ece 476 power system analysis
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Lecture 24Transient Stability
Professor Tom OverbyeDepartment of Electrical and
Computer Engineering
ECE 476
POWER SYSTEM ANALYSIS
2
Announcements
Be reading Chapter 10 and 13– Including CH 10 article about zone 3 relays, CH 13
DSA and blackout articles
HW 11 is not turned in but should be done before final. HW 11 is 13.1, 13.7, 13.8, 13.18, and SP1
Final is Wednesday Dec 12 from 1:30 to 4:30pm in EL 269 (note room change). Final is comprehensive. One new note sheet, and your two old note sheets are allowed
3
In the News: Dallman Accident
Cause of CWLP (Springfield, IL) Dallman generator accident is still under investigation, but after the generator tripped the main turbine steam stop valve did not close. With the generator’s electrical output at zero (i.e., it was disconnected) the turbine/rotor continued to accelerate up to over 5000 rpm (3600 normal). Then it suddenly stopped.
– 95 MW unit, $60 million to repair damage/replace unit
On a more positive note for CWLP their Dallman 4 plant is on schedule for a January 2010 start.
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After the Dallman Accident
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Outside of Dallman
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2007 Energy Bill
Congress and President Bush are currently discussing an energy bill, with a key sticking point being whether to require investor owned utilities to provide 15% of their electricity from renewables by 2020.
– There is a big difference between 15% of capacity versus 15% of energy from renewables
A major issue is lack of wind capacity in the southeast U.S.
7
US Wind Resource Map
8
Transient Stability Analysis
For transient stability analysis we need to consider three systems
1. Prefault - before the fault occurs the system is assumed to be at an equilibrium point
2. Faulted - the fault changes the system equations, moving the system away from its equilibrium point
3. Postfault - after fault is cleared the system hopefully returns to a new operating point
9
Transient Stability Solution Methods
There are two methods for solving the transient stability problem
1. Numerical integration 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
2. 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
10
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.
11
SMIB Example, cont’d
Simplified prefault system
1
The prefault system has two
equilibrium points; the left one
is stable, the right one unstable
sin M th
a
P XE
12
SMIB Example, Faulted System
During the fault the system changes
The equivalent system during the fault is then
During this fault nopower can be transferredfrom the generator to the system
13
SMIB Example, Post Fault System
After the fault the system again changes
The equivalent system after the fault is then
14
SMIB Example, Dynamics
eDuring the disturbance the form of P ( ) changes,
altering the power system dynamics:
1sina th
Mth
E VP
M X
15
Transient Stability Solution Methods
There are two methods for solving the transient stability problem
1. Numerical integration 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
2. 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
16
Numerical Integration of DEs
0 0
0
Assume we have a problem of the form
( ) with (t )
This is known as an initial value problem since the
initial value of is given at some value of time, t .
We then need to determine (t) for futu
x f x x x
x
x
re time.
Except for special cases, such as linear systems, no
analytic solution is possible. We must use numerical
technqiues.
17
Examples
1 2
0
1 2
0
2
Example 1: Exponential Decay
A simple example with an analytic solution is
x with x(0) x
This has a solution x(t) x
Example 2: Mass-Spring Syste
or
x
1
m
t
kx gM Mx Dx
x
x k x gMM
x
e
D x
18
Euler’s Method
The simplest technique for numerically integrating
these equations is known as Euler's method. Key idea
dis to approximate ( ( )) as
dt tThen
( ) ( ) ( ( ))
In general the smaller the ti
t
t t t t t
x xx f x
x x f x
me step, , the better the
approximation.
t
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Euler’s Method Algorithm
0
0 0
end
Set t = t (usually 0)
(t ) =
Pick the time step t, which is problem specific
While t t Do
( ) ( ) ( ( ))
End While
t t t t t
t t t
x x
x x f x
20
Euler’s Method Example 1
0
0
Consider the Exponential Decay Example
x with x(0) x
This has a solution x(t) x
Since we know the solution we can compare the accuracy
of Euler's method for different time steps
t
x
e
21
Euler’s Method Example 1, cont’d
t xactual(t) x(t) t=0.1 x(t) t=0.05
0 10 10 10
0.1 9.048 9 9.02
0.2 8.187 8.10 8.15
0.3 7.408 7.29 7.35
… … … …
1.0 3.678 3.49 3.58
… … … …
2.0 1.353 1.22 1.29
22
Euler’s Method Example 2
1 2
2 1
1 2
1
Consider the equations describing the horizontal
position of a cart attached to a lossless spring:
x
Assuming initial conditions of (0) 1 and x (0) 0,
the analytic solution is x ( ) cos .
We
x
x x
x
t t
can again compare the results of the analytic and
numerical solutions
23
Euler's Method Example 2, cont'd
1 1 2
2 2 1
Starting from the initial conditions at t =0 we next
calculate the value of x(t) at time t = 0.25.
(0.25) (0) 0.25 (0) 1.0
(0.25) (0) 0.25 (0) 0.25
Then we continue on to the next time step, t
x x x
x x x
1 1 2
2 2 1
= 0.50
(0.50) (0.25) 0.25 (0.25)
1.0 0.25 ( 0.25) 0.9375
(0.50) (0.25) 0.25 (0.25)
0.25 0.25 (1.0) 0.50
x x x
x x x
24
Euler's Method Example 2, cont'd
t x1actual(t) x1(t) t=0.25
0 1 1
0.25 0.9689 1
0.50 0.8776 0.9375
0.75 0.7317 0.8125
1.00 0.5403 0.6289
… … …
10.0 -0.8391 -3.129
100.0 0.8623 -151,983
25
Euler's Method Example 2, cont'd
t x1(10)
actual -0.8391
0.25 -3.129
0.10 -1.4088
0.01 -0.8823
0.001 -0.8423
Below is a comparison of the solution values for x1(t)at time t = 10 seconds
26
Transient Stability Example
A 60 Hz generator is supplying 550 MW to an infinite bus (with 1.0 per unit voltage) through two parallel transmission lines. Determine initial angle change for a fault midway down one of the lines.H = 20 seconds, D = 0.1. Use t=0.01 second.
Ea
27
Transient Stability Example, cont'd
a
e
We first need to determine the pre-fault values.
Since P = 550 MW (5.5 pu) I = 5.5
E 1.0 0.1 5.5 1.141 28.8
Next to get P ( ) we need to determine the
thevenin equivalent during the fault looking
j
into
the network from the generator
0.05 0.05 0.1 0.08333
0.3333 0th
th
Z j j j j
V
28
Transient Stability Example, cont'd
prefaulte
m
faultede
1 2
1.141 1.0Therefore prefault we have P ( ) sin
0.1and P 5.5 (0) 28.8 (0) 0.50265 radians
1.141 0.3333and during the fault P ( ) sin
0.08333Let x and x . The equations to integ
1 2
2 1 2
1 2
rate are
1 1.141 0.33335.5 sin 0.1
20/ 60 0.08333
(0) 0.50265 (0) 0.0
x x
x x x
x x
29
Transient Stability Example, cont'd
1 2
2 1 29.425 5.5 4.564sin 0.1
0.50265(0)
0
With Euler's Method we get
0.50265 0 0.50265(0.01) 0.01
0 31.11 0.3111
0.50265 0.3111 0.50576(0.02) 0.01
0.3111 30.82 0.
x x
x x x
x
x
x
6193
30
Transient Stability Example, cont'd
0 0.5 1 1.5 2
Simulation time in seconds
0
60
120
180
240G
en
erat
or
ang
le in
deg
rees
clearing at 0.3 seconds
clearing at 0.2 seconds
clearing at 0.1 seconds
31
Equal Area Criteria
The goal of the equal area criteria is to try to determine whether a system is stable or not without having to completely integrate the system response.
System willbe stable afterthe fault ifthe DecelArea is greaterthan the Accel. Area
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