-
A Novel Approach for the Steady-State Analysis of a Three-Phase
Self Excited Induction
Generator Including Series Compensation Mohammad Naser Hashemnia
Electrical Engineering Dept.
Sharif University of Technology Tehran, Iran
Email: [email protected]
Ali Kashiha Electrical Engineering Dept.
Islamic Azad University Kermanshah Science and
Research Branch Kermanshah, Iran
Email: [email protected]
Kourosh Ansari
Faculty of Engineering Ferdowsi University of
Mashhad Mashhad, Iran
Email: [email protected]
Abstract A new method to evaluate the steady state
performance of a three-phase self excited induction generator
based on conductance minimization is proposed. It can be simply
used to take series compensation into account. Among the priorities
of this method are the absence of convergence problem and
flexibility. Simple methods to find the frequency and magnetizing
reactance have been proposed. Simulation results show the
efficiency of this method..
Keywords Induction Generator; Self Excitation; Series
Compensation; Wind Energy
I. INTRODUCTION Use of an induction machine as a generator
is
becoming popular for harnessing the renewable energy resources.
Traditionally, synchronous generators have been used for power
generation but induction generators are increasingly being used
these days because of their relative advantageous features over
conventional synchronous generators. These features are brushless
and rugged construction, low cost, maintenance and operational
simplicity, self-protection against faults, good dynamic response,
and capability to generate power at varying speed. For its
simplicity, robustness, and small size per generated kW, the
induction generator is favored for small hydro and wind power
plants. The need of external reactive power, to produce a rotating
flux wave limits the application of an induction generator as a
stand-alone generator. However, it is possible for an induction
machine to operate as a self excited induction generator (SEIG) if
capacitors are connected to the stator terminals to supply
sufficient reactive power [1].
Self excitation in a self excited induction generator (SEIG)
occurs when its rotor is revolved by a prime mover and sufficient
capacitance is connected across its terminals. The residual flux in
rotor core induces voltages at stator and a phase leading current
will pass through the capacitor bank. The remnant magnetic field
will thus be enhanced by the reactive power of capacitor bank and
the
voltage will therefore increase. This positive feedback
phenomenon will continue till the voltages and currents of the
machine reach a stable operating point due to saturation [2].
In recent years, induction generators have gained attraction in
power generation specially renewable energy sources. Self
excitation in a self excited induction generator (SEIG) occurs when
its rotor is revolved by a prime mover and sufficient capacitance
is connected across its terminals. The residual flux in rotor core
induces voltages at stator and a phase leading current will pass
through the capacitor bank. The remnant magnetic field will thus be
enhanced by the reactive power of capacitor bank and the voltage
will therefore increase. This positive feedback phenomenon will
continue till the voltages and currents of the machine reach a
stable operating point due to saturation.
II. EQUIVALENT CIRCUIT ANALYSIS The per phase steady state
equivalent circuit of a SEIG
is shown in Fig.1, in which 1R , 2R and loadR are stator, rotor
and load resistance and 1X , 2X , mX , cX and
loadX are stator, rotor, magnetizing, capacitance and load
reactance respectively, all referred to stator and at base
frequency. a and b are per unit frequency and per unit speed
respectively.
Malik and Haque [3] tried to solve the equivalent circuit by
loop impedance method while Mcpherson [4] used the nodal admittance
method.
In the loop impedance method, the impedance of the equivalent
loop is equated to zero as follows:
0looploop =IZ . (1)
where:
2010 IEEE Symposium on Industrial Electronics and Applications
(ISIEA 2010), October 3-5, 2010, Penang, Malaysia
978-1-4244-7647-3/10/$26.00 2010 IEEE 371
-
))(||(
)||)((
221
1
2loadload
loop
baRjXjXjX
aR
aXjjX
aRZ
m
c
+++
++=. (2)
Fig. 1. SEIG equivalent circuit
loopI can not be zero since there are voltages and currents in
the circuit. Hence, loopZ is equal to zero or its real and
imaginary parts are zero. Equating the real and imaginary parts,
two nonlinear simultaneous equations containing two unknowns ( a
and mX ) have to be solved. Numerical methods can be used for this
computation.
In the nodal admittance method, the admittance across the airgap
branch is considered. According to the energy conservation law, the
generated active/reactive power by the rotor part of the equivalent
circuit must be equal to the consumed active/reactive power of the
stator part of the equivalent circuit. Considering:
rr gEP2|| = . (3)
ss gEP2|| = . (4)
where E is the airgap branch voltage and rg and
sg are stator and rotor conductances respectively, a fifth order
polynomial wth respect to a is attained for pure resistive loads. a
can be calculated using numerical methods. After a s calculated, mX
can be attained by equating the produced reactive power of the
rotor to the consumed reactive power of the stator.
Although both of the above mentioned methods are applicable for
evaluation of machines performance, they both have the deficiency
that they involve long and tedious algebraic calculations. On the
other hand, the details of the coefficients changes with equivalent
circuit. For instance, load change from R-L to R-C or including
series compensation will change the order of equations. Actually,
these methods dont have the required flexibility. Therefore, an
iterative method is proposed in
[5] whose advantages are generality (possibility of taking RC
loads, core loss and series compensation into account) and not
having to be involved in the tedious task of computing the
polynomial coefficients. Although fast convergence is claimed, (3
to 6 iterations to gain a precision of 10e-06 in per unit frequency
for light loads and up to 8 iterations for heavy loads), more
number of iterations might be needed for heavy loads and conditions
of unsuccessful voltage build-up. As Table. I shows, more
iterations are needed when the load resistance decreases (generator
loading increases) and it might become impossible to generate any
voltage if there is a very little load resistance.
Although the conventional methods are effective in simulating
the SEIG performance, they have common disadvantages. They can be
listed out as:
1. All the coefficients of the non-linear equations or a higher
order polynomial need to be derived manually. The mathematical
manipulations are tedious, time-consuming and prone to human
errors.
2. The expressions for the coefficients are very long and
complicated, which require tremendous human effort for accurate
programming and debugging.
3. The model lacks flexibility as the coefficients are valid
only for a given circuit configuration. For example, inclusion of
the core-loss resistance or the addition of compensation capacitive
reactance will change the order of the equations.
III. MINIMUM REQUIRED CAPACITANCE
The core of a SEIG can have a magnetizing reactance between and
max,mX where max,mX pertains to the linear region while 0X pertains
to a completely saturated core. When the magnetizing reactance is
at its maximum, minimum reactive power is absorbed by mX , so the
produced reactive power by the excitation capacitance should also
be at its minimum. Therefore,
CX is maximum and C is minimum. It can also be gathered that the
more the core saturates, the more the required capacitance will
be.
The value of the capacitance should be chosen with care; A value
of capacitance much higher than its minimum ( minCC >> )
results in a decrease in frequency, increment of stator and rotor
currents (due to the reduction of capacitive reactance), more ohmic
losses and thus less efficiency. It can also deteriorate the
insulation of the machine because of the increase in the airgap and
the terminal voltages (due to production of more reactive power).
On the other hand, a high value of capacitance has some advantages
such as more current capability of the generator and better voltage
regulation. A compromise is thus needed in a real
implementation.
TABLE I
372
-
NUMBER OF ITERATIONS FOR DIFFERENT LOADING CONDITIONS
)(puRload
0.01
0.1
1 2 5 7 9 20
100
Iterations 18 16 9 7 7 6 6 6 6 Voltage build-up
+ + + + + +
IV. THE PROPOSED METHOD A glance at the aforementioned methods
reveals that
there is need to solve two nonlinear simultaneous equations in
the loop impedance method, while there is capability of decoupling
the unknown parameters in the nodal admittance method. Therefore,
nodal admittance will be used in the proposed method. Suppose it is
desired to analyse the equivalent circuit having the parameters of
the machine (except mX ), speed and excitation capacitance. It
should first be noted that if it is looked into the circuit through
the magnetizing branch, KCL will result: 0net =mVY where netY is
the total admittance observed through the magnetizing branch
and
mV is its voltage. To have successful voltage buildup, 0mV , so
0net =Y or 0net =g and 0net =b where
netg and netb refer to the observed conductance and susceptance
of the magnetizing branch. By varying the per unit frequency (a)
between a fraction of b (0.6b for instance) and a value close to b
(0.999b for instance),
netg is calculated. The calculations are as follows:
ba
RjXZ
+= 22rot . (5)
rotZ
Y 1rot = . (6)
)( rotrot Yrealg = . (7)
loadload jXaRZl += . (8)
2|| ajXZZ clc = . (9)
cstat ZjXaRZ ++= 11 . (10)
stat
stat ZY 1= . (11)
)real(Ystat=statg . (12)
rotstatnet ggg += . (13)
The per unit frequency which makes netg minimum is the operating
frequency. The min command in Matlab environment has been used to
calculate the minimum element of the conductance matrix. After a is
calculated, mX can be attained upon equating netb to zero. A
similar approach can be used to obtain the minimum capacitance.
V. SEIG WITH COMPENSATION A SEIG has a weaker voltage regulation
than a same
rating synchronous generator [3]. The series compensation method
is a simple and still effective way which results in a flatter
voltage profile using a negative feedback phenomenon. Actually, the
voltage tends to decrease after loading the generator but the
current passing through the series capacitance will increase the
produced reactive power and the voltage will increase as a result.
The equivalent circuits for long and short compensations are shown
in Figs. 2 and 3 respectively.
1jX
mjX
2jX
baR
2loadjX
aRload
2aXj c
aR 1
2aXj cs
Fig. 2. SEIG with long compensation
1jX
mjX
2jX
baR
2loadjX
aRload
2aXj c
aR 1
2aXj cs
Fig. 3. SEIG with short compensation
If we are to analyze the equivalent circuit using the
loop impedance method, the coefficients of the nonlinear
equations will become too complicated due to the
373
-
presence of the term 2aX cs . On the other hand, the
order of the polynomial will change from 7 to 9 (for R-L loads)
[4]. In [5], the iterative method is extended to a SEIG with
compensation. Using the proposed method in this paper, it is
sufficient to insert for the stator impedance in long compensation
(equation (10)):
211
stat aXjZjX
aRZ csc ++= . (14)
and for the short compensation (equation (8)):
2loadload
a aXjjXRZ csl += . (15)
The rest of the procedure is as before and mX can be easily
attained by equating the susceptance of the shunt branch to
zero.
VI. STEADY-STATE CALCULATIONS
Having 1E , a , mX , cX , b and the load ( loadR and
loadX ) the unknown quantities of Fig. 1 can be obtained as:
2load
1
||
,)(
aXjZjXR
jXRjXar
EI
cLL
LLlss
s
=
++=
. (16)
lrjXbaREI+
=2
12 . (17)
c
sc
jXajXaRIjX
I+
= 2loadload
load . (18)
)( loadloadload ajXRIVt += . (19)
2load || IIXQ scc = . (20)
22
2 ||3. RIabbPin
= . (21)
2load
22load
load2||XaRRVP tout +
= . (22)
in
out
PP
= . (23)
)||
|||(|32
22
22loss
c
gss R
VrIRIP ++= .(24)
VII. SIMULATION RESULTS The same machine as reference [5] is
used for simulation and the results are compared with other methods
to validate the effectiveness of this method. Fig. 4 shows the
variation of per-unit frequency with load resistance. It is seen
that the frequency increases with the increase of load resistance.
Fig. 5 shows the variation of magnetizing reactance with load
resistance. It is observed that the magnetizing reactance decreases
(and accordingly the voltage increases) with the increase of load
resistance. The proposed method is compared with the other methods
mentioned in [3,4,5] which shows good agreement. Fig. 6 shows the
variation of the terminal voltage with the load current (known as
the external characteristic of the machine). It is seen that there
is a point in which both the voltage and current decrease with the
decrease of load resistance, much like self excited DC generators.
Fig. 7 shows the variation of terminal voltage with load resistance
for pure SEIG and SEIG with log and short compensations. It shows
that the long compensation has an increasing voltage profile while
the pure SEIG has a decreasing one and the short compensation has a
flat one.
2 4 6 8 10 12 14 16 18 200.95
0.96
0.97
0.98
0.99
1
Load Resistance (pu)
Gen
erat
ed fr
eque
ncy
(pu)
Suggested methodRoots method (Mr.Mcpherson)Iterative method
(Mr.Chan)
Fig. 4. Variation of the generated frequency with load
resistance
0.5 1 1.5 2 2.5 3 3.5 4 4.5 51
2
3
4
5
6
Load resistance(pu)
Mag
netiz
ing
reac
tanc
e(pu
)
Suggested methodRoots method (Mr.Mcpherson)Iterative method
(Mr.Chan)
374
-
Fig. 5. Variation of the magnetizing reactance with load
resistance
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.2
0.4
0.6
0.8
1
1.2
1.4
Load Current (pu)
Ter
min
al V
Olta
ge (
pu)
Fig. 6. Variation of the terminal voltage with load current
2 3 4 5 6 7 8 9 101
1.5
2
2.5
3
3.5
Load Resistance (pu)
Ter
min
al V
olta
ge (
pu)
Variation of Terminal Voltage with Load Resistance (xcs= 0.1
pu)
no compensationlong compensationshort compensatin
Fig. 7. Variation of terminal voltage with load resistance
VIII. CONCLUSION A simple and novel method to analyze the steady
state performance of a self excited induction generator including
series compensation was presented. Among the priorities of this
method over the more common methods are the ability to be augmented
to different kinds of loads and absence of the convergence problem.
This approach can be used in windmill systems to predict the steady
state performance of the system. A future work will demonstrate in
detail the application of this method to SEIGs with core loss.
REFERENCES [1] S. Mahley, Steady state analysis of three-phase
self-excited
induction generator, Thesis submitted in partial fulfillment of
the requirements for the award of degree of Master of Engineering
in Power Systems & Electric Drives, Thapar University, Patiala,
June 2008, pp.1-3.
[2] N. Hashemnia and H. Lesani, A novel method for steady state
analysis of the three-phase self excited induction generators, 18th
International Conference on Electrical Machines, ICEM 2008,
September 2008, pp.1 4.
[3] N.H.malik and S.E.Hague,Steady state analysis and
performance of an isolated self-excited induction Generator, IEEE
Trans. On Energy Conversion,Vol.Ec1,No.3,pp.134-139 , September
1986.
[4] L.Quazene and G.Mcpherson, Jr, Analysis of the isolated
induction generator, IEEE Trans. on P.A.S, Vol.Pas 102, No 8,
pp.2793-2798.
[5] T.F.Chan, Analysis ofsSelf-excited induction generator using
an iterative method, IEEE Trans. on Energy conversion,Vol.10, No.3,
pp.502-507 , September 1995.
375
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