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Modelling of multimachine transients during unbalanced power system disturbances
ELS EVI E R
ELECTRIC3 POUJER SWSTEITI$ RESEnRCkl
Electric Power Systems Research 42 (1997) 17-20
Modelling of multimachine transients during unbalanced power system disturbances E. Akpinar *, P. Pillay, S.M.A. Sabur Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148, USA Received 14 February 1995
Abstract
The d,q model has been widely used to study single machine transients but it is difficult to use for multimachines during unbalanced faults when the effect of source impedance is taken into account. In this paper, a hybrid model (ABC/dq) of an induction motor has been used in order to analyze multimachines connected to the same bus. In this model, the algorithm is simpler and applicable for the analysis of a motor drive connected to the same bus with the induction motors. © 1997 Published by Elsevier Science S.A.
Keywords: Induction motor; Multimachine transients
1. Introduction
2. Mathematical model of the induction motor
Multimachines connected to a c o m m o n bus through a line exhibit a cross-coupling in the machine models. The solution of these equations in the conventional d,q frame during bus transfer or autoreclose is difficult since the actual abe variables have been transformed. The d,q model has been used for a multimachine connected to the same bus [1] by computing the armature terminal voltage of the machines with a delay of one time step. In this paper, a d q / A B C model previously used to model slip energy recovery drives [2] is modified to a ABC/dq model to study multimachine transients and their interaction during bus transfer. The error of the calculation of terminal voltages one time step late is thus eliminated. The remainder of this paper is organized as follows: Section 2 presents the mathematical model of the induction machine while Section 3 has the mathematical model for the multimachine. Sections 4 and 5 have the results and conclusions respectively.
A hybrid (ABC/dq) model, which retains the actual stator phase variables but transforms the rotor only, is used for the induction applied to the rotor states; then a c o m m u t a t o r transformation is used such that the inductance matrix is not a function of the rotor angle. In the model, it is assumed that the d-axis coincides with phase A of the stator while the q axis leads the d axis by 90 ° (electrical). The nonlinear differential equations of the ABCdq model can be obtained by applying two transformations in cascade. The orthogonal transformation matrix given below is used to transform the balanced three phase rotor windings to a two phase equivalent.
C1 = X/~ b
* Corresponding author. Tel.: + 1 504 2866650. 0378-7796/97/$17.00 © 1997 Published by Elsevier Science S.A. All rights reserved. PIt S0378-7796(96)011 71-6
1
0
-1 2 -1
2 -,~5
2
2
(1)
18
E. Akpinar et aL /Electric Power Systems Research 42 (1997) 17-20
The second transformation matrix transforms rotor variables into a reference frame stationary relative to the stator. This matrix can be obtained as follows: cos 0 Cz--- _ s i n 0
sin 0 cos0
(2)
where 0 is the angle between the rotor phase a and the stator phase A. The impedance matrix, ZM of the ABC/dq model is obtained by performing the following operation:
ZM=~
C02T~ CIT0ZIIz21ZI2Iz22 0 O1~
02
ZM=
0 Mp - Mco~
0 R, + L~p 0 l Mp + ~ corM --5 ~-~ Mp + ~1 Mco~
11 "~-[[al ibl icl fdl
o
Z=
i,3 ZL ZM2
ZL ZL
ZL
ZL
ZM3
Mp
0
- ½MP R r + Lrp --triOs
-~/2~ Mp L ~o ~ l< +
Mp
ra + L~p 0 0 0 0
ZL= 3. Mathematical model of multimachine
(5)
where V1 = [esa esb es,. 0 0]T
(6)
V2 = [es, esb esc 0 0]T
(7)
113 = [esa esb esc 0 0]T
(8)
The voltage vector for one machine contains the source voltages, e,a, e,b, esc and the rotor terminal voltages, Va and Vq as defined above. The rotor terminal voltages are zero for the squirrel cage induction motor. This
(12)
where Z M , ZM2 and ZM3 a r e the impedance matrices of the induction motors in the hybrid reference frame, ZL is the matrix consisting of the source parameters.
where Rs is the sum of the one phase source resistance and the induction motor stator phase resistance, Ls is the sum of the one phase source inductance and the induction motor stator self inductance.
IV1 V2 113]x = Z[II 12 I3]x
(11)
i 3]
ZMI ZL
-
Fig. 1 shows the connection of the motors to the infinite bus through the source impedance. The power source is represented by a Thevenin equivalent (i.e., an independent voltage source in series with an equivalent source impedance). In this paper, only three machines have been considered in the computer simulation program which can be used for any number of machines. The nonlinear model of the system in compact form is written below.
(9) (10)
I3 = [io3
0 Rs + L~p - M I MP -5 ~ C°s Mp + ½
iql]T
/2 =
(3)
where Zll, Z12, Z2t and Z22 are the corresponding component impedance matrix of a 3-phase ABC/abc induction motor in a 'natural' reference frame as given in [2]. I is the identity matrix. CtT, and CZT are the transpose of C 1 and (£'2, respectively. The complete motor equations obtained by performing matrix multiplication in (3) are as follows: R~ + L,p 0
voltage vector repeats itself for the other machines. The current vector for each induction motor is given below. The motor phase currents are defined as the state variables.
(4)
0 rb + Lbp 0 0 0
0 0 r,.+Ld9 0 0
0 0 0 0 0
0 0 0 0 0
(13)
The mechanical motion is incorporated into the torque equation in (14) and the electromagnetic torque is formulated in (15) for an induction motor at the hybrid reference frame. The friction and windage loss is neglected in the torque balance equation in (14). Te+ TL+ eSo.
J dt ~
L~
iA
1/
Ho,chlne i
YI
t/
Mo,chlne 2 Nochlne
Fig. 1. Multimachinesystem.
3
E. Akpinar et al./Electric Power Systems Research 42 (1997) 17-20 s0 ~ Amp
8
1.8
.5
Figure 288 ~ Amp
2a.
Stator
1.5
phase
. . . . qIIIIIIIIIII[IIIIIIIIUlIIIltIIIIIIIUltl$11IIIIIIII$11,11""' -es~ -~88, Figure
2.8
A current
(5 Hp)
r"
.5 I a 1.'5 2b. Stator phase A current
Time(s) 2.'e (20 Hp)
I~88-] Amp
-SO~
-a~88/
,
8
,5
Figure
,
1.8
. r,~
,
~._
2.8
2c. Stator phase A current
(50 Hp)
Amp
888
S
,5
Figure
1,.8
1.5
2.'8
2d. Line current
Fig. 2. (a) Stator phase A current (5 Hp); (b) stator phase A current (20 Hp); (c) stator phase A current (50 Hp); (d) line current.
~ --iaib-~ iqib ~/3idic iqic'~ Te=2e M ( --i~iq+--2 2 2+~)
(15)
TL1 = 0.0001529~02 t
(16)
T m = 0.0005666~o22
(17)
TL3 = 0.001639o9~3
(18)
4. Results In this section, the simulation results for the three induction motors connected to the same bus have been presented. The source has been disconnected from the motors for 134 ms (8 cycles) and reclosed. The action of the circuit breaker in practice is represented by increasing the source resistance from its original value to the value of 1000 ft. This change of resistance is carried out at the zero level of each phase current. The induction motors having the following parameters in the table are connected to the infinite bus voltage of 480 V. The number os poles of each machine is 4.
Fig. 2 a - c show the stator phase A currents of the 5, 20 and 50 Hp machines respectively while Fig. 2(d) shows the line current of phase A. A 1000 ~ resistance is inserted into the line (source resistance is increased from its original value to 1000 f~ at the zero level of the current) at 1.5 s and it is reclosed after 134 ms. During this period the line current is zero, as is shown from the Fig. 2(d). The motor phase currents, as can be observed from Fig. 2 a - c , are not zero during this opening period because some amount of power circulates between the motors. When the motors are reconnected to the power bus, the motor phase currents increase to high values. The current magnitude depends on the phase difference between the source and armature terminal voltage as well as the difference between the magnitudes. In order to minimize the transient currents, the phase difference is detected by a monitor and the reclosion operation is carried out at an appropriate time. Fig. 3 a - c show the rotor speeds of the 5, 20 and 50 Hp machines respectively in terms of electrical rad s - ]. When the three phase source is disconnected at 1.5 s for 134 ms, the rotor speeds decrease, but the reclosing of the power to the motors after 134 ms forces the rotor speed to the previous steady state operating point. During this failure of the power the maximum change of the rotor speed appears in the 5 Hp machine because it has the lowest moment of inertia constant among these three machines. Fig. 4 a - c show the electromagnetic torques of the 5, 20 and 50 Hp machines respectively. When the three phase source is disconnected, the electromagnetic torque does not go to zero because stator currents still circulate between the motors and the rotor currents are non-zero. The electromagnetic torques of all three machines reach high negative values after the reclosing of Rad/s 311t8 200
188
Time
Ll 0.271 0.0789 0.035
Lr 0.277 0.0797 0.035
M 0.213 0.62 0.028
R1 1.73 0.29 0.087
Rr 1.21 0.27 0.228
J 0.07 0.44 2.5
8
.5
Figure
I .'8
3a. R o t o r
1 .'5
speed
2 .~8
(5 Hp)
Rad / s 388 288
18
Time
/ 8
.5
I .'8
3b. R o t o r
I ,'5
speed
(s) 2 .'~
(20 Hp)
4sa R a d / s 208
18
Time B
where L 1 and R 1 a r e the stator self inductance and stator winding resistance respectively. The source resistance and inductance are 0.001 f~ and 0.001 H respectively.
(s)
8
Figure
Hp 5 20 50
19
.~ Figure
i.'8 3c. R o t o r
(s)
*.'5 e.'8 speed (50 Hp)
Fig. 3. (a) Rotor speed (5 Hp); (b) rotor speed (20 Hp); (c) rotor speed (50 Hp).
E. Akpinar et al./ Electric Power Systems Research 42 (1997) 17 20
20
investigate the behaviour of induction machines in the process control industry during the supply disturbances.
N-M
4~ 2 -Ze
ime
.5
e
l.'B 4a. E l e c t r o m a g n e t i c
Figure
1 .'5
torque
(s)
Z.'B (5 Hp)
esa, Csb, Csc
7?
~,~,~ ~,~
-lee -2se
Time Figure
Appendix A. Nomenclature
4b. E l e c t r o m a g n e t i c
torque
(s)
(20 Hp)
Lr
N-M
]i -2
me fl
.5
Figure
t .'8
4c. E l e c t r o m a g n e t i c
I .d5
torque
(s)
2.'8 (50 Hp)
Fig. 4. (a) Electromagnetic torque (5 Hp); (b) electromagnetic torque (20 Hp); (c) electromagnetic torque (50 Hp).
the power to the motors. This high negative torque can be controlled by limiting the motor phase currents with the use of phase detector.
Ls M P P Rr Rs J (D e
0 5. Conclusions In this paper, a hybrid model (ABC/dq) of an induction machine is used to analyze multimachine transients during power system disturbances. The source impedance is included into the model in order to investigate the interaction of the machines. It is observed that, when the power source is disconnected, the large rated machine feeds the power stored in its inertia to the small machines. In the detailed simulation model, the source parameters include the series reactances of the supply transformers employed in the plant. This model is useful to
infinite bus voltages stator phase currents, A rotor direct and quadrature axis current, m per phase rotor self inductance (sum of the per phase motor mutual inductance and rotor leakage inductance), H sum of per phase stator self inductance and source inductance, H x//~/3) (motor mutual inductance per phase), H
d/dt number of poles per phase rotor resistance, sum of per phase stator resistance and source resistance, f~ moment of inertia constant rotor speed (p0) in electrical rad s -l electrical angle between stator and rotor phase A, rad
References [1] J.S. Mayer and O. Wasynczuk, An efficient method of stiffly connected power systems with stator and network transients included, 1EEE/PES 1991 Winter Meeting, New York, February 3 7, 1991. [2] E. Akpinar, P. Pillay and A. Ersak, Starting transients in slip energy recovery induction motor drives Part h Formulation and modelling, IEEE Trans. Energy Con., 7 (1) (1992) 238-244. [3] E. Akpinar and P. Pillay, Bus transfer model of a drive and an induction motor on the same bus using an ABCdq model, IEEE/PES 1994 Summer Meeting.
ELECTRIC3 POUJER SWSTEITI$ RESEnRCkl
Electric Power Systems Research 42 (1997) 17-20
Modelling of multimachine transients during unbalanced power system disturbances E. Akpinar *, P. Pillay, S.M.A. Sabur Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148, USA Received 14 February 1995
Abstract
The d,q model has been widely used to study single machine transients but it is difficult to use for multimachines during unbalanced faults when the effect of source impedance is taken into account. In this paper, a hybrid model (ABC/dq) of an induction motor has been used in order to analyze multimachines connected to the same bus. In this model, the algorithm is simpler and applicable for the analysis of a motor drive connected to the same bus with the induction motors. © 1997 Published by Elsevier Science S.A.
Keywords: Induction motor; Multimachine transients
1. Introduction
2. Mathematical model of the induction motor
Multimachines connected to a c o m m o n bus through a line exhibit a cross-coupling in the machine models. The solution of these equations in the conventional d,q frame during bus transfer or autoreclose is difficult since the actual abe variables have been transformed. The d,q model has been used for a multimachine connected to the same bus [1] by computing the armature terminal voltage of the machines with a delay of one time step. In this paper, a d q / A B C model previously used to model slip energy recovery drives [2] is modified to a ABC/dq model to study multimachine transients and their interaction during bus transfer. The error of the calculation of terminal voltages one time step late is thus eliminated. The remainder of this paper is organized as follows: Section 2 presents the mathematical model of the induction machine while Section 3 has the mathematical model for the multimachine. Sections 4 and 5 have the results and conclusions respectively.
A hybrid (ABC/dq) model, which retains the actual stator phase variables but transforms the rotor only, is used for the induction applied to the rotor states; then a c o m m u t a t o r transformation is used such that the inductance matrix is not a function of the rotor angle. In the model, it is assumed that the d-axis coincides with phase A of the stator while the q axis leads the d axis by 90 ° (electrical). The nonlinear differential equations of the ABCdq model can be obtained by applying two transformations in cascade. The orthogonal transformation matrix given below is used to transform the balanced three phase rotor windings to a two phase equivalent.
C1 = X/~ b
* Corresponding author. Tel.: + 1 504 2866650. 0378-7796/97/$17.00 © 1997 Published by Elsevier Science S.A. All rights reserved. PIt S0378-7796(96)011 71-6
1
0
-1 2 -1
2 -,~5
2
2
(1)
18
E. Akpinar et aL /Electric Power Systems Research 42 (1997) 17-20
The second transformation matrix transforms rotor variables into a reference frame stationary relative to the stator. This matrix can be obtained as follows: cos 0 Cz--- _ s i n 0
sin 0 cos0
(2)
where 0 is the angle between the rotor phase a and the stator phase A. The impedance matrix, ZM of the ABC/dq model is obtained by performing the following operation:
ZM=~
C02T~ CIT0ZIIz21ZI2Iz22 0 O1~
02
ZM=
0 Mp - Mco~
0 R, + L~p 0 l Mp + ~ corM --5 ~-~ Mp + ~1 Mco~
11 "~-[[al ibl icl fdl
o
Z=
i,3 ZL ZM2
ZL ZL
ZL
ZL
ZM3
Mp
0
- ½MP R r + Lrp --triOs
-~/2~ Mp L ~o ~ l< +
Mp
ra + L~p 0 0 0 0
ZL= 3. Mathematical model of multimachine
(5)
where V1 = [esa esb es,. 0 0]T
(6)
V2 = [es, esb esc 0 0]T
(7)
113 = [esa esb esc 0 0]T
(8)
The voltage vector for one machine contains the source voltages, e,a, e,b, esc and the rotor terminal voltages, Va and Vq as defined above. The rotor terminal voltages are zero for the squirrel cage induction motor. This
(12)
where Z M , ZM2 and ZM3 a r e the impedance matrices of the induction motors in the hybrid reference frame, ZL is the matrix consisting of the source parameters.
where Rs is the sum of the one phase source resistance and the induction motor stator phase resistance, Ls is the sum of the one phase source inductance and the induction motor stator self inductance.
IV1 V2 113]x = Z[II 12 I3]x
(11)
i 3]
ZMI ZL
-
Fig. 1 shows the connection of the motors to the infinite bus through the source impedance. The power source is represented by a Thevenin equivalent (i.e., an independent voltage source in series with an equivalent source impedance). In this paper, only three machines have been considered in the computer simulation program which can be used for any number of machines. The nonlinear model of the system in compact form is written below.
(9) (10)
I3 = [io3
0 Rs + L~p - M I MP -5 ~ C°s Mp + ½
iql]T
/2 =
(3)
where Zll, Z12, Z2t and Z22 are the corresponding component impedance matrix of a 3-phase ABC/abc induction motor in a 'natural' reference frame as given in [2]. I is the identity matrix. CtT, and CZT are the transpose of C 1 and (£'2, respectively. The complete motor equations obtained by performing matrix multiplication in (3) are as follows: R~ + L,p 0
voltage vector repeats itself for the other machines. The current vector for each induction motor is given below. The motor phase currents are defined as the state variables.
(4)
0 rb + Lbp 0 0 0
0 0 r,.+Ld9 0 0
0 0 0 0 0
0 0 0 0 0
(13)
The mechanical motion is incorporated into the torque equation in (14) and the electromagnetic torque is formulated in (15) for an induction motor at the hybrid reference frame. The friction and windage loss is neglected in the torque balance equation in (14). Te+ TL+ eSo.
J dt ~
L~
iA
1/
Ho,chlne i
YI
t/
Mo,chlne 2 Nochlne
Fig. 1. Multimachinesystem.
3
E. Akpinar et al./Electric Power Systems Research 42 (1997) 17-20 s0 ~ Amp
8
1.8
.5
Figure 288 ~ Amp
2a.
Stator
1.5
phase
. . . . qIIIIIIIIIII[IIIIIIIIUlIIIltIIIIIIIUltl$11IIIIIIII$11,11""' -es~ -~88, Figure
2.8
A current
(5 Hp)
r"
.5 I a 1.'5 2b. Stator phase A current
Time(s) 2.'e (20 Hp)
I~88-] Amp
-SO~
-a~88/
,
8
,5
Figure
,
1.8
. r,~
,
~._
2.8
2c. Stator phase A current
(50 Hp)
Amp
888
S
,5
Figure
1,.8
1.5
2.'8
2d. Line current
Fig. 2. (a) Stator phase A current (5 Hp); (b) stator phase A current (20 Hp); (c) stator phase A current (50 Hp); (d) line current.
~ --iaib-~ iqib ~/3idic iqic'~ Te=2e M ( --i~iq+--2 2 2+~)
(15)
TL1 = 0.0001529~02 t
(16)
T m = 0.0005666~o22
(17)
TL3 = 0.001639o9~3
(18)
4. Results In this section, the simulation results for the three induction motors connected to the same bus have been presented. The source has been disconnected from the motors for 134 ms (8 cycles) and reclosed. The action of the circuit breaker in practice is represented by increasing the source resistance from its original value to the value of 1000 ft. This change of resistance is carried out at the zero level of each phase current. The induction motors having the following parameters in the table are connected to the infinite bus voltage of 480 V. The number os poles of each machine is 4.
Fig. 2 a - c show the stator phase A currents of the 5, 20 and 50 Hp machines respectively while Fig. 2(d) shows the line current of phase A. A 1000 ~ resistance is inserted into the line (source resistance is increased from its original value to 1000 f~ at the zero level of the current) at 1.5 s and it is reclosed after 134 ms. During this period the line current is zero, as is shown from the Fig. 2(d). The motor phase currents, as can be observed from Fig. 2 a - c , are not zero during this opening period because some amount of power circulates between the motors. When the motors are reconnected to the power bus, the motor phase currents increase to high values. The current magnitude depends on the phase difference between the source and armature terminal voltage as well as the difference between the magnitudes. In order to minimize the transient currents, the phase difference is detected by a monitor and the reclosion operation is carried out at an appropriate time. Fig. 3 a - c show the rotor speeds of the 5, 20 and 50 Hp machines respectively in terms of electrical rad s - ]. When the three phase source is disconnected at 1.5 s for 134 ms, the rotor speeds decrease, but the reclosing of the power to the motors after 134 ms forces the rotor speed to the previous steady state operating point. During this failure of the power the maximum change of the rotor speed appears in the 5 Hp machine because it has the lowest moment of inertia constant among these three machines. Fig. 4 a - c show the electromagnetic torques of the 5, 20 and 50 Hp machines respectively. When the three phase source is disconnected, the electromagnetic torque does not go to zero because stator currents still circulate between the motors and the rotor currents are non-zero. The electromagnetic torques of all three machines reach high negative values after the reclosing of Rad/s 311t8 200
188
Time
Ll 0.271 0.0789 0.035
Lr 0.277 0.0797 0.035
M 0.213 0.62 0.028
R1 1.73 0.29 0.087
Rr 1.21 0.27 0.228
J 0.07 0.44 2.5
8
.5
Figure
I .'8
3a. R o t o r
1 .'5
speed
2 .~8
(5 Hp)
Rad / s 388 288
18
Time
/ 8
.5
I .'8
3b. R o t o r
I ,'5
speed
(s) 2 .'~
(20 Hp)
4sa R a d / s 208
18
Time B
where L 1 and R 1 a r e the stator self inductance and stator winding resistance respectively. The source resistance and inductance are 0.001 f~ and 0.001 H respectively.
(s)
8
Figure
Hp 5 20 50
19
.~ Figure
i.'8 3c. R o t o r
(s)
*.'5 e.'8 speed (50 Hp)
Fig. 3. (a) Rotor speed (5 Hp); (b) rotor speed (20 Hp); (c) rotor speed (50 Hp).
E. Akpinar et al./ Electric Power Systems Research 42 (1997) 17 20
20
investigate the behaviour of induction machines in the process control industry during the supply disturbances.
N-M
4~ 2 -Ze
ime
.5
e
l.'B 4a. E l e c t r o m a g n e t i c
Figure
1 .'5
torque
(s)
Z.'B (5 Hp)
esa, Csb, Csc
7?
~,~,~ ~,~
-lee -2se
Time Figure
Appendix A. Nomenclature
4b. E l e c t r o m a g n e t i c
torque
(s)
(20 Hp)
Lr
N-M
]i -2
me fl
.5
Figure
t .'8
4c. E l e c t r o m a g n e t i c
I .d5
torque
(s)
2.'8 (50 Hp)
Fig. 4. (a) Electromagnetic torque (5 Hp); (b) electromagnetic torque (20 Hp); (c) electromagnetic torque (50 Hp).
the power to the motors. This high negative torque can be controlled by limiting the motor phase currents with the use of phase detector.
Ls M P P Rr Rs J (D e
0 5. Conclusions In this paper, a hybrid model (ABC/dq) of an induction machine is used to analyze multimachine transients during power system disturbances. The source impedance is included into the model in order to investigate the interaction of the machines. It is observed that, when the power source is disconnected, the large rated machine feeds the power stored in its inertia to the small machines. In the detailed simulation model, the source parameters include the series reactances of the supply transformers employed in the plant. This model is useful to
infinite bus voltages stator phase currents, A rotor direct and quadrature axis current, m per phase rotor self inductance (sum of the per phase motor mutual inductance and rotor leakage inductance), H sum of per phase stator self inductance and source inductance, H x//~/3) (motor mutual inductance per phase), H
d/dt number of poles per phase rotor resistance, sum of per phase stator resistance and source resistance, f~ moment of inertia constant rotor speed (p0) in electrical rad s -l electrical angle between stator and rotor phase A, rad
References [1] J.S. Mayer and O. Wasynczuk, An efficient method of stiffly connected power systems with stator and network transients included, 1EEE/PES 1991 Winter Meeting, New York, February 3 7, 1991. [2] E. Akpinar, P. Pillay and A. Ersak, Starting transients in slip energy recovery induction motor drives Part h Formulation and modelling, IEEE Trans. Energy Con., 7 (1) (1992) 238-244. [3] E. Akpinar and P. Pillay, Bus transfer model of a drive and an induction motor on the same bus using an ABCdq model, IEEE/PES 1994 Summer Meeting.