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What do you think about this system response?
Time
Rotor Angle
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How about this response?
Time
Rotor Angle
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Compare these two responses
Time
Rotor Angle
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What about these responses?
Time
Rotor Angle
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Compare these instabilities
Time
Rotor Angle
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Steady-state = stable equilibrium
things are not changing concerned with whether the system
variables are within the correct limits
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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
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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?
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Main Concern: synchronism of system synchronous machines
Instability => at least one rotor angle becomes unbounded with respect tothe rest of the system
Also referred to as "going out of step" or "slipping a pole"
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Additional Concerns: limits on other system variables
Transient Voltage Dips Short-term current & power limits
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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
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Primary components to be modeled
Synchronous generators
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Traditional control options
Generation based control exciters, speed governors, voltage
regulators, power system stabilizers
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Traditional Transmission Control Devices
Slow changes modeled as a constant value
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FACTS Devices
May respond in the 1-30 second time frame
modeled as active devices
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May be used to help control transient stability problems
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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
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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
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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
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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
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Mass-Spring Analogy
Mass-Spring System
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Equations of motion
Newton => F = Ma = Mx’’ Steady-state = Stable equilibrium
= Pre-fault
F = -K x - D x’ + w = Mball x’’ = 0 Can solve for x
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Fault-on system
New equation of motion
F = -K x - D x’ + (Mball + Mbird)g = (Mball + Mbird) x’’
Initial Conditions? x = xss x’ = 0
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How do we determine x(t)?
Solve directly Numerical methods
(Euler, Runge-Kutta, etc.) Energy methods
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Simulation of the Pre-fault & Fault-on system responses
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Post-fault system
"New" equation of motion
F = -K x - D x’ + w = Mball x’’ Initial Conditions? x = xc x’ = xc’
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Simulation of the Pre-fault, Fault-on, and Post-fault system responses
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Transient Stability?
Does x tend to become unbounded? Do any of the system variables
violate limits in the transition?
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Power System Equations
Start with Newton again ....T = I
We want to describe the motion of the rotating masses of the generators in the system
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The swing equation
2H d2 = Pacc
o dt2
P = T = d2/dt2, acceleration is the second
derivative of angular displacement w.r.t. time
= d/dt, speed is the first derivative
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Accelerating Power, Pacc
Pacc = Pmech - Pelec
Steady State => No acceleration Pacc = 0 => Pmech = Pelec
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Classical Generator Model
Generator connected to Infinite bus through 2 lossless transmission lines
E’ and xd’ are constants is governed by the swing equation
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Simplifying the system . . .
Combine xd’ & XL1 & XL2
jXT = jxd’ + jXL1 || jXL2
The simplified system . . .
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Recall the power-angle curve
Pelec = E’ |VR| sin( ) XT
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Use power-angle curve
Determine steady state (SEP)
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Fault study
Pre-fault => system as given Fault => Short circuit at infinite bus
Pelec = [E’(0)/ jXT]sin() = 0
Post-Fault => Open one transmission line XT2 = xd’ + XL2 > XT
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Power angle curves
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Graphical illustration of the fault study
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Equal Area Criterion
2H d2 = Pacc
o dt2
rearrange & multiply both sides by 2d/dt
2 d d2 = o Pacc d dt dt2 H dt
=>d {d}2 = o Pacc ddt {dt } H dt
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Integrating,
{d}2 = o Pacc d{dt} H dt
For the system to be stable, must go through a maximum => d/dt must go through zero. Thus . . . m
o Pacc d = 0 = { d2
H { dt } o
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The equal area criterion . . .
For the total area to be zero, the positive part must equal the negative part. (A1 = A2)
Pacc d = A1 <= “Positive” Area
Pacc d = A2 <= “Negative” Area
cl
o
m
cl
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For the system to be stable for a given clearing angle , there must be sufficient area under the curve for A2 to “cover” A1.
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In-class Exercise . . .
Draw a P- 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?
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Clearing at 80 degrees
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Clearing at 120 degrees
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What would plots of vs. t look like for these 2 cases?