Problem 2 Consider a single-machine infinite-bus (SMIB). The phasor voltage at the infinite bus and the internal generator node are denote by E20 and VZ8, respectively (E and V are constant). XL is the line reactance. The mechanical power input into the machine is denoted by p(t), while the frequency w(t) d8(t)/dt denotes the rate of change of the phase angle 8(t). The SMIB dynamics is EZO given by: = VZ8 oÏÏ ooo XL G $(t) = w(t) w(t)=2w (t) + p(t) - (EV/XL) 8(t) Assume zero initial conditions, i.e., 8(0) = 0, w(0) = 0, and V = E = 1, XL = 0.1. a) In transient stability analysis, one needs to closely monitor and control the phase angle 8(t) of the plant, with the help of the mechanical power input p(t). From the above dynamics equations, using p(t) as input and 8(t) as output, show that the input-output transfer function is: H(s) = 1 A(s) P(s) s² + 2s + 10 = where A(s) and P(s) denote, respectively, the Laplace transforms of 8(t) and p(t). b) Assume a unit step input, i.e., p(t) = u(t). Use inverse Laplace transform (and partial fractions expansion) to compute 8(t) (in time-domain). Hint: You may find the above tip useful.

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Useful Tip: You may often find the following partial fractions expansion technique useful:
1
(s²+as+b-s² — as) /b 1
s (s² +as+b)
s (s² +as+b)
bs
for any pair of values of a and b / 0.
=
s+a
b(s²+as+b)
Transcribed Image Text:Useful Tip: You may often find the following partial fractions expansion technique useful: 1 (s²+as+b-s² — as) /b 1 s (s² +as+b) s (s² +as+b) bs for any pair of values of a and b / 0. = s+a b(s²+as+b)
Problem 2
Consider a single-machine infinite-bus (SMIB). The phasor voltage at the infinite
bus and the internal generator node are denote by E20 and VZ8, respectively (E and V are constant).
XL is the line reactance. The mechanical power input into the machine is denoted by p(t), while the
frequency w(t) d8(t)/dt denotes the rate of change of the phase angle 8(t). The SMIB dynamics is
EZO
given by:
=
V28
oÏÏ
ooo
XL
G
$(t) = w(t)
w(t)=2w (t) + p(t) - (EV/XL) 8(t)
Assume zero initial conditions, i.e., 8(0) = 0, w(0) = 0, and V = E= 1, XL = 0.1.
a) In transient stability analysis, one needs to closely monitor and control the phase angle 8(t) of the
plant, with the help of the mechanical power input p(t). From the above dynamics equations,
using p(t) as input and 8(t) as output, show that the input-output transfer function is:
H(s) =
A(s)
P(s)
=
1
s² + 2s + 10.
where A(s) and P(s) denote, respectively, the Laplace transforms of 8(t) and p(t).
b) Assume a unit step input, i.e., p(t) = u(t). Use inverse Laplace transform (and partial fractions
expansion) to compute 8(t) (in time-domain). Hint: You may find the above tip useful.
Transcribed Image Text:Problem 2 Consider a single-machine infinite-bus (SMIB). The phasor voltage at the infinite bus and the internal generator node are denote by E20 and VZ8, respectively (E and V are constant). XL is the line reactance. The mechanical power input into the machine is denoted by p(t), while the frequency w(t) d8(t)/dt denotes the rate of change of the phase angle 8(t). The SMIB dynamics is EZO given by: = V28 oÏÏ ooo XL G $(t) = w(t) w(t)=2w (t) + p(t) - (EV/XL) 8(t) Assume zero initial conditions, i.e., 8(0) = 0, w(0) = 0, and V = E= 1, XL = 0.1. a) In transient stability analysis, one needs to closely monitor and control the phase angle 8(t) of the plant, with the help of the mechanical power input p(t). From the above dynamics equations, using p(t) as input and 8(t) as output, show that the input-output transfer function is: H(s) = A(s) P(s) = 1 s² + 2s + 10. where A(s) and P(s) denote, respectively, the Laplace transforms of 8(t) and p(t). b) Assume a unit step input, i.e., p(t) = u(t). Use inverse Laplace transform (and partial fractions expansion) to compute 8(t) (in time-domain). Hint: You may find the above tip useful.
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