ICSET 2008
Application of Interval Computation Technique to Fixed Speed Wind
Energy Conversion System
R.K. Thakur1, and V. Agarwal2, Senior Member, IEEE
Abstract—Any wind energy conversion system (WECS)
experiences uncertainty in wind velocity, tip-speed ratio, power
co-efficient and stiffness of the wind turbine shaft. Also, the
magnetizing inductance of the induction generator varies during
self-excitation process. This makes the analysis and control of
WECS, a difficult proposition. This paper presents an interval
analysis based computation method for the calculation and
analysis of various machine parameters and variables in WECS.
The usefulness of the proposed method in the design aspects of a
WECS is discussed. The realistic (dependent) uncertainties in
the equations are solved to analyze the behaviour of a selfexcited induction generator under balanced operating
conditions. Experimental results are included to highlight the
behaviour of the induction generator voltage during transient
phase.
energy is, for many reasons, one of the most
promising renewable energy sources. The wind energy
can be harnessed using a Wind Energy Conversion System
(WECS), as shown in Fig. 1, comprising a wind turbine, an
electric generator, a power electronic converter and the
corresponding control system [1]. This is a highly interactive,
complex, multiple input multiple output system which
exhibits non-linear, non-minimum phase dynamics. The
objective is to convert the wind energy (at varying wind
velocity) to electric energy while maximizing the power and
safety of operators [2]. Wind is one of the most abundant
renewable sources of energy in nature. It is an erratic,
uncertain and uncontrolled variable. It is a function of many
factors such as temperature, pressure, geographical topology,
terrain, region etc. The available wind energy is converted
into mechanical power by wind turbines [3]. The power can
be captured at two different tip-speed ratios. Accordingly,
there are two possible regions of turbine operation, namely
the high- and low- speed regions. High-speed operation is in
general bounded by the speed limit of the machine. Lowspeed region is not bounded by the speed limit of the
machine, but in this region the system has non-linear, nonminimum phase dynamics [4]. To analyze WECS we need a
computation method which can yield results with the user
defined accuracy and the method of calculation should be
reliable. In this context, it is noted that the interval
IND
Manuscript received April 30, 2008.
R. K. Thakur is with the College of Military Engineering, Pune, India and
is presently pursuing Ph.D. in System & Control Engineering, IIT-Bombay,
Powai, Mumbai-400 076, India (e-mail: [email protected]).
V. Agarwal is with the Department of Electrical Engineering, IITBombay, Powai, Mumbai-400 076, India (e-mail: [email protected],
Fa x: +91-22-25723707).
Pwind
Pmech
Gear Box
Induction
generator
Capacitor bank
Fig. 1 Schematic diagram of a wind energy conversion system (WECS)
The paper is organized as follows:
Section II provides the modeling of a WECS. Section III
describes the interval computation technique. Section IV
presents analysis and simulation of induction generator at
different operating condition. Experimental results are
included in section V. The major conclusions of the work are
included in Section VI.
II. MATHEMATICAL MODEL OF A WECS
In this section, mathematical model of a wind energy
conversion system comprising wind turbine and its associated
parameters and an induction generator are developed.
A. Wind turbine [2 - 4]
The wind power is converted into mechanical power as per
the aerodynamic power-speed characteristics of the wind
turbine. The mechanical power is a function of wind speed
(V), rotor speed (ωr) and the tip-speed ratio (Ȝ), and is given
by the following equation:
Pmech = 12 ρ Ad RC p (λ )V 3
(1)
Wind speed varies drastically and hence averaging is used
to determine the correlation between wind speed and
mechanical power. So, the calculation of mechanical power is
not unique rather it is valid over a range. Also, power is
proportional to cube of the wind velocity. Hence, average
value of wind speed is used to reflect the total energy
available for harnessing, whereas actually the total energy
available for harnessing varies over a range.
Tip-speed ratio (Ȝ) is a function of linear velocity of tip of the
blade and the wind velocity and is given by the following
equation:
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c 2008 IEEE
978-1-4244-1888-6/08/$25.00 Pel
computation technique is able to provide reliable results.
I. INTRODUCTION
W
Vt
Wind Turbine
λ = ωr R V
(2)
As the wind velocity varies from top to bottom of the swept
area of the solid disc of rotating blade, the tip-speed ratio
value, Ȝ also varies accordingly and hence can be reflected by
a range of its maximum and minimum values. It is an
important parameter which determines the quantum of energy
harnessing from the wind and is used for the maximum power
point tracking.
The power co-efficient (Cp) is a non-linear function of the
tip-speed ratio (Ȝ), pitch angle (ȕ) and blade design. It is
given by the following equation:
ª π (λ − 3) º
C p = (0.44 − 0.0176β )sin «
» − 0.00184(λ − 3) β
¬15 − 0.3β ¼
(3)
Fig. 2 D-q axis equivalent circuit of SCIG at no load corresponding to:
(a) d axis (b) q-axis.
It is sensitive to dirt on the blade surface. In most of the
studies, Cp is regarded as a constant for all rotor speeds which
is aerodynamically not true and its variation at different rotor
speeds must be accounted for during investigations. Hence, it
is more appropriate to consider a range of Cp and not a
constant value. This parameter is maximized for harnessing
the maximum energy from wind.
Electric power, Pel = η Pmech
(4)
where, η is a function of ωr.
For a stationary reference frame (Ȧ = 0) and before excitation,
all terminal voltages are considered to be zero and the
generator equation is given by the following equation [6, 7].
ªv qsº §Rs +pLs +1/ pC
pLm
0
0 · ªi qs º
«vds » ¨ 0
Rs +pLs +1/ pC
pLm ¸ «i ds »
0
«v qr»=¨ pLm
−ωr Lm
Rr +pLr −ωr Lr ¸ «i qr »
¸
«v » ¨© ωr Lm
pLm
ωr Lm Rr +pLr¹ «i »
¬ dr¼
¬ dr¼
(6)
B. Mathematical modeling of induction generator
The d-q axis equivalent circuit of a Squirrel Cage
Induction Generator (SCIG) is shown in Fig. 2. The
equivalent circuit represents the dynamic characteristics of
SCIG [3, 4]. The following assumptions are made in the
modeling of SCIG: The rotor and stator resistances and
inductances are assumed to be constant; saturation in rotor
and stator iron cores and space harmonics arising from
saturation of stator and rotor teeth are ignored. The voltage
equation for SCIG is obtained from the d-q axis equivalent
circuit for a rotating reference frame. It gives the dynamic
model of the SCIG in terms of the machine parameters and
direct and quadrature axis current and voltage variables
respectively. It is given by the following equation:
ªv qsº §Rs +pLs +1/ pC Ȧ Ls
pLm
Ȧ Lm · ªi qs º
«vds » ¨ -Ȧ Ls Rs +pLs +1/ pC Ȧ Lm pLm ¸ «i ds »
«v qr»=¨ pLm
(ω−ωr)Lm Rr +pLr (ω−ωr)Lr¸ «i qr »
¸
«v » ¨© (ωr −ω)Lm
pLm
(ωr −ω)Lm Rr +pLr ¹ «i »
¬ dr¼
¬ dr¼
(5)
The model (5) describes the dynamic characteristics of the
induction generator in rotating reference frame.
The electromagnetic torque is given by:
3 p Lm
Tel =
( λ ds λ qr − λ qs λ dr )
2 2 K
(7)
The dynamic equation of motion is written as:
ω r =
1
J
(Tm ech − Tel )
(8)
where, J is the equivalent inertia constant of wind turbine and
induction generator combined.
III. INTERVAL COMPUTATION TECHNIQUE
Interval computation technique [8] is a mathematical
analysis method requiring a computer – manual computing
will be too tedious. The successful and accurate
implementation of interval analysis depends significantly on
the understanding of the relevant mathematical analysis in the
various applications in engineering and sciences. It translates
the theorems and techniques of the application into sharper
versions. It transforms the qualitative reasoning to a
quantitative aspect. It can be used for the solution of large
linear and non-linear equations with interval coefficients. It
gives error bounds for singularly perturbed systems of
ordinary differential equations.
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The squirrel cage induction generator is described by
a large system of ordinary differential equation with saturated
hysteresis non-linearity. For self-excitation process in a
squirrel cage induction generator, a minimum air gap flux
linkage is required [9]. This value of minimum flux linkage is
dependent on machine parameters and capacitance values for
stable steady-state operation. The self-excitation process is
associated with bifurcation theory i.e. the emergence of two
distinct solutions when a critical value of parameter is
reached and exceeded. At the critical parameter value, a
radical qualitative change takes place in the system dynamics
i.e. either buildup or collapse of the steady-state operating
points. SCIG self-excitation exhibits a saddle-node
bifurcation in which the terminal voltage builds-up from zero
equilibrium point and the critical parameter is magnetizing
inductance value [10].
The self excitation process in an SCIG is described by a
non-linear differential equation of the following type:
.
X = f ( X , Λ)
(9)
where, X represents the set of state variables and Λ represents
the parameters of the machine which governs the dynamic
behaviour of the SCIG system. By setting the derivative term
of X equal to zero, the equilibrium point of SCIG is
determined which yields the following equations:
f ( X , Λ ) = 0 → A(Λ ) X = 0
(10)
In the beginning the state variables are zero, so the trivial
equilibrium point is X = 0 . The other feasible solutions of the
system exist when the determinant of ( sI − A) is set to zero
[9, 10]. The resulting equation yields expressions relating the
rotor speed to the angular speed of generated voltage in terms
of machine parameters, magnetizing inductance and
excitation capacitance values. The describing function of (6)
yields a characteristic equation of order 6. The coefficients of
the characteristic equation are in the form of some range with
their infimum and supremum values. The coefficients are
itself the polynomials of these variables with structured
uncertainty which makes otherwise the solution further
difficult. Also, there exists truncation and rounding off errors
in the computation using existing method. This leads to an
erroneous result which gives unreliable values of the design
parameters. Thus, interval computation technique is used to
solve such equations for reliable solution.
det [sI-A] = C
2
Rr + ω r X 1 ))
2
2 6
X1 s
2
5
+ 2C X 1 X 2 s + (C
+ 2CL2 X 1 ) s
3
4
+
2
2
(X2
+ X 1 (2 Rs
2
(2( Rs ( Rr X 2 + L2ω r X 1 )C
2
2
+ ( L2 X 2 + Rr X 1 )C ) s + (C Rs X 3 + 2C ( Rs Rr L2
2
+ω r L2 X 1 )
+
2
2
L2 ) s
+ 2( Rs CX 3 + Rr L2 ) s + X 3
where, L1, L2, X1, X2 and X3 are
+
2
Rr X 2
(11)
L1 = L s + L m ; L 2 = L r + L m ; X 1 = L1 L 2 − L2m ;
X 2 = R s L 2 + R r L1 ; X 3 = R r2 + ω r2 L22 .
Equation (11) is solved using interval method for a pair of
complex poles on the imaginary axis of the s-plane. The
solution of the characteristic equation gives all possible
values of the design parameters at a given operating point.
The describing function for the same induction generator is
also obtained with the resistive load. The characteristic
equation with the resistive load yields an equation of order 6.
Solving this 6th order equation by interval method for the
condition of self-excitation process yields the capacitance
value required at different loading conditions.
IV. RESULTS AND SIMULATION ANALYSIS
An induction generator with the following specification
[7], is simulated using Matlab/Simulink software: 3-φ, Yconnected, 220 V, 4.8A, 60Hz, parameters in pu are, Rs =
0.0946, Rr = 0.0439, Xs= Xr = 0.0865, Xm (max) = 2.12. The
rotor speed of induction generator and machine parameters
other than magnetizing inductance are assumed to be constant
during self-excitation. The magnetizing inductance is set to a
value between 0 and the unsaturated value which ascertains
the structured uncertainties and this translates the system
dynamics into a sharper version enabling the quantitative
analysis of the highly non-linear system. It also ensures the
condition to capture all possible solutions.
The solution of (11) for the above mentioned induction
generator under the stated assumptions yields the capacitance
value and the corresponding magnetizing inductance value.
The results are summarized in Table 1.
TABLE 1
VALUE OF MAGNETIZING INDUCTANCE CALCULATED BY INTERVAL
METHOD AT RESPECTIVE
SPEED=300 RAD/SEC
Sr no.
1
2
3
4
CAPACITANCE
Range of Magnetizing
Inductance value, H
VALUE
AND
ROTOR
Range of Capacitance
Value, ȝF
0.2678 - 0.27174
37.89 - 40
0.1483 - 0.1496
68.27 - 70
0.1164 - 0.1168
89.06 - 90
0.0983 - 0.09845
104.4 - 110
Clearly, the use of higher capacitance value drives the
induction generator into deep saturated region. This condition
of the system results in transients with large magnitude during
self-excitation process.
The plots showing the results obtained from the calculation
method with and without load are shown in Figs. 3, 4 and 5.
Clearly, Fig. 3 shows that the use of large capacitance value
drives the induction generator into deeper saturated region
which causes the transients with large magnitude. It is clear
from the graph in Fig. 4 that an increase in load resistance
requires a corresponding higher capacitance value to induce
the self-excitation process in different operating conditions in
the induction generator. It is evident from Fig. 5 that as the
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load resistance is increased the system is driven into deep
saturated region causing increased transient amplitude in the
voltage build-up.
results for respective capacitance value are summarized in
Table 2. The magnetizing inductance values obtained from
simulation are close to the value obtained using the interval
computation technique corresponding to the respective
capacitance value. Thus, the method of calculation using
interval computation technique is verified. The method is
found to give the solutions for machine parameters which are
accurate to the user defined value. Thus, it can be concluded
that the proposed calculation method is reliable.
TABLE 2
VALUE OF MAGNETIZING INDUCTANCE OBTAINED FROM THE
SIMULATIONS AS SHOWN IN FIGS. 6(a), 6(b), 6(c), and 6(d)
Sr no.
Magnetizing Inductance
Capacitance Value, ȝF
value, H
Fig. 3 Plot of magnetizing inductance value versus capacitance value.
1
2
3
4
0.265
0.145
0.115
0.098
40
70
90
110
Fig. 4 Plot of load resistance versus the corresponding minimum capacitance
value
(a)
Fig. 5 Plot of magnetizing inductance versus load resistance
The information regarding load resistance and
corresponding capacitance and magnetizing inductance values
in Figs. 3, 4, and 5 are useful in the design of different types
of controller such as adaptive controller, deadbeat controller,
model reference based predictive controller, quantitative
feedback theory based controller, sliding mode controller,
supervisory control and data acquisition systems etc. The
results obtained from the proposed calculation method, and
listed in Table 1 are verified through the simulation of the
induction generator. The saturation non-linearity in the
induction generator is included in the simulation by adopting
five segments, piece-wise linearization of magnetizing
inductance in the function block of Simulink.
The magnetizing inductance value is obtained from the
simulation of the induction generator for C = 40 μF. The
simulation result is shown in Fig. 6 (a). The magnetizing
inductance value is 0.265H. Similarly for other capacitance
values: C = 70μF, C = 90μF, and C = 110 μF, the induction
generator is set to self-excitation process and the simulation
results are shown in Figs. 6(b), 6(c) and 6(d). The
magnetizing inductance values obtained from simulation
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(b)
(c)
(d)
Fig. 6 Plots showing the variation of magnetizing inductance for different
capacitance values: (a) 40μF (b) 70μF (c) 90μF (d) 110μF
Further the simulation results also verify the fact that the use
of large capacitance value drives the induction generator
deeper into saturation and thus change the transient
characteristics of system dynamics with increased transient
amplitudes and faster speed of response.
V. EXPERIMENTAL RESULTS
In this section, experimental results on induction generator
with the following specifications are presented: 3-φ, Yconnected, 220 V, 5.1A, 50Hz, parameters are, Rs =5.7ȍ, Rr
= 4.6ȍ, Ls= 0.021H, Lr = 0.01987H, Lm (max) = 0.319H.
The induction generator of the above specification is
driven by a dc motor at different speeds. The stator terminals
of the induction generator are connected with the 3-φ,
capacitor bank of desired value to induce self-excitation in the
induction generator. The experimental results of the voltage
build-up during self-excitation are shown in Figs. 7, 8 and 9.
The figures are obtained using Agilent’s Infiiniium
oscilloscope, with a multiplying factor of 20 on vertical axis.
It is observed that with the rotor speed increased from 301.5
rad/sec to 314 rad/sec, keeping the capacitance value fixed for
C = 70 μF, there is negligible transient amplitude in Fig. 7
whereas there is noticeable transient amplitude during selfexcitation in Fig. 8. The speed of response has improved as is
obvious in Fig. 8 in comparison to Fig. 7. In Fig. 9, the speed
of response has further improved with the capacitance value
increased from C = 70 μF to C = 90 μF keeping the rotor
speed constant to 314 rad/sec but with larger transient
amplitudes.
Thus, it is experimentally verified that the transients
during the self-excitation process depend largely on the
capacitance value used and magnetizing inductance value of
the induction generator. The speed of response improves with
an increase in the capacitance value but that increases the
transient amplitudes. Thus, the knowledge of all possible
solutions in the search region of capacitance value and
magnetizing inductance value permits the trade-offs between
the specifications of transient characteristics of the induction
generator.
Fig. 7 Output voltage: Capacitance value=70 μF, Rotor speed =301rad/sec,
X-Axis:1small division = 0.5sec.,Y-Axis: 1small division = 5*20 volt
Fig. 8 Output voltage: Capacitance value=70 μF, Rotor speed =314rad/sec,
X-Axis:1small division = 0.5sec., Y-Axis:1small division = 5*20 volt
Fig. 9 Output voltage: Capacitance value=90 μF, Rotor speed =314rad/sec,
X-Axis:1small division = 0.5 sec.,Y-Axis: 1 small division = 5*20 volt
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VI. CONCLUSION
A direct method based on interval analysis of computation
for solution of characteristics equations of induction generator
has been proposed. This method predicts the range in which
lies the magnetizing inductance value and capacitance value
required to induce the self-excitation in an induction
generator. The use of the interval of magnetizing inductance
leads to an accurate prediction of whether or not selfexcitation will occur for various capacitance values and
captures all possible solutions. The proposed method is fast
and can directly be used for transient analysis investigation of
voltage build-up. It also predicts the range of capacitance
value in which there is no solution and so self-excitation will
not occur. With the decrease in magnetizing inductance value,
the terminal voltage increases, representing a stable operation.
ACKNOWLEDGMENT
The authors wish to thank Prof. P. S. V. Nataraj, IITBombay, for his help during the course of this work.
List of symbols:
Pm ech =M echanical pow er
λ = T ip-speed ratio
R = R adius of the w ind turbine
V =W ind velocity
Pel = Electrical pow er
Ad =D isc area
ρ =A ir density
η =Efficiency
Vt = T erm inal voltage
R s , R r = Per phase stator and rotor (re ferred to stator) resistance
L s , L r = Per phase stator and rotor (referred to stator) inductance.
L m = P er phase m agnetizing inductance (referred to stator).
C = Per phase term inal excitation capacitance
iqs , i ds = S tator quadrature and direct axis currents
iqr , i dr = R otor quadrature and direct axis currents
v qs , v ds = S tator quadrature and direct axis voltage
REFERENCES
[1]
M.G.Simoes, F. A. Farret, “Renewable energy systems design and
analysis with induction generators”, CRC Press, New York, 2004.
[2] E.S. Abdin and Wilson Xu, “Control design and dynamic performance
analysis of a wind turbine induction generator unit”, IEEE Trans. EC,
vol. 15, no. 1, pp. 91-96, March 2000.
[3] C. Grantham, D. Sutanto, and B. Mismail,“Steady-state and transients
analysis of self-excited induction generators”, Proc. IEE., vol. 136, pt.
B, no. 2, pp. 61-68,. 1989.
[4] N.R.Ullah and T. Thiringer, “Variable speed wind turbines for power
system stability enhancement”, IEEE Trans. EC, vol. 22, no 1, pp. 5260, March 2007.
[5] D. Seyoum, C. Grantham and F. Rahman, “The dynamics of an isolated
self-excited induction generator driven by a wind turbine”, Proceedings
of IECON, vol. 2, pp. 1364-69, Dec. 2001.
[6] D. Seyoum, M.F. Rahman and C. Grantham, “Terminal voltage control
of a wind turbine driven isolated induction generator using stator
oriented field control”, IEEE Applied Power Electronics Conference
(APEC), Florida, Feb. 9-13, 2003, pp. 846-852.
[7] Li.Wang, Ching-Huei Lee, “A novel analysis on the performance of an
isolated self-excited induction generator”, IEEE trans. EC, vol. 12, No.
2, pp. 109-117, June 1997.
[8] R.E. Moore, “Methods and Application of Interval Analysis”, SIAM
studies in Applied Mathematics, 1979.
[9] O. Ojo, “Dynamics and system bifurcation in autonomous induction
generators”, IEEE trans. Industrial Applications, vol. 31, no. 4, pp. 918924, Jul/Aug. 1995.
[10] O. Ojo, “Minimum air gap flux linkage requirement for self-excitation
in stand-alone induction generators”, IEEE trans. EC, vol. 10, no. 3, pp.
484-492, Sept. 1995.
v qr , v dr = R otor quadrature and direct axis voltage
ω r = Angular speeds of rotor
ω = Angular speeds of synchronous, reference fram e
p = D erivative w ith tim e
β = Blade pitch angle
Tel =Electrom agnetic torque
Tm ech =M echanical torque
λ ds , λ qs = Stator direct and quadrature axis flux linkage
λ dr , λ qr = R otor direct and quadrature axis flux linkage
L1
=
Ls + Lm
L2
=
Lr + Lm
2
K = L1 L 2 − L m
A = System m atrix
Λ = G enerator param eter
X = S tate variables
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