104 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 2, MAY 1980
Computer Simulation of Generator-ThyristorNetworks in Power SystemsPRADIPTA K. DASH, R. W. MENZIES, SENIOR MEMBER, IEEE,
AND R. M. MATHUR, SENIOR MEMBER, IEEE
Abstract-The digital computer simulation of a generator-thyristornetwork in power systems is described in this paper. The digital modelof the synchronous generator with either controlled or uncontrolledrectifier bridge deals efficiently with the complex commutation condi-tions when the source impedance is significant. From the equationsdeveloped in this paper, the performance of synchronous generatorswith or without damper windings, connected to controlled or uncon-trolled bridge rectifier loads, can be predicted from given loading data,field current, load current, generator speed, and bridge delay angle. Thesimulation technique is used to evaluate the performance of a salient-pole synchronous generator during both steady-state and fault conditions.
INTRODUCTIONC ONTROLLED rectifier bridges fed by synchronous gen-
erators are commonly used in ac exciter systems andHVDC transmission systems. AC exciters, loaded with diodesor thyristor bridges, are used in modern excitation systems tosupply generator field current. In some HVDC systems [1] -[31, isolated generators are used to supply the total converterpower. In most HVDC systems, filter networks are used toreduce the harmonic content of the machine line currents, andthe generator load approaches a normal sinusoidal load currentcondition. This paper assumes that no filter circuits are con-nected to the generator-thyristor system and that phase cur-rents, therefore, are nonsinusoidal.Recently, several authors [4], [5] have studied the perfor-
mance of a synchronous generator loaded with thyristorbridges. The method of their study lies in the development ofcertain algebraic equations based on the assumption of aconstant field flux linkage. This paper, however, develops adigital computer solution based on Kron's method of tensoranalysis and bridge equations as outlined in [61 and [7] . Thisapproach is highly efficient for significant time-varying sourceimpedance and complex commutation conditions. The digitalmodel is used to predict the performance of the generator withor without damper windings, connected to controlled or un-controlled bridge rectifier loads from given loading data, fieldcurrent, load current or impedance, generator speed, orbridge delay angle. Also, a realistic power system with severalsix-pulse bridges and varying transformer connections couldbe analyzed for faults at the transformer terminals or on theHVDC converter.
Manuscript received June 8, 1977; revised November 1, 1979. Thework of P. K. Dash was supported by the Department of Science andTechnology, Government of India.
P. K. Dash is with the Regional Engineering College, Rourkela, India.R. W. Menzies and R. M. Mathur are with the Department of Electri-
cal Engineering, University of Manitoba, Winnipeg, Man., Canada.
CIRCUIT DESCRIPTION AND ANALYSIS
A salient-pole star-connected three-phase synchronous gen-erator, with the field winding energized by a constant voltageVj, and having short-circuited damper windings along directand quadrature axes, is connected to an SCR bridge as shownin Fig. 1. An average direct current If which flows in the fieldwinding and the rotor pole position is measured from the Aphase winding. The field rotates at w rad/s. The three-phase SCR bridge consists of six rectifier elements, and thecommutation of currents between the thyristors occursasymetrically between pairs of thyristors in a prescribedsequence. To facilitate the analysis of the generator-thyristornetwork, saturation and hysteresis in the magnetic circuits ofthe generator are neglected.
In order to develop a meaningful simulation procedure for agenerator-thyristor system, it is first necessary to examine theoperation of the three-phase bridge using the technique out-lined in [6].
However, if the thyristors have negligible operational imped-ances, the impedance matrix Zi is reduced to
ZL ZL 03X2ZL ZL
ZL ZL
02X3 102X2
(3)
where the load impedance ZL is given by
ZL =R + pL p = d/dt (two-differential operator).In a thyristor bridge system, only a certain number of thy-ristors conduct. Therefore, a new set of current is definedwhich is related to the preceding ones by a transformationmatrix. When thyristors 5, 1, and 6 conduct, the equations ofconstraint are
i2 = 0 4 = 0 is = ° (4)
Since il + i2 + i3 + 4 + i5 + i6 = 0, two independent currentsia and ip defme the network. A transformation matrix be-tween the old and new sets is defined as
1e011 -O I-
2 o 0
[Cn] = 3 1 -1 . (5)
4 0 0
5 _0 0jAfter the commutation period with the two thyristors con-
ducting is over, the transformation matrix [Cn I is changed to
a
1
2
[C.] = 3
4
5
0
1
0
The formation of the matrix C,, can be automatically com-
puted by having a square dummy matrix C,, of dimension(5 X 5), 5 being the number of independent currents in [Ijand being initially filled with zero elements. The conversionof [C,,] is controlled by a system state array [SI which con-
tains 0 or 1 depending on whether or not the individual
Fig. 3. Three-phase thyristor bridge.
thyristors are conducting. For the case with 5, 1, 6 conducting
[S] =[1 0 0 0 1 1]. (7)
The independent set of variables [VI], [IiJ, and [Zi] are re-lated to the new set after a topology change as
[hi = [CnJ [In] [Vnj = [Cnf" [R
[Zn] = [Cn]T [ZiJ [Cn ] = [Rn ] +PjLnJ] (8)
To determine the thyristor voltages, a mesh sum voltage Vx isfound as
[KI] = [SX] [VtIwhere matrix Sx and voltage vector are given by
F-i 0000o-1 vt11I 0 - Vt2
[SXl=[lo 0 0 0 -1 Vt400 0 1 0 -1
t
LVt5J
If thyristor 6 is conducting Vt6 = 0, and the remaining voltagesare determined. However, if thyristor 6 is nonconducting, itis necessary to determine the state of other thyristors in orderto calculate voltage Vt6.
GENERATOR-THYRISTOR MODELINGTo facilitate computation of the detailed interaction be-
tween the synchronous generator and the thyristor bridge (Fig.3), the synchronous generator (Fig. 2) is modeled in directphase quantities. The following equations are used to describethe generator-thyristor network in the power system:
[VPJ [ tsrL PIp1 [(WL:8R5s ) wLsrl rIPL41T Lr JpIr. i I TQy -R,..JLvrJ sr Lrr] LPr rs)T -rr LIrJ
(10)where the inductance and resistance matrices are given inAppendix III.
105
ZL
[Zil =ZL
ZLL
106 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 2, MAY 1980
The rotor voltage [VTr], and current vector I,, respectively,are given by
[Vr] = [Ir] $ikd (11)
where Ifo is the constant field current.After establishing the synchronous machine equations, the
bridge equations could be combined with them in the follow-ing manner:
[V] [°c J[2. ¼:r] [OR ] [2Ir(12)
and
[I2 [° U [2] [2V] [CT ] [2VEquations through (12) are manipulated to yield
[Vn ] [CT CT Ls Cl Cn Cn CT LsjJ [Ph:]
Vr srr CL Cn rr Pir
I Ls - Rss) C1 CnQ {CnC' wLr}] [ in]+Ww)T (C
a
Cn) -Rrr Ij
(13)In(13)
[Vn) [Rn +PLn 0] I[In] (14)
Combining (13) and (14), the rate of change of rotor currentsand rotor bridge modified mesh current is
In (15), the matrix [VrJ is given by
[Efo + Xad if][IV] = O , where Efo =XadIfo
and Ifo is the initial value of field current.
(17)
DIGITAL COMPUTER SOLUTIONWhen the generator field is at rotor angle 0 = 6 + 3ir/2, the
armature voltages Vb and J' are equal. The displacementangle 6 accounts for the translation of the voltage intersectioncaused by armature magnetomotive force (MMF). For a givendelay angle a, the commutation from phase b to c will com-mence at 0 = 6 + a + 3ir/2. During commutation, the rotorangleD = S + a- + 3 + 37r/2 and armature currentsii =IL, ib =-IL + i, and ic = -i. Commutation commences at = 0 withi =0 and is completed at 3==y with i = .I.The performance of the generator-thyristor system is ob-
tained by integrating (15) by a fourth-order Runge-Kuttanumerical integration technique. This has the advantage thatthe local truncation error is available. A satisfactory steplength for the numerical integration is found from the eigen-value, and this usually varies between 0.1 and 1 ms. The ac-curacy of the solution is determined by monitoring the localtruncation error; and, if this is outside the defined limits, thestep length is reduced. A computer flowchart outlining theprogram development is shown in Fig. 7. One of the chiefdifficulties is to determine the value of ifo, ikde, ikq0' andtheir initial derivatives. These quantities along with S and yare determined from the commutation voltage constraintsV1 = V, at j3 = 0, and Va = Vb at 7= r/3 - a and from thefollowing equations:
J8 +7r113 d f0 +Ir/3
ifdO =
0 oikd dO =JO+t/
0,ikq dO = 0.
cT7cT (Lss-Rss)c1 C
-Rn Cn
LT cT L
Lrr
-Ln)
Lrs Cl Cn
(18)
The preceding formulation is general and the method is cap-able of handling commutation angles greater than 7r/3, whenthree or four thyristor conduction modes are likely to occur.
0 For a synchronous machine operating with diodes, the operat-ing equations are modified by setting the delay angle ax = 0,Vr while computing the value of 0.
SIMULATION OF TRANSIENT CONDITIONS(15) During disturbances on the system, the speed of the syn-
chronous machine will vary. If Pm is the mechanical powerinput, and pO is the instantaneous speed, the mechanicaltorque Tm is given by Pm IpO. The equation of motion ofthe synchronous machine can then be expressed as
(16)
which may be compared with standard form y = f(x, y) re-
quired for a numerical integration process.
p20 = (1H)(Tm - Teo)=P26
where
6 =0 - 0O * t.
(19)
(20)
The electrical torque T, is given in terms of the armature
[PI]
[A-']
and
[A-1] =
T T ICn' CILsr
Lrr I
DASH et al.: SIMULATION OF GENERATOR-THYRISTOR NETWORKS
1.1- .6
o-0
-1.1
3
- 00
-o--S
_/
4Th565 5
/ V
(a)
(a)
I. -I.6
.55 .8a
E&L. ItoF
a
0
6.
(b)
A
o l. _295 385 475 565 855
9 degrees
(c)
(d)Fig. 4. Performance of a 15-kVA salient-pole synchronous generatorwith thyristor bridge (IL = 1.1 per unit): - Increased field resis-tance: --Original field resistance. (a) Phase voltage and current of
a generator. (b) Field and damper winding current. (c) Load voltageand current. (d) Thyristor voltage.
phase quantities as
2
Te = [Pa(ib - ic)+ 4b(ic - ia)+ 4PcQa i) (21)
where the flux linkages Oa, Ob, and 'pc can be computedfrom the phase currents and inductance matrices of the gen-erator given in preceding sections.The steady-state solutions of generator and converter volt-
ages and currents are used as initial values for disturbancessuperimposed on the normal mode of operation. Equations(15) and (19)-(21) are solved to yield the generator-thyristorsystem performance during transient conditions. The neces-
0
0o
derees
(c)
Fig. 5. Performance of a 15-kVA generator and bridge at 2.0-per unitload current. (a) Phase voltage and current waveforms. (b) Field anddamper winding currents. (c) Load voltage waveform.
sary changes to simulate faults or system abnormalities can beincorporated at any stage of the study by a few simple instruc-tions in the program.
DIGITAL SIMULATION RESULTSThe simulation technique developed in the preceding sec-
tions has been used to determine the performance of a 15-kVA50-Hz 400-V 1000-r/min salient-pole synchronous generator,the parameters of which are given in Appendix II. The 15-kVAgenerator-thyristor system performance has been evaluatedfor load current of 1.1 and 2.0 per unit with a load consistingof 0.1-per-unit resistance and 2.0-per-unit inductance. Al-though a large number of studies have been performed withdifferent values of various parameters, such as firing angle a,subtransient saliency, i.e., the magnitude of (xdj - X4'), andfield winding and damper winding resistances, etc., only alimited set of illustrative results is included in this paper.Observations derived from the remaining studies are also de-scribed. The figures in the paper are all for a = 15°.Results pertaining to the steady-state operation for the two
values of load current are shown in Figs. 4 and 5, respectively.
107
---- IL
-1.61
108 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 2, MAY 1980
a
4.C
3.0-
aj 2.0
10
-e degree
(a)
-0 degrees
(b)
O
0-
-1.5
-e degrees
475 565
(c) STOPFig. 6. Performance of a 15-kVA synchronous generator for a fault ona phase. (a) Phase a current. (b) (z) dc load voltage: (ii) dc load cur- Fig. 7. Flowchart of the digital simulation program: N = number ofrent. (c) Phase b voltage. thyristors conducting at one time.
From these figures it can be seen that both the phase voltageand the dc load voltage output of the rectifier are distorted,and this distortion increases with the increase of the load.The voltage distortion was also found to depend on the sub-
transient saliency. In simulation studies, variation of sub-transient saliency was achieved by varying mutual inductancebetween the armature and rotor windings. The studies showthat increasing the subtransient saliency increased the voltagedistortion.Increasing the field resistances by as much as three times
showed no significant change in the magnitude of variations inthe phase and load voltages, as shown by the dotted curves inFig. 4. However, the perturbations in the field current and
commutation angle y changed by a small amount. Increasingthe resistance of the quadrature-axis damper winding to threetimes its normal value, the maximum excursion in the magni-tude of the quadrature-axis damper winding current wasreduced from nearly 0.3 to 0.18 per unit. However, therewas no significant change in the magnitudes of the phasevoltage, load voltage, and field current. Thus by changing thedamper winding parameters, the magnitude of current excur-sions in these windings can be altered, although the changes inthe machine load angle 6 and a are quite minimal.A single-phase-to-ground fault is simulated by slightly modi-
fying the program shown in the flowchart. The blocks show-ing the computations of initial values of field and damper
i)
1.5
DASH et al.: SIMULATION OF GENERATOR-THYRISTOR NETWORKS
wmding currents are deleted from Fig. 7 for simulating thefault. Results for a single-phase-to-ground fault on a phase areshown in Fig. 6. This figure shows that there is a rapid in-crease in both dc load current and ac line current in the eventof a fault. Also, the generation of harmonics and an increasein magnitudes are noticed in both the unfaulted phase voltage(Fig. 6(c)) and dc load voltage (Fig. 6(b)).
CONCLUSIONSThis paper presents a computer-aided analysis of a generator-
thyristor network in power systems from a given loading data,speed, and bridge firing delay angle. Detailed knowledge ofvarious equivalent circuits and operating modes during opera-tion is not required, and both normal and abnormal conditionsare easily handled. The simulation technique brings out thedynamic interaction between the synchronous generator andthe thyristor network, and shows the influence on theseinteractions by the parameters of the generator. Also, unlike
i.the earlier methods, the simulation technique can take intoaccount the winding resistances, and the concepts of constantflux linkage can be dispensed with. The digital simulationresults highlight the procedure presented in this paper.
APPENDIX I
IL Load current.Ia Ib , I Instantaneous three-phase line currents.
1i, 12, 13, 14, is Instantaneous independent mesh currents.icy, is Instantaneous independent currents of re-
duced network.if, ikd, ikq Transient field and damper winding currents.
Lao, a2 ,M0 Inductance coefficient of the armature.Maq Mutual inductance between armature and
q-axis damper winding.f System frequency.
r Resistance.Va, Vb, VC Armature phase voltages.
Vf Field voltage.VL Load voltage.a Controlled bridge delay angle.8 Rotor angle measured from start of com-
mutation interval.y Commutation angle.0 Field axis with respect to phase axis.w Angular rotor speed.& Flux linkage.p d/dt (operator).
SYSTEM PARAMETERSThe parameters in per-unit form for a 15-kVA 400-V
three-phase 50-Hz salient-pole generator are as follows:
w = 314 rad/s = 1.0 per unit
Ld = 0.24, Lq = 0.17
Maf=0.36, Makd = 0.176
Makq = 0.09, Lff = 0.42, Lkkd = 0.23
Lkkq = 0.08, Mfkd =O.32,
rkq = 0.044,
Re = 0.2,
rf = 0.0007
ra = 0.002, Ifo = 2.5 A
Le =2.0
1 per unit of voltage = 230 V, 1 per unit of current = 21.5 A.
APPENDIX III
-Lao - La2 cos 20 -MSO - La2 cos (20 - 2rr/3)[LSS] =-MO - La2 cos (20 - 27T/3) -Lao - La2 COS (20 + 2i43)
_.MSO - La2 cos (20 + 2/3) -MSO - La2 cos 20
Maf Cos 0
[ =sr]- Maf cos (0 - IO3)LMaf cos (0 + 27r/3)
[Lrs] = [Lsr]T
-MSO La 2 cos (20 + 2:
MS- La2 cos 20
-Lao - La2 cos (20 -2
Makd COS 0 -Ma kq SinO
Makd cos (0 - 2>f/3) -Makq sin (0 - 21r/3)Makd cos (0 + I13) -Makq sin (0 + 27r/3)
LFf MfLkd L I
[Lrr] =Mfksd LJd °
O O Lkkq _
and
Lao = (Ld + Lq)12 La2 = (Ld - Lq)12 Mso = 05Lao
109
110 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 2, MAY 1980
La2 sin 20 La2 Sin (20 - 2r/3) La2 sin (20 + 27r/3)
[LssJ La2 sin (20 - 2i1/3) La2 sin (20 + 2Xr/3) La2 sin 20
LLa2 sin (20 + 2i1/3) La 2 sin 20 La2 sin (20 - 271/3)[ Maf sin 0 Makd sin 0 -Makq COS 0 1
[L;r] = -Maf sin (0 - 27r/3) -Makd sin (0 - 224/3) -Makq cos (0 - 2i1/3)
L-Maf sin (0 + 27r/3) -Makd sin (0 + 27r/3) -Makq cos (0 + 271/3)[Lrs] = [(Ls;r)T] [L;r] = [03x I ] (null matrix).
REFERENCES
[1] N. G. Hingorani, J. L. Hay, and R. E. Crosbie, "Dynamic simula-tion of HVDC transmission system on digital computers," Proc.Inst. Elec. Eng., vol. 113, p. 793, 1966.
[2] W. Shepherd, and J. S. Htsui, "Method of digital computation ofthyristors switching circuits," Proc. Inst. Elec. Eng., vol. 118,p. 993, 1971.
[3] J. L. Hay, and N. G. Hingorani, "Dynamical simulation of multi-converter, HVDC systems by digital computer," IEEE Trans.Power App. Syst., vol. PAS-89, p. 218, 1970.
[4] W. J. Bownick and V. H. Jones, "Performance of a synchronous
generator with a bridge rectifier," Proc. Inst. Elec. Eng., vol. 119,no.9, p. 1338, 1972.
[5] P. W. Franklin, "A theoretical study of the three-phase salient poletype generator with simultaneous ac and dc bridge rectified out-put: Part I," IEEE Trans. Power App. Syst., vol. PAS-92, p. 543,1973.
[6] S. Williams and I. R. Smith, "Fast digital computation of 3-phasethyristor bridge circuits," Proc. Inst. Elec. Eng., vol. 120, no. 7,p. 791, July, 1973.
[71 S. Williams and I. R. Smith, "Prediction of performance of syn-chronous generator/transformer/bridge convertor installation(summary)," Proc. Inst. Elec. Eng., vol. 122, p. 1286, Nov. 1973.
Electronic Analog Slip Calculator for
Induction Motor Drives
R. VENKATARAMAN, BELLAMKONDA RAMASWAMI, AND JOACHIM HOLTZ
Abstract-For some schemes of variable speed control of squirrel-cage induction motor fed by a current-source inverter, accurate evalua-tion of motor slip frequency is essential for obtaining optimum torqueoutput. For evaluating slip accurately, a digital speed transducer coupledto the shaft is generally required. Such a transducer and the associateddigital circuits make the system complex and expensive. An analogspeed transducer which is relatively cheap could be used only if theslip could be obtained accurately by some other means. This paperpresents an inexpensive and accurate method of obtaining an analogsignal proportional to the slip by using a simple calculator circuit thatuses the dc link current, dc link voltage, and inverter frequency as itsinputs. The slip calculator described here is capable of giving the correctoutput only under steady-state conditions. The design of the slip cal-
Manuscript received July 24, 1978; revised April 6, 1979. Facilitiesfor this research were supplied by the Indian Institute of Technology,Madras, India.R. Venkataraman and B. Ramaswami are with the Department of
Electrical Engineering, Indian Institute of Technology, Madras-600 036,India.
J. Holtz is with Bergische Universitat, Elektrische Maschinen undAntriebe, 5600 Wuppertal 1, Germany.
culator is illustrated in the Appendix with theexample.
aid of a numerical
I. INTRODUCTIONN CLOSED-LOOP speed control of an induction motor,where the speed of motor is controlled by varying its
frequency through an inverter, it is essential to operate themotor under all speed and load conditions at the nominalflux defined by the machine nameplate voltage and frequency.Operating the motor at constant nominal air-gap flux valueresults in the same pullout torque at all supply frequenciesand hence permits constant full-load torque capability of themotor at all frequencies. Sensing the air-gap flux with the aidof special sensors such as Hall-effect probes calls for custom-made motors and also makes the scheme quite complex.The flux can be maintained very nearly at a constant level bykeeping the motor voltage to frequency ratio constant providedthe frequency is not very low. A voltage-fed inverter for wide-range voltage control should be pulsewidth modulated or one
