Transcript Introductions - Kansas State University

What do you think about this system
response?
Rotor
Angle
Time
How about this response?
Rotor
Angle
Time
Compare these two responses
Rotor
Angle
Time
What about these responses?
Rotor
Angle
Time
Compare these instabilities
Rotor
Angle
Time
Steady-state = stable equilibrium


things are not changing
concerned with whether the
system variables are within the
correct limits
Transient Stability




"Transient" means changing
The state of the system is
changing
We are concerned with the
transition from one equilibrium to
another
The change is a result of a
"large" disturbance
Primary Questions


1. Does the system reach a new
steady state that is acceptable?
2. Do the variables of the system
remain within safe limits as the
system moves from one state to
the next?
Main Concern: synchronism of
system synchronous machines


Instability => at least one rotor
angle becomes unbounded with
respect to the rest of the
system
Also referred to as "going out of
step" or "slipping a pole"
Additional Concerns: limits on other
system variables


Transient Voltage Dips
Short-term current & power
limits
Time Frame

Typical time frame of concern
1 - 30 seconds



Model system components that
are "active" in this time scale
Faster changes -> assume
instantaneous
Slower changes -> assume
constants
Primary components to be modeled

Synchronous generators
Traditional control options

Generation based control
exciters, speed governors, voltage
regulators, power system stabilizers
Traditional Transmission Control
Devices


Slow changes
modeled as a constant value
FACTS Devices


May respond in the 1-30 second
time frame
modeled as active devices
May be used to help control
transient stability problems
Kundur's classification of methods
for improving T.S.




Minimization of disturbance
severity and duration
Increase in forces restoring
synchronism
Reduction of accelerating torque
by reducing input mechanical
power
Reduction of accelerating torque
by applying artificial load
Commonly used methods of
improving transient stability

High-speed fault clearing, reduction of
transmission system impedance, shunt
compensation, dynamic braking, reactor
switching, independent and single-pole
switching, fast-valving of steam
systems, generator tripping, controlled
separation, high-speed excitation
systems, discontinuous excitation
control, and control of HVDC links
FACTS devices = Exciting control
opportunities!


Deregulation & separation of
transmission & generation
functions of a utility
FACTS devices can help to
control transient problems from
the transmission system
3 Minute In-Class Activity




1. Pick a partner
2. Person wearing the most blue
= scribe Other person = speaker
3. Write a one-sentence
definition of "TRANSIENT
STABILITY”
4. Share with the class
Mass-Spring Analogy

Mass-Spring System
Equations of motion


Newton => F = Ma = Mx’’
Steady-state = Stable equilibrium
= Pre-fault
 SF = -K x - D x’ + w = Mball x’’ = 0

Can solve for x
Fault-on system

New equation of motion
 SF = -K x - D x’ + (Mball + Mbird)


= (Mball + Mbird) x’’
Initial Conditions?
x=
xss
x’ = 0
How do we determine x(t)?


Solve directly
Numerical methods
(Euler, Runge-Kutta, etc.)

Energy methods
Simulation of the Pre-fault & Faulton system responses
Post-fault system

"New" equation of motion
 SF = -K x - D x’ + w = Mball x’’


Initial Conditions?
x=
xc
x’ = xc’
Simulation of the Pre-fault, Faulton, and Post-fault system
responses
Transient Stability?


Does x tend to become
unbounded?
Do any of the system variables
violate limits in the transition?
Power System Equations
Start with Newton again ....
T=Ia
We want to describe the motion of the
rotating masses of the generators in
the system
The swing equation


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
2H d2 d = Pacc
wo dt2
P=Tw
a = d2d/dt2, acceleration is the second
derivative of angular displacement
w.r.t. time
w = dd/dt, speed is the first derivative


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Accelerating Power, Pacc
Pacc = Pmech - Pelec
Steady State => No acceleration
Pacc = 0 => Pmech = Pelec
Classical Generator Model

Generator connected to Infinite bus
through 2 lossless transmission lines

E’ and xd’ are constants
d is governed by the swing equation

Simplifying the system . . .

Combine xd’ & XL1 & XL2
jXT = jxd’ + jXL1 || jXL2

The simplified system . . .

Recall the power-angle curve

Pelec = E’ |VR| sin( d )
XT
Use power-angle curve

Determine steady state (SEP)
Fault study


Pre-fault => system as given
Fault => Short circuit at infinite bus
Pelec = [E’(0)/ jXT]sin(d) = 0

Post-Fault => Open one transmission line
XT2 = xd’ + XL2 > XT
Power angle curves
Graphical illustration of the fault study
Equal Area Criterion



2H d2 d = Pacc
wo dt2
rearrange & multiply both sides by 2dd/dt
2 dd d2d
dt dt2
=>
= wo Pacc dd
H
dt
d {dd}2 = wo Pacc dd
dt {dt }
H
dt
Integrating,


{dd}2 = wo Pacc dd
{dt}
H
dt
For the system to be stable, d must go
through a maximum => dd/dt must go
through zero. Thus . . .
dm
wo Pacc dd = 0 = { dd }2
H
{ dt }



do
The equal area criterion . . .

For the total area to be zero, the positive
part must equal the negative part. (A1 =
A2)
dcl

do
Pacc dd = A1 <= “Positive” Area
dm

dcl
Pacc
dd = A2 <= “Negative” Area
For the system to be stable for a given clearing angle d,
there must be sufficient area under the curve for A2 to
“cover” A1.
In-class Exercise . . .

Draw a P-d curve

For a clearing angle of 80 degrees
is the system stable?
what is the maximum angle?

For a clearing angle of 120 degrees
is the system stable?
what is the maximum angle?
Clearing at 80 degrees
Clearing at 120 degrees
What would plots of d vs. t look like for
these 2 cases?