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
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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)
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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!
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Example Oscillations
• This graph shows oscillations on the Michigan/Ontario Interface on 8/14/03
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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
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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
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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)
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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
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Example: One Signal
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Verification: Linear Trend Line Only
22
Verification: Linear Trend Line + One Mode
23
Verification: Linear Trend Line + Two Modes
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Verification: Linear Trend Line + Three Modes
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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
Measurement-Based Modal Analysis
• 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
Measurement-Based Modal Analysis
• 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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AUSTIN 2
PASADENA 3
ARLINGTON 1
M CKINNEY 3
JACKSONVILLE 1
KYLE
M ANSFIELD
GREENVILLE 1
HOUSTON 4
CYPRESS 1
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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!
PWDVectorGrid Variables
Time (Seconds)151050
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
Original Value Reproduced Value
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)
PWDVectorGrid Variables
Time (Seconds)161412108642
Valu
es
6059.9959.9859.9759.9659.9559.9459.9359.9259.9159.9
59.8959.8859.8759.8659.8559.8459.8359.8259.8159.8
Original Value Reproduced Value
54
2000 Bus System Example, Two Signals
PWDVectorGrid Variables
Time (Seconds)161412108642
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
Original Value Reproduced Value
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 2 7
F O R T W O R T H 2 9
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
S A N A N T O N I O 3 2
SAN ANTONIO 37
S A N A N T O N I O 4 0
S A N A N T O N I O 4 7
S A N A N T O N I O 5 0
S A N A N T O N I O 5 1
S A N A N T O N I O 5 2
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
Simulation Time (Seconds)300250200150100500
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
Simulation Time (Seconds)300250200150100500
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
Simulation Time (Seconds)302520151050
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
72
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
PWDVectorGrid Variables
Time (Seconds)2015105
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
Original Value Reproduced Value
PWDVectorGrid Variables
Time (Seconds)2015105
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
Original Value Reproduced Value
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)
76
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