Power System Application of Measurement-Based Modal Analysis

2020PSERCSummerTutorials

PowerSystemApplicationofMeasurement-BasedModalAnalysis

ThomasJ.OverbyeErleNye‘59ChairforEngineering

ExcellenceTexasA&[email protected]

2

Acknowledgments

•  Work presented here has been supported by a variety of sources including PSERC, the Texas A&M Smart Grid Center, DOE, ARPA-E, NSF, EPRI, BPA, and PowerWorld. Their support is gratefully acknowledged!

•  Slides also include contributions from many of my students, postdocs, staff and colleagues at TAMU, UIUC, other PSERC schools, and PowerWorld –  Special thanks to Bernie Lesieutre, Alex Borden and Jim

Gronquist!

3

It is a Great Time to be a Power and Energy Engineer!

•  Electric grids are in a time of rapid transition, with lots of positive developments and lots of engineering challenges!

•  It is good to keep in mind the essence of engineering, which is defined by Merriam- Webster’s as “The application of science and mathematics by which the properties of matter and the sources of energy in nature are made useful to people.”

4

Overview

•  To meet the challenges of today, we need to widely leverage tools from other domains and make them useful

•  This tutorial presents one such tool, the application of measurement-based modal analysis techniques for large-scale electric grids

5

A Few Initial Thoughts

•  “If I have seen further, it is by standing on the shoulders of giants.” –  Isaac Newton 1676

•  The grid we inherited from the past was smart; our challenge to make it smarter!

Left: control center in early 1900’s, right: ISO New England control center

6

A Few Initial Thoughts, cont.

•  While the grid of 2000 was named the top engineering technology of the 20th century, the grid of 2020 is even more complex –  There is question of whether anyone really fully

understands it! •  My passion is to do research and develop tools to

make large-scale electric grid analysis as easy as possible… But it can still be quite complex!!

•  Today’s focus is to show how measurement-based modal analysis can be a part of every day power systems engineering analysis

7

Modeling Cautions!

•  "All models are wrong but some are useful," George Box, Empirical Model-Building and Response Surfaces, (1987, p. 424) –  Models are an approximation to reality, not reality, so

they always have some degree of approximation –  Box went on to say that the practical question is how

wrong to they have to be to not be useful •  A good part of engineering is deciding what is the

appropriate level of modeling, and knowing under what conditions the model will fail

8

Signals

•  Throughout the talk I’ll be using the term “signal,” which has several definitions

•  A definition from Merrian-Webster is –  “A detectable physical quantity or impulse by which

messages or information can be transmitted.” •  A common electrical engineering definition is “any

time-varying quantity” •  Our focus today is on such time-varying signals,

particularly associated with oscillations

9

Oscillations

•  An oscillation is just a repetitive motion that can be either undamped, positively damped (decaying with time) or negatively damped (growing with time)

•  If the oscillation can be written as a sinusoid then

•  The damping ratio is

( ) ( )( ) ( )cos sin cos

where and tan

t t

2 2

e a t b t e C t

bC A Ba

α αω ω ω θ

θ

+ = +

−⎛ ⎞= + = ⎜ ⎟⎝ ⎠

2 2

αξα ω−=+

The percent damping is just the damping ratio multiplied by 100; goal is sufficiently positive damping

10

Types of Oscillations

•  There are several different types of oscillations, including simple ones with just a single frequency; under-damped oscillations have zero frequency

Speed, Gen Bus 1 #1gfedcb Speed, Gen Bus 2 #1gfedcb

54.543.532.521.510.50

60.560.4560.4

60.3560.3

60.2560.2

60.1560.1

60.0560

59.9559.9

59.8559.8

59.7559.7

59.6559.6

59.5559.5

11

Power System Oscillations

•  Power systems can experience a wide range of oscillations, ranging from highly damped and high frequency switching transients to sustained low frequency (< 2 Hz) inter-area oscillations affecting an entire interconnect

•  Types of oscillations include –  Transients: Usually high frequency and highly damped –  Local plant: Usually from 1 to 5 Hz –  Inter-area oscillations: From 0.15 to 1 Hz –  Slower dynamics: Such as AGC, less than 0.15 Hz –  Subsynchronous resonance: 10 to 50 Hz (less than

synchronous)

12

Example Oscillations

•  The below graph shows an oscillation that was observed during a 1996 WECC Blackout

The electric grid and electric grid modeling has changed substantially since 1996!

13

Example Oscillations

•  This graph shows oscillations on the Michigan/Ontario Interface on 8/14/03

14

More General Signal Analysis

•  More generally we may wish to better understand the dynamic behavior of the power grid, either following a disturbance or during ambient conditions –  Events are more common in studies

Image Source: M. Venkatasubramanian, “Oscillation Monitoring System”, June 2015 http://www.energy.gov/sites/prod/files/2015/07/f24/3.%20Mani%20Oscillation%20Monitoring.pdf

Bus 2376 (PKNOBDUM) Frequency

Time181614121086420

Bus

2376

(PKN

OBD

UM

) Fre

quen

cy

6059.9959.9859.9759.9659.9559.9459.9359.9259.9159.9

59.8959.8859.8759.8659.8559.8459.8359.8259.8159.8

59.79

Bus 2376 (PKNOBDUM) Frequency

15

Small Signal Analysis and Measurement-Based Modal Analysis •  Small signal analysis has been used for decades

to determine power system frequency response –  It is a model-based approach that considers the

properties of a power system, linearized about an operating point

•  Measurement-based modal analysis determines the observed dynamic properties of a system –  Input can either be measurements from devices (such

as PMUs) or dynamic simulation results –  The same approach can be used regardless of the

measurement source

16

Ring-down Modal Analysis

•  Ring-down analysis seeks to determine the frequency and damping of key power system modes following some disturbance

•  There are several different techniques, with the Prony approach the oldest (from 1795)

•  Regardless of technique, the goal is to represent the response of a sampled signal as a set of exponentially damped sinusoidals (modes)

( )( ) cosi

qt

i i ii 1

y t Ae tσ ω φ=

= +∑ Damping (%) i2 2i i

100αα ω−= ×+

17

Where We Are Going: Extracting the Modes from Signals •  The goal is to gain information about the electric

grid by extracting modal information from its signals –  The frequency and damping of the modes is key

•  The premise is we’ll be able to reproduce a complex signal, over a period of time, as a set a of sinusoidal modes –  We’ll also allow for linear

detrending

0.1𝑡+cos�(2𝜋2𝑡) 

18

Example: The Summation of two damped exponentials

•  This example was created by going from the modes to a signal

•  We’ll be going in the opposite direction (i.e., from a measured signal to the modes)

19

Some Reasonable Expectations

•  “Trust but verify” (going back to Reagan using a Russian proverb) –  We should be able to show how well the modes match

the original signal(s) •  Flexible to handle between one and many signals

–  We’ll go up to simultaneously considering 40,000 signals

•  Fast –  What is presented will be, with a discussion of the

computational scaling •  Easy to use

–  This is software implementation specific

20

Example: One Signal

21

Verification: Linear Trend Line Only

22

Verification: Linear Trend Line + One Mode

23

Verification: Linear Trend Line + Two Modes

24

Verification: Linear Trend Line + Three Modes

25

Verification: Linear Trend Line + Four Modes

26

Verification: Linear Trend Line + Five Modes

It is hard to tell a difference on this one, illustrating that modes manifest differently in different signals

27

A Larger Example We’ll Finish With Applying the developed techniques to the response of all 43,400 substation frequencies from an 110,000 bus electric grid(20 million plus values)

28

Measurement-Based Modal Analysis

•  There are a number of different approaches •  The idea of all techniques is to approximate a

signal, yorg(t), by the sum of other, simpler signals (basis functions) –  Basis functions are usually exponentials, with linear and

quadratic functions used to detrend the signal –  Properties of the original signal can be quantified from

basis function properties •  Examples are frequency and damping

–  Signal is considered over time with t=0 as the start

•  Approaches sample the original signal yorg(t)

29


•  Vector y consists of m uniformly sampled points from yorg(t) at a sampling value of DT, starting with t=0, with values yj for j=1…m –  Times are then tj= (j-1)DT –  At each time point j, the approximation of yj is

1

i

i+1 1

ˆ (t , ) ( , )

where is a vector with the real and imaginary eigenvalue components,

with ( , ) for corresponding to a real eigenvalue, and

( , ) cos( t ) and ( )

i j

i j

n

j j i i ji

ti j

ti j j i

y b t

t e

t e

α

α

φ

φ α

φ α φ

=

+

=

=

=

∑α α

α

α

α α i+1sin( t )

for a complex eigenvector value

i jtjeα α=

30


•  Error (residual) value at each point j is

•  The closeness of the fit can be quantified using the Euclidean norm of the residuals

•  Hence we need to determine a and b –  Recall

ˆ( , ) ( , )j j j j jr t y y t= −α α

222

1

1 1ˆ( ( , )) ( )2 2

m

j j jjy y t

=

− =∑ α r α

1

ˆ (t , ) ( , )n

j j i i ji

y b tφ=

=∑α α

31

Sampling Rate and Aliasing

•  The Nyquist-Shannon sampling theory requires sampling at twice the highest desired frequency –  For example, to see a 5 Hz frequency we need to

sample the signal at a rate of at least 10 Hz •  Sampling shifts the frequency spectrum by 1/T

(where T is the sample time), which causes frequency overlap

•  This is known as aliasing, which can cause a high frequency signal to appear to be a lower frequency signal –  Aliasing can be reduced by fast sampling and/or low

pass filters Image: upload.wikimedia.org/wikipedia/commons/thumb/2/28/AliasingSines.svg/2000px-AliasingSines.svg.png

32

One Solution Approach: The Matrix Pencil Method

•  There are several algorithms for finding the modes. We’ll use the Matrix Pencil Method –  This is a newer technique for determining modes from

noisy signals (from about 1990, introduced to power system problems in 2005); it is an alternative to the Prony Method (which dates back to 1795, introduced into power in 1990 by Hauer, Demeure and Scharf)

•  Given m samples, with L=m/2, the first step is to form the Hankel Matrix, Y such that

Refernece: A. Singh and M. Crow, "The Matrix Pencil for Power System Modal Extraction," IEEE Transactions on Power Systems, vol. 20, no. 1, pp. 501-502, Institute of Electrical and Electronics Engineers (IEEE), Feb 2005.

1 2 L 1

2 3 L 2

m L m L 1 m

y y yy y y

y y y

+

+

− − +

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

Y

KL

M M O ML

This not a sparse matrix

33

Algorithm Details, cont.

•  Then calculate Y’s singular values using an economy singular value decomposition (SVD)

•  The ratio of each singular value is then compared to the largest singular value sc; retain the ones with a ratio > than a threshold –  This determines the modal order, M –  Assuming V is ordered by singular

values (highest to lowest), let Vp be then matrix with the first M columns of V

= TY UΣV

The computational complexity increases with the cube of the number of measurements!

This threshold is a value that can be changed; decrease it to get more modes.

34

Aside: The Matrix Singular Value Decomposition (SVD)

•  The SVD is a factorization of a matrix that generalizes the eigendecomposition to any m by n matrix to produce where S is a diagonal matrix of the singular values

•  The singular values are non-negative, real numbers that can be used to indicate the major components of a matrix (the gist is they provide a way to decrease the rank of a matrix)

= TY UΣVThe original concept is more than 100 years old, but has found lots of recent applications

35

Aside: SVD Image Compression Example

Image Source: www.math.utah.edu/~goller/F15_M2270/BradyMathews_SVDImage.pdf

Images can be represented with matrices. When an SVD is applied and only the largest singular values are retained the image is compressed.

36

Aside: SVD and Principle Component Analysis (PCA)

•  The previous image compression example demonstrates PCA, which reduces dimensionality –  Extracting the principle components

•  The principle components are associated with the largest singular values –  This helps to extract the key features of the data and

removes redundancy •  PCA can be used to do facial recognition •  The Matrix Pencil Method is similar; that is,

retaining only the largest singular values from the Hankel matrix

37

Matrix Pencil Algorithm Details, cont.

•  Then form the matrices V1 and V2 such that –  V1 is the matrix consisting of all but the last row of Vp –  V2 is the matrix consisting of all but the first row of Vp

•  Discrete-time poles are found as the generalized eigenvalues of the pair (V2

TV1, V1TV1) = (A,B)

•  These eigenvalues are the discrete-time poles, zi with the modal eigenvalues then ln( )ii

zT

λ =Δ

The log of a complex number z=r�� is ln(r) + j�

If B is nonsingular (the situation here) then the generalized eigenvalues are the eigenvalues of B-1A

38

Matrix Pencil Method with Many Signals

•  The Matrix Pencil approach can be used with one signal or with multiple signals

•  Multiple signals are handled by forming a Yk matrix for each signal k using the measurements for that signal and then combining the matrices

,k ,k ,k

,k , ,k

,k ,k ,k

1 2 L 1

2 3 k L 2k

m L m L 1 m

1

N

y y yy y y

y y y

+

+

− − +

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Y

YY

Y

KL

M M O ML

M

The required computation scales linearly with the number of signals

39

Matrix Pencil Method with Many Signals

•  However, when dealing with many signals, usually the signals are somewhat correlated, so vary few of the signals are actually need to be included to determine the desired modes

•  Ultimately we are finding

•  The a is common to all the signals (i.e., the system modes) while the b vector is signal specific (i.e., how the modes manifest in that signal)

1(t , ) ( , )

n

j j i i ji

y b tφ=

=∑α α

40

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

( )

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

tji

tji i j ji

e

e t

α

α

α

α

+

+

=

=

Φ =

Φ = Φ

y Φ α b

b Φ α yΦ α

α

α

( )1 1( ) sin( )

for a complex eigenvalue; 1

Finally, ( ) is the pseudoinverse of ( )

i jti j

j

e t

t j T

α α+ +

+

=

= − Δ

α

Φ α Φ α

Where m is the number of measurements and n is the number of modes

41

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 UΣVA VΣ U

42

Aside: Pseudoinverse Least Squares Matrix Example

•  Assume we wish to fit 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 12 1 4 so = 2 16 1 4 6 1

mb

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

A

43

Aside: Pseudoinverse Least Squares Matrix Example

•  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 0.9260.219 0.976 0 1.012 0.765 0.543 0.345

0.143 0.071 0.2140.762 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

44

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

45

Iterative Matrix Pencil Method

•  When there are a large number of signals the iterative matrix pencil method works by –  Selecting an initial signal to calculate the a vector –  Quickly calculating the b vectors for all the signals, and

getting a cost function for how closely the reconstructed signals match their sampled values

–  Selecting a signal that has a high cost function, and repeating the above adding this signal to the algorithm to get an updated a

An open access paper describing this is W. Trinh, K.S. Shetye, I. Idehen, T.J. Overbye, "Iterative Matrix Pencil Method for Power System Modal Analysis," Proc. 52nd Hawaii International Conference on System Sciences, Wailea, HI, January 2019; available at scholarspace.manoa.hawaii.edu/handle/10125/59803

46

Demonstrations Using Large-Scale Synthetic Electric Grids

•  The following examples demonstrate the approach using large-scale synthetic grids –  Synthetic grids are designed to mimic the complexity of

the actual grids, but are fictional so they contain no CEII, allowing them to be publicly disseminated

–  For those who are interested, PSERC project S-91 (Generating Value from Detailed, Realistic Synthetic Electric Grids) has just started. Additional industrial advisors are certainly welcome to join the team!

•  More details on this project are available at overbye.engr.tamu.edu/pserc-project-s-91

•  Many synthetic grids, including the ones used here, are available at electricgrids.engr.tamu.edu

47

Texas 2000 Bus System Example

•  This synthetic grids serves an electric load on the ERCOT footprint

•  We’ll use the Iterative Matrix Pencil Method to examine its modes –  The contingency is the loss of two large 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 [email protected].

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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22%A

Am ps

29%A

Am ps

29%A

Am ps

15%A

Am ps

62%A

Am ps

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

812981308131

607860796080

563 MW 563 MW 563 MW 660 MW 660 MW 660 MW

ClosedClosedClosedClosedClosedClosed

The measurements will be the frequencies at all 2000 buses

48

2000 Bus System Example, Initially Just One Signal

•  Initially our goal is to understand the modal frequencies and their damping

•  First we’ll consider just one of the 2000 signals; arbitrarily I selected bus 8126 (Mount Pleasant)

Simulation Time (Seconds)20181614121086420

Bus

Freq

uenc

y (H

z)

60

59.98

59.96

59.94

59.92

59.9

59.88

59.86

59.84

59.82

59.8

59.78

Frequency, Bus 2127 (MIAMI 0)gfedcb

Frequency, Bus 1079 (ODESSA 1 8)gfedcbFrequency, Bus 7042 (VICTORIA 2 0)

gfedcbFrequency, Bus 5260 (GLEN ROSE 1 0)

gfedcbFrequency, Bus 8082 (FRANKLIN 0)gfedcb

Frequency, Bus 7159 (HOUSTON 5 0)gfedcbFrequency, Bus 6226 (BLANCO 0)

gfedcbFrequency, Bus 4192 (BROWNSVILLE 1 0)

gfedcbFrequency, Bus 4195 (OILTON 0)gfedcb

Frequency, Bus 8126 (MOUNT PLEASANT 1 0)gfedcb

49

Some Initial Considerations

•  The input is a dynamics study running using a ½ cycle time step; data was saved every 3 steps, so at 40 Hz –  The contingency was applied at time = 2 seconds

•  We need to pick the portion of the signal to consider and the sampling frequency –  Because of the underlying SVD, the algorithm scales

with the cube of the number of time points (in a single signal)

•  I selected between 2 and 17 seconds •  I sampled at ten times per second (so a total of

150 samples)

50

2000 Bus System Example, One Signal

•  The results from the Matrix Pencil Method are

Trust but verify results

Calculated mode information

PWDVectorGrid Variables

Time (Seconds)161412108642

Value

s

6059.9959.9859.9759.9659.9559.9459.9359.9259.9159.9

59.8959.8859.8759.8659.8559.8459.8359.8259.8159.8

59.79

Original Value Reproduced Value

51

Some Observations

•  These results are based on the consideration of just one signal

•  The start time should be at or after the event!



Valu

es

6059.9959.9859.9759.9659.9559.9459.9359.9259.9159.9

59.8959.8859.8759.8659.8559.8459.8359.8259.8159.8

59.79


The results show the algorithm trying to match the first two flat seconds; this should not be done!!

If it isn’t then…

52

2000 Bus System Example, One Signal Included, Cost for All

•  Using the previously discussed pseudoinverse approach, for a given set of modes (a) the bk vectors for all the signals can be quickly calculated

–  Recall that the dimensions of the pseudoinverse are the number of modes by the number of sample points for one signal

•  This allows each cost function to be calculated •  The Iterative Matrix Pencil approach sequentially

adds the signals with the worst match (i.e., the highest cost function)

( )k k+=b Φ α y

53

2000 Bus System Example, the Worst Match (Bus 7061)



Valu

es

6059.9959.9859.9759.9659.9559.9459.9359.9259.9159.9

59.8959.8859.8759.8659.8559.8459.8359.8259.8159.8


54

2000 Bus System Example, Two Signals



Valu

es

6059.9959.98

59.9759.9659.9559.94

59.9359.9259.9159.9

59.89

59.8859.8759.8659.8559.84

59.8359.8259.81


The new match on Bus 7061 is quite good!

With two signals

With one signal

55

2000 Bus System Example, Iterative Matrix Pencil

•  The Iterative Matrix Pencil intelligently adds signals until a specified number is met –  Doing ten iterations takes about four seconds

56

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

–  The number of sample points can be automatically determined from the highest desired frequency (the Nyquist-Shannon sampling theory requires sampling at twice the highest desired frequency)

–  Determining how all the signals are manifested in the modes is quite fast!!

57

Getting Mode Details •  An advantage of this approach is the contribution of

each mode in each signal is directly available

This slide shows the mode with the lowest damping, sorted by the signals with the largest magnitude in the mode

58

A Couple of Comments on Damping

•  How damping is defined seems to depend on prior industry experience –  Folks familiar with eigenvalue analysis will tend to define

it in terms of the eigenvalues

–  Multiplying this value by 100 gives a damping percentage

( ) ( )( ) ( )cos sin cos

where and tan

t t

2 2

e a t b t e C t

bC A Ba

α αω ω ω θ

θ

+ = +

−⎛ ⎞= + = ⎜ ⎟⎝ ⎠ 2 2

αξα ω−=+

59

A Couple of Comments on Damping

•  However, it can also be defined more graphically, in terms of a decrease in a signal from one peak to the next (see below for SPP) –  In SPP, to be considered “damped”, one of the following

two requirements must be met •  Peak to peak magnitude decreased 5% over one cycle •  Peak to peak decreases by 22.6% over 5 cycles

www.spp.org/documents/28859/spp%20disturbance%20performance%20requirements%20(twg%20approved).pdf

60

A Couple of Comments on Damping: An Easy Conversion Between the Two •  Assume we want 5% drop peak to peak

•  0.95= 𝑒↑(𝑡− 𝑇↓𝑠𝑡𝑎𝑟𝑡 )  •  Time for one cycle is 1/freqà[𝑡− 𝑇↓𝑠𝑡𝑎𝑟𝑡 = 1/𝑓 ] •  0.95= 𝑒↑λ/𝑓 à ln(0.95)= λ/𝑓  à λ=ln(0.95)𝑓

•  Plug this into Damping Ratio calculation •  Damping Ratio= −ln(0.95)𝑓 /√� [ln(0.95)𝑓 ]↑2 + (2𝜋𝑓)↑2    •  The frequency cancels out in this equation •  Damping Ratio= −ln(0.95) /√� [ln(0.95)]↑2 + (2𝜋)↑2   

=0.0081633

61

Visualizing the Mode

•  If the grid has embedded geographic coordinates, the contributions for the mode to each signal can be readily visualized

•  One approach is to utilize Geographic Data Views –  T.J. Overbye, E.M. Rantanen, S. Judd, "Electric power

control center visualizations using geographic data views," Bulk Power System Dynamics and Control -- VII. Revitalizing Operational Reliability -- 2007 IREP Symposium, Charleston, SC, August 2007, pp1-8; available at ieeexplore.ieee.org/document/4410539

•  The GDVs will be used to show the geographic location of the magnitude and angle of the contribution of the mode in each signal

62

Texas 2000 Bus Substation GDV

ABILENE 1

ALBANY 1

ALEDO 1

ALLEN 1

A L L E N 2

ALVIN

ARL INGTON 1

A R L I N G T O N 1 1

A R L I N G T O N 4

ARLINGTON 6

ARM STRONG 1

AUSTIN 1

AUSTIN 2

A U S T I N 2 4

AUSTIN 27

AUSTIN 3

A U S T I N 6

A U S T I N 8

B A L C H S P R I N G S

BASTROP

BAYTOWN 1

BAYTOWN 2B A Y T O W N 3

BELTON

BOYD

BREM OND

BRENHAM

BRIDGEPORT

BROWNSVILLE 1

B R O W N S V I L L E 2

B R O W N S V I L L E 3

B R O W N W O O D

BRYAN 1

BRYSON 1

BURL ESON

C A L D W E L L

CEDAR CREEK 1

C E D A R P A R K

CHANNELVIEW 1C H A N N E L V I E W 2

C H R I S T I N E

C I B O L O

C L E B U R N E 1

C L E B U R N E 2

C O N R O E 5

COPPERAS COVE

CORPUS CHRISTI 1C O R P U S C H R I S T I 1 4

CORPUS CHRISTI 3

CORSICANA 2

C U S H I N G 1

CYPRESS 1

DALLAS 1

D A L L A S 1 1

D A L L A S 1 4

DALLAS 19

DAL L AS 2

D A L L A S 2 0

D A L L A S 2 4

D A L L A S 2 6

D A L L A S 2 7

DALLAS 3

D A L L A S 4 0

D A L L A S 4 4

DALLAS 5

D A L L A S 9

DAYTON

DEER PARK

D E L R I O

DENTON 1DENTON 3

DONNA

ELMENDORF

ENNIS

E U L E S S 2

FAIRFIELD 1

F A I R F I E L D 2

FANNIN

F O R T W O R T H 2

F O R T W O R T H 2 0



F O R T W O R T H 8

FRANKLIN

F R E E P O R T 1

FREEPORT 2

F R I E N D S W O O D

FRISCO 2

GARLAND 1

GL EN ROSE 1

GOLDTHWAITE 1

GRAHAM

GRANBURY 1

GRANBURY 2

G R A N B U R Y 3

G R A N D P R A I R I E 1

G R A N D P R A I R I E 3

GREENVIL L E 1

GREGORY

H A R L I N G E N 1

H E R M L E I G H

H O U S T O N 1 0

H O U S T O N 1 3

H O U S T O N 1 4

H O U S T O N 1 8

H O U S T O N 1 9

H O U S T O N 2 0

H O U S T O N 3 2

H O U S T O N 3 3

HOUSTON 39

HOUSTON 4H O U S T O N 4 5

H O U S T O N 4 8

HOUSTON 5

HOUSTON 6

H O U S T O N 6 7

H O U S T O N 6 9

H O U S T O N 7

H O U S T O N 7 3

H O U S T O N 7 8

H O U S T O N 8 3H O U S T O N 8 8

H O U S T O N 9

HOUSTON 90

HURST

H U T C H I N S

I R V I N G 3

JACKSBORO 1

J A C K S O N V I L L E 1

J E W E T T 1

JOSHUA

KATY 1

KATY 2KATY 3

KELLER 1KELLER 2

K E N E D Y

K I L L E E N 3

L A GRANGE

L A P O R T E

L A K E J A C K S O N

L APORTE

L A R E D O 4

L E A G U E C I T Y

L E A N D E R 1

L E W I S V I L L E 1

L E W I S V I L L E 2

L U F K I N 3

M A G N O L I A 1

MANSF IEL D

M ARBLE FALLS 2

MARION 1

M C C A M E Y 1

M CKINNEY 1M C K I N N E Y 3

M E R K E L 1

M E S Q U I T E 2

M E S Q U I T E 3

M IAM I

M IDLOTHIAN 1

MISSION 1

M I S S I O N 2

M I S S I O N 3

MISSION 4

M I S S O U R I C I T Y 2

M O N A H A N S 1

M OUNT PLEASANT 1

M T. ENTERPRISE

NEM O

NEW BRAUNFELS 1

NEW BRAUNFELS 2

NURSERY

ODESSA 1

OILTON

O L M I T O

OL NEY 1

PAL O PINTO 1

P A N H A N D L E 2

P A N H A N D L E 4

P A R A D I S E

PARIS 1

PARIS 2

PASADENA 1PASADENA 2

PASADENA 3P A S A D E N A 4

P E A R S A L L

P F L U G E R V I L L E

PLANO 1P L A N O 2P L A N O 7

POINT COM FORT 1

POINT COM FORT 2

POOLVILLE

R A L L S 1

RHOM E

RICHARDSON 2

R I C H M O N D 1

RIESEL 1

R O A N O K E

ROCKDALE 1

ROSCOE 5

R O U N D R O C K 3

R O W L E T T 1

R O W L E T T 2

S A C H S E

SAN ANTONIO 1

SAN ANTONIO 12

S A N A N T O N I O 1 3

SAN ANTONIO 2


SAN ANTONIO 37






S A N A N T O N I O 8

S A N A N T O N I O 9

S A N B E N I T O

SAN JUAN

S A N M A R C O S

S A N P E R L I T A

S A R I T A 1

SAVOY

S E B A S T I A N 1

SEGUIN 1

S H E R M A N 1

SHIRO

SILVER

SPRING 2

SPRING 7

S P R I N G 8

STAFFORD

STEPHENVIL L E

S T E R L I N G C I T Y 1

SUGAR LAND 1

SUGAR L AND 2

SUGAR LAND 3

S U N N Y V A L E

TAFT 1

TEM PLE 1

THOMPSONS

T R I N I D A D 1

TYLER 4

TYLER 7

VICTORIA 1

VICTORIA 2

V I C T O R I A 3

WACO 1

W A C O 2

WADSWORTH

WHARTON 1

W I C H I T A F A L L S 1

WIL L IS 1

W I L L I S 2

WINCHESTER

WINGATE

Size is proportional to the substation MW throughput, while the color is based on the amount of substation generation; we’ll use the same substation GDV to display damping

63

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

IRAA

N 2

PRES

IDIO

1

SAND

ERSO

N

MON

AHAN

S 2

GRA

NDF

ALLS

MAR

FA

GA

RD

EN C

ITY

ODE

SSA

4

NOTR

EES M

IDLA

ND 4

BIG

SP

RIN

G 1

O D ONNEL L 2

ODE

SSA

6

BIG

SP

RIN

GS

MID

LAND

2

COAHOMA

MID

LAN

D 3

ALPI

NE

FORT

DAV

IS MC

CAM

EY 1

BIG

SP

RIN

G 4

CRAN

E

ODE

SSA

5FO

RT

STOC

KTO

N 1

AND

REW

S

FORS

AN

BIG

LA

KE

MID

LAND

5

OZO

NA

MON

AHAN

S 1

STAN

TON

ODONNE LLLE

NOR

AH

IRAA

N 3

BIG

SP

RIN

G 6

ODES

SA 3

BIG

SP

RIN

G 3

BIG

SP

RIN

G 7

MID

LAN

D 1

IRAA

N 1

ODES

SA 1

FOR

T ST

OCKT

ON

2

FOR

T ST

OCKT

ON

3

BIG

SP

RIN

G 2

KERM

ITPECO

S

SHEF

FIE

LD

MC

CAM

EY 2

LAMES

A

GO

LDSM

ITH

RALLS 2 PARIS 3SAVOY 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

LINDSAY ECTORSUNSET

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

HOWEARCHER 2

COLLINSVILLE

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 ALSTYNETIOGAVALLEY VIEW

MIAMI

TRENTON

HASKELL

SAINT JO

KERR

VILLE

SWEETWATER 4

LEAKEY

HAMLIN

ME RKEL 3

COL EMAN

SO

NO

RA

CAMP WOOD

LLAN

O

WILLOW CITY

A BILENE 4

KING SLAND

ROSCOE 6

BL ACKWELLC

HRIS

TOV

AL

LUEDERS

COL ORAD O CITY

ST AMF ORD

ELDO

RADO

SAN SABA

ABILENE 2

LOMETA

SAN A N GE

LO 1

TUSCOLAABILENE 7

SNYDE R 1

CHE ROK EE

R ICH LAN D SPR IN GS

ROSCOE 2

BALLINGER

SILVE R

ROSCOE 5

ANSON

DEL

RIO

HUNT

WINGATE

STER

LING C

ITY 1

ROSCOE 4

BRADY

TRENT 1

OVALO

NOLAN

M ILES

BRACKETTVILLE 2

M ASO N

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

SABINALSWEETWATER 2

LORAINE 2

ROSCOE 3

SA

N A

NG

ELO

3

TUSOCOLA

EDEN

SNYDE R 2

SWEETWATER 1

M ENARD

WINTERS

GOLDSBORO

SWEET

WA TER 3

GOODFEL

LOW AFB

SANTA ANNA

ROWE NA

TRENT 2

JUNC

TION

LAMPASAS

LAUGHLIN A F B

EDINBURG 3

LAREDO 7

BRUNI 2

MCALLEN 2

POTEET

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 GROVEPORTLAND

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 2DALLAS 21

PLANO 2

THORNTON

DENTON 5

EARLY

DECATUR

KELLER 2

PONDERJACKSBORO 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

LORENAFORT HOOD

ARLINGTON 5

MCKINNEY 3

DALLAS 38ARLINGTON 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 5GRAND 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 30AZLE

HUTCHINS

VALLEY MILLS

DENTON 6

MESQUITE 3

GRANBURY 1

DALLAS 2

MOUNT CALM

RIESEL 2

DALLAS 17ARLINGTON 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 11GRAND 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 37FORT 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

VERS

E

SAN ANTONIO 46

BASTROP

MAR

ION 2

PA IGE

SAN ANTONIO 42

WAELDER

ATAS

COSA

SAN ANTONIO 40

CIB

OLO

A US 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

A US TIN 3

SAN ANTONIO 48

SAN ANTONIO 19

SAN ANTONIO 2BE

RGH

EIM

NEW BRAUNFELS 2

D HANIS

SAN ANTONIO 24

A US TIN 7

BRENHAM

BERTRAM

MAR

ION 1

L A GRANGE

M AR BL E FA LLS 1

ROU ND RO CK 1

WIM B ER LEY

GON

ZALE

S

SAN ANTONIO 11

SAN ANTONIO 14

HUTTO

LULI NGSAN ANTONIO 21

A US 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

A US TIN 5

NEW BRAUNFELS 3

SUTHERLAND SPRINGS

AUS TIN 34

CAMERON

SAN ANTONIO 35

MED

INA

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 RO CK

JARRELL

SAN ANTONIO 1

SAN ANTONIO 36

SCH

ULEN

BURG

SAN ANTONIO 22

DRIPPING SPRINGS

SAN ANTONIO 26

M AR BL E FA LLS 3

BOE

RNE 1

WA SH ING T ON

SCH

ERTZ

A US TIN 8

SEGU

IN 1

SAN ANTONIO 45

SAN ANTONIO 52

CUERO 2

SAN ANTONIO 10

A US TIN 16

AUSTIN 39

SAN ANTONIO 51

SAN ANTONIO 41

LA VER

NIA

MANOR

SAN ANTONIO 38

GID DING S

AUS TIN 32AUS TIN 38

AUS TIN 30AUS TIN 28

COL

UMB

US

SHINER

SAN ANTONIO 3

C ALDWELL

AUS TIN 18IND

USTRYDAL E

SAN ANTONIO 37

SAN ANTONIO 18

SAN ANTONIO 33

MAR

TIND

ALE

SAN ANTONIO 17SAN ANTONIO 23

A US TIN 29

LEXI NGTON

LEANDE R 2AUS 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

C AN YO N L AK 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 HEDWIGSAN ANTONIO 16

BOE

RNE 2

SAN ANTONIO 32

A US 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

A US TIN 33

BULV

ERDE

G EO RG ET O WN 3

SAN ANTONIO 9

FLORESVILLE

SAN ANTONIO 29

SAN ANTONIO 7

BUCKHOLTS

SOM ERVIL LE

WE

IMAR

COM

FOR

T

NORDHEIM

AUS TIN 9

AUS TIN 22

HELO

TES

SAN ANTONIO 47

BAN

DERA

SAN ANTONIO 30

ROCKD ALE 2

CEDAR CREEK 2

NIXO

N

C HA PPE LL 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

HO USTON 80

HOUS TON 1

HOUS TON 9

HO USTON 73

NURSERY

BACL IFF

SP RING 1

HOUSTON 72

PORT LAVACA

CONROE 5

SUGAR L AND 3

DAYTON

VICTORIA 2

KATY 1

M AGNOL IA 2

HO USTON 29

M ISS OU R I C ITY 1RICH MO ND 1

HO USTON 11

HO USTON 33

SP RING 8

POINT COMFORT 2

HO USTON 28

EL C

AMP

O

HO USTON 12

PORT O CONNOR

DICKINSONGALVESTON 1

HO USTON 10

BAYTOWN 2

LEAGUE CITY

LIVERPOOL

HO USTON 16SUGAR L AND 1

RICHMO

ND 2

HO USTON 82

PINEHURST

PE ARLAND 2

HO USTON 47

HO USTON 79

HO USTON 49

WADSWORTH

HO USTON 18

HOUSTON 88

TOM BALL 1

HO USTON 22

DEER P ARKWAL LI S

HO USTON 67

SP RING 4

SO U TH HO U ST ON

HOUSTON 3

SEADRIFT

HO USTON 34

SUGAR L AND 2

CONROE 7

LOLITA

LAPORTE

NEW CANEY

HO USTON 14

HOUS TON 2

HOUSTON 78

HO USTON 40

BLOOMINGTON

WILLIS 2

HO USTON 51

PALA

CIOS

CONROE 4

HO USTON 52

SP RING 2

KING WOOD 1HO USTON 5

TEXAS CITY 1

KING WOOD 2

HOUS TON 8

SP RING 7HO USTON 41

FRESNO

HO USTON 48

HO USTON 58PATTISON HOUSTON 77

M ISS OU R I C ITY 2

HOUSTON 4

TOM BALL 2CYPRESS 1

HEM PSTEAD

WEST COLUMBIA

BAYTOWN 1

SANTA FE 1

HO USTON 27

SP RING 5

HO USTON 46

HO USTON 23

M ONTGO MERY

HUN

GER

FORD

HOUSTON 84

HOUSTON 83

HO USTON 24

HO USTON 76

HO USTON 43

HO USTON 81

HO USTON 35

HO USTON 90

BEASLEY

HOUS TON 7

LA MARQUE

WALLE R

PORTE R

CYPRESS 2

HO USTON 30

FUL SHEAR

LIBERTY

WINNIE

HO USTON 63

HO USTON 70

HO USTON 55

CROSBY

MANVEL

HO USTON 54HO USTON 89

DAMON

HOUSTON 74

HO USTON 26

HO USTON 32

HOCKLEY

ANGLETON

CONROE 3

HO USTON 39

HO USTON 59

PE ARLAND 1

HO USTON 56

HO USTON 13

C HA NN EL VIE W 2

BAY C

ITY

PASADENA 3

BROOKSHIRE

HO USTON 25

TE XAS C IT Y 2

VICTORIA 3

HO USTON 36

GANADO

M ONT BEL VIEU

BAYTOWN 3

SP RING 6 HUMBLE 2HO USTON 69

SANTA FE 2

PORT BO LIVAR

HO USTON 42

HOUSTON 85

HO USTON 50

HO USTON 62

EAG

LE LAKE

CONROE 2

WEBSTE R

SWEEN

Y

BLESS

ING

WH

AR

TO

N

1

LAKE

JACK

SON

BELL VIL L E

KATY 4

HO USTON 38 FR IEN D SWO O D

HO USTON 17

HO USTON 66

FREEP ORT 1

NEW

GU

LF

HO USTON 60

PASADENA 1STAFF ORD

EDN

A

HO USTON 21

NEED

VILLE

SE ALY KATY 3

HO USTON 37

HO USTON 19

THO MPSONS

HO USTON 61

GALVESTON 3

EAST

BER

NAR

D

HO USTON 20

C HA NN EL VIE W 1

FREEP ORT 3

HO USTON 15

HO USTON 31

ROSHARON

HO USTON 44

HO USTON 75

HOUSTON 71

HUMBLE 1

HO USTON 68

WILLIS 1

BAYTOWN 4

VAN

VLE

CK

HUFFMAN

HO USTON 53HOUSTON 91

ROSENBERG

PASADENA 5

HO USTON 45

HOUS TON 6

PASADENA 4

HO USTON 57

GALVESTON 2

POINT COMFORT 1

KATY 2

SP RING 3

DANBURY

WH

ARTO

N 2

VICTORIA 1

HO USTON 64

HOUSTON 65

LA PORTE

M AGNOL IA 1

A LVIN

HO USTON 86

BRA

ZORIA

HOUSTON 87

FREEPORT 2

CONROE 6

MADISONV ILLE

NORMANGEE

ARP

CENTERVILLE

LUFKIN 3

FLINTBULLARD

CHANDLER

PL AN TE RS VILL E

MARQUEZ

TENNESSEE COL ONY

ANDERSON

MINEOLA

LOVELADY

HUNT INGTON

BRYAN 4

FRANKSTON

TYLER 4

BEN WHEELER

GARRISON

ETOILELUFKIN 1

PALESTINE 1

LINDALE

WHITEHOUSEOVERTON

GRAND SALINE

SHIRO

MIDWAY

SULPHUR SPRINGS

VAN

MURCHISON

NACOGDOCHES 2

DIBOLL

FAIR FIELD 1

TYLER 8ATHENS 1

BUFFALO

RUSK

WINONA

FAIR FIELD 2 CUSHING 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

C OL LEG E ST AT ION 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

C OL LEG E ST AT ION 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 to dynamically prune the display using the GDV Options, Pruning command

64

Visualization of 0.63 Hz Mode with Pruning and Some Color

The display was pruned so only one arrow per geographic region is shown; the size of the arrow is proportional to its magnitude, and a color mapping is used for the angle

65

Application to a Larger System

•  The following few slides show an application to a larger, real system

•  The examples are from PSERC Project S-92G, which is currently looking at the dynamic aspects of interconnecting the North American Eastern and Western grids

•  There are many cross-cutting issues associated with this, and additional PSERC industrial advisor involvement is welcomed!!!

66

Some Preliminary Results from S-92G Germane to Modal Analysis •  The project is primarily looking at the dynamic

aspects of interconnecting the grids, but is also considering static power flow and contingency analysis considerations –  There is a public synthetic model analysis, and a not

public consideration of the actual grid models •  The actual grid model was created by merging the

East and West models •  It has 110,000 buses, 14,000 generators, 37,000

dynamic model devices with 243 different model types –  Integrations are solved using a ½ cycle time step

67

Model 1: Heavy Load Conditions with 828 GW of Load; Substation GDV with Generation

Sized and Colored by MW Value

G r a n g s t o n

H a n g in g R o c k E n e r g y F a c ilit y

K e y s t o n e

W o lf H ills

C lin c h R iv e r

Rockpor t ( INM I)

Amos

Gavin

Donald C. Cook

Kam m er

M ount aineer

Ti l ly

Churchill Falls

M a n ic o u a g a n

La Grande 2ALa Grande 3

N e w L a u d e r d a le

Independence

White Bluff

Silverhawk

North Anna

Aut augaville

G r e e n s v ille P o w e r S t a t io n

Kyr ene

Wansley

Nelson

M O A P A

Conem augh

Arkansas Nuclear One

Chalk Point

D e lt a P o w e r P la n t

C o p p e r M o u n t a in

Wat t s Bar Nuclear

Cent r alia (TRAENE)

Caledonia

Bad Creek

Sout h Bend

R o u n d B u t t e

Lim er ick 500 kV sub

Ca lv e rt Cl i ffs

C o y o t e S p r in g s

Salem (PSEGN)

Brunswick Count y

Lower Granit e

Bowen

Clover

Hunt er st own

Dever s

Corona

C P V C u n n in g h a m C r e e k

Coronado

Shawnee (TVA)

R o w a n C o u n t y E n e r g y C o m p le x

Mi l le r (AL AP)

L o w e r M o n u m e n t a l

Susquehanna

H o p e C r e e k ( P S E G N )

St er lingt on

Cum ber land (TVA)

Oconee

Sequoy ah (Tv a )

Vict or J. Daniel

Diab lo Cany on

Yukon

Raccoon M ount ain

Four Corner s (AZPS)

Tenaska Georgia

G r a s s la n d

Grand Coulee

Conasauga

Peace CanyonGordon M. Shrum

Char les Lenzie

M o s s L a n d in g ( D U E N N O )

M e t c a lf E n e r g y C e n t e r

M cNary

Napavine

New M adr id

Joseph M . Far ley

Possum Point

C h o c t a w G a s G e n e r a t io n

Cot t onwood Energy

St eel Cit y

Im per ial Valley

APEX

B a x t e r W ils o n

Mica

Harquahala Valley

Lakeover

Cryst al River

R e d B lu f f

Palo Verde

R e lia n t E n e r g y C h o c t a w C o u n t y

C o lo r a d o R iv e r

For t M ar t in (M ONG)

Ba th County

P lu m P o in t E n e r g y S t a t io n

Cit r us Energy

R ic h m o n d P la n t ( C P L C )

Harrison

Paradise (TVA)

B ig C a j u n 2

Revelstoke

Keyst one (RRI)

Redhawk 1 & 2

Lit t le Goose Ph

Longview Power

B o a r d m a n ( P G E )

Keephi l ls

Alv in W. Vogtle

Per r yville

SHIC

Front Royal

Dar lingt on (OPG)

Wa te rfo rd

Grand Gulf

M agnolia (M AGNEN)

Ft. Drum

Chief Joseph

Ar lingt on Valley

H e r m is t o n P o w e r P r o j e c t

Peach Bottom

Surry

McGuire

Bruce

Martin (F L PL )

Scherer

Ronco

M esquit e Power

Genesee

M ount St orm

D w o r s h a k

Browns Ferry

Jocassee

El Dorado

M ayo

L a c k a w a n n a

West Count y Energy Cent er

Bull Run (TVA)

Hyder

Co lstrip

Edwin I . Hat ch

Har r y Allen

Hot Spr ings

Cl i ffside

John Day Ph

Ashe

Cholla

Convoy

Clif t y Creek

E d g e w a t e r ( W P L )

Nebraska Cit y - NEB

Dolet Hills

M a d is o n G e n e r a t in g S t a t io n

Callaway (UNIEL)

Cayuga

Cane Run

Hawt horn

S m it h ( O M U )

Jef f r ey Energy Cent er

Holcom b

Blackst one (ANP)

M ansf ield (FIRGEN)

Pet e 1 ( IP&L)

Duane Arnold

Bonanza

Point Beach

P r a ir ie S t a t e E n e r g y C a m p u s

Pawnee

Gent lem an

Tuco

Avon Lake

River side (PSOK)

B lu e L a k e

Beav e r Va l le y

Rose ton

Point Lepreau

Rocky Reach

O c e a n S t a t e P o w e r

M iam i For t

F e rm i

Buckner

Labadie

M id d le t o w n ( N R G )

Leland Olds

CPV Valley

Linden (PSEGF)

Seabrook

Gowanus

M id la n d C o g e n e r a t io n V e n t u r e ( M C V )

Bridger

W. H. Zim m er

T r a c y & C la r k M o u n t a in

Dresden

East BendCof f een

Clint on (AM ERGEN)

Belle River

Schahfer

Hem pst ead

Nelson

L a k e f ie ld U t ilit ie s

C o r d o v a E n e r g y

Newm an

Ot t um wa - IES

N e w in g t o n E n e r g y C e n t e r

Nor t heast ern

Bever ly

To lk

Redbud Power Plant

Spurlock

W a t e r f o r d E n e r g y C e n t e r

Bayonne

Braidwood

Gibson (PSI)

Nor t h Valm y

O n e t a E n e r g y C e n t e r

R e n a is s a n c e P o w e r P r o j e c t

H o b b s ( S W P S )

Rush I sland

Raun

AES Som erset

Tap

Oliver Wind I I

Sam m is

L in d e n C o g e n P la n t ( E C O A S T )

Rober t M oses Niagara

P o w e r t o n G e n e r a t in g S t a t io n

Lake Road (LAROGE)

Scriba

Ia tan

M ic h ig a n C it y

Pir key

H u g o ( W E F A )

Greenwood

Per r y (FINUOP)

Holland Energy

Canal (M IRNE)

S a r p y C o u n t y

D u c k C r e e k

Glen Canyon

L a cy gne

Indian Point 3

Bergen (PSEGF)

Oswego Harbor

H ic k o r y

F lin t C r e e k ( S W E P )

N in e M ile P o in t ( C O O P S E )

I n d e p e n d e n c e S t a t io n ( S it h e )

Phill

Fr esh Kills

Millstone

Colum bia (WPL)

M ont icello (NM C)

Dan E. Ka rn

Musk ogee

G a r d n e r P a r k ( P r o p o s e d )

A t h e n s G e n e r a t in g P la n t

C u r r a n t C r e e k

Joliet 29

Homer City

St uar t (DP&L)

J. K. Sm it h

M ir a n t Z e e la n d G e n e r a t in g P la n t

G r a n d R iv e r D a m ( G R D A )

King

Soone r

Sherburne

Newt on

L u n a E n e r g y F a c ilit y

Oak Creek Nor t h

Moses

I n v e r H ills

Em ery

Sir Adam Beck 1

Hunt er Plant

Sioux

L e b r o c k

Hunt ingt on

Sem inole (OKGE)-1

Tr im ble Count y (LGEC)

W o o d s d a le

M onroe (DETED)

Car roll Count y

Cogen

Thom as Hill

S t F r a n c is

Wilkes

St. Cla ir

Wolf Creek (WCNOC)

Lordst own

Saukville

W a u k e g a n ( M I D G E N )

Reynolds

Lawrenceburg

Laram ie River

Welsh (SWEP)

Reid

San Juan 345

Louisa (M IDAM )

Nor t hf ield M ount ain

E m p o r ia P e a k in g P la n t

Tidd

C la r k C o u n t y

Lakeside

La Salle

Coyot e

Baldwin Energy Com plex

Kyger Creek

Ast or ia Gas TurbinesRainey

Council Bluf f s

Quad Cit ies (EXGEN)

B e a v e r C r e e k

G o r d o n E v a n s

Cr aig (TSGT)

F o x E n e r g y C e n t e r ( K a u k a u n a )

Brown (SIGE)

Sugar Creek

Blenheim -Gilboa

Cov e rt

By ron (EXGEN)

Ludington

Int erm ount ain Generat ing

S t . J o s e p h E n e r g y C e n t e r

B r u n o t I s la n d

Ghent

K e n d a ll C o u n t y P r o j e c t

St ought ons

C a s s C o u n t y

Ant elope Valley (BEPC)

J. H. Cam pbell (CEC)

Jam es A Fit zpat r ick

Palisades (NM C)

Prair ie I sland

Joppa St eam

Yarm out h

S t o n y B r o o k

Lem oyne

M ill Creek (LGEC)

My stic

O k la u n io n ( N o r t h H V D C )

C h u t e - d e s - P a s s e s

Com anche

Edwardspor t

B e lle d u n e ( N B P O )

Conesv i l le

Cooper St at ion

Kinca id

Brown (KUC)

L a f o r g e 1 [ L A 1 ]

M a n ic 1

M a n ic 5 - P a

Saint e-M arguer it e-3

Br isay

B e r s im is 2

L a R o m a in e 3

O u t a r d e s 2

M anic 5

L a R o m a in e I I

B e a u p r T

Le Plat

B e r s im is 1

O u t a r d e s 3

East m ain-1

La Grande 1

Kem ano

Bedding

EDD7

Bat t le River

Sheerness

Sundance

SS

Edm onst on

Wax

WCE-Sugar

Lauderdale

Cocodr ie

Thelm a

C la y B o s w e ll E n e r g y C e n t e r

Clar k (NEVP)

H a y n e s G e n e r a t in g S t a t io n

River Bend

H e lls C a n y o n

Turk ey Po int

A g u a F r ia

But ler

AES Granit e Ridge

Catawba

East Towanda

W e s t P h o e n ix C C 4 & C C 5

Cape Canaveral

C r o c k e t t C o g e n

Ft. Myers

B a t e s v ille G e n e r a t io n F a c ilit y

Edgem oor

Plant Franklin

S a f e H a r b o r

Hay Road

Alt a

R a n d o lp h

Cast aic

I s le M a lig n e

Wyodak

For t St . Vrain

D e lt a E n e r g y C e n t e r ( C P N )

Cope

E a s t w o o d

Rad isson

He lm

William s-St

Apache

Langley

Shast a

S e n e c a - C E I

Hayden

Shawville

Colb

Jovit a

R ic h a r d R u s s e ll

Wayne Lee

S w if t C u r r e n t

L lo y d m in s t e r

B ig B e n d P r o j e c t

Wells (DOPD)

F o r t C h u r c h ill

Ba rry (AL AP)

Arcogen

S e w a r d ( R R I )

Shand

Seven M ile

CrossSantan

The Dalles Ph

S u n d a n c e ( A Z P S )

Pr iest Rapids

Big St one

M ount ainview Power

A v e n u e / K in g b ir d

Por t Everglades

P la in s E n d

B r a n d y B r a n c h G e n e r a t in g S t a t io n

G r if f it h E n e r g y P r o j e c t

Penny Hill

Long Spruce

S u n r is e P o w e r P r o j e c t

The Geyser s

N ic h o ls S t a t io n

Four River

Wat eree (SOCG)

Kelsey

D e lt a P u m p s

Po lk

Riv ie ra

Robinson

La Rosit a

Gant t

P la n t X ( S W P S )

Oahe

K o o t e n a y C a n a l

Blyt he 2

Bighorn

FordAA2-115

Lingan

Colgat e

Johnst on

M c I n t o s h - C A E S

S h ilo h I I

Beauharnois-Ouest

D e a r b o r n I n d u s t r ia l G e n e r a t io n L L C

Vist a

Shady Hills

O x b o w ( I D P C )

Ninem ile Point (ELA)

PLAYA

C le v e la n d N a t u r a l G a s

Gorgas

Ocot illo

G a lla g h e r

Bonneville

Sut t on

Manatee

Tap

Sewaren

Debary

McDonough

Sout h River

Montour

G r a n d R a p id s ( M H )

AES I r onwood

M organt own

Davis

M ar shall (DUPC)

Buck (DUPC)

M ust ang St at ion

Asheville

Little Gypsy

M o s s y r o c k

M uddy Run

Exxon

M e r c h a n t

B e a c o n

Har r is (CPLC)

Taf t Proj ect

Hines Energy Com plex

Cr ist

E s c a la n t e

CPCWest

D o w S t . C h a r le s

Bear Garden

Gat eway (PG)

Chest er f ield

Big Bend

N a u g h t o n

Boundary

Cherokee (PSCO)

M ill Creek St at ion

C o s u m n e s ( S M U D )

Gerald Andrus

K it t a t in n y

El Segundo

H a r d e e P o w e r S t a t io n - S E C 1

Boundary Dam

B r id g e p o r t

Acad ia

Nor t hside

Jasper Count y

M e lo n e s

Belm ont

AES Red Oak

St ant on Energy Cent er I

Om ar

Ot ay M esa

M ar shland

P. L. Bar t ow

B r o a d R iv e r E n e r g y C e n t e r

Nipawin

L o s E s t e r o s

PPL Brunner I sland

Anclot e

East Shore

G ilb e r t ( R R I )

Lincolnt on

P it t s b u r g ( M I R )

Lewis

Poplar River

L e w is C r e e k

Pot r ero

W u s k w a t im

P r e s id e n t e J u a r e z ( R o s a r it o )

H u n g r y H o r s e

Car t er s

W a lla c e D a m

Vandolah

Pr int z

L a n s in g S m it h ( G U P C )

R o b e r t M o s e s P o w e r D a m

R o c k in g h a m P o w e r P la n t

Gar r ison

Gast on (ALAP)

R o c k y M o u n t a in E n e r g y C e n t e r

Deser t Basin

Rocky M ount ain

C e r r o P r ie t o I I

V3

L a T u q u e

Coal Creek

T e n o r o c

Sab ine

S c a t t e r g o o d

Dynegy

Brandon Shores

Roxboro (CPL C)

Rodem acher

Sanf ord (FLPL)

K e a r n y ( P S E G F )

S e m in o le G e n e r a t in g S t a t io n

L a n c a s t e r

GEC

Winyah

F o r t R a n d a ll

Rainey

Jones St at ion

U r q u h a r t - S C E G

Dicker son

Past or ia

Har r ingt on

H a r t w e ll E n e r g y L im it e d P a r t n e

T ig e r B a y

PPG

Brownlee

Swif t

Lim est one (M H)

Ivanpah

S u t t e r P o w e r P la n t

CPCEast

Brunswick (CPLC)

Essex

N o x o n R a p id s

Yat es

E d w a r d H y a t t

Walnut

Conowingo

Sout haven

McIntosh

R E P O W E R

Hoover NVHoover AZ

L a s A g u ila s

F r e d o n ia ( P S P L )

M o n r o e ( P V I )

Be lews Creek

S p in d le H ill E n e r g y

D r y F o r k S t a t io n

C h a m b e r s

St . Lucie (FLPL)

Beaver

P r im m

L ib b y - P A C I F

E s c o n d id o P e a k e r

M u r r a y E n e r g y F a c ilit y

C u n n in g h a m

Bear Swam p

High Deser t

C o llie r v ille

C a r t a n z a

G r o s M o r n e

Sum m er

E d d y s t o n e

F a ir f ie ld P s

Saeger st own Rd.

DOW Cogen

Cit r us Spr ings

Cane Island

A r v a h B H o p k in s

E . B . C a m p b e ll

H. L. Culbreat h Bayside

Wanapum

D e s J o a c h im s

Kipling

R. H. Saunder s

Pickering

L o w e r N o t c h

C a r g ill S a lt I n c

N ia g a r a W e s t

W e lls ( G L H I F )

Halt on

A b it ib i C a n y o n

Henvey

Chout eau (ASEC)

Cooper

Wilson (TVA)

Gallat in (PRI)

Lansing

Alm a

Yucca

S t a t e lin e ( E M D E )

John Sevier

C a n n o n F a lls

F o n t a n a ( T V A )

Paradise

Genoa

Allen (TVA)

Por t -Alf r ed

Pickwick

B u ll C r e e k

Kingston

M oselle

B u ll S h o a ls

M c C o r m ic k

M o r g a n E n e r g y C e n t e r

N o r t h e a s t ( K C P L )

M ar shallt own

M a r io n ( S I P C )

D e c a t u r E n e r g y C e n t e r

W h e e le r ( T V A )

Nor t h Om aha

S . H a r r is o n v ille

S ik e s t o n

S o u t h w e s t I I

P r a ir ie C r e e k

9 SubCheswick

M a c t a q u a c

Concord

U n iv e r s it y

Joliet 9

Culley

SCS 138

Dallm an

Dan River

D e e r h a v e n

Edwards

Pr it ch

D r e s d e n E n e r g y C e n t e r

E . F . B a r r e t t

V a lle y ( W E P )

E a s t R iv e r

War r ick

G ib s o n C it y

G e r m a n t o w n

T a u m S a u k

G r a n d T o w e r

H e n n e p in

Holt sville

H o r s e s h o e L a k e

Vernon Boulevard

M eram ec

W h it in g R e f in e r y ( W C E )

M u s t a n g

Northport

O a k G r o v e

Neenah

Par is

T u f t s C o v e

P r e s q u e I s le

W. Frem ont

R iv e r s id e E n e r g y C e n t e r

Vir ginia Cit y

Alpine

W ill C o u n t y

S h e lb y v ille

V e n ic e ( U N I E L )

C a it h n e s s

S o u t h w e s t e r n

Conoco

Plym out h

Spr ingdale

Sm it h M ount ain

Havana

E x x o n M o b il

Pulliam

P o r t W a s h in g t o n ( W E P )

P o r t J e f f e r s o n

Ar senal Hill

T u ls a ( P S O K )

Y o r k S t .

A n a d a r k o ( A N D )

Car illon

Dean

JuddT r e n t o n C h a n n e l

C h e m o lit e

M a s s p o w e r

M ilf ord (CPVI)

PSE

Rise

K e n d a ll S q u a r e

J o h n H K e r r

J a y H y d r o

B u c k s p o r t

T e r r e ll C r e e k

FRSQ

JM C

D a n s k a m m e r

F a c e R o c k

B la c k h a w k

Black Dog

B e t h le h e m E n e r g y C e n t e r

B e r k s h ir e P o w e r

M PLPA n g u s A n s o n

Ginna

I c e H a r b o r

Far ibalt

D e e r R iv e r

L o s M e d a n o s E n e r g y C e n t e r

M c M e e k in

R io G r a n d e

R iv e r s id e ( N S P )

M PS

T r a c y ( T R V A P O )

T o w a n t ic E n e r g y C e n t e r

High Br idge

A E S C a y u g a

R u m f o r d

W e s t o n 4

S . O . P u r d o m

Edgar

N o r t h e n d

B r a n d o n

Lee St at ion (DUPC)

K la m a t h F a lls

68

Rockpor t ( INM I)

Amos

Gavin

Donald C. Cook

Kam m er

M ount aineer

Ti l ly

Churchill Falls

M a n ic o u a g a n

L a Grande 2ALa Grande 3

Independence

Whit e Bluf f

S ilv e r h a w k

North Anna

Wansley

Nelson

A t t a la E n e r g y C e n t e r

Ar kansas Nuclear One

Wat t s Bar Nuclear

Cent r alia (TRAENE)

Bad Creek

Lim er ick 500 kV sub

Ca lv e rt Cl i ffs

Salem (PSEGN)

B r u n s w ic k C o u n t y

S h a w n e e ( T V A )

Mi l le r (AL AP)

Susquehanna

H o p e C r e e k ( P S E G N )

S t e r lin g t o n

Cum ber land (TVA)

Oconee

Sequoy ah (Tv a )

Vict or J. Daniel

Diab lo Cany on

Raccoon M ount ain

Four Corner s (AZPS)

Grand Coulee

G o r d o n M . S h r u m

Char les Lenzie

M o s s L a n d in g ( D U E N N O )

M c N a r y

Napavine

N e w M a d r id

Joseph M . Far ley

Cot t onwood Energy

Im per ial Valley

APEX

B a x t e r W ils o n

M ica

L a k e o v e r

Palo Verde

R e lia n t E n e r g y C h o c t a w C o u n t y

For t M ar t in (M ONG)

Bath County

P lu m P o in t E n e r g y S t a t io n

C it r u s E n e r g y

R ic h m o n d P la n t ( C P L C )

Harrison

Paradise (TVA)

Keyst one (RRI)

Ocot illo

Longview Power

Keephi l ls

Alvin W. Vogt le

Per r yville

Dar lingt on (OPG)

Wa te rfo rd

Grand Gulf

F t. Drum

C h ie f J o s e p h

Ar lingt on Valley

H e r m is t o n P o w e r P r o j e c t

Peach Bottom

Surry

McGuire

Bruce

Scherer

Genesee

D w o r s h a k

Browns Ferry

J o c a s s e e

E l Dorado

Co lstrip

Edwin I . Hat ch

John Day Ph

Ashe

Clif t y Creek

E d g e w a t e r ( W P L )

N e b r a s k a C it y - N E B

Callaway (UNIEL)

Cayuga

Cane Run

M ansf ield (FIRGEN)

Pet e 1 ( IP&L)

Duane Arnold

B o n a n z a

Point Beach

P r a ir ie S t a t e E n e r g y C a m p u s

Pawnee

G e n t le m a n

Beav e r Va l le y

Point Lepreau

R o c k y R e a c h

M iam i For t

F e rm i

Labadie

Leland Olds

CPV Valley

Seabrook

M id la n d C o g e n e r a t io n V e n t u r e ( M C V )

Br idger

W. H. Zim m er

Dresden

East BendC o f f e e n

Clint on (AM ERGEN)

Belle River

S c h a h f e r

H e m p s t e a d

N e w m a n

Ot t um wa - IES

N e w in g t o n E n e r g y C e n t e r

Tolk

Redbud Power Plant

Spur lock

W a t e r f o r d E n e r g y C e n t e r

Braidwood

Gibson (PSI)

H o b b s ( S W P S )

Rush I sland

Raun

AES Som erset

Oliver Wind I I

Sam m is

Rober t M oses Niagara

Scriba

Iat an

M ic h ig a n C it y

Pir key

H u g o ( W E F A )

Per r y (FINUOP)

Canal (M IRNE)

D u c k C r e e k

Glen Canyon

Ind ian Po int 3

B e r g e n ( P S E G F )

F lin t C r e e k ( S W E P )

N in e M ile P o in t ( C O O P S E )

I n d e p e n d e n c e S t a t io n ( S it h e )

Phill

Millstone

Colum bia (WPL)

M ont icello (NM C)

Dan E. Karn

M ont ville

G a r d n e r P a r k ( P r o p o s e d )

C u r r a n t C r e e k

Hom er Cit y

M e a d o w

King

Sherburne

Oak Creek Nor t h

Moses

Em ery

Sir Adam Beck 1

H u n t e r P la n t

Sioux

W o o d b in e

T r im b le C o u n t y ( L G E C )

M onroe (DETED)

Car roll Count y

Cogen

T h o m a s H ill

Wilkes

St. Cla ir

Wolf Creek (WCNOC)

Lordst own

Lawrenceburg

Laram ie River

Welsh (SWEP)

Reid

F o w le r R id g e

San Juan 345

L o u is a ( M I D A M )

Davis-Besse

TiddLakeside

La Salle

Coyot e

B a ld w in E n e r g y C o m p le x

Kyger Creek

A s t o r ia G a s T u r b in e s

Council Bluf f s

Quad Cit ies (EXGEN)

Craig (TSGT)

Sugar Creek

Blenheim -Gilboa

By ron (EXGEN)

Ludington

I n t e r m o u n t a in G e n e r a t in g

S t . J o s e p h E n e r g y C e n t e r

B r u n o t I s la n d

Ghent

St ought ons

A n t e lo p e V a lle y ( B E P C )

J. H. Cam pbell (CEC)

Jam es A Fit zpat r ick

Palisades (NM C)

Prair ie I sland

J o p p a S t e a m

E a s t S h o r e

M ill C r e e k ( L G E C )

O k la u n io n ( N o r t h H V D C )

C h u t e - d e s - P a s s e s

Com anche

E d w a r d s p o r t

B e lle d u n e ( N B P O )

Cooper St at ion

Kincaid

M a n ic 1

M a n ic 5 - P a

Saint e-M arguer it e-3

Br isay

B e r s im is 2

L a R o m a in e 3

M a n ic 5

L a R o m a in e I I

B e a u p r T

Le Plat

B e r s im is 1

O u t a r d e s 3

East m ain-1

L a G r a n d e 1

Kem ano

Bedding

B a t t le R iv e r

S h e e r n e s s

Sundance

SS

Edm onst on

Wax

Thelm a

C la y B o s w e ll E n e r g y C e n t e r

River Bend

H e lls C a n y o n

Tur key Point

AES Granit e Ridge

Catawba

C a lif o r n ia F la t s

C r o c k e t t C o g e n

Plant Fr anklin

S a f e H a r b o r

Cast aic

I s le M a lig n e

Wyodak

D e lt a E n e r g y C e n t e r ( C P N )

E a s t w o o d

Radisson

Helm

W illia m s - S t

Apache

Shast a

S e n e c a - C E I

Hayden

Colb

C a b in e t G o r g e

W a y n e L e e

W e lls ( D O P D )

Bar r y (ALAP)

A r c o g e n

S e w a r d ( R R I )

Cross

The Dalles Ph

P r ie s t R a p id s

B ig S t o n e

M o u n t a in v ie w P o w e r

Por t Everglades

P la in s E n d

G r if f it h E n e r g y P r o j e c t

Long Spruce

The Geyser s

Kelsey

Po lk

Robinson

L a R o s it a

Gant t

Oahe

J o h n s t o n

Beauharnois-Ouest

M e r r im a c k

Ninem ile Point (ELA)

Gorgas

B o n n e v ille

M cDonough

Montour

G r a n d R a p id s ( M H )

M o r g a n t o w n

Davis

G le n r o c k

Buck (DUPC)

Lit t le Gypsy

M uddy Run

Har r is (CPLC)

T a f t P r o j e c t

CPCWest

D o w S t . C h a r le s

C h e s t e r f ie ld

N a u g h t o n

Boundary

C h e r o k e e ( P S C O )

K it t a t in n y

B o u n d a r y D a m

Acadia

N o r t h s id e

Jasper Count y

M e lo n e s

Br illiant

O t a y M e s a

P. L. Bar t ow

P P L B r u n n e r I s la n d

P it t s b u r g ( M I R )

Lewis

Poplar River

L e w is C r e e k

Pot r ero

L a n s in g S m it h ( G U P C )

R o b e r t M o s e s P o w e r D a m

Rocky M ount ain

C e r r o P r ie t o I I

L a T u q u e

F o r t P e c k

Coal Creek

T e n o r o c

Sab ine

B r a n d o n S h o r e s

Rodem acher

Sanf ord (FLPL)

S e m in o le G e n e r a t in g S t a t io n

L a n c a s t e r

Winyah

Rainey

J o n e s S t a t io n

Past or ia

Har r ingt on

PPG

Swif t

Lim est one (M H)

Brunswick (CPLC)

E d w a r d H y a t t

Conowingo

McIntosh

H o o v e r N VHoover AZ

F r e d o n ia ( P S P L )

St . Lucie (FLPL)

L ib b y - P A C I F

E s c o n d id o P e a k e r

M u r r a y E n e r g y F a c ilit y

High Deser t

C o llie r v ille

G r o s M o r n e

Sum m erF a ir f ie ld P s

Saeger st own Rd.

C a lif o r n ia V a lle y S o la r R a n c h

DOW Cogen

C it r u s S p r in g s

C a n e I s la n d

A r v a h B H o p k in s

E . B . C a m p b e ll

H . L . C u lb r e a t h B a y s id e

W a n a p u m

D e s J o a c h im s

R. H. Saunder s

Pickering

O t t o H o ld e n

A b it ib i C a n y o n

H e n v e y

C h o u t e a u ( A S E C )

Cooper

Lansing

Alm a

Paradise

Allen (TVA)

Por t -Alf r ed

M c C o r m ic k

M o r g a n E n e r g y C e n t e r

S ik e s t o n

S o u t h w e s t I I

P r a ir ie C r e e k

C h e s w ic k

M a c t a q u a c

Culley

Dan River

D r e s d e n E n e r g y C e n t e r

C o u r t e n a y B a y

War r ick

T a u m S a u k

V e r n o n B o u le v a r d

N o r t h p o r t

P r e s q u e I s le

Vir ginia Cit y

Conoco

Plym out h

Spr ingdale

Car illon

R iv e r R o u g e

PSE

J o h n H K e r r

T e r r e ll C r e e k

F a c e R o c k

B e t h le h e m E n e r g y C e n t e r

Pequonic

Ginna

D e e r R iv e r

L o s M e d a n o s E n e r g y C e n t e r

R u m f o r d

W e s t o n 4

EdgarK la m a t h F a lls

Nixon

Drake

Rawhide

Model 2: Light Load Conditions with 408 GW of Load; Substation GDV with Generation

Sized and Colored by MW Value

69

Bus Frequency Results for a Generator Outage Contingency

Image shows the frequencies at all 110,000 buses; it was run for 80 seconds just to demonstrate the system stays stable


Freq

uenc

y (Hz

)

59.995

59.99

59.98559.98

59.975

59.97

59.965

59.96

59.955

59.9559.945

59.94

59.935

59.93

59.925

59.9259.915

59.91

59.905

Five minute East-West simulations


Volta

ge M

agni

tude

(PU

)

1.07

1.065

1.06

1.055

1.05

1.045

1.04

1.035

1.03

1.025

1.02

1.015

1.01

1.005

1

0.995

For modal analysis we’ll be looking at the first 20 second

70

Spatial Frequency Contour (Movies Can Also be Easily Created)

Transient Stability Time (Sec): 3.000

71

Bus Frequency Results for a Generator Outage Contingency


Freq

uenc

y (H

z)

60

59.98

59.96

59.94

59.92

59.9

A few selected results for the first 30 seconds

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Iterative Matrix Pencil Method Applied to 43,400 Substation Signals

Processing all 43,400 signals took about 75 seconds (with 20 seconds of simulation data, sampling at 10 Hz)

73

Iterative Matrix Pencil Method Applied to 43,400 Substation Signals

Trust but verify results



Valu

es

60

59.99

59.98

59.97

59.96

59.95

59.94

59.93

59.92

59.91

59.9

59.89

59.88




Valu

es

60.0005

60

59.9995

59.999

59.9985

59.998

59.9975

59.997

59.9965

59.996

59.9955

59.995

59.9945

59.994


Matching for a large deviation example

The worst match (out of 43,400 signals); note the change in the y-axis

74

Large System Visualization of a Mode using GDVs

75

Large System GDV Visualization of Another Mode (Same Arrow Scale)

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And a Third, Perhaps Less Familiar Mode (with 2x magnification)

77

Results with a Light Load

•  Below are the results for the light load case. Modal analysis allows different conditions to be compared.

78

Summary

•  The tutorial has covered the power system application of measurement-based modal analysis

•  Techniques are now available that can be readily applied to both small and large sets of power system measurements, either from the actual system or from simulations

•  The result is measurement-based modal analysis is now be a standard power system analysis tool

•  Large-scale system results can also be readily visualized

79

Questions? [email protected]

Power System Application of Measurement-Based Modal Analysis

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