Panel-06-4 Voltage Stability and Voltage Recovery

8/6/2019 Panel-06-4 Voltage Stability and Voltage Recovery

1/43

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


2/43

IEEE PES Power Systems Conference and Exposition PSCE 2006Atlanta, GA, Oct. 29 Nov. 1, 2006

2

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


3/43


3

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


4/43


4

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


5/43


5

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


6/43


6

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)


7/43


7

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


8/43


8

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


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


9/43


9

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)


10/43


10

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


11/43


11

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


12/43


12

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


13/43


13

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


14/43


14

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


15/43


15

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


Rotorreactance(%

ofstandstillvalue)

Standstill

Synchronous speed

0 20 40 60 80 10080

82

84

86

88

90

92

94

96

98

100


Rotorresistance(%

ofstandstillvalue)

.

.

Standstill

Synchronous speed


16/43


16

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


Torque(p.u.) Estimation

Procedure


17/43


17

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


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


18/43


18

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 =


19/43


19

Model Estimation: User interface


20/43


20

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


21/43


21


(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


22/43


22


(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


23/43


23



24/43


24

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 =


25/43


25

Synchronous Generator Modeling


26/43


26

Exciter, Turbine-Governor Modeling


27/43


28/43


28

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


29/43


29

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


30/43


30

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


31/43


31

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


32/43

Q i t ti A l i N i l E l


33/43


33

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


514.2


5.263 k


31.56


7.644 M


4.660 M


161.0 m


Q i t ti A l i N i l E l


34/43


34

Quasi-static Analysis Numerical Example:Line-to-line fault

0.00 0.15 0.30 0.45 0.60

5.375 k


623.3


5.350 k


29.69


8.152 M


4.633 M


27.98 m


0.00 0.15 0.30 0.45 0.60

5.589 k


837.6


5.669 k


8.529


8.522 M


4.238 M


-1.444


O ti l All ti f St ti d D i


35/43


35

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



36/43


36


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


37/43


37


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



38/43


38


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



39/43


39


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.


40/43

Sensitivity Analysis Costate Method


41/43


41

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


42/43


42

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


43/43


43

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

Panel-06-4 Voltage Stability and Voltage Recovery

Documents