panel-06-4 voltage stability and voltage recovery
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PSERC
IEEE PES Power Systems Conference and Exposition PSCE 2006
Atlanta, GA, October 29 November 1, 2006
Voltage Stability and Voltage
Recovery: Load Dynamics andDynamic VAR Sources
Sakis Meliopoulos, George Cokkinides, and
George StefopoulosSchool of Electrical and Computer EngineeringGeorgia Institute of Technology
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Outline
Motivation
Basic research focus Electric load dynamics modeling
Induction motor representation
Synchronous generator representation
Quadratization and quadratic integration method
Example results Optimal allocation of static and dynamic VAR sources
Conclusions
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Introduction
The objective of this project is to formulate and solve
the optimal allocation problem of static and dynamicVAR sources in electric power systems
The proposed research takes into consideration
both steady-state and dynamic system behavior The proposed research assumes both static
(capacitor banks) and dynamic VAR sources
The issues of system modeling are extensivelyaddressed, with particular emphasis on loadmodeling
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Basic Concepts: Voltage Phenomena
Voltage recovery Rate of return to normal voltage level after a disturbance,
fault, etc.
Voltage stability Ability of a power system to maintain acceptable voltages
at all system buses under normal conditions and afterdisturbances
Voltage collapse
Phenomenon in which a relatively fast sequence of eventsafter voltage instability leads to a voltage decay tounacceptably low values in general a non-recoverablesituation
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Voltage Recovery: Typical Phenomena
Typically motors willstall if their terminal
voltage sags below 90%for too long (e.g. morethan 20 cycles)
The voltage recovery, followingthe clearing of a fault, maybe slow for weak systems withheavy induction motor loads
The voltage recovery, following
the clearing of a fault, maybe slow for weak systems withheavy induction motor loads
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.00 0.50 1.00 1.50
Seconds
2.00
Voltage
(pu)
Motors will tripif voltage sagsfor too long
-0.50-1.00
Fault Fault Cleared
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Modeling Approach
ThreeThree--phase physically based modelsphase physically based models
Explicit load model dynamicsExplicit load model dynamics
TwoTwo--axes generator model with exciter andaxes generator model with exciter andgovernorgovernor
Steady state (Quadratic power flow)Steady state (Quadratic power flow)
Transient analysis (Quadratic integrationTransient analysis (Quadratic integrationmethod)method)
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Important Issue: Electric Load Modeling
Static load representation
Constant impedance load Constant current load
Constant power load
Voltage/Frequency dependent load models
Cannot capture allvoltage phenomena
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Characteristics of Induction Motor Loads
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Speed (% of synchronous)
PowerFa
ctor(%)
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
Speed (% of synchronous)
Torque,
Power,
Current(p.u.)
Reactive power
Motor curre nt
Active power
Mechanical loadSlip-torque characteristic
Operating point
Induction motor operating conditions fordifferent operating speed values
Steady State Operation: Intersection of Mech-Load/Electric Torque
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Effects of Induction Motor Loads (steady-
state)Voltage profile of the 24-bus RTS after a line contingency
(a) constant power load representation(b) induction motors (50%)
(a) (b)
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Effects of Induction Motor Load (transient)
CommentReactive power absorption is VERY sensitive to motor speed
Comment
Reactive power absorption is VERY sensitive to motor speed
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Effects of Induction Motor Load (transient)
Contingency simulation:Effects of load dynamics
Contingency simulation:Effects of load dynamics
50% Induction motors
2% Slowdown during faultVmax=1.01, Vmin=0.82
No induction motorsVmax=1.046, Vmin=0.908
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High Fidelity Power System Simulator
Electric load representation Loads by types (power, impedance, motors, etc.) Load dynamics and controls
Generator model Two-axes model
Exciter models Turbine-governor models
Three phase circuit models Three-phase physically based network modeling
Model quadratization A simple procedure of introducing new variables to create a
model consisting of linear and quadratic equations withoutapproximations
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Induction Motor NEMA Designs
0 20 40 60 80 1000
0.5
1
1.5
2
2.5
3
3.5
4
Tor
que
(p.u.
)
Speed (% of rated)
NEMA DESIGN A, B, C, D for AC INDUCTION MOTORS
Design A
Design B
Design C
Design DDeep-bar squirrel- cage
motors
Double-cage rotors
Using slip-dependentmotor parameters thetorque-speed motor
characteristics areaccurately represented
Using slip-dependentmotor parameters thetorque-speed motor
characteristics areaccurately represented
Slip-dependentrotor parameters
Slip-dependent
rotor parameters
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Slip-Dependent Rotor Impedance
2
2 )( scsbasr ++=
sedsx +=)(2
I~
r1 jx1 r2(s) jx2(s)
r2(s)( 1- s )
sjxmE
~
BUS k
This model can capture the behavior of any motor typeby appropriate selection of the model parameters
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Slip-Dependent Rotor Impedance
Rotor resistance Rotor reactance
0 10 20 30 40 50 60 70 80 90 100100
100.5
101
101.5
102
102.5
103
103.5
104
104.5
105
Speed (% of synchronous)
Rotorreactance(%
ofstandstillvalue)
Standstill
Synchronous speed
0 20 40 60 80 10080
82
84
86
88
90
92
94
96
98
100
Speed (% of synchronous)
Rotorresistance(%
ofstandstillvalue)
.
.
Standstill
Synchronous speed
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Induction Motor Model Estimation:
Numerical Example
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
4
Speed (% of synchronous)
Torque(p.u.) Estimation
Procedure
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Model Estimation: Formulation
Least-squares estimation:
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
Speed (% of synchronous)
Torque
(p.u.)
Measured speed-torque curve
(m measured points)
WrrrwJT
m
i
ii ===1
2min
s.t.
1~~
0
~~))((
~)(0
~~~0
~~0
/
~)(
~)(
~
22
1111
*
+=
+++++=
=
=
=
++=
nnn
nnnmk
nnn
nn
srem
ssss
YsjxYr
sWEbbjgVjbg
EYW
WWU
UsrT
EjbgVjbgI
=
tmeasuremencurrentaisif,)(
tmeasurementorqueaisif,)(
,
,,
iIpI
iTpTr
imeasuredi
imeasurediem
i
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Model Estimation: Solution
Solution process
Gauss-Newton-type method
Need for global convergence strategies (line search,trust region)
Need for proper state and equation scaling
[ ]Tmmss edcbagxrxp =
( ) )()()()(1
1 nTnnTnnn prWpHpHWpHpp =
+
[ ]npp
TTT
em
npIpTpH
=
= //)(
[ ]Trrmmss rxgxrxp =
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Model Estimation: User interface
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3-Phase Quadratic Motor Model
(steady-state)Steady-state operating modes: Constant slip mode
Predefined slip value Operating point at specific speed Linear model
Constant torque equilibrium mode
Predefined constant value of mechanical torque Operating point at torque equilibrium Nonlinear model (quadratic) Slip computed via the power flow solution
Slip-dependent torque equilibrium mode Predefined slip-dependent mechanical load model Operating point at torque equilibrium Nonlinear model (quadratic)
Slip and torque computed via the power flow solution
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3-Phase Quadratic Motor Model
(steady-state)rs jxs rr jxr
jxmrr
1-snV1
I1
E1 sngm
rrsn-1
2-SnV2
I2
E2 jxm
rs jxs rr jxr
gm
(0)
120
1~~ITIabc
=
120
1 ~~0 VTVabc
=)
~~)((
~0 111 VEjbgI ss ++=
)~~
)((~
0 222 VEjbgI ss ++=
0000
~)(
~0 VjbgI +=
20 nnm cbaT =
(2)
(1)
rs + rr
V0
I0jxs + jxr
nmsmsss sWEbbjggVjbg 111~~
))((~
)(0 ++++++=
)2(~~
))((~
)(0 222 nmsmsss sWEbbjggVjbg ++++++=
1~~
0 11 += YsjxYr nrr1
~)2(
~0 22 += YsjxYr nrr
rnrnsm rsUrsUT )2(0 21 +=
111
~~~0 EYW =
222 ~~~0 EYW =1
*
11
~~0 UWW =
2
*
22
~~0 UWW =
Slip-dependent torque model
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3-Phase Quadratic Motor Model
(quasi-steady-state) Augmentation of the steady-state equation set with the
swing equation of the rotor motion
Constant torque mode or slip-dependent torque mode
)()()(
tTtTdt
tdJ Lm
n =
ssnn s =0
constTL =2
nnL cbaT ++= Model suitable for small motor representation or
aggregate models of a number of small motors
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3-Phase Quadratic Motor Model
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Synchronous Generator Modeling
2-axis quadratized model
O
d-axis
q-axi
s
reference
Id
B
C
A
~
Iq~
Ig~ rIg
~Vg~
E~
jxqIq
jxdId
~
~
AB = jxqIg
BC = j(xd - xq)Id
)()()()()()()()()(~~~~
2121 tctstttwtwtztztTIIEVx aqdgT
=
qdg III~~~
+=
qqddqdg IjxIjxIIrVE~~
)~~
(~~
0 ++++=
diidrr IEIE +=0)()(0 tcEtsE ir =
qirqri IEIE =0
.
220 specir EEE +=
( ) ( ) ( )sqidiqrdrma tDIIzIIztTtT +++++= )(33)()(0 21rEtz
=)(0 1
iEtz = )(0 2
)()()()(0
)()()()(0
2
1
tsttstw
tcttctw
s
s
+=
+=
stdttd = )(/)(
)(/)( tTdttdJ a=
)(/)( 1 twdttds =
)(/)( 2 twdttdc =
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Synchronous Generator Modeling
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Exciter, Turbine-Governor Modeling
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Quadratic Integration Method
3-stage, implicit, Runge-Kutta method based
on collocation 2nd member of the Lobatto family methods
(IIIA)
3 collocation points (two endpoints of theinterval and the midpoint)
4th order accurate A-stable
Free of fictitious numerical oscillations
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Quadratic Integration: System Solution
ConnectivityConstraints
Newtons
Method x(t)x(t)
Component Model
kkTk
kkTk
kk
k
bxFx
xFx
xYi
+=
M
2
1
0
),(0
02
1
cT
T
uxGbxFx
xFx
Yx =
+=
M
System Model
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Quasi Steady-State Analysis
Analysis through time using simplified, yet
realistic, dynamic models Consideration of only essential dynamic
characteristics of power systems components
(ignore fast electric phenomena)
Sinusoidal steady-state network conditions
Simulation times up to a few seconds
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Quasi-static analysis example1
2
1
2
1 2
1
2
1 2
1Ph
1Ph
1Ph
SOURCE01
BUS01 BUS01-L3
BUS01-L1BUS01-L2
BUS02-L1 BUS02
SOURCE02
BUS02-L2
BUS04-L2
BUS04 BUS04-TBUS04-L3
BUS03-L2
BUS03 BUS03-L3BUS03-L1
BUS03-T
BUS04-L4BUS04-L1
BUS05-L1
BUS05
BUS05-L2
BUS05-TThree-Phase, Breaker-Oriented Model
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Q i t ti A l i N i l E l
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Quasi-static Analysis Numerical Example:Line-to-line fault
0.00 0.15 0.30 0.45 0.60
4.830 k
7.588 kBUS05-T_PHASE_A (V)
470.4
7.692 kBUS05-T_PHASE_B (V)
4.835 k
7.870 kBUS05-T_PHASE_C (V)
18.68
98.87MOTOR_SPEED (%)
6.974 M
39.40 MMOTOR_ACTIVE_POWER (W)
4.272 M
58.76 MMOTOR_REACTIVE_POWER (VA)
73.25 m
3.715MOTOR_TORQUE (p.u.)
0.00 0.15 0.30 0.45 0.60
5.283 k
7.580 kBUS05-T_PHASE_A (V)
514.2
7.683 kBUS05-T_PHASE_B (V)
5.263 k
7.861 kBUS05-T_PHASE_C (V)
31.56
97.81MOTOR_SPEED (%)
7.644 M
33.78 MMOTOR_ACTIVE_POWER (W)
4.660 M
50.77 MMOTOR_REACTIVE_POWER (VA)
161.0 m
3.112MOTOR_TORQUE (p.u.)
Q i t ti A l i N i l E l
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Quasi-static Analysis Numerical Example:Line-to-line fault
0.00 0.15 0.30 0.45 0.60
5.375 k
7.581 kBUS05-T_PHASE_A (V)
623.3
7.685 kBUS05-T_PHASE_B (V)
5.350 k
7.861 kBUS05-T_PHASE_C (V)
29.69
96.73MOTOR_SPEED (%)
8.152 M
34.10 MMOTOR_ACTIVE_POWER (W)
4.633 M
46.23 MMOTOR_REACTIVE_POWER (VA)
27.98 m
3.164MOTOR_TORQUE (p.u.)
0.00 0.15 0.30 0.45 0.60
5.589 k
7.590 kBUS05-T_PHASE_A (V)
837.6
7.699 kBUS05-T_PHASE_B (V)
5.669 k
7.871 kBUS05-T_PHASE_C (V)
8.529
89.46MOTOR_SPEED (%)
8.522 M
38.45 MMOTOR_ACTIVE_POWER (W)
4.238 M
30.60 MMOTOR_REACTIVE_POWER (VA)
-1.444
3.569MOTOR_TORQUE (p.u.)
O ti l All ti f St ti d D i
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Optimal Allocation of Static and Dynamic
VAR Sources How can voltage problems be controlled
Planning for adequate VAR support Addition of dynamic VAR sources for fast
response
Develop methodology for the selection of theoptimal mix and placement of static and dynamicVAR resources in large power systems, toimprove voltage recovery and dynamicperformance
Optimal Allocation of Static and Dynamic
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Optimal Allocation of Static and Dynamic
VAR Sources Formulation of criteria for acceptable voltage recovery.
Such criteria can include, but are not limited to, speed of
voltage recovery, avoidance of unnecessary relayoperations, avoidance of motor stalling and avoidance ofsystem voltage collapse
Development of suitable simulation models that capture:
Dynamics of the electric load
Relay response during voltage recovery dynamics
Optimal use of available means for fast voltage control
Formulate optimization methodologies for determiningoptimal mix of static and dynamic VAR sources tomeet criteria
VAR Source Allocation Problem
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VAR Source Allocation Problem
Formulation Given:
A power system comprising of generating units,
transmission network, existing VAR sources and loads withcertain load composition
Expected daily variations of the electric load
A number of candidate buses for VAR source placement(computed via static/trajectory sensitivity analysis),k=1,2,,K
Capacitor modules of Xk,I MVAr at Yk,I kV level, i=1,2,,M,
at cost Cc,i Dynamic VAR sources of capacity Dmin,I, Dmax,I MVAr at at
Yk,I kV level, i=1,2,,M, at cost Cd,I
Voltage limits and voltage recovery criteria
VAR Source Allocation Problem
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VAR Source Allocation Problem
Formulation Compute:
The optimal selection of Xi,k and Dmin,I, Dmax,I at bus k
(k=1,2,,K) that observe voltage limits and meet voltagerecovery criteria
VAR Source Allocation Problem
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VAR Source Allocation Problem
FormulationObjective: Minimize the sum of the cost of the static VAR
sources and the cost of the dynamic VAR sources
Constraints: The usual operating constraints plus the voltagerecovery rate criteria/constraints
Solution process:
Identify buses and/or circuits in which to place static anddynamic VAR sources, as well as the amount of reactive
compensation
Decision variablesTrajectory sensitivity methods are utilized to linearize the
above optimization problem.
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Sensitivity Analysis Costate Method
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Sensitivity Analysis Costate Method
Used to form state space, by selectingadditions based on static performance criteria
),(0 uxG=
1
=
x
G
x
hx
T
u
Gx
u
h
du
dh T
=
),( uxhJ=
Power Flow Equations
Performance Function
Trajectory Sensitivity Analysis Numerical
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Trajectory Sensitivity Analysis Numerical
Computation Used to form state space, by selecting
additions based on dynamic performancecriteria
),,,(0
),,,()(
uyxtg
uyxtf
dt
tdx
=
=
),(0 uXG=
Numerical
Integration 1
=
X
G
X
hX
T
u
GX
u
h
du
dh T
=
),,,( uyxthJ=
Synopsis
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Synopsis
Impact of load representation and load dynamics onvoltage recovery phenomena
An analysis approach has been presented thatcaptures the phenomena with high fidelity based on: Physically based three phase power system model Electric motor speed dependent models
Synchronous machine representation with exciter andgovernor models Model quadratization Quadratic integration of system dynamics
The presented analysis is utilized for optimalallocation of dynamic and static VAR resources viasuccessive linearization or dynamic programmingmethods
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