A Power System Example
Starrett Mini-Lecture #3
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
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Generator connected to Infinite bus through 2
lossless transmission lines
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E’ and xd’ are constants
d is governed by the swing equation
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Simplifying the system . . .
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Combine xd’ & XL1 & XL2
jXT = jxd’ + jXL1 || jXL2
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The simplified system . . .
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Recall the power-angle curve
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Pelec = E’ |VR| sin( d )
XT
Use power-angle curve
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Determine steady state (SEP)
Fault study
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Pre-fault => system as given
Fault => Short circuit at infinite bus
Pelec = [E’(0)/ jXT]sin(d) = 0
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Post-Fault => Open one transmission line
XT2 = xd’ + XL2 > XT
Power angle curves
Graphical illustration of the fault
study
Equal Area Criterion
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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,
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{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 }
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do
The equal area criterion . . .
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For the total area to be zero, the positive part
must equal the negative part. (A1 = A2)
dcl
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Pacc dd = A1 <= “Positive” Area
do
dm
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Pacc dd = A2 <= “Negative” Area
dcl
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 . . .
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Draw a P-d curve
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For a clearing angle of 80 degrees
is the system stable?
what is the maximum angle?
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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?