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Estimation of maximum power point of a double diode model photovoltaic module

2017; Institution of Engineering and Technology; Volume: 10; Issue: 6 Linguagem: Inglês

10.1049/iet-pel.2016.0632

1755-4543

Himanshu Sekhar Sahu, Sisir Kumar Nayak,

Solar Radiation and Photovoltaics

IET Power ElectronicsVolume 10, Issue 6 p. 667-675 Research ArticleFree Access Estimation of maximum power point of a double diode model photovoltaic module Himanshu Sekhar Sahu, Corresponding Author Himanshu Sekhar Sahu himanshu.sahu@iitg.ernet.in Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039 IndiaSearch for more papers by this authorSisir Kumar Nayak, Sisir Kumar Nayak Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039 IndiaSearch for more papers by this author Himanshu Sekhar Sahu, Corresponding Author Himanshu Sekhar Sahu himanshu.sahu@iitg.ernet.in Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039 IndiaSearch for more papers by this authorSisir Kumar Nayak, Sisir Kumar Nayak Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039 IndiaSearch for more papers by this author First published: 01 May 2017 https://doi.org/10.1049/iet-pel.2016.0632Citations: 8AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study presents a robust technique to estimate the maximum power point (MPP) of a double-diode model (DDM) photovoltaic (PV) module. The MPP of the DDM PV module under different environmental conditions (DECs) are estimated by using the simple and computationally efficient Levenberg–Marquardt method. The variation of double diode PV module parameters with the change of temperature and irradiation is analysed. The MPP of the DDM PV array at non-standard test condition obtained from the proposed method is verified with the MPP obtained experimentally and by MATLAB simulation. A comparative study of MPP of a DDM PV module estimated from the proposed method with different existing methods is performed for different operating conditions. The maximum measurement error in power generation of a DDM PV module for different methods under DEC is presented which indicates better performance of the proposed method for the estimation of MPP of a DDM PV module. 1 Introduction Photovoltaic (PV) source is more promising as compared to other renewable energy sources to fulfil the present energy demand [1, 2]. Nowadays, the installation of PV plants in residential and commercial use is promoted by the development of technology [3, 4]. Proper modelling of the PV cells and the modules are required for understanding of the behaviour of the PV systems and also it is essential for the grid integration. In the literature, many researchers have designed PV cells and modules by taking the single-diode model (SDM) and double-diode model (DDM). The SDM of the PV module is most widely used because of its accuracy and low computational cost. However, since the recombination losses are not taken into account, its accuracy at open-circuit voltage (OCV) highly diminishes at low solar irradiation level [5, 6]. The DDM PV module is used to improve the modelling accuracy by connecting another diode in parallel with SDM. Since, DDM takes into account the recombination losses and hence it is more accurate than the SDM at low solar irradiation for representing the cell behaviours [5, 7-11]. Therefore, the DDM circuit is most useful for PV module and it is considered for this study. However, the estimation of PV parameters of the DDM is a challenging problem because of solving the number of non-linear equations. To reduce the complexity of solving non-linear equations, many researchers have assumed that the diode quality factors of two diodes with a reasonable accuracy based on Shockley's diffusion theory are and [5, 12, 13]. Maximum power point tracking (MPPT) of a PV array is required for the integration of solar power to the grid. Since, the output current and voltage characteristic of a PV array is non-linear in nature and it varies with solar irradiation and temperature and hence MPPT of a PV array is not simple. Many researchers have used three different methods such as direct, artificial intelligence, and indirect methods [14, 15] to track the maximum power point (MPP) of a PV array. The direct methods [16-19] are used for online tracking of MPP of a PV module. The artificial intelligent methods are neural fuzzy logic, neural network, genetic algorithm (GA) [20, 21] and pattern search (PS) algorithm [22] but the practical implementation of these methods may not be the first choice due to computational demanding. The indirect methods for the estimation of MPP of a PV array [15] are based on mathematical functions derived from empirical data and these methods are fractional open-circuit voltage (OCV) and fractional short-circuit current (SCC) [23]. These methods are very simple, however, the accuracy of these methods is not good enough. The MPP of a PV module is estimated by considering the linear MPP locus [24]. However, the linear approximation is not valid in the low irradiation level as well as the assumptions made on irradiation independence on the OCV may not be valid. An analytical method for estimation of MPP based on mean value theorem (MVT) has been proposed in [25] which does not require any iteration scheme. However, the periodic measurement of the OCV by power interruption is still needed, which is the disadvantage of this method. An analytical expression of MPP voltage and current for an ideal SDM PV module based on several assumptions for the estimation of maximum power under different environmental conditions (DECs) has been proposed by Saloux et al. [26]. For the direct MPP calculation of the SDM under different operating conditions, an accurate and computationally efficient explicit expression in terms of five parameters of the SDM using the Lambert W function has been proposed [27]. However, the explicit expression of MPP voltage and current of the DDM using the Lambert W function method is difficult due to the complexity of the DDM. In this study, the Levenberg–Marquardt (LM) algorithm for the estimation of MPP of a DDM PV module is categorised into the indirect methods. A non-linear least square (NLS) method based on trust region reflective algorithm has been proposed to estimate the PV module parameters [28], but it may fail to converge at an optimum point. In recent a contribution, Gauss–Seidel (GS) method is used for the estimation of the PV parameters of an SDM of a module using datasheet values, and the MPP of a PV array under DEC is estimated by using modified GS method [29]. However, modified GS method may converge at the wrong point in some cases. Considering the aforementioned limitations, this paper presents the estimation of MPP of a DDM PV module based on its characteristic equation by using the LM method for DEC, assuming the diode quality factors of two diodes as 1 and 2, respectively. Since the LM algorithm is the combination of steepest descent (SD) and Gauss–Newton (GN) algorithms, it is stable and has fast response and hence it is a robust algorithm [30, 31]. A comparison of the LM method with six different existing MPPT methods such as GA, PS, MVT, Saloux, NLS, and GS is presented under DECs. It is observed that the accuracy of the LM method is more as compared to other methods and the time taken to obtain MPP () of a DDM PV module is low for different operating conditions but slightly more than the two explicit methods such as MVT and Saloux. However, the accuracy of the estimation of the MPP of a module in MVT and Saloux methods is less as compared to other methods. Thus, the performance of the proposed method to estimate the MPP of a DDM PV module is more accurate and computationally efficient. In Section 2, the parameters of a DDM PV module at STC (standard test condition) and for DEC are estimated. The effect of solar irradiation and temperature on the PV parameters is analysed in this section. The estimation of MPP of a DDM PV module is presented in Section 3. The details of modelling of a PV array are explained and the simulation results of MPP of a PV array are validated with the experimental results under DEC and presented in Section 4. Finally the conclusion is presented in Section 5. 2 Parameters estimation of a DDM PV module In this paper, a DDM is presented which accounts the recombination loss in the depletion region of a PV module to estimate the MPP accurately. Fig. 1 shows an electrical equivalent circuit of the DDM PV module. Fig. 1Open in figure viewerPowerPoint Equivalent circuit of DDM PV module The output current of the PV module is given as follows: (1)where I is the current generated by the module, is the photo-generated current, and are the dark saturation currents, and are the diode quality factors of two diodes, V is the output voltage, and are the parallel and series resistances, respectively, is the number of cells connected in series, is the junction thermal voltage. Where k is the Boltzman's constant, T is the junction operating temperature, and q is the charge of an electron. In this work, to reduce the computational time and complexity for calculation of non-linear equations, the values of diode quality factor of two diodes are approximated as and . Thus, seven parameters such as , , , , , , and of the DDM PV module are reduced to five parameters DDM and those five parameters are estimated from the datasheet values provided by the manufacturer as presented in Table 1. Three important points such as (, 0), (0, ), and (, ) in the (voltage–current) V–I curve [32] of the PV module are used to estimate the five parameters by using the iterative method. , , , and are OCV, SCC, MPP voltage, and MPP current, respectively. Table 1. PV parameters (a) Datasheet values Module parameters Actual values 8.76 A 37.30 V 30.30 V 8.25 A 60 0.05% % (b) Estimated values Module parameters Estimated values 8.76 A 0.2687 nA Substituting the values of and and putting the aforementioned three important points in (1), (2)–(4) are obtained as follows: (2) (3) (4)To obtain five parameters, two more equations are required. The power generation of the PV module at any point on the V–I curve is given as follows: (5)The derivative of power with respect to (w.r.t.) voltage at MPP is as follows: (6)Taking the derivative of power w.r.t. voltage, (5) can be written as follows: (7)Substituting (6) into (7), yield (8)Also the derivative of current in (1) w.r.t. voltage can be written as follows: (9)Substituting the value of from (8) into (9), the resulting equation can be written as follows: (10)Substituting from (2) into (3) and (4), the equations are obtained as follows: (11) (12)Four unknown PV parameters such as , , , and are presented in three equations such as (10)–(12). To estimate the four PV parameters, the equation derived from the derivative of current w.r.t. voltage at short-circuit point in the V–I curve is considered and it is expressed as follows [7]: (13)Putting (13) into (9) and taking some assumption [33], the equation obtained is given as follows: (14)Solving (10)–(12), and (14) using Newton–Raphson (NR) method, the unknown PV parameters such as , , , and are obtained. Then the value of is obtained by using the estimated PV parameters value in (2). The datasheet values of the PV module and the estimated PV parameters value of the module at STC (STC, i.e., temperature and solar irradiation ) are given in Table 1. 2.1 Temperature and solar irradiation dependence PV parameters The SCC and OCV of the PV module vary linearly with temperature [34] and the relationships are given as follows: (15) (16)The dark saturation currents and of two diodes mainly depend on the temperature and these are given as follows [35]: (17) (18)where is the band gap energy of the material at STC and for silicon cell its value is 1.121 eV. depends on cell temperature and is given as follows: (19)The and of the PV module at all operating conditions are given as follows [36]: (20) (21)The photo-generated current of the PV modules depends on the temperature and solar irradiation and is given as follows [34, 37]: (22)where is the photo-generated current at STC and is the temperature coefficient of the PV module. The SCC depends on solar irradiation and is given as follows: (23)where is the SCC at STC. For solar irradiation dependence, the OCV of the double-diode PV module from (2) is written as follows: (24)For the estimation of under different irradiation conditions, NR method is used to solve the above non-linear equation. 3 Estimation of MPP of a DDM PV module The estimation of MPP of the module at a given environmental conditions is required for integration of PV power with the power grid. In this paper, a novel approach is used to estimate the MPP of a DDM PV module at given environmental conditions by using the estimated PV parameters such as, , , , and . For the estimation of MPP of a DDM PV module, first the and under given environmental condition are estimated. To estimate the and at a particular solar irradiation and temperature, two equations in terms of unknown variables and are required. Since the values of and are very small as shown in Table 1(b), the Jacobian matrix may close to singular. Therefore, by taking the approximation and in the PV module [7], (11) can be expressed as follows: (25)The expressions of and obtained by solving (25) and (12) are given as follows: (26) (27)where and . Substituting and obtained from (26) and (27) into (10) and (14), the equations are obtained as follows: (28)where and (29)where , , and The above two non-linear (28) and (29) are solved by using the LM algorithm to obtain two unknown parameters, i.e. and . The LM algorithm [31, 38] to obtain the unknown parameters is expressed as follows: (30)where is the present weight, is the next weight, () Hessian matrix evaluated at , is the error vector, and (>0) is the combination coefficient. A flowchart for the LM algorithms is shown in Fig. 2. In this flowchart, and are the present and the last total errors, respectively. Fig. 2Open in figure viewerPowerPoint Algorithm for estimation of MPP of a DDM PV module by using LM method The LM method behaves as the combination of SD and GN based on different orders of gradient. The LM method begins with the SD to take the advantage of its low sensitivity to initial values and when the calculated value closes to the final solution, it behaves as GN for the fast convergence. The automatic switching sequence of LM method from SD to GN is controlled by the parameter . When the LM method behaves as SD, the parameter plays a crucial rule and the Hessian matrix in (30) becomes dominant in diagonal, but when the LM method behaves as GN stage, the parameter changes automatically and takes small value, this ensures that the Hessian outweighs the matrix . If the error is decreased after updating the algorithm given in (30), then is decreased by a factor of 10 for decreasing the gradient descent. Otherwise, if the error is increased, then is increased by a factor of 10 for increasing the gradient. Once the algorithm is converged, it will give voltage and current at MPP, in return, it will give the maximum power () generation of a DDM PV module. 4 Results and discussion In a large PV plant, an array is formed by connecting the modules in series and parallel [39] and the parameters of an array are estimated from the parameters of the DDM PV modules. In this analysis, a 60 kW, 10 × 24 PV array (number of modules in series, i.e. , number of strings in parallel, i.e. ) is designed by taking 240 modules of the same rating as given in Table 1. The array parameters estimated from the module parameters at STC are presented in Table 2. Table 2. PV array parameters are estimated from module parameters Datasheet values Estimated values 4.1 MPP estimation of a PV array The present LM algorithm is validated by comparing the simulation result with an experimental result for a 5 × 5 PV array of rating 75 W at fixed temperature with different irradiations. Further, the present algorithm is utilised to simulate 60 kW PV array based on DDM at standard temperature with different irradiations and at standard irradiation with different temperatures. The voltage–current (V–I) and voltage–power (V–P) characteristics of a 60 kW PV array under DECs are analysed, and an actual MPP of a PV array is estimated from the above characteristics. By using the proposed algorithm, the MPP of the same array is also estimated and compared with an actual MPP. The peak value of power in V–P characteristics of a PV array is taken as an actual maximum power () and the MPP of a PV array under DECs is estimated using the LM algorithm by taking the irradiation and temperature dependant and . The V–I and V–P characteristics of a 60 kW PV array for varying irradiations from 200 to 800 W/m2 at standard temperature of are presented in Figs. 3a and b, respectively. It is observed that the estimated maximum power () of a PV array by using the LM algorithm varies from 11.411 to 48.064 kW and the obtained from the curve varies from 11.510 to 49.150 kW with the increase of solar irradiation as given in Table 3. The V–I and V–P characteristics of the same PV array for varying temperatures from 20 to 80°C at standard irradiation of 1000 W/m2 are presented in Figs. 3a and b, respectively. It is observed that the and vary from 61.776 to 49.429 kW and 63.150 to 49.770 kW, respectively, with an increment of temperature as given in Table 3. Table 3. MPP of a PV array under DEC Different irradiation at a temperature of 25°C Different temperature at a standard irradiation of 1000 W/m2 Irradiation, W/m2 , V , A , kW , kW (%) Error of Temperature, °C , V , A , kW , kW (%) Error of 200 293.97 38.82 11.411 11.510 0.86 20 308.76 200.08 61.776 63.150 2.17 400 301.05 78.79 23.719 23.890 0.71 40 289.44 198.62 57.488 59.190 2.87 600 302.06 118.78 35.878 36.260 1.05 60 271.41 197.05 53.481 54.030 1.016 800 302.79 158.74 48.064 49.150 2.20 80 252.58 195.70 49.429 49.770 0.68 Fig. 3Open in figure viewerPowerPoint Simulation results of a 60 kW PV array (a) V–I and (b) V–P curves of a PV array at standard temperature () with varying irradiations and at an irradiance of 1000 W/m2 with varying temperatures, respectively The percentage (%) error in w.r.t. of a PV array under aforementioned environmental conditions is calculated as follows: (31)It is observed from Table 3 that the relative errors of the by the proposed method w.r.t. the of a PV array under DEC are minimal. To evaluate the accuracy of the proposed method, the estimation of MPP of a PV array is carried out for different operating conditions, i.e. non-STC. For this analysis, the solar irradiation is varied from 100 to 900 W/m2 with the increment of 100 W/m2 and the temperature is varied from 20 to 140°C with the increment of 20°C. The deviation of from of a PV array under different operating conditions is presented in Fig. 4. It is observed from Fig. 4 that the closely matches with of an array under different operating conditions. From Table 4, it is observed that the % error of w.r.t. of an array is minimal under different operating conditions. Therefore, the proposed algorithm is robust and accurate to estimate the MPP of an array for any operating condition. Table 4. (%) Error of in the proposed method under different operating conditions Irradiation, W/m2 (%) Error at 20°C (%) Error at 40°C (%) Error at 60°C (%) Error at 80°C (%) Error at 100°C (%) Error at 120°C (%) Error at 140°C 100 1.22 1.13 1.08 1.11 0.93 1.07 0.98 200 0.83 0.93 0.95 1.07 0.73 0.88 0.87 300 1.08 1.18 1.22 0.78 0.94 0.87 0.58 400 1.13 1.30 1.30 1.47 1.28 1.20 1.35 500 0.98 1.07 0.87 1.12 1.19 1.18 0.85 600 1.33 1.46 1.10 1.64 1.52 1.65 1.28 700 1.16 1.35 1.43 1.27 1.18 1.33 1.03 800 2.22 2.13 2.00 1.95 1.76 1.96 1.81 900 1.86 1.74 1.61 1.53 1.37 1.60 1.42 Fig. 4Open in figure viewerPowerPoint Comparison of using the proposed method with under different operating conditions A comparison of the LM method with different existing methods such as GA, PS, MVT, Saloux, NLS, and GS in terms of optimal solution, i.e. MPP and execution time to obtain the MPP () are carried out by Matlab simulation for a 250 W DDM PV module as given in Table 1. The simulation results of above different methods for the MPP of a 250 W DDM PV module at a temperature of with varying irradiations are presented in Table 5. The of 176.80, 125.70, and 74.20 W under different solar irradiations such as, 700, 500, and 300 W/m2, respectively, are obtained from the V-P characteristics of a DDM PV module at a temperature of . Considering the population size of 100, a crossover fraction of 0.845 and tournament function in GA method, the maximum power of 173.21, 123.17, and 70.89 W of a DDM PV module are achieved after time duration, i.e. of 36.67, 43.08, and 56.64 s with an irradiations of 700, 500, and 300 W/m2, respectively as shown in Fig. 5. From Table 5, it is observed that the GA method has highest computational cost among all the aforementioned existing methods. The explicit methods such as MVT and Saloux require less execution time but high % error of among all the methods. However, it is also observed that the maximum power obtained from the DDM PV module by the LM method is more or less same as the actual maximum power of the same module for different solar irradiations. The proposed method has low which is slightly more than the of MVT and Saloux methods. In Fig. 6, the V–P characteristic of the module is plotted for the solar irradiation of 700 W/m2 and the for an aforementioned methods are denoted in the V–P characteristic for further study of the accuracy of the LM method. This figure shows that the using the LM method is more or less same as whereas there is a significant deviation of w.r.t. in other methods. Therefore, the estimation of MPP of a DDM PV module by the proposed method under DEC is more accurate and computationally efficient. Table 5. Comparison of MPP of a 250 W DDM PV module at a temperature of with varying irradiations Different MPPT methods Irradiation, W/m2 , V , A , W (%) Error of w.r.t. , s GA 300 27.80 2.55 70.89 4.46 56.64 500 29.12 4.23 123.17 2.01 43.08 700 29.21 5.93 173.21 2.03 36.67 PS 300 28.81 2.48 71.44 3.71 2.7961 500 29.37 4.19 123.06 2.10 3.0284 700 29.77 5.84 173.85 1.66 2.8589 MVT 300 29.00 2.44 70.76 4.63 0.01135 500 29.54 4.10 121.11 3.65 0.01337 700 29.92 5.77 172.63 2.35 0.01012 Saloux 300 30.15 2.59 78.08 5.22 0.00981 500 30.28 4.29 129.90 3.34 0.00866 700 30.37 6.02 182.82 3.40 0.00852 NLS 300 29.68 2.45 72.71 2.00 0.400136 500 30.04 4.11 123.46 1.78 0.410115 700 30.16 5.78 174.27 1.43 0.37728 GS 300 29.12 2.47 71.92 3.07 0.17425 500 29.60 4.13 122.24 2.75 0.18316 700 30.10 5.79 174.27 1.43 0.15924 LM 300 29.87 2.45 73.18 1.37 0.01978 500 30.21 4.11 124.16 1.22 0.02018 700 30.29 5.78 175.07 0.97 0.01979 Fig. 5Open in figure viewerPowerPoint Maximum power generation of a 250 W DDM PV module in GA method under DECs Fig. 6Open in figure viewerPowerPoint using different methods at an irradiation of 700 W/m2 and temperature of 25°C 4.2 Experimental validation To verify the simulation results, an experiment is performed in the Department of Electronics and Electrical Engineering, IIT Guwahati by taking the same rating of 3 W PV modules, two current and voltage sensors, one rheostat (1000 Ω, 2 A), and the transparent papers of different thickness to set the different intensities of solar irradiance as shown in Fig. 7a. The datasheet values of a 3 W PV module provided by the manufacturer are , , , , and , and . The power generated from a 5 × 5 PV array is measured by considering the current and voltage sensor data. Fig. 7Open in figure viewerPowerPoint Experimental validation of the proposed method for 75 W PV array (a) Laboratory scale experimental setup, and (b) V–P curve of a PV array at a temperature of 38°C with varying irradiations The level of solar irradiance while performing the experiment (10:30 AM, 9 April 2016) is calculated by using (23) and the data sheet values of the module provided by the manufacturer. The temperature was measured by the temperature sensor (i.e., ) at the time of experiment. The calculated solar irradiance while conducting the experiment is 810 W/m2, and other different levels of solar irradiance such as, 605, 520, and 380 W/m2 are obtained by using different thickness of transparent paper on the modules. The V–P characteristics of a 5 × 5 PV array of 75 W rating are presented in Fig. 7b. It is seen from Fig. 7b that there is a deviation between the simulated and measured V–P curve of a PV array for lower voltage range from 0 to 30 V and high voltage range from 40 to 50 V. The reasons for the deviation between the two curves are as follows: (i) the solar irradiation may be changing slightly while performing an experiment, (ii) at lower voltage range, i.e. 0–30 V before MPP, the variation in load resistance (rheostat) is not smooth whereas, at the MPP region, the resistance is varied precisely to obtain the correct MPP of an array, and (iii) since the current changes significantly with small change in voltage at higher voltage range, i.e. 40–50 V after MPP in the V–I curve, the variation in electrical load, i.e. rheostat is not smooth enough to collect the current and voltage experimental data precisely. However, if the V–I curve of the PV array is obtained using an electronic load (6063B 250 W DC Electronic Load by Agilent) instead of electrical load, then the measured V–I curve can closely match with the simulated V–I curve of an array. Hence, the V–P curves obtained experimentally will be more or less same as the simulated V–P curve. The deviation of the in LM method from and measured maximum power () of a PV array are minimal compared to other methods as given in Table 6. The % error of w.r.t. and in LM method is always less as compared with other existing methods for different irradiation conditions as shown in Fig. 8. Table 6. Comparison of with and of a 5 × 5 PV array under different irradiations at a temperature of G, W/m2 in GA, W in PS, W in MVT, W in Saloux, W in NLS, W in GS, W in LM, W , W , W 380 22.78 22.85 22.46 25.06 23.10 23.02 23.57 23.38 23.75 520 30.56 30.75 30.54 34.48 31.44 30.88 32.79 31.95 33.91 605 36.78 37.27 36.99 41.40 38.11 37.00 39.28 38.42 40.35 810 52.05 52.69 52.01 57.21 53.09 52.74 54.51 53.68 55.52 Fig. 8Open in figure viewerPowerPoint (%) Error of w.r.t. and in different methods for different irradiation conditions The small variation in maximum power is due to the little variation in solar irradiance and ambient temperature while performing the experiment and discrepancy in the PV cell parameters. Fluke model 175 multimeters which have an accuracy of 1% in DC current and 0.15% in DC voltage has been used for the measurement of current and voltage of an array. Therefore, the total measurement error of ±1.15% is introduced in the present measurement of power generation of a PV array under DEC. Considering the error of ±1.15%, the measured power of an array at an irradiation of 380 W/m2 will vary from 23.11 to 23.64 W as given in Table 7. The power generation of an aforementioned PV array in GA, PS, MVT, Saloux, NLS, GS, and LM methods by MATLAB SIMULINK will vary from 22.55 to 23.00 W, 22.58 to 23.11 W, 22.20 to 22.71 W, 24.77 to 25.34 W, 22.83 to 23.36 W, 22.78 to 23.25 W, and 23.33 to 23.80 W, respectively, as given in Table 7. Hence, the maximum measurement error introduced in the present analysis by considering the maximum measured power of 23.64 W with minimum estimated power of 22.55 W in GA, 22.58 W in PS, 22.20 W in MVT, 22.94 W in NLS, and 22.78 W in GS are 4.61, 4.48, 6.09, 3.42, and 3.63%, respectively. Similarly, the maximum measurement error of 8.80% is introduced in Saloux method by considering the maximum estimated power of 25.34 W in Saloux with minimum measured power of 23.11 W. However, the maximum measurement error of 2.89%, which is well within the limit, is introduced in the LM method by considering the maximum estimated power of 23.80 W in LM with minimum measured power of 23.11 W. In the same way, the maximum measurement error introduced in aforesaid methods for the solar irradiation of 520, 605, and 810 W/m2 can be estimated. Table 7. Maximum measurement error in the power generation of a 5 × 5 PV array in different methods for the irradiation of 380 W/m2 (%) Uncertainty in measurement in GA, W in PS, W in MVT, W in Saloux, W in NLS, W in GS, W in LM, W , W +1.15 corrected 22.55 22.58 22.20 24.77 22.83 22.78 23.33 23.11 actual measurement 22.78 22.85 22.46 25.06 23.