that the equations involved ina model fitting or curve fitting are always redundantin comparison to the number of the unknowns. In ourinvestigations, we have used 102 equations in thecase of the two-function approach, i.e. 51 points foreach transfer function. In the case of thethree-function approach, 153 equations are used. Wedo not think that there is any need to limit thenumber of equations to the number of the unknownparameters as stated by Prof. Keyhani.

We agree with Dr. Keyhani that the d-axisoperational impedance Z (s) is highly noisy at lowfrequencies. Many difdculties are encountered inconducting the SSE'R test and particularly in this lowfrequency range (reference 6 of the paper). Theproblem of the calculation of the d-axis operationalinductance, L (s) in this low frequency range couldbe overcome bgvarious techniques. one way, which we

have used with success in our experimentalinvestigation on the microalternator, is to apply thefitting technique to the measurements of the lowfrequency part of the transfer function in order toobtain an approximate fitted function for Ld(s) in

this range. Another practical method is to elminateR directly during the measurement process [l].dwever, in the simulations investigated in ourpaper, the noise was added to the data of Ld(s)instead of 2 (s) since the objective was just toverify the ne& of a third transfer function for thecomplete and accurate identification of the d-axismodel.

Professor R.M. Saunders has made someinteresting comments concerning the models ofsynchronous machines which are obtained by the SSE'R

tests and asked for our reaction to them. In theSSFR tests, the frequency response measurements areincremental in nature. Thus, the parameters obtainedfrom them are likely to give more accurate results inthe theoretical analysis of small disturbancesituations than it can be expected from theparameters obtained by the standard short-circuittests. In the case of small disturbance situations,the models of synchronous machines are linear. Theaccuracy of these models depends on the variousassumptions which have to be introduced to developthem from the theory of synchronous machines.However, the effect of any manufacturing tolerancesor defects on these models will be taken care of inthe measured data of the SSFR test.

REFERENCES

[l] International Electrotechnical Comnission (IEC),"First Supplement to Publication 34-4 (1967)

Unconfirmed Test Methods for DeterminingSynchronous Machine Quantities", Publication NO.34-4Ar 1972.

Manuscript received September 1 , 1989 .

a “three transfer functionsn approach for the standstill

Documents