New Formula to Determine the Minimum

New Formula to Determine the
Minimum Capacitance Required
for Self-Excited Induction
Generator
Ali Mohamed Ahmed El-Tamaly
Electrical Engineering Department, Faculty of Engineering
Elminia University, Elminia, Egypt
Self Excited Induction Generator Equipped
with SCR Rectifier / LCI Inverter
Io
Rectifier
UG
VI
VR
IG
LCI
100
80
60
40
20
0
0
i (t ) =
0.5k
1.0k
1.5k
2.0k
2.5k


5
7
11
13
I o  cos (ωt ) − cos (5ωt ) − cos (7ωt ) +
cos (5ωt ) + cos (13ωt ) 
...... 
2 3 
1
1
1
1
π
Utility interfacing of SCIG via diode
rectifier and PWM inverter
Rectifier
PWM
VR
Electric
Utility
VI
IG
Induction Generator Equipped with PWM
Rectifier / LCI Inverter
Variable frequency
PWM Converter
S1
Ia
S3
a
IG
S2
Ib b
S4
DC Link
S5
Ic
S6
C
LCI Inverter
Electric
Utility
Most of previous work uses numerical
iterative method to determine this
minimum
capacitor.
But
the
numerical iteration takes long time
and divergence may be occurs. For
this reason it cannot be used online.
A new simple formula for the
minimum
self-excited
capacitor
required for induction generator is
presented here. By using this formula
there is no need for iteration and it
can be used to obtain the minimum
capacitor required online. Complete
mathematical analysis for induction
generator to drive this new formula is
presented. The result from this new
formula is typical as the results from
iterative processes
The IEEE recommended equivalent circuit
of three-phase induction generator.
Vt
RL
-jXc
jXr
jXs
Rs
Is
Rr
s
jXm
jXL
Where the slip S is shown in
equation (1)
N sa − N a a − b
=
S=
N sa
a
a:
a is Pu frequency fa/fr and
speed Na/Nsr.
b
is Pu
The loop equation for IS of Fig.2 can be
written as :-
IS Z = 0
(3)
Where Z is the loop impedance seen by the
current, IS and can be obtained as in (4)
Z=Zm+ZLC+Zs
(4)
Zm = ( Xm )
 Rr 


 a −b 
 - jX C   RL

Z LC =  2  
+ jX L 

 a   a
Rs
Zs =
+ jX S
a
In steady state operation IS ≠ 0
otherwise there is no generated
voltage , Then
Z=Zm+ZLC+Zs =0
(4)
By equating both the real and
imaginary parts of (3) by zero we get
two nonlinear equations ((5) and (6)) in
function of Xm and a. Solving (5) and
(6) together yields the values of Xm and
a.
P ( X m , a ) = ( A1 X m + A2 ) a 3 + ( A3 X m + A4 ) a 2
+ ( A5 X m + A6 ) a + ( A7 X m + A8 ) = 0
Q( Xm , a) = (B1 Xm + B2 )a4 + (B3 Xm + B4 ) a3
+ (B5 Xm + B6 ) a2 + (B7 Xm + B8 ) a + B9 = 0
it is possible to compute Cmin by using
the following iterative procedure
If we use the value of Xm in the calculation
of Cmin then we can modifie (5) and (6) to
be as shown in (7) and (8) are function in a
and XC.
− a1a3 + a2 a2 + (a3 X C + a4 )a − a5 X C = 0
(7)
− b1a 4 + b2 a 3 + (b3 X C + b4 )a 2
(8)
+ (b5 X C + b6 )a − b7 X C = 0
Then by separation of XC in
(7) and (8)
a1a 3 − a2 a 2 − a4 a
XC =
a3 a − a5
b1a 4 − b2 a 3 − b4 a 2 − b6 a
XC =
b3 a 2 − b5 a − b7
(9)
(10)
By equating the right hand sides in (9) and (10)
then we have the following equation:-
(a1 b3 − a3 b1 ) a 4 − (a2 b3 + a1 b5 − a3 b2 − a5 b1 ) a 3
(12)
+ (a 2 b5 + a3 b4 − a 4 b3 − a1 b7 − a5 b2 ) a 2
− (a3 b6 + a5 b4 − a 4 b5 − a 2 b7 ) a + (a5 b6 + a 4 b7 ) = 0
From (12) the frequency can be calculated
and then substitute this frequency in (9) or
(10) to calculate XC and Cmin.
NEW FORMULA TO CALCULATE CMIN
BY USING NODAL ANALYSIS
YL
YC
Rs
a
Vt/a
RL
a
Yin
JXc
a2
Xr
Xs
Is
Xm
JXL
Vt
Yt = 0
a
Where Yt =Yin +YL +YC
(13)
Rr
a-b
C4 = X L2 Rr (L2 L3 − L1 ) + X L2 RS L22 + RL L12
C3 = X L2 Rr v (L1 − L2 L3 ) − 2v( X L2 RS L22 + R L L12 )
(
C2 = RL2 (Rr L22 − Rr L1 + Rr L2L3 + XL2Rs Rr2 + L22v2
)
+ 2 * RL Rr Rs (L2 L3 − L1 ) + RL (L12v2 + Rr2L3 + Rs2L22
C1 = RL2 Rr v (L1 − L2 L3 ) − 2v RL Rs L22 (RS + RL )
(
)
C0 = Rr2 + L22 v2 RL Rs (RL + Rs )
1From the imaginary part we can drive a simple
formula for the minimum value of terminal capacitor
as shown in (17).
M4 
1  XLa

Cmin = 
+ 2
2
2π  M3 M1 + M2 
M 1 = Rs Rr − f ( f − v) L1
M2 = Rr f L3 + Rs ( f −v) L2
M3 = R + X f
2
L
2
L
2
M4 =Rr *M2 − L2 f ( f −v)M1
100
80
60
40
20
0
C min
50
100
150
%speed
200
Variation of Cmin with rotational speed at RL=1pu
and unity power factor.
100
80
60
40
20
0
50
100
150
%speed
200
Variation of Cmin with rotational
speed at no load.
CONCLUSIONS
In this paper a new formula for the
minimum capacitance required for selfexcited induction generator is presented.
This new formula is simple and it does
not need numerical iteration. For this
reason this new formula helps to
determine the minimum capacitance
required for self excited induction
generator on line. The new formula gives
typical results as the results obtained
from iterative technique without any
iteration or divergence problem.

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