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CHAPTER 5

FIBONACCI SERACH METHOD FOR MAXIMUM POWER

POINT TRACKING

5.1 INTRODUCTION

It has been shown in previous chapters that partial shading of

SPVA is one of the main causes for reduced energy yield of many SPV

arrays. Use of anti parallel diodes with series connected modules may produce

multiple power peaks in the P-V characteristics. For instance, partial shading

on a SPVA will result in different irradiance on the array’s individual

modules. This again will result in different optimal operating voltages and

currents for the individual modules. There are in principle two approaches to

handle this issue:

One solution is to cascade each module with a separate power

converter (Lindergen 1999). This allows the modules to

operate at different voltage and current while maintaining the

same current output after the power converter. This technique

is known as “module-converter approach”.

If the PV array is to be connected to one single power

converter, multiple MPPs on the power-voltage curve will

occur. Conventional MPPTs tend to lock to a local MPP,

which may not be the global optimum. One solution to ensure

the operating point corresponds to the global optimum is to

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periodically scan the entire operating range and search for the

global MPP (Patel and Agarwal 2008 b).This is known as

“scanning MPP approach”.

If we assume lossless power converters, the module-converter

approach is able to harness more power. This is because the scanning MPP

approach can only set one operating point for the entire SPV array; therefore,

all modules connected in parallel must have the same voltage, whereas all

modules connected in series must carry the same current. Individual modules

may therefore not operate at their own optimum, although the SPV array as

such is at its optimum operating point. However, in reality lossless power

converters do not exist and from the point of view of converter efficiency, a

larger power rating is beneficial. Due to relatively large power loss in smaller

converters, the power loss under normal and shaded operating conditions is

higher than that by using a central or string inverter. This shortcoming is

particularly pronounced when partial shading conditions are rare. However, in

situations where partial shading would be the norm, such as the building

integrated systems, the module converter approach may be beneficial. When

the module converter approach is used, a scanning MPP algorithm is needed

at the module level, since an individual module may also have multiple

maxima in case of partial shading. To avoid such scanning, each bypass diode

in the module would have to be paired with one power converter. Such an

approach is feasible in principle, but would suffer even more from the

relatively large power loss resulting from small power rating of individual

converters. Therefore, in practice, a scanning MPP algorithm is necessary

whether string or module-converter approaches are used.

A very little attention has been drawn in the literature on assessing

the performance of MPPT under partial shaded conditions (scanning MPP

approach) due to the complexity and extensive measurement equipments

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required for this purpose. Against this background, different MPPT schemes

proposed in the literature under non uniform irradiance conditions have been

reviewed and an improved MPP approach has been presented.

V/I ratio for the load is decided on other considerations. However

V/I ratio for the SPVA is governed by MPP of the array. To match these

different requirements, a dc-dc converter is interposed between SPVA and the

load. V/I ratio on the input and output sides of the dc-dc converter will be the

same whereas V and I values observed on the two sides may be different. The

situation is a kin to a transformer used in AC to match the impedances.

5.2 MPPT SCHEMES

Several MPPT schemes are available in the literature. A few

popular schemes are:

Perturb and observe (P&O) or Hill climbing methods (Hua et

al 1998, Fernia et al 2005).

Incremental conductance method (Hussein and Muta 1995)

Short circuit current method (Noguchi et al 2002)

Open-circuit voltage method (Dorofte et al 2005)

Ripple correlation method (Esram et al 2006)

Modified techniques to minimize hardware and to improve the

performance (Kasa et al 2000, Veerachary et al 2001, Dorofte

et al 2005, Sera et al 2006, Kasa et al 2005).

The tracking schemes mentioned above are effective and time

tested under uniform solar irradiance, where the P-V curve of SPV module

exhibits only one MPP for a given temperature and irradiance. But partial

shading results in a deformation of the overall P-V curve. That is P-V curves

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often exhibit multiple local maxima at different locations, which may also

result in quite odd ratios between global MPP (GMPP) voltage and open-

circuit voltage. These factors can present a considerable hindrance to the

accurate operation of a MPPT. Some of the popular literatures which present

the methods to address this issue are presented here:

Solodovnik et al (2004) have suggested a state space- based

approach to search the GMPP. This method is fast and

accurate but is system specific, is complex, and requires more

sensors.

Miyatake et al (2004) have reported an MPPT scheme that

uses Fibonacci sequence to track the GMPP under partially

shaded conditions. However, the method does not guarantee

GMPP tracking under all conditions.

Testing of various commercially available inverters in

partially shaded conditions and power loss due to shading has

been presented by Bruendlinger et al (2006).

Intelligent PV modules have been proposed by Roman et al

(2006).

Two stage MPPT scheme for partially shaded SPVA has been

proposed by Kobayashi et al (2006).

Behaviour of SPVA under partial shaded conditions has been

presented by Xiao et al (2007).

An analytical model, based on Lambert function and its

properties, has been presented by Petrone et al (2007).

Patel and Agarwal (2008 b) has developed a GMPP tracking

scheme with P&O as a subroutine.

In this chapter the modifications are made in Fibonacci method so

as to track GMPP always. The results are checked against binary search

method. The results are compared and validated.

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5.3 IMPORTANT OBSERVATIONS OF SPVA UNDER

PARTIAL SHADED CONDITIONS

The partial shade has more impact on series connected modules.

The multiple peaks are mainly introduced by series connection in any

configuration. The simulation of series connected SPVA array characteristics

under partial shaded condition is shown in Figure 5.1 using the generalized

program discussed in Chapter 4.

Figure 5.1 Characteristics of SPVA under three different shading

patterns A, B and C

It may be concluded from the simulation results that:

V-I curves under partially shaded conditions have multiple

steps, while the V-P curves have multiple peaks. Maximum

number of peaks equals to the number of shading patterns.

In addition to irradiance and temperature, the magnitude of

GMPP, and the voltage at which it occurs are also dependent

on the shading pattern and array configuration.

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The GMPP may lie on the left side of the load line (Rmp) too.

That is Rmp > Ractual (Ractual = Vactual/Iactual). So, the two stage

technique will not be able to track this point.

5.4 MODIFIED FIBONACCI SEARCH ALGORITHM

The Fibonacci search technique is a method of searching a sorted

array using a divide and conquer algorithm that narrows down possible

locations with the help of Fibonacci numbers. It continuously narrows down

the range with the optimal point always lying within that range. The shifting

can be either to the left or to the right. The direction of shifting is based on the

values of that function at two check points, V1 and V2. It is a mathematical

search. The Fibonacci search is based on the sequence of Fibonacci numbers

defined by the Equations (5.1) to (5.2)

1F,0F 10 (5.1)

2n1nn FFF for n = 2,3, … (5.2)

Hence the Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21…

Consider the points x1, x2, x3 and x4 as shown in Figure 5.2. x3 and

x4 are the extremities. The range is given by Equation (5.3)

iiii Aaba (5.3)

The two check points are x1 and x2. Since the value of the function

at x2 is greater than that at x1, the range is shifted to the right. x1 now becomes

x3. x2 becomes x1. A new value of x2 is generated using the Fibonacci search

formula as given in Equation (5.4)

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3432 xx)n(F

)1n(Fxx (5.4)

where F (n) is the nth

Fibonacci number. From Equation (5.5) and Figure 5.2,

it is seen that the range is narrowed.

Figure 5.2 Fibonacci Search algorithm

iiii1i1i1i aAbaaba (5.5)

Now in the new range the value of the function at x1 is greater than

that at x2. Hence the range shifts to the left. x2 becomes x4 , x1 becomes x2.

The new value of x1 is given by the formula as given by Equation (5.6).

3441 xx)n(F

)1n(Fxx (5.6)

Here xi and F (xi) represent voltage and power respectively.

Initially two voltage values are generated. The power values at these voltages

are measured. They are compared and accordingly the range is shifted either

to the left or to the right. Another voltage value is generated and the process is

continued till the MPP is reached. The number of iterations and the voltage

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values are decided with the help of Fibonacci numbers. Hence the names

Fibonacci search. The flowchart for Fibonacci search based GMPP tracking is

shown in Figure 5.3.

Algorithm

Step 1 : Initialize the variables.

Step 2 : Measure power.

Step 3 : If the power has changed from the previous settled

(previous MPP) and if the power difference is greater

than the power ripple, then sudden irradiance change

is said to have occurred and hence goto step 4 else

goto step 3.

Step 4 : Two voltage values V1 and V2 are generated.

Step 5 : The power values P1 and P2 are measured.

Step 6 : If (P1>P2) then shift left else shift right.

Step 7 : A voltage value either new V1 or new V2 is generated

accordingly.

Step 8 : Power value is accordingly measured. The other

voltage remains the same.

Step 9 : Go to step 6

The original method proposed in literature (Miyatake et al 2004)

does not guarantee the GMPPT tracking under all positions of the initial

operating point. The proposed method can find GMPPT by doing wide range

search irrespective of the initial position of operating point. That is,

irrespective of the Global peak's position in the P-V characteristics, the

Fibonacci search algorithm tracks the Global peak thus making the solar panel

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to operate at the voltage where it delivers the maximum power. Figure 5.4

indicates the GMPP tracking of Fibonacci search algorithm when it occurs at

different locations. M-file code for Improved Fibonacci search method is

given in Appendix A3.1.

Figure 5.3 Flow chart for Fibonacci Search algorithm

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Figure 5.4 P-V characteristics with tracked MPP for different cases

5.5 DESIGN AND IMPLEMENTATION OF MPPT WITH

PROPOSED ALGORITHM

MPPT can be realized using different converter topologies. In this

work, boost converter is used due to its high efficiency. The MPPT design is

explained below. The DC voltage transfer function is given by Equation (5.9)

D1

1

V

V

s

o (5.9)

where Vo = Output voltage, Vs = Source voltage, D = Duty cycle

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The critical values of the inductance and capacitance can be

calculated using the Equation (5.10) and Equation (5.11).

f2

DRD1L

2

b (5.10)

fR2

DCmin (5.11)

f = frequency 10 kHz. The inductance and capacitance are calculated as

L=100uH, C=220uF. The value of resistance used is 100

Fibonacci search algorithm implementation is done in embedded

MATLAB function block available in MATLAB. Most of the cases the

requirement of load voltage is greater than source voltage, because long series

strings in SPVA may result in more power loss under partial shaded

conditions. Hence the boost converter is used to match the source impedance

with the load impedance thus delivering maximum power to the load. Figure

5.5 shows the block diagram for implementing the improved Fibonacci search

algorithm with boost converter. The embedded function block provides the

reference voltage, which is in turn compared with the actual voltage at which

the solar panel is operating. The error signal is fed to the PI controller block

and the resultant signal is finally compared with the triangular wave to

produce gating signal to the switch. MATLAB-SIMULINK model for the

system is shown in Appendix A3.2. PI controller is tuned using Zeigler-

Nichol first method. The value of Kp is 0.005 and Ki is 20.

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Figure 5.5 Block diagram of Fibonacci algorithm based MPPT method

The simulation results shown in Figure 5.6 are obtained when both

the panels receive uniform irradiance of 1000 W/m2 up to a time of 0.3 sec.

After 0.3 sec the input to the second panel is suddenly changed to 500 W/m2

whereas the first panel receives the same irradiance of 1000 W/m2 thus

indicating partial shaded condition. Step input is applied to simulate the

sudden irradiance changes. The results shown in Figure 5.7 are obtained when

both the panels receive uniform irradiance of 200 W/m2 up to a time of 0.3

sec after that the input to the second panel is changed to 800 W/m2

whereas

the first panel receives the same irradiance of 200 W/m2 thus indicating

partial shaded condition.

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Figure 5.6 Input and output characteristics of the Boost converter

(Figure 5.5). Characteristics (a) – (f) show input current,

output current, input voltage, output voltage, input power

and output power of the converter when solar insolation

suddenly changes from 1000W/m2

to 500W/m2

after 0.3 sec

for the second panel only

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Figure 5.7 Input and output characteristics of the Boost converter

(Figure 5.5). Characteristics (a) – (f) show input current,

output current, input voltage, output voltage, input power

and output power of the converter when solar insolation

suddenly changes from 200W/m2

to 800W/m2

after 0.3 sec

for the second panel only

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5.6 HARDWARE IMPLEMENTATION OF PROPOSED

ALGORITHM FOR GMPP TRACKING

The schematic diagram of the proposed system is shown in Figure

5.8. In Direct Coupled system the solar panel is directly connected across the

load. The maximum power is not transferred to the load because the load and

the source resistances do not match. DC-DC converter (step up/step down)

serves the purpose of transferring maximum power from the SPV module to

the load. The DC-DC converter acts as an interface between the SPV module

and the load. n order to move the operating point of the solar panel to the

MPP, a closed loop system must be implemented to sense the voltage and

current. After sensing the two parameters, an algorithm must be implemented

to generate the error signal. The error signal in digital form is given to the

DAC (0808) which converts it to the corresponding analog signal. This signal

is then compared with a high frequency triangular wave of 10 kHz. The pulse

generated is given to the gate of the power semi conductor device (MOSFET-

IRF 460), thereby changing the duty cycle of the converter. This generated

pulse must be able to trigger the MOSFET of the power circuit. Thus the

source impedance is matched with the load impedance and maximum power

is transferred. The hardware set up of the proposed system is shown in Figure

5.9. The hardware results are shown in Figure 5.10 without shading, that is,

two panels in series receive same illumination. The same for shaded

conditions are shown in Figure 5.11. In Figure 5.10 and Figure 5.11, the

waveforms are traced using single phase clamp on power quality analyzer to

measure the input and output powers directly. In this display, the values in

watts and VA are the rounded-off values by the meter. Figure 5.12 shows the

pulses produced by PIC microcontroller before and after shading are

introduced.

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Figure 5.8 Hardware Schematic of the proposed MPPT system

Figure 5.9 Hardware set-up of the MPPT system

SPV Panel boardStep down circuit for PIC controller

PIC controllerDAC,

Comparator,Op-amp, optocoupler Boost

Regulator

Power supplycircuit

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Figure 5.10 I-V characteristics of two panels connected in series without

shading

Input

Output

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Figure 5.11 I-V characteristics of two panels connected in series with

partial shading

Input

Output

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Figure 5.12 CRO screens showing gate pulse and output voltage before

and after tracking

The results for other shading patterns are tabulated in Table 5.1.

The values in % indicate the % of ripple content. From the tabular column it

can observed that the ripple percentage in most of the cases lies below 1%

Table 5.1 Results for different shading patterns

G1

(W/m2)

G2

(W/m2)

VSPV

(V)

ISPV

(A)

PSPV

(W)

Vo

(V)

Io

(A)

Po

(W)

1000 80035

(0.8%)

1.76

(0.8%)

62.1

(0.7%)

77.5

(0.4%)

0.78

(0.8%)

60.1

(0.9%)

1000 50035

(0.9%)

1.12

(0.7%)

39.25

(0.5%)

61.6

(0.3%)

0.62

(0.6%)

38

(0.8%)

800 20017.48

(2.2%)

1.43

(0.9%)

24.92

(0.2%)

49.19

(0.4%)

0.49

(0.2%)

24.2

(0.8%)

500 50033.28

(0.9%)

1.03

(0.4%)

34.28

(0.4%)

57.58

(0.3%)

0.58

(0.3%)

33.15

(0.8%)

1000 30017.84

(2.8%)

1.72

(0.1%)

32.5

(0.3%)

54.98

(0.4%)

0.6

(0.1%)

30.12

(0.1%)

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5.7 CONCLUSION

Since the efficiency of the solar panel is only 13%, it is necessary

to operate it at its maximum power point. If the solar panel operates at a point

other than the point where it delivers maximum power, the power transferred

to the load can be as low as 70% of the maximum value. Hence it is necessary

to the track the GMPP under partial shaded conditions. In this chapter, the

Fibonacci search algorithm has been discussed to track the GMPP under

partial shaded conditions. It is proved that the Fibonacci search algorithm

tracks the GMPP when multiple peaks exist in the P-V characteristics using

MATLAB software and hardware. The GMPP tracking algorithm can be

further improved by using intelligent optimization techniques. These

improvements are discussed in Chapter 6 and Chapter 7.