 Open Access
 Authors : Kritika Rana
 Paper ID : IJERTV9IS060955
 Volume & Issue : Volume 09, Issue 06 (June 2020)
 Published (First Online): 04072020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Mathematical Analysis of ThreeDiode Model with P&O MPPT using MATLAB/Simulink
Kritika Rana
Student, Dept. of ECE,
Indira Gandhi Delhi Technical University for Women, New Delhi, India
Abstract: With the growing need for renewable energy resources, photovoltaic based energy generation has gained significant importance due to its high reliability owing to the abundant availability of sunlight and the direct conversion of light into electricity. But the overall efficiency of conversion is still low as compared to cost input in the system. Hence for designing highly efficient systems, successful simulation studies are required. The research paper insights the mathematical modeling of a three diode model and its performance are analyzed in comparison with a single diode and twodiode model. The model's performance is also analyzed under different atmospheric conditions with MPPT.
Keywords Renewable energy, Photovoltaic (PV), Three diode model, maximum power point tracking, boost converter, MATLAB

INTRODUCTION
Intending to lower the carbon emissions and reduce the dependency on nonrenewable energy resources there has been significant technological development for utilizing renewable energy resources, such as Wind, Solar, Hydro, and many more.
photovoltaic effect, electronhole pairs are generated when incident photons have energy higher than the bandgap energy of the material. This generation and recombination are responsible for electricity generation in the device [3]. In general, a PV module is modeled based on different series and parallel combinations of solar cells concerning the power requirement. With a series connection, the voltage profile of the module is improved and with parallel the current profile. The output from the PV module is dependent on operating temperature, irradiance, angle of irradiance, and resistance (series and parallel).

Single Diode Model
In this model, a single diode is connected in parallel with a current source, a parallel resistance, and a series resistance. The output current equation is:
=
where,
= [ + ()] (/1000)
+
But for a long time, the focus has been on solarbased energy generation mainly due to high availability and easy conversion
= [exp (
+
)]
of light to electricity. PV energy conversion is static and does not require any rotating parts in the system for conversion. The
=
Where I
PV is the photon current generated due to incident light,
system can be designed with basic semiconductor devices including diodes and transistors. But even with these benefits, the solarbased generation suffers from one major drawback, i.e., low efficiency in comparison to high cost for setting up the system. Hence various models have been presented for solar cell modeling including single diode, twodiode, and three diode model. The paper covers the performance analysis of all three models for varying atmospheric conditions. There is non linearity in the IV and PV characteristics during the operation, due to varying atmospheric conditions [1]. To maximize the output from the PV module it is operated at the maximum power point, yielding maximum output concerning atmospheric conditions. The most used method for maximum power point tracking is the PerturbandObserve method due to its high efficiency and easy employability. The MPPT is used to control the duty cycle for the switching element of the boost converter under varying conditions. The main reason for using the boost converter is to avoid using transformers, to reduce the overall power loss of the circuit and limitation of switching frequency associated with it [2].


PV MODULE
A photovoltaic cell is a PN junction diode that is responsible for generating electricity when exposed to sunlight. This generation is mainly due to the recombination of electronhole pairs, which are generated due to the photovoltaic effect. In the
Io is the reverse saturation current of the diode, VT is the thermal voltage of diode and is the diode ideality factor. The other parameters Rs (series resistance) and RP (parallel resistance) are adjusted as per the requirement. The single diode equivalent model is given in Figure1a.
Fig 1a: Single Diode PV model

Two diode model
In the two diode model, there are two diodes (both in saturation), series resistance, parallel resistance, and current source. The mathematical model for a twodiode model of a solar cell is given as,
= 1 2
where,
= [ + ()] (/1000)
+
1 = 1 [exp ( )]
1 1
+
2 = 2 [exp ( )]
=
+
2 2
Where IPV is the photon current generated due to incident light, Io1 and Io2 are the reverse saturation current of the diodes (which in general are kept equal), VT1 and VT2 are the thermal voltages of the diodes and 1 and 2 are the ideality factors for the diodes. The twodiode equivalent model is given in Figure1b and its extraction parameters are taken from the article [1].
Fig 1b: TwoDiode PV model
It has already been established that the performance of a two diode model is similar to single diode model provided, the ideality factors are set such that 1 is unity and 2 is between 1 2 1.2, hence the performance analysis done here is for twodiode model and threediode model

Threediode model
In the three diode model, there are three diodes (both in saturation), series resistance, parallel resistance, and current source. The mathematical model for a threediode model of a solar cell is given as,
= 1 2 3
Where,
= [ + ()] (/1000)
+
Fig 1c: ThreeDiode PV model
Table1: ThreeDiode model parameters
Parameter
Value
Open circuit voltage (Voc)
21.1 V
Short Circuit Current (Isc) Series Resistance (Rs)
3.8 A 0.205
Shunt Resistance (R)
578.38
ki
0.065
kv
0.08
Io1
4.98e08 A
Io2
7.24e10 A
Io3
1.42e07 A
1
1.282
2
1.8043
3
1.4364


PERFORMANCE OF DIFFERENT DIODE MODELS
The performance analysis for twodiode model and threediode model is given in Figures 2a, 2b, 2c, and 2d.
1 = 1 [exp ( )]
1 1
+
2 = 2 [exp ( )]
2 2
+
3 = 3 [exp ( )]
3 3
Figure 2a: IV characteristics at Irradiance of 500 W/m2
=
+
Where IPV is the photon current generated due to incident light, Io1, Io2, and Io3 are the reverse saturation current of the diodes (which in general are kept equal), VT1, VT2 and VT3 are the thermal voltages of the diodes and 1, 2 and 3 are the ideality factors for the diodes.
The threediode equivalent model is given in Figure1c and its extraction parameters are taken from the article [4], given in Table 1
Figure 2b: IV characteristics at Irradiance of 1000 W/m2
Figure 2c: PV characteristics at Irradiance of 500 W/m2Figure 2d: PV characteristics at Irradiance of 1000 W/m2
It is evident that with an increase in the number of diodes in the circuit the opencircuit voltage decreases, provided the ideality factors are maintained constant [5]. But when the ideality factor of diodes is varied then the results obtained are given in Figure 3, and the values for the respective ideality factor are given in table 2.
Table2: Variation of open circuit voltage with varying ideality factors of diode
Case
1
2
3
Open circuit Voltage (V)
1.
1.282
1.282
1.282
20.4
2.
1.282
1.282
1.382
21.32
3.
1.282
1.382
1.382
21.33
4.
1.282
1.382
1.484
21.75
Figure3: IV characteristic variation with varying ideality factor.

P&O MPPT
Due to varying atmospheric conditions, there is a nonlinearity in PV characteristics hence distinct maximum power points, so there is a need to maximize the power output from the PV module during these varying conditions to increase the efficiency [6]. Hence, we use maximum power point tracking
along with PV modules to get maximum power for current atmospheric conditions. There are many algorithms for maximum power point tracking, but the most preferred algorithm is Perturb and Observe algorithm. The algorithm is represented via the flow chart in figure 4. The algorithm is easy to implement and requires a low number of parameters to function. The major drawback of the algorithm is the oscillations around the maximum power point, for which different controllers are used with this.
Figure 4: P&O Algorithm

BOOST CONVERTER
The DCDC converter is used to regulate the output voltage with the help of highfrequency switching, inductors, and capacitor with respect to the unregulated input voltage. The output voltage is varied based on the duty cycle of the switching element and other devices are used to minimize signal noise that occurs in the output due to the presence of nonlinear devices. The converters can either scale up the voltage (boost converter) or scale down the voltage (buck converter). In the paper, a boost converter is used to scale up the voltage from the PV module to support the load. The switching element used is an IGBT due to its higher efficiency compared to MOSFET and BJT. The Simulink model of boost converter used in the paper is given in figure 5
Figure 5: Boost converter
The duty cycle of IGBT is controlled via MPPT, to regulate the duty cycle under varying atmospheric conditions for maximum output. The switching frequency of IGBT is kept such that to avoid high switching losses [7]. Ideal boost converter equation is given as:
= 1
8. REFERENCE

Basha, C. H., Rani, C., Brisilla, R. M., & Odofin, S. (2020). Mathematical Design and Analysis of Photovoltaic Cell Using MATLAB/Simulink. In Soft Computing for Problem Solving (pp. 711726). Springer, Singapore.

RosasCaro, J. C., Ramirez, J. M., Peng, F. Z., & Valderrabano, A. (2010). A DCDC multilevel boost converter. IET Power Electronics, 3(1), 129137
1

Babu, B. C., Cermak, T., Gurjar, S., Leonowicz, Z. M., & Piegari,
L. (2015, June). Analysis of mathematical modeling of PV module
where Vout is the output voltage and Vin is the input DC voltage. D is the duty cycle for the pulses given to the switching element. With advancing technology multilevel DCDC converters are used with inverters in PV based generation systems


PERFORMANCE WITH MPPT
This section covers the performance analysis of the twodiode model and threediode model with P&O based MPPT. The parameters for the module are used as in [5]. All the analysis is done using a mathematical model of the module in MATLAB/Simulink.
Figure 6: Output voltage variation with MPPT

CONCLUSION
It is evident from the analysis that if diodes are increased in the photovoltaic module, the effective output voltage will decrease. But if the ideality factor of the diodes is optimized the performance of the circuit can be improved. Further research can be done to provide an algorithm that can optimize the ideality factor ratio of the diodes for optimum performance, and use of different controllers with MPPT circuit to prevent oscillations and provide a smooth curve till steady state is achieved.
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