Transcript Slides

ECE 476
POWER SYSTEM ANALYSIS
Lecture 23
Power System Protection and Transient Stability
Professor Tom Overbye
Department of Electrical and
Computer Engineering
Announcements

Design Project has firm due date of Dec 4
–
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Potentially useful article: T.J. Overbye, “Fostering
Intuitive Minds for Power System Design,” IEEE Power
and Energy Magazine, July-August 2003
Be reading Chapter 13.
HW 10 is 8.3, 8.5, 9.1,9.2 (bus 3), 9.13, 9.53 is due
on Thursday Dec 4.
Final is Tuesday Dec 16 from 7 to 10pm in EL 165
(note this is NOT what the web says). Final is
comprehensive. One new note sheet, and your two
old note sheets are allowed
1
Fault Calculation Example
The zero, positive and negative sequence bus impedance matrixes for a
three bus, three phase power system are given below. Determine the per
unit fault current (sequence values only) for a double line to ground fault
involving phases "B and C" at bus 2. The prefault voltage at all buses is
1.0 per unit. Assume the fault impedance is zero.
0
0.1 0
Z0  j  0 0.2 0 


0 0.1
 0
0.12 0.08 0.04 
Z   Z   j  0.08 0.12 0.06 


0.04 0.06 0.08
2
Directional Relays

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Directional relays are commonly used to protect
high voltage transmission lines
Voltage and current measurements are used to
determine direction of current flow (into or out of
line)
Relays on both ends of line communicate and will
only trip the line if excessive current is flowing into
the line from both ends
–
–
line carrier communication is popular in which a high
frequency signal (30 kHz to 300 kHz) is used
microwave communication is sometimes used
3
Impedance Relays

Impedance (distance) relays measure ratio of
voltage to current to determine if a fault exists on a
particular line
Assume Z is the line impedance and x is the
normalized fault location (x  0 at bus 1, x  1 at bus 2)
V1
V1
Normally
is high; during fault
 xZ
I12
I12
4
Impedance Relays Protection Zones

To avoid inadvertent tripping for faults on other
transmission lines, impedance relays usually have
several zones of protection:
–
–
–

zone 1 may be 80% of line for a 3f fault; trip is
instantaneous
zone 2 may cover 120% of line but with a delay to prevent
tripping for faults on adjacent lines
zone 3 went further; most removed due to 8/14/03 events
The key problem is that different fault types will
present the relays with different apparent
impedances; adequate protection for a 3f fault gives
very limited protection for LL faults
5
Impedance Relay Trip Characteristics
Source: August 14th 2003 Blackout Final Report, p. 78
6
Differential Relays

Main idea behind differential protection is that
during normal operation the net current into a
device should sum to zero for each phase
–

transformer turns ratios must, of course, be considered
Differential protection is used with geographically
local devices
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buses
transformers
generators
I1  I 2  I3  0 for each phase
except during a fault
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Other Types of Relays

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In addition to providing fault protection, relays are
used to protect the system against operational
problems as well
Being automatic devices, relays can respond much
quicker than a human operator and therefore have
an advantage when time is of the essence
Other common types of relays include
–
–
–
under-frequency for load: e.g., 10% of system load must
be shed if system frequency falls to 59.3 Hz
over-frequency on generators
under-voltage on loads (less common)
8
Sequence of Events Recording
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During major system disturbances numerous relays
at a number of substations may operate
Event reconstruction requires time synchronization
between substations to figure out the sequence of
events
Most utilities now have sequence of events
recording that provide time synchronization of at
least 1 microsecond
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Use of GPS for Fault Location
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Since power system lines may span hundreds of
miles, a key difficulty in power system restoration is
determining the location of the fault
One newer technique is the use of the global
positioning system (GPS).
GPS can provide time synchronization of about 1
microsecond
Since the traveling electromagnetic waves propagate
at about the speed of light (300m per microsecond),
the fault location can be found by comparing arrival
times of the waves at each substation
10
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
–
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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
11
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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Example of Transient Behavior
13
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
14
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
15
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
16
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 
18
Generator Swing Equation
This equation is known as the generator swing equation
Pm  Pe ( )  M 
Adding damping we get
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
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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

1.
2.
3.
For transient stability analysis we need to consider
three systems
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
23
Transient Stability Solution Methods

1.
There are two methods for solving the transient
stability problem
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
24
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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SMIB Example, Post Fault System
After the fault the system again changes
The equivalent system after the fault is then
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SMIB Example, Dynamics
During the disturbance the form of Pe ( ) changes,
altering the power system dynamics:
1
 
M


EaVth
 PM  X sin  


th
29
Transient Stability Solution Methods

1.
There are two methods for solving the transient
stability problem
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
30