voltage recovery and optimal allocation of var … recovery and optimal allocation of var support...

Voltage Recovery and Optimal Allocation of VAr Support via Quadratic Power System Modeling and Simulation

George StefopoulosPh.D. CandidateSchool of Electrical and Computer EngineeringGeorgia Institute of Technology

PSERC Tele-seminar

May 5th, 2009

2

Presentation outline Introduction

Understanding basic voltage phenomena

Modeling of factors affecting voltage problems Electric-load modeling

Power-system analysis and simulation Quasi-steady-state analysis Full transient analysis

Optimization and reactive support Control of voltage and reactive power Optimal allocation and operation of static and

dynamic VAR sources

Conclusions

3

Basic Concepts: Voltage Phenomena

Voltage recovery following faults 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 after disturbances

Voltage collapse Phenomenon in which a relatively fast

sequence of events after voltage instability leads to a voltage decay to unacceptably low values – in general a non-recoverable situation

Introduction → Modeling → Simulation → Optimization → Synopsis

4

Voltage Recovery following faults: Effects of Load

Typically motors will stall if their terminal voltage sags below 90% for too long (e.g. more than 20 cycles)

The voltage recovery, followingthe 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 1.50Seconds

2.00

Volta

ge (p

u)

Motors will tripif voltage sagsfor too long

-0.50-1.00

FaultFault Cleared


5

Voltage Recovery: Field Recordings

Phase A (V) Phase B (V) Phase C (V)

Voltage Recovery SlowLoad SheddingOvervoltageLoad RestorationVoltage SagVoltage collapse


There exist many field recording of fault delayed voltage recovery events

Duration of events may last from less than a second up to several minutes

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Active Power P (p.u.)

Vol

tage

V (p

.u.)

Voltage Stability: P-V Curves

Point of voltage collapse

Stable region

Unstable region


7

Voltage Stability: System Oscillations

During transient swings voltage collapse may occur near the center of oscillation. Below

example shows a snapshot of a transient swing


8

Voltage Collapse: Effects of Reactive Support


9

Analysis of Voltage Related Phenomena Dynamic in nature phenomena

Small-disturbance stability Static steady-state analysis (PV curves) Continuation power flow Linearizations

V-Q sensitivity analysis Q-V modal analysis

Large-disturbance stability Non-linear dynamic analysis Dynamic simulation

Transient time frame Long-term time frame


10

Presentation outline

IntroductionModeling of factors affecting

voltage problems Power-system analysis and

simulationOptimization and reactive

supportConclusions

Network/Load Modeling Existing power-flow analysis tools typically do not

represent load dynamics Load dynamics may introduces asymmetries and

imbalances Three-phase models vs. single-phase equivalents

Positive sequence analysis – Balanced and symmetric operation

Symmetrical component analysis – Unbalanced operation symmetric conditions

Phase domain analysis – Unbalanced operation, asymmetric conditions, physically-based analysis

Proposed approach Power system modeling, with emphasis on dynamic loads

(induction motor loads) Physically-based, three-phase, high fidelity system

representation Quadratized component modeling

11Introduction → Modeling → Simulation → Optimization → Synopsis

12

Electric Load Modeling

Induction motors VAR requirements vary drastically with

operating conditions (motor speed) Dynamics

Cold load pick-up High currents Motor restart/stalling Transformer inrush currents


13

Traditional Electric Load Modeling

Static load representation: Constant impedance load Constant current load Constant power load Voltage/frequency-dependent load

models

Cannot capture all the phenomena


14

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)

Pow

er F

acto

r (%

)

Induction motor operating conditions for different operating speed values

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8


Torq

ue, P

ower

, Cur

rent

(p.u

.)

Reactive power

Motor current

Active power

Mechanica l loadSlip-torque characteristic

Operating po in t


15

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)


16

Contingency simulation:

Effects of load dynamics 50% Induction motors2% Slowdown during fault

Vmax=1.01, Vmin=0.8250% Induction motors

Vmax=1.046, Vmin=0.908

Effects of Induction Motor Loads (Transient)


17

Electric Load Modeling Static load representation:

Constant impedance load Constant current load Constant power load Voltage/frequency-dependent load models

Dynamic load representation Induction motors Generalized dynamic load models

Three-phase and single-phase modelsIssue: In general the compositionof the load is not known. Need realdata to define model


18

Estimation of Electric Load Composition

Obtain time recordings of transient events

Identify load composition Specific signature of each load type

Identify load model parameters

Issue:

Identification of load parameters


19

3-Phase Induction Motor Models:NEMA Design Motor Models

Design A Design B

Design C Design D


20

Slip-Dependent Rotor Impedance

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

Torq

ue (p

.u.)

Speed (% of rated)

NEMA DESIGN A, B, C, D for AC INDUCTION MOTORSDesign A

Design B

Design C

Design D Deep-bar squirrel-cage motors

Double-cage rotors

Using slip-dependentmotor parameters the torque-speed motor characteristics are accurately represented

Slip-dependentrotor parameters


21

Slip-Dependent Rotor Impedance

Idk~

r1 jx1 r2(s) jx2(s)

r2(s)( 1- s )

sjxmEn

~

BUS k

sedsxscsbasr

⋅+=⋅+⋅+=

)()(

2

22

This model can capture the behavior of any motor type by appropriate selection of the model parameters

Figure shows single-phase equivalent


22

Induction Motor Model Estimation Realistic representation of motor loads Accurate representation of motors of

various classes Correct identification of motor equivalent

circuit Practical estimation of slip-dependent

rotor models

No-load test Blocked-rotor test Field measurements


23

Induction Motor Model Estimation

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5


Torq

ue (p

.u.)

WrrrwJ Tm

iii == ∑

=1

2min

Measured speed-torque curve(m measurement points)

Least-squares estimation

−−

=.)(

,)(

,

,,

imeasuredi

imeasurediemi IpI

TpTr


Solution process Gauss-Newton-type method

( ) )()()()( 11 nTnnTnnn pWrpHpWHpHpp −+ −=

[ ]npp

TTTem

n pIpTpH=

∂∂∂∂= //)(

Need for global convergence strategies (line search, trust region) Need for proper state and equation scaling

Single-Phase Induction Motor Model: Operational Characteristics


Split phase

Capacitor start

Capacitor start, capacitor run

Permanent split capacitor

Model Quadratization Description of a system using linear or quadratic

equations without introducing any approximations Introduction of additional state variables

25

)),(),(( 0 )),(),(()(

ttytxgttytxftx

==

[ ] )(...)()()(

)()()(00

)( 1

321 tbXFX

XFX

tztytx

tAtAtAtx

nT

T

+

+

⋅=

( )

( )

+

++=

Tnn

T

TT

FFX

FFXAJ ...

11


26





supportConclusions

27

Types of Analysis

Steady-state analysis

Quasi-steady-state analysis

Full transient analysis


28

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 Computation of RMS values of electrical

quantities at fundamental frequency Consideration of motor dynamics only


Three-Phase Induction Motor Model: Steady-State, 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

29

jxsrs jxr

jxm

rr

1-ss

( )gm rrV1 E1

I1

jxsrs jxr

jxm

rr

s-12-s

( )gm rrV2 E2

I2

j(xs+xr)rs+rr

V0

I0


)()()(

tTtTdt

tdJ Lm

n −=ω

ssnn s ωωω −−=0

constTL =2

nnL cbaT ωω ++=

Single-Phase Induction Motor Model

30

+

_+

+ +

+

_

_

_

_C CsV2

V1

Vmain

VauxIauxImain

Vc VcsVs

I1

I2Ic Ics

Steady-state Frequency-domain analysis of equivalent circuit

Quasi-steady-state Frequency-domain analysis of equivalent circuit Inclusion of mechanical equations for rotor movement Switching model


Distribution Feeder Voltage Recovery: Quasi-Steady-State Analysis




0.0 0.5 1.0 1.5

17.92

493.0

968.1

1.443 k

1.918 k

2.393 k

2.869 k

3.344 k

3.819 k

4.294 k

4.769 k

5.244 k

5.719 k

6.194 k

6.669 k

7.145 k

7.620 k

8.095 k PXFMR__Bus_SUB2__Voltage_Phase_A (V)

0.0 0.5 1.0 1.5

243.9 m

38.79

77.34

115.9

154.4

193.0

231.5

270.1

308.6

347.2

385.7

424.3

462.8

501.4

539.9

578.5

617.0

655.6 Current__Line_FDR1-POLE13__Phase_A (A)Current__Line_FDR2-POLE1__Phase_A (A)PXFMR__Bus_SUB2__Current_Phase_A (A)



0.0 0.5 1.0 1.5

386.2 m

16.63

32.87

49.12

65.36

81.60

97.85

114.1

130.3

146.6

162.8

179.1

195.3

211.6

227.8

244.0

260.3

276.5 IndMotor_MCC-P2:_Voltage_Phase_B (V)IndMotor_MCC-P3:_Voltage_Phase_B (V)IndMotor_MCC-P8A:_Voltage__Phase_B (V)IndMotor_MCC1:_Voltage_Phase_B (V)IndMotor_MCC3:_Voltage_Phase_B (V)IndMotor_MCC5:_Voltage_Phase_B (V)IndMotor_MCC6:_Voltage__Phase_B (V)IndMotor_MCC8:_Voltage__Phase_B (V)

0.0 0.5 1.0 1.5

1.501

7.251

13.00

18.75

24.50

30.25

36.00

41.75

47.50

53.25

59.00

64.75

70.50

76.25

82.00

87.75

93.50

99.25 IndMotor_MCC-P2:_Speed (%)IndMotor_MCC-P3:_Speed (%)IndMotor_MCC-P8A:_Speed (%)IndMotor_MCC1:_Speed (%)IndMotor_MCC3:_Speed (%)IndMotor_MCC5:_Speed (%)IndMotor_MCC6:_Speed (%)IndMotor_MCC8:_Speed (%)



0.0 0.5 1.0 1.5

2.857

92.98 k

186.0 k

278.9 k

371.9 k

464.9 k

557.9 k

650.9 k

743.8 k

836.8 k

929.8 k

1.023 M

1.116 M

1.209 M

1.302 M

1.395 M

1.488 M

1.581 M IndMotor_MCC-P2:_Real_Power (W)IndMotor_MCC-P3:_Real_Power (W)IndMotor_MCC-P8A:_Real_Power (W)IndMotor_MCC1:_Real_Power (W)IndMotor_MCC3:_Real_Power (W)IndMotor_MCC5:_Real_Power (W)IndMotor_MCC6:_Real_Power (W)IndMotor_MCC8:_Real_Power (W)

0.0 0.5 1.0 1.5

-3.748

111.3 k

222.6 k

333.9 k

445.2 k

556.6 k

667.9 k

779.2 k

890.5 k

1.002 M

1.113 M

1.224 M

1.336 M

1.447 M

1.558 M

1.670 M

1.781 M

1.892 M IndMotor_MCC-P2:_Reactive_Power (VA)IndMotor_MCC-P3:_Reactive_Power (VA)IndMotor_MCC-P8A:_Reactive_Power (VA)IndMotor_MCC1:_Reactive_Power (VA)IndMotor_MCC3:_Reactive_Power (VA)IndMotor_MCC5:_Reactive_Power (VA)IndMotor_MCC6:_Reactive_Power (VA)IndMotor_MCC8:_Reactive_Power (VA)

35

Full transient analysis

Time domain analysis using high-fidelity models

Inclusion of fast dynamics Computation of actual time domain

waveform Consideration of waveform harmonic

distortion Consideration of motor dynamics and

transformer inrush currents


Three-Phase Induction Motor Model: Transient Analysis

36

ReferenceStator phase A magnetic axis

vAs(t)

vNs(t)

iAs(t)

iNs(t)

vBs(t)iBs(t)

vCs(t)iCs(t)

Rotor phase A magnetic axis

vAr(t)

θm(t)

ωm(t)

iAr(t)

iBr(t)vBr(t)

vCr(t)

vNr(t)

iCr(t)

iNr(t)

ias(t)

ibs(t)ics(t)

ibr(t)

iar(t)

icr(t)

Reference

Stator phase A magnetic axis

vAs(t)iAs(t)

vBs(t)iBs(t)

vCs(t)iCs(t)

Rotor phase A magnetic axis

vAr(t)

θm(t)

ωm(t)

iAr(t)

iBr(t)vBr(t)

vCr(t)iCr(t)

ias(t)

ibs(t)

ics(t)

ibr(t)

iar(t)

icr(t)


Time domain analysis of mutually coupled windings Inclusion of mechanical equations for rotor movement

Single-Phase Induction Motor Model

37

+

_+

+ +

+

_

_

_

_C CsV2

V1

Vmain

VauxIauxImain

Vc VcsVs

I1

I2Ic Ics

Full transient Time domain analysis of mutually coupled windings Inclusion of mechanical equations for rotor movement Switching model


Modeling of Inrush Currents: Saturable-Core Reactor Model

38

v1(t)

v2(t)

i1(t)

i2(t)

iL(t)

)()()(21 tvtv

dttd

−=λ

( ))()()(0

01 tsigntitin

λλλ

⋅⋅=

)()( titi 12 −=


Modeling of Inrush Currents: Saturable-Core Transformer Model

39

v1(t)

v2(t)

i1(t)

i2(t)

gL

igL(t)iL(t)

r1 L1 r2L2

v3(t)

i3(t)

v4(t)

i4(t)

+

-

e1(t) te1(t)

1:t

Im(t)

)()(1 te

dttd=

λ

)()()()()( 11

11121 tedt

tdiLtirtvtv ++=−

)()(

)()()( 13

23243 ttedt

tdiLtirtvtv ++=−

( ) )()()()( 10

0 tegtsigntiti L

n

m ⋅+⋅⋅= λλλ

0)()()( 31 =−+ tittiti m

)()( titi 12 −=

)()( 34 titi −=


Distribution Feeder Voltage Recovery: Full Transient Analysis


42





supportConclusions

43

Control of Voltage and Reactive Power

Generator VAR outputs Regulating transformers (ULTC) Shunt capacitors/reactors

Fixed capacitor banks (FC) Mechanically switched capacitor banks

(MSC)

Series capacitors Synchronous condensers


44

Control of Voltage and Reactive Power Static VAR (shunt) compensators (SVC)

Thyristor-switched reactor (TSR) Thyristor-controlled reactor (TCR) Thyristor-switched capacitor (TSC) Static synchronous compensator (STATCOM)

Static series compensators Thyristor-switched series capacitor (TSSC) Thyristor-controlled series capacitor (TCSC) GTO Thyristor-controlled series capacitor (GCSC) Static synchronous series compensator (SSSC) Phase angle regulator (PAR)

Combined compensators Unified power flow controller (UPFC)


45

Static VAR Systems

Static VAR compensator at CERN laboratories, Switzerland (www.triumf.ca)

Combination of SVC and switched shunt capacitors/reactors with coordinated outputs


46

SuperVAR Machine: Superconducting Synchronous Condenser

American Superconductor Corp. and Tennessee Valley Authority

Installed in Nov. 2005, in the TVA network, at Gallatin, TN (Hoeganaes Corp. steel plant)

(IEEE Spectrum, Jan. 2006)

www.amsuper.com


47

Mitigation of Voltage Recovery Problems

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 the optimal mix and placement of static and dynamic VAR resources applicable in large power systems, to improve voltage recovery and dynamic performance


Mitigation of Voltage Recovery Problems

Optimal allocation of new static and dynamic reactive resources (planning problem) Selection of location Selection of size Determination of optimal mix

(static/dynamic)

Optimal operation of existing static and dynamic reactive resources (operational problem)


Performance Indices

49

Voltage Security Index

∑

−=

j

n

j,step

j,avejjV V

VVWJ

2

( )dttzztzJt

tthreshold∫ −=

2

1

)())((

Voltage Recovery Index

dtdu

tdzdu

tzdJ t

t∫−=2

1

)())((

( )

+

++= ∑∑

==

n

iiRR

n

iidipdipiVV JfWJfWJfWXCWXJ

1,4

1,3,21 )()(

12))(( tttvJdip −=Voltage Dip Index

Annualized cost of additions

Performance penalties


Reactive Resource Allocation –Planning Problem

Selection of a limited number if states at each stage

Evaluation of cost for each state at each stage

Application of dynamic programming recurrence to define optimal transitions

Determination of optimal trajectory by backtracking

50

Cost*(X0,k+3)Cost*(X0,k+2)Cost*(X0,k+1)

Cost*(X1,k+3)Cost*(X1,k+2)Cost*(X1,k+1)Cost*(X1,k)

Cost*(XN,k)

State 0X0,k

Cost0,k

State 1X1,k

Cost1,k

State 2X2,k

Cost2,k

.

.

.

State NXN,k

CostN,k

States

State 0X0,k+1

Cost0,k+1

State 1X1,k+1

Cost1,k+1

State 2X2,k+1

Cost2,k+1

.

.

.

State NXN,k+1

CostN,k+1

Stages(years in the

planning horizon)

...

0 Timek k+1 k+2 k+3

State 0X0,k+2

Cost0,k+2

State 1X1,k+2

Cost1,k+2

State 2X2,k+2

Cost2,k+2

.

.

.

State NXN,k+2

CostN,k+2

State 0X0,k+3

Cost0,k+3

State 1X1,k+3

Cost1,k+3

State 2X2,k+3

Cost2,k+3

.

.

.

State NXN,k+3

CostN,k+3

Base Case

Cost*(XN,k+1) Cost*(XN,k+2) Cost*(XN,k+3)

Cost*(X2,k+3)Cost*(X2,k+2)Cost*(X2,k+1)Cost*(X2,k)

Cost*(X0,k)

[ ])()(min)( 1,,,*

1,*

++ →+= kikjkjjki XXTXCostXCost


Evaluation of Cost for Specific State at Specific Stage

Solution of base case optimal power flow

Selection of critical contingencies Optimal simulation of each critical

contingencies (operational problem) Computation of performance


Reactive Resource Allocation

Assume N resources (VAr modules or financial amount) and P candidate locations

Allocate the resources in the specific locations such that the combined performance index is minimized

Total number of cases

52

−−+

11

PPN


Reactive Resource Allocation Solution via

dynamic programming

Stage: Location State: Remaining

units of resource to be allocated

Optimal value function

Recurrence relation

Boundary conditions

Answer:53

n = Np1(N)

f1(N)

States(remaining units

of resource)

n = 0pk(0)

fk(0)

n = 1pk(1)

fk(1)

n = 2pk(2)

fk(2)

.

.

.

n = Npk(N)

fk(N)

Stages(location)

...

2 k k+1 P

n = 0pk+1(0)

fk+1(0)

n = 1pk+1(1)

fk+1(1)

n = 2pk+1(2)

fk+1(2)

.

.

.

n = Npk+1(N)

fk+1(N)

n = 0pp(0)

fP(0)

n = 1pp(1)

fP(1)

n = 2pp(2)

fP(2)

.

.

.

n = Npp(N)

fP(N)

...

1

n = 0pk(0)

fk(0)

n = 1pk(1)

fk(1)

n = 2pk(2)

fk(2)

.

.

.

n = Npk(N)

fk(N)


NiPN

i2)2(

1+− ∑

=

)(),( npnf kk

( )[ ])](|[min)( 1,...,1,0 kkknnk nNpnJnfk

−= +=

)(),( 11 NpNf

Operational Problem Assume specific dynamic VAr sources in a system Compute the optimal reactive injections--------------------------------------------------------------------- Classical optimal control problem

54

( )( )tptutytxg

tptutytxftx,),(),(),( 0,),(),(),()(

==

[ ]FFF ttytxJ ),(),( ϕ=

[ ]FFF tptytx ,),(),( 0 ψ=

[ ] [ ]{ }dtfxgvJ F

IF

t

t

TTt

T ∫ −+−+= λµψϕˆ

( ) ul htptutytxhh ≤≤ ,),(),(),(

Direct Transcription ( )

0)( ..,)(=

=Xcts

yxXF MMφ


Static and Dynamic Sensitivity Analysis

Static sensitivity Determine favorable locations based on

steady state criteria Rank system contingencies

Trajectory sensitivity Determine favorable locations based on

dynamic performance penalties


Static Sensitivity Analysis

56

)0.0,(),( 0 =−=∆ iinew uxJuxJJ

: initial state0xnewx : post-installation state

Complete nonlinear approach

PI Sensitivity Approach

ii

ududJJ ∆⋅=∆

( )0.0,, oo =−

∆+=∆ iii

i

uxJuududxxJJ

State Sensitivity Approach

∂

∂−

∂∂

=i

iT

i

i

i uuxgx

uuxJ

dudJ ),(ˆ),(

1),(),(ˆ−

∂∂

∂∂

=x

uxgx

uxJx iiT

PI Sensitivity

∂

∂

∂∂

−=−

i

ii

i uuxg

xuxg

dudx ),(),( 1

State Sensitivity


Trajectory Sensitivity Analysis

57

)()()()(tDxtCytBxtAx

uu

uu

+⋅=+⋅=

),,,(0),,,()(

uyxtguyxtftx

==

dudy

yh

dudx

xh

uh

dudh

⋅∂∂

+⋅∂∂

+∂∂

=

),,,( uyxthJ =

Numerical

Integration

),(0 uXG= 1

ˆ−

∂∂

⋅∂∂

=XG

XhX T

uGX

uh

dudh T

∂∂⋅−

∂∂

= ˆ


Synopsis

Understanding and characterizing voltage problems and develop mitigation techniques

Simulation of voltage recovery phenomena in transmission and distribution networks

Optimal allocation and operation of static and dynamic reactive support resources

Development and implementation of appropriate simulation tools Modeling Simulation Decision making


voltage recovery and optimal allocation of var … recovery and optimal allocation of var support...

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