EE532 Power System Dynamics and Transients

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EE532 Power System Dynamics and Transients
EUMP Distance Education Services

• Satish J Ranade
• Classical Analysis
• First-swing transient stability
• Lecture 2

2

• Angle Stability Revisited-
• Machine Connected to Infinite Bus

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8
P
KE Builds up
P
Excess KE Needs to be removed
Fault makes generator electric Power
zero Generator accelerates. Can generator get
back to constant --synchronous speed? Only if it
can get rid of excess KE Excess KE needs to go
into the infinite bus through the line? Will it?
What happens if it cant? Stability means
returning to synchronous speed
SPEED
Line Real Power P
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Power System Stability

• First swing stability-Background needed

P
KE Builds up
P
Excess KE Needs to be removed

• Power flow characteristics in
• Network
• What governs power flow in a line?
• Dynamics of Turbine generator
• How does a generator change speed?
• How does generator dynamics affect
• Power flow?

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Power System Stability

• Modeling

P
KE Builds up
P
Excess KE Needs to be removed
Mechanical Model --- Swing Equation Network
Model --- Power Angle Equation
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Power System Stability

• Modeling Assumptions

P
KE Builds up
P
Excess KE Needs to be removed
Mechanical power is constant Rotor speed changes
SLOWLY as compared to 60 Hz Voltage and current
can be represented by slowly varying phasors
Real power can be calculated from phasor
models System remains balanced (Some Unbalanced
faults can be handled)
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• SLOWLY VARYING PHASOR

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• SLOWLY VARYING PHASOR
• Original amplitude and phase modulated wave

Write as a cosine at ?o and a tome varying phase
?(t)
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• SLOWLY VARYING PHASOR

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• SLOWLY VARYING PHASOR

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Power System Stability
First swing stability-Generator Electrical

• Classical model
• The generator reactance X Xd
• E is the phasor induced voltage
• It is assumed that the magnitude
• of E freezes at the value
• just prior to disturbance

I
Vt/0
E/d
E/d Vtpf/0 Ipf (RajXd)
Vtpf , Itpf are terminal voltage and current
before the fault
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Power System Stability
First swing stability-Power Transmission

• The N- node network can be represented by the
  admittance matrix equation
• I Y V
• The current injected into node m
• Im ?Nm1 Ymn Vn
• The Complex power into node m is
• SmVm (?Nm1 Ymn Vn)

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Power System Stability
First swing stability-Power Transmission

• Let VmVm/?m VnVn/?n YmnYmn/?mn
• Starting with
• SmVm (?Nm1 Ymn Vn)
• Can show
• PmRe(Sm)Vm ?nm1 Ymn Vn cos (?m - ?n -
  ?mn )
• QmIm(Sm)Vm ?nm1 Ymn Vn sin (?m - ?n -
  ?mn )

Sm
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Power System Stability
First swing stability-Power Transmission

• Real Power PmRe(Sm)Vm ?Nm1 Ymn
  Vn cos (?m - ?n - ?mn )
• Real power flow is controlled by voltage phase
  angles ?m, ?n
• For a line ( or generator with resistance and
  capacitance neglected, i.e., pure inductive
  reactance
• Vm/?m Pmn Vn/?n

jX


Y
1/jX
-1/jX
1/jX
-1/jX
Pmn Vm Vn sin (?m - ?n ) /X
Real power goes from higher phase angle to lower
phase angle To make power go down a line need to
make sending end phase angle larger
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Power System Stability
First swing stability-Dynamics
Pm Pe

• Generator dynamics

Tm Te
Tm Te J d?m /dt
where ?m mechanical speed
If mechanical torque Tm gt Electrical torque Te
speed ?m goes up If mechanical torque Tm lt
Electrical torque Te speed ?m goes down
Normally write this in terms of Power, speed in
electrical rad/s and units of PU
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First swing stability-Dynamics

Pm Pe

• Mechanical position
• and speed

Rotor spins CCW at speed
Tm ?m Te
?m
?m Mechanical position with respect to
stationary reference (rad.) ?m Mechanical
speed with respect to stationary
reference(rad/sec) ?m d ?m / dt ?m ? ?m dt
?m
Fixed Reference
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First swing stability-Dynamics

• In steady state synchronous Machines run at
  synchronous speed

Pm Pe
P number of poles ?msyn (2/P) ?syn ?syn
2 p f Electrical frequency in rad/s f
electrical frequency in Hz Position and speed
can be measured in mechanical (actual) or
electrical units ? (P/2)?m electrical radians
per second d (P/2) dm electrical radians per
second
Tm ?m Te
?m
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First swing stability-Dynamics

• In steady state synchronous Machines run at
  synchronous speed

Pm Pe
Tm ?m Te
?m
4 poles at 60 Hz ?syn 377 el. Rad/sec
?msyn 188.5 Rad/sec (1800 rpm)
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First swing stability-Dynamics

Pm Pe

• Mechanical position
• and speed

Tm ?m Te
?m
dm Mechanical position with respect to
reference rotating at synchronous speed
?msyn ?m Mechanical speed with respect to
rotating reference (rad/sec) ?m - ?msyn d
dm / dt
?m
?msyn
Reference rotates At Synchronous Speed
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First swing stability-Dynamics

• Mechanical position
• and speed

Pm Pe
?m - ?msyn 0 gt dm constant ?m - ?msyn 0 gt
dm increases ?m - ?msyn 0 gt dm
decreases Think about watching the hash mark on
the rotor under a strobe. If the rotor turns
faster than the strobe the hash advances at the
difference speed
Tm ?m Te
?m
?m
?msyn
Reference rotates At Synchronous Speed
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First swing stability-Dynamics

Pm Pe

• Back to
• Generator dynamics

Tm Te
J d2dm /dt2 Tm Te dd / dt ?m- ?msyn
Using PT? and dividing by Base MVA SB (?m/SB)
J d2dm /dt2 Pm - Pe Pa per unit
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First swing stability-Dynamics
Pm Pe

• Generator dynamics

Tm Te
(2?m/?msyn2) ((1/2 J?msyn2)/SB) J d2 /dt Pm -
Pe
(2?pu/?syn) ((1/2J?syn2 )/SB) J d? /dt Pm - Pe
Pa
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First swing stability-Dynamics
Pm Pe

• Generator dynamics

Tm Te
(2?m/?msyn2) ((1/2 J?msyn2 ) / SB) d2dm /dt2
Pm - Pe
(2?m/?msyn2) H d2dm /dt2 Pm - Pe
H(1/2) J?syn2/SB KE at synchronous speed/ Base
MVA H is called machine inertia in seconds
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First swing stability-Dynamics
Pm Pe

• Generator dynamics

Tm Te
(2?m/?msyn2) H d2dm /dt2 Pm - Pe
Multiply all angles and speeds by P/2 to go to
electrical units
Note ?/?syn ?pu
(2?pu/?syn) H d2d /dt2 Pm - Pe
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First swing stability-Dynamics
First swing stability-Dynamics
Pm Pe

• Generator dynamics

Tm Te
(2?pu/?syn) H d2d /dt2 Pm - Pe
This is not necessary but for now assume ?pu1
for simplicity
(2H/?syn) d2d /dt2 Pm - Pe
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First swing stability-Dynamics
Pm Pe

• Generator dynamics

Finally, since ?syn 2pf d? /dt ( pf/H )(Pm
Pe) dd / dt ? ?syn Equivalently, d2 d
/ dt2 Pm Pe dd / dt ? ?syn These are
called the SWING EQUATIONS Note ? and d are
NOT in pu f is electrical frequency in HZ
Tm Te
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First swing stability-Swing Equation

Pm Pe

• A generator connected to an infinite bus through
  a line
• Initially PmPe

jXL
jXd
Pm

Fixed (Infinite Bus)
Pe
V/0
E/d
If d changes Pe changes Pe E V sin (d)
/(XXL) If Pm or Pe changes speed changes
Pm Pe (pf/H) d? /dt
How are the two related?
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First swing stability-Swing Equation
Pm Pe

• A generator connected to an infinite bus through
  a line
• Initially PmPe Suppose we increase Pm then
  speed increases

jXL
jXd
Pm

Fixed (Infinite Bus)
Pe
V/0
E/d
If the speed of the generator increases the phase
angle d of induced voltage increases
Inf. Bus 60
Gen 60.6
d
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First swing stability-Swing Equation
Pm Pe

• A generator connected to an infinite bus through
  a line
• Initially PmPe Suppose we increase Pm then
  speed increases

If the speed of the generator increases the phase
angle d of induced voltage increases
Inf. Bus 60
Gen 60.6
d
d can be constant only if the frequencies are
identical -- Synchronous
d d/dt ?-?syn
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First swing stability-Swing Equation
SUMMARY Lecture 2

• A generator connected to an infinite bus through
  a line
• Initially PmPe

Pm Pe
jXL
jXd
Stability is governed by the Swing Equation
d2d/dt2 (pf/H) (Pm-Pe)
Swing Equation Power Angle Equation
dd /dt ?-?syn
For our system Pe E V sin (d) /(XXL)
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Power System Stability
First swing stability-Swing Equation

• Lecture 3
• Qualitative Analysis
• Examples of developing Swing Equations and
  Power Angle Curves
• Equal Area Criterion of Stability