Lecture 24Transient Stability

Professor Tom OverbyeDepartment of Electrical and

Computer Engineering

ECE 476

POWER SYSTEM ANALYSIS

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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

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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.

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US Wind Resource Map

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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

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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

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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

1

The prefault system has two

equilibrium points; the left one

is stable, the right one unstable

sin M th

a

P XE

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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 nopower can be transferredfrom 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

eDuring the disturbance the form of P ( ) changes,

altering the power system dynamics:

1sina th

Mth

E VP

M X

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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