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" DEVELOPING SWING EQUATION
= Derivation of swing equation
8 Om = Wemtt 5m, Wem is the constant angular velocity
= take the derivative of 0,,, we obtain —
do,
a
dt dt
= take the second derivative of 6,,, we obtain —
@O, ds,
dt dt
= substitute into the law of rotation
{Spat =1,-7,
multiplying w,, to obtain power equation
GS, yy PS,
5 T= OT — OT, = FF
dt dt . .
Je, DEVELOPING SWING EQUATION
erivation of swing equation
= swing equation in terms of inertial constant M
e 2s, pap
dae” €
« relations between electrical power angle 6 and
mechanical power angle 6,, and electrical speed and
mechanical speed
6-2
2
64, @= 5 @, wherep is pole number
» swing equation in terms of electrical power angle 6
A the
Jute pp
pat .
=» converting the swing equation into per unit system
2H dd 2H
oe ae = Pam — Pay WhereMf = a DEVELOPING SWING EQUATION
2M xq AS
>, a 5, Ge = Paco or Peay
W,. kinetic energy in MJ at rated speed H
S, machine power rating in MVA
2H do
Ta det men —Fecpin
H do 5
af we = Pau — Papuy (radians)
H dé
Teo F qe a mm Fac (degrees)
130 f dt? EXAMPLE I
A 3-phase, 60-Hz, 500-MVA, 15-kV, 32-pole hydrolelctric
generating unit has an H constant of 2.0 p.u.-s.
(a) Determine syn and Wmsyn.
(b) Write the swing equation of the unit.
(c) The unit is initially operating at pm.pu = pe.pu = 1.0, @ =
@syn , and 6 = 10 degrees when a 3-phase-to-ground
bolted short circuit at the generator terminals causes pe.pu
to drop to zero. Determine the power angle 3 cycles after
the shoty circuit commences. Assume pe.pu remains
constant at 1.0 p.u. Also assume G)p.u(t) = 1.0 in the swing
equation. EXAMPLE III
« A3-phase, 60-Hz, 400-kV,13.8-kV, 4-pole steam turbine
generating unit has an H constant of 5.0 p.u.-s. The generating
unit is initially operating at pm.pu = pe.pu = 0.7 p.u., @& = Wsyn,
6 = 12 degrees when a fault reduces the generation of electrical
power by 70%.
= Determine the power angle 6 five cycles after the fault
commences. Assume the the accelerating power remains
constant during the fault. Also assume wp.ut) = 0 1.0 in the
swing equation.
= Repeat (b) fora bolted 3-phase fault at the generator terminals
that reduces the electrical power output to zero. Compare the
power angle with that determined in (a). Example II
A Power plant has two 3-phase, 60-Hz generating
units with the following ratings:
Unit 1:500-MVA, 15-kV, 0.85 power factor, 32-
pole, Hi = 2.0 p.u.-s
Unit 2:300-MVA, 15-kV, 0.90 power factor, 32-
pole, H1 = 2.5 p.u.-s
Determine the per unit swing equation of each
unit on a 100-MVA system base.
If the units are assumed to “swing together” that
is, 6.(t) = 3.(t), combine the two swing equations
into one equivalent swing equations.
