Available online at www.sciencedirect.com
Electric Power Systems Research 78 (2008) 777–784
A novel maximum power point tracking for photovoltaic
applications under partially shaded insolation conditions
Nabil A. Ahmed a,∗ , Masafumi Miyatake b
a
Department of Electrical Engineering, Faculty of Technological Studies, Public Authority of Applied Education and Training, Shuwaikh, Kuwait
b Department of Electrical and Electronics Engineering, Sophia University, Tokyo, Japan
Received 9 June 2006; received in revised form 22 May 2007; accepted 31 May 2007
Available online 25 September 2007
Abstract
In this paper, a novel strategy of maximum power point tracking is presented for photovoltaic power generation systems based on Fibonacci
search algorithm to realize simple control system to track the real maximum power point even under non-uniform or for rapidly changing insolation
conditions. The Fibonacci search technique was modified in order to apply to time-variant P–V characteristics of the PV array.
It is proved experimentally and through simulation results that the algorithm has good performances and very fast response even for partial
shaded PV modules. The proposed technique can be applied to various types of PV power conditioners including dc–dc and dc–ac converters. It
can be commonly used instead of the conventional MPPT techniques. This method requires neither the measurement of temperature and insolation
level nor prior knowledge about the model of the used photovoltaic panel. The proposed method can track the global maximum power point in
most cases by doing a wide-range search and more power can be extracted with partial shaded PV modules as compared to the other methods.
Simulation and experimental results are presented to prove the feasibility of the proposed technique.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Photovoltaic generation system; Maximum power point tracking; Local maximum; Partially shaded array; Rapidly changeable insolation; Fibonacci
search algorithm
1. Introduction
The ever-increasing demand for conventional energy sources
like coal, natural gas and crude oil is driving society towards the
research and development of alternate energy sources. Many
such renewable energy sources like wind energy and solar or
photovoltaic (PV) are now well developed, cost effective and
are being widely used, while some others like fuel cells are in
their advanced developmental stage. These energy sources are
preferred for being environmental friendly. Photovoltaic energy
has become one of the most promising sources of energy due to
the fact that PV energy is free and sustainable. Besides this, PV
is scaleable from very small to very large and easy to integrate
with existing power converters [1].
Although the rapid development of PV materials technology and the increased demands for PV materials have led to
∗
Corresponding author.
E-mail address: na.ahmed@paaet.edu.kw (N.A. Ahmed).
0378-7796/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsr.2007.05.026
a reduction of the PV module costs [2], the capital costs of
PV systems are still very high. Therefore, there is a necessity
to design a power converter that is not only high in efficiency
but also optimizes the energy production of the PV material.
The power produced by a PV module is dependent upon the
solar irradiance and temperature of the PV panel. The output
power of a PV module also varies as a function of its operating
point because of the inherent non-linear current voltage (I–V)
relationship of a typical PV cell. Therefore, a maximum power
point tracking (MPPT) algorithm is commonly used to maximize
the power drown from PV modules under varying atmospheric
conditions. Among the MPPT techniques, the perturbation and
observation (P and O) method is the most popular because of the
simplicity of its control structure and few measured parameters
for the power tracking [3]. The name of algorithm itself reveals
that it operates by periodically perturbing the control variable
and comparing the instantaneous PV output power after perturbation with that before. Therefore, continuous perturbation or
oscillation occurs around the MPP in steady state. Incremental
conductance (IncCond) method tracks the MPP accurately by
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N.A. Ahmed, M. Miyatake / Electric Power Systems Research 78 (2008) 777–784
comparing the incremental conductance dI/dV and chord conductance I/V [4]. It depends on the fact that at the MPP, the
derivative of PV output power with respect to the PV voltage is
zero. This algorithm has the advantage of no oscillation occurs
around the MPP in steady state. The drawback of this method is
the complexity of the design controller. The mountain climbing
method is also often used for MPPT of PV systems. It is simple
but the convergence is slow. Voltage- and current-based MPPT
methods have been developed in [11,12].
However, these methods cannot readily track immediate and
rapid changes in environmental conditions or partial shading of
the PV modules. Tracking the maximum power point is difficult
when a PV array is partially shaded or is installed in a rapidly
changeable insolation conditions, because two or more local
maximum power points may appear. The PV power–voltage
(P–V) characteristic sometimes shows multi-local maximum
operating points under non-uniform isolation conditions or
under partially shading PV modules. In this case the operating
point of the PV system can tend to converge to a local maximum
operating point which is not the real maximum one on the P–V
curve. A few papers have been reported trying to avoid this difficulty. However, most of those control systems become rather
complicated [5–9].
The authors have proposed a new simple MPPT technique
using the Fibonacci line search. It enables intuitive adjustment
of control parameters and fast response. The method was already
verified with simulations and experiments. However, the method
cannot always find the global maximum point when the part of
PV array is partially covered with shadow [10,11].
In this paper, a novel strategy of MPPT is proposed for photovoltaic power generation systems based on improving the
original Fibonacci search algorithm to realize simple control
system to track the real maximum power point even under nonuniform insolation conditions or for rapidly changing insolation.
The Fibonacci search technique was modified in order to apply
to time-variant P–V characteristics of the PV array. A new initialization function has been introduced to initializing the search
condition when sudden or partial change in insolation is detected
and doing a wide search, which leads to the real maximum power.
It is proved experimentally and through simulation results
that the algorithm has good performances and very fast response
even for partial shaded PV modules. The proposed technique can
be applied to various types of PV power conditioners including
dc–dc and dc–ac converters. It can be commonly used instead of
the conventional MPPT techniques. This method requires neither the measurement of temperature and insolation level nor
prior knowledge about the model of the used PV array. The
proposed method can track the global maximum power point
in most cases by doing a wide-range search and more power
can be extracted with partial shaded PV modules as compared
to the other methods. Simulation and experimental results are
presented to prove the feasibility of the proposed technique.
The Fibonacci numbers are defined in the following manner:
c0 = 0,
c1 = 1,
cn = cn−2 + cn−1
for n ≥ 2
(1)
Based on (1), the Fibonacci numbers are calculated as
c2 = 1,
c3 = 2,
c4 = 3,
c7 = 13
c5 = 5,
c6 = 8,
(2)
The Fibonacci line search is generally used for optimization
techniques applied to one variable functions. This method iteratively restricts and shifts the searching range so as to contain
optimal point in the range. The direction of the search shifting
is decided by the value of the function at two check points in
the range. The process of restricting and shifting is illustrated in
Fig. 1.
2.1. Bracketing search method
An approach for finding the minimum of a function f(x) in a
given interval is to evaluate the function many times and search
for a local minimum. To reduce the number of function evaluations it is important to have a good strategy for determining
where f(x) is to be evaluated. Two efficient bracketing methods are the golden ratio and Fibonacci searches. To use either
bracketing method for finding the minimum of f(x), a special
condition must be met to ensure that there is a proper minimum
in the given interval.
2. Line search algorithm with Fibonacci sequence
This section describes how to use Fibonacci search algorithm
for the MPPT array.
Fig. 1. The decision process of Fibonacci search algorithm in one search operation: (a) f(c0 ) ≤ f(d0 ) and f(c0 ) > f(d0 ).
N.A. Ahmed, M. Miyatake / Electric Power Systems Research 78 (2008) 777–784
2.2. Definition of unimodal function
3. Application of Fibonacci search algorithm for MPPT
of photovoltaic
The function f(x) is unimodal in an interval I = [a, b], if there
exists a unique number p ∝ I such that
f (x) is decreasing on [a, p]
and
f (x) is increasing on [p, b].
In the Fibonacci search two function evaluations are made at
the first iteration and then only one function evaluation is made
for each subsequent iteration. The search is terminated at the
kth subinterval, provided that |bk − ak | ≤ δ or |f(bk ) − f(ak )| ≤ ε
where δ and ε are the predetermined tolerances.
Assume we are given a function f(x) that is unimodal on
the interval [a0 , b0 ]. A value r0 (0.5 < r0 < 1) is selected so
that two of the interior points c0 and d0 will be used in
the next subinterval and there will be only one new function evaluation. In the classical search methods as golden ratio
search two function evaluations are made at the first iteration
and then two function evaluations are made for each subsequent iteration and the value of r0 remains constant on each
subinterval.
The Fibonacci search method differs from the golden ratio
method in that the value of r0 is not constant on each subinterval.
The (k + 1)st subinterval is obtained by reducing the length of
the kth subinterval by a factor of rk = cn − 1 − k /cn − k . The length
of the last subinterval is (a0 − b0 )/cn . Additionally, the number
of subintervals (iterations) is predetermined and based on the
specified tolerances.
If f(c0 ) ≤ f(d0 ), then the minimum must occur in the subinterval [a0 , d0 ] and we replace a1 = a0 and b1 = d0 and continue the
search in the new subinterval [a1 , b1 ] = [a0 , d0 ]. If f(c0 ) > f(d0 ),
then the minimum must occur in the subinterval [c0 , b0 ] and we
replace a1 = c0 and b1 = b0 and continue the search in the new
subinterval [a1 , b1 ] = [c0 , b0 ]. These choices are shown in Fig. 1.
The relation of the variables ai and bi at any iteration interval,
which are illustrated in Fig. 1, and the Fibonacci sequence cn
are expressed as
ai = cn+1 ,
bi = cn
(3)
In the algorithm implementation, the Fibonacci numbers are
given explicitly (e.g. as constants in the frame program). Eq. (3)
shows that the operation is based on the Fibonacci sequence and
at each iteration the length of the interval is chosen according to
the Fibonacci rule. The value of ai+1 and bi+1 can be decided in
the next iteration as
ai+1 = cn (= bi ),
bi+1 = cn−1
779
(4)
The search is terminated when the variable n of the number cn becomes zero or when the condition |bk − ak | ≤ δ or
|f(bk ) − f(ak )| ≤ ε is satisfied.
When the Fibonacci search algorithm is applied to the MPPT
of PV panel, the variable x can be considered as voltage, current
of the PV array or duty cycle of the power converter (duty cycle D
is selected in this paper) and the function f(x) can be regarded as
the output power. However, the original search algorithm cannot
be directly applied to time-variant function such as P–V characteristics of PV generation system. Hence, the method should be
modified for real-time optimization.
3.1. Modification for sudden insolation change
The Fibonacci sequence numbers used in this method is limited between c1 and cN , where cN is the last term in the sequence,
and the method restricts search range so as to contain the MPP.
The proposed search method iteratively restricts and shifts the
searching range so as to contain the optimal point in the range.
The direction of the shifting is decided by the value of the function at two check points in the range. The process of restricting
and shifting is illustrated in Fig. 2. However, the MPP may move
to be outside of the search range due to the sudden change in the
insolation. In this case, the search range is being widening as
shown in Fig. 2(b) and this is realized by reversing the process
of Fig. 1. Fig. 2(b) shows the operation of widening the search
range to chase the maximum power point. If the optimal point is
assumed to be out of the search range owing to sudden change
of insolation, the search range must continue to shift in the same
direction. The situation is detected by counting the number of
shifts. The counting is initialized when the range shifts to the
other direction. If it shifts to the same direction more than one
time (it is set by the parameter M), the range was not narrowed
but widen.
The widening process is practically implemented when the
range is shifted to the same direction more than M times and the
Fibonacci sequence does not reach the last term cN . The value
of ai+1 and bi+1 will be decided in the next iteration as
ai+1 = cn+2 ,
bi+1 = cn+1 (= ai )
(5)
The adjustment of control parameters is done with the constants
M and N, which influence the speed of response and the oscillation at the steady state. Where, M counts for the number of
searching in the same direction and N for the maximum number
of restricting and widening the search.
3.2. Modification for gradual insolation change
Fig. 3 shows how to cope with the slow change of insolation. The output power is measured twice at two different points
(x1 and x2 ) during each control cycle. The measurement should
be done at the identical timing, because the insolation may
change between the consecutive measurements. If the insolation changes gradually, it causes miss tracking of the position of
MPP due to the time difference between measurements. Therefore, the power measured firstly is corrected by using the power
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N.A. Ahmed, M. Miyatake / Electric Power Systems Research 78 (2008) 777–784
measured at the end of the cycle by assuming linear change
of output power as shown in Fig. 3. The correction uses the
following equation:
Pi (x2−d ) = 2Pi (x2−d ) − Pi−1 (x1+d )
(6)
where the d represents the direction of shifting, Pi (x2−d ) and
Pi−1 (x1+d ) is the measured power at the beginning (T/2) and the
middle of the control cycle for each iteration, Pi (x2−d ) is the
estimated power for each iteration.
Fig. 3 shows the measured power at two consecutive points
and how this power is corrected to be used in Fibonacci search.
When the PV array is partially shaded, the P–V curve has
two or more maximum points. The algorithm may find a local
maximum which is not always the same as the global maximum
and fails to track the actual global maximum. In the following
subsection, a new correction function is proposed which enables
to find the global maximum point.
3.3. New function to guarantee global maximum power
A new function is proposed to improve the search algorithm
by the initializing the search condition when sudden or partial
change of insolation is detected. This new function leads the
operating point to the global maximum point, because a widerange search can be realized.
The following equation is introduced for the detection of such
sudden or partial change of insolation.
|Pi (x2−d ) − Pi−1 (x1+d )|
<r
Pi (x2−d )
(i)
(i)
Fig. 2. Comparison of f (x1 ) and f (x2 ) to restrict or widen the search range:
(a) restricting the search range and (b) widening the search range.
(7)
If Eq. (7) is not satisfied, the shaded portion of Fig. 3 is preceded and the condition is initialized. The constant r was decided
by experimental trials according to the fact that if r is large, initialization often occurs even if the insolation is not changed and
if r is very small, no initialization occurs. The optimal value of
the constant r in Eq. (7) is decided practically and it is set to be
0.2.
The flowchart of the proposed real-time endless optimizing approach based on Fibonacci search algorithm proposed in
this section and using the suggested function to improve the
Fibonacci search for the proposed study for partially shaded PV
panel is shown in Fig. 4.
Fig. 5 shows simulation results of the maximum power
and generated power under two extreme studying cases
when the irradiance is sinusoidal changing and when a step
change is occurred. A very fast response and accurate MPPT
are achieved even for rapidly and continuously changing
insolation.
4. Experimental verifications
Fig. 3. Correction of measured power.
The fundamental results of simulations and experiments are
given in [10]. New experimental results of partial shading effect
are mentioned in this paper.
N.A. Ahmed, M. Miyatake / Electric Power Systems Research 78 (2008) 777–784
781
Fig. 4. Flowchart of the MPPT control technique with initialization function.
4.1. Configuration of experimental system
The configuration of the experimental system is shown in
Fig. 6. The experimental set up is composed of a PV panel connected to an electronic load through a PWM boost converter.
The rated output power of the PV array is 300 W which consist of six series PV modules (Fuji Electric Co. ELR-615-160Z,
50 W) shown in Fig. 7. A one leg of a 6in1 intelligent power
module (IPM) is used as a boost converter. Gating signals of the
IPM switching device are generated by a digitals signal processor (DSP). A TMS320LF2407A DSP digital control platform
is implemented to obtain the control and PWM functions via
software that maximizes the output power. The DSP realizes the
complicated algorithm and controls the boost converter switch
according to the control algorithm so that the output power can
track the maximum power at each sampling instant. Output current and voltage of the boost converter are continuously on line
sensed instead of those of the PV array in order to maximize the
output power including the losses of the power converter. The
control program runs on the DSP is developed with C compiler
on a PC. An electronic load is used as a 200 V constant dc voltage load. Control period is set to 0.1 s. The circuit parameters
are illustrated in Table 1.
The boost converter voltage and current are continuously
sensed and the duty cycle of the boost converter switching device
is calculated by the DSP according to the proposed MPPT using
the Fibonacci search algorithm presented in this paper. The virtual variable xi of the Fibonacci sequence cn is converted to the
duty cycle of the boost converter D (0 ≤ D ≤ 1).
D=
xi
256 − 1
(8)
The number 256 which decide the resolution of the duty cycle
is derived from the use of 8 bit integer. The parameters M and N
are set to 1 and 8, respectively. The initial conditions are selected
as follows:
(0)
x1 = 255−c11 −c10 = 111,
(0)
x2 = 255−c11 = 166
(9)
Fig. 8 shows the relation between the variable x and Fibonacci
sequence at initial conditions.
Table 2 shows the relation between the Fibonacci numbers,
shift direction and the estimated duty cycle during a complete
control period.
4.2. Results and discussion
Partial shading is made artificially by covering the PV array
with a sheet instantaneously. The tested PV array is composed of
six series modules with rated output power of 300 W. Four modules are partially covered to produce insolation of 0.7 kW/m2
and two modules are partially covered to produce insolation of
0.2 kW/m2 . The simulated and measured P–V characteristics at
the above conditions are as shown in Fig. 9. The P–V characTable 1
Design specifications and circuit parameters
Item
PV array rating, P (W)
Boost inductor, L (mH)
Smoothing capacitor, C (␮F)
Output voltage, Vr (V)
Switching frequency, fs (kHz)
Control cycle, T (s)
Value
300
60
1000
200
10
0.1
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Table 2
Relation of x and, shift direction and duty cycle
Iteration no.
Case
x3
x1
x2
x4
d
Direction
D
1
2
3
4
5
6
7
f(x1 ) ≥ f(x2 )
f(x1 ) ≥ f(x2 )
f(x1 ) ≥ f(x2 )
f(x1 ) ≥ f(x2 )
f(x1 ) ≥ f(x2 )
f(x1 ) ≥ f(x2 )
f(x1 ) ≥ f(x2 )
22
111
166
166
166
166
174
111
166
200
187
179
174
179
166
200
221
200
187
179
179
255
255
255
221
200
187
187
1
1
1
0
0
0
1
Right
Right
Right
Left
Left
Left
Right
0.4353
0.6510
0.7S43
0.7333
0.7020
0.6824
0.7020
teristic under partial shading has a global maximum point and a
local maximum point. The difference of output power and voltage at two maximum points is relatively high. Most conventional
MPPTs cannot find such global maximum point. The points of
global maximum power and local power are located at 60 V,
125 W and 110 V, 80 W, respectively.
The step response of the proposed MPPT algorithm is examined experimentally compared to the original method without
the proposed function given in Eq. (7). The generated output
Fig. 6. Configuration of experimental system.
Fig. 7. Photovoltaic solar panel for experiments.
Fig. 5. Simulation results at sudden change of insolation: (a) continuous sinusoidal change and (b) step change.
Fig. 8. Relation of x and Fibonacci sequence at initial conditions.
N.A. Ahmed, M. Miyatake / Electric Power Systems Research 78 (2008) 777–784
Fig. 9. Simulated and measured P–V characteristics under partial shading: (a)
simulated and (b) measured.
783
Fig. 10. Experimental response of MPPT with sudden partial shading: (a) proposed method (b) original method.
power of the proposed and original search algorithms are shown
in Fig. 10(a) and (b), respectively. The working duty cycle are
also shown in Fig. 11(a) and (b) for the two methods, respectively. The oscillation in the output power and the duty cycle
shown in Figs. 10(a) and 11(a) are due to the wide search range
of the proposed method to arrive the real global maximum power
point.
Fig. 10(b) indicates that the original method could not able to
change the duty cycle to track the real maximum power point due
to the rapid change in the insolation shown in Fig. 9. Therefore,
the original method cannot track the global maximum power
point and the generated power is 80 W (local maximum) with a
duty cycle of 0.32 as shown in Figs. 10 and 11(b)Figs. 10(b) and
11(b). On the contrary, the new proposed method can track well
the global maximum power point by doing wide-range search as
shown in Figs. 10(a) and 11(a). The generated power is 125 W
at a duty cycle of 0.67.
Considering the well-known steady-state voltage conversion
characteristic of the typical PWM boost converter, which generally represented by
vout =
Vin
1−D
(10)
and the output load voltage was kept constant to 200 V by the
electronic load. The operating PV voltages corresponding to the
duty cycles of 0.32 and 0.67 are 110 and 60 V, respectively,
which correspond to the local and global maximum points of
Fig. 9.
The total output power with partial shading is increased by
a factor of about 36% using the proposed correction function.
Fig. 11. Duty cycle variations to sudden partial shading: (a) proposed method
and (b) original method.
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N.A. Ahmed, M. Miyatake / Electric Power Systems Research 78 (2008) 777–784
The saving value in the output power depends on the number
of shaded modules in the PV array. In most cases, the proposed method has the guarantee to find the global maximum
power point under partial shading or sudden shading conditions
independent on the circuit configuration. However, there is a
fluctuation in the output power and duty cycle due to the initialization of the search conditions at the beginning of each search
when sudden or partial change of insolation is detected.
5. Conclusions
In this paper a novel maximum power point tracking method
based on Fibonacci search algorithm has been improved and
adopted for the MPPT of partial shading or rapidly changing
insolation conditions of PV arrays. The line search method with
Fibonacci sequence has been employed and modified to apply
for time-variant power–voltage characteristics of PV arrays to
realize simple control system to track the real maximum power.
This method requires neither the measurement of temperature
and insolation level nor prior knowledge about the model of
the used PV panel. The proposed method can track the global
maximum power point in most cases by doing a wide-range
search and more power can be extracted with partial shaded PV
modules as compared to the other methods.
The proposed algorithm has been verified experimentally
and with simulation results. A very fast response, good performance and accurate MPPT have been achieved even for
PV modules under partial shading or rapidly changing insolations. The proposed approach can be commonly applied
instead of the conventional MPPT techniques to any type of
PV power conditioners including dc–dc and dc–ac interface
systems.
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