Transcript Distributed PV, Small Signal Stability

ECE 576 – Power System
Dynamics and Stability
Lecture 25: Distributed PV,
Small Signal Stability
Prof. Tom Overbye
University of Illinois at Urbana-Champaign
[email protected]
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Announcements
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•
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Read Chapters 8 and 9
Homework 8 should be completed before final but need
not be turned in
Final Exam is Wednesday May 14 at 7 to 10pm in
classroom. Closed book, closed notes, your two
previous note sheets and one new note sheet allowed,
simple calculators allowed
2
Status of Nuclear Power Worldwide
In the USA, the five
reactors under
construction (about
1200 MW each) are
1) two units at
the Vogtle plant in
Georgia (2017)
2) two units in South
Carolina (2017/9)
3) TVA's Watts Bar
Unit 2 (2015)
Source: Fortune Magazine, April 2014
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Distributed PV System Modeling
•
PV in the distribution system is usually operated at
unity power factor
– There is research investigating the benefits of changing this
– IEEE Std 1547 prevents voltage regulation, but would allow
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non-unity power factor
– A simple model is just as negative constant power load
An issue is tripping on abnormal frequency or voltage
conditions
– IEEE Std 1547 says, "The DR unit shall cease to energize the
Area EPS for faults on the Area EPS circuit to which it is
connected.” (note EPS is electric power system)
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Distributed PV System Modeling
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An issue is tripping on abnormal frequency or voltage
conditions
– IEEE Std 1547 says, "The DR unit shall cease to energize the
Area EPS for faults on the Area EPS circuit to which it is
connected.” (note EPS is electric power system)
– This is a key safety requirement!
– Small units (less than 30kW) need to disconnect if the voltage
is < 0.5 pu in 0.16 seconds, and in 2 seconds if between 0.5
and 0.88 pu; also in 1 second if between 1.1 and 1.2, and in
0.16 seconds if higher
– Small units need to disconnect in 0.16 seconds if the
frequency is > 60.5 Hz, or less than 59.3 Hz
– Reconnection is after minutes
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Distributed PV System Modeling
•
Below is a prototype model for distributed solar PV
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Oscillations
•
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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 it has
the form  t
e  a cos t   b sin t    e t C cos t   
 b 
where C  A  B and   tan  
 a 
2
•
2
And the damping ratio is defined as (see Kundur 12.46)


 
2
2
The percent damping is just the
damping ratio multiplied by 100
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Power System Oscillations
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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
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The below graph shows an oscillation that was
observed during a 1996 WECC Blackout
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Example Oscillations
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The below graph shows oscillations on the
Michigan/Ontario Interface on 8/14/03
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Small Signal Stability Analysis
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Small signal stability is the ability of the power system
to maintain synchronism following a small disturbance
– System is continually subject to small disturbances, such as
•
•
changes in the load
The operating equilibrium point (EP) obviously must be
stable
Small system stability analysis (SSA) is studied to get a
feel for how close the system is to losing stability and to
get additional insight into the system response
– There must be positive damping
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Small Signal Stability Analysis
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Model based SSA is performed by linearizing about an
EP, and then calculating the associated eigenvalues (and
other properties) of the linearized system
With the advent of PMUs, measurement based
techniques are becoming increasingly common; this
approach is typically broken into two types
– Ringdown analysis is performed after the power system has
experienced a significant disturbance that has moved it away
from its EP
– Ambient analysis is performed when the power system is
operating in quasi-steady state
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An On-line Oscillation Detection Tool
Image source: WECC Joint Synchronized Information Subcommittee Report, October 2013
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Model Based SSA
•
Assume the power system is modeled as in our standard
form as
x  f  x, y 
0 = g(x, y)
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The system can be linearized about an equilibrium point
Δx = AΔx  BΔy
0 = CΔx + DΔy
•
If there are just classical generator
models then D is the power flow
Jacobian;otherwise it also includes
Eliminating Dy gives the stator algebraic equations
Δx =  A  BD-1C  Δx  A sys Δx
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Model Based SSA
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The matrix Asys can be calculated doing a partial
factorization, just like what was done with Kron
reduction
SSA is done by looking at the eigenvalues (and other
properties) of Asys
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SSA Two Generator Example
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Consider the two bus, two classical generator system
from lectures 18 and 20 with Xd1'=0.3, H1=3.0,
Xd2'=0.2, H2=6.0
Bus 2
GENCLS
Bus 1
GENCLS
X=0.22
slack
11.59 Deg
1.095 pu
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0.00 Deg
1.000 pu
Essentially everything needed to calculate the A, B, C
and D matrices was covered in lecture 20
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SSA Two Generator Example
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The A matrix is calculated differentiating f(x,y) with
respect to x (where x is d1, D1, d2, D2)
dd 1
 D1. pus
dt
d D1, pu
1

PM 1  PE1  D1D1. pu 

dt
2H 1
dd 2
 D2. pus
dt
d D2 , pu
1

PM 2  PE 2  D2 D1. pu 

dt
2H 2
PEi   EDi2  EDiVDi  Gi   EQi2  EQiVQi  Gi   EDiVQi  EQiVDi  Bi
EDi  jEQi  Ei  cos d i  j sin d i 
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SSA Two Generator Example
•
Giving
376.99
0
0 
 0
 0.761

0
0
0

A
 0
0
0
376.99 


0
0

0
.
389
0


•
B, C and D are as calculated previously for the implicit
integration, except the elements in B are not multiplied
by Dt/2
0
0
0 
 0
 0.2889 0.6505

0
0

B
 0
0
0
0 


0
0
0
.
0833
0
.
3893


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SSA Two Generator Example
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The C and D matrices are
 3.903
 1.733
C
 0

 0
•
0
0
0
0
0 4.671
0
1.0
0
7.88
0
4.54 
 0
 7.88

0 
0
4
.
54
0

, D
 0
0
4.54
0
9.54 



0
4
.
54
0

9
.
54
0


Giving
A sys
376.99
0
0 
 0
 0.229

0
0
.
229
0

 A - BD-1C  
 0
0
0
376.99 


0
.
114
0

0
.
114
0


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SSA Two Generator
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Calculating the eigenvalues gives a complex pair and
two zero eigenvalues
The complex pair, with values of +/- j11.39 corresponds
to the generators oscillating against each other at 1.81
Hz
One of the zero eigenvalues corresponds to the lack of
an angle reference
– Could be rectified by redefining angles to be with respect to a
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reference angle (see book 226) or we just live with the zero
Other zero is associated with lack of speed dependence
in the generator torques
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SSA Two Generator Speeds
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The two generator system response is shown below for
a small disturbance
60.5
Notice the
actual response
closely
matches the
calculated
frequency
60.45
60.4
60.35
60.3
60.25
60.2
60.15
60.1
60.05
60
59.95
59.9
59.85
59.8
59.75
59.7
59.65
59.6
59.55
59.5
0
0.5
1
1.5
b
c
d
e
f
g
2
2.5
Speed, Gen Bus 1 #1 g
b
c
d
e
f
3
3.5
4
4.5
5
Speed, Gen Bus 2 #1
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SSA Three Generator Example
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The two generator system is extended to three
generators with the third generator having H3 of 8 and
Xd3'=0.3
GENCLS
Bus 1
Bus 2
X=0.2
GENCLS
slack
3.53 Deg
1.0500 pu
X=0.2
Bus 3
GENCLS
X=0.2
0.00 Deg
1.0000 pu
-3.53 Deg
1.050 pu
200 MW
0 Mvar
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SSA Three Generator Example
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Using SSA, two frequencies are identified: one at 2.02
Hz and one at 1.51 Hz
We next
discuss
modal
analysis to
determine
the contribution
of each
mode to
each signal
60.0100
60.0090
60.0080
60.0070
60.0060
60.0050
60.0040
60.0030
60.0020
60.0010
60.0000
59.9990
59.9980
59.9970
59.9960
0
0.5
b
c
d
e
f
g
1
1.5
Speed_Gen Bus 1 #1 g
b
c
d
e
f
2
2.5
3
Speed_Gen Bus 2 #1 g
b
c
d
e
f
3.5
4
4.5
5
Speed_Gen 3 #1
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Large System Studies
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The challenge with large systems, which could have
more than 100,000 states, is the shear size
– Most eigenvalues are associated with the local plants
– Computing all the eigenvalues is computationally
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challenging, order n3
Specialized approaches can be used to calculate
particular eigenvalues of large matrices
– See Kundur, Section 12.8 and associated references
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Single Machine Infinite Bus
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A quite useful analysis technique is to consider the
small signal stability associated with a single generator
connected to the rest of the system through an
equivalent transmission line
Driving point impedance looking into the system is
used to calculate the equivalent line's impedance
– The Zii value can be calculated quite quickly using sparse
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vector methods
Rest of the system is assumed to be an infinite bus with
its voltage set to match the generator's real and reactive
power injection and voltage
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Small SMIB Example
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As a small example, consider the 4 bus system shown
below, in which bus 2 really is an infinite bus
Bus 1
Bus 2
X=0.2
GENCLS Bus 4
Infinite Bus
X=0.1
slack
11.59 Deg
1.0946 pu
6.59 Deg
1.046 pu
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Bus 3
X=0.1
X=0.2
4.46 Deg
1.029 pu
0.00 Deg
1.000 pu
To get the SMIB for bus 4, first calculate Z44
Ybus
 25
 0
 j
 10

 10
0 10
10 
1 0
0 
 Z 44  j0.1269

0 15
0

0 0
13.33 
Z44 is Zth in
parallel with
jX'd,4 (which is
j0.3) so Zth is
j0.22
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Small SMIB Example
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The infinite bus voltage is then calculated so as to
match the bus i terminal voltage and current
Vinf  Vi  Z i I i
*
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 Pi  jQi 
where 
  Ii
 Vi 
In the example we have
*
While this was demonstrated
on an extremely small system
for clarity, the approach works
the same for any size system
*
 P4  jQ4   1  j0.572 

 
  1  j0.328
 V4
  1.072  j0.220 
Vinf   1.072  j0.220   ( j0.22)  1  j0.328 
Vinf  1.0
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Calculating the A Matrix
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The SMIB model A matrix can then be calculated either
analytically or numerically
–
•
The equivalent line's impedance can be embedded in the
generator model so the infinite bus looks like the "terminal"
This matrix is calculated in PowerWorld by selecting
Transient Stability, SMIB Eigenvalues
– Select Run SMIB to perform an SMIB analysis for all the
generators in a case
– Right click on a generator on the SMIB form and select Show
SMIB to see the Generator SMIB Eigenvalue Dialog
– These two bus equivalent networks can also be saved, which
can be quite useful for understanding the behavior of individual
generators
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Example: Bus 4 SMIB Dialog
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On the SMIB dialog, the General Information tab shows
information about the two bus equivalent
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Example: Bus 4 SMIB Dialog
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On the SMIB dialog, the A Matrix tab shows the Asys
matrix for the SMIB generator
In this example A21 is showing
D4 , pu
d 4
1  PE ,4


2H 4  d 4
 0.3753

1

 1 

   
  1.2812 cos  23.94   
 6    0.3  0.22 


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Example: Bus 4 SMIB Dialog
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On the SMIB dialog, the Eigenvalues tab shows the Asys
matrix eigenvalues and participation factors (which
we'll cover shortly)
Saving the two bus SMIB equivalent, and putting a
short, self-cleared fault at the terminal shows the 1.89
Hz, undamped response
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Example: Bus 4 with
GENROU Model
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•
The eigenvalues can be calculated for any set of
generator models
This example replaces the bus 4 generator classical
machine with a GENROU model
– There are now six eigenvalues, with the dominate response
coming from the electro-mechanical mode with a frequency
of 1.83 Hz, and damping of 6.92%
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