# Maximum Power Exploitation of Photovoltaic System under Fast-Varying Solar Irradiation and Local Shading

Authors: Yi-Jui Chiu; Bi Li; Chin-Ling Chen; Shui-Yang Lien; Ding Chen; Ji-Ming Yi; Yung-Hui Shih
Journal: International Journal of Photoenergy
 (2022)

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## Abstract

The photovoltaic (PV) model commonly used in engineering has difficulty in accurately predicting the actual power generation. In this study, a PV model commonly used in engineering was used to establish a PV simulation model and improve it based on experimental data. The influence of the PV output power characteristics and local shading on the power generation efficiency of the PV system was analyzed using MATLAB and the improved model. Aiming at the problem that most maximum power point tracking (MPPT) algorithms have difficulty quickly tracking the maximum power point (MPP) under fast-varying solar irradiation; a polynomial fitting-MPPT (PF-MPPT) algorithm and a simple fitting-MPPT (SPF-MPPT) algorithm based on polynomial fitting were proposed to track the maximum power point under the fast-varying solar irradiation. Finally, an improved MPPT system with the PF-MPPT algorithm was proposed to solve the problem of the significant reduction of the output power of a PV system under fast-varying solar irradiation and local shading. The simulation results showed that under fast-varying solar irradiation, the power-tracking abilities, and stabilities of the proposed two algorithms were similar to those of the perturb and observe (P&O) algorithm, but the tracking speed was over 2.58 times that of the constant voltage tracking (CVT) algorithm and four times that of the P&O algorithm. In addition, under fast-varying solar irradiation and local shading, the speed, ability, and stability of the improved MPPT system with the PF-MPPT algorithm when tracking the maximum power were 9.52, 1.32, and 1.84 times of the MPPT system with the P&O algorithm and 2.18, 1.41, and 2.00 times of the MPPT system with the particle swarm optimization algorithm, respectively.


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## Body

## 1. Introduction

In recent years, an increasing number of researchers are recognizing the importance of photovoltaic power generation technology. Dobakhshari [1] proposed a maximum power point tracking (MPPT) optimization algorithm that limited the voltage search space. Their study showed that the algorithm could optimize the existing MPPT algorithm and that it exhibited good dynamic performances. Li [2] proposed a variable-weather-parameter MPPT algorithm and found that the performance of this method was better than those of the perturb and observe (P&O) algorithm and fuzzy algorithm under different irradiances and loads. Following this, Peng et al. [3] proposed an improved P&O algorithm to solve the power-tracking problem of solar cells under fast-varying solar irradiation. The results showed that the algorithm had better accuracy, speed, and efficiency under fast-varying solar irradiation than the other methods. Yilmaz et al. [4] proposed a new MPPT method, demonstrating that their method exhibited excellent performances under sudden environmental changes, and the speed to reach steady state was four times that of the P&O algorithm. Ko and Chao [5] proposed an improved Quine–McCluskey (QM) algorithm, and LabVIEW and PSIM were used for simulation experiments. The results showed that the proposed algorithm exhibited better performances under a step change in the environment than the P&O and QM algorithms. Mohanty et al. [6] proposed an improved gray wolf optimization algorithm to overcome the limitations of the traditional P&O algorithm, such as its low tracking efficiency and steady-state oscillations. The simulation results showed that this algorithm had a better tracking ability than the P&O algorithm. Kofinas et al. [7] proposed a direct numerical control (DNC) MPPT algorithm based on neural control. The results showed that the algorithm exhibited better performances in a harsh environment than the traditional P&O method. Alik and Jusoh [8] proposed a control method based on the traditional P&O and Check algorithms. Simulink results showed that the proposed control method had a higher tracking efficiency under local shading than the P&O algorithm. Eltamaly [9] proposed an improved particle swarm optimization (PSO) algorithm based on the genetic and PSO algorithms. Simulink and PSIM results showed that the algorithm could more accurately track the global maximum power point than the fuzzy logic control algorithm. Hou et al. [10] proposed an MPPT control method based on the glow worm swarm optimization algorithm. Simulink results showed that the algorithm had a better tracking ability under sudden irradiance than the P&O algorithm. Giraldo et al. [11] proposed an MPPT algorithm based on fuzzy control. Simulink results showed the algorithm had better stability and a lower power consumption than the P&O algorithm. Bingol and Ozkaya [12] proposed five different PV array structure designs for PV systems that were vulnerable to the negative impact of local shading. Simulink results showed that the PV array structure had a significant impact on the PV efficiency, and the full cross structure exhibited the best performance. Macaulay and Zhou [13] proposed an improved P&O algorithm with a variable step size based on a fuzzy algorithm. The results showed that the algorithm had a faster response speed and a higher stability under rapidly changing solar irradiation than the P&O algorithm. Huang and Wang [14] proposed an algorithm to analyze the degradation factors of a PV module. A large quantity of PV data was used to study the influencing factors of the PV module aging. The results showed that the main cause of the power loss was the optical degradation of a PV module. Li and Qi [15] proposed an improved gravity search algorithm. The results showed that the algorithm improved the tracking time and accuracy significantly compared to the PSO algorithm and gravity search algorithm. Nabipour et al. [16] proposed an MPPT algorithm based on a fuzzy adaptive algorithm; the test results showed that the algorithm exhibited better steady-state and dynamic tracking performances than the traditional fuzzy algorithm. Teng et al. [17] proposed an MPPT algorithm based on parameter estimation. Simulation and experimental results show that the algorithm exhibited better performances than the P&O algorithm. Wjya et al. [18] proposed a new method for a solar microgrid based on long short-term memory and demand response to solve the problem of an unstable solar output voltage. The results showed that the method can meet the power requirements of electrical appliances while ensuring voltage stability. Sher et al. [19] proposed an improved P&O method. The short-circuit current method was used to determine the initial operating point of a PV system, and then traditional P&O technology was implemented. The results showed that the algorithm had a better tracking ability and stability than the P&O algorithm. Fares et al. [20] proposed an improved MPPT method based on the squirrel search algorithm (ssa) and compared it with the conventional ssa and pso algorithms. The results showed that the proposed algorithm has faster convergence and lower power oscillations. Shams et al. [21] proposed an improved MPPT algorithm based on a butterfly optimization algorithm and experimentally validated it on a SEPIC converter topology. The results showed that the proposed method achieved an average steady-state efficiency of 99.85% and the response speed improved by 86.15%. Pervez et al. [22] proposed an MPPT algorithm based on the most valuable player algorithm (MVPA) and compared it with the PSO algorithm, and the results showed that the proposed algorithm has a faster tracking speed and higher power tracking efficiency. Shams et al. [23] proposed an improved MPPT algorithm based on the team game optimization algorithm and conducted experiments under different shading conditions. The experimental results showed that the average MPPT efficiency of the proposed method was 99.78%, the average tracking time was 0.9 s, and the convergence speed was 115% of the original one. Premkumar and Sowmya [24] proposed an improved MPPT algorithm based on the bionic whale optimization (WO), and the simulation results showed that the proposed method has a faster convergence speed and tracking efficiency. Manoharan et al. [25] proposed an improved MPPT algorithm based on the P&O algorithm and verified the experiments via simulations using MATLAB/Simulink. The results showed that the proposed algorithm can accurately track the MPP under various operating conditions. Premkumar and Sumithira [26] proposed an MPPT algorithm based on the P&O and Whale Optimization (WOA) algorithms, which was validated through modeling and simulation using MATLAB/Simulink. The simulation results showed that the proposed algorithm improves the output power of the photovoltaic system under partial shading. Premkumar et al. [27] proposed an MPPT algorithm based on the SALP group and the P&O algorithm and simulated it with MATLAB/Simulink simulation tool. The results showed that the proposed algorithm has higher tracking performance than the P&O algorithm. Priyadarshi et al. [28] proposed an intelligent fuzzy particle swarm optimization (FPSO) based the MPPT algorithm, which was validated using MATLAB/Simulink. The results showed that the proposed MPPT algorithm has high tracking efficiency and excellent dynamic control. Priyadarshi et al. [29] proposed an MPPT method based on the Flower Pollination Algorithm (FPA), and the results showed that the proposed method has better power tracking capability and faster convergence speed. Priyadarshi et al. [30] proposed a hybrid MPPT method based on adaptive neural and particle swarm optimization, and the results showed that the proposed method has better drive control and can obtain high-quality inverter currents. Padmanaban et al. [31] proposed an MPPT algorithm based on adaptive neural and artificial bee colony, and the results showed that the proposed algorithm can effectively improve the performance of grid-connected PV systems. Tayagaki et al. [32] analyzed the power generation of solar cells at different positions on the vehicle body. The results show that the power generated by a PV module installed on the side was less than one quarter of the level. When the inclination angle was 40°, the solar modules had a better power generation efficiency. Zakaria et al. [33] proposed a solar azimuth tracking method based on artificial vision. The collected images were used to identify the current solar azimuth and then control the orientation of the solar cells. The experimental results showed that the proposed tracking method could improve the efficiency of PV power generation.In this study, a PV model commonly used in engineering was used to establish a PV simulation model. Based on experimental data, the model was improved, and the output characteristics of the PV module were analyzed by using the improved model and MATLAB. To solve the problem of the low power generation efficiency of the traditional MPPT method under fast-varying solar irradiation, the PF-MPPT algorithm and SPF-MPPT algorithm based on a polynomial fitting algorithm were proposed to track the maximum power point in the current environment. The performances of the PF-MPPT algorithm, SPF-MPPT algorithm, P&O algorithm, and constant voltage tracking (CVT) algorithm under fast-varying solar irradiation was simulated and analyzed. To solve the problem that the output power of a traditional PV system decreases significantly under fast-varying solar irradiation and local shading, an improved MPPT system with the PF-MPPT algorithm was proposed and compared with the traditional MPPT system with the PSO and P&O algorithms.


## 2. Contributions of Previous Work

There are many scientific papers on improving the efficiency of photovoltaics [1–33]. These papers usually include descriptions of the studies, proposed solutions for optimization algorithms, and feasibility validation of new algorithms. However, most of these papers focus on analysis under fast-varying radiance or local shading, often ignoring the possible coexistence of fast-varying radiance and local shading. Table 1 shows a comparison of the differences between these previous studies.Table 1
Comparison of the differences between previous studies.
AuthorYearObjectivesTechnologyApplicable environmentQiu et al.(this paper)2022MPPTA PF-MPPT algorithm and an improved MPPT systemFast-varying solar irradiation and local shadingDobakhshari [1]2018MPPTA MPPT optimization algorithmFast-varying solar irradiationLi [2]2019MPPTA variable-weather-parameter MPPT algorithmDifferent irradiancesPeng et al. [3]2018MPPTAn improved P&O algorithmFast-varying solar irradiationYilmaz et al. [4]2019MPPTA new MPPT methodFast-varying solar irradiationKo and Chao et al. [5]2012MPPTAn improved Quine–McCluskey (QM) algorithmFast-varying solar irradiationMohanty et al. [6]2015MPPTAn improved gray wolf optimization algorithmFast-varying solar irradiationKofinas et al. [7]2015MPPTA direct numerical control (DNC) MPPT algorithmFast-varying solar irradiationAlik and Jusoh [8]2017MPPTA control method based on the traditional P&O and check algorithmsLocal shadingEltamaly [9]2015MPPTAn improved PSO algorithm based on the genetic algorithmsLocal shadingHou et al. [10]2016MPPTAn MPPT control method based on the glowworm swarm optimization algorithmFast-varying solar irradiationGiraldo et al. [11]2017MPPTAn MPPT algorithm based on fuzzy controlFast-varying solar irradiationBingol and Ozkaya [12]2018Overcoming the negative effects of local shadingFive different PV array structure designsLocal shadingMacaulay and Zhou et al. [13]2018MPPTAn improved P&O algorithm with a variable step sizeFast-varying solar irradiationHuang and Wang [14]2018Study on influencing factors of PV module agingAn algorithm to analyze the degradation factors of a PV moduleNatural environmentLi and Qi [15]2018MPPTAn improved gravity search algorithmLocal shadingNabipour et al. [16]2018MPPTAn MPPT algorithm based on a fuzzy adaptive algorithmFast-varying solar irradiationTeng et al. [17]2016MPPTAn MPPT algorithm based on parameter estimationFast-varying solar irradiationWjya et al. [18]2021Overcoming the problem of unstable output voltageA new method of solar micro grid based on long-term short-term memory and demand responseNatural environmentSher et al. [19]2015MPPTAn improved P&O methodFast-varying solar irradiationFares et al. [20]2021MPPTAn improved MPPT method based on the squirrel search algorithmLocal shadingShams et al. [21]2020MPPTAn improved MPPT algorithm based on a butterfly optimization algorithmFast-varying solar irradiationPervez et al. [22]2021MPPTAn MPPT algorithm based on the most valuable player algorithmLocal shadingShams et al. [23]2020MPPTAn improved MPPT algorithm based on the team game optimization algorithmLocal shadingPremkumar and Sowmya [24]2019MPPTAn improved MPPT algorithm based on the bionic whale optimizationLocal shadingManoharan et al. [25]2020MPPTAn improved MPPT algorithm based on the P&O algorithmFast-varying solar irradiationPremkumar and Sumithira [26]2018MPPTA MPPT algorithm based on P&O and Whale Optimization algorithmsLocal shadingPremkumaret al. [27]2021MPPTA MPPT algorithm based on SALP group and P&O algorithmFast-varying solar irradiationPriyadarshi et al. [28]2018MPPTAn intelligent fuzzy particle swarm optimization based MPPT algorithmLocal shadingPriyadarshi et al. [29]2018MPPTAn MPPT method based on Flower pollination algorithmLocal shadingPriyadarshi et al. [30]2019MPPTA hybrid MPPT method based on adaptive neural and PSOLocal shadingPriyadarshi et al. [31]2019MPPTA MPPT algorithm based on adaptive neural and artificial bee colonyLocal shadingTayagaki et al. [32]2019To improve power generation efficiencyStudy on the relationship between PV module inclination and power generation efficiencyNatural environmentZakaria et al. [33]2015To improve power generation efficiencyA solar azimuth tracking method based on artificial visionNatural environmentBased on the previous literature review, to solve the problem that the output power of a traditional PV system decreases significantly under fast-varying solar irradiation and local shading, an improved MPPT system with the PF-MPPT algorithm was proposed.


## 3. PV Model and Experiment

### 3.1. PV Model

The PV model commonly used in engineering is as follows:(1)I=Isc1−1−ImIsce−Um/Um/Uoc−1/ln1−Im/IscUoceU/Um/Uoc−1/ln1−Im/IscUoc−1,where I is the PV output current, U is the PV output voltage, Isc is the short-circuit current, Im is the peak current, Uoc is the open-circuit voltage, and Um is the peak voltage.IfC2=Um/Uoc−1/ln1−Im/Isc and C1=1−Im/Isce−Um/C2Uoc are substituted into (1), the following can be obtained:
(2)I=Isc1−C1eU/C2Uoc−1Because the model is obtained in a fixed environment and ignores the effects of different external environments, the parametersIsc, Im, Uoc, and Um need to be compensated. The compensation formula for Isc is as follows:
(3)Isc=IscSSref1+aΔT,where S is the current solar irradiance, Sref is the standard solar irradiance, ∆T is the temperature difference of the PV module between the current and standard working conditions, and a is the temperature compensation coefficient of the current.The compensation formula forIm is as follows:
(4)Im=ImSSref1+aΔT.The compensation formula forUoc is as follows:
(5)Uoc=Uoc1−cΔTlne+bΔS,where c is the temperature compensation coefficient of the voltage, e is the Euler number, ∆S is the solar irradiance difference between the current and standard conditions, and b is the compensation coefficient of the solar irradiance.The compensation formula forUm is as follows:
(6)Um=Um1−cΔTlne+bΔS.The values ofa, b, and c are often set to 0.0005 °C−1, 0.2 m2/W, and 0.00288°C−1, respectively. A PV simulation model was established in Simulink, as shown in Figure 1.Figure 1
Photovoltaic simulation model.


### 3.2. Experiment

To obtain the relationship between the output voltage and current of the PV module under different environments, PV module output characteristic experiments were conducted. The experiments were carried out under external environmental conditions. The relevant parameters of PV panels and PV converters are shown in Table2. The experimental arrangement is shown in Figure 2. Solar irradiation was measured with a solar irradiation instrument. The temperature of the PV module was measured with a temperature measuring instrument. Sliding rheostats in the range of 0–1000 Ω and 0–500 Ω were used to simulate different loads. The output voltage and current of the PV module was measured with two multimeters, and experimental data were processed with a computer. The PV module was connected in series with the load (sliding rheostat).Table 2
Relevant parameters of photovoltaic (PV) panel and PV converter.
PV panelDimensions:1956∗992∗40 mmMaximum nominal power350 WPeak voltage:36 VPeak current:9.72 AOpen circuit voltage:43.2 VShort circuit current:10.69 APV converterDimensions:156∗81∗43mmPower:400 WInput voltage range:12-50 VOutput voltage range:28-95 VFigure 2
Experimental setup. (a) Experimental environment. (b) Experimental tool rack. (c) Experimental circuit.
(a)(b)(c)


### 3.3. Improved PV Model

The experimental and simulated environmental conditions are shown in Table3. The environmental conditions for the simulation and improved simulation settings are the same.Table 3
Environmental conditions of experiment and simulation.
GroupEnvironment conditionGroupEnvironment conditionExperiment 1100±10 W/m2, 27–31 °CSimulation 1100 W/m2, 29 °CExperiment 2300±20 W/m2, 28–38 °CSimulation 2300 W/m2, 33 °CExperiment 3680±30 W/m2, 38–51 °CSimulation 3680 W/m2,45 °CExperiment 41000±50 W/m2, 43–59 °CSimulation 41000 W/m2,51 °CThe relationship between the mathematical model and experimental data is shown in Figure3. Under four different environmental conditions, the open-circuit voltages of the experimental data were Uoc_e1=41.5V, Uoc_e2=42.7V, Uoc_e3=43.2V, and Uoc_e4=44.2V. Thus, Uoc_e1<Uoc_e2<Uoc_e3<Uoc_e4. The open-circuit voltage of the simulation data was small, and Uoc_s2<Uoc_s1<Uoc_s4<Uoc_s3, which did not agree with the experimental data. Under the same conditions, the short-circuit current in the experiment was less than that of the simulation.Figure 3
Mathematical model and experimental data.Based on the analysis of Figure3, the photovoltaic model was improved. The improved PV model is as follows:
(7)I=KiIsc1−C1eKvU/C2Uoc−1,where the current scale factor Ki=0.78, the voltage scale factor Kv=0.86, a=0.00255°C−1, b=0.55m2/W, and c=0.00288°C−1.The relationship between the improved mathematical model and experimental data is shown in Figure4. Under the four different environmental conditions, the open-circuit voltages of the improved simulation data were Uoc_is1=39.66V, Uoc_is2=41.59V, Uoc_is3=44.19V, and Uoc_Is4=46.5V, which were similar to the corresponding experimental data. Under the same conditions, the short-circuit currents of the improved simulation data were Isc_is1=842.3mA, Isc_is2=2552mA, Isc_is3=5959mA, and Isc_Is4=8891mA, which were close to the experimental values of 850 mA, 2580 mA, 5960 mA, and 9370 mA, respectively. In conclusion, the improved model was more in line with the output characteristics of the PV module.Figure 4
Improved mathematical model and experimental data.


## 3.1. PV Model

The PV model commonly used in engineering is as follows:(1)I=Isc1−1−ImIsce−Um/Um/Uoc−1/ln1−Im/IscUoceU/Um/Uoc−1/ln1−Im/IscUoc−1,where I is the PV output current, U is the PV output voltage, Isc is the short-circuit current, Im is the peak current, Uoc is the open-circuit voltage, and Um is the peak voltage.IfC2=Um/Uoc−1/ln1−Im/Isc and C1=1−Im/Isce−Um/C2Uoc are substituted into (1), the following can be obtained:
(2)I=Isc1−C1eU/C2Uoc−1Because the model is obtained in a fixed environment and ignores the effects of different external environments, the parametersIsc, Im, Uoc, and Um need to be compensated. The compensation formula for Isc is as follows:
(3)Isc=IscSSref1+aΔT,where S is the current solar irradiance, Sref is the standard solar irradiance, ∆T is the temperature difference of the PV module between the current and standard working conditions, and a is the temperature compensation coefficient of the current.The compensation formula forIm is as follows:
(4)Im=ImSSref1+aΔT.The compensation formula forUoc is as follows:
(5)Uoc=Uoc1−cΔTlne+bΔS,where c is the temperature compensation coefficient of the voltage, e is the Euler number, ∆S is the solar irradiance difference between the current and standard conditions, and b is the compensation coefficient of the solar irradiance.The compensation formula forUm is as follows:
(6)Um=Um1−cΔTlne+bΔS.The values ofa, b, and c are often set to 0.0005 °C−1, 0.2 m2/W, and 0.00288°C−1, respectively. A PV simulation model was established in Simulink, as shown in Figure 1.Figure 1
Photovoltaic simulation model.


## 3.2. Experiment

To obtain the relationship between the output voltage and current of the PV module under different environments, PV module output characteristic experiments were conducted. The experiments were carried out under external environmental conditions. The relevant parameters of PV panels and PV converters are shown in Table2. The experimental arrangement is shown in Figure 2. Solar irradiation was measured with a solar irradiation instrument. The temperature of the PV module was measured with a temperature measuring instrument. Sliding rheostats in the range of 0–1000 Ω and 0–500 Ω were used to simulate different loads. The output voltage and current of the PV module was measured with two multimeters, and experimental data were processed with a computer. The PV module was connected in series with the load (sliding rheostat).Table 2
Relevant parameters of photovoltaic (PV) panel and PV converter.
PV panelDimensions:1956∗992∗40 mmMaximum nominal power350 WPeak voltage:36 VPeak current:9.72 AOpen circuit voltage:43.2 VShort circuit current:10.69 APV converterDimensions:156∗81∗43mmPower:400 WInput voltage range:12-50 VOutput voltage range:28-95 VFigure 2
Experimental setup. (a) Experimental environment. (b) Experimental tool rack. (c) Experimental circuit.
(a)(b)(c)


## 3.3. Improved PV Model

The experimental and simulated environmental conditions are shown in Table3. The environmental conditions for the simulation and improved simulation settings are the same.Table 3
Environmental conditions of experiment and simulation.
GroupEnvironment conditionGroupEnvironment conditionExperiment 1100±10 W/m2, 27–31 °CSimulation 1100 W/m2, 29 °CExperiment 2300±20 W/m2, 28–38 °CSimulation 2300 W/m2, 33 °CExperiment 3680±30 W/m2, 38–51 °CSimulation 3680 W/m2,45 °CExperiment 41000±50 W/m2, 43–59 °CSimulation 41000 W/m2,51 °CThe relationship between the mathematical model and experimental data is shown in Figure3. Under four different environmental conditions, the open-circuit voltages of the experimental data were Uoc_e1=41.5V, Uoc_e2=42.7V, Uoc_e3=43.2V, and Uoc_e4=44.2V. Thus, Uoc_e1<Uoc_e2<Uoc_e3<Uoc_e4. The open-circuit voltage of the simulation data was small, and Uoc_s2<Uoc_s1<Uoc_s4<Uoc_s3, which did not agree with the experimental data. Under the same conditions, the short-circuit current in the experiment was less than that of the simulation.Figure 3
Mathematical model and experimental data.Based on the analysis of Figure3, the photovoltaic model was improved. The improved PV model is as follows:
(7)I=KiIsc1−C1eKvU/C2Uoc−1,where the current scale factor Ki=0.78, the voltage scale factor Kv=0.86, a=0.00255°C−1, b=0.55m2/W, and c=0.00288°C−1.The relationship between the improved mathematical model and experimental data is shown in Figure4. Under the four different environmental conditions, the open-circuit voltages of the improved simulation data were Uoc_is1=39.66V, Uoc_is2=41.59V, Uoc_is3=44.19V, and Uoc_Is4=46.5V, which were similar to the corresponding experimental data. Under the same conditions, the short-circuit currents of the improved simulation data were Isc_is1=842.3mA, Isc_is2=2552mA, Isc_is3=5959mA, and Isc_Is4=8891mA, which were close to the experimental values of 850 mA, 2580 mA, 5960 mA, and 9370 mA, respectively. In conclusion, the improved model was more in line with the output characteristics of the PV module.Figure 4
Improved mathematical model and experimental data.


## 4. PV Output Power Characteristic Analysis

Under standard solar irradiance, the voltage-power-temperature characteristics from the simulation obtained using the improved PV model are shown in Figure5. The maximum power Pm varied from 295 to 319 W, and the open-circuit voltage Uoc varied between 38 and 43 V. The smaller the temperature T was, the smaller the power change, the greater the maximum power Pm, the greater the peak voltage Um and the greater the open-circuit voltage Uoc became. Overall, the power output curve first increased and then decreased rapidly with the increase in voltage. When the output voltage U was less than 30 V, the output power increased linearly with the increase in temperature, and the output power increased with the increase in the output voltage at the same temperature. When the output voltage U was greater than 30 V and less than 40 V, the output power increased first and then decreased with the increase in temperature. When the output voltage U was greater than about 40 V, the output power decreased parabolically with the increase in temperature.Figure 5
Voltage-power-temperature characteristics.Under standard temperature conditions, the simulation results of the voltage-power-solar irradiance characteristics obtained using the improved PV model are shown in Figure6. The maximum power Pm varied between 25 and 396 W, and the open-circuit voltage Uoc varied between 40 and 53 V. The smaller the solar irradiance S was, the smaller the power change, the smaller the maximum power Pm, the smaller the peak voltage Um, and the smaller the open-circuit voltage Uoc became. Overall, the power output curve first increased and then decreased rapidly with the increase in voltage. When the output voltage U was less than 50 V, the output power increased linearly with the increase in the solar irradiance, and the output power increased with the increase in the output voltage under the same solar irradiance. When the output voltage U was greater than 50 V, the power increased linearly with the increase in the solar irradiance, but a greater solar irradiance was required to generate power. Overall, the greater the solar irradiation was, the faster the power rise became.Figure 6
Voltage-power-solar irradiance characteristics.


## 5. Improved MPPT Algorithm

At present, the approaches used to improve the algorithms to increase the PV efficiency are to improve P&O algorithms by changing the step size or relevant parameters or to integrate and improve a variety of intelligent algorithms (neural network, particle swarm optimization, gray wolf optimization, fuzzy control, and genetic algorithm). The research on effective improved algorithms based on the polynomial fitting algorithm is lacking at present.Aiming at the problem that the traditional MPPT methods have low power generation efficiencies under fast-varying solar irradiation, an MPPT algorithm based on the polynomial fitting is proposed to predict the bestUm value in the current environment.Based on many PV module output data and the MATLAB curve fitting tool, the relationship between the solar irradianceS, temperature T, and peak voltage Um was fitted. In the fitting formulas below, the temperature T is denoted as x, the solar irradiance S is denoted as y, and the peak voltage Um is denoted as z. The fitted functions were as follows:
(1)
The powers ofx and y were set to one:(8)Fx,y=34.94−0.1082x+0.008274y.The results of this fitted formula are shown in Figure7. The coefficient of determination R2=0.9953>0.99, showing that the fitting effect was good.
(2)
The power ofx was set to two and the power of y was set to one:(9)Fx,y=31.46−0.09423x+0.00952y+2.19×10−5x2−2.491×10−5xy.Figure 7
Curve fitting with Equation (8).
(a)
Fitting effect(b)
Fitting residualsThe results of this fitted formula are shown in Figure8. R2=0.9994>0.9953, showing that the fitting effect of the algorithm was better than that of Equation (8).
(3)
The power ofx was set to one, and the power of y was set to two:(10)Fx,y=33.88−0.09204x+0.01059y−2.491×10−5xy−8.242×10−7y2.Figure 8
Curve fitting with Equation (9).
(a)
Fitting effect(b)
Fitting residualsThe results of the fitted formula are shown in Figure9. R2=0.9997>0.9994, showing that the fitting effect of the algorithm was better than that of Equation (9).
(4)
The powers ofx and y were set to higher numbers. Using this formula for fitting, R2≤0.9996. Thus, the number of calculations was increased, but the fitting effect was not improvedFigure 9
Curve fitting with Equation (10).
(a)
Fitting effect(b)
Fitting residualsOverall, when the power ofx is set to 1 and the power of y is set to 2, the resulting algorithm has a better fitting effect and is defined as the polynomial fitting-MPPT (PF-MPPT) algorithm as follows:
(11)Fx,y=33.88−0.09204x+0.01059y−2.491×10−5xy−8.242×10−7y2.When the power ofx and y is set to one, the resulting algorithm is less computationally intensive and is defined as the simple polynomial fitting-MPPT (SPF-MPPT) algorithm, as follows:
(12)Fx,y=34.94−0.1082x+0.008274y.


## 6. Simulation Analysis of the MPPT Algorithm

The variations of the solar irradiance and temperature with time are shown in Figure10. Signal builder was used to represent the changes of the solar irradiance and temperature over time as input signals.Figure 10
Variations of solar irradiance and temperature.The improved PV model was used to build the MPPT system model (CVT algorithm as an example), as shown in Figure11. The average power and amplitude of the PF-MPPT algorithm, SPF-MPPT algorithm, traditional CVT algorithm, and P&O algorithm are shown in Table 4. The simulation results of each algorithm at the operating point are shown in Figure 12.Figure 11
Maximum power point tracking (MPPT) system model.Table 4
Average power and amplitude of algorithms at the operating point (unit: W).
TimePF-MPPTSPF-MPPTAverage powerAmplitudeAverage powerAmplitude0.0–0.5 s182.510.5181.5120.5–1.0 s305.529.6308.530.31.0–1.5 s127.04.5126.64.51.5–2.0 s182.712.1182.512.52.0–2.5 s330.225.9334.326.92.5–3.0 s180.59.5180.712.1TimeCVTP&OAverage powerAmplitudeAverage powerAmplitude0.0–0.5 s157.018.2157.018.20.5–1.0 s153.831.7153.831.71.0–1.5 s129.02.9129.02.91.5–2.0 s154.618.4154.618.42.0–2.5 s148.332.1148.332.12.5–3.0 s153.917.5153.917.5Figure 12
Output power of simulation system.
(a)
PF-MPPT algorithm(b)
SPF-MPPT algorithm(c)
Constant voltage tracking (CVT) algorithm(d)
Perturb and observe (P&O) algorithmBy setting an appropriate target value, the time when the output power curve of the analysis algorithm increased from 0 W to the target value was calculated. The less time was consumed, the faster the power increased, and the faster the algorithm tracked the power.As shown in Table4, the total average power of the PF-MPPT algorithm was 218.06 W, and the total average amplitude was 15.35 W. The total average power of the SPF-MPPT algorithm was 219.0 W, and the total average amplitude was 15.88 W. The total average power of the CVT algorithm was 149.43 W, and the total average amplitude was 20.12 W. The total average power of the P&O algorithm was 219.3 W, and the total average amplitude was 15.05 W. Thus, the following conclusions can be made: (1) The power-tracking abilities of the proposed PF-MPPT and SPF-MPPT algorithms were similar to that of the P&O algorithm and were 1.46 times that of the CVT algorithm. (2) The stabilities of the tracking power of the two algorithms were similar to that of the P&O algorithm, and they were 1.26 times that of the CVT algorithm.As shown in Figure12, if the target power was set to 150 W, the PF-MPPT algorithm required 0.00925 s, the SPF-MPPT algorithm required 0.0087 s, the CVT algorithm required 0.0239 s, and the P&O algorithm required 0.03715 s. Thus, the speeds of the tracking power of the PF-MPPT and SPF-MPPT algorithms were similar. The speed of the tracking power of the PF-MPPT algorithm was over 2.58 times that of the CVT algorithm and four times that of the P&O algorithm.In conclusion, under fast-varying solar radiation, the proposed algorithm has high tracking speed and output power. However, there are some limitations. For example, the algorithm is based on the input and output data of photovoltaic modules; that is, the algorithm is more suitable for customized development than for large-scale promotion.


## 7. Output Characteristics of the PV System under Local Shading

The PV system under local shading will produce a hot-spot effect. Bypass diodes were used to provide a low impedance path for each module to protect the PV module. The photovoltaic system composed of four PV modules in series was affected by local shading, as shown in Figure13. Case 1 was the case of a PV module under normal light, case 2 was the case of a PV module under two kinds of shading, and case 3 was the case of a PV module under three kinds of shading, and the shading was more severe.Figure 13
PV system affected by local shading. (a) A schematic diagram of a PV system under local shading. (b) The irradiance of the PV module under three shading conditions.
(a)(b)Simulink was used to build a PV simulation model under local shading, as shown in Figure14. The results of the model are shown in Figure 15. The output power curve of case1 has only one global maximum power point (G1) of 1424 W. The output power curve of case2 had two local maximum power points (G21 and G22) and a global maximum power point (G23) of 794.7 W. The output power curve of case 3 had three local maximum power points (G21, G33, and G34) and one global maximum power point (G32) of 529 W. Thus, with the increase in the shading degree, the output power of the PV module decreased significantly, and there were multiple extreme values.Figure 14
PV simulation model under local shading.Figure 15
Output characteristics of PV system under local shading.


## 8. Optimization of the PV System under Fast-Varying Solar Irradiation and Local Shading

Previous PV studies focused on either fast-varying solar irradiation or local shading. Various improvements on the traditional MPPT algorithm have been proposed, including an adaptive step size improvement of the P&O, an adaptive improvement of the inertia weight or learning factor of the PSO, a gray wolf optimization and salp swarm algorithm complementary improvement, a firefly optimization, and a CVT algorithm complementary improvement. There are also ways to overcome local shading by changing the structure of the PV array.The traditional MPPT system is shown in Figure16. There are some problems with the traditional MPPT system:
(1)
The traditional MPPT algorithm control center cannot process multiple groups of environmental data at the same time, and thus, it cannot meet the requirements of the proposed PF-MPPT algorithm(2)
The problem of a sharp power reduction of a PV system under local shading cannot be avoidedFigure 16
Traditional MPPT system.To solve the problems of the traditional MPPT system, an improved MPPT system was proposed, as shown in Figure17. The system has the following advantages:
(1)
It can ensure the series relationship of solar modules(2)
An algorithm control center is provided for each PV module. This system can maximize the power generation efficiency of each PV module, overcome the problem of significant power decline under local shading, and meet the requirements of the PF-MPPT algorithmFigure 17
Improved MPPT system.


### 8.1. Irradiation Was Varied and Temperature Was 25 °C

Signal builder was used to generate the solar irradiance and temperature variation over time as input signals. The solar irradiance variations of PV modules with time are shown in Figure18. Within 0–0.5 s, the PV system was in a local shading situation similar to that in case 2. After the irradiation increased rapidly at 0.5 s, the PV system was under normal lighting conditions similar to those in case 1. After the irradiation dropped rapidly at 2 s, the PV system was in a local shading situation similar to that in case 3. The simulation results are shown in Figure 19. The average power and amplitude values of the improved MPPT system with the PF-MPPT algorithm, the MPPT system with the PSO algorithm, and the MPPT system with the P&O algorithm at the operating point are shown in Table 5.Figure 18
Solar irradiance variations of PV modules.Figure 19
Simulation results of output power.Table 5
Average power and amplitude at the operating point (unit: W).
TimePF-MPPTPSOP&OAverage powerAmplitudeAverage powerAmplitudeAverage powerAmplitude0–1 s899.134.8702.0101.9649.964.01–2 s1162.074.0922.3120.61086.5149.02–3 s654.025.2452.036.2431.436.8As shown in Table5, under the local shadows in case 2 and case 3, the average output power values of the improved MPPT system with the PF-MPPT algorithm were 899.1 and 654.0 W, respectively, exceeding the global maximum power points G23 (794.7 W) and G32 (529 W) of the ordinary PV system. This showed that the improved MPPT system with the PF-MPPT algorithm could overcome the problem of the significant power reduction of the PV system under local shading.As shown in Figure19, when the target power was set to 600 W, the MPPT system with the PSO algorithm required 0.03259 s, the MPPT system with the P&O algorithm required 0.02323 s, and the improved MPPT system with the PF-MPPT algorithm required 0.009748 s. The tracking power speed of the improved MPPT system with the PF-MPPT algorithm was more than 3.34 times that of the MPPT system with the PSO algorithm and 2.38 times that of the MPPT system with the P&O algorithm.The total average power of the improved MPPT system with the PF-MPPT algorithm was 905.03 W, and the total average amplitude was 44.67 W. The total average power of the MPPT system with the PSO algorithm was 692.1 W, and the total average amplitude was 86.23 W. The total average power of the MPPT system with the P&O algorithm was 722.6 W, and the total average amplitude was 83.27 W. Thus, the following conclusions can be made:(1)
The power-tracking ability of the improved MPPT system with the PF-MPPT algorithm was 1.31 times that of the MPPT system with the PSO algorithm and 1.25 times that of the MPPT system with the P&O algorithm(2)
The stability of the tracking power of the improved MPPT system with the PF-MPPT algorithm was 1.93 times that of the MPPT system with the PSO algorithm and 1.86 times that of the MPPT system with the P&O algorithm


### 8.2. Irradiation and Temperature Were Varied

Most previous studies on the MPPT algorithm only considered the influence of solar irradiation on the output power of the PV system but ignored the influence of the module temperature. Signal builder was used to generate the changes of the solar irradiance and temperature over time as input signals. The solar irradiance variations of PV modules with time are shown in Figure18. The relationship between the PV module temperature T, external temperature Te, and solar irradiance S is generally as follows:
(13)T=Te+0.028S.The temperature variations of PV modules with time are shown in Figure20. The average power and amplitude of the improved MPPT system with the PF-MPPT algorithm, the MPPT system with the PSO algorithm, and the MPPT system with the P&O algorithm at the operating point are shown in Table 6. The simulation results are shown in Figure 21.Figure 20
Temperature variations of photovoltaic modules.Table 6
Average power and amplitude values at the operating point (unit: W).
TimePF-MPPTPSOP&OAverage powerAmplitudeAverage powerAverage powerAmplitudeAverage power0–1 s892.350.8634.7892.350.8634.71–2 s1139.078.0838.11139.078.0838.12–3 s644.125.5421.6644.125.5421.6Figure 21
Simulation results of output power.As shown in Table6, under the local shadows in case 2 and case 3, the values of the average output power of the improved MPPT system with the PF-MPPT algorithm were 892.3 and 644.1 W, respectively, exceeding the global maximum power points G23 (794.7 W) and G32 (529 W) of the ordinary PV system. This showed that the improved MPPT system with the PF-MPPT algorithm could overcome the problem of the significant power reduction of PV systems under local shading.As shown in Figure21, when the target power was set to 500 W, the MPPT system with the PSO algorithm required 0.0767 s, the MPPT system with the P&O algorithm required 0.01759 s, and the improved MPPT system with the PF-MPPT algorithm required 0.00806 s. The tracking power speed of the improved MPPT system with the PF-MPPT algorithm was 9.52 times that of the MPPT system with the PSO algorithm and 2.18 times that of the MPPT system with the P&O algorithm.The total average power of the improved MPPT system with the PF-MPPT algorithm was 891.8 W, and the total average amplitude was 51.43 W. The total average power of the MPPT system with the PSO algorithm was 632.47 W, and the total average amplitude was 103.30 W. The total average power of the MPPT system with the P&O algorithm was 674.0 W, and the total average amplitude was 94.7 W. Thus, the following conclusions can be made:(1)
The power-tracking ability of the improved MPPT system with the PF-MPPT algorithm was 1.41 times that of the MPPT system with the PSO algorithm and 1.32 times that of the MPPT system with the P&O algorithm(2)
The stability of the tracking power of the improved MPPT system with the PF-MPPT algorithm was 2.00 times that of the MPPT system with the PSO algorithm and 1.84 times that of the MPPT system with the P&O algorithmCompared with the simulation data at a fixed temperature, the module temperature had a significant impact on the output power and stability of the PV system. The total average power of the improved MPPT system with the PF-MPPT algorithm was reduced by 13.32 W, and the total average amplitude was increased by 6.76 W. The total average power of the MPPT system with the PSO algorithm was reduced by 59.63 W, and the total average amplitude was increased by 58.63 W. The total average power of the MPPT system with the P&O algorithm was reduced by 48.6 W, and the total average amplitude was increased by 11.43 W. The improved MPPT system with the PF-MPPT algorithm had better resistance to the negative effects of temperature.In conclusion, it is necessary to introduce variations of the PV module temperature in simulation experiments so that they are more in line real systems. Compared with the MPPT system with the PSO or P&O algorithm, the improved MPPT system with the PF-MPPT algorithm can greatly improve the efficiency of PV systems in harsh environments.


## 8.1. Irradiation Was Varied and Temperature Was 25 °C

Signal builder was used to generate the solar irradiance and temperature variation over time as input signals. The solar irradiance variations of PV modules with time are shown in Figure18. Within 0–0.5 s, the PV system was in a local shading situation similar to that in case 2. After the irradiation increased rapidly at 0.5 s, the PV system was under normal lighting conditions similar to those in case 1. After the irradiation dropped rapidly at 2 s, the PV system was in a local shading situation similar to that in case 3. The simulation results are shown in Figure 19. The average power and amplitude values of the improved MPPT system with the PF-MPPT algorithm, the MPPT system with the PSO algorithm, and the MPPT system with the P&O algorithm at the operating point are shown in Table 5.Figure 18
Solar irradiance variations of PV modules.Figure 19
Simulation results of output power.Table 5
Average power and amplitude at the operating point (unit: W).
TimePF-MPPTPSOP&OAverage powerAmplitudeAverage powerAmplitudeAverage powerAmplitude0–1 s899.134.8702.0101.9649.964.01–2 s1162.074.0922.3120.61086.5149.02–3 s654.025.2452.036.2431.436.8As shown in Table5, under the local shadows in case 2 and case 3, the average output power values of the improved MPPT system with the PF-MPPT algorithm were 899.1 and 654.0 W, respectively, exceeding the global maximum power points G23 (794.7 W) and G32 (529 W) of the ordinary PV system. This showed that the improved MPPT system with the PF-MPPT algorithm could overcome the problem of the significant power reduction of the PV system under local shading.As shown in Figure19, when the target power was set to 600 W, the MPPT system with the PSO algorithm required 0.03259 s, the MPPT system with the P&O algorithm required 0.02323 s, and the improved MPPT system with the PF-MPPT algorithm required 0.009748 s. The tracking power speed of the improved MPPT system with the PF-MPPT algorithm was more than 3.34 times that of the MPPT system with the PSO algorithm and 2.38 times that of the MPPT system with the P&O algorithm.The total average power of the improved MPPT system with the PF-MPPT algorithm was 905.03 W, and the total average amplitude was 44.67 W. The total average power of the MPPT system with the PSO algorithm was 692.1 W, and the total average amplitude was 86.23 W. The total average power of the MPPT system with the P&O algorithm was 722.6 W, and the total average amplitude was 83.27 W. Thus, the following conclusions can be made:(1)
The power-tracking ability of the improved MPPT system with the PF-MPPT algorithm was 1.31 times that of the MPPT system with the PSO algorithm and 1.25 times that of the MPPT system with the P&O algorithm(2)
The stability of the tracking power of the improved MPPT system with the PF-MPPT algorithm was 1.93 times that of the MPPT system with the PSO algorithm and 1.86 times that of the MPPT system with the P&O algorithm


## 8.2. Irradiation and Temperature Were Varied

Most previous studies on the MPPT algorithm only considered the influence of solar irradiation on the output power of the PV system but ignored the influence of the module temperature. Signal builder was used to generate the changes of the solar irradiance and temperature over time as input signals. The solar irradiance variations of PV modules with time are shown in Figure18. The relationship between the PV module temperature T, external temperature Te, and solar irradiance S is generally as follows:
(13)T=Te+0.028S.The temperature variations of PV modules with time are shown in Figure20. The average power and amplitude of the improved MPPT system with the PF-MPPT algorithm, the MPPT system with the PSO algorithm, and the MPPT system with the P&O algorithm at the operating point are shown in Table 6. The simulation results are shown in Figure 21.Figure 20
Temperature variations of photovoltaic modules.Table 6
Average power and amplitude values at the operating point (unit: W).
TimePF-MPPTPSOP&OAverage powerAmplitudeAverage powerAverage powerAmplitudeAverage power0–1 s892.350.8634.7892.350.8634.71–2 s1139.078.0838.11139.078.0838.12–3 s644.125.5421.6644.125.5421.6Figure 21
Simulation results of output power.As shown in Table6, under the local shadows in case 2 and case 3, the values of the average output power of the improved MPPT system with the PF-MPPT algorithm were 892.3 and 644.1 W, respectively, exceeding the global maximum power points G23 (794.7 W) and G32 (529 W) of the ordinary PV system. This showed that the improved MPPT system with the PF-MPPT algorithm could overcome the problem of the significant power reduction of PV systems under local shading.As shown in Figure21, when the target power was set to 500 W, the MPPT system with the PSO algorithm required 0.0767 s, the MPPT system with the P&O algorithm required 0.01759 s, and the improved MPPT system with the PF-MPPT algorithm required 0.00806 s. The tracking power speed of the improved MPPT system with the PF-MPPT algorithm was 9.52 times that of the MPPT system with the PSO algorithm and 2.18 times that of the MPPT system with the P&O algorithm.The total average power of the improved MPPT system with the PF-MPPT algorithm was 891.8 W, and the total average amplitude was 51.43 W. The total average power of the MPPT system with the PSO algorithm was 632.47 W, and the total average amplitude was 103.30 W. The total average power of the MPPT system with the P&O algorithm was 674.0 W, and the total average amplitude was 94.7 W. Thus, the following conclusions can be made:(1)
The power-tracking ability of the improved MPPT system with the PF-MPPT algorithm was 1.41 times that of the MPPT system with the PSO algorithm and 1.32 times that of the MPPT system with the P&O algorithm(2)
The stability of the tracking power of the improved MPPT system with the PF-MPPT algorithm was 2.00 times that of the MPPT system with the PSO algorithm and 1.84 times that of the MPPT system with the P&O algorithmCompared with the simulation data at a fixed temperature, the module temperature had a significant impact on the output power and stability of the PV system. The total average power of the improved MPPT system with the PF-MPPT algorithm was reduced by 13.32 W, and the total average amplitude was increased by 6.76 W. The total average power of the MPPT system with the PSO algorithm was reduced by 59.63 W, and the total average amplitude was increased by 58.63 W. The total average power of the MPPT system with the P&O algorithm was reduced by 48.6 W, and the total average amplitude was increased by 11.43 W. The improved MPPT system with the PF-MPPT algorithm had better resistance to the negative effects of temperature.In conclusion, it is necessary to introduce variations of the PV module temperature in simulation experiments so that they are more in line real systems. Compared with the MPPT system with the PSO or P&O algorithm, the improved MPPT system with the PF-MPPT algorithm can greatly improve the efficiency of PV systems in harsh environments.


## 9. Conclusions

The PV model commonly used in engineering was used to establish a PV simulation model, and the model was improved based on experimental data. The improved model was used to analyze the PV output power characteristics and the influence of local shading on the PV system power generation efficiency. The results showed that the output power curve of the PV module first increased and then decreased rapidly with the increase in the voltage. With the increase in the shading degree, the output power of the PV system decreased significantly, and there were several extreme values.Aiming at the problem that most MPPT algorithms have difficulty tracking the maximum power point quickly under fast-varying solar irradiation, a PF-MPPT algorithm and an SPF-MPPT algorithm based on polynomial fitting were proposed to track the maximum power point in the current environment. The MPPT simulation system was used to analyze the performances of the PF-MPPT algorithm, SPF-MPPT algorithm, P&O algorithm, and CVT algorithm under fast-varying solar irradiation. The simulation results are as follows:(1)
The speed of the tracking power of the proposed PF-MPPT algorithm was similar to that of the SPF-MPPT algorithm, which was over 2.58 times that of the CVT algorithm and four times that of the P&O algorithm(2)
The power-tracking abilities and stabilities of the proposed PF-MPPT and SPF-MPPT algorithms were similar to those of the P&O algorithm, and they were 1.46 and 1.16 times those of the CVT algorithm, respectively(1)
Aiming at the problem that the output power of the traditional MPPT system decreases under fast-varying solar irradiation and local shading; an improved MPPT system with the PF-MPPT algorithm was proposed to track the maximum power point in the current environment. The simulation results showed the following:(2)
Under fast-varying solar irradiation and local shading, the speed, ability, and stability of the improved MPPT system with the PF-MPPT algorithm when tracking the maximum power were 9.52, 1.41, and 2.00 times those of the MPPT system with the PSO algorithm and 2.18, 1.32, and 1.84 times those of the MPPT system with the P&O algorithm, respectively(3)
Under the local shadows in case 2 and case 3, the average output power values of the improved MPPT system with the PF-MPPT algorithm were 892.3 and 644.1 W, respectively, exceeding the global maximum power points G23 (794.7 W) and G32 (529 W) of the photovoltaic system. This showed that the improved MPPT system with the PF-MPPT algorithm could overcome the problem of the significant power reduction of photovoltaic systems under local shading(4)
The module temperature had a significant impact on the output power and stability of the photovoltaic system. The improved MPPT system with the PF-MPPT algorithm had better resistance to the negative effects of the temperatureIn this paper, the effects of radiation and temperature on the output power of the PV model were investigated. In the future, more factors such as radiation, temperature, haze, dust, and wind will be studied on the PV model output power.



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*Source: 2064216-2022-08-05.xml*