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
• Numerical Solution Multi-machine Systems
• Lecture 5

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First swing stability-Numerical Solution

Pm Pe

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

Stability is governed by the Swing Equation
Swing Equation Power Angle Equation
d2d/dt2 (pf/H) (Pm-Pe)
dd /dt ?-?syn
Pe E V sin (d) /(XXL)
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First swing stability-Numerical Solution
Nonlinear ODE in state variable form
d ? /dt (pf/H) (Pm-Pmax sin d)
dd /dt ?-?syn
PmaxEV/(XdXL)
Usually cannot get a closed form solution
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First swing stability-Numerical Solution
Numerical solution find d(t) and ?(t)
d ? /dt (pf/H) (Pm-Pmax sin d)
dd /dt ?-?syn
Divide time into intervals tn-2,tn-1,tn, Predic
t dn from dn-1, d n-2,
Step size
dn
dn-1
dn-2
tn-2 tn-1 tn t
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First swing stability-Numerical Solution
Numerical solution find d(t) and ?(t)
d ? /dt (pf/H) (Pm-Pmax sin d)
dd /dt ?-?syn
Euler Method Uniform time step h dn dn-1 h
dd/dtttn-1
dd/dtttn-1
dn-1
h
tn-1 tn-2 tn1 t
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First swing stability-Equal Area
CriterionApplication

• Establish initial conditions
• Define sequence of events and network for each
  event
• Develop Power angle curves
• Apply EAC

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First swing stability-Equal Area Criterion
Example 1
Stability under small change in mechanical power
A 10 MVA, 0.8 pf lagging, 4160 V, 60Hz,
three-phase generator supplies 50 rated power
at .8 pf lagging to a 4160 V infinite bus.
Determine if the generator is first-swing stable
if the prime mover power is increased by 10
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First swing stability-Numerical Solution-Small
change in Pm
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First swing stability-Numerical Solution-Small
change in Pm
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Equal Area Criterion-Small change in mechanical
power
EAC
Remember
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First swing stability-Numerical Solution-Small
change in Pm
The EAC in the previous slide says angle swings
to 9.77 deg and then swings back Oscillates
around the new equilibrium of 8.949 deg
( Step size of 0.001 is a little big
oscillation is growing Due to numerical
instability)
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First swing stability-Numerical Solution-Small
change in PmEffect of Damping Damper windings
provide relative speed damping. Other effects
provide absolute damping.This will make swing
settle
Swing equation with relative speed damping
d ? /dt (pf/H) (Pm-Pmax sin d-D (?- ?syn)
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Example 2-Fault
First swing stability-Numerical Solution-Fault
2
1
8
3
The infinite bus receives 1 pu real power at 0.95
power factor lagging
A fault at bus 3 is cleared by opening lines from
1-3 and 2-3 when the generator power angle
dReaches 40 deg. Is the system first swing
stable?
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Example 2-Fault
Apply EAC
P
Pe
Pm
d
dm
do
dcl
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First swing stability-Numerical Solution-Fault
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First swing stability-Numerical Solution-Fault
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Equal Area CriterionExample 2-Fault
Rotor swings to 55 degrees then swings back-
STABLE
P
Pe
Pm
d
do
dcl
dm
55 40 24
23.95 40 55
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First swing stability-Numerical Solution-Fault
tgttrict.35
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Equal Area CriterionExample 2-Fault
Rotor swings past 156 degrees UN STABLE
P
Pe
Pm
d
do
dcl
dm
156 120 24
23.95 40 55
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Equal Area CriterionExample 2-Fault- Critical
Clearing
Rotor swings past 156 degrees UN STABLE
P
Pe
Pm
d
d0 d1
dmp- d1
dcl112.9
23.95
156 112 24
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Stability of Numerical Solutions

• Can become unstable due to
• Roundoff
• Approximation

dd/dtttn-1
Approximation Error
dn-1
h
tn-1 tn-2 tn1 t
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Stability of Numerical Solutions

• Can control
• Roundoff ( Reduce airthmetic operations)
• Approximation(Use more advanced method, but will
  increase arithmetic operations)
• Need to chose a reasonable step size ( or
  adaptively vary step size)
• Numerical stability Means the the global error
  remains bounded
• (See Crows Text for EE531/ Also see Kundur)

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Stability of Numerical Solutions
For the small change in Pm(Example 1) Euler is
unstable even with h.001
Note x- axis units n h time in seconds
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Additional Methods

• Taylor Series-based

First order approximation gives the Euler method
See Crow Text
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Additional Methods

• Taylor Series-based Second Order

Approximate the second derivative to get a second
order method
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Additional Methods

• Taylor Series-based Second order method

We do not know f(x(tn1),tn1) so approximate it
by the Euler formula
Euler formula for x(tn1)
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Additional Methods

• Taylor Series-based Second order method

Kundur and GS call this modified Euler. It is
also a second order Runge Kutta. Our texts write
the formula as follows
At time tn we know x(tn) Calculate
derivative dx/dtf(x(tn), tn) Now estimate
x(tn1), cal the estimate x x(tn1) x(tn)h
f(x(tn), tn), h tn1 tn Next estimate a new
value for the derivative, call it dx/dt
dx/dtf(x (tn1), tn1) Average the derivatives
dx/dt and dx/dt Finally, the next value
x(tn1) x(tn)h (dx/dt dx/dt)/2
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Additional Methods

• Taylor Series-based Second order method

First Estimate
Slope 1
Slope 2
Final Estimate
Estimated
Actual
h
Average
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Additional Methods

• Taylor Series-Fourth Order Runge Kutta

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Additional Methods

• Taylor Series-Fourth Order Runge Kutta

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Additional Methods

• Multistep

Assume a polynomial fit, e.g.,
Then
Next for ttn
And for ttn1 tnh
So
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Additional Methods

• Multistep

Assume a polynomial fit, e.g.,
If we approximate f(x(tn1),tn1) as before we
get the modified Euler or RK2
Alternatively, if we solve for x(tn1) we call
the method The Trapezoidal rule. Nonlinear
case, must solve iteratively Also called an
Implicit Method
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Additional Methods

• Trapezoidal Rule

Trapezoid approximates Area under the curve
h
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Next

• More on Numerical Solution
• Multimachine System