2010 2nd IEEE International Symposium on Power Electronics for Distributed Generation Systems
A New Approach to Achieve Maximum Power
Point Tracking for PV System with a Variable
Inductor
Longlong Zhang*, William Gerard Hurley**, and Werner Wölfle**
* Institute of Power Electronics, Zhejiang University, Hangzhou, P. R. China
** Department of Electrical & Electronic Engineering, National University of Ireland, Galway, Ireland
where D is the duty cycle of the buck converter, and
RL is the load resistance of the buck converter.
Abstract—Maximum Power Transfer in solar
photovoltaic applications is achieved by impedance
matching with a dc-dc converter with maximum power
point tracking by the incremental conductance method.
Regulation and dynamic control is achieved by operating
with continuous conduction. It can be shown that under
stable operation, the required output inductor has an
inductance versus current characteristic whereby the
inductance falls off with increasing current, corresponding
to increasing incident solar radiation. This paper describes
how a variable sloped air-gap inductor whereby the
inductor core progressively saturates with increasing
current meets this requirement and has the advantage of
reducing the overall size of the inductor by 60% and
increases the operating range of the overall tracker to
recover solar energy at low solar levels.
iP
+
vP
C1
(a)
iP
+
r
−
VOC '
+
vP
RLR
−
I. INTRODUCTION
vP = iP ⋅ RLR
vP = VOC'−iP ⋅ r
There have been renewed interests in solar PV in
recent years and thus led to further studies in Maximum
Power Point Tracking (MPPT) [1-3]. MPPT in solar
photovoltaic (PV) systems is normally achieved by
either the Perturb and Observe method (P&O) or by the
Incremental Conductance Method (ICM). In the ICM
approach, the output resistance of the PV panel is equal
to the load resistance as expected from the celebrated
max power transfer theorem; this may be shown by
linearising the I-V output characteristic of a PV panel
about the operating point as illustrated in Fig. 1. Thus
the equivalent resistance r at the maximum power
point should meet the following equation,
(b)
iP
1
RLR
I SC
I P1
1
r
IP2
VP1
VP2
VOC
(1)
VOC '
(c)
Fig. 1 (a) Maximum Power Transfer in a PV Module
(b) Thevenin Equivalent Circuit
(c) MPPT Based on Impedance Matching
RLR is the regulated resistance in order to
achieve MPPT, VP and I P are the PV voltage and
where
current at the MPP.
The actual load resistance is matched to RLR by a
buck converter through the control of the duty cycle D ,
the regulated resistance R LR is
1
(2)
RLR = 2 RL
D
The equivalent circuit of a PV module is shown in
Fig. 2. The main equations are summarized in the
Appendix
Consider two levels of illumination intensity at
points
and
in Fig. 1(b), the current at the MPP
decreases going from 1 to point 2, which changes the
value of the PV resistance at the MPP. In order to
①
This work was supported by the China Scholarship Council (CSC),
File No. 2009102634.
978-1-4244-5670-3/10/$26.00 ©2010 IEEE
RL
D1
−
Index Terms—MPPT, Variable Inductor, Impedance
Matching, Photovoltaic.
∆V
V
−r = −
= RLR = P
∆I
IP
iL
LV
M1
948
②
achieve MPPT, the regulated resistance
adjusted by changing the duty cycle
ID
I ph
VD
Conversely, the higher value of inductance required at
light loads may be achieved without increasing the
volume of the inductor. Variable inductance may be
achieved using a sloped air-gap (SAG), whereby the
inductor core progressively saturates with increasing
current [5]. Alternatively, a powered iron core may be
used so that it is progressively saturated with increasing
current to yield the L-i characteristics of Fig. 4. The use
of a reconfigurable inductor in a boost circuit for PV
applications is described in [6].
RLR should be
D in (2).
I
RS
D
Fig. 2 Equivalent Circuit of a PV Module
The buck converter should work in the Continuous
Current Mode (CCM) in order to satisfy (2). In
discontinuous conduction mode this relationship is not
valid and the stable operation of the converter is more
complex. In continuous conduction, for a load power
change the duty cycle changes temporarily during a
transient but it reverts to Vout/Vin in the steady state. On
the other hand in discontinuous conduction the power is
a function of the dead time and therefore a different
control strategy is required that involves dual control
moving from CCM to DCM and vice versa.
II. CHARACTERISTICS OF VARIABLE INDUCTOR
The Inductance versus current (L-i) characteristic
of the variable inductor is shown in Fig. 4(a). The
variable inductor is based on a sloped air-gap (SAG)
and the L-i characteristic of the inductor is controlled by
the shape of the air-gap [5]. A typical construction is
shown in Fig. 4(b).
L
The minimum inductance in a buck converter in
CMM is given by
R ⋅ (1 − D )
(3)
Lmin = L
2 fs
i
f s is the switching frequency.
where
The minimum inductance may be restated by
combining (1), (2) and (3) to yield
D 2 ⋅ (1 − D ) ⋅ VP
(4)
Lmin =
2 fs ⋅ IP
The PV voltage is relatively constant over the full
range of solar intensity [4] ( VP = 41.6V in the example
to follow), thus the minimum inductance is a function
of duty cycle D and the output current of the PV
panel I P under a constant switching frequency
(fs=20kHz). The characteristics of the minimum
inductance under different duty cycles are shown in
Fig. 3.
Fig. 4 Characteristics of the Variable Inductor
The role of the variable inductor in the stable
operation of the buck converter is explained by
reference to Fig. 5. Continuous conduction can only be
achieved with inductance values above the dashed line
in Fig. 5 (the shaded area is off limits). The lower limit
of load current (corresponding to low solar insolation) is
given by IO1 as long as the inductance is greater than L1.
Evidently, at higher currents (and higher insolaton
levels), say IO2, a smaller inductor L2 would suffice,
with the added advantage of a reduced volume occupied
by the inductor. Conversely, setting the inductance at L2
would limit the lower load range to values of current
(and solar insolation) greater than IO2. The variable
inductor with the L-i characteristic shown in Fig. 4 has
the advantages of increasing the load range and
reducing the inductor volume by up to 60% [5].
200
D=0.87
D=0.5
D=0.67
Lmin(uH)
150
100
L
50
L1
0
0
1
2
3
4
5
Inductor Current (A)
L2
Fig. 3 The Characteristics of the Minimum Inductance under Different
Load Conditions
0 I o1
Evidently the minimum inductance to achieve
CCM falls off as the solar intensity increases.
Io2
IO
Fig. 5 Comparison of CCM Conditions in an MPPT DC/DC Converter
with a Variable Inductance.
949
The voltage across an inductor is related to its flux
linkage and this in turn is related to the current, the
dependence of the inductance on its current must be
taken into account:
dλ ,
λ = L(i ) ⋅ i ;
V=
dt
di
dL ,
= L +i
dt
dt
dL  di ,

= L +i ⋅
di  dt

di
= Leff ⋅ .
dt
The variation in internal resistance at the maximum
power point is shown in Fig. 6(a) and the corresponding
value Lmin in shown in Fig. 6(b)
TABLE I
PARAMETERS UNDER DIFFERENT LOAD CONDICTIONS
Internal
Maximum
Insulation VP
Resistance
IP(A)
D
Power
(W/cm2) (V)
Output(W)
( )
800
41.3
4.1
169
0.89
10.07
600
41.4
3.1
128
0.77
13.35
400
41.6
2.0
83.2
0.627
20.49
200
41.6
1.0
41.6
0.44
40.78
Ω
(5)
Lmin
(µH)
21.3
45.2
75.1
111
The converter used for the experimental validation
of the role of the variable inductor is shown in Fig. 7(a)
and the circuit schematic in Fig. 7(b).
Leff in (5) is readily found from the L/i characteristic of
the inductor. The Leff versus current characteristic is
more insightful. For the purposes of this paper we
shall use Leff for characterizing the inductor
III. SIMULATION RESULTS AND DISCUSSION
Simulation studies were carried out on a 210W
Sanyo HIP photovoltaic module. The solar insolation
was varied from 200 W/m2 to 800 W/m2. The results are
summarized in Table 1 for the PV panel voltage Vp and
current Ip at the maximum power point as well as the
internal resistance and the required duty cycle D for
continuous conduction at 20 kHz.
45
Internal Resistance r (Ohm)
40
(a) Prototype Converter
35
30
LVR
25
20
D1
15
RL
C1
10
5
0
0
200
400
600
800
1000
Solar Insolation(W/m2)
(a)
(b)
Fig. 7 (a) Prototype Converter and (b) Equivalent Circuit.
120
140
80
120
100
60
Lmin(uH)
Lmin(uH)
100
40
20
80
60
40
0
20
0
200
400
600
800
1000
0
Solar Insolation(W/m2)
0
(b)
1
2
3
4
5
Inductor Current (A)
Fig. 6 (a) Internal Resistance
(b) Minimum Inductance as a Function of Solar Insolation.
Fig. 8 Mesaured Inductance and Mimimum Inductance.
950
(a) 21.3 µH 800 W/m2
(a) 21.3 µH 800 W/m2
(b)21.3 µH at 200 W/m2
(b) 21.3 µH 200 W/m2
(c) 111 µH at 200 W/m2
(c) Variable Inductor of 111uH at 200 W/m2
Fig. 10 Experimental Results of Inductor Current under Different
Conditions
Fig. 9 Simulation Results of Inductor Current under Different
Conditions
951
The effective inductance of the variable inductor
was measured and the results are shown in Fig. 8.
Fig. 9(a) shows the inductor current for 21.3 µH, at
800 W/m2 (point a in Fig. 5) and the current is continuous.
Fig. 9(b) shows the inductor current for 21.3 µH, at 200
W/m2 (point b in Fig. 5) and as expected the converter is
operating in discontinuous mode. Repeating the
simulation for a 111 µH, at 200 W/m2 (point c in Fig. 5)
in Fig. 9(c) shows that continuous conduction has been
restored. Finally, the inductor current for 111µH at 800
W/m2 is shown in Fig. 9(a) with a smaller ripple as
expected. The above simulation results are validated by
the experiments which are shown in Fig. 10.
Fig. 10(a) and Fig. 10(b) use a fixed inductor to
illustrate the effect of CCM at 800 W/m2 and DCM at
200W/m2 respectively.
In Fig. 10(c), the variable
inductor was used and the onset of DCM is shown as
expected at 200 W/m2. The same variable inductor was
used at 800W/m2 and the converter was observed to
operate in CCM, however it was not at the boundary of
DCM because the inductance value was approximately 45
µH (see Fig. 8) and this is higher than the critical value of
21µH given in Table I.
IV. CONCLUSION
This paper presents a new topology of an MPPT
controller for solar power applications that incorporated a
variable inductance versus current characteristic. The
converter achieves CCM over a wide range of solar
insolation. The new inductor occupies a smaller volume
than the traditional fixed-gap inductor used in this
converter. MPPT is achieved down to lower levels of
solar insolation.
The solar module output characteristics are
represented by an equivalent circuit model in Fig. 2 with
the following equations, the parameter values given are
for the Sanyo HIP 210 W module:
The output current is given by:
I = I ph − I sat⋅(e
− 1)
(A1)
scref
Gref
isc
cell
e
−1
The author would like to thank for the financial
support of China Scholarship Council (CSC), File No.
2009102634.
REFERENCES
[1]
[2]
[8]
[9]
952
(A3)
ACKNOWLEDGMENT
[7]
Iscref Short circuit current at standard conditions (5.61A)
ph
qV
nKTcell N s
The open circuit voltage, Voc, is given as:
(A4)
VOC = Vocref + α VOC (Tcell − Tref )
Vocref Open circuit voltage at standard operating
conditions (51.6V)).
αvoc
Open circuit voltage temperature coefficient (80mV/˚C).
[6]
ref
I
I sat =
[4]
[5]
q
Charge on an electron (q = 1.6022 x 10-19 C)
K Boltzmann constant (K=1.38*10-23 m².kg.s-2 °K)
n
Ideality factor
(n=1.5)
I
Output current
(calculated in (A1))
Iph Photo-generated current (calculated in (A2))
Isat Saturation current
(calculated in (A3))
V
Output Voltage
Ns Number of cells in series (82)
Rs Series resistance
(0.004 Ω)
Tcell Solar Panel temperature (°K)
The photo generated current Iph is given by:
G
(A2)
I =I
⋅ [1 + α (T − T )]
ph
α
[3]
APPENDIX
qV
nKTcell N s
G Solar irradiance
Gref Reference solar irradiance at standard conditions
(1000W/m²)
isc Short circuit current temperature coefficient
(1.67mA/˚C)
Tref Reference temperature at standard conditions
(298˚K)
The saturation current, Isat, is given by:
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