International Journal of Soft Computing and Engineering


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 30 No. 4 (2023) 

 

 

101 
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Mathematical Modelling Non Linearity Characteristic Analysis and 

Efficiency Enhancement Strategies for Hybrid Solar Photovoltaic 

Energy System 

 

Naval Kishor Jain1, Shobhit Srivastava2 
1 PhD Scholar, Department of Mechanical Engineering, Maharishi University of Information Technology, Lucknow, 

India   
2 Associate Professor, Department of Mechanical Engineering, Maharishi University of Information Technology, 

Lucknow, India  Email: nvl.jain@gmail.com 

 

Article History: 

Received: 26-08-2023 

Revised: 01-10-2023 

Accepted: 30-10-2023 

Abstract:  

As the world continues to grapple with the challenges posed by climate change and the 

depletion of conventional energy sources, renewable energy systems such as solar 

photovoltaic (PV) technology have gained significant prominence. Hybrid solar PV 

systems, which combine multiple energy sources and storage solutions, offer a promising 

avenue to improve the reliability and efficiency of renewable energy generation. This 

research focuses on the mathematical modeling, non-linearity characteristic analysis, and 

the development of efficiency enhancement strategies for hybrid solar PV energy systems. 

The study begins with the development of comprehensive mathematical models that 

accurately represent the complex interactions within hybrid solar PV systems. These models 

consider various factors, including solar irradiance, temperature variations, load profiles, 

and the dynamic behavior of energy storage components. The incorporation of non-linear 

characteristics, often overlooked in conventional models, allows for a more realistic 

representation of system performance. To enhance the efficiency of hybrid solar PV 

systems, a range of strategies are proposed and evaluated. These strategies encompass 

advanced control algorithms, optimized sizing and placement of energy storage elements, 

and the integration of emerging technologies such as artificial intelligence and machine 

learning. The goal is to mitigate non-linear effects, maximize energy utilization, and 

improve system response to dynamic operating conditions. 

The findings from this research provide valuable insights into the design, operation, and 

performance optimization of hybrid solar PV energy systems. By addressing non-linearity 

characteristics and developing efficient strategies, this study contributes to the advancement 

of renewable energy technologies and fosters the transition towards a more sustainable and 

resilient energy future.  

Keywords: Solar Photovoltaic System, Maximum Power Point Tracking, Soft Computing 

Techniques, Meta Heuristic Search, Particle Swarm Optimization, Cuckoo Search 

Optimization, PV-T System,  Adaptive Neuro Fuzzy Inference System. 

1. INTRODUCTION 

A Solar panels or solar cells collect solar energy and turn the light energy into electricity. PV cells, 

sometimes referred to as "solar cells," employ the photovoltaic effect to store the solar energy that 

results in a current flowing between two oppositely charged layers. The solar cell's conversion 

efficiency is calculated by dividing the ratio of solar energy (irradiation) that strikes on the area of 

solar cell to the electrical energy output of the cell. The semiconductor material used in solar cells has 



Communications on Applied Nonlinear Analysis 

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been doped to create p-n junction which absorbs the sunlight and with the help of photovoltaic effect 

it and transform it into direct current.  To represent the behavior of solar cell a single-diode model is 

commonly used because it is straightforward and precisely represents the characteristics of p-n 

junction. [1] 

 

Figure 1.1: Equivalent Circuit of a Single Diode Solar Cell 

The solar cells are connected in series or parallel to create the proper voltage and current. When the 

cells are connected in series, a significant output voltage is created; however, when the cells are 

connected in parallel, a huge output current is generated. The mathematical analogous circuit 

architecture for a single-diode PV panel is shown in Figure 1.1 and consists of parallel and series 

resistances, a current source, and a diode. The mathematical model is important to understand and 

calculate the I-V and P-V characteristics of the PV cell under various operational conditions. [2] 

The mathematical modeling is also important to understand the dynamic performance of solar 

photovoltaic system under different operational conditions. It shall be noted that under situations of 

constant irradiance and varying temperature or vice versa the I-V and P-V characteristics changes and 

the performance of solar cell is hence influenced by the operational conditions.                        

Mathematical Modelling of Solar Cell 

In Figure 3.1, the PV equivalent circuit is shown. It shall be noted that a potential difference is 

produced when light strikes photovoltaic cells, and this voltage varies linearly with solar insolation. It 

is feasible to model the ideal solar cell as a current source. Current leakage proportional to solar cell 

terminal voltage is provided by shunt resistance (Rp). Series resistance is used to depict the losses due 

to semiconductor and metal contacts (Rs). Parallel diodes are used to simulate the p-n junctions of PV 

cells in order to calculate the current generated by light impinging on a PV cell. The solar cell behavior 

is provided by the equation given below. The I-V relationship of the PV system defines the modeling 

of the PV cell as follows:[22] 

𝐼 = 𝐼𝑝𝑣 − 𝐼𝑆 (exp [
𝑞(𝑉 + 𝑅𝑠𝐼)

𝑁𝑠𝑘𝑇𝑎
] − 1) −

𝑉 + 𝑅𝑠𝐼

𝑅𝑝

(1.1)

𝐼𝑝𝑣 = (𝐼𝑝𝑣,𝑛 + 𝐾𝐼Δ𝑇)
𝐺

𝐺𝑛
                                             (1.2)

𝐼𝑆 =
𝐼𝑆𝐶,𝑛 + 𝐾𝐼Δ𝑇

ex p(𝑉𝑂𝐶,𝑛 + 𝐾𝑉Δ𝑇) /𝑎(𝑁𝑠𝑘𝑇/𝑞) − 1
            (1.3)

 



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 𝐼𝑃𝑉 = 𝐼𝑃ℎ − 𝐼𝐷 − 𝐼𝑃                                            (1.4) 

𝐼𝑝 =
(𝑉𝑝+𝑅𝑠𝐼)

𝑅𝑝
                                    (1.5) 

𝐼𝑝 - photo current, 𝐼𝐷 - Diode current 

𝐼𝐷 = 𝐼0 [ex p
((𝑉𝑝+𝑅𝑠𝐼)

(𝑛𝐾𝐵𝑇))−1
]                                                     (1.6) 

Now substituting the value of 𝐼𝐷 

𝐼𝑃𝑉 = 𝐼𝑃ℎ − 𝐼0 [ex p
(𝑉𝑝+𝑅𝑠𝐼)

(𝑛𝐾𝐵𝑇)−1
] − 𝐼𝑃                                  (1.7) 

𝐼𝑃𝑉 = 𝐼𝑃ℎ − 𝐼0 [exp
(𝑉𝑝 + 𝑅𝑠𝐼)

(𝑛𝐾𝐵𝑇)
− 1] −

(𝑉𝑝 + 𝑅𝑠𝐼)

𝑅𝑝

(1.8)
 

𝑉𝑇 =
(𝐾𝐵𝑇𝑐)

𝑞𝑒
                                                                     (1.9) 

𝐼𝑎 =
𝑛𝑠𝐴𝑓𝐾𝐵𝑇𝑐

𝑞
= 𝑛𝐴𝑉𝑇                                                    (1.10) 

𝐼𝑃𝑉 = 𝐼𝑃ℎ ref − 𝐼0 ref [exp (
(𝑉𝑝)

𝑎ref 
) − 1]                                (1.11) 

𝐼𝑠𝑐.𝑟𝑒𝑓 = 𝐼Ph.ref − 𝐼0.𝑟𝑒𝑓 [ex p (
(0)

𝑎𝑟𝑒𝑓
) − 1] = 𝐼Phref           (1.12) 

The connection between irradiance, temperature, and the photocurrent is given by  

𝐼𝑝𝑣 =
𝐺

𝐺ℝ𝕖𝕗
(𝐼𝑝ℎ.𝑟𝑒𝑓 + 𝜇𝑠𝑐 ⋅ Δ𝑇)                                        (1.13) 

 Where, 

G-Irradiance𝑊/𝑚2 

𝐺ref  - Irradiance at STC (1000 W/m2) 

ΔT = 𝑇𝑐 − 𝑇𝑐,𝑟𝑒𝑓                                                                                (1.14) 

𝜇𝑠𝑐-Coefficient temperature of Short circuit and  𝐼0 is given by the  

I0 = Iscex p (
−Voc.rff

a
) (

Tc

Tc.ref
)

3

Xex p [(
q∈G

A⋅K
) (

1

 Tc.ref
−

1

 Tc
)]                     (1.15) 

 

The mathematical model can be employed to derive the characteristic of solar photovoltaic cell, 

modules and arrays/ The given topology can be used for implementation of partial shading and variable 

irradiation characteristic analysis. The characteristics has been analyzed and presented with help of 

simulation of solar pane. The paper has been divided into five segments. The first segment is related 

to introductory concept of solar photovoltaic modules. Next section related to survey of research 

papers. Introduction of maximum power point system has been discussed in next section which has 



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been followed by proposed methodology and its implementation. Simulation and results has been 

shown in next part of the paper. 

2. RELATED WORKS 

Tanuj Sen et al. (2018), in their work have proposed modified PSO algorithm owing to the 

incapacity of traditional MPPT techniques to track global maximum power of the PV characteristics 

having multiple peaks and validations through simulations. The authors have also pointed the 

advantage of reduced steady state oscillations [1]. 

G. Dileep et al. (2017), in their research have penned about the adaptive PSO algorithm so as to 

obtain improved overall speed and competency of the system and for the same they utilized two unalike 

shading conditions for the validation of proposed approach. The results thus obtained clearly proves 

that the discussed approach can obtain the global point of maximum power point in all the cases [2]. 

R.Nagarajan et al. (2018) has described the method to enlarge  the output value of voltage from 

PV system by making use of maximum power point tracking approach namely Particle Swarm 

Optimization .In their work they they have used PI controller in addition to PSO for boost convertor 

to convert DC to DC voltage[3]. 

Kashif Ishaque et al. (2012) have depicted a modified Particle Swarm optimization for improved 

version of maximum power point tracking. This paper also suggests that the proposed method can be 

used to tracking of power even in the altering environmental conditions with the advantage of reduced 

oscillations in steady state after the MPP is located [4]. 

Faiza  Belhachat et al. (2018) have given a review on the techniques of maximum power point 

tracking  ranging from old age lesser used techniques to modern times advanced ones so that users can 

make a good choice based on the performances of each method while making any system[5] 

Ali M. Eltamaly et al. (2020)  have discussed  certain problems of PSO including long convergence 

time by already updating the initial values of the duty ratio of converter. In addition to this, they also 

gave comparisons between the ability of different methods to find GP under dynamic shading 

conditions [6]. 

Makbul A. et al. (2017) have given a description of various maximum power point techniques of 

the PV systems during normal climatic condition and partial shading conditions. The authors have 

mainly focused on partial shading conditions since the last decade owing to increased requirement of 

output [7]. 

Zhu Liying et al. (2017)  have  penned about all maximum power point tracking technique used in 

extracting peak power output during varying shading conditions and the limitations of conventional 

methods over particle swarm optimization methods[8]. 

Rozana Alik et al. (2017), in their research work had demonstrated the unwanted impacts of partial 

shading in PV system and an improved perturb and observe algorithm. The authors have quoted about 

the various merits of this method like lower cost, simplicity and accuracy. The proposed method is 

advancement over conventional P and O method which makes system unstable. The author discussed 

all related flowcharts and algorithms [9]. 



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Mingxuan Mao et al. (2017) proposed a novel method of tracking maximum power point 

simultaneously reducing steady state oscillations. The methodology being incorporated in the paper 

ensured faster and more accurate searching of global maximum which in turn explains its superiority 

over conventional methods [10]. 

Gomathi B et al. (2016) proposed an incremental conductance algorithm based Solar Maximum 

Power Point Tracking System makes vivid illustrations about the incremental conductance technique. 

The paper describes about its steady state accuracy and higher efficiency. The authors have 

systematically modelled the PV module and solar radiation using basic equations, flowchart of the 

algorithm, DC-DC Converter of all the three types and their comparison. The results proved that Boost 

and Cuk converter provides the best results and have lower ripples [11]. 

Mr. M. Rupesh et al. (2018) presented a detailed investigation of the two MPPT approaches 

namely P and O and incremental conductance. Here, in this work fundamental quantities like current 

and voltage are being followed to mimic the described algorithms. The paper also discusses the 

modelling of PV cell, I-V and P-V graphs of the solar array being obtained at different irradiations.The 

complete setup of PV system as shown includes boost converter and MPPT controller. The voltage 

profile of boost converter with both the algorithms is also given [12]. 

S. Manna et al. (2021) have thrown light on the drift free perturb and observe method which 

incorporates current in addition to voltage and power as used in conventionally used perturb and 

observe. The drift algorithm is named so because it efficiently solves the problem of drift which is 

caused due to the altering environmental conditions mainly sudden to change in insulation levels 

during cloudy days and hence the authors have performed a test of both the algorithms for variation in 

insulation level and proved the percentage increase in power during the time of drift in the method 

discussed which improves its efficiency and accuracy levels [13]. 

Saad Motahhir et al. (2018) extracted the parameters for modelling the PV panel and further 

explained the consequences of temperature and radiation on PV array. The authors have also explained 

why the conventional theorems behave inaccurately when temperature and radiations are increased 

and hence for the solution of the same, they have discussed a modified incremental conductance 

theorem which can successfully reduce the steady state oscillations [22]. 

Abul kalam Azad et al. (2016) discussed about the application of perturb and observe and 

incremental conductance in the PV system. Here, output has been directly connected to the grid so as 

to make it run like a solar generator on cloudy days. The simulation results from both the algorithms 

are compared under same conditions and the author concluded that the P & O is not very effective 

under varying atmospheric conditions while the later works accurately [23]. 

Afshan Ilyas et al. (2017) have proposed a detailed explanation of incremental conductance 

algorithm. Here, the PV module has been integrated with dc-dc converter. The paper also includes 

modelling of solar PV cell. The authors used a real time reading of the parameters and concluded that 

incremental conductance has a higher tracking speed and accuracy [24]. 

Jubaer Ahmed et al. (2017) had given explanation about the difference between partial shading 

and uniform radiance. This work discusses the two renowned method namely particle swarm 



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optimization and perturb and observe and also evaluates their performance during partial shading and 

dynamic type shading conditions [25]. 

Ehtisham Lodhi et al. (2017) explained about the consequences of unalike shading on the 

photovoltaic system. This paper explains the presence of multiple peaks during the times of partial 

shading i.e when the sun is not coming constantly and there are clouds and hence gives explanation of 

the particle swarm optimization method in finding the global peak among the various peaks present 

during shaded conditions. The authors have given a detailed flowchart of the algorithm which explains 

the step by step procedure to use the algorithm in searching the maximum peak point during shading 

conditions of weather. The paper concludes that this algorithm has higher convergence rate and 

tracking efficiency than the conventionally used methods [26]. 

T. Diana et al. (2019)  had proposed one of the best known optimization technique so as to extract 

the maximum power from solar system i.e. particle swarm optimization. This method utilises an 

objective function.  The author has tested the algorithm under different conditions of temperature and 

radiation level so as to justify the efficiency level of its algorithm in contrast to the conventionally 

known tracking algorithms. The author have also presented various graphs to justify the work [27]. 

R Sridhar et al. (2017) shown the increment of output of PV system in case of variable 

environmental conditions. The paper discusses about the widely known particle swarm optimization 

and the simulations are carried out which speaks about the efficiency of this method. The author has 

explained the Particle swarm optimization method in detail. They have penned about the detailed 

analysis of characteristics and modelling of PV array using mathematical equations and the PV and IV 

curves are plotted to further explain the working of PV system. [28]. 

Nadia Hanis et al. (2016) in their work explained the need to popularise the renewable energy 

sources and their dependence on temperature and radiation. The mathematical modelling has been 

discussed in detail using solar PV equations of voltage and current. In addition to this, Particle based 

optimization based MPPT is also given. The flowchart of the above mentioned algorithm further makes 

the explanation easier by explaining the direction of the algorithm. The characteristics curves are 

obtained under varying temperature and irradiance conditions are also given to study the convergence 

of the theorems under different conditions of environment [29]. 

Ahmed Hossam El-din et al. (2017) compared the two widely known algorithms namely perturb 

and observe and PSO under uniform temperature conditions. The Particle swarm algorithm has a high 

level speed and can work under varying parameters of irradiance and temperature. The paper also 

shows the module performance of the system. The research is carried out on a 100 KW grid connected 

PV system under different conditions of environment [30]. 

Malik Sameeullah et al. (2016)  discussed about various MPPT schemes and their implementation. 

The authors compares the features, cost, control strategy of all the discussed methodologies so that one 

can easily opt for a good algorithm according to his area of research.  This paper includes al the 

techniques from current/ voltage feedback technique to modern day  hybrid MPPT technique which 

are useful in different types of environmental conditions from fine day to varying climatic 

conditions.[31]. 



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Arti Pandey et al. (2019) discussed the various techniques of MPPT in tabular form and the urgent 

need to move towards the renewable energy sources as the cost of fossil fuels is continuously rising 

and the emission level of carbon dioxide from non-renewable energy sources is quite high and 

dangerous for the environment and the life of humans..It gives a brief about all the majorly known 

MPPT methods and their merits and demerits. Furthermore, advantages and disadvantages of all the 

methods have been listed in points [32]. 

3. MATHEMATICAL MODELING OF SYSTEM 

PV arrays are created by connecting PV modules in a series-parallel configuration. The aggregate 

output of the PV array will be the same as the total power produced by all of the modules. As a result, 

even small adjustments to one PV module can affect the entire system and might result in issues with 

further PV modules. Sometimes situational, sometimes natural, shading is a phenomenon that cannot 

always be avoided. Figure 3.1 contains a symbolic description of the shading of solar photovoltaic 

panels. A PV array is made up of PV modules that are linked in parallel and series to provide the 

necessary voltage. It is critical to address this issue because under various lighting setups, modules 

have heat dependent losses influencing the power they generate under standard illumination. Shading 

has an impact on photovoltaic (PV) panels since they are made of crystalline silicon cells coupled to 

one another. 

1. 𝑆𝑜𝑙𝑎𝑟 𝑝𝑎𝑛𝑒𝑙 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛: 

[𝜂PV =
𝑃out

𝑃in

] 

2. 𝑆𝑜𝑙𝑎𝑟𝑐𝑒𝑙𝑙𝑐𝑢𝑟𝑟𝑒𝑛𝑡 − 𝑣𝑜𝑙𝑡𝑎𝑔𝑒𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠ℎ𝑖𝑝(𝑆ℎ𝑜𝑐𝑘𝑙𝑒𝑦𝑑𝑖𝑜𝑑𝑒𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛): 

[𝐼PV = 𝐼ph − 𝐼0 (𝑒
𝑉PV
𝑛𝑉𝑡 − 1)] 

3. 𝑆𝑜𝑙𝑎𝑟 𝑐𝑒𝑙𝑙 𝑝𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛: 

[𝑃out = 𝑉PV ⋅ 𝐼PV] 

4. 𝑀𝑎𝑥𝑖𝑚𝑢𝑚𝑝𝑜𝑤𝑒𝑟𝑝𝑜𝑖𝑛𝑡𝑡𝑟𝑎𝑐𝑘𝑖𝑛𝑔(𝑀𝑃𝑃𝑇)𝑎𝑙𝑔𝑜𝑟𝑖𝑡ℎ𝑚: 

[𝑃out = 𝑉MPPT ⋅ 𝐼MPPT] 

5. 𝑁𝑜𝑛 − 𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦 𝑖𝑛𝑑𝑒𝑥 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑠𝑜𝑙𝑎𝑟 𝑝𝑎𝑛𝑒𝑙: 

[𝑁𝐿 =
𝑃out,max − 𝑃out,min

𝑃out,max + 𝑃out,min

] 

6. 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓 𝑎 𝐷𝐶 − 𝐷𝐶 𝑏𝑜𝑜𝑠𝑡 𝑐𝑜𝑛𝑣𝑒𝑟𝑡𝑒𝑟: 

[𝜂boost =
𝑉out ⋅ 𝐼out

𝑉in ⋅ 𝐼in

] 

7. 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓 𝑎 𝐷𝐶 − 𝐷𝐶 𝑏𝑢𝑐𝑘 𝑐𝑜𝑛𝑣𝑒𝑟𝑡𝑒𝑟: 

[𝜂buck =
𝑉out ⋅ 𝐼out

𝑉in ⋅ 𝐼in

] 



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8. 𝐵𝑎𝑡𝑡𝑒𝑟𝑦 𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔 𝑎𝑛𝑑 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦: 

[𝜂batt =
𝐸out

𝐸in

] 

9. 𝐸𝑛𝑒𝑟𝑔𝑦 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑎 ℎ𝑦𝑏𝑟𝑖𝑑 𝑃𝑉 𝑠𝑦𝑠𝑡𝑒𝑚: 

[𝐸in = 𝐸PV + 𝐸grid + 𝐸batt] 

10. 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓 𝑎𝑛 𝑖𝑛𝑣𝑒𝑟𝑡𝑒𝑟: 

[𝜂inverter =
𝑃out,AC

𝑃in,DC

] 

11. 𝑃ℎ𝑜𝑡𝑜𝑣𝑜𝑙𝑡𝑎𝑖𝑐 𝑐𝑒𝑙𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛: 

[𝑇cell = 𝑇ambient +
𝐼PV ⋅ 𝑅cell

𝐴cell ⋅ ℎcell

] 

12. 𝐿𝑎𝑚𝑏𝑒𝑟𝑡𝑊𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛(𝑢𝑠𝑒𝑑𝑖𝑛𝑚𝑜𝑑𝑒𝑙𝑖𝑛𝑔𝑑𝑖𝑜𝑑𝑒𝑖𝑑𝑒𝑎𝑙𝑖𝑡𝑦𝑓𝑎𝑐𝑡𝑜𝑟, 𝑛): 

[𝑛 = −
𝑉𝑡

𝐼0
⋅ 𝑊 (−

𝐼0

𝐼PV

⋅ 𝑒
−

𝑉PV
𝑉𝑡 )] 

13. 𝑁𝑜𝑛 − 𝑙𝑖𝑛𝑒𝑎𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠ℎ𝑖𝑝 𝑓𝑜𝑟 𝑃𝑉 𝑐𝑒𝑙𝑙𝑠: 

[𝑉PV = 𝑉oc − 𝐼PV ⋅ 𝑅𝑠] 

14. 𝑆ℎ𝑜𝑟𝑡 − 𝑐𝑖𝑟𝑐𝑢𝑖𝑡 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑃𝑉 𝑐𝑒𝑙𝑙: 

[𝐼sc =
𝑉oc

𝑅𝑠
] 

15. 𝑂𝑝𝑒𝑛 − 𝑐𝑖𝑟𝑐𝑢𝑖𝑡 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑜𝑓 𝑎 𝑃𝑉 𝑐𝑒𝑙𝑙: 

[𝑉oc = 𝑛 ⋅ 𝑉𝑡 ⋅ ln (
𝐼ph

𝐼0
+ 1)] 

16. 𝐼𝑟𝑟𝑎𝑑𝑖𝑎𝑛𝑐𝑒 − 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛: 

[𝐼ph = 𝐺 ⋅ 𝐴cell ⋅ 𝐼sc,STC ⋅ (
𝑇cell

𝑇STC

)] 

17. 𝑆𝑜𝑙𝑎𝑟 𝑐𝑒𝑙𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡: 

[𝛼T =
𝐼sc,STC

𝐼ph,STC

⋅
𝑇STC − 𝑇cell

𝑇STC

] 

18. 𝐷𝑖𝑜𝑑𝑒 𝑖𝑑𝑒𝑎𝑙𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛: 

[𝑛𝑇 = 𝑛0 + 𝛼𝑛 ⋅ (𝑇cell − 𝑇STC)] 

 

 



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19. 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑚𝑒𝑛𝑡 𝑤𝑖𝑡ℎ 𝑏𝑖𝑓𝑎𝑐𝑖𝑎𝑙 𝑠𝑜𝑙𝑎𝑟 𝑝𝑎𝑛𝑒𝑙𝑠: 

[𝜂bifacial =
2 ⋅ 𝜂PV

1 +
ℎfront

ℎrear

] 

20. 𝑀𝑜𝑑𝑒𝑙𝑖𝑛𝑔 𝑡ℎ𝑒 𝑖𝑚𝑝𝑎𝑐𝑡 𝑜𝑓 𝑠ℎ𝑎𝑑𝑖𝑛𝑔 𝑜𝑛 𝑃𝑉 𝑎𝑟𝑟𝑎𝑦𝑠: 

[𝐼shaded = 𝐼unshaded ⋅ (1 −
𝐴shaded

𝐴total

)] 

21. 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑜𝑓𝑝𝑜𝑤𝑒𝑟(𝑃 − 𝑇𝑐𝑢𝑟𝑣𝑒): 

[
𝑑𝑃out

𝑑𝑇
= −

𝑃out

𝑇cell

] 

22. 𝐵𝑎𝑡𝑡𝑒𝑟𝑦 𝑐ℎ𝑎𝑟𝑔𝑒 − 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑚𝑜𝑑𝑒𝑙𝑖𝑛𝑔: 

[𝜂batt =
𝐸discharge

𝐸charge

] 

23. 𝐸𝑛𝑒𝑟𝑔𝑦 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑒𝑛𝑡𝑖𝑟𝑒 ℎ𝑦𝑏𝑟𝑖𝑑 𝑠𝑦𝑠𝑡𝑒𝑚: 

[𝐸in = 𝐸PV + 𝐸wind + 𝐸grid + 𝐸batt] 

24. 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑙𝑜𝑠𝑠𝑒𝑠 𝑖𝑛 𝑤𝑖𝑟𝑖𝑛𝑔 𝑎𝑛𝑑 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑜𝑛𝑠: 

[𝑃loss = 𝐼2 ⋅ 𝑅loss] 

25. 𝑆𝑜𝑙𝑎𝑟 𝑐𝑒𝑙𝑙 𝑓𝑖𝑙𝑙 𝑓𝑎𝑐𝑡𝑜𝑟 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛: 

[Fill Factor (FF) =
𝑉MPPT ⋅ 𝐼MPPT

𝑉oc ⋅ 𝐼sc

] 

 

Figure 3.1: Impact of Shading on Characteristics of PV System 

A conventional PV panel has solar cells linked in series to produce a high voltage, but all of the cells 

share the same current when they are connected in series. If the PV module or PV cells are shaded, 

they may be forced to operate in a reverse-biased zone and function as a load rather than a power 

supply. The panel can sustain permanent damage if the temperature of the cell rises considerably and 



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causes a thermal breakdown or second breakdown. The second breakdown phenomena occurs when 

the temperature of a reversely biased cell rises over a certain point, leading to a drop in the magnitude 

of the reverse voltage and an increase in the cell's current value. In this instance, the P-N junction 

temperature substantially rises, resulting in irreparable cell damage. [17] 

 

Figure 3.2 : Normal Operation of PV String 

Figure 3.2 depicts a photovoltaic string operating normally. It should be emphasized that each 

photovoltaic cell in a panel will produce the same amount of electrical power, or around 0.5 volts, 

provided that the quantity of sunlight reaching its surface remains constant. When the sun is shining 

strongly, a 2 watt PV cell, for instance, will provide a continuous current of roughly 4 amperes (0.5 x 

4 = 2 watts).However, if a cell is externally shadowed in any manner, it will cease producing electrical 

energy and begin functioning more like a semi conductive resistance, greatly limiting the total amount 

of energy the solar panel can produce. Let's use the example of three series-connected 0.5 volt 

photovoltaic cells that each get 1 kW/m2 of solar irradiation as our example. Due to the series 

connection of the three PV cells, the output current (I) generated will be the same. Given that the 

current is common and constant, the I-V characteristic curves of the three cells may be summed along 

the voltage (horizontal) axis, and the resultant total output voltage, VT, is equal to the sum of the 

individual cell voltages (V1 + V2 + V3 = 0.5V + 0.5V + 0.5V = 1.5V). If we were to use the 2 watt 

cell example from before, the maximum power point for this series string would be 6 watts (1.5V x 

4A = 6W). Let's now assume that Solar Cell No. 2 in the string is either entirely or partially shaded, 

although the other two cells in the series-connected string have not—i.e., they still receive full sun. 

The output of the string with a series connection will afterwards sharply decline, as demonstrated. In 

this scenario, the shaded cell ceases producing electrical energy and behaves more like a semi 

conductive resistance.  

As was already noted, if one of the PV cells is partially blocked by snow, leaves, or other debris, it 

will no longer be able to generate any electrical energy, as was seen above. Consequently, the bypass 

diode will take over and turn on as illustrated. 

In this situation, cell number two ceases generating electrical energy when it is shaded and starts to 

behave like the semi-conductive resistance we previously mentioned. As seen by the green arrows 

above, the shaded cell generates reverse power, which forward biases the parallel-connected bypass 

diode (i.e., turns it "ON") and directs current flow from the two healthy cells via itself. By giving the 



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produced current an electrical channel to follow, the bypass diode connected across the shaded cell 

keeps the other two PV cells running. Figure 3.4 illustrates how the bypass diode functions and 

improves the performance. 

 

Figure 3.3 :  Impact of Shading on Characteristics of PV System 

 

Figure 3.4 : Connection of Bypass Diode in PV System 

Cells 1 and 3 continue to create energy, albeit at a slower pace, despite the fact that one cell (cell 2 in 

this case) is shaded. As a consequence, the output would be 4 watts when utilizing the same 2 watt cell 

as in the previous example and assuming no losses through the bypass diode. 

When forward biased, or conducting, parallel linked bypass diodes have a forward voltage drop of 

roughly 0.6 volts, which restricts any excessive reverse negative voltage generated by the shaded cell 

and, as a result, reduces hot spot temperature conditions and prevents cell breakdown. When the 

shading is removed, this enables the cell to go back to its original state. 

It would be too expensive and challenging to install to include a bypass diode across each and every 

cell, as we have done in our simple example. In actuality, bypass diodes are often installed on the rear 

of PV cell groups or sub-strings (typically 16 to 24 cells), or in the junction boxes of solar modules. 

Charge controllers have an algorithm built in to get the maximum power out of PV modules. Peak 

voltage is the voltage where it produces the most power (Pmax). Temperature and sunlight insolation 

rate both affect maximum power. [4]. 

 Figure 3.5 indicates the significance of maximum power point tracking on the performance of solar 

photovoltaic system. MPPT compares the voltage, current, and battery out of the system. When it's 

chilly outside or there are clouds in the sky, MPPT is incredibly efficient and can get the most out of 



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a PV module. When a battery is deeply drained, MPPT technology can improve current flow and speed 

up battery recharge. The amount of battery input current from a PV module may be maximized using 

a charge controller integrated with the MPPT algorithm. The following are the primary attributes of 

an MPPT solar charge controller: 

• It fixes fluctuations in PV cell voltage and current characteristics brought by varying illumination 

conditions. 

• It enables the usage of voltage greater than the battery system's operational voltage and compels the 

PV module to generate electricity at its MPP. 

• It makes the system less complicated and makes it more effective. 

 

Figure 3.5: Significance of MPPT on Power Output of Solar PV System 

All around the world, there is a growing need for clean, renewable energy. Making effective and 

efficient PV systems is always urgently needed given the rising popularity of solar power. The process 

to select effectively specific voltage and current parameters are met, when the power is at its peak, so 

that the solar energy system's energy conversion rate reach a high level. Maximum Power Point is the 

name of this operational point (MPP). A PV panel's nonlinear power-voltage characteristic is 

influenced by both the temperature of the environment and the amount of sunlight received. When 

compared to sunshine irradiation, the temperature-related change in voltage and power is less 

substantial. 

The power output of a PV panel varies during the day since the amount of sunshine is not consistent. 

Additionally, the MPP changes when the amount of sunshine and the temperature of the atmosphere 

alter. In order to obtain the maximum power at any irradiance and temperature, MPP must be 

maintained. Maximum Power Point Tracking refers to keeping a PV panel's operating point at MPP 

regardless of temperature and irradiance (MPPT). Handling partial shadowing conditions is a 

significant issue with solar power generating systems. The sunlight's irradiance varies throughout the 

panel when there is partial shadowing. A significant number of PV panels are linked in series to 

generate the desired amount of electricity in a PV power production system. The PV panels are exposed 

to non-uniform irradiance when partially shaded, and in this case, the power-voltage characteristics 

show several power peaks. 

 



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1. Photovoltaic Cell Current-Voltage Relationship: 

𝐼 = 𝐼𝑝ℎ − 𝐼0 (𝑒
𝑣|𝑉+𝐼𝑅𝑥|

𝑛+𝑇 − 1) −
𝑉 + 𝐼𝑅𝑥

𝑅𝑥ℎ
 

where 𝐼 is the cell current, 𝐼𝑝ℎ is the photo-generated current, 𝐼0 is the dark saturation current, 𝑉 is the 

cell voltage, 𝑅𝑠  is the series resistance, 𝑅𝑠ℎ  is the shunt resistance, 𝑛 is the ideality factor, 𝑘 is the 

Boltzmann constant, 𝑇  is the cell temperature, and 𝑞  is the charge of an electron. 

2. Maximum Power Point (MPP) Tracking: 

𝑑𝑃

𝑑𝑉
= 0 

where 𝑃 is the power, and 𝑉 is the voltage. This condition is used to find the maximum power point. 

3. Fill Factor (FF): 

𝐹𝐹 =
𝑉𝑚𝑝𝐼𝑚𝑝

𝑉𝑠𝑐𝐼𝑥𝑐
 

where 𝑉𝑚𝑝 and 𝐼𝑚𝑝 are the voltage and current at the maximum power point, respectively, and 𝑉𝑂𝐶 and 

𝐼𝐵𝐶  are the open-circuit voltage and short-circuit current, respectively. 

4. Solar Cell Efficiency: 

𝜂 =
𝑃max

𝑃m
=

𝑉min𝐼min

𝐴 ⋅ 𝐺
 

5 Where 𝑃max is the maximum power output, 𝑃𝑖𝑛 is the input power, 𝐴 is the area of solar cell, 

and G is the irradiance. 

6 The Temperature Effect on Photovoltaic Efficiency: 

𝜂(𝑇) = 𝜂𝑆𝑇𝐶 − 𝛽(𝑇 − 𝑇𝑆𝑇𝐶) 

where 𝜂(𝑇)  is the efficiency at temperature 𝑇, 𝜂𝑆𝑇𝐶  is the efficiency at Standard Test 

Conditions (STC), 𝛽  is the temperature coefficient, and 𝑇𝑆𝑇𝐶  is the STC temperature. 

6. Irradiance Effect on Photocurrent: 

𝐼𝑝ℎ(𝐺) = 𝐼𝑝ℎ,𝑆𝑇𝐶

𝐺

𝐺𝑆𝑇𝐶
 

where 𝐼𝑝ℎ,𝑆𝑇𝐶 is the photocurrent at STC, 𝐺 is the actual irradiance, and 𝐺𝑆𝑇𝐶 is the irradiance at STC. 

7. Power Output of PV Module: 

𝑃out = 𝑁cell ⋅ 𝑉𝑚𝑝 ⋅ 𝐼𝑚𝑝 

where 𝑁cell  is the number of solar cells in the module. 

8. Hybrid System Efficiency: 

where 𝜂𝑃𝑉 and 𝜂𝑜𝑡ℎ𝑒𝑥 are the efficiencies of the PV system and the other energy system (e.g., wind, 

thermal) respectively, 𝑃𝑃𝑉 and 𝑃other  are their respective power outputs. 

 



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9. Battery Charge Equation: 

𝑄new = 𝑄old + 𝐼charge Δ𝑡 − 𝐼dincharge Δ𝑡 

where 𝑄new  and 𝑄𝑜𝑙𝑑  are the new and old charge states of the battery, 𝐼charge  and 𝐼discharge  are the 

charging and discharging currents, and Δ𝑡 is the time interval. 

10 Energy Stored in Battery: 

𝐸 = 𝑄 ⋅ 𝑉lot  

where 𝐸  is the energy, 𝑄  is the charge, and 𝑉𝑏𝑎𝑡  is the battery voltage. 

11. Overall System Energy Balance: 

𝐸in = 𝐸𝑃𝑉 + 𝐸other = 𝐸out + 𝐸loss  

where 𝐸𝑖𝑛  is the total energy input, 𝐸𝑃𝑉  and 𝐸other  are the energies from the PV system and other 

sources, 𝐸out  is the energy output, and 𝐸loss  are the losses. 

12. Converter Efficiency: 

𝜂conv =
𝑃ous , ami 

𝑃intanx 

 

where 𝑃out , conv  and 𝑃int,conv  are the output and input powers of the converter. 

13. Inverter Efficiency: 

𝜂ine =
𝑃𝐴𝐶

𝑃𝐷𝐶
 

where 𝑃𝐴𝐶  is the AC  output power and 𝑃𝐷𝐶  is the DC  input power to the inverter. 

14. Load Demand Satisfaction: 

∑  𝑃out 

∑  𝑃lend 

× 100% 

where ∑𝑃out  is the total power output from the system and ∑𝑃loud  is the total load demand. 

15. Capacity Factor: 

𝐶𝐹 =
 Actual Energy Output 

 Maximum Possible Energy Output 
 

16 Energy Payback Time: 

EPBT =
 Energy Invested 

 Annual Energy Production 
 

17 Levelized Cost of Energy (LCOE): 

 LCOE =
 Total Lifetime Cost 

 Total Lifetime Energy Production 
 

 

 



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18 Net Present Value (NPV): 

NPV = ∑  

𝑛

𝑡=1

𝐺𝑡

(1 + 𝑟)𝑡
 

where 𝐶𝑡  is the cash flow in year 𝑡, 𝑛  is the project lifetime, and 𝑟  is the discount rate. 

19. Internal Rate of Return (IRR): 

0 = ∑  

𝑛

𝑡=1

𝐶𝑡

(1 + IRR)𝑡
 

20 Return on Investment (ROI): 

 ROI =
 Net Profit 

 Investment Cost 
× 100% 

21 Carbon Footprint Reduction: 

CFR = Emission Factor × Energy Produced 

22 System Reliability: 

𝑅(𝑡) = 𝑒−𝜆𝑡 

where 𝑅(𝑡)  is the reliability function, 𝜆  is the failure rate, and 𝑡  is the time. 

23. Maintenance Cost Over Time: 

𝐶maint (𝑡) = 𝐶initial + ∑  

𝑛

𝑖=1

𝐶yearly ⋅ (1 + 𝑟)−𝑖 

where 𝐶initial  is the initial maintenance cost, 𝐶yearly  is the annual maintenance cost, and 𝑟 is the 

discount rate. 

24. Thermal Model for PV Temperature: 

𝑇𝑃𝑉 = 𝑇ambient +
𝑁𝑂𝐶𝑇 − 20

800
⋅ 𝐺 

where 𝑇𝑃𝑉 is the PV module temperature, 𝑇ambient  is the ambient temperature, 𝑁𝑂𝐶𝑇 is the Nominal 

Operating Cell Temperature, and 𝐺  is the solar irradiance. 

25. Wind Energy Conversion (if part of the hybrid system): 

𝑃wind =
1

2
𝜌𝐴𝑣3𝐶𝑝 

where 𝑃wind  is the power generated by wind, 𝜌 is the air density, 𝐴 is the area swept by the wind 

turbine blades, 𝑣 is the wind speed, and 𝐶𝑝 is the power coefficient of the by the wind turbine blades, 

v is the wind speed.  These equations provide a comprehensive overview of the various aspects of 

modeling and analyzing a hybrid solar photovoltaic energy system. They can be used for performance 

assessment, optimization, and designing strategies for efficiency enhancement. Global Power Peak is 

the name of this power peak's maximum (GPP). Only when a PV system is run at GPP can its power 

output under partial shade conditions reach its maximum. In order to get the most power out of a 



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partially shaded PV system, the operating point should be kept at GPP under partial shading conditions. 

[1 

4. PROPOSED METHODOLOGY 

Inspired by the brood parasitism of some cuckoo species, which deposit their eggs in the nests of other 

species as hosts, Xin-She Yang and Susah Deb developed a novel optimization approach in 2009 

dubbed Cuckoo Search (CS). It's more versatile and efficient than Particle Swarm Optimization (PSO) 

and the Genetic Algorithm (GA) in solving optimization problems (Yang and Deb 2014). The cuckoo's 

reproductive strategy provided as inspiration for the heuristic search algorithm used in CS. The term 

"nature inspired computation" refers to a class of computing algorithms that is influenced by studies 

of natural systems. Potential solutions to an optimization issue are analogous to individuals in a 

population, with the fitness function serving as the criterion for success. Cuckoo eggs often hatch 

before host eggs. The first cuckoo chick to emerge from its egg will immediately begin throwing host 

eggs out of the nest. This results in a larger share of the host bird's diet for the cuckoo chick. A cuckoo 

egg represents an unexplored avenue of inquiry. The idea is to swap out mediocre nesting strategies 

for new, maybe better ones (cuckoos) (Sakthi & Nedunchezhian 2014). 

The CS algorithm is inspired by brood parasitism in cuckoo species, the Levy flying behavior of birds, 

and fruit flies. It is possible that certain species of cuckoo lay their eggs in group nests. When the host 

bird discovers the eggs aren't its own, it either throws them away or leaves the nest to start a new one 

elsewhere. For the sake of simplicity, the explanation of CS is based on three idealised rules:  

• The cuckoo lays its single egg in a nest chosen at random;  

• The best nests produce high-quality offspring;  

• The number of host nests is fixed, and the cuckoo egg is detected by the host with probability 

[0, 1]. More calculations are done to identify and eliminate the poorest nests.  

The CS algorithm is a fast-convergence optimization method. Its original release date was 2009. The 

algorithm was conceived as a nod to the cuckoo bird's parasitic reproduction strategy. This bird does 

not build its own nest and instead prefers to use the nests of other species. It uses a strategy to choose 

a suitable host nest that involves randomly visiting several nests until it finds one with the highest 

chances of producing healthy offspring. Cuckoos will occasionally remove the host bird's eggs from 

the nest to increase the chances of hatching their own. To lessen the chances of being found, certain 

cuckoo species may alter the shape of their eggs so that they are similar to those of the host bird. If the 

host bird figures out the cuckoo's trick, it may abandon the nest or discard the cuckoo's eggs. The CS 

algorithm is inspired by the foraging behaviour of cuckoos. The random steps and Lévy flight 

characteristics that CS uses during its search boost the global search and may even hasten convergence. 

Even though the original CS (OCS) method was designed to deal with multi-variable problems with 

various objectives, it is effective for monitoring MPPT of PV systems due to its lengthy convergence 

time and high oscillations under steady-state situations. This problem is handled in the next part by 

introducing the enhanced CS (ICS) method. On the other hand, the OCS uses the Lévy flight to 

randomly move a number of searching agents whose initial values are inside the searching area's 



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borders and to update those values as they move. When a new generation is produced, the OCS requires 

a step back to the prior site, as shown by Equation (5.14): 

𝑑𝑖+1
𝑘 = 𝑑𝑖

𝑘 + 𝛼 ⋅
|𝑢|

𝑣1/𝛽
⋅ (𝑑best − 𝑑𝑖

𝑘)                             (4.1) 

where I is the generation number (i=,1,2,......it), k is the order of searching agents in the swarm 

(k=1,2,....ss), ss is the swarm size, is the step size (which can be determined depending on the problem, 

though it is generally recommended that =1), and u and v are matrices with uniform distribution - their 

values can be determined as shown in Equation (4.1). Pseudo-code for the CS algorithm is provided 

in Figure 4.1. 

𝑢 ≈ 𝑁(0, 𝜎𝑢
2) and 𝑣 ≈ 𝑁(0, 𝜎𝑣

2)                                    (4.2) 

where the variance of 𝑢 and 𝑣 can be obtained from : 

𝜎𝑢 = (
Γ(1+𝛽)⋅si n(𝜋⋅𝛽/2)

Γ(
1+𝛽

2
)⋅𝛽⋅2

(
𝛽−1

2
)

)  and 𝜎𝑣 = 1                                  (4.3) 

 

Figure 4.1: Simulation Model of Proposed System with Cuckoo Search Algorithm 

The enhanced CS (ICS) suggested in this study enhances the OCS's tracking mechanism to more 

efficiently track PV systems' GP for uniform irradiance and PSC with the shortest convergence time, 

lowest failure rate, and fewest steady-state oscillations possible without adding complexity. The ICS 

suggested in this research attracts the worst particle with values close to the global best, and the stages 

following them respectively. By adding the difference between the worst cuckoo position and the best 

cuckoo position after multiplying this value by random to the worst cuckoo position, the software was 

able to replace the worst cuckoo with the one that was close to the best one. The findings from the 

simulation and experimental work sections shown a significant decrease in convergence time and 



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oscillations at steady state, demonstrating the ICS' superiority to the original cuckoo search method 

and other optimization strategies under investigation. The Levy flight function is the fundamental 

determinant of the CSO MPPT method convergence, which was inspired by the parasitic swarm 

intelligence of cuckoo birds. The PV system is initially subjected to a variety of duty cycles at random, 

and the generated voltage and currents are utilized to estimate the power. The duty cycle is changed 

until it performs at its peak efficiency and level of fitness.  The following stages are used to demonstrate 

this logic: 

Step 1: Initialize the particles' positions 𝑎0
i:SS  and send it to the PV system to determine the 

corresponding power𝑃0
1:5𝑠. 

Then determine the maximum power 𝑃best  and its corresponding duty ratio, dbest. 

Step 2: Determine the worst particle power, 𝑃averst,  its order, 𝑘worst,  and its corresponding duty 

ratio, 𝑑worst  

Step 3: Check if rand > 𝑝𝑎. If so, go to Step 4; otherwise, go to Step 7. 

Step 4: Attract the worst nest to the best nest using 𝑑𝑖
𝑘wass = 𝑑worst + ran d(𝑑best − 𝑑worst), 

Step 5: Send the new value of 𝑑𝑖
𝑘warss  to the PV system to determine the corresponding power 𝑃𝑖

𝑘woost , 

then check if 𝑃𝑖
𝑘worst > 𝑃best, , then 𝑃best = 𝑃𝑖

𝑘wast  and 𝑑best = 𝑑𝑖
𝑘warst . 

Step 6: Check the stopping criteria as shown in Equation (8). If it is valid, go to Step 1; otherwise, go 

to Step 2. 

Step 7: Add a step to each nest using Lévy flight by using this equation 𝑑𝑖
𝑘 = 𝑑𝑖

𝑘 + 𝐾 ⋅
|𝜇|

𝑣
1
𝛽

⋅

(𝑑bent − 𝑑𝑖
𝑘), then check if 𝑑𝑖

𝑘 > 𝑑max, 𝑑𝑖
𝑘 = 𝑑max, otherwise, if 𝑑𝑖

𝑘 < 𝑑min,𝑑𝑖
𝑘 = 𝑑min  

Step 8: Send the duty ratio 𝑑𝑖
𝑘 to the PV system to determine the corresponding power 𝑃𝑖

𝑘, then check 

if 𝑃𝑖
𝑘 > 𝑃best , then 𝑃best = 𝑃𝑖

𝑘 and 𝑑best = 𝑑𝑖
𝑘 Step 9: Check if 𝑘 < SS. If so, go to Step 7; otherwise, 

go to Step 6. The parameters used for simulation of improved cuckoo search algorithm is explained in 

Table 4.1. 

Table 4.1: Parameter used in Cuckoo Search based Algorithm 

PARAMETERS VALUE 

No. of particles ( N ) 10 

No. of dimensions ( D ) 2 

Maximum velocity ( Vmax ) 2.70 

No. of iterations ( Itermax) 80 

Levi distribution factor (β) 3/2 

Acceleration factor (K) .8 

 ∑V 1 

 

 



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5. RESULTS AND DISCUSSIONS 

Water is used as coolant. Water is made to flow on the panel at natural or gravitational flow. A pipe 

of  56cm with  10 no of holes is placed at the top of the panel. Water is allowed to flow at three different 

rates such as 1L/minute, 1.5L/minute and 2L/minute. Output of the panel at three different water flow 

rates are compared. Flow rate of 2L/minute is found to be most effective. 

 

Figure 5.1 Front Surface Cooling by water 

The results of this research work can be listed as follows: 

• Performance Analysis of Power at Various Temperature. 

• Analysis of Cooling System for Tempearature Regulation of Solar Panels 

• Power Output Analysis of Cooling System Coupled Solar PV System 

 

Fig. 5.2 Simulink Diagram of Proposed System with Parametric Variation 

 



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Fig 5.3  Power Voltage Waveform of Photovoltaic System without Cooling 

Neglecting the relevance of solar accessories such as inverters, MPPTs, and charge controllers in the 

plant has a significant impact on the plant's efficiency. The wrong accessory layout reduces the PV 

plant's performance. All of these factors reduce the plant's efficiency, lengthening the payback period, 

which contributes to solar energy's lack of popularity. The research conducted is effective in improving 

the system's performance under high-temperature conditions. 

 

Fig.5.4  Power Voltage Waveform of Photovoltaic System with grass at back Surface 

 

Fig. 5.5 Power Voltage Curve of Photovoltaic System with Front Cooling System 



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Fig.5.6  Comparative Analysis of Three Cases for Validation of Cooling System 

 

Figure 5.7: General Simulation Model of Proposed System 

 

 

Figure 5.8 Implementation of Heuristic MPPT for Electrical Characteristic and Performance 

Analysis of SPV System 



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Figure 5.9 Implementation of Conventional MPPT 

 

Figure 5.10 Output of SPV System with P and O MPPT 

 

Figure 5.11 Output of SPV System in Normal Configuration with Heuristic MPPT 

 

Figure 5.12 Output of SPV System in PVT Configuration with Heuristic MPPT 

 



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Table 5.4 Comparison of PV/T and PV Power Output with MPPT 

Type of Configuration P & O MPPT INC MPPT Proposed 

Normal PV System 140.5 Watt 150 Watt 151 Watt 

PVT System 170.9 Watt 172 Watt 180 Watt 

 

 

Figure 5.13 Output of SPV System in PVT Configuration with Conventional MPPT 

The analysis of Figure proves the effectiveness of proposed system with complex operational 

conditions. It is evident from the plot that proposed CSA hybrid methodologies have minimum 

oscillations and it has been able to track the maximum power point of the system during transient 

condition of irradiation and temperature. The analysis proves the effectiveness of maximum power 

point tracking on operational efficiency of solar photovoltaic system. 

6. CONCLUSIONS 

The PV panel's temperature is transforming efficiency functions, which can influence photovoltaic 

strength. The solar PV/T system eliminates solar panel heat losses. Hybrid Solar PV/T collector 

technology is proposed in this work to increase energy efficiency per unit area. In this study are 

discussed the effects of combining the PV and Thermal Systems into a single device and its 

performance analysis. With the PV/T technique, the experimental findings have shown that the 

electrical performance of the PV module has greatly improved. The findings showed that the combined 

PV/T system's electric and heat output is much more than PV alone A novel Maximum power point 

tracking is introduced in this research as a means through which solar photovoltaic systems can 

maximize their output. This research explains the dynamic nature of the maximum power point system. 

The research focuses on the conventional and soft computing methods that have been used to the 

construction of maximum power point algorithms, covering both their theory and their practical 

applications. For maximum power monitoring, a CSA algorithm has been developed. Comprehensive 

investigation of the suggested algorithm in normal, complicated, and partially shaded modes of 

operation demonstrates the algorithm's efficacy in enhancing the operational efficiency of the solar 

system in these settings. Improved cuckoo search optimization technique improved the tracking speed 

and it tracked the maximum power point under complex operating conditions improving the 

operational efficiency of the system compared to conventional technique. 

 

 

 



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