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Simulation of the transient response of singly-excited electrical machines
Cornput.& Elect Engng,Vol.3, pp. 271-279.PergamonPress, 1976.Printedin Grea!Britain
SIMULATION OF THE TRANSIENT SINGLY-EXCITED ELECTRICAL
RESPONSE OF MACHINES
J. R. SMITH,J. PENMANand H. AnD-ALLAH Department of Electrical Engineering, University of Southampton, Southampton, England (Received 24 November 1975; in revised/otto 15 March 1976)
Abstraet--A method is presented for the simulation of singly-excited, alternating current, polyphase electrical machines under large-scale disturbance conditions. The method, based on the two-axis theory of machines, is modified to allow for the frequency dependence of the effective rotor parameters. The simulated results included in the paper, for a representative range of machines, are compared with those obtained from practical tests.
INTRODUCTION
Reliable and accurate simulation methods are necessary in predicting the transient behaviour of induction machines when subjected to various disturbance conditions [1-3]. The analysis may be associated with the selective study of important drives, where the simulation accuracy must be high, or conversely, an overall system study concerning the effect of transmission system faults on numerous industrial complexes, where the simulation accuracy is less restrictive. It is with the former type of study that the present paper is concerned. To a large extent the complexity of the mathematical models employed in simulations depends upon the study to be undertaken, and to predict accurately the starting current and torque, or the speed-recovery transient in response to a system voltage fluctuation, it is necessary to employ a detailed modelling procedure in which some account is taken of the inherent non-linear nature of the machine equations. The method described in the paper is suitable for time-domain dynamic simulation of wound-rotor induction machines, squirrel-cage induction machines and reluctance machines[4,5]. The principal non-linearities associated with these types of electrical machine are included in the analysis. The mathematical analysis, using 2-axis equations[6], is outlined. The program capability is illustrated, and simulation accuracy verified, by comparing the transient response analysis for both a squirrel-cage induction motor and a reluctance machine for which data and practical test-results were made available. SINGLY-EXCITED ELECTRICAL MACHINES
Induction machines are used for many industrial drives where a simple, reliable and robust machine is the first requirement and where the use of a non-synchronous speed drive is of no disadvantage. These machines are manufactured in sizes up to about 12 MVA but are used predominantly in the smaller sizes. In running up to speed, these machines demand a reactive current which depresses the supply system voltage, but since motors are usually started individually this will not create excessive drops unless large motors are started on relatively weak supplies, or overlapping sequential starts of several machines are required. However, when recovering from a supply system fault condition all the connected induction machines make a simultaneous demand for reactive current. In these circumstances the voltage drop may be so large as to cause certain motors to lose speed and eventually stall. It is principally in this area of induction motor performance, that interest in the dynamic properties of the induction motor is focussed. Predictive studies of post-disturbance transient response are necessary to ensure overall system stability and reliability. In contrast, the reluctance machine, in many respects similar in physical configuration to the induction machine, is a constant speed device and, as a consequence, finds application where the drive speed must remain constant during load variations. The majority of reluctance motors are of small rating, only exceptionally exceeding 50kVA. The ability of the machine to run synchronously is a function of the design of the rotor, in which the polar and interpolar flux paths 271
272
J.R. SMnH et al.
are constructed to provide regions of low and high reluctance respectively. The reluctance torque so produced pulls the motor into step with the rotating field of the stator. The machines are started by the induction motor action of the rotor circuits; the subsequent synchronous performance is then heavily dependent upon the design of the magnetic circuits. Transient response analysis provides a means of optimising the rotor parameters for increased performance and assists in predicting the performance limits of a given machine. MACHINE PERFORMANCE EQUATIONS
The basis of analysis for the electrical machines considered here shown in schematic form in Figs. l(a) and (b), is the matrix "
Ida
s
Vb s
R°" +pL,/ pM,.
pM., pM., R." +pLb ~ pMb,
p Maa
'
pM,~
(1) v',,
R "., + pL ;',. •
pM.,.
in which R is the winding resistance, L is the winding self-inductance and M the mutual-inductance between windings. The phase voltage and currents are respectively given by the column vectors [v] and [i] in eqn (1). The inductance contained in the coefficient matrix of eqn (1) are functions of angular position and hence of time. To simplify these equations and to facilitate an economical simulation a two-axis transformation, of the stator and rotor currents and voltages, is performed such that the resulting equations have constant coefficients. The two-axis linear electrical equations of the machine may now be written in the general form
( ,)IiI=ILr'{H- L.I+I I/til}. Vi~
O- ~
(2)
(G) ' ~8
s
3-ph. reluctance m / c
3-ph. induction t a l c
/
\
Transformation
Transformation I
2
\
1 Derived 2-axis machine
Fig. I.
(bl
Simulation of the transient response of singly-excited electrical machines
273
The current vector [i] contains the transformed currents ias, i~s, ia,, iqr, where ia, and iq, represent currents in the wound or squirrel-cage winding on the rotor. Similarly, the voltage vector [v] contains the axis coil voltages vas, vq~, vdr, vq,, and for the general case of a singly-excited machine, with short-circuited rotor windings, vdr = pq, =0. The vectors [i] and [v] are transformations of the actual terminal current and voltage vectors of the machine. The transformation from the six-coil primititive induction machine, for which stationary axes are specified for the stator and rotating axes specified for the rotor, to the four-coil derived machine is shown schematically in Figs. l(a) and (c). Similarly, the transformation from the primitive reluctance machine to the derived machine is shown schematically in Figs. l(b) and (c). Transformations [4] may be chosen such that the derived machine has stationary axes, rigidly fixed in either the stator or rotor, or rotating reference axes. The equations formulated in a stationary reference frame attached to the stator bear a close resemblance to the physical systems of the machine. However, all quantities vary at the supply frequency in the steady state, and because of this, small time-intervals are required during the numerical integration process, resulting in increased computational time. For the steady-state vectors [i] and [v] to be independent of time a synchronously rotating reference frame, to which both the stator and rotor quantities are referred, must be specified. In general the reference frame in which the equations for solution are formulated depends upon the form of solution required and the variation of specific stator or rotor parameters during a particular solution sequence. In the present case, the study of the effect and variation of the rotor parameters is of major importance for both the induction and reluctance machines and consequently reference axes, rigidly fixed to the rotating member of the machine, are chosen. This formulation has the added advantage of closely corresponding to the well-established method [7] adopted for synchronous machine analysis. In practice, the supply to the motor is taken through a transmission circuit as illustrated in Fig. 2. In this case the parameters of the transformed stator circuits are modified to account for this.
_ooooo
Fig.2.
The coefficient matrices of eqn (2) are defined as: (i) A diagonal resistance matrix [R ] = diag (Rds + R,, Rq~ + R,, Rd,, Rqs )
(3)
(ii) An inductance matrix
ds qs dr qr ds IULa*+ L, Ma,a, ] qs ] Lq, + L, Mq,qr [L] = dr Mara, La, qr L M~,~, L~,
(4)
and (iii) A modified inductance matrix
ds ds
[G]=qs[L,
qs - L ' ] +[Gm]
(5)
274
J. R. SMITH et al.
where ds [G,, ] = qs
qs
- Ld~
dr
- Md,q,
qr
" (6)
Finally, the equation of motion of the shaft is T = [i]'[Gml[i]+J~or
(7)
where J is the inertia of the machine and rotating load. Under transient conditions the variation of the load torque, T, is dependent on the type of motor drive. Generally, T is taken to remain either independent of rotor speed, or proportional to rotor speed, or, as in the case of pump drives, proportional to the square of rotor speed. A more accurate representation of the load would be to treat the mechanical system as consisting of two rigid inertias coupled by a flexible shaft, and solve the equations along with those of the electrical system. Inherent machine nonlinearities In the formulation of an acceptable mathematical model for digital simulations, the reliability and accuracy of the specified data is a crucial factor in determining the detail to be included in the digital model. However, it will be shown that for specific cases a purely linear representation of an induction machine very often leads to optimistic results. It is therefore expedient to make compromises in the modelling detail to an extent that the improved representation included in the digital model is commensurate with the computational effort expended on the one hand, and the improvement in response obtained on the other. This, in turn, affects the extent to which the machine nonlinearities are included in the model, and for present purposes their inclusion has been confined to the slip-dependence of the rotor parameters. The variations in the magnetising reactance, due to the saturation of the main flux paths, may be represented in a way similar to that adopted for synchronous machines[8], however, average values may be taken for the magnetising inductances and these are used throughout the simulation. One point that has emerged from a previous investigation fl] concerns the inclusion of the motor stray load loss in the calculations. The torque produced by the motor is less than that calculated from the equivalent circuit of Fig. 3 because of this loss, which varies approximately as the square of the current. For a typical large motor this loss is between 0.5 and 0.7% and its effect is included in subsequent simulations. Slip-dependence of rotor parameters When conductors are contained in slots the leakage flux causes the current to crowd towards the top of the bar adjacent to the air-gap. Because this effect is a function of frequency, the effective resistance and reactance of a rotor bar will change during speed variations, especially during the run-up period. Evolving from solutions of the field equations, expressions have previously been derived [9, 10] for the rotor parameters as functions of slip. For the simplest case of a rectangular bar of depth x in an open slot, the resistance Ro, of the bar in terms of the resistance at zero frequency Ro is given as R~ = Rox sinh (2x/8) + sin (2x/8) (8) cosh (2x/8) + cos (2x/8)
Fig. 3. Induction m/c equivalent circuit.
275
Simulation of the transient response of singly-excited electrical machines
in which 8, the eddy-current skin depth, is given by
8=
•]
2 iaJO-~o
(9)
where t7 is the conductor conductivity and/~o the permeability of free space. The reactance X, is given by
X~= Ro___xsinh(2x/8)-sin (2x/6)÷ ~ ~ sins (.~!)coth 8 cosh (2x/8) - cos (2x/8)
(2~rnx/t~)
2~rnx/8
(10)
The summation term in eqn (10) represents the effect of the slot opening and it is not usually necessary to consider more than ten terms in the series. Another bar shape that is often used is the " T " bar, which makes use of the deep-bar effect to increase the rotor resistance at stating. It can be shown that for a " T " bar, neglecting insulation thickness, a one-dimensional solution yields acceptable results. For this bar the impedance is given as R~+jX~=J
X (a + c) cosh a(b + d ) - (a - c) cosh a(b - d) trc (a + c) sinh a ( b + d) + (a - c) sinh a ( b - d)
(11)
where a = 2j/82, and the dimensions of the bar are as shown in Fig. 4. For rotor conductors of irregular cross section or for double-cage induction machines, and where an approximation to a rectangular or " T " bar configuration would undoubtedly lead to large errors in the calculated values of R~ and X~, Bruges [11] had devised a method of assessing the impedance of a conductor of any shape embedded in an open slot by the evaluation of ladder-type networks. One means of including relationships of the kind given by eqns (9)-(11) into the machine model representation, is to derive from them multiplying factors by which the zero frequency values of rotor resistance and leakage reactance are multiplied to give the effective parameter values at any rotor slip frequency. Figure 5 shows the computed per-unit resistance and reactance variations for the rectangular sectioned bar of a 3500 h.p. motor. Resistance
Reactance 0.29
0.064
=. Q.
¢ o
0.19
O.05
cz
tY
0.036 t7
Fig. 4. A rotor "T" bar.
=i
o
I I
2
0.09
Time, sec
Fig. S. Variation of rotor bar resistance and reactance with slip. O V E R V I E W OF THE C O M P U T E R PROGRAM
The program, which is available on request, is for single machine studies and consists of a main program and six subroutines. Initially the main program calls subroutine ALLDATA which reads and assigns data and specifies the solution sequence, 'disturbance type and duration of disturbance. The load characteristic is selected and the form of results printout and interval are also selected. This information is carded through to the subroutines by common statements. This is immediately followed by a call for subroutine INITIAL which' calculates the machine pre-disturbance initial conditions for either a standstill or steady-state running condition. The initial calculated values of all variables are printed at this stage. The main program now enters the integration cycle and calls the integration subroutine RK. A Runge-Kutta fourth order algorithm with Gills coefficients is used. This algorithm accepts
J.R. SMITHet
276
al.
numerical values for the integrable variables, as formed from the machine electrical and mechanical equations calculated in subroutine DERIVS, and calculates corresponding solution values from them. For the studies conducted, the maximum step-length consistent with the stability of the numerical integration process was found to be 0.0005 sec when simulating large scale disturbances, such as machine run-up and system fault conditions and 0.01 sec for small scale disturbances such as perturbations about a steady-state operating point.
Program performance guide Core size required: 7000 Input medium: 80 column cards for program and data Compile time: 0.5 sec }C.D.C. 7600 computer. Run Time: 4000 iterations per sec
EXAMPLES Two examples illustrating the use of the program are given. The transient responses obtained for an induction and reluctance motor are compared with results obtained from test recordings. In addition further simulation results are presented to illustrate the effect that transmission systems may have on the behaviour of an induction machine.
Example 1: Induction motor The principal data for the test machine is given in Table 1. Figure 6 shows the responses of speed, per-unit torque and current for the run-up, load application and a short-circuit on the terminals of this motor. The results obtained from test relate to a reduced voltage condition, of 87% nominal voltage, and correspond to the run-up from standstill only. I00 O0
~.-I iApplicationof load
l,: ~. Runup
•~~ ~~ sooo
"-
I
"~k
I i
(o) iTerminal3-ph, sc.
ifault removed
:d"
-
000
:
} (b)
-~ ~
4°°t I Il ~
~ ~ ooor llrl o"
f
,
,
,
,
L~
4oo~ 8 O0
/
• -Testpoints
ooo
±~K
" 'i i
i
I
i' ,
i!" ll'",
,
i
I'l
(c)
I
300 ]
I
!1|1~~11111111,,"":
J,,IIUIHI~,,,,,, ~: ooo |tWt~NINWWIIIUttI~k,,..... L,,,,,,,,,,,.,,,,,,,,,' , ,,,
(d)
.................................
''"" ...................... i l',ll~m,,""..............................
3oo
Test points [e)
-~
ooo O0
Fig. 6. Inductionmotor transient response.
277
Simulation of the transient response of singly-excited electrical machines Table 1. Principal data for induction motor Rating: Stator
3500 h . p .
6 K.V.
resistance Rss
50 Hz
pole
12
..........
..
Rotor resistance (start) Rrr ...... Stator leakage reactance Xss ...... Rotor leakage reactance (start) Xrr Magnetising reactance
..
Xm . . . . . . . . . .
Rectangular rotor slots dimensions . . . . Inertia constant KVA/KWsec . . . . . . . .
0.0068
pu
0.0176 0.096
pu pu
0.I05
pu
2.89
pu 7ram
25ram x 1.216
The effect of neglecting the machine nonlinearities in the formulation is shown in Fig. 7 for an 880 h.p. induction motor pump drive. These curves relate to the speed recovery-transient when the machine is subjected to a terminal short-circuit of 22 msec duration. An important aspect of the simulation of induction machines is concerned with the prediction of the speed recovery of the machine following severe system voltage fluctuations or faults. A variety of system conditions[2] such as the effect of fault-level at the point of machine connection and the fault-clearing time effect the behaviour of the machine and must be accounted for in the simulation. The effect of progressively reducing the ratio of fault-level to machine capacity is illustrated in Fig. 8. In this way a simulation may be used to lead to an estimation of the critical fault-level below which the motor recovery following a system fault is unlikely. Figure 8 also illustrates the effect of a range of fault-clearing times on the motor recovery. Ideally this effect should be studied in close relation to the possible variations in fault-levels. 48
Fault cleared 16
32
.o. N B
16
4
O0
0.22 0.4
0.8
I.'2
116
O0
I
2
S
i 4
3
i 5
S
Fig. 7. Speed recovery of induction motor pump drive. ©, Test curve; A, non-linear model; x, linear model.
Fig. 8. Speed recovery of an induction motor water pump drive for various fault infeed conditions and clearing times. ©
Infeed 10 MVA
A x [] •
20 30 30 30
MVA MVA MVA MVA
Fault clearance 0.22 s 0.22 s 0.22 s 0.40 s 0.10 s
Example 2: Reluctance machine The data for the test machine is given in Table 2, and Figs. 9-11 relate to various loading conditions imposed upon the machine as it runs up from standstill with zero initial conditions. In each case a computed and a measured result are given for comparison. In all cases the angle between the rotor direct axis and the rotating frame, defined by the peak of the stator mmf wave of phase "a", is taken as zero degrees at the moment of switching on the supply to the machine. Table 2. Principal data for reluctance motor Rating:
3 h.p.;
415 VL;
50 Hz;
4
pole
resistance Rss Rotor resistances Rdr
........
0.0469
........
0.0413
pu pu
Rqr
........
0.0533
pu
........
Stator
Magnetising inductances Lmd
....
Lmq
....
pu 0.00084 pu 0.0034 pu 0.00102 pu 0.0033 pu 0.00056 pu
KVA/KWsec
....
0.0257
Stator
inductances Lds
Lqs . . . . . . . .
Rotor inductances Ldr
........
Lqr
........
Inertia constant
0.0035
278
J. R. SMm~ et al. :5
o.
3o r
~--~\
E vlOO
80 i ~ S p e e d
160
240
k,,...,.~/
rds/sec
Fig. 9(a)
0~ v
80
16Q
240
Speed, rds/sec
Fig. 9(b).
c~
Or
f
6
32
48
6'4
Time, sec x 10
Fig. 10(a).
Time,
sec
Fig. lO(b)
O0
16
3
Time, s e c x 10
'
Fig. IRa).
Time, sec
Fig, ll(b).
Figure 9 illustrates the nature of the torque-speed curve for the unloaded machine with no added inertia. This is a severe test for a reluctance machine and if the machine is stable for this condition it will in general be stable for load conditions within the output capacity of the machine. Figure 10 shows the torque-time curve for the machine as it runs up against its nominal full-load. For this test the inertia of the load was six times that of the rotor. Figure 11 relates to a run-up against 1.33 x nominal full load with the same inertial conditions. In this case the machine fails to
Simulation of the transient response of singly-excited electrical machines
279
synchronise and pole-slipping occurs. The computed results are shown in Figs. 9(a), 10(a) and 1l(a) and the measured results in Figs. 9(b), 10(b) and 1l(b). The measurement equipment used for these tests had an upper frequency limit of 55 Hz. Due to this, the double frequency components, correctly present in the computed results for the on-load machine simulations, do not appear in the measured results. D I S C U S S I O N AND C O N C L U S I O N S
The method described in the paper, for determining the transient behaviour of singly-excited electrical machines, leads to accurate simulations of response to various balanced disturbance conditions. The simulated results, compared wherever possible with those obtained from practical tests for a representative range of machine sizes, substantiate the simple approach adopted. The program may be used with equal convenience for instructional purposes or as an aid in the prediction of individual machine behaviour when subjected to certain loading and system conditions. REFERENCES 1. S. S. Kaisi and B, Adkins, Transient stability of power systems containing both synchronous and induction machines. Proc. IEE 118(1) (1971). 2. W. D. Humpage, K. E. Durrani and V. F. Carualho, Dynamic-response analysis of interconnected synchronousasynchronous-machine groups, Proc. lEE, 116(12) (1969). 3. I. R. Smith and A. Sriharan, Transients in induction machines with terminal capacitors. Proc. IEE 115(4) 0968). 4. A.J.O. Cruickshank, A. F. Anderson and R. W. Menzies, Stability of reluctance motors from freely accelerating torque speed cures. Trans. A.I.E.E. (P.A.$.) 91 (1972). 5. V. B. Honsinger, Stability of reluctance motors. Trans. A.I.E.E. (P.A.S.) 91 (1972). 6. P. C. Krause, Simulation of symmetdcai induction machinery: Trans. IEEE FAS-84(11) (1965). 7. G. J. Rogers and J. R. Smith, Mathematical models of synchronous electrical machines. Int. J. Num. Meth. Engng 6 (1972). 8. G. Shackshaft, A general purpose turbo-alternator model. Proc. lEE 110 (1963). 9. P. E. Jones, N. Mullineux, J. R. Reed and R. L. Stoll, Solid rectangular and T-shaped conductors in semi-closed slots. J. Engng Math. 3 (1969). 10. S. A. Swarm and W. Salmon, Effective resistance and reactance of a rectangular conductor placed in a semi-closed slot. Proc. IEE 110 (1963). I I. W. E. Bruges, Evaluation and application of certain ladder-type networks, Proc. Roy. Soc. Edihburgh 62, II (1946).
SIMULATION OF THE TRANSIENT SINGLY-EXCITED ELECTRICAL
RESPONSE OF MACHINES
J. R. SMITH,J. PENMANand H. AnD-ALLAH Department of Electrical Engineering, University of Southampton, Southampton, England (Received 24 November 1975; in revised/otto 15 March 1976)
Abstraet--A method is presented for the simulation of singly-excited, alternating current, polyphase electrical machines under large-scale disturbance conditions. The method, based on the two-axis theory of machines, is modified to allow for the frequency dependence of the effective rotor parameters. The simulated results included in the paper, for a representative range of machines, are compared with those obtained from practical tests.
INTRODUCTION
Reliable and accurate simulation methods are necessary in predicting the transient behaviour of induction machines when subjected to various disturbance conditions [1-3]. The analysis may be associated with the selective study of important drives, where the simulation accuracy must be high, or conversely, an overall system study concerning the effect of transmission system faults on numerous industrial complexes, where the simulation accuracy is less restrictive. It is with the former type of study that the present paper is concerned. To a large extent the complexity of the mathematical models employed in simulations depends upon the study to be undertaken, and to predict accurately the starting current and torque, or the speed-recovery transient in response to a system voltage fluctuation, it is necessary to employ a detailed modelling procedure in which some account is taken of the inherent non-linear nature of the machine equations. The method described in the paper is suitable for time-domain dynamic simulation of wound-rotor induction machines, squirrel-cage induction machines and reluctance machines[4,5]. The principal non-linearities associated with these types of electrical machine are included in the analysis. The mathematical analysis, using 2-axis equations[6], is outlined. The program capability is illustrated, and simulation accuracy verified, by comparing the transient response analysis for both a squirrel-cage induction motor and a reluctance machine for which data and practical test-results were made available. SINGLY-EXCITED ELECTRICAL MACHINES
Induction machines are used for many industrial drives where a simple, reliable and robust machine is the first requirement and where the use of a non-synchronous speed drive is of no disadvantage. These machines are manufactured in sizes up to about 12 MVA but are used predominantly in the smaller sizes. In running up to speed, these machines demand a reactive current which depresses the supply system voltage, but since motors are usually started individually this will not create excessive drops unless large motors are started on relatively weak supplies, or overlapping sequential starts of several machines are required. However, when recovering from a supply system fault condition all the connected induction machines make a simultaneous demand for reactive current. In these circumstances the voltage drop may be so large as to cause certain motors to lose speed and eventually stall. It is principally in this area of induction motor performance, that interest in the dynamic properties of the induction motor is focussed. Predictive studies of post-disturbance transient response are necessary to ensure overall system stability and reliability. In contrast, the reluctance machine, in many respects similar in physical configuration to the induction machine, is a constant speed device and, as a consequence, finds application where the drive speed must remain constant during load variations. The majority of reluctance motors are of small rating, only exceptionally exceeding 50kVA. The ability of the machine to run synchronously is a function of the design of the rotor, in which the polar and interpolar flux paths 271
272
J.R. SMnH et al.
are constructed to provide regions of low and high reluctance respectively. The reluctance torque so produced pulls the motor into step with the rotating field of the stator. The machines are started by the induction motor action of the rotor circuits; the subsequent synchronous performance is then heavily dependent upon the design of the magnetic circuits. Transient response analysis provides a means of optimising the rotor parameters for increased performance and assists in predicting the performance limits of a given machine. MACHINE PERFORMANCE EQUATIONS
The basis of analysis for the electrical machines considered here shown in schematic form in Figs. l(a) and (b), is the matrix "
Ida
s
Vb s
R°" +pL,/ pM,.
pM., pM., R." +pLb ~ pMb,
p Maa
'
pM,~
(1) v',,
R "., + pL ;',. •
pM.,.
in which R is the winding resistance, L is the winding self-inductance and M the mutual-inductance between windings. The phase voltage and currents are respectively given by the column vectors [v] and [i] in eqn (1). The inductance contained in the coefficient matrix of eqn (1) are functions of angular position and hence of time. To simplify these equations and to facilitate an economical simulation a two-axis transformation, of the stator and rotor currents and voltages, is performed such that the resulting equations have constant coefficients. The two-axis linear electrical equations of the machine may now be written in the general form
( ,)IiI=ILr'{H- L.I+I I/til}. Vi~
O- ~
(2)
(G) ' ~8
s
3-ph. reluctance m / c
3-ph. induction t a l c
/
\
Transformation
Transformation I
2
\
1 Derived 2-axis machine
Fig. I.
(bl
Simulation of the transient response of singly-excited electrical machines
273
The current vector [i] contains the transformed currents ias, i~s, ia,, iqr, where ia, and iq, represent currents in the wound or squirrel-cage winding on the rotor. Similarly, the voltage vector [v] contains the axis coil voltages vas, vq~, vdr, vq,, and for the general case of a singly-excited machine, with short-circuited rotor windings, vdr = pq, =0. The vectors [i] and [v] are transformations of the actual terminal current and voltage vectors of the machine. The transformation from the six-coil primititive induction machine, for which stationary axes are specified for the stator and rotating axes specified for the rotor, to the four-coil derived machine is shown schematically in Figs. l(a) and (c). Similarly, the transformation from the primitive reluctance machine to the derived machine is shown schematically in Figs. l(b) and (c). Transformations [4] may be chosen such that the derived machine has stationary axes, rigidly fixed in either the stator or rotor, or rotating reference axes. The equations formulated in a stationary reference frame attached to the stator bear a close resemblance to the physical systems of the machine. However, all quantities vary at the supply frequency in the steady state, and because of this, small time-intervals are required during the numerical integration process, resulting in increased computational time. For the steady-state vectors [i] and [v] to be independent of time a synchronously rotating reference frame, to which both the stator and rotor quantities are referred, must be specified. In general the reference frame in which the equations for solution are formulated depends upon the form of solution required and the variation of specific stator or rotor parameters during a particular solution sequence. In the present case, the study of the effect and variation of the rotor parameters is of major importance for both the induction and reluctance machines and consequently reference axes, rigidly fixed to the rotating member of the machine, are chosen. This formulation has the added advantage of closely corresponding to the well-established method [7] adopted for synchronous machine analysis. In practice, the supply to the motor is taken through a transmission circuit as illustrated in Fig. 2. In this case the parameters of the transformed stator circuits are modified to account for this.
_ooooo
Fig.2.
The coefficient matrices of eqn (2) are defined as: (i) A diagonal resistance matrix [R ] = diag (Rds + R,, Rq~ + R,, Rd,, Rqs )
(3)
(ii) An inductance matrix
ds qs dr qr ds IULa*+ L, Ma,a, ] qs ] Lq, + L, Mq,qr [L] = dr Mara, La, qr L M~,~, L~,
(4)
and (iii) A modified inductance matrix
ds ds
[G]=qs[L,
qs - L ' ] +[Gm]
(5)
274
J. R. SMITH et al.
where ds [G,, ] = qs
qs
- Ld~
dr
- Md,q,
qr
" (6)
Finally, the equation of motion of the shaft is T = [i]'[Gml[i]+J~or
(7)
where J is the inertia of the machine and rotating load. Under transient conditions the variation of the load torque, T, is dependent on the type of motor drive. Generally, T is taken to remain either independent of rotor speed, or proportional to rotor speed, or, as in the case of pump drives, proportional to the square of rotor speed. A more accurate representation of the load would be to treat the mechanical system as consisting of two rigid inertias coupled by a flexible shaft, and solve the equations along with those of the electrical system. Inherent machine nonlinearities In the formulation of an acceptable mathematical model for digital simulations, the reliability and accuracy of the specified data is a crucial factor in determining the detail to be included in the digital model. However, it will be shown that for specific cases a purely linear representation of an induction machine very often leads to optimistic results. It is therefore expedient to make compromises in the modelling detail to an extent that the improved representation included in the digital model is commensurate with the computational effort expended on the one hand, and the improvement in response obtained on the other. This, in turn, affects the extent to which the machine nonlinearities are included in the model, and for present purposes their inclusion has been confined to the slip-dependence of the rotor parameters. The variations in the magnetising reactance, due to the saturation of the main flux paths, may be represented in a way similar to that adopted for synchronous machines[8], however, average values may be taken for the magnetising inductances and these are used throughout the simulation. One point that has emerged from a previous investigation fl] concerns the inclusion of the motor stray load loss in the calculations. The torque produced by the motor is less than that calculated from the equivalent circuit of Fig. 3 because of this loss, which varies approximately as the square of the current. For a typical large motor this loss is between 0.5 and 0.7% and its effect is included in subsequent simulations. Slip-dependence of rotor parameters When conductors are contained in slots the leakage flux causes the current to crowd towards the top of the bar adjacent to the air-gap. Because this effect is a function of frequency, the effective resistance and reactance of a rotor bar will change during speed variations, especially during the run-up period. Evolving from solutions of the field equations, expressions have previously been derived [9, 10] for the rotor parameters as functions of slip. For the simplest case of a rectangular bar of depth x in an open slot, the resistance Ro, of the bar in terms of the resistance at zero frequency Ro is given as R~ = Rox sinh (2x/8) + sin (2x/8) (8) cosh (2x/8) + cos (2x/8)
Fig. 3. Induction m/c equivalent circuit.
275
Simulation of the transient response of singly-excited electrical machines
in which 8, the eddy-current skin depth, is given by
8=
•]
2 iaJO-~o
(9)
where t7 is the conductor conductivity and/~o the permeability of free space. The reactance X, is given by
X~= Ro___xsinh(2x/8)-sin (2x/6)÷ ~ ~ sins (.~!)coth 8 cosh (2x/8) - cos (2x/8)
(2~rnx/t~)
2~rnx/8
(10)
The summation term in eqn (10) represents the effect of the slot opening and it is not usually necessary to consider more than ten terms in the series. Another bar shape that is often used is the " T " bar, which makes use of the deep-bar effect to increase the rotor resistance at stating. It can be shown that for a " T " bar, neglecting insulation thickness, a one-dimensional solution yields acceptable results. For this bar the impedance is given as R~+jX~=J
X (a + c) cosh a(b + d ) - (a - c) cosh a(b - d) trc (a + c) sinh a ( b + d) + (a - c) sinh a ( b - d)
(11)
where a = 2j/82, and the dimensions of the bar are as shown in Fig. 4. For rotor conductors of irregular cross section or for double-cage induction machines, and where an approximation to a rectangular or " T " bar configuration would undoubtedly lead to large errors in the calculated values of R~ and X~, Bruges [11] had devised a method of assessing the impedance of a conductor of any shape embedded in an open slot by the evaluation of ladder-type networks. One means of including relationships of the kind given by eqns (9)-(11) into the machine model representation, is to derive from them multiplying factors by which the zero frequency values of rotor resistance and leakage reactance are multiplied to give the effective parameter values at any rotor slip frequency. Figure 5 shows the computed per-unit resistance and reactance variations for the rectangular sectioned bar of a 3500 h.p. motor. Resistance
Reactance 0.29
0.064
=. Q.
¢ o
0.19
O.05
cz
tY
0.036 t7
Fig. 4. A rotor "T" bar.
=i
o
I I
2
0.09
Time, sec
Fig. S. Variation of rotor bar resistance and reactance with slip. O V E R V I E W OF THE C O M P U T E R PROGRAM
The program, which is available on request, is for single machine studies and consists of a main program and six subroutines. Initially the main program calls subroutine ALLDATA which reads and assigns data and specifies the solution sequence, 'disturbance type and duration of disturbance. The load characteristic is selected and the form of results printout and interval are also selected. This information is carded through to the subroutines by common statements. This is immediately followed by a call for subroutine INITIAL which' calculates the machine pre-disturbance initial conditions for either a standstill or steady-state running condition. The initial calculated values of all variables are printed at this stage. The main program now enters the integration cycle and calls the integration subroutine RK. A Runge-Kutta fourth order algorithm with Gills coefficients is used. This algorithm accepts
J.R. SMITHet
276
al.
numerical values for the integrable variables, as formed from the machine electrical and mechanical equations calculated in subroutine DERIVS, and calculates corresponding solution values from them. For the studies conducted, the maximum step-length consistent with the stability of the numerical integration process was found to be 0.0005 sec when simulating large scale disturbances, such as machine run-up and system fault conditions and 0.01 sec for small scale disturbances such as perturbations about a steady-state operating point.
Program performance guide Core size required: 7000 Input medium: 80 column cards for program and data Compile time: 0.5 sec }C.D.C. 7600 computer. Run Time: 4000 iterations per sec
EXAMPLES Two examples illustrating the use of the program are given. The transient responses obtained for an induction and reluctance motor are compared with results obtained from test recordings. In addition further simulation results are presented to illustrate the effect that transmission systems may have on the behaviour of an induction machine.
Example 1: Induction motor The principal data for the test machine is given in Table 1. Figure 6 shows the responses of speed, per-unit torque and current for the run-up, load application and a short-circuit on the terminals of this motor. The results obtained from test relate to a reduced voltage condition, of 87% nominal voltage, and correspond to the run-up from standstill only. I00 O0
~.-I iApplicationof load
l,: ~. Runup
•~~ ~~ sooo
"-
I
"~k
I i
(o) iTerminal3-ph, sc.
ifault removed
:d"
-
000
:
} (b)
-~ ~
4°°t I Il ~
~ ~ ooor llrl o"
f
,
,
,
,
L~
4oo~ 8 O0
/
• -Testpoints
ooo
±~K
" 'i i
i
I
i' ,
i!" ll'",
,
i
I'l
(c)
I
300 ]
I
!1|1~~11111111,,"":
J,,IIUIHI~,,,,,, ~: ooo |tWt~NINWWIIIUttI~k,,..... L,,,,,,,,,,,.,,,,,,,,,' , ,,,
(d)
.................................
''"" ...................... i l',ll~m,,""..............................
3oo
Test points [e)
-~
ooo O0
Fig. 6. Inductionmotor transient response.
277
Simulation of the transient response of singly-excited electrical machines Table 1. Principal data for induction motor Rating: Stator
3500 h . p .
6 K.V.
resistance Rss
50 Hz
pole
12
..........
..
Rotor resistance (start) Rrr ...... Stator leakage reactance Xss ...... Rotor leakage reactance (start) Xrr Magnetising reactance
..
Xm . . . . . . . . . .
Rectangular rotor slots dimensions . . . . Inertia constant KVA/KWsec . . . . . . . .
0.0068
pu
0.0176 0.096
pu pu
0.I05
pu
2.89
pu 7ram
25ram x 1.216
The effect of neglecting the machine nonlinearities in the formulation is shown in Fig. 7 for an 880 h.p. induction motor pump drive. These curves relate to the speed recovery-transient when the machine is subjected to a terminal short-circuit of 22 msec duration. An important aspect of the simulation of induction machines is concerned with the prediction of the speed recovery of the machine following severe system voltage fluctuations or faults. A variety of system conditions[2] such as the effect of fault-level at the point of machine connection and the fault-clearing time effect the behaviour of the machine and must be accounted for in the simulation. The effect of progressively reducing the ratio of fault-level to machine capacity is illustrated in Fig. 8. In this way a simulation may be used to lead to an estimation of the critical fault-level below which the motor recovery following a system fault is unlikely. Figure 8 also illustrates the effect of a range of fault-clearing times on the motor recovery. Ideally this effect should be studied in close relation to the possible variations in fault-levels. 48
Fault cleared 16
32
.o. N B
16
4
O0
0.22 0.4
0.8
I.'2
116
O0
I
2
S
i 4
3
i 5
S
Fig. 7. Speed recovery of induction motor pump drive. ©, Test curve; A, non-linear model; x, linear model.
Fig. 8. Speed recovery of an induction motor water pump drive for various fault infeed conditions and clearing times. ©
Infeed 10 MVA
A x [] •
20 30 30 30
MVA MVA MVA MVA
Fault clearance 0.22 s 0.22 s 0.22 s 0.40 s 0.10 s
Example 2: Reluctance machine The data for the test machine is given in Table 2, and Figs. 9-11 relate to various loading conditions imposed upon the machine as it runs up from standstill with zero initial conditions. In each case a computed and a measured result are given for comparison. In all cases the angle between the rotor direct axis and the rotating frame, defined by the peak of the stator mmf wave of phase "a", is taken as zero degrees at the moment of switching on the supply to the machine. Table 2. Principal data for reluctance motor Rating:
3 h.p.;
415 VL;
50 Hz;
4
pole
resistance Rss Rotor resistances Rdr
........
0.0469
........
0.0413
pu pu
Rqr
........
0.0533
pu
........
Stator
Magnetising inductances Lmd
....
Lmq
....
pu 0.00084 pu 0.0034 pu 0.00102 pu 0.0033 pu 0.00056 pu
KVA/KWsec
....
0.0257
Stator
inductances Lds
Lqs . . . . . . . .
Rotor inductances Ldr
........
Lqr
........
Inertia constant
0.0035
278
J. R. SMm~ et al. :5
o.
3o r
~--~\
E vlOO
80 i ~ S p e e d
160
240
k,,...,.~/
rds/sec
Fig. 9(a)
0~ v
80
16Q
240
Speed, rds/sec
Fig. 9(b).
c~
Or
f
6
32
48
6'4
Time, sec x 10
Fig. 10(a).
Time,
sec
Fig. lO(b)
O0
16
3
Time, s e c x 10
'
Fig. IRa).
Time, sec
Fig, ll(b).
Figure 9 illustrates the nature of the torque-speed curve for the unloaded machine with no added inertia. This is a severe test for a reluctance machine and if the machine is stable for this condition it will in general be stable for load conditions within the output capacity of the machine. Figure 10 shows the torque-time curve for the machine as it runs up against its nominal full-load. For this test the inertia of the load was six times that of the rotor. Figure 11 relates to a run-up against 1.33 x nominal full load with the same inertial conditions. In this case the machine fails to
Simulation of the transient response of singly-excited electrical machines
279
synchronise and pole-slipping occurs. The computed results are shown in Figs. 9(a), 10(a) and 1l(a) and the measured results in Figs. 9(b), 10(b) and 1l(b). The measurement equipment used for these tests had an upper frequency limit of 55 Hz. Due to this, the double frequency components, correctly present in the computed results for the on-load machine simulations, do not appear in the measured results. D I S C U S S I O N AND C O N C L U S I O N S
The method described in the paper, for determining the transient behaviour of singly-excited electrical machines, leads to accurate simulations of response to various balanced disturbance conditions. The simulated results, compared wherever possible with those obtained from practical tests for a representative range of machine sizes, substantiate the simple approach adopted. The program may be used with equal convenience for instructional purposes or as an aid in the prediction of individual machine behaviour when subjected to certain loading and system conditions. REFERENCES 1. S. S. Kaisi and B, Adkins, Transient stability of power systems containing both synchronous and induction machines. Proc. IEE 118(1) (1971). 2. W. D. Humpage, K. E. Durrani and V. F. Carualho, Dynamic-response analysis of interconnected synchronousasynchronous-machine groups, Proc. lEE, 116(12) (1969). 3. I. R. Smith and A. Sriharan, Transients in induction machines with terminal capacitors. Proc. IEE 115(4) 0968). 4. A.J.O. Cruickshank, A. F. Anderson and R. W. Menzies, Stability of reluctance motors from freely accelerating torque speed cures. Trans. A.I.E.E. (P.A.$.) 91 (1972). 5. V. B. Honsinger, Stability of reluctance motors. Trans. A.I.E.E. (P.A.S.) 91 (1972). 6. P. C. Krause, Simulation of symmetdcai induction machinery: Trans. IEEE FAS-84(11) (1965). 7. G. J. Rogers and J. R. Smith, Mathematical models of synchronous electrical machines. Int. J. Num. Meth. Engng 6 (1972). 8. G. Shackshaft, A general purpose turbo-alternator model. Proc. lEE 110 (1963). 9. P. E. Jones, N. Mullineux, J. R. Reed and R. L. Stoll, Solid rectangular and T-shaped conductors in semi-closed slots. J. Engng Math. 3 (1969). 10. S. A. Swarm and W. Salmon, Effective resistance and reactance of a rectangular conductor placed in a semi-closed slot. Proc. IEE 110 (1963). I I. W. E. Bruges, Evaluation and application of certain ladder-type networks, Proc. Roy. Soc. Edihburgh 62, II (1946).