#### Transcript Lecture 23

```ECE 476
Power System Analysis
Lecture 23: Transient Stability
Prof. Tom Overbye
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
[email protected]
Announcements
• HW 10 is 11.1, 11.4, 11.12, 11.19, 11.21; quiz on Dec
1 (hence it will not be turned in)
•
We will be dropping your lowest two HW and/or Quiz scores
• Chapter 6 Design Project 1 is assigned. It will count as
three regular home works and is due on Dec 3.
–
For tower configurations assume a symmetric conductor spacing, with the distance
in feet given by the following formula: (Last two digits of your EIN+150)/10.
Example student A has an UIN of xxx65. Then his/her spacing is (65+150)/10 =
21.50 ft.
• Final exam is on Monday December 12, 1:30-4:30pm
1
Power System Time Scales and
Transient Stability
Image source: P.W. Sauer, M.A. Pai, Power System Dynamics and Stability, 1997, Fig 1.2, modified
2
Example of Frequency Variation
• Figure shows Eastern Interconnect frequency
variation after loss of 2600 MWs
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Example of Dynamics Behavior
Source: August 14th 2003 Blackout Final Report
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Power System Dynamics Motivation:
Frequency Decline September 2011 Blackout
Image Source: Arizona-Southern California Outages on September 8, 2011 Report, FERC and NERC,April 2012
5
Power Grid Disturbance Example
Figures show the frequency change as a result of the sudden loss of a
large amount of generation in the Southern WECC
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Green is bus quite close to
location of generator trip while
blue and red are quite distant.
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Time in Seconds
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Frequency Contour
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Recap: Power Flow
• The power flow is used to determine a quasi steadystate operating condition for a power system
–
Goal is to solve a set of algebraic equations
•
–
–
g(y) = 0 [y variables are bus voltage and angle]
Models employed reflect the steady-state assumption
Using a power flow, after a contingency occurs (such as
opening a line), the algebraic equations are solved to
determine a new equilibrium
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Power Flow vs. Dynamics
• Dynamics simulations is used to determine whether
following a contingency the power system returns to a
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Goal is to solve a set of differential and algebraic equations,
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•
–
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dx/dt = f(x,y) [y variables are bus voltage and angle]
g(x,y) = 0
[x variables are dynamic state variables]
Models reflect the transient stability time frame (up to dozens
of seconds)
•
•
Slow Values  Treat as constants
Ultra Fast States  Treat as algebraic relationships
8
Power System Transient Stability
• In order to operate as an interconnected system all
of the generators (and other synchronous machines)
must remain in synchronism with one another
–
synchronism requires that (for two pole machines) the
rotors turn at exactly the same speed
• Loss of synchronism results in a condition in which
no net power can be transferred between the
machines
• A system is said to be transiently unstable if
following a disturbance one or more of the
generators lose synchronism
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Generator Transient Stability Models
• In order to study the transient response of a power
system we need to develop models for the
generator valid during the transient time frame of
several seconds following a system disturbance
• We need to develop both electrical and mechanical
models for the generators
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Generator Electrical Model
• The simplest generator model, known as the
classical model, treats the generator as a voltage
source behind the direct-axis transient reactance;
the voltage magnitude is fixed, but its angle
changes according to the mechanical dynamics
VT Ea
Pe ( ) 
sin 
'
Xd
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Generator Mechanical Model
Generator Mechanical Block Diagram
Tm  J m  TD  Te ( )
Tm  mechanical input torque (N-m)
J  moment of inertia of turbine & rotor
 m  angular acceleration of turbine & rotor
TD  damping torque
Te ( )  equivalent electrical torque
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Generator Mechanical Model, cont’d
In general power = torque  angular speed
Hence when a generator is spinning at speed s
Tm
 J m  TD  Te ( )
Tm s  ( J m  TD  Te ( )) s
Pm
Pm
 J ms  TDs  Pe ( )
Initially we'll assume no damping (i.e., TD  0)
Then
Pm  Pe ( )  J ms
Pm is the mechanical power input, which is assumed
to be constant throughout the study time period
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Generator Mechanical Model, cont’d
Pm  Pe ( )
m
m
m
 J  ms
 st  
 rotor angle
d m

  m  s  
dt
 m  
Pm  Pe ( )  J s m  J s
J s
 inertia of machine at synchronous speed
Convert to per unit by dividing by MVA rating, S B ,
Pm Pe ( )
J s 2s


SB
SB
S B 2s
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Generator Mechanical Model, cont’d
Pm Pe ( )
J s 2 s


SB
SB
S B 2 s
Pm  Pe ( )

SB
J  s2 1

2S B  f s
J  s2
Define
2S B
H  per unit inertia constant (sec)
(since  s  2 f s )
All values are now converted to per unit
H
H
Pm  Pe ( ) 

Define M 
 fs
 fs
Then
Pm  Pe ( )  M 
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Generator Swing Equation
This equation is known as the generator swing equation
Pm  Pe ( )  M 
Pm  Pe ( )  M   D
This equation is analogous to a mass suspended by
a spring
k x  gM  Mx  Dx
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Single Machine Infinite Bus (SMIB)
• To understand the transient stability problem we’ll
first consider the case of a single machine
(generator) connected to a power system bus with
a fixed voltage magnitude and angle (known as an
infinite bus) through a transmission line with
impedance jXL
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SMIB, cont’d
Ea
Pe ( ) 
sin 
'
Xd  XL
M   D
Ea
 PM  '
sin 
Xd  XL
18
SMIB Equilibrium Points
Equilibrium points are determined by setting the
right-hand side to zero
Ea
M   D  PM  '
sin 
Xd  XL
Ea
PM  '
sin   0
Xd  XL
Define X th  X d'  X L
1  PM
X th 
  sin 

E

a 
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Transient Stability Analysis
• For transient stability analysis we need to consider
three systems
1.
2.
3.
Prefault - before the fault occurs the system is assumed
to be at an equilibrium point
Faulted - the fault changes the system equations,
moving the system away from its equilibrium point
Postfault - after fault is cleared the system hopefully
returns to a new operating point
Actual transient stability studies can have
multiple events
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Transient Stability Solution Methods
• There are two methods for solving the transient
stability problem
1.
Numerical integration
•
2.
this is by far the most common technique, particularly for
large systems; during the fault and after the fault the power
system differential equations are solved using numerical
methods
Direct or energy methods; for a two bus system this
method is known as the equal area criteria
•
mostly used to provide an intuitive insight into the transient
stability problem
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SMIB Example
• Assume a generator is supplying power to an
infinite bus through two parallel transmission lines.
Then a balanced three phase fault occurs at the
terminal of one of the lines. The fault is cleared by
the opening of this line’s circuit breakers.
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SMIB Example, cont’d
Simplified prefault system
The prefault system has two
equilibrium points; the left one
is stable, the right one unstable
1  PM
X th 
  sin 

E

a 
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SMIB Example, Faulted System
During the fault the system changes
The equivalent system during the fault is then
During this fault no
power can be transferred
from the generator to
the system
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