Abstract: Two of the most frequent causes of power system voltage instability are voltage recovery following faults, and voltage collapse during transient swings.
Voltage Stability and Voltage Recovery: Load Dynamics and Dynamic VAR Sources A. P. Sakis Meliopoulos, George Cokkinides, and George Stefopoulos School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia 30332 - 0250
Abstract: Two of the most frequent causes of power system voltage instability are voltage recovery following faults, and voltage collapse during transient swings. We propose methods for the identification and mitigation of these phenomena by appropriate usage of control devices such as dynamic VAR sources. We also propose a method for the reliable simulation of these phenomena, called “the time continuation method”. This method combines the efficiency of power flow techniques and the capabilities of transient stability models. It is based on a quadratic model of the power system, including generators, voltage regulators and dynamic loads consisting mainly of induction motors. This model explains recently observed voltage stability and voltage recovery phenomena. Numerical experiments with visualizations are presented for the purpose of quantifying the phenomena and identifying the major affecting parameters. It is shown that proper combinations of static and dynamic reactive power sources can alleviate the risk of voltage instability. Finally, we propose a methodology for selecting the location and size and mix of static and dynamic VAR resources for the purpose of alleviating the risk of voltage instability and slow voltage recovery. Index Terms— Dynamic load modeling, Induction motor model, Load flow analysis, Voltage recovery, Quadratic Integration.
I. INTRODUCTION
T
WO phenomena that may lead to voltage instability are: (a) the recovery rate of the voltage following a fault and (b) the voltage collapse at the center of oscillation during a transient swing. It is well-known that in both phenomena, the voltage recovery may be delayed by load dynamics (such as the dynamics of induction motors, etc.), especially when not enough fast reacting reactive resources (dynamic VAR sources) exist [1-9]. At the same time, the slow voltage recovery may trigger the operation of protective relays disconnecting electric loads and subsequent creation of overvoltages. The phenomena are well known to utilities and they are typically studied with full scale transient simulations. Traditional power flow methodologies do not capture these phenomena. Real time tools are almost exclusively based on traditional load flow models and they are not capable of capturing the dynamic nature of voltage recovery phenomena. This practice leads to discrepancies between the analytical models and the real behavior of the system. The issue of load modeling and the effects of dynamic loads on voltage phenomena have been studied to a significant extent in the literature [1-18]. In [1] the issues of voltage dips
142440178X/06/$20.00 ©2006 IEEE
124
in 3-phase systems after symmetric or asymmetric faults and the accurate modeling of voltage recovery are addressed. In [2,3] the voltage recovery phenomena and the effect of induction motor loads are studied from a practical point of view, based on actual events from utility experience. References [4,5] study the voltage recovery of wind turbines after short-circuits. The issue of mitigating the delayed voltage recovery using fast VAR resources is addressed in [6-8]. The impact of induction motor loads on voltage phenomena has also been studied on a more general research basis. Reference [9] addresses the topic of voltage oscillatory instability caused by induction motors, in particular in isolated power systems, while [10] refers to the impact of induction motor loads in the system loadability margins and in the damping of inter-area oscillations. Finally, references [11-20] are indicative of current research approaches and issues in induction motor load modeling in power systems. This paper introduces a new approach to the study of voltage recovery phenomena that can take into consideration the key dynamic characteristics of the load, while avoiding the task of performing full-scale, time-domain, dynamic simulations. The approach basically uses an advanced load flow modeling for the electric network, which is assumed to operate at quasi-steady state, coupled with quasi-dynamic models of generating units and dynamic loads. This allows a more realistic representation of load dynamics. While the methodology is capable of handling various classes of dynamic electric loads, we focus our attention on induction motor loads, which represents the majority of electric loads, and specifically of dynamic loads. The induction motor nonlinearities depend on the slip and cause singularities as the slip approaches zero. To avoid numerical problems, the proposed solution method is based on quadratization of the induction motor model [19-20]. This model is interfaced with the quadratized power flow model to provide a robust solution method for a system with induction motors. In addition, this model is a more realistic representation of a power system with moderate increase of the complexity of the power flow equations [20]. Furthermore the methodology makes use of an advanced numerical integration scheme with improved numerical stability properties, which provides a means of overcoming observed numerical problems with other methods [21-22]. Finally, the paper proposes a methodology for optimally selecting the location and size of static and dynamic VAR resources for the purpose of alleviating the risk of voltage
PSCE 2006
instability and slow voltage recovery. The problem is formulated as a nonlinear optimization problem. Trajectory sensitivity methods are used to linearize the optimization problem which in turn is solved via appropriate linear programming techniques.
0.95 0.90 Voltage (pu)
II.
1.00
PROBLEM STATEMENT
The problem of transient voltage sags during .disturbances and recovery after the disturbance has been removed is quite well known. The importance of the problem has been well identified; its significance is increasing especially in modern restructured power systems that may frequently operate close to their limits under heavy loading conditions. Furthermore, the increased number of voltage-sensitive loads and the requirements for improved power system reliability and power quality are imposing more strict criteria for the voltage recovery after severe disturbances. It is well known that slow voltage recovery phenomena have secondary effects such as operation of protective relays, electric load disruption, motor stalling, etc. For example, some motors may trip if the voltage remains below 90% for more than 25 cycles. Many sensitive loads may have stricter settings of protective equipment. A typical situation of voltage recovery following a disturbance is illustrated in Figure 1. Note there is a fault during which the voltage collapses to a certain value. When the fault clears, the voltage recovers quickly to another level and then slowly will build up to the normal voltage. The last period of slow recovery is mostly affected by the load dynamics and especially induction motor behavior. The objective of the paper is to present a method that can be used to study voltage recovery events after a disturbance. More specifically the problem is stated as follows: Assume a power system with dynamic loads. A fault occurs at some place in the system and it is cleared by the protection devices after some period of time. The objective it to study the voltage recovery after the disturbance has been cleared at the buses where dynamic or other sensitive loads are connected and also determine how these loads affect the recovery process. The paper proposes a hybrid approach to the study of voltage recovery that is based on static load flow techniques taking also into account the essential dynamic features of the generators and dynamic loads. More specifically, the power network is assumed to be operating in sinusoidal quasi-steady state and only the dynamics of generating units and loads are considered. This approach provides a more realistic tool compared to traditional static load flow, avoiding however the full scale transient simulation which requires detailed system and load dynamic models. The proposed model is presented next followed by examples.
0.85 0.80 0.75 0.70 Motors will trip if voltage sags for too long
0.65 0.60 Fault -1.00
-0.50
Fault Cleared 0.00
0.50
1.00
1.50 2.00 Seconds
Figure 1. Possible behavior of voltage recovery during and after a disturbance.
III.
QUASI-DYNAMIC SINGLE-PHASE QUADRATIZED POWER FLOW ANALYSIS
A. Overview of Single Phase Quadratized Analysis The proposed system modeling is based on a single phase quadratized power flow (SPQPF) model. The basic idea of the SPQPF is to express the network equations as a set of linear and quadratic equations. This is achieved in a systematic way by developing the models of individual components in this form and then applying connectivity constraints to obtain the network equations. The general form of the model for any component k , at each time step, is as in (1)
ªx k T F k x k º « T 1 » ªi k º k k k k k, « » Y x « x F2 x » b » « ¬0 ¼ » « ¼ ¬
(1)
where i k is the current through the component, x k is the vector of the component states and b k the driving vector for each component, which may contain past history terms, as k well, in the case of dynamical models. Matrix Y models the linear part of the component and matrices Fi k the nonlinear (quadratic) part. This model can refer to a passive component of the system (no dynamical equations) or a dynamic component of the system, i.e. a component that is described with algebraic and differential equations. The form (1) results from (a) the quadratization of the equations and (b) the integration of the differential equations. The integration of the differential equations is described in the next section. Two examples of this modeling approach are given in subsequent paragraphs (induction motor and synchronous generator). Application of the connectivity constraints (Kirchoff’s current
125
C. Induction Motor Model Typically induction motors are represented in power system studies as constant power loads. Although this is a valid representation for steady state operation, induction motors do not always operate under constant power, especially when large deviations of voltage occur. In reality induction motors in steady state operate at a point where the electro-mechanical torque of the motor equals the mechanical torque of the electric load. As the voltage at the terminals of the induction motor changes, the operating point will change, the motor will accelerate or decelerate and during transients the operating point will not be at the intersection of the electrical torque curve and the mechanical load torque curve. We present a model here that captures this behavior. The model is in quadratic form and it is integrated into the single phase quadratized power flow model. The induction motor model is integrated in the proposed formulation by recasting the equations in a quadratic form. Specifically, a quadratic induction machine model has been developed [19-20] based on the typical steady state equivalent circuit of the induction motor. This model has been extended to a quasi-dynamic model, shown in Figure 2, by appending the differential equations that describe the motor rotor dynamics. The model equations of the induction motor are:
law) at each bus yields a set of quadratized equations for the whole system:
ª 0º « 0» ¬ ¼
ª X T F1 X º « » Y X « X T F2 X » b « » ¬ ¼
G( X ) ,
(2)
where X : system state vector, Y : linear term coefficient matrix (admittance matrix), Fi : quadratic term coefficient matrix,
b : driving vector. The solution to the quadratic equations is obtained using the Newton-Raphson iterative method:
Xn
X n 1 J ( X n 1 ) 1 G ( X n 1 )
(3)
where : iteration step, n J ( X n 1 ) : Jacobian matrix at iteration n 1 . The iterative procedure terminates when the norm of the equations is less than a defined tolerance. This iterative procedure is repeated at each time step. If the system exhibits no dynamical behavior (i.e. is completely static), the analysis is equivalent to the load flow analysis and the solution of the above system of equations provides the steady state solution of the system.
~ I dk 0
B. Quadratic integration method[21-22] A new numerical integration scheme is employed for the solution of the dynamical equations. It relies on a collocationbased implicit Runge-Kutta method (Lobatto family) and is Astable and order 4 accurate [22]. The method is based on the following two innovations: (a) the nonlinear model equations (differential or differential-algebraic) are reformulated to a fully equivalent system of linear differential and quadratic algebraic equations, by introducing additional state variables, as described in the previous section, and (b) the system model equations are integrated assuming that the system states vary quadratically within a time step (quadratic integration). Assuming the general nonlinear, non-autonomous dynamical system: (4) x f (t , x) ,
h h 5h f (tm , xm ) f (t , x(t )) x(t h) f (t h, x(t h)) 3 24 24 2h h h x (t ) f (tm , xm ) f (t , x(t )) x(t h) f (t h, x(t h)) 3 6 6
~ ~ jbm E n E n
sn ~ ~ ( g1 jb1 )(Vk E n ) r2 jx 2 sn
(6)
An additional equation links the electrical state variables to the mechanical torque produced by the motor. This equation is derived by equating the mechanical power (torque times mechanical frequency) to the power consumed by the variable resistor in the equivalent circuit of Fig. 4.
0
2 ~ En sn r2 TemZs r2 jx2 sn
(7)
where s n : induction motor slip,
Tem : mechanical torque produced by motor,
Zs : synchronous mechanical speed. The dynamic equation describing the motor rotor dynamics
h,
is:
(5)
2 H dZ m (t ) dt Zs
the algebraic equations at each integration step of length resulting from the quadratic integration method, are: xm
~ ~ ( g1 jb1 )(Vk E n )
Above equations are put in the matrix form of equation (1). Two examples of the modeling methodology are given next.
Tem TL ,
(8)
2 0 TL a bZ n cZ n ,
(9)
0
126
Z n s nZ s Z s ,
(10)
~ I dk
BUS k 1 r1 + jx1
~I
1 jxm
dk
1 r2 + jx2
= g1+jb1
~ ~ ( g1 jb1 )Vk ( g1 jb1 ) E n
dZ n ( t ) Z s Tacc (t ) 2H dt 0 Tacc (t ) Tem (t ) TL (t ), 0
= jbm = g2+jb2
0 TL (t ) a bZn (t ) cZn (t ) r1
r2
jx1
~
En
~ ~* 0 Zn (t ) sn (t )Z s Z s , 0 WnWn U n ~ ~ ~ 0 ( g1 jb1 )Vk ( g1 j (b1 bm )) E n Wn s n ~ ~ ~ ~~ 0 r2Yn jx 2 s nYn 1, 0 Wn Yn E n
jx2 r2
jxm
( 1- sn ) sn
(14) Application of the integration method described in section B on the differential equation will make the above model of the form expressed in equation (1). Note that this model contains real and complex state variables. The complex state variables are expanded in real and imaginary parts to get the final model in terms of real state variables in the form of equation (1). The details are omitted. The final form of the model contains 14 equations and fourteen state variables.
Figure 2. Induction motor equivalent circuit.
where H : Zm :
inertia constant (s), rotor mechanical speed (rad/s),
Zs : synchronous mechanical speed (rad/s), Tem : electrical torque in p.u., TL
:
mechanical load-torque in p.u..
D. Generator Model A two-axis quadratized synchronous generator model is developed for the representation of generators. The basic phasor diagram is presented in Figure 3. The angle of rotor position (d-axis) T (t ) and the rotor angular velocity Z (t ) are defined and then the rotor angle is defined as:
This simplified transient model can capture the behavior of the induction motor during voltage variations. The electrical transients in the motor are neglected, as they do not have significant effect in the network solution, especially for the time scales of interest, which are very long compared to the time scales of the electrical transients. Phasor representation is therefore used for the electrical quantities. The elimination of stator electrical transients makes it possible to interface the motor model with the quasi-steady-state network model. Note that this model is nonlinear and not quadratic since the equations contain high order expressions of state variables. The model is quadratized with the introduction of three ~ ~ additional state variables, namely Yn , Wn , U n defined as
G (t ) T (t ) Z s t
Un
introducing additional state variables, so that the differential equations are linear and the algebraic at most quadratic, the model equations in compact form are:
(11) i q-ax
(12) (13)
x
>V~
B
~ Iq
k
Zn (t ) Tacc (t ) Tem (t ) TL (t ) sn (t ) U n
@
~ ~ ~ En Yn Wn .
O
The quadratic model equations are:
\
~ jxqIq ~ jxdId
I
~ Id
~ Ig
~ Vg
~A
rIg
AB = jxqIg BC = j(xd - xq)Id
Fig. 3. Two-axis synchronous machine phasor diagram
~ Ig
127
s
~ C E
The state vector is defined as: T
(15)
2
is d-ax
~ Wn
1 r2 jx2 sn ~~ Yn En ~ ~* WnWn
S,
where G (t ) S is the angle difference between the rotor (d2 axis), rotating at speed Z (t ) , and a synchronously rotating reference frame at speed Zs . After quadratizing the model and
follows:
~ Yn
Tem (t )Z s U n s n (t ) r2 2
~ ~ Id Iq
G
reference
0
~ ~ ~ ~ ~ ~ E V g r ( I d I q ) jx d I d jx q I q
0
E r I dr E i I di
0
E r s (t ) E i c (t )
0
Ei I qr E r I qi
0
E r2 Ei2 E spec.
0
Ta ( t ) Tm ( t ) 3 z1 I dr I qr 3 z 2 I di I qi D Z ( t ) Z s
0 0
z1Z (t ) E r z 2Z (t ) Ei
0
w1 (t ) c(t ) Z (t ) Z s c (t )
IV.
The proposed approach for voltage recovery with dynamic load representation is based on simulating the dynamical equation describing the motor or generator rotor dynamics and solving along with the network load flow equations and the additional internal device equations. We refer to this procedure as the time-continuation single phase quadratized power flow. Therefore, following a disturbance, the electrical torque produced by the motor will change, due to the terminal voltage variation, causing a deviation in the torque balance between motor torque and load torque. The rotor speed will transiently change. Since there is an imbalance between the load torque and the motor torque, the rotor speed of the machines will change in accordance to the equation of motion. More specifically a typical scenario consists of the following phases: 1) Pre-fault phase: The system is operating at steady state condition. The solution is obtained by load-flow analysis using the motor at torque equilibrium mode. 2) During-fault phase: When a fault (or a disturbance in general) takes place the motor enters a transient operating condition. Typically, the motor is supplied by a considerably reduced voltage resulting in a decrease in the motor electromechanical torque. Subsequently the motor decelerates since the mechanical load will be higher than the electro-magnetic torque. Depending on the voltage level and on the mechanical load characteristics (the load may be constant torque, which is the worst case, or it may depend on the speed) the motor will decelerate and most likely will stall unless the fault is cleared and the voltage is restored in time. The deceleration of the induction motor is computed with the time-continuation single phase quandratized power flow as described earlier. Specifically at each time step the electromechanical torque and the mechanical load torque are computed and the deceleration of the motor over the time step is computed. Then the process is repeated at the new operating point. The timecontinuation procedure is applied throughout the fault duration. The final operating condition at the end of the fault period provides the initial conditions for the post-fault period. 3) Post-fault phase: The time-continuation single phase quadratized power flow approach is also applied to the postcontingency system. The procedure provides the voltage recovery transient at each bus without using full-scale transient simulation during the longer post-fault period. As mentioned before, the final operating condition at the end of the fault period is the initial conditions of the post fault system. The proposed approach is illustrated with two example test systems that will help demonstrate the methodology and clarify the concepts. The examples are kept relatively simple; however, the methodology is applicable to more realistic systems and scenarios, as well.
0 w2 ( t ) s ( t ) Z ( t ) Z s s ( t ) dG (t ) Z (t ) Zs dt dZ ( t ) Z s Ta (t ) 2H dt ds ( t ) w1 (t ) dt dc ( t ) w2 ( t ) dt where ~ I g : armature current (positive direction is into the r
xd
generator), : armature resistance, : direct-axis synchronous reactance,
xq : quadrature-axis synchronous reactance, ~ Vg : terminal voltage,
Z s : synchronous speed, H : inertia constant of the generator, Tm : mechanical power supplied by a prime-mover (in p.u.), TD : damping torque (p.u.), which can be approximated by TD D (Z Z s ) with D constant, ~ E Ee jG is the internal generator voltage; it is an input to the model and its magnitude is specified by an exciter system. The state vector of the model is defined as: xT x1T x 2T , with ~ ~ ~ ~ x1T >V g E I d I q @,
>
x 2T
>Ta (t )
@
z 1 (t )
z 2 (t )
w1 (t )
METHODOLOGY FOR VOLTAGE RECOVERY STUDY
w2 (t ) G (t ) Z (t ) s (t ) c (t ) @ .
Again the same procedure is followed here as for the motor to cast above model in the form o equation (1). Specifically, the differential equations are integrated. The resulting equations from the integration are grouped with the other equations of the model to form a set of equations in the structure of equation (1).
128
V.
EXAMPLE RESULTS
faulted line. During the post-fault period the analysis is again performed using the time-continuation single phase quadratized power flow. At each time step the motor accelerates as determined by the dynamic model of the induction motor. Results from the test case are presented in Figures 5 through 8. Figure 5 shows the three motor speeds throughout the period of study. Note that the deceleration rate is different for each motor. Figure 6 illustrates the voltage recovery at the three motor terminal buses. Figures 7 and 8 present the motor active and reactive power during the pre-fault, fault and postfault phases. Due to the different electrical and mechanical characteristics of the motors 2 and 3 compared to motor 1, and due to the system topology the recovery is considerably slower at BUS04 and BUS05, compared to BUS03.
The example power system 1 is illustrated in Figure 4. Induction motors exist at three buses (BUS03, BUS04, BUS05) and generators at two buses. The rest of the system loads are constant power and constant impedance loads. The system data are given in the Appendix. A bolted three phase fault is assumed in the middle of the line BUS03 - BUS04. The fault is cleared in 0.2 seconds by tripping the line. The motor mechanical loads are modeled as speed dependent loads with quadratic equations (15). During the prefault phase, each load torque is equal to 1 p.u.
TL
a bZ c Z 2
Where: TL
(15)
: mechanical load torque (p.u.),
Z a , b, c
: angular velocity (p.u. of Zs ),
1
: model coefficients.
0.98
The mechanical load of motor at BUS03 has a strong linear dependence on speed; the mechanical load at BUS04, has a strong quadratic dependence, and, the load at BUS05, has a constant torque load. The mechanical load model parameters are given in the appendix. The combined motor-load inertia constants for each motor are: H 1 1.5 sec , H 2 0.5 sec and H 3 1.5 sec , for motors 1, 2 and 3 respectively.
Motor speed (p.u.)
0.96
The following study phases are defined: 1) Pre-fault period: The system is operating at steady state condition. The solution provides an operating point where each motor operates at equilibrium (EM torque equals Mechanical Load torque). G
0.94 Motor 1 (H=1.5s) 0.92 Motor 3 (H=1.5s)
0.9
Motor 2 (H=0.5s)
0.88 0.86 0.84 0.82 -0.1
0
0.1
0.2
SOURCE01
BUS01
BUS02 +SEQ
+SEQ
+SEQ
+SEQ
BUS03
S
BUS04
Z +SEQ
+SEQ
S
0.5
0.6
0.7
0.8
The example power system 2 is illustrated in Figure 9. This system is vulnerable to transient oscillations between the two generating substations and the voltage collapse at the middle substation during these oscillations. This voltage collapse does not recover fast enough because of the presence of the induction motor load at this substation. Figure 9 illustrates one snapshot of the system at which time the voltage has collapsed to 56.35 kV (49%). During the presentation of the paper, the phenomena will be illustrated with animations.
+SEQ
M
0.4
Figure 5. Motor speed as a function of time after a 3-phase line fault.
SOURCE02
S
0.3
Time (s)
G
M
BUS05
Z
VI.
CONCLUSIONS
A practical method for realistically studying the effects of load dynamics and especially induction motor loads on voltage recovery phenomena was presented in the paper. The approach is based on the single phase quadratized power flow method with realistic modeling of induction motor loads. This method models the power system as a set of at most quadratic equations, reducing thus the degree of nonlinearity. Similarly, the induction motor loads are modeled as a set of quadratic equations that are solved simultaneously with the power flow equations. The quadratization approach provides a good approach for avoiding the stiffness of the problem when the
M
Figure 4. Single-line diagram of the example system.
2) During-fault period: A three phase fault is assumed in the middle of the line BUS03 - BUS04, at time t 0 . The motor terminal voltage and its electrical torque are reduced due to the fault. The three motors decelerate at a rate depending on their inertias and the terminal voltage. The fault clearing time is 0.2 sec, i.e., 12 cycles (on 60Hz period). It is important to emphasize that each motor decelerates at different rates. 3) Post-fault period: The fault is cleared by the removal of the
129
0.5
slip approaches zero. The trajectory of the voltage recovery (or collapse) is computed with the time-continuation single phase quadratized power flow. At each time step the motors at the various busses of the system accelerate or decelerate depending on whether the electro-mechanical torque is greater than the mechanical load torque. The proposed methodology combines the simplicity of the standard power flow methods and the load dynamics that can be found only in full-scale time-domain simulation models. Preliminary results from several test cases on a simple power system were presented to demonstrate the process and to establish the feasibility of the approach.
Motor 2
Motor reactive power (p.u. @ 100MVA)
0.45 Motor 3 0.4
Motor 1
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.1
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
Motor terminal voltage (p.u.)
0.9
Figure 8. Motor absorbed reactive power during pre-fault, fault, and post-fault periods.
0.8 0.7
1 Vab
BUS03
Vab = 99.20 kV NORTHBUS2
BUSG1
BUSG1H
Vab = 99.20 kV
kV Vab = 112.3 kV Vab = 56.35 kV
BUST1A
G
BUST2A
BUST2M
1 =299.20 Vab VabkV =1
BUS04
0.6
Vab = 112.3 kV NORTHBUS1
=215.14 VabkV = 112.3
BUSG3H
1 =215.14 Vab VabkV = 112.3 kV Vab = 112.3 kV BUSG2
BUSG2H
BUSG3
Vab = 99.20 kV
BUST1B
IM
ANGSPEED
G
BUST2B
G
BUS05 0.5
Vab = 112.3 kV
Vab = 99.20 kV
SOUTHBUS1
SOUTHBUS2
0.4 0.3 Vab = 59.11 VabkV = 59.11 kV BUST3A
0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
BUST3C
Time (s)
Figure 6. Voltage recovery at motor terminal buses after a 3-phase line fault. Motor 1 0.5
Motor active power (p.u. @ 100MVA)
BUST3B
Vab = 59.11 kV
Motor 2
Vab = 59.11 kV BUST3D
Figure 9. Example Power System 2 for Demonstrating the Combined Phenomena of Voltage Collapse During Transient swings and Voltage Recovery. Figure illustrates one snapshot of system.
Motor 3 0.4
VII.
APPENDIX
The appendix contains the data of the test system. 0.3
Table I. Line Data (values at 100 MVA base) From BUS 01 01 03 02 04 04 01
0.2
0.1
0 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
To BUS 03 03 04 04 05 05 02
Nom. V (kV) 115 115 115 115 115 115 115
R (p.u.) 0.012255 0.024335 0.024335 0.024335 0.012255 0.012255 0.012255
X (p.u.) 0.161871 0.210308 0.210308 0.210308 0.161871 0.161871 0.161871
B/2 (p.u.) 0.038661 0.030095 0.030095 0.030095 0.038661 0.038661 0.038661
Time (s)
Table II. Transformer Data (values at 100 MVA base)
Figure 7. Motor absorbed active power during pre-fault, fault, and post-fault periods.
From
To
SOURC E 01 SOURC E 02
BUS 01 BUS 02
Ratio (kV/kV) 15/115 15/115
R (p.u.) 0.0046 7 0.0046 7
X (p.u.) 0.0546 7 0.0546 7
Core conductance 0.0075
Core susceptance 0.0075
0.0075
0.0075
Table III. Induction Motor Data Power Rating
130
Nominal Voltage
Stator
Rotor
Magnetizing Reactance
(MVA)
20 20 20
(kV)
115 115 115
(p.u.) R (p.u.) 0.01 0.01 0.01
X (p.u.) 0.06 0.06 0.06
R (p.u.) 0.02 0.01 0.01
X (p.u.) 0.06 0.08 0.08
[11] 3.50 3.50 3.50
[12]
Generator Data: SOURCE01: Slack generator, Voltage 1.02 p.u. SOURCE02: PV controlled, Volatge 1.02 p.u., P=50 MW.
[13]
Load Data: BUS03: Constant power: 10 MW, 7 MVAr Constant impedance: 15 MW, 5MVAr at nominal voltage BUS04: Constant power: 10 MW, 3 MVAr BUS05: Constant power: 9 MW, 9 MVAr Constant impedance: 15 MW, 5MVAr at nominal voltage
[14]
[15]
[16]
[17]
[18]
Mechanical Load Data: Motor 1: TL 0.1 0.85Z 0.07234 Z 2 (p.u.)
Motor 2: TL Motor 3: TL
[19]
2
0.05 0.3Z 0.66926 Z (p.u.) 1.0 (p.u.) [20]
VIII. REFERENCES [1]
M. H. J. Bollen, “Voltage recovery after unbalanced and balanced voltage dips in three-phase systems,” IEEE Trans. on Power Systems, vol. 18, issue 4, Oct. 2003, pp. 1376-1381. [2] B. R. Williams, W. R. Schmus and D. C. Dawson, “Transmission voltage recovery delayed by stalled air conditioner compressors,” IEEE Trans. on Power Systems, vol. 7, no. 3, Aug. 1992, pp. 1173-1181. [3] L. Y. Taylor, and S. -M. Hsu, “Transmission voltage recovery following a fault event in the Metro Atlanta area,” Proceedings of the 2000 IEEEPES Summer Meeting, July 16-20, 2000, pp. 537-542. [4] T. Sun, Z. Chen and F. Blaabjerg, “Voltage recovery of grid-connected wind turbines after a short-circuit fault,” Proceedings of the 29th Annual Conference of the IEEE Industrial Electronics Society (IECON ‘03), vol. 3, Nov. 2-6, 2003, pp. 2723-2728. [5] T. Sun, Z. Chen and F. Blaabjerg, “Voltage recovery of grid-connected wind turbines with DFIG after a short-circuit fault,” Proceedings of the 35th Annual IEEE Power Electronics Specialists Conference (PESC ‘04), vol. 3, June 20-25, 2004, pp. 1991-1997. [6] L. Haijun and H. W. Renzhen, “Preventing of transient voltage instability due to induction motor loads by static condenser,” Proceedings of the 1994 IEEE Conference on Industrial Technology, Dec. 5-9, 1994, pp. 827-831. [7] A. E. Hammad and M. Z. El-Sadek, “Prevention of transient voltage instabilities due to induction motor loads by static VAR compensators,” IEEE Trans. on Power Systems, vol. 4, no. 3, Aug. 1989, pp. 1182-1190. [8] I.A. Hamzah and J. A. Yasin, “Static VAR compensators (SVC) required to solve the problem of delayed voltage recovery following faults in the power system of the Saudi electricity company, western region (SEC-WR),” Proceedings of the 2003 IEEE PowerTech Conference, vol. 4, Bologna, Italy, June 23-26, 2003. [9] F. P. de Mello, and J. W. Feltes, “Voltage oscillatory instability caused by induction motor loads,” IEEE Trans. on Power Systems, vol. 11, no. 3, Aug. 1996, pp. 1279-1285. [10] N. Martins, S. Gomes Jr., R. M. Henriques, C. B. Gomes, A. de Andrade Barbosa, and A. C. B. Martins, “Impact of induction motor loads in system loadability margins and damping of inter-area modes,”
131
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
Proceedings of the 2003 IEEE-PES General Meeting, Toronto, Canada, July 13-17, 2003. J. Undrill, A. Renno, and G. Drobnjak, “Dynamics of a large induction motor load system,” Proceedings of the 2003 IEEE-PES General Meeting, Toronto, Canada, July 13-17, 2003. K. Morison, H. Hamadani, and L. Wang, “Practical issues in load modeling for voltage stability studies,” Proceedings of the 2003 IEEEPES General Meeting, Toronto, Canada, July 13-17, 2003. K. Tomiyama, S. Ueoka, T. Takano, I. Iyoda, K. Matsuno, K. Temma, and J. J. Paserba, “Modeling of Load During and After System Faults Based on Actual Field Data,” Proceedings of the 2003 IEEE-PES General Meeting, Toronto, Canada, July 13-17, 2003. I. R. Navarro, O. Samuelsson, and S. Lindahi, “Automatic determination of parameters in dynamic load models from normal operation data,” Proceedings of the 2003 IEEE-PES General Meeting, Toronto, Canada, July 13-17, 2003. I. R. Navarro, O. Samuelsson, and S. Lindahi, “Influence of normalization in dynamics reactive load models,” IEEE Trans. on Power Systems, vol. 18, issue 2, May 2003, pp. 972-973. C. –J. Lin, A. Y. –T. Chen, C. –Y. Chiou, C. –H. Huang, H. –D. Chiang, J. –C. Wang and L. Fekih-Ahmed, “Dynamic load models in power systems using the measurement approach,” IEEE Trans. on Power Systems, vol. 8, issue 1, Feb. 1993, pp. 309-315. D. J. Hill, “Nonlinear dynamic load models with recovery for voltage stability studies,” ,” IEEE Trans. on Power Systems, vol. 8, no. 1, Feb. 1993, pp. 166-176. D. Karlsson and D. J. Hill, “Modelling and identification of nonlinear dynamic loads in power systems,” IEEE Trans. on Power Systems, vol. 9, no. 1, Feb. 1994, pp. 157-166. A. P. Sakis Meliopoulos, Wenzhong Gao, Shengyuan Li, G. J. Cokkinides and Roger Dougal, “Quadratized induction machine model for power flow analysis,” Proceedings of the Second IASTED International Conference, EuroPES, Crete, Greece, pp 194-199, June 25-28, 2002. G. K. Stefopoulos and A. P. Meliopoulos, “Induction motor load dynamics: Impact on voltage recovery phenomena,” to be presented at the 2005-2006 IEEE PES T&D Conference and Exposition, Dallas, TX, May 21-26, 2006. A. P. Sakis Meliopoulos, G. J. Cokkinides, and G. K. Stefopoulos, “Symbolic integration of dynamical systems by collocation methods,” Proceedings of the 2005 IEEE-PES General Meeting, San Francisco, CA, June 12-16, 2005, pp. 2387-2392. A. P. Sakis Meliopoulos, G. J. Cokkinides, and G. K. Stefopoulos, “Quadratic integration method,” Proceedings of the 8th Int. Conference on Power System Transients (IPST 05), Montréal, Canada, June 19-23, 2005. M. J. Laufenberg and M. A. Pai, “A new approach to dynamic security assessment using trajectory sensitivities,” IEEE Trans. on Power Systems, vol. 13, no. 3, Aug. 1998, pp. 953-958. I. A. Hiskens and M. Akke, “Analysis of the Nordel power grid disturbance of January 1, 1997 using trajectory sensitivities,” IEEE Trans. on Power Systems, vol. 14, no. 3, Aug. 1999, pp. 987-994. I. A. Hiskens and M. A. Pai, “Trajectory sensitivity analysis of hybrid systems,” IEEE Trans. on Circuit and Systems – Part I, vol. 47, no. 2, Feb. 2000, pp. 204-220. T. B. Nguyen, M. A. Pai, and I. A. Hiskens, “Direct computation of critical clearing time using trajectory sensitivities,” Proceedings of the 2000 IEEE-PES Summer Meeting, July 16-20, 2000, vol. 1, pp. 604-608. I. A. Hiskens and M. A. Pai, “Power system applications of trajectory sensitivities,” Proceedings of the 2002 IEEE-PES Winter Meeting, New York, NY, Jan. 27-31, 2002, vol. 2, pp. 1200-1205. M. Zima and G. Andersson, “Stability assessment and emergency control method using trajectory sensitivities,” Proceedings of the 2003 IEEE PowerTech Conference, vol. 2, Bologna, Italy, June 23-26, 2003.
