2622 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 7, JULY 2008
A Variable Step Size INC MPPT
Method for PV Systems
Fangrui Liu, Shanxu Duan, Fei Liu, Bangyin Liu, and Yong Kang
Abstract—Maximum power point tracking (MPPT) techniques
are employed in photovoltaic (PV) systems to make full utilization
of PV array output power which depends on solar irradiation
and ambient temperature. Among all the MPPT strategies, the
incremental conductance (INC) algorithm is widely used due to
the high tracking accuracy at steady state and good adaptability
to the rapidly changing atmospheric conditions. In this paper,
a modified variable step size INC MPPT algorithm is proposed,
which automatically adjusts the step size to track the PV array
maximum power point. Compared with the conventional fixed
step size method, the proposed approach can effectively improve
the MPPT speed and accuracy simultaneously. Furthermore, it is
simple and can be easily implemented in digital signal processors.
A theoretical analysis and the design principle of the proposed
method are provided and its feasibility is also verified by simu-
lation and experimental results.
Index Terms—Incremental conductance (INC), maximum
power point tracking (MPPT), variable step size.
I. INTRODUCTION
P
HOTOVOLTAIC (PV) generation is becoming increas-
ingly important as a renewable source since it exhibits
many merits such as cleanness, little maintenance and no noise.
The output power of PV arrays is always changing with weather
conditions, i.e., solar irradiation and atmospheric temperature.
Therefore, a maximum power point tracking (MPPT) control
to extract maximum power from the PV arrays at real time
becomes indispensable in PV generation systems.
In recent years, a large number of techniques have been
proposed for tracking the maximum power point (MPP)
[1]–[12]. Fractional open-circuit voltage and short-circuit cur-
rent [1], [2] strategies provide a simple and effective way to
acquire the maximum power. However, they require periodical
disconnection or short-circuit of the PV modules to measure
the open-circuit voltage or short-circuit current for reference,
resulting in more power loss. Hill climbing and perturb and
observe (P&O) methods are widely applied in the MPPT con-
trollers due to their simplicity and easy implementation [3]–[5].
The P&O method involves a perturbation in the operating volt-
age of the PV array, while the hill climbing strategy introduces
Manuscript received July 13, 2007; revised February 18, 2008. This work
was supported in part by the Delta Power Electronics Science and Education
Development Fund under Grant DREK200501 and in part by the National
Natural Science Foundation of China under Grant 50777025.
The authors are with the College of Electrical and Electronic Engi-
neering, Huazhong University of Science and Technology, Wuhan 430074,
China (e-mail: fangruihust@163.com; dshanxu@263.com; dyj_lf@163.com;
lby@smail.hust.edu.cn; ykang@mail.hust.edu.cn).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2008.920550
a perturbation in the duty ratio of the power converter [5] and
is more attractive due to the simplified control structure [6].
Nevertheless, steady-state oscillations always appear in both
methods due to the perturbation. Thus, the power loss may
be increased. Incremental conductance (INC) method, which
is based on the fact that the slope of the PV array power versus
voltage curve is zero at the MPP, has been proposed to improve
the tracking accuracy and dynamic performance under rapidly
varying conditions [7], [8]. The steady state oscillations would
be eliminated in theory since the derivative of the power with
respect to the voltage vanishes at MPP. However, null value of
the slope of the PV array power versus voltage curve seldom oc-
curs due to the resolution of digital implementation. Although
the INC method is a little more complicated compared with the
P&O/hill climbing strategy, it can be easily implemented due to
the advancements of digital signal processors (DSPs) [9].
Moreover, fuzzy and neural network methods [10], [11] that
focus on the nonlinear characteristics of PV array provide a
good alternative for the MPPT control. Since the output charac-
teristics of the PV array should be well ascertained to create the
MPPT control rules, the versatility of these methods is limited.
The INC MPTT algorithm usually uses a fixed iteration step
size, which is determined by the accuracy and tracking speed re-
quirement. Thus, the corresponding design should satisfactorily
address the tradeoff between the dynamics and steady state os-
cillations. To solve these problems, a modified INC MPPT with
variable step size is proposed in this paper. The step size is auto-
matically tuned according to the inherent PV array characteris-
tics. If the operating point is far from MPP, it increases the step
size which enables a fast tracking ability. If the operating point
is near to the MPP, the step size becomes very small that the
oscillation is well reduced contributing to a higher efficiency. In
the following, the design principle of the modified variable step
size INC MPPT is presented on the basis of uniform irradiance
for PV array. Both simulation and experimental design exam-
ples are then provided, and the corresponding results confirm
that the proposed method can effectively improve the dynamic
performance and steady state performance simultaneously.
II. PV A
RRAY MPPT
A. PV Array Characteristics
Generally, a PV module comprises of a number of PV cells
connected in either series or parallel and its mathematical
model can be simply expressed as [12]–[14]
I
o
= n
p
I
ph
− n
p
I
rs
exp
K
o
V
n
s
− 1
(1)
0278-0046/$25.00 © 2008 IEEE
Authorized licensed use limited to: S. Abbas Taher. Downloaded on December 28, 2008 at 03:41 from IEEE Xplore. Restrictions apply.
LIU et al.: VARIABLE STEP SIZE INC MPPT METHOD FOR PV SYSTEMS 2623
Fig. 1. Variation of the normalized power and slope of power versus voltage
curves.
where I
o
denotes the PV array output current, V is the PV
output voltage, I
ph
is the cell photocurrent that is proportional
to solar irradiation, I
rs
is the cell reverse saturation current
that mainly depends on the temperature, K
o
is a constant, n
s
and n
p
are the numbers of series strings and parallel strings in
the PV array, respectively. The corresponding PV output power
and slope of output power versus output voltage curves can be
obtained as shown in Fig. 1.
B. Variable Step Size INC MPPT Algorithm
The step size for the INC MPPT method is generally fixed.
The power drawn from the PV array with a lager step size
contributes to faster dynamics but excessive steady state oscilla-
tions, resulting in a comparatively low efficiency. This situation
is reversed while the MPPT is running with a smaller step
size. Thus, the MPPT with fixed step size should make a sat-
isfactory tradeoff between the dynamics and oscillations. Such
design dilemma can be solved with variable step size iteration
[14]–[16].
However, all these strategies were proposed for P&O/hill
climbing MPPT method and the derivation of the essential
parameters of variable step size were not provided [14], [16]. In
this paper, a modified variable step size algorithm is proposed
for the INC MPPT method and is dedicated to find a simple and
effective way to improve tracking accuracy as well as tracking
dynamics.
In most applications, the MPP tracker is achieved by con-
necting a dc-dc converter between the PV array and load
[17]–[19]. The PV output power is used to directly control the
power converter duty cycle to reduce well the complexity of the
system [6]. The flowchart of the modified variable step size INC
MPPT algorithm is shown in Fig. 2, where the converter duty
cycle iteration step size is automatically tuned.
The PV output power is employed to directly control the
converter duty cycle, contributing to a simplified control system
[6]. Note that V (k) and I(k) are the PV array output voltage
and current at time k. In addition, D (k) and step are the duty
Fig. 2. Flowchart of the variable step size INC MPPT algorithm.
cycle and change of duty cycle (step size), respectively. The
variable step size adopted to reduce the problem mentioned
above is shown as follows [15]:
D(k)=D(k − 1) ± N ∗
dP
dV
(2)
where coefficient N is the scaling factor which is tuned at the
design time to adjust the step size. The variable step size can
also be realized from the slope of the P–D curve in [16] for
P&O MPPT as
D(k)=D(k − 1) ± N ∗
∆P
∆D
(3)
where ∆D is the step-change in duty cycle in the previous
sampling period. As shown in Fig. 1, the derivative of power
to voltage (dP/dV ) of a PV array can be seen to be varying
smoothly and is recommended in [15] as a suitable parameter
for determining the variable step size of the P&O algorithm.
Thus, |dP |/|dV | is also employed herein to determine the
variable step size for the INC MPPT algorithm. The update rule
for duty cycle can be obtained as follows:
D(k)=D(k − 1) ± N ∗
P (k) − P (k − 1)
V (k) − V (k − 1)
. (4)
Scaling factor N essentially determines the performance of
the MPPT system. Manual tuning of this parameter is tedious
and the obtained optimal results may be valid only for a
given system and operating condition [15]. A simple method
to determine the scaling factor is proposed here. Comparatively
large step size ∆D
max
for fixed step size MPPT operation is
initially chosen. With such value, the dynamic performance is
good enough, while the steady-state performance may not be
satisfactory. The steady-state value instead of dynamic value in
Authorized licensed use limited to: S. Abbas Taher. Downloaded on December 28, 2008 at 03:41 from IEEE Xplore. Restrictions apply.
2624 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 7, JULY 2008
Fig. 3. MPPT system.
the startup process [15] of the derivative of PV array output
power to voltage can be evaluated under the fixed step size
operation with ∆D
max
, which will be chosen as the upper
limiter as the variable step size INC MPPT method. It is known
that |dP |/|dV | is almost at its lowest value around the PV
MPP. To ensure the convergence of the MPPT update rule, the
variable step rule must obey the following:
N ∗
dP
dV
fixed step=∆D
max
< ∆D
max
(5)
where |(dP/dV )|
fixed step=∆D
max
is the |dP |/|dV | at fixed
step size operation of ∆D
max
. The scaling factor can therefore
be obtained as
N<∆D
max
/
dP
dV
fixed step=∆D
max
. (6)
If (6) cannot be satisfied, the variable step size INC MPPT
will be working with a fixed step size of the previously set
upper limiter ∆D
max
. Equation (6) provides a simple guidance
to determine the scaling factor N of the variable step size INC
MPPT algorithm. With the satisfaction of (6), larger N exhibits
a comparatively faster response than a smaller N, which will be
further discussed in Section III. The step size will become tiny
as dP/dV becomes very small around the MPP.
III. S
IMULATION AND EXPERIMENTAL EVA LUAT IO N
A simple MPPT PV system shown in Fig. 3 is developed
to test the feasibility of the proposed method. A push-pull
converter is used as the power interface between the PV array
and the load to achieve maximum power. Assuming that the
turns of the two primary windings are the same, the output
voltage of the converter can be expressed as [17]
V
o
=2mDV (7)
where m is the turn ratio of the secondary winding to the
primary winding and D is the duty cycle. It can be seen that
the input dc voltage can be easily shifted to a high level. This
converter is suitable for a lower PV output voltage and higher
desirable dc-link voltage case, where electrical isolation is also
required.
TAB LE I
GFM-120 C
RYSTALLINE SILICON PV MODULE SPECIFICATIONS
Fig. 4. PV array output power with fixed step size INC MPPT. (a) Fixed step
size of 0.01. (b) Fixed step size of 0.05.
A. Simulation Results
To verify the performance of the proposed modified vari-
able step size INC MPPT algorithm, a MATLAB-SIMULINK
model of the PV system shown in Fig. 3 is initially developed.
GFM-120 Crystalline Silicon PV module is used for the PV
array model in simulation and experiment and the specifications
are listed in Table I.
To compare the performance of the variable step size INC
MPPT method with the ordinary fixed step size INC MPPT
method, the simulations are configured under exactly the same
conditions to compare the performances. The PV array in
simulation is composed of one PV module, and the sampling
period [3], [18] used for MPPT algorithm is chosen as 0.025 s.
The duty cycle command is therefore updated every 0.025 s.
The output power performance of INC MPPT with fixed step
size of 0.01 and 0.05 under irradiation step change conditions
are shown in Fig. 4. The irradiation was suddenly changed from
1000 to 400 W/m
2
at 0.4 s and changed back to 1000 W/m
2
at 1.4 s. For the comparative purpose, the allowable maximum
duty size ∆D
max
[referred to (6)] is set as 0.05 for the proposed
variable step size INC MPPT method. The corresponding PV
Authorized licensed use limited to: S. Abbas Taher. Downloaded on December 28, 2008 at 03:41 from IEEE Xplore. Restrictions apply.
LIU et al.: VARIABLE STEP SIZE INC MPPT METHOD FOR PV SYSTEMS 2625
Fig. 5. PV array output power and step size with variable step size INC MPPT
(N =0.06). (a) PV output power. (b) Step size.
output power and step size under N =0.06 and N =0.12 are
shown in Figs. 5 and 6, respectively. The tracking performance
under both fixed and variable INC MPPT methods are presented
in Table II. Compared with the MPPT with fixed step size
of 0.01 [Fig. 4(a)], the MPPT with fixed step size of 0.05
[Fig. 4(b)] exhibits a good dynamic performance but larger
steady state oscillations. The tracking time with fixed step size
of 0.05 under irradiation step change conditions is only several
MPPT sampling periods and the tracking ability can be further
improved with larger step size. However, it is achieved at the
sacrifice of MPPT efficiency. The PV array average output
power with fixed step size of 0.05 is 114.5 W and decreased
by 3.3% compared with the output power of 118.4 W with step
size of 0.01. The proposed variable step size method solves
the dilemma as evident from Figs. 5 and 6. The oscillations at
stead state in these two figures are almost eliminated due to the
very small |dP |/|dV | and the PV array output power is above
119.3 W. Moreover, the dynamic performance is obviously
faster than that of fixed step size of 0.01. It also can be seen
that the proposed strategies with N =0.12 [refer to Fig. 6(a)]
shows a faster dynamic response than that of N =0.06 [refer
to Fig. 5(a)]. A bigger N [but still with the satisfaction of (6)]
can be chosen to achieve a faster response.
B. Experimental Results
The operation of the variable step size INC MPPT method
has also been evaluated by experiment. A prototype of the
Fig. 6. PV array output power and step size with variable step size INC MPPT
(N =0.12). (a) PV output power. (b) Step size.
MPPT system depicted in Fig. 3 is constructed and the push-
pull converter specifications are chosen as follows:
1) dc capacitance: 470 µF (PV side), 47 µF (filter);
2) filter inductance: 0.35 mH;
3) transformer turn ratio: 8/38 (primary to secondary);
4) switching frequency: 20 kHz.
In the experiment, three PV modules with specifications
illustrated in Table I are connected in series. The control system
is implemented in a TMS320LF2407 DSP.
The start waveforms with variable step size INC MPPT al-
gorithm are shown in Fig. 7. When the system approaches near
the MPP, the step size becomes very small, resulting in a smooth
power curve. However, the PV current and power increase with
large steps due to the large step size at the beginning (referred
to Fig. 2). This can be overcome by adding a simple constant
voltage tracking (CVT) start program as shown in Fig. 8. The
MPP voltage has been reported to be nearly 78% of the open
voltage [6]. The preset voltage V
set
is set as 0.8V
oc
to enable the
converter duty cycle to increase linearly to approach MPP. Once
the PV output voltage goes lower than V
set
, the control unit
switches to the variable step size INC MPPT algorithm. Thus,
the PV system reaches the MPPT very smoothly as illustrated in
Fig. 9. A variable resistive load was directly connected the PV
arrays as well to test the maximum power. The maximum power
difference between the PV array could be produced and the
array outputs with the proposed variable step size INC MPPT
method is within several watts. Thus, the MPPT efficiency of
Authorized licensed use limited to: S. Abbas Taher. Downloaded on December 28, 2008 at 03:41 from IEEE Xplore. Restrictions apply.
2626 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 7, JULY 2008
TAB LE II
T
RACKING PERFORMANCE COMPARISON BETWEEN FIXED AND VARIABLE STEP SIZE INC MPPT METHODS
Fig. 7. Start waveforms of PV output voltage, current, and power with
variable step size INC MPPT method.
Fig. 8. CVT start program.
the proposed method under the current environment is about
99.2%, while the efficiency for fixed step size INC MPPT
strategy is 98.9% with the same experimental setup.
The MPPT efficiency difference is not obvious mainly due
to the small step size chosen for the fixed step size INC MPPT
algorithm. The purpose of this paper is to improve the dynamic
response as well and will be further illustrated in the following
figures.
It is recommended in [3] and [18] that the whole system in
one MPPT cycle should reach the steady state before another
begins. The MPPT sampling interval chosen here for the ex-
periment is comparatively as large as 0.25 s to investigate how
Fig. 9. Start waveforms with variable step size MPPT algorithm and CVT
start program.
the proposed MPPT method functions under dynamic working
conditions. The PV system may suffer rapidly changing irradi-
ation under practical operation. According to the characteristics
of PV modules, there is a severe variation in the maximum
output power while the MPP voltage changes little. A switch
is introduced to parallel with one of three series-connected PV
modules to simulate the effect of the insulation on the PV
system. When the switch is turned on or turned off, both the
output power and output voltage of the PV array will suffer
a step change, simulating a worse working condition for the
MPPT control. When the switch is turned off, PV modules
number is changed from two to three. The corresponding PV
system output voltage, current and power curves with the
proposed variable step size INC MPPT algorithms are shown
in Fig. 10(a), while Fig. 10(b) illustrates those waveforms for
the PV module numbers is suddenly changed from three to two.
The PV array output waveforms with fixed step size INC MPPT
under PV module number suddenly changing conditions are
shown in Fig. 11. The sampling periods used by both MPPT
methods are the same. A small fixed step size is chosen to
achieve almost same steady-state accuracy as the variable step
size method. From these figures, it can be seen that the PV sys-
tem with variable step size gets to the MPP within 1.5 s while
it takes 7.5 s for the fixed step size method to track the MPP
when the PV output power is suddenly changed. Nevertheless,
the tracking time is long, the dynamic process is finished within
6 MPPT sampling periods. It is evident that the PV system with
variable step size INC MPPT algorithm has a good dynamic
performance. Due to the inherent iteration characteristic, the
Authorized licensed use limited to: S. Abbas Taher. Downloaded on December 28, 2008 at 03:41 from IEEE Xplore. Restrictions apply.