ECEN 667 Power System Stability

Lecture 21: Modal Analysis

Prof. Tom Overbye

Dept. of Electrical and Computer Engineering

Texas A&M University

overbye@tamu.edu

Special Guest Lecture by TA Hanyue Li

1

Announcements

• Read Chapter 8

• Homework 5 is due on Tuesday Nov 12

2

Quickly Determining the b Vectors

• A key insight is from an approach known as the

variable projection method (from Borden, 2013) that

for any signal k

A. Borden, B.C. Lesieutre, J. Gronquist, "Power System Modal Analysis Tool Developed for Industry Use," Proc. 2013

North American Power Symposium, Manhattan, KS, Sept. 2013

i

1

ˆ ( ) ( )

And then the residual is minimized by selecting ( )

where ( ) is the m by n matrix with values

( ) if corresponds to a real eigenvalue,

and ( ) cos( ) and

i j

i j

k k

k k

t

ji

t

ji i j

e

e t

y α Φ α b

b Φ α y

Φ α

α

α

1 1 ( ) sin( )

for a complex eigenvalue; 1

Finally, ( ) is the pseudoinverse of ( )

i jt

ji i j

j

e t

t j T

α

Φ α Φ α

Where m is the number of measurements

and n is the number of modes

3

Aside: Pseudoinverse of a Matrix

• The pseudoinverse of a matrix generalizes concept of a

matrix inverse to an m by n matrix, in which m >= n

– Specifically this is a Moore-Penrose Matrix Inverse

• Notation for the pseudoinverse of A is A+

• Satisfies AA+A = A

• If A is a square matrix, then A+ = A-1

• Quite useful for solving the least squares problem since

the least squares solution of Ax = b is x = A+ b

• Can be calculated using an SVD T

T

A UΣV

A VΣ U

4

Least Squares Matrix Pseudoinverse Example

• Assume we wish to fix a line (mx + b = y) to three

data points: (1,1), (2,4), (6,4)

• Two unknowns, m and b; hence x = [m b]T

• Setup in form of Ax = b

1 1 1 1 1

2 1 4 so = 2 1

6 1 4 6 1

m

b

A

5

Least Squares Matrix Pseudoinverse Example, cont.

• Doing an economy SVD

• Computing the pseudoinverse

0.182 0.7656.559 0 0.976 0.219

0.331 0.5430 0.988 0.219 0.976

0.926 0.345

T

A UΣV

0.976 0.219 0.152 0 0.182 0.331 0.926

0.219 0.976 0 1.012 0.765 0.543 0.345

0.143 0.071 0.214

0.762 0.548 0.310

T

T

A VΣ U

A VΣ U

In an economy SVD the S matrix has dimensions of

m by m if m < n or n by n if n < m

6

Least Squares Matrix Pseudoinverse Example, cont.

• Computing x = [m b]T gives

• With the pseudoinverse approach we immediately see

the sensitivity of the elements of x to the elements of b

– New values of m and b can be readily calculated if y changes

• Computationally the SVD is order m2n+n3 (with n < m)

10.143 0.071 0.214 0.429

40.762 0.548 0.310 1.71

4

A b

7

2000 Bus System Example

• In PowerWorld open the case

ECEN460_TSGC_GenDrop

– This is a 2000 bus synthetic electric grid on the ERCOT

footprint that we use in our undergraduate and graduate classes

to study larger power systems

• Run the predefined transient stability contingency

– Response following the loss of two generators This is a synthetic power system model that does NOT represent the actual grid. It was developed as part of the US ARPA-E Grid Data research project and contains no CEII. To reference the model development approach, use:

For more information, contact abirchfield@tamu.edu.

A.B. Birchfield, T. Xu, K.M. Gegner, K.S. Shetye, and T.J. Overbye, "Grid Structural Characteristics as Validation Criteria for Synthetic Networks," IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 3258-3265, July 2017.

Potential Coal Plant RetirementsStatusMax MWBus Number

Note: this grid is fictitious and doesn't

represent the real Texas grid

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WACO 1

AUSTIN 2

PASADENA 3

ARLINGTON 1

M CKINNEY 3

JACKSONVILLE 1

KYLE

M ANSFIELD

GREENVILLE 1

HOUSTON 4

CYPRESS 1

HOUSTON 90

POINT COM FORT 2

LAKE JACKSON

SILVER

ROSCOE 2

TRINIDAD 1

AUSTIN 3

EL CAM PO

LEAGUE CITY

COPPERAS COVE

CEDAR CREEK 1

COLLEGE STATION 2

PFLUGERVILLE

CEDAR PARK

ANGLETON

SHERM AN 1

WINCHESTER

LEANDER 1

ABILENE 2

KENEDY

GRAND PRAIRIE 3

SAN ANTONIO 2

M IDLOTHIAN 1

BAY CITY

CUERO 2

FAIRFIELD 1

TEXAS CITY 1

SAN ANTONIO 37

TAFT 1

SPRING 2

POOLVILLEALLEN 1

VICTORIA 1

SNYDER 2

SEGUIN 1SUGAR LAND 2

ALVIN

M AGNOLIA 1

BRYSON 1

KILLEEN 4

COLUM BUS

CALDWELL

LAREDO 4

CONROE 5

LAPORTE

FANNIN

TYLER 4

PARIS 2

CORPUS CHRISTI 1

SAN JUAN

M ISSION 4

DENTON 1

M ERKEL 1

WACO 2

LAREDO 1

BREM OND

DAYTON

FLUVANNA 2

BRYAN 1

KILLEEN 3

STEPHENVILLE

WINGATE

FREEPORT 1

M ISSOURI CITY 2

NEWGULF

GRANBURY 2

BURNET

GRANBURY 1

SUGAR LAND 3

WILLIS 1

SAN ANTONIO 50

KATY 2

NURSERY

NEW BRAUNFELS 1

KELLER 2

CHANNELVIEW 1

ALEDO 1

SAN ANTONIO 22

GARLAND 1

JACKSBORO 1

PANHANDLE 2

ROCKDALE 1

SAN MARCOSHOUSTON 5

BAYTOWN 1

ODESSA 1

SAN ANTONIO 1

ELMENDORF

WILLIS 2

SAVOY

PANHANDLE 4

WICHITA FALLS 1

WADSWORTH

MARBLE FALLS 2

O DONNELL 1

MCCAMEY 1

PEARSALL

CORPUS CHRISTI 3

SHIRO

PORT LAVACA

FAIRFIELD 2

SARITA 1

ARMSTRONG 1

FLORESVILLE

VICTORIA 2

GEORGETOWN 3

GLEN ROSE 1

OLNEY 1

MISSION 1

GREGORY

GOLDTHWAITE 1

CORSICANA 2

CUSHING 1

MOUNT PLEASANT 2

DALLAS 1

GRAHAM

RICHARDSON 2

BROWNWOOD

TYLER 7

MOUNT PLEASANT 1

MT. ENTERPRISE

JEWETT 1

SAN PERLITA

AUSTIN 1

MONAHANS 1

FRANKLIN

OILTON

CHRISTINE

CHRISTOVAL

WHARTON 1

MIAMI

KERRVILLE

PALO PINTO 1

STERLING CITY 1

ROSCOE 5

DEL RIO BOERNE 2

MARION 1

BRENHAM

GALVESTON 1

LA GRANGE

BASTROP

PARIS 1

YOAKUM

RALLS 1

TEMPLE 1

LAREDO 7

SPRING 8

MCKINNEY 1

EAGLE PASS

LUFKIN 3

FREEPORT 2

HONDO

HERMLEIGHABILENE 1

ALBANY 1

ENNIS

RIESEL 1

BRIDGEPORT

KATY 3

THOMPSONS

8129

8130

8131

6078

6079

6080

563 MW

563 MW

563 MW

660 MW

660 MW

660 MW

Closed

Closed

Closed

Closed

Closed

Closed

8

2000 Bus System Example

• On the Transient Stability page select Results

from RAM, view the Bus page display just the bus

frequencies, right-click and select Modal Analysis

All Columns We’ll work with frequencies

in Hz rather than per unit

9

2000 Bus System Example

• Initially our goal is to understand the modal

frequencies and their damping

• Initially we’ll consider just one of the 2000 signals

Toggle the Include to

“No” for all the 2000

signals, except set

the first one to “Yes”

Once only the first

signal is selected,

click the Do Modal

Analysis

10

2000 Bus System Example, One Signal

• Note the modes and their damping. The signal

dialog indicates that we have fairly accurately

reproduced the first signal

Comparison of frequency for

the first bus

Calculated mode

information

11

2000 Bus System Example, Comparing the Other Signals

• Sort on the cost function to see the worst match

• Now we will rerun it including this bus frequency

signal as well; that is, now a total of two signals

Bus 8078 has the worst match

12

2000 Bus System Example, Now With Two Signals

• The left image shows the original match for bus 8078

(when not included), and the right when included

13

Updated Modes

• The calculated modes and damping have changed

slightly with the additional signal

14

2000 Bus System Example, Iterative Matrix Pencil

• Now run the Iterative Matrix Pencil method

15

2000 Bus System Example, Iterative Matrix Pencil

• With the inclusion of just ten signals (out of 2000)

quite accurate modal information is obtained for all

2000 frequency signals

This is the worst match (at bus 1072)

16

Takeaways So Far

• Modal analysis can be quickly done on a large

number of signals

– Computationally is an O(N3) process for one signal,

where N is the number of sample points; it varies linearly

with the number of included signals

– Number of sample points is automatically determined

from the highest desired frequency (The Nyquist-

Shannon sampling theory requires sampling at twice the

highest desired frequency)

– Determining how signals are manifested in modes is fast

(done when Include Reproduced is Yes)

17

Getting Mode Details

• Detailed information about each mode is available by

right-clicking on mode and selecting Show Dialog

18

Quickly Visualizing Results: Geographic Data Views (GDVs)

• GDVs were introduced into PowerWorld about twelve

years back to provide a quick way of visualizing

geographic information; they are ideally suited for

modal results

• To create a GDV oneline of the substations, go to Case

Information, Aggregation, Substations. Then right-

click in any column (Gen MW here) and select

Geographic Data View, Select Column then

Geographic Data View

• This displays the Geographic Data View Customization

Dialog

19

GDV Customization Dialog

• Select Browse to Open an Existing Oneline and select

NorthAmerica_Blank

• The Fields and

Attributes page can

be used to customize

the GDV appearance

• As an example of a

GDV showing the

substation gen MW,

select the Total Area field

• Select Add Geographic Data View Objects…

20

GDV Display Customization

• As created the GDV shows substation generation,

but we will shortly modify it to show modal

information

• Save the display with a

different name – say

ECEN460_SubGen

• To modify a GDV oneline,

just right-click on any

GDV display object and

select View Geographic

Data View Options

O DONNELL 1

BIG SPRING 1

O DONNELL 2

BIG SPRINGS

M CCAM EY 1

M ONAHANS 1

BIG SPRING 3

IRAAN 1

ODESSA 1

GOLDSM ITH

PANHANDLE 2

PARIS 2

WICHITA FALLS 1

VERNON 1

ARCHER 2

RALLS 1

PARIS 1

PANHANDLE 1

M IAM I

BLACKWELL

CHRISTOVAL

ABILENE 2

SNYDER 1

ROSCOE 2

ROSCOE 5

WINGATE

STERLING CITY 1

TRENT 1

NOLAN

HERM LEIGH

M ERKEL 1

ABILENE 1

FLUVANNA 2

SYNDER

ROSCOE 1

ROSCOE 3

SARITA 3

CHRISTINEFANNIN

SEBASTIAN 2

GREGORY

M ISSION 1

CORPUS CHRISTI 3

SANTA ROSA 1

TAFT 2

SARITA 1

ARM STRONG 1

CORPUS CHRISTI 1

SARITA 2

BRUNI 1

TAFT 1

SAN PERLITA

BROWNSVILLE 1

SEBASTIAN 1

DALLAS 1

GRAHAM

ARL INGTON 1

DENTON 1BRYSON 1

POOL VIL L E

OLNEY 1

GOLDTHWAITE 1

GLEN ROSE 1

TEM PLE 1

MCKINNEY 1

GRANBURY 1

DALLAS 2

RIESEL 1

BRIDGEPORT

ENNIS

ALBANY 1

GRANBURY 2

PALO PINTO 1

M IDLOTHIAN 1

BASTROP

AUSTIN 3

SAN ANTONIO 2

MARION 1

LA GRANGE

WINCHESTER

ROCKDAL E 1

SAN ANTONIO 1

SEGUIN 1

M ARBLE FALLS 2

ELMENDORF

AUSTIN 2

CEDAR CREEK 1

NEW BRAUNFELS 1

AUSTIN 1

PASADENA 2

NURSERY

VICTORIA 2

POINT COM FORT 2

BAYTOWN 2

WADSWORTH

DEER PARKLAPORTE

HOUSTON 5

HOUSTON 4

BAYTOWN 1PASADENA 3

BAYTOWN 3

WHARTON 1

FREEPORT 1

NEWGULF

PASADENA 1

THOMPSONS

CHANNEL VIEW 1

WIL L IS 1

HOUSTON 6

POINT COM FORT 1VICTORIA 1

LA PORTE

ALVIN

FREEPORT 2

FAIRFIELD 1

JEWETT 1

M T. ENTERPRISE

FRANKLIN

JACKSONVILLE 1

M OUNT PLEASANT 1

The GDV Styles can also

be shown by selecting

Onelines, List Display,

Geo Data View Styles

21

Transient Stability Toolbar

• One application of GDVs is wide area visualization

of time-sequenced results using the transient

stability toolbar

• These displays can then be made into a movie by

either

– Capturing the screen as the contours are creating using

screen recording software such as Camtasia

– Or having Simulator automatically store the contour

images as jpegs and then creating a movie using software

such as Microsoft Movie Maker

22

Transient Stability Toolbar

• The toolbar uses stored transient stability results,

and hence it is used only after the transient stability

solution has finished

– It can be used with results that are either saved RAM or

saved to the hard drive

– It has recently been updated to show PMU and modal

analysis data

• It requires having a oneline with objects associated

with the desired results (such as buses, generators,

substations, etc)

– Again these can be auto-created with GDVs

23

Showing the Toolbar

• The toolbar is shown by either

– Selecting Add-ons, Stability Case Info, Show Transient

Contour Toolbar

– On the Transient Stability Analysis Form select the Show

Transient Contour Toolbar button at the bottom of the form

• The toolbar is only fully enabled if there are results to

contour

• We’ll demonstrate it using the previous GDV, except

changing the object size to be almost not visible

The oneline is saved as ECEN460_SubSmall

24

Toolbar

Use to control

time

Only objects

with associated

time-varying

data should be

selected

Use options to

control the contour

and the storage of

images for movies

25

Toolbar: Voltage Deviation

Sometimes the best

field to visualize is

the deviation from the

initial values (here the

figure shows voltage

magnitude deviation)

Designing synthetic

grids helps to

highlight the

complexities in

designing the actual

grid!!

26

Visualizing Mode Magnitude and Angle Components

• GDVs can be used to quickly visualize information

about the modes

• To visualize mode magnitude and angles, we’ll use the

substation custom fields to store the mode values

• Rerun modal analysis using the substation average

frequency field. That is,

– Start from the Results From RAM, Substation display

– Show just Frequency Average field

– Right-click to select “Modal Analysis All Columns” to show

the Modal Analysis Form

– Select the Iterative Matrix Pencil method and click on Do

Modal Analysis; view the Mode Details dialog for 0.63 Hz

27

Visualizing Mode Magnitude and Angle Components

• We’ll use the custom fields

to tell the substation objects

about this mode’s data

– Select the Angle field, set the

Custom Floating Point Field to

1; click Transfer Results

– Repeat, except click on the

Magnitude Unscaled field and set the Custom

Floating field to 2

• Create a new GDV display using the

ECEN460_SubSmall display (the customizations

are on the next slide)

28

GDV Visualization of Mode Magnitude and Angle Components

• Set the GDV Style as

– On the General Display Options page set the Style to

Arrows and the Text to Show field to None.

– On the Fields and Attributes page

• Total Area uses the Custom Float 2 field (magnitude); a ballpark

largest size should be 50000.

• Rotation Angle uses the Custom

Float 1 field (angle)

• Line Thickness uses the Custom

Float 2 field (magnitude)

• Once a oneline has been setup,

it can be saved for repeated use

• Save the oneline as

ECEN460_Modes

29

Visualization of 0.63 Hz Mode

ODES

SA 2

PRES

IDIO

2

O D ONNEL L 1

BIG

SP

RIN

G 5

VAN

HO

RN

IRAAN 2

PRES

IDIO

1

SANDERSO

N

MONAHAN

S 2

GRA

NDF

ALLS

MARFA

GA

RD

EN C

ITY

ODESS

A 4

NO

TREES M

IDLAND 4

BIG

SP

RIN

G 1

O DONNEL L 2

ODESS

A 6

BIG

SP

RIN

GS

MID

LAND 2

COAHOMA

MID

LAN

D 3

ALPI

NE

FORT D

AVIS M

CCA

MEY

1

BIG

SP

RIN

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CRANE

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BIG

LA

KE

MID

LAND 5

OZO

NA

MONAHAN

S 1

ST

AN

TO

N

ODONNE LL

LEN

ORAH

IRAAN 3

BIG

SP

RIN

G 6

ODES

SA 3

BIG

SP

RIN

G 3

BIG

SP

RIN

G 7

MID

LAN

D 1

IRAAN 1

ODES

SA 1

FOR

T ST

OCKT

ON

2

FOR

T ST

OCKT

ON

3

BIG

SP

RIN

G 2

KERM

IT

PECOS

SHEF

FIE

LD

MC

CAM

EY 2

LAMES

A

GO

LDSM

ITH

RALLS 2PARIS 3

SAVOY HONEY GROVE

MEMPHIS

IOWA PARKVERNON 2

PANHANDL E 2

CHILDRESS

FORESTBURG

SHERMAN 3

PANHANDL E 4

WICH ITA FALL S 6

PARIS 2

WICH ITA FALL S 4

WHEELER

DEPORT

WICH ITA FALL S 3

SILVERTON

BONHAMMATADOR

ARTHUR CITY

LEONARD

GORDONVILLE

KNOX CITY

WICH ITA FALL S 7

LINDSAYECTOR

SUNSET

CLARKSVILLE

PANHANDL E 3

SADLERWHITESBORO

SHEP PARD AFB

POTTSBORO

BELLS

CROSB YT ON

ELECTRA

PATTONVILLE

HENRIETTA

PANHANDL E 6

BOWIE

WICH ITA FALL S 1

WHITE DEER

ROXTON

VERNON 1

DENISON 2 SUMNERGAINESVILLE

ASPERMONT

WICH ITA FALL S 5

WINDTHORST

HOWE

ARCHER 2COLLINSVILLE

JAYTON

MUENSTER 1

PADUCAH

NOCONA

DODD CITY

WICH ITA FALL S 2

MONTAGUE

SPUR

ARCHER 1 ANNONA

SEYMOUR

DENISON 1

PANHANDL E 5

RALLS 1

SHERMAN 1WHITEWRIGHT

SHERMAN 2

ROCHESTER

BURKB URNETT

MUENSTER 2

ARCHER CITY

PETROLIA

PARIS 1

BAGWELLTELEPHONE

PANHANDL E 1

VAN ALSTYNE

TIOGAVALLEY VIEW

MIAMI

TRENTON

HASKELL

SAINT JO

KE

RR

VIL

LE

SWEETWATER 4

LEAKEY

HAMLIN

ME RKEL 3

COL EMAN

SO

NO

RA

CAMP WOOD

LLAN

O

WILLOW CITY

ABILENE 4

KING SLAND

ROSCOE 6

BLAC

KWELL

CHR

IST

OVAL

LUEDERS

COL ORAD O CITY

ST AMF ORD

ELDORADO

SAN SABA

ABILENE 2

LOMETA

SAN

AN

GELO

1

TUSCOLAABILENE 7

SNYD

E R 1

CHE ROK EE

R ICH LAN D SPR IN GS

ROSCOE 2

BALLI

NGER

SILVE R

ROSCOE 5

ANSON

DEL R

IO

HUNT

WINGATE

STER

LIN

G C

ITY

1

ROSCOE 4

BRADY

TRENT 1

OVALO

NOLAN

MILE

S

BRACKETTVILLE 2

M ASON

BRACKETTVILLE 3

ABILENE 6

HERMLEIGH

ME RK EL 1

LORAINE 1ABILENE 1

ROCKSPRINGS

ME RKEL 2

FLUVANNA 2

DYE SS AFB

HAWLEY

BU C HA NA N D A M 2

ABILENE 3

SWEETWATER 5

BU C HA NA N D A M 1

INGR

AM

KEMPNER

FLU VANNA 1

STER

LIN

G C

ITY

2

SAN

AN

GELO

2

ROTAN

SY NDE R

BRACKETTVILLE 1

ROSCOE 1

UVALDE

FRED

ERICK

SBUR

G

ABILENE 5

SABINAL

SWEETWATER 2

LORAINE 2

ROSCOE 3

SA

N A

NG

ELO

3

TUSOCOLA

EDEN

SNYDE R 2

SWEETWATER 1

MEN

ARD

WINTERS

GOLDSBORO

SWEET

WA TER 3

GOODFEL

LOW AFB

SANTA ANNA

ROWE NA

TRENT 2

JUNCTIO

N

LAMPASAS

LAUGHLIN A F B

EDINBURG 3

LAREDO 7

BRUNI 2

MCALLEN 2

POT

EET

SARITA 3HEBBRONVILLE

PROGRESO

EDINBURG 2

EAGLE PASS

CHARLOTTE

CORPUS CHRISTI 4

WOODSBORO

MERCEDES

CORPUS CHRISTI 16

EDCOUCH

ENCINAL

MOORE

CHRISTINE

MCALLEN 3

FANNIN

SAN YGNACIO

KINGSVILLELAREDO 6

HIDALGO

CORPUS CHRISTI 18

CRYSTAL CITY

SINTON

BEEVILLE

MISSION 4SAN JUAN

FALCON HEIGHTS 2

FALCON HEIGHTS 1

LA JOYASEBASTIAN 2

BISHOP

ARMSTRONG 2

SKIDMORE

CORPUS CHRISTI 9

GREGORYCORPUS CHRISTI 2

WESLACOHARLINGEN 1

ELSA

CORPUS CHRISTI 17

BROWNSVILLE 2

CORPUS CHRISTI 10

MISSION 1

CORPUS CHRISTI 8

MISSION 3

CORPUS CHRISTI 3

RIO HONDO

CORPUS CHRISTI 5

PHARRSANTA ROSA 1

ODEM

CORPUS CHRISTI 13

LAREDO 3

OLMITO

EDINBURG 1

TAFT 2

PORT ISABEL

DONNA

INGLESIDE

CORPUS CHRISTI 11

LOS FRESNOS HARLINGEN 2

RIVIERA

LAREDO 1

JOURDANTON

GEORGE WEST

ROMA

CORPUS CHRISTI 12

SANTA MARIA

MCALLEN 1

CORPUS CHRISTI 7

FALFURRIAS

PETTUS

BROWNSVILLE 3

ALICE

RIO GRANDE CITY

PREMONT

FREER

PEARSALL

SANDIA

SARITA 1

DILLEY

ORANGE GROVE

PORTLAND

LAREDO 4

THREE RIVERS

ROCKPORT

CORPUS CHRISTI 15

ARMSTRONG 1

PENITASSULLIVAN CITY

CORPUS CHRISTI 6

MISSION 2

CORPUS CHRISTI 1

GRULLA

SARITA 2

HARGILL

CORPUS CHRISTI 14

MATHIS

ARANSAS PASS

SAN BENITO

PLEASANTON

REFUGIO

LYTLE

LAREDO 2

BRUNI 1

GOLIAD

SAN DIEGOPORT ARANSAS

TAFT 1

SAN PERLITA

ZAPATA

SANTA ROSA 2

COTULLA

QUEMADO

CARRIZO SPRINGS

LAREDO 5

RAYMONDVILLE

BROWNSVILLE 1

LEMING

SEBASTIAN 1

GARLAND 5

KERENS

LIPAN

DALLAS 24

DENTON 4

ALBANY 2

DALLAS 21

PLANO 2

THORNTON

DENTON 5

EARLY

DECATUR

KELLER 2

PONDER

JACKSBORO 1

ALEDO 1

GAT ESV ILL E 4

CLEBURNE 2

FORT WORTH 1 4

FORT WORTH 2 3

FORT WORTH 1

TROY

SPRINGTOWN

PRINCETON

DALLAS 27DALLAS 1

ROYSE CITY

BOYD

ST EPHE NV ILLE

MANSFIELD

GREENVILLE 1PLANO 4

GROESBECK

GRAHAM

MABANK 2

ARLINGTON 1DALLAS 41

WACO 3

DALLAS 19

BANGS

PROSPER

EVANT

TERRELL 1

AUBREY

WACO 4

LORENA

FORT HOOD

ARLINGTON 5

MCKINNEY 3

DALLAS 38

ARLINGTON 9

MILFORD

CLIFTON 1

DALLAS 15

FROST

GRANDVIEW

HASLET

BARDWELL

LAKE DALLASPLANO 1

EULESS 2

FORT WORTH 2 4

FLOWER MOUND 2

DALLAS 11

COMANCHE

RICHARDSON 2

FORT WORTH 8

IRVING 1

DALLAS 44LANCASTER 1

PALO PINTO 2

ADDISON

HICO

KRUM

KAUFMAN

CADDO MILLSPLANO 7

FORT WORTH 5

BAIRD

PERRIN

BROWNWOOD

DALLAS 6

MARLIN

ARLINGTON 6

DALLAS 8

JONESBORO

DALLAS 33

GO LDTH WAITE 2

PLANO 5

BELTON

TOLAR

BRECKE NRIDGE

EASTLAND

COOPER

WACO 1

DALLAS 42

GAT ESV ILL E 1

ANNA

MOODY

FORT WORTH 2

WACO 6

GRAND PRAIRIE 1

ROWLETT 2

PLANO 6

IRVING 5DALLAS 7

BRYSON 2

MORGAN

SACHSE

PARADISE

RIO VISTA

STRAWN

MAY

DESOTO

MESQUITE 2

ROSEBUD

DENTON 2

DALLAS 4

AXTELL

NOLANV ILLE

DENTON 1

DALLAS 39

DUBLIN 1

MERTENS

WEATHERFORD 1

PLANO 3

MALONE

DENTON 3

GRAFORD

ARLINGTON 8

ROCKWALL 1

CORSICANA 2

GARLAND 2

GAT ESV ILL E 2

CARROLLTON 3

FORT WORTH 2 6

RHOME

VENUS

OLNEY 2

CARROLLTON 1

DALLAS 20

GORDON

MERIDIAN

KILLEEN 4

DUNCANVILLE 1

ARGYLE

BRYSON 1

FORT WORTH 1 2

MINERAL WELLS

COOLIDGE

RICE

DALLAS 35

POOLVILLE

WAL NUT SPRINGS

BEDFORD 2

HUBBARD

FERRIS

GARLAND 4

DALLAS 32

PILOT POINT

FORT WORTH 6

N OR TH RIC H LAN D H ILL S 2

JUSTIN

SCURRY

SANGER

KILLEEN 1

MABANK 1

MEXIA

CARROLLTON 2

NEVADA

ZEPHYR

TERRELL 2DALLAS 14

DALLAS 13

DALLAS 28

HAMILTON

ARLINGTON 7

EULESS 1

FRISCO 1

SALADO

OLNEY 1

GO LDTH WAITE 1

FORT WORTH 2 2

GRAPEVINE

GLEN ROSE 2KOPPERL

BEDFORD 1FORT WORTH 2 9 DALLAS 5

GRAND PRAIRIE 2

DALLAS 22

FLOWER MOUND 1ROANOKE

MILLSAP

FORRESTONITALY

FORNEY

LAVON

GLEN ROSE 1DE LEON

KELLER 1

CEDAR HILL

MC GREGOR

COMMERCE

MIDLOTHIAN 2

WEST

BARTLETT

DALLAS 16

COPPELL

WILMER

DAWSON

MART

LITTLE E LM

TEMPLE 2TEMPLE 1

QUINLAN

CHICO

DUBLIN 2

ARLINGTON 2

WOLFE CITY

DALLAS 26

WHITNEY

FORT WORTH 2 5FORT WORTH 2 7

JOSHUA

DALLAS 12

MCKINNEY 1ALLEN 1

ALEDO 2

THE COLONY

SUNNYVALEDALLAS 10

FORT WORTH 4 DALLAS 30

AZLE

HUTCHINS

VALLEY MILLS

DENTON 6

MESQUITE 3

GRANBURY 1

DALLAS 2

MOUNT CALM

RIESEL 2

DALLAS 17

ARLINGTON 4

CELINA

FORT WORTH 1 6

RED OAK

CLYDE

IRVING 2ROWLETT 1

JACKSBORO 2

GRANBURY 3

SOUTHLAKE

DALLAS 29

LEWISVILLE 1

DALLAS 36

HILLSBORO

FORT WORTH 1 5

CISCO

ITASCA

LANCASTER 2

GARLAND 1DALLAS 9

DUNCANVILLE 2

MESQUITE 1

MAYPEARL

GARLAND 3

ALLEN 2

RIESEL 1

BRIDGEPORT

LEWISVILLE 2

SEAGOVILLE

DALLAS 31HALTOM CITY

BALCH SPRINGSFORT WORTH 1 3DALLAS 43

KILLEEN 2

WACO 5

FORT WORTH 9

ENNIS

DALLAS 3

RICHARDSON 1

FORT WORTH 2 1

WACO 2

ROGERS

CHINA S PRING

ALBANY 1

MCKINNEY 2

GRANBURY 2

CLIFTON 2

FARMERSVILLE

ARLINGTON 10

KILLEEN 3

WYLIE

PALO PINTO 1

FORT WORTH 1 0

JOSEPHINELEWSIVILLE

DALLAS 40

KEMPWAX AHACHIE 2

HURSTFORT WORTH 1 8

WEATHERFORD 2

ROCKWALL 2WEATHERFORD 3

FORT WORTH 3

NEWARK

SANTO

DALLAS 18

ELM MOTT

FORT WORTH 1 7

WAX AHACHIE 1

ARLINGTON 3FORT WORTH 2 0

FORT WORTH 7FORT WORTH 1 1

MIDLOTHIAN 1

HEWITT

ARLINGTON 11

GRAND PRAIRIE 3

MELISSA

COPPERAS COVE

CLEBURNE 1

GREENVILLE 2

ALVORD

GORMAN

COLLEYVILLE

HOLLAND

TEMPLE 3

DALLAS 34

IRVING 4

FRISCO 2

GAT ESV ILL E 3

DALLAS 25DALLAS 37

FORT WORTH 1 9

WOODWAY

MINGUS

THRO CKMO RTON

NEMO

ABBOTT

ALVARADO

CORSICANA 1

HARKER H EI GHTS

IRVING 3

BURLESON

FORT WORTH 2 8

DALLAS 23

N OR TH RIC H LAN D H ILL S 1

CROSS PLA INS

BL OOM ING GROVE

CON

VE

RS

E

SAN ANTONIO 46

BASTROP

MAR

ION 2

PA IGE

SAN ANTONIO 42

WAELDER

ATAS

COSA

SAN ANTONIO 40

CIB

OLO

AUS TIN 6

AUS TIN 25

H OR SE SH O E BA Y

LEANDE R 1

AUS TIN 10

SAN ANTONIO 31

SAN ANTONIO 49

ELGIN

AUS TIN 31

RUNGE

YOAKUM

AUS TIN 37

PFLUG ER VILL E

SAN ANTONIO 12

AUS TIN 3

SAN ANTONIO 48

SAN ANTONIO 19

SAN ANTONIO 2BE

RGH

EIM

NEW BRAUNFELS 2

D HANIS

SAN ANTONIO 24

AUS TIN 7

BRENHAM

BERTRAM

MAR

ION 1

LA G

RANGE

M AR BL E FA LLS 1

ROU ND RO CK 1

WIM

BER

LEY

GON

ZALE

S

SAN ANTONIO 11

SAN ANTONIO 14

HUTTO

LULI NGSAN ANTONIO 21

AUS TIN 12

SAN ANTONIO 15

ADK

INS

WIN

CHES

TER

ROCKDALE 1

NAT

ALIA

AUS TIN 20

SAN ANTONIO 43

SAN ANTONIO 5

TAYLOR

SMI THVIL LE

CEDAR PARK

SAN ANTONIO 25

HALLETTSVILLE

AUS TIN 5

NEW BRAUNFELS 3

SUTHERLAND SPRINGS

AUS TIN 34

CAMERON

SAN A

NTONIO

35

MED

IN

A

M ANCHACA

SPRIN

G B

RAN

CH

AUS TIN 19

FLATONIA

GRANGER

FAYE

TTEVIL

LE

G EO RG ET O WN 2

AUS TIN 35

PIPE CR

EEK

RED ROCK

JARRELL

SAN A

NTONIO

1

SAN ANTONIO 36

SCH

ULEN

BURG

SAN ANTONIO 22

DRIPPING SPRINGS

SAN ANTONIO 26

M AR BL E FA LLS 3

BO

ERN

E 1

WA

SHING

TON

SC

HER

TZ

AUS TIN 8

SEGU

IN 1

SAN ANTONIO 45

SAN ANTONIO 52

CUERO 2

SAN ANTONIO 10

AUS TIN 16

AUSTIN 39

SAN ANTONIO 51

SAN ANTONIO 41

LA VER

NIA

MANOR

SAN ANTONIO 38

GIDDING

S

AUS TIN 32

AUS TIN 38

AUS TIN 30

AUS TIN 28

COL

UMB

US

SHIN

ER

SAN ANTONIO 3

CALDWELL

AUS TIN 18IND

USTRYDAL E

SAN ANTONIO 37

SAN ANTONIO 18

SAN ANTONIO 33

MAR

TIND

ALE

SAN ANTONIO 17

SAN ANTONIO 23

AUS TIN 29

L EXI NGTON

LEANDE R 2

AUS TIN 15

MICO

M AR BL E FA LLS 2

AUS TIN 36

LOCKHART

AUS TIN 17

SAN ANTONIO 13

FALLS CITY

AUS TIN 27

AUS TIN 23

SAN ANTONIO 27

BLAN

CO

AUS TIN 24

SAN

MA

RCO

S

BUDA

DEL VALLE

AUS TIN 4

SAN ANTONIO 50

BURLINGTON

AUS TIN 13

ELM

ENDO

RF

SAN ANTONIO 6

CAN

YON

LAK

E

JO H NS ON CIT Y

YORKTOWN

AUS TIN 2

LACKLAND A F B

L IBERTY HIL L

KENEDY

CUERO 1

SAN ANTONIO 20

VON

ORM

Y

KYL E

SEGU

IN 2

UNIVERSAL CITY

SPICEWOO D

SAINT HEDWIGS

AN ANTONIO 16

BOE

RNE 2

SAN ANTONIO 32

AUS TIN 26

AUS TIN 21

SAN ANTONIO 44

CEDAR CREEK 1

NEW BRAUNFELS 1

AUS TIN 11

SAN ANTONIO 39

HON

DO

SAN ANTONIO 4

NEW

ULM

SAN ANTONIO 8

AUS TIN 33

BU

LVER

DE

G EO RG ET O WN 3

SAN ANTONIO 9

FLORESVILLE

SAN ANTONIO 29

SAN A

NTONIO

7

BUCKHOLTS

SOMERVIL LE

WE

IMAR

COM

FOR

T

NORDHEIM

AUS TIN 9

AUS TIN 22

HELO

TES

SAN ANTONIO 47

BA

NDE

RA

SAN ANTONIO 30

ROCKD ALE 2

CEDAR CREEK 2

NIXO

N

CHA

PPELL H

ILL

R OU N D R OC K 3

AUS TIN 14

OILTON

AUS TIN 1

SAN ANTONIO 28

ROU ND RO CK 2

ROU ND RO CK 4

SAN ANTONIO 34

DEVINE

G EO RG ET O WN 1

BURNE T

INEZ

PASADENA 2

PRAIRIE VI EW

CONROE 1

SIMONTO

N

HOUSTON 80

HOUS TON 1

HOUS TON 9

HOUSTON 73

NURSERY

BACL IFF

SP RING 1

HOUSTON 72

PORT LAVACA

CONROE 5

SUGAR L AND 3

DAYTON

VICTORIA 2

KATY 1

MAGNOL IA 2

HOUSTON 29

MISS

OUR

I CITY

1RICHMO

ND 1

HOUSTON 11

HOUSTON 33

SP RING 8

POINT COMFORT 2

HOUSTON 28

EL C

AMP

O

HOUSTON 12

PORT O CONNOR

DICKINSONGALVESTON 1

HOUSTON 10

BAYTOWN 2

L EAGUE CITY

LIVERPO

OL

HOUSTON 16

SUGAR L AND 1

RICHMO

ND 2

HOUSTON 82

PINEHURST

PE ARLAND 2

HOUSTON 47

HOUSTON 79

HOUSTON 49

WADSWORTH

HOUSTON 18

HOUSTON 88

TOMBALL 1

HOUSTON 22

DEER P ARKWALLI S

HOUSTON 67

SP RING 4

SOU

TH HO

UST

ON

HOUSTON 3

SEADRIFT

HOUSTON 34

SUGAR L AND 2

CONROE 7

LOLIT

A

LAPORTE

NEW CANEY

HOUSTON 14

HOUS TON 2

HOUSTON 78

HOUSTON 40

BLOOMINGTON

WILLIS 2

HOUSTON 51

PALA

CIOS

CONROE 4

HOUSTON 52

SP RING 2

KINGW

OOD 1HO

USTON 5

TEXAS CITY 1

KINGW

OOD 2

HOUS TON 8

SP RING 7HOUSTON 41

FRESNO

HOUSTON 48

HOUSTON 58PATTISON HO

USTON 77

MISS

OUR

I CITY

2

HOUSTON 4

TOMBALL 2

CYPRESS 1

HEMPSTEAD

WEST COLUMBIA

BAYTOWN 1

SANTA FE 1

HOUSTON 27

SP RING 5

HOUSTON 46

HOUSTON 23

MONTGO

MERY

HUN

GER

FORD

HOUSTON 84

HOUSTON 83

HOUSTON 24

HOUSTON 76

HOUSTON 43

HOUSTON 81

HOUSTON 35

HOUSTON 90

BEASLEY

HOUS TON 7

LA MARQUE

WALLE R

PORTE R

CYPRESS 2

HOUSTON 30

FUL SHEAR

LIBERTY

WINNIE

HOUSTON 63

HOUSTON 70

HOUSTON 55

CROSBY

MANVEL

HOUSTON 54HO

USTON 89

DAM

ON

HOUSTON 74

HOUSTON 26

HOUSTON 32

HOCKLEY

ANGLETON

CONROE 3

HOUSTON 39

HOUSTON 59

PE ARLAND 1

HOUSTON 56

HOUSTON 13

CHA

NNEL

VIEW

2

BAY C

ITY

PASADENA 3

BROOKSHIRE

HOUSTON 25

TE XAS C IT Y 2

VICTORIA 3

HOUSTON 36

GAN

ADO

MONT BEL VIEU

BAYTOWN 3

SP RING 6 HUMBLE 2HO

USTON 69

SANTA FE 2

PORT BOLIVAR

HOUSTON 42

HOUSTON 85

HOUSTON 50

HOUSTON 62

EAG

LE LAKE

CONROE 2

WEBSTE R

SWEEN

Y

BLESS

ING

WHAR

TON 1

LAKE

JACK

SON

BELL VIL L E

KATY 4

HOUSTON 38 FR

IEND

SWO

OD

HOUSTON 17

HOUSTON 66

FREEP ORT 1

NEW

GU

LF

HOUSTON 60

PASADENA 1STAFF ORD

EDN

A

HOUSTON 21

NEED

VILLE

SEALY KATY 3

HOUSTON 37

HOUSTON 19

THOMPSONS

HOUSTON 61

GALVESTON 3

EAST

BER

NAR

D

HOUSTON 20

CHA

NNEL

VIEW

1

FREEP ORT 3

HOUSTON 15

HOUSTON 31

ROSHARON

HOUSTON 44

HOUSTON 75

HOUSTON 71

HUMBLE 1

HOUSTON 68

WILLIS 1

BAYTOWN 4V

AN V

LECK

HUFFMAN

HOUSTON 53HO

USTON 91

ROSENBERG

PASADENA 5

HOUSTON 45

HOUS TON 6

PASADENA 4

HOUSTON 57

GALVESTON 2

POINT COMFORT 1

KATY 2

SP RING 3

DAN

BUR

Y

WH

ARTO

N 2

VICTORIA 1

HOUSTON 64

HOUSTON 65

LA PORTE

MAGNOL IA 1

ALVIN

HOUSTON 86

BRA

ZORIA

HOUSTON 87

FREEPORT 2

CONROE 6

MADISONV ILLE

NORMANGEE

ARP

CENTERVILLE

LUFKIN 3

FLINT

BULLARD

CHANDLER

PLAN

TERS

VILLE

MARQUEZ

TENNESSEE COL ONY

ANDERSON

MINEOLA

LOVELADY

HUNT INGTON

BRYAN 4

FRANKSTON

TYLER 4

BEN WHEELER

GARRISON

ETOILELUFKIN 1

PALESTINE 1

LINDALE

WHITEHOUSE

OVERTON

GRAND SALINE

SHIRO

MIDWAY

SULPHUR SPRINGS

VAN

MURCHISON

NACOGDOCHES 2

DIBOLL

FAIR FIELD 1

TYLER 8ATHENS 1

BUFFALO

RUSK

WINONA

FAIR FIELD 2CUSHING 2

NACO GDOCH ES 3

FAIR FIELD 3

TRINIDAD 2

DIKE

JEWETT 1

PALESTINE 2

EMORY

MT. ENTERPRISE

TYLER 9

FRANKLIN

TYLER 3

ELKHART

WORTHAM

OAKWOOD

COOKVILLE

COL

LEGE ST

ATION

1

EDGEWOOD

CUSHING 1

TYLER 2

ALTO

CALVERT

JEWETT 2

NAVASOTA

MOUNT P LEASANT 2

SCROGGINS

WILLS POINT

ATHENS 2

POINT

LUFKIN 2

JACKS ONV ILLE 1

CROCKETT

HENDERSON 1

BRYAN 3

JACKS ONV ILLE 2

BRYAN 5

MALAKOFF

CUMBY

NACO GDOCH ES 1

MOUNT P LEASANT 1

TEAGUE

TYLER 7

TYLER 6

YANTIS

POLLOK

TYLER 1

COL

LEGE ST

ATION

2

HEARNE

MOUNT VERNON

IOLA

STREETMAN

COMO

HENDERSON 2

TRINIDAD 1

ZAVALLA

GRAPELAND

BRYAN 2

CANTON TYLER 5

BREMOND

BRYAN 1

In this display the

arrows show the

magnitude and angle

(direction) for the

mode at each

substation.

However, the

problem is there are

too many arrows!

The solution it so

dynamically prune

the display using the

GDV Options,

Pruning command

30

GDV Display Pruning

• This page allows GDVs to

be selectively displayed

• Select Do Pruning to

modify the oneline so an

invisible grid is

added to the display

and only one GDV arrow

is visible in each grid area

• I also added some color, using a circular color map

to highlight the angles

31

Visualization of 0.63 Hz Mode

Again save the

oneline, now as

ECEN460_Modes;

it can then be

used to quickly

visualize the other

modes

To show other

modes, just go to the

Mode Details dialog

for a different

frequency and again

transfer the angle and

magnitude

32

Visualization of 0.76 Hz Mode

33

Visualizing the Source of Oscillations

• For showing the source of sustained oscillations

assume zero damping and recognizing that the source

of the oscillation will have a leading phase angle, shift

all the fk,j so it is zero at the bus with the largest Ak,j

and set t=0. Then

• Then approximate the mode j power flow on all lines

between buses m and n as

• The location of the oscillation has positive net power

, , ,cosk j k j k jA f

,m ,m ,n ,n

1( cos cos )mn k k k k

mn

P A AX

f f

34

Visualizing the Source of Oscillations in PowerWorld

• Now done mostly automatically in PowerWorld

• To apply the approach with transient stability

– Run the transient stability storing the bus angles

– Do modal analysis on the bus angles; technique is only

applicable if there are lightly or negatively damped

modes

– Select the desired mode to display the Mode Analysis

Mode Details Dialog

– When analyzing just bus angles a Visualization

Source of Oscillations button appears; click to

transfer flows to line and substation custom float1

35

Visualizing the Source of Oscillations in PowerWorld Example

As an example, a 1Hz sustained oscillation was introduced

into the 2000 bus system

Transient Stability Time

(Sec): 15.000

36

Visualizing the Source of Oscillations in PowerWorld Example

37

Visualizing the Source of Oscillations in PowerWorld Example

This button only

appears when doing

modal analysis on bus

angles. Clicking on it

1) calculates the mode

power flow on each

line and places the

value in the line’s

custom float 1 field, 2)

calculates the net mode

power injection at each

substation and stores it

in the substation’s

custom float 1 field.

38

Visualizing the Source of Oscillations in PowerWorld Example

DALLAS 1

BRIDGEPORT

BASTROP

M ARION 1

LA GRANGE

WINCHESTER

SAN ANTONIO 1

SEGUIN 1

ELM ENDORF

NEW BRAUNFELS 1

AUSTIN 1

PASADENA 2

BAYTOWN 2

WADSWORTH

DEER PARK

SUGAR LAND 2

LAPORTE

HOUSTON 5

BAYTOWN 1

PASADENA 3

WHARTON 1

FREEPORT 1

PASADENA 1

THOMPSONS

CHANNELVIEW 1

WILLIS 1

HOUSTON 6

POINT COM FORT 1

LA PORTE

ALVIN

FREEPORT 2

M T. ENTERPRISE

The source of the

forced oscillation

is correctly

identified (bus

7356) at the

Thompsons

substation

This is a substation GDV, in

which size of objects is

proportional to the custom float

1 field; a red/blue color map

shows the direction (red is the

source)

39

Visualizing the Source of Oscillations in PowerWorld Example

33%A

Amps

22%A

Amps

30%A

Amps

42%A

Amps

6%A

Amps

27%A

Amps

5%A

Amps

9%A

Amps

22%A

Amps 7%

A

Amps

32%A

Amps

27%A

Amps

9%A

Amps

31%A

Amps

21%A

Amps

29%A

Amps

30%A

Amps 3%

A

Amps

39%A

Amps

28%A

Amps

3%A

Amps

26%A

Amps

32%A

Amps

6%A

Amps

31%A

Amps

3%A

Amps

39%A

Amps

2%A

Amps

31%A

Amps

17%A

Amps

35%A

Amps

28%A

Amps

26%A

Amps

HOUSTON 78

HOUSTON 44

HOUSTON 51

HOUSTON 45

PASADENA 2

HOUSTON 70

HOUSTON 63

M ANVEL

FULSHEAR

LA M ARQUE

HOUSTON 7

HOUSTON 13

HOUSTON 56

PEARLAND 1

HOUSTON 25

BROOKSHIRE

CHANNELVIEW 2

HOUSTON 59

HOUSTON 26

HOUSTON 74

HOUSTON 39 HOUSTON 32

HOUSTON 48

FRESNO

HOUSTON 77

HOUSTON 52

HOUSTON 8

HOUSTON 43

HOUSTON 24

BEASLEY

HOUSTON 35

HOUSTON 81

HOUSTON 83

HOUSTON 46

HOUSTON 27

HOUSTON 84

HOUSTON 23

HOUSTON 68

ROSENBERG

HOUSTON 91

HOUSTON 53

HOUSTON 31

HOUSTON 15

HOUSTON 20

HOUSTON 71

ROSHARON

PASADENA 5

HOUSTON 87

LA PORTE

HOUSTON 65

PASADENA 4

HOUSTON 6

HOUSTON 64

HOUSTON 85

SANTA FE 2

HOUSTON 62

HOUSTON 50

M ONT BELVIEU

HOUSTON 36

TEXAS CITY 2

BAYTOWN 3

WEBSTER

NEEDVILLE

HOUSTON 21

STAFFORD

HOUSTON 61

HOUSTON 19

HOUSTON 37

PASADENA 1

FRIENDSWOOD

HOUSTON 38

KATY 4

HOUSTON 60

HOUSTON 17

HOUSTON 3

HOUSTON 82

RICHM OND 2

SOUTH HOUSTON

M ISSOURI CITY 1

KATY 1

HOUSTON 12

SIM ONTON

SUGAR LAND 1

HOUSTON 34

HOUSTON 79

HOUSTON 88

RICHM OND 1

HOUSTON 18

HOUSTON 33

HOUSTON 49

DEER PARK

HOUSTON 22

HOUSTON 47

PEARLAND 2

HOUSTON 9

HOUSTON 73

HOUSTON 1

DICKINSON

HOUSTON 14

HOUSTON 10

HOUSTON 72

HOUSTON 16

BACLIFF

BAYTOWN 2

PASADENA 3

HOUSTON 4

HOUSTON 90

LEAGUE CITY

TEXAS CITY 1

SUGAR LAND 2

ALVIN

LAPORTE

MISSOURI CITY 2

SUGAR LAND 3

KATY 2

CHANNELVIEW 1

BAYTOWN 1

GALVESTON 1

KATY 3

THOMPSONS

Visualizing the lines’ Custom Float1 fields shows

the flow of the oscillation