2011 International Conference on Recent Advancements in Electrical, Electronics and Control Engineering
Transient Stability Studies in SMIB System with Detailed Machine Models Dr. S. Kalyani
M. Prakash
Professor Dept. of Electrical & Electronics Engineering Kamaraj College of Engineering & Technology Virudhunagar - 626001 kal_yani_79@yahoo.co.in
Assistant Professor Dept. of Electrical & Electronics Engineering Kamaraj College of Engineering & Technology Virudhunagar – 626001 prakash13688@gmail.com
G. Angeline Ezhilarasi Research Scholar Department of Electrical Engineering Indian Institute of Technology Madras Chennai - 600036 angel.ezhil@gmail.com
I.
— This paper presents a detailed transient stability Abstract analysis using various machine models. It highlights on the accuracy of the approximated models of synchronous machines for power system transient stability analysis. The transient stability simulation is performed on a SMIB system considering four different detailed models of generators and the classical model. The exciter dynamics is represented by IEEE Type 1 Exciter model in the detailed analysis. Exclusive comparison of performance using various models is done. Transient Stability Analysis and Critical Clearing Time (CCT) is evaluated for specified fault at various locations. locations.
I NTRODUCTION NTRODUCTION
Stability of power systems continues to be of a major concern in system operation [1]. This arises from the fact that in steady state, the average electrical speed of all the generators must remain the same anywhere in the system. This is termed as the synchronous operation of a system. Any disturbance small or large can affect the synchronous operation [2]. The stability of a system determines whether the system can settle down to a new or original steady state after the transient disappears.
Keywords: transient stability, SMIB system, exciter, detailed models, critical clearing time
Although stability is an integral property of the system, for the purpose of analysis, it is divided into two classes, viz., Steady-State or Small Signal Stability and Transient Stability [3]. A power system is ‘ steady state stable’ stable’ for a particular stead y state s tate operating co ndition if, i f, following a ny small disturbance, it reaches a steady state operating condition which is identical or close to the pre-disturbance operating condition. A small disturbance is the one for which system dynamics can be analyzed from linearized equations. The small (random) changes in the load or generation belong to the class of small disturbances. A power system is ‘transiently stable’ for a particular steadystate operating condition and for a particular large disturbance or sequence of disturbances if, following that disturbance(s), it reaches an acceptable steady-state operating condition. Faults like three phase short circuits, resulting in sudden voltage dip are large disturbances and require remedial action in the form of clearing of fa ult.
Nomenclature Nomenclature
δ ω ω s
Rotor angle in degrees Angular speed of machine in rad/sec Synchronous speed of machine in rad/sec F System frequency in Hz H Inertia constant of the machine in sec Pm Mechanical power input Pe Electrical power output R s Stator resistance of the synchronous machine Vt Terminal Voltage of the synchronous machine Xd(Xq) Direct (Quadrature) axis synchronous reactance X’d(X’q) Direct (Quadrature) axis transient reactance E’d(E’q) Direct(Quadrature) axis components of transient internal voltage Id(Iq) Direct (Quadrature) axis components of terminal current I1d(I2q) Current Components of the Direct (Quadrature) axis damper circuit. ( ) Linkage Components of the Direct ψ 1d ψ 1d 2q 2q Flux (Quadrature) axis damper circuit T’do(T’qo)Open-Circuit direct (Quadrature) axis transient time constant T”do(T”qo)Open-Circuit direct (Quadrature) axis subtransient time constant Efd Exciter output voltage (applied to generator field) Ifd Generator field current SE(Efd) Exciter Saturation function R f f Exciter Rate Feedback VR Output Voltage of the Regulator Vref Reference Voltage TA,TF,TE Time constant of amplifier, stabilizing transformer and exciter. K F,K A, K A, E Gain of stabilizing transformer, amplifier and exciter TR Regulated input filter time constant
It is important to note that, steady-state stability is a function of only the operating condition, whereas transient stability is a function of both the operating condition and the disturbance(s). This complicates the analysis of transient stability considerably. Transient Stability requires repeated analysis for different disturbances that are being considered. Transient stability analysis in a power system is concerned with the power system's ability to remain synchronously stable following a serious disturbance such as a three-phase short-circuit. Present day transient stability analysis of power system is mainly performed by simulations. Calculation of critical clearing time (CCT) for a fault is a major assessment in stability studies. This paper presents the mathematical formulation of different models of synchronous machine used in the transient stability simulation. Synchronous machines are, in general, expressed by d, q axis Park model in the simulations [4]. There are several types of approximated models. In this paper, we have considered the simple
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2011 International Conference on Recent Advancements in Electrical, Electronics and Control Engineering '
classical model and four detailed models of machine representation. The detailed model includes Model 1.0 (Flux-Decay Model) having no damper circuit, Model 1.1 (Two-Axis Model) with one q-axis damper circuit, Model 2.1 having one damper circuit both in d-axis and q-axis, Model 2.2 which has one d-axis damper circuit and two qaxis damper circuits. Stator transients are neglected in all of the four models considered. The exciter dynamics is represented by the IEEE Type 1 Exciter model in the simulation [5]. The IEEE Type 1 excitation system represents a majority of the excitation systems in service and is widely used. The accuracy of different synchronous machine models are evaluated by performing simulations for three phase short circuit fault in a model SMIB system. II.
dE q dt
=
1 ' d 0
T
( − E − ( X ' q
=
dt
− X d' ) ( I d − I1d ) + E fd )
'
dE d
d
1 ' q0
T
( − E
(3)
+ ( X q − X q' ) ( I q − I 2 q ) )
' d
(4)
d ψ 1 d
1
=
dt
'' d 0
T
( −ψ
1d
+ E q' −
(X
− X ls ) I d
' d
)
(5)
d ψ 2 q
POWER SYSTEM MODELS
dt
This section describes in brief the differential equations governing the dynamic behavior of the system generator and the excitation system [1-6].
=
1 '' q0
T
( −ψ
2q
− E d' − ( X q' − X ls ) I q ) (6)
where
A. Generator Model
I1d =
The highest model (Model 2.2) of the synchronous machine includes the field circuit and the damper circuit in the dq coordinates. With certain valid approximations it reduces to classical model
X d '− X d ''
( X d '− X ls )
I 2 q =
A.1 Model 2.2 The dynamic circuit of the synchronous machine in the sub-transient mode is shown in Fig. 1. This is the sixth order machine model, which includes field circuit (fd) and one damper circuit (1d) on d - axis and two damper circuits (1q, 2q) on the q-axis [6]. This is the highest accurate model considering dynamics in the rotor circuits of both d-axis and q-axis, neglecting only the effect of stator transients. The differential equations governing this model are represented by Equations (1) to (6). This model is preferred to s tudy the damping contribution of power system stabilizers used for transient stability improvement.
2
(ψ + ( X 1d
− Xls ) Id − E q' ) (7)
X q '− X q ''
( X
' d
q '− X ls )
2
(ψ
2q
+ ( X q' − X ls ) I q + E d' )
ψ1d =
X d '− X ls ⎛
( X d ''− X ls ) ' ⎞ E q ⎟ ⎜ψ d + X d '' I d − ( X d '− X d '') ⎝ ( X d '− X ls ) ⎠ (9)
ψ 2q =
X q '− X ls ⎛
( X
⎜ψ q + X q '' I q ⎝
⎜ q '− X q '' )
( X + (X
⎞ E d ' ⎟ ⎟ ' X − ) q ls ⎠
q
''− X ls )
A.2 Model 2.1 This is the fifth order model which includes two rotor circuits on d - axis (a field circuit and a damper circuit) and one damper circuit (1q) on q-axis. The differential equations governing this model are represented by equations (1) to (5). The I2q component in equation (4) is absent as the dynamics created by 2q damper circuit are neglected in this model. It is also a sub-transient model, the synchronous machine dynamic circuit being same as shown in Fig. 1 Fig. 1 Synchronous machine sub-transient dynamic circuit
d t
dω dt
= ω − ω s
=
π f H
( Pm − P e )
A..3 Model 1.1 This model includes only the rotor circuit on d – axis (field winding, fd ) and one rotor circuit on q – axis (damper winding, 1q). It is the simpler and commonly used detailed model representation in stability studies. It is a fourth order model including equations from (1) to (4) neglecting the quantities pertaining to 1d and 2q circuits. Exciter dynamics can be easily incorporated in this model. This model is also referred to as two-axis model in literature. The synchronous machine dynamic circuit is the
(1)
(8)
(10)
d δ
(2)
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2011 International Conference on Recent Advancements in Electrical, Electronics and Control Engineering
same as shown in Fig. 1 with a single modification. As this model is a transient model, the X d’’ in the dynamic circuit of Fig. 1 is replaced by transient reactance, X d’, of the machine.
but with some modifications can also represent the static exciters. The first block (T R ) represents the voltage setting circuit, the second block (K A) represents the automatic voltage regulator, the third (K E) with attendant saturation function represents the self-exciter main exciter and finally the derivative stabilizing loop (K F). In most instances, the time constant T F is small enough to be neglected. The dynamics of exciter, as seen from Fig. 2, is represented by differential equations given by equations (11) to (13). The output of the regulator, V R , is limited. It is to be noted that the limits on VR also imply limits on E fd. The saturation function SE = f (Efd) represents the saturation of exciter.
A.4 Model 1.0 This model is a further approximation of the twoaxis model described above. This model is derived based on the assumption that the term T q0’ is sufficiently small and hence the 1q damper dynamics on the q-axis also being neglected. Therefore, this is a third order model containing only the field circuit (fd) dynamics on the d – axis and none in the q – axis. The system behavior in this model representation is governed by equations (1) to (3). The machine dynamic circuit being the same as Fig. 1, with X d’’ replaced by X d’. This model is suitable for operational planning studies like selective contingency analysis, where the time factor is more crucial. This model is commonly referred as one-axis flux decay model in literature.
dE fd dt d R f dt
A.5 Classical Model
=
=
1 T E
( − ( K
E
+ S E ( E fd ) E fd ) + V R )
(11)
1 ⎛ T F
⎞ K F − + R E ⎜ f fd ⎟ T F ⎠ ⎝ (12)
It is the simplest model of the synchronous machine. In this model, the synchronous generator is represented by a constant voltage source behind d-axis transient reactance. In this model, the voltage is assumed constant, only its phase angle changes. It is obvious that the changes in the flux linkages and the transient saliency are neglected. This model is very useful for the first swing transient stability study, involving a short period of study say one second or less. It is a second order model governed by two dynamic state variables ( δ, ω ) represented by equations (1) and (2). Exciter dynamics cannot be incorporated in this model.
dV R dt
=
1⎛
⎞ KAKF Efd + KA (Vref − Vt ) ⎟ ⎜ −V R + KARF − T A ⎝ TF ⎠ (13)
with the limit constraints
V Rm in ≤ V R ≤ V Rm a x III.
;
E
m in fd
≤ E fd ≤ E
m ax fd
SIMULATION RESULTS
Fig. 3 shows a SMIB model system used for simulation study, in which a synchronous generator is delivering power to infinite bus through a double circuit transmission line taken from [2]. The generator and exciter parameters and system data are also shown in Fig. 3. A remote power station connected to a load centre through a long transmission line can be approximated by a SMIB system. This simplification of a SMIB system enables one to gain insights into the dynamic behavior of a synchronous generator. The simulation is performed for a 3LG fault (three phase short circuit fault) occurring at different locations on line L2 at t = 1 sec and the fault is cleared by tripping the faulted line. Table I shows the initial steady state (pre-fault) values obtained for generator and exciter variables determined for different dynamic machine models for the given system parameters. The generator variables and exciter variables, describing the system dynamic behavior at any time instant, are called the state variables. The pre-fault values for these state variables are obtained by equating the system dynamic equations for the corresponding model to zero and solving them with known data. The numerical integration technique adopted for solving the system dynamic equations in each model is the 4th order RK method.
B. Excitation System Model The excitation system of synchronous generators and motors has an extreme effect on the system stability. The main objective of the excitation system is to control the field current of the synchronous machine. The field current is controlled so as to regulate the terminal voltage of the machine. The basic schematic of IEEE Type 1 Excitation system is shown in Fig. 2.
Fig. 2 IEEE Type I Excitation System
The IEEE Type 1 excitation system [5] represents a majority of excitation systems and is widely used in dynamic study. It essentially represents the rotating exciters
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2011 International Conference on Recent Advancements in Electrical, Electronics and Control Engineering
1sec in the middle of the line L 2 (X = 50%) and cleared by line tripping. As seen from Fig. 4, when the fault is cleared at 1.170 sec, the system loses stability. Hence, the Critical Clearing Time is predicted as 1.160 sec, where the system is oscillatory stable. The responses of the power angle and terminal voltage of the generator obtained by numerical simulation for each detailed model stated above is shown in Fig. 5. The curves in Fig. 5 are drawn for the 3LG fault located at X = 10% from send end of Line L 2 and fault clearing time being the Critical Clearing time as shown in Table II for the concerned model. Fig. 6 and Fig. 7 shows the time response of generator dynamic variables ( δ, ω , E d’, Eq’) and exciter dynamic variables (V t, Efd, R f, Vr ) respectively for the model SMIB system obtained by simulation for a 3LG fault at X=25% from sending end bus of Line L2. The fault clearing time is assumed as the Critical Clearing Time (CCT) equal to 1.140 sec, seen from T able II.
Fig. 3 SMIB Model System
Generator Data
Xd = 1.7572;Xd’ = 0.4245; Xd” = 0.23;Xq = 1.5845;Xq’ = 1.04; Xq” = 0.50; Tdo’= 6.66s; Tdo” = 0.03s;Tqo’= 0.44s;Tqo” = 0.06s;R a = 0.00327;H = 3.542s; Exciter Data
K A=400;K E=1.00;K F = 0.05;T A = 0.025;T E = 0.35;T F = 0.35; Efdmax = 6.0;Efdmin = -6.0;V r max = 0.2;V r min = -0.2; Saturation Characteristics
Asat = 0.03; Bsat = 0.693;K sd = K sq =1;S(Efd) = Asat e (All quantities are specified in pu)
Bsat Efd
Table I Initial Conditions (Pre-fault Values) of the SMIB System Generator Variables
Classical
Model 1.0
Model 1.1
Ed’ = 1.0864, δ0 = 34.550
Vd = 0.6726, V q = 0.8063, I d = 0.3823, Iq = 0.4253 Ed’ = 0.2315, E q’ = 0.9700 , δ0 = 61.480, Pe0 = 0.6011 E’ = 1.0882
Model 2.1
Model 2.2
E’’ = 1.0649, ψ 1d = 0.8077
ψ 2q = 0.6738
Exciter Variables
Efd = 1.4795, Ifd = 1.4795 Vr = 1.6073, R f = 0.2114 Vref = 1.0540
Fig. 4 Swing Curves of the Model SMIB System (Classical Model)
Table II shows the results of the transient stability simulation performed on the model SMIB system shown in Fig. 3 for different machine models considered. The fault is assumed to occur at various locations from the sending end bus of Line L2. The critical clearing time (CCT), as shown in Table II, is obtained by repeating the simulation for many fault clearing times and observing the nature of the swing curve. Critical clearing time gives a picture of the maximum allowable time up to which the fault can persist in the system, without creating loss of synchronism. At a particular fault clearing time instant, the rotor oscillations of the system generator increases continuously, indicating loss of synchronism. One time step before this fault clearing time is recorded as Critical Clearing Time (CCT). All the values in Table I and Table II ar e represented in pu.
Fig. 5 Swing Curves and Terminal Voltage Curves of M odel SMIB System (with Detailed Models)
The swing of rotor angle oscillations simulated for different fault clearing times considering the simple classical machine model without exciter dynamics is shown in Fig. 4. The three phase fault is applied at a time instant of
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2011 International Conference on Recent Advancements in Electrical, Electronics and Control Engineering
Table II Transient Stability Simulation Result of SMIB System under Different Machine Models Fault
Model Type
Loc
CCT (secs)
(X)
0 = X
% 0 1 = X
% 0 5 = X
% 0 8 = X
% 0 9 = X
δ
ω
Ed’
Eq’
(deg)
(r/s)
(pu)
(pu)
ψ 1d (pu)
Exciter Variables at CCT
ψ 2q (pu)
Efd
R f
Vr
(pu)
(pu)
(pu)
Classical
1.100
40.73
316.55
-
-
-
-
-
-
-
Model 1.0
1.080
65.22
316.02
-
0.9641
-
-
-1.9314
0.2231
-18.075
Model 1.1
1.150
78.63
318.15
0.2559
0.9735
-
-
-6.0000
0.24
-29.129
Model 2.1
1.130
72.46
317.35
0.2497
0.9715
0.8061
-
-6.0000
0.2487
-42.925
Model 2.2
1.110
67.65
316.55
0.2815
0.9713
0.8063
-0.681
-6.0000
0.2383
-36.592
Classical
1.120
42.73
316.74
-
-
-
-
-
-
-
Model 1.0
1.110
67.91
316.40
-
0.9749
-
-
3.7663
0.2045
8.8071
Model 1.1
1.100
67.84
316.37
0.2023
0.9727
-
-
3.7709
0.2045
8.8692
Model 2.1
1.150
73.16
317.06
0.195
0.9743
0.8082
-
5.3815
0.2005
9.1456
Model 2.2
1.120
68.58
316.41
0.1567
0.9733
0.8083
-0.663
4.5945
0.2022
10.341
1.140
44.64
316.84
-
-
-
-
-
-
-
Model 1.0
1.130
69.13
316.37
-
0.9748
-
-
3.4712
0.2057
5.9907
Model 1.1
1.140
71.57
316.66
0.2013
0.9738
-
-
3.7324
0.2052
5.3151
Model 2.1
1.210
79.64
317.28
0.1946
0.9758
0.8082
-
5.0927
0.2022
3.6256
Model 2.2
1.170
72.82
316.61
0.1464
0.9747
0.8083
-0.661
4.6820
0.2028
5.8373
Classical
1.160
46.65
316.92
-
-
-
-
-
-
-
Model 1.0
1.150
70.25
316.32
-
0.9754
-
-
3.4423
0.206
4.6614
Model 1.1
1.170
73.73
316.64
0.1998
0.9745
-
-
3.7022
0.2056
3.6255
Model 2.1
1.250
82.30
317.11
0.1941
0.9767
0.8082
-
4.7843
0.2034
1.1538
Model 2.2
1.190
72.82
316.33
0.1448
0.9751
0.8083
-0.661
4.542
0.2034
4.4962
Classical
1.150
47.11
316.90
-
-
-
-
-
-
-
Model 1.0
1.140
69.63
316.33
-
0.9759
-
-
3.7921
0.2049
6.2487
Model 1.1
1.150
71.90
316.56
0.1978
0.974
-
-
4.0787
0.2043
5.6542
Model 2.1
1.220
79.90
317.17
0.1910
0.9761
0.8083
-
5.5957
0.2009
4.1777
Model 2.2
1.170
71.84
316.39
0.1423
0.9747
0.8083
-0.660
4.9753
0.2019
6.7031
Classical
1.130
44.92
316.77
-
-
-
-
-
-
-
Model 1.0
1.120
68.27
316.31
-
0.9757
-
-
3.997
0.204
8.5919
Model 1.1
1.120
69.40
316.45
0.1975
0.9733
-
-
4.2189
0.2035
8.3634
Model 2.1
1.170
74.76
317.03
0.191
0.9749
0.8083
-
5.8725
0.1994
8.5393
Model 2.2
1.140
70.03
316.44
0.1449
0.9739
0.8084
-0.661
5.1386
0.2010
9.7602
Classical % 5 2 = X
Generator Variables at CCT
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2011 International Conference on Recent Advancements in Electrical, Electronics and Control Engineering
taking a model SMIB system. The performance and accuracy of each detailed model are compared in terms of critical clearing time. The classical model gives more error, because the decay of field flux is neglected in this model. Model 2.2, although more accurate model including the effect of all damper circuits, has resulted in more unstable responses than Model 1.1 showing a contradictory result. Hence, it is concluded that Model 1.1 (Two Axis Model) is more suitable to analyze transient stability of synchronous machines.
REFERENCES
Fig. 6 Response of Generator Variables ( δ, ω , Ed’, Eq’) of SMIB System (Fault at X=25%, cleared at CCT=1.140 sec)
Fig. 7 Response of Exciter Variables (V t, Efd, R f, Vr ) of SMIB System (Fault at X=25%, cleared at CCT=1.140 sec)
IV. CONCLUSION This paper presents the mathematical formulation of various approximated models of detailed machine representation. The dynamics of exciter is incorporated in all the detailed models. The excitation system considered in this paper is IEEE Type 1 excitation system. The dynamics of a synchronous generator is illustrated by
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[4]
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[5]
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