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1 A New Optimization Approach for Maximizing the Photovoltaic Panel Power Based on Genetic Algorithm and Lagrange Multiplier Algorithm Mahdi M. El-Arini 2 Ahmed M. Othman 3 Ahmed Fathy Electrical Power & Machine Dept. Faculty of Engineering Zagazig University Egypt ABSTRACT In the recent years the solar energy has become one of the most important alternative sources of electric energy so it is important to operate Photovoltaic PV) panel at the optimal point to obtain the possible maximum efficiency. This paper presents a new optimization approach in order to maximize the output electrical power of the PV module. The proposed optimization technique is based on an objective function which represents the output power of PV module and some constraints which are either equality or inequality constraints. First the dummy variables that affect the output power are classified into two categories which are independent and dependant variables. The new approach is multi-stage technique in the first stage Genetic Algorithm is used to obtain the best initial population at optimal solution and then this initial is fed to the second stage which is Lagrange Multiplier algorithm to find the final optimal solution. The proposed technique is applied on Helwan city at latitude Egypt. The results showed that the proposed constrained optimization method to maximize the output power of PV module is applicable. Keywords: Maximum Power Point Tracking MPPT) Objective Function Optimization Approach Photovoltaic system and Solar Energy. 1. INTRODUCTION In the last years global warming and energy policies have become a hot topic on the international agenda. Developed countries are trying to reduce their greenhouse gas emissions. Renewable energy sources are considered as a technological option for generating clean energy. Among them Photovoltaic PV) system has received a great attention as it appears to be one of the most promising renewable energy sources. Photovoltaic power generation has an important role to play due to the fact that it is a green source. The only emissions associated with PV power generation are those from the production of its components. However the development for improving the efficiency of the PV system is still a challenging field of research. MPPT algorithms are necessary in PV applications because the MPP of a solar module varies with the irradiation and temperature as shown in Fig. 1 and Fig.2. So the use of MPPT algorithms is required in order to obtain the maximum output power from a solar Module [1]. One can define the optimization as the process that finds values of variables that minimize or maximize the objective function while satisfying the constraints. The optimization problems are centered on three main factors which are: a. b. c. Objective function which is to be minimized or maximized; for example in manufacturing we want to maximize the profit or minimize cost. A set of unknown or variables that affect on the objective function; or example in manufacturing the variables are amount of resources used for the time spent. A set of constraints that allows the unknown to take on a certain values but exclude others; for example in manufacturing one constraint is that all time variables to be non-negative. Fig 1: PV module voltage Power at different irradiance levels Fig 2: PV module voltage current at different temperature levels In the literature the most popular optimization technique of the PV power is based on the usage of the Maximum Power Point Tracking MMPT). The MPPT control method which uses one estimate processes between every two Perturb processes in search for the maximum PV output EPP) is proposed by C. Liu et al. An intelligent approach for MPPT DC/DC Boost converter focused on P&O algorithm and compared to a 79

2 designed fuzzy logic controller is presented by Mayssa Farhat et al. A comparative study of two type of Maximum Power Point Tracking MPPT) which are Perturb and Observe P&O) and incremental conductance method is introduced by B. Amrouche et al. The implementation of fuzzy logic controller based on the change of power and change of power with respect to change of voltage is studied by C. S. Chin et al. fuzzy determines the size of the perturbed voltage. The performance of fuzzy logic with various membership functions MFs) is tested to optimize the MPPT. Fuzzy logic can facilitate the tracking of maximum power faster and minimize the voltage variation. A novel intelligent fuzzy logic controller for MPPT in grid-connected photovoltaic systems based on boost converter and single phase grid-connected inverter is introduced by Guohui Zeng et al. An intelligent control method for MPPT of a photovoltaic system under variable temperature and insulation conditions which uses a fuzzy logic controller applied to a DC-DC converter device is proposed by M.S. Aït Cheikh et al. A fuzzy logic control FLC) is proposed by Pongsakor Takun et al. to control MPPT for a PV system. The analysis of the optimal operation of PV panels as a function of the weather conditions solar irradiation temperatures) and the design of a PV system provided with MPPT command ensuring instantaneously optimal operation of photovoltaic panels is presented by M. El Ouariachi et al. The various obtained results showed that the optimal photovoltaic panels electrical properties voltage current and power) depend on the solar irradiation and the panels association parallels or series). Maximum Power Point Tracking MPPT) of photovoltaic PV) system using particle swarm optimization PSO) algorithm is presented by Kashif Ishaque et al. The algorithm is employed on a buck-boost converter and tested experimentally using a PV array simulator. The design and implementation of an optimized stand-alone solar pumping system is presented by Tomás Perpétuo Corrêa et al. It investigates the validity of an alternative to reduce the payback time of photovoltaic standalone pumping systems which optimizes not only the photovoltaic conversion efficiency using a Maximum Power Point Tracking algorithm but also the losses in the induction motor. In this paper we present a new optimization approach in order to maximize the output electrical power of the PV module. The proposed optimization technique is based on an objective function which represents the output power of PV module and some constraints which are either equality or inequality constraints. First the dummy variables that affect on the output power are classified into two categories which are independent and dependant variables. The Genetic Algorithm GA) is used to obtain the initial population at optimal solution and then this initial is fed to Lagrange Multiplier algorithm to find the final optimal solution. The proposed technique is applied to Helwan city at latitude Egypt. The results showed that the proposed constrained optimization method to maximize the output power of PV module is applicable. 2. MATHEMATICAL MODEL OF PV MODULE The solar flux which strikes a collector is resolved to three components; direct-beam radiation that passes in a straight line through the atmosphere to the receiver diffuse radiation that has been scattered by molecules and aerosols in the atmosphere and reflected radiation that has bounced off the ground or other surface in front of the collector as shown in Fig. 3. Fig 3: Solar radiation striking a collector IC is a combination of direct beam IBC diffuses IDC and reflected IRC. The total rate of radiation on a clear day [1]: sin cos m Where strikes a collector 1) cos cos ) sin ) 2) 3) m: Air mass β: Altitude angle. : Solar azimuth angle. : PV module azimuth angle ve for east south and ve for west south). : PV module tilts Angle. : Reflection factor with range from about 0.8 for fresh snow to about 0.1 for a bituminousand-gravel roof with a typical default value for ordinary ground or grass taken to be about 0.2. : Sky diffuse factor and is given by the following equation [1] sin[ 100)] 4) are dependent on the day number and can be obtained by the following eqns. [1]. 80

3 275) sin / 100)] sin[ ) 5) ): The cell reverses saturation current at reference temperature The band gap of semi conductor used in the cell. The cell short-circuit current at reference temperature and radiation The short circuit current temperature coefficient The solar radiation strikes a tilted module in W/m2. : : 6) Ki : Where d is the day Number. A PV cell can be simulated by a real diode in parallel with an ideal current source which depends on impinging radiation. The generalized equivalent circuit of the PV cell including both series and parallel resistances is shown in Fig. 4 [12 13]. : The PV module consists of of series cells and of parallel branches as shown in Fig. 5. Fig 4: The equivalent circuit for a PV cell One can derive the following equation for current and voltage in one diode model: exp 7). Fig 5: The equivalent circuit for a PV module. 1 9) ) M 8) The PV module s current I under arbitrary operating conditions can be described as in [17] as follows: ) The reverse saturation current is dependent on the temperature and is given by the following eqn. ) ) ) ) Where : 10) The short circuit current depends on the solar radiation and cell temperature as follows: 11) Cell output current : Short Circuit Current : Reverse diode saturation current : Cell output voltage : Cell series resistance Ω). : Cell parallel resistance Ω). : Diode ideality factor. : Boltzmann constant 1.38e-23). : Cell junction temperature C). : The reference temperature of the PV cell exp. ) 1 and 3. PROPOSED TECHNIQUE. 12) 13) 3.1 The Proposed Optimization Problem The mathematical model of any continuous optimization problem can be described as follows: ) 14) ) is called the objective function to be minimized or maximized; and is subjected to some equality and inequality constraints: a. ) ) 81

4 b. b. 12 c. Where ℎ ) ) ) 17) : is a vector of state dependant) variables and has f-dimension. : is a vector of control independent) variables and has r-dimension. First the dummy variables that affect on the output power are classified into two categories which are independent U) and dependant variables X); [ [ ]And ] ) ) The proposed generalized objective Function is expressed in the following form: ) 1 ) ) ) ) ) ) ) ) 18) The proposed parametric constrained can be described in this section by the following eqns. < < < < < < < < < < The simplest form of genetic algorithm involves three types of operators: selection crossover single point) and mutation. Selection This operator selects chromosomes in the population for reproduction. The fitter the chromosome the more times it is likely to be selected to reproduce. Crossover This operator randomly chooses a locus and exchanges the subsequences before and after that locus between two chromosomes to create two offspring. For example the strings and could be crossed over after the third locus in each to produce the two offspring and The crossover operator roughly mimics biological recombination between two single chromosome haploid) organisms. 19) 20) The limits of independent variables are selected according to the following points: a. 3.2 Genetic Algorithms Genetic algorithms GAs) are gradient-free parallel optimization algorithms that use a performance criterion for evaluation and a population of possible solutions to the search for a global optimum. GAs is capable of handling complex and irregular solution spaces and they have been applied to various difficult optimization problems [14]. GAs is inspired by the biological process of Darwinian evolution where selection mutation and crossover play a major role. The manipulation is done by the genetic operators that work on the chromosomes in which the parameters of possible solutions are encoded. In each generation of the GA the new solutions replace the solutions in the population that are selected for deletion. The main elements of GAs are as follows: populations of chromosomes selection according to fitness crossover to produce new offspring and random mutation of new offspring. A simple GA flow chart is shown in Fig. 6. Mutation This operator randomly flips some of the bits in a chromosome. The proposed equality constraint is given as ) The solar azimuth angle is positive for east of south and becomes negative for west of south; so the limits are selected as ± 45. When 0 the module becomes flat plate and produce power while when 90; the module becomes perpendicular to the mounting surface and produces zero power; so the selected limits are assumed between 0 to The Method of Lagrange Multipliers Lagrange multipliers play a vital role in the study of constrained optimization. Lagrange multiplier can be interpreted as the rate of change in the objective function with respect to changes in the associated constraint function [15]. The main formula of this method can be derived as follows: ) 21) Where and are the respective gradients. Constant λ in Eqn. 21) are called the Lagrange multipliers of the constrained problem. 82

5 By solving eqn. 21 one can get the value of λ and then obtaining the value of U vector and X vector at optimal solution and then the optimal power extracted from the PV module. Start Generate a population of chromosomes of size N: x1 x2 xn Our analysis uses a real data for solar radiation and ambient temperature measured by solar radiation and meteorological station located at National Research Institute of Astronomy and Geophysics Helwan Cairo Egypt which is located at latitude N and longitude E. The station is over a hil top of about 114 m height above sea level. Calculate the fitness function of each chromosome fx1) fx2) fxn) a. b. c. d. Using the real data for solar radiation and ambient temperature as an input to Genetic Algorithm program that obtain the optimal output power from the PV module. Studying the effect of changing the initial population on the value of fitness function in order to obtain the optimal maximum) possible power from the PV module. The initial population at the optimal PV output power is fed to the Lagrange multiplier algorithm to find the final optimal power. A comparison between the two methods GA and Lagrange Multiplier algorithm) is performed. Fig. 7 shows these proposed technique steps. Yes Is the termination criterion is satisfied? The proposed technique is based on the following main steps: No Select a pair of chromosomes for mating With the cross over probability pc exchange parts of the two selected chromosomes and create two offspring With the mutation probability pm randomly change the gene values in the two offspring chromosomes Place the resulting chromosomes in the new population No Is the size of the new population equal to N? Yes Replace the current chromosome population with the new population Stop Fig 6: A simple GA flow chart 83

6 Solar radiation GA Program U0 X0) at maximum power Ambient temperature Lagrange Multiplier Algorithm Changing the Initial Population U0 X0) radiation and ambient temperature as inputs to GA program. These data is measured at Helwan city for the sunny day of 10 June 2012 and starts from hour 6:10 AM to hour 5:50 PM and is shown in Fig. 10. Optimal PV Output Power Fig 7: The proposed Analysis steps 4. NUMERICAL ANALYSIS The proposed technique in this paper uses bpsx150 PV module which has 72 cells connected in series and its electrical characteristic is shown in Table 1 [16]. Table 1: The electrical characteristic of bpsx150 PV module Maximum power Pmax) Voltage at Pmax Vmp) Current at Pmax Imp) Warranted minimum Pmax Short-circuit current Isc) Open-circuit voltage Voc) Maximum system voltage Temperature coefficient of Isc Temperature coefficient of Voc Temperature coefficient of power NOCT Fig 9: Detailed configuration of PV Module subsystem 150W 34.5V 4.35A 140W 4.75A 43.5V 600V 0.065±0.015)%/ C 160±20)mV/ C 0.5±0.05)%/ C 47±2 C The bpsx150 PV module is simulated by Matlab/Simulink toolbox as shown in Fig. 8 and the power voltage characteristic at different solar radiation is shown in Fig. 1 also the current voltage characteristic at different solar radiation is shown in Fig. 2. Fig 8: Simulink Model of the PV module The detailed configuration of the PV module shown in Fig. 8 is given in Fig. 9. The analysis of the proposed algorithm is performed on measured solar Fig 10: Measured solar radiation and ambient temperature The initial population of GA program which contains ) plays an important role in the optimal value of fitness function. We start the analysis by changing the initial tilt angle at acceptable limits and fixing the initial collector azimuth angle and the initial module voltage at acceptable values until the maximum power is obtained. Table 2 shows the effect of changing the initial tilt angle on the optimal power at 6:10 AM. The optimal output power is equal to W at initial tilt angle equal to 45. Then the initial tilt angle at optimal power is fixed at 45 and then changing the initial collector azimuth angle at acceptable limits with fixed acceptable initial module voltage until the optimal power is obtained. Table 3 shows the effect of changing the initial collector azimuth angle on the optimal power at 6:10 AM. The optimal output power is equal to W at initial collector azimuth angle equal to 41. Then the initial tilt angle at optimal power is fixed at 45 and the initial collector azimuth angle is fixed 41 and the initial module voltage is changed until the optimal power is obtained. 84

7 Table 4 shows the effect of changing the initial module voltage on the optimal power at 6:10 AM. The optimal output power is equal to W at initial voltage equal to 42V. Then the optimal power obtained by GA at 6:10 AM is equal to W and is occurred at initial population equal to ) V). The curve that shows the change of output power with three variables of initial population is shown in Fig. 11. to The final population at optimal solution is equal and The GA optimal power and the corresponding tilt angle or studying hours of the sunny day 10 June 2012 is shown in Fig. 13. Table 5 summarizes the optimal power and the corresponding tilt angle of the GA program for studying hours of the sunny day 10 June Once the PV module used in study has a fixed NS72 and fixed NP1 then they do not yet variables. The GA program response of the analysis of that hour is shown in Fig. 12. The values of constant dependant values at optimal solution are δ β m and ω 60. After performing the GA analysis it s important to study another solving method to ensure the validity of our proposed technique. The other method used is the Lagrange multiplier. The initial condition at optimal power which is obtained from the GA is fed to the Lagrange multiplier algorithm to find the optimal power. Fig 11: The optimal PV module power at changing the initial population in GA at 6:10 AM Fig 13: The optimal power and the corresponding tilt angle for 10 June The Matlab function fmincon is used to perform the Lagrange multiplier analysis. The Lagrange multiplier algorithm response is shown in Fig. 14. A comparison between the GA and Lagrange multiplier algorithm is performed and is showed in Fig. 14 and Table 6. From Table 6 one can derive that the difference between both techniques is very small which is considered as an acceptable error. Fig 12: The optimal PV module power at changing the initial population in GA at 6:10 AM 85

8 Fig 14 : The Lagrange multiplier response 5. CONCLUSION In this work a new optimization approach that maximizes the PV module output power is presented. A new proposed generalized objective function of PV module power and constraints are also presented in this paper. Genetic Algorithm GA) gives the initial population at optimal power and then feeds it to Lagrange Multiplier algorithm that finds the final optimal PV module power. Our analysis is based on real measured data of solar radiation and ambient temperature at Helwan city at latitude Egypt. The analysis is done for complete sunny day and clear sky of 10 June 2012 and the results showed that the error between two proposed solution techniques is acceptable. The results showed that the proposed constrained optimization method to maximize the output power of PV module is applicable. Table 2: Effect of changing the initial tilt angle on the module 6:10 AM Initial solution [Σ0 ϕc0 VM0] U0 [0 5 30] U0 [3 5 30] U0 [6 5 30] U0 [9 5 30] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [45 530] U0 [48 530] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [765 30] U0 [ ] Independent Variables Σ φ V V) Dependant Variables G Kw/m I A) * * * * * * * * * * * * * * * * * *10-6 T C) I A) P W)

9 Table 3: Effect of changing the initial collector azimuth angle on the module 6:10 AM Initial solution [Σ0 ϕc0 VM0] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] Independent Variables Dependant Variables V) Kw/m A) A) W) C) * * * * * * * * * * * * * * * * * * * * Table 4: Effect of changing the initial Module voltage on the module 6:10 AM Initial solution [Σ0 ϕc0 VM0] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] U0 [ ] Independent Variables Dependant Variables V) Kw/m A) W) * * * * * * * * * * * * * * *

10 Table 5: The optimal power and the corresponding tilt angle by GA for 10 June Hour Initial Condition at optimal solution [Σ0 ϕc0 VM0] Tilt power Optimal power W) 6:10 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM 11:00 AM 12:00 PM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 5:50 PM [454142] [543226] [273042] [423614] [213242] [70036] [183042] [ ] [ ] [3532] [ ] [ ] [ ] Fig 15: A comparison between the GA and Lagrange multiplier algorithm responses Table 6: A comparison between the GA and Lagrange multiplier algorithm Hour Genetic Algorithm Solution Lagrange Multiplier Algorithm Results Popt Error in Optimal Tilt Angle Error in Optimal Power Σopt Popt Σopt 6:10 AM E-04 7:00 AM :00 AM :00 AM :00 AM :00 AM :00 PM 1:00 PM :00 PM :00 PM :00 PM :00 PM :50 PM

11 modeling of electrical energies produced by the photovoltaic panels and systems Revue des Energies Renouvelables Vol. 14 No pp ACKNOWLEDGEMENT Cordial thanks and deep respect are offered to the Solar & space Research Dept. National Research Institute of Astronomy and Geophysics Helwan Egypt for supporting this work and providing us the required real data. [10] Kashif Ishaque Zainal Salam Amir Shamsudin and Muhammad Amjad A direct control based maximum power point tracking method for photovoltaic system under partial shading conditions using particle swarm optimization algorithm Applied Energy Journal Vol pp [11] Tomás Perpétuo Corrêa Seleme Isaac Seleme Jr. and Selênio Rocha Silva Efficiency optimization in stand-alone photovoltaic pumping system Renewable Energy Journal Vol pp [12] N. Hidouri L.sbita Water Photovoltaic Pumping System Based on DTC SPMSM Drives Journal of Electric Engineering: Theory and Application Vol.1 No. 2 pp REFERENCES [1] Gilbert M. Masters Renewable and Efficient Electric Power Systems John Wiley & Sons Inc. Hoboken New Jersey [2] C. Liu B. Wu and R. Cheung Advanced algorithm for MPPT control of photovoltaic systems Canadian Solar Buildings Conference Montreal August [3] Mayssa Farhat and Lassâad Sbita Advanced fuzzy MPPT control algorithm for photovoltaic systems Science Academy Transactions on Renewable Energy Systems Engineering and Technology Vol. 1 No. 1 March [4] [5] [6] [7] B. Amrouche M. Belhamel and A. Guessoum Artificial intelligence based P&O MPPT method for photovoltaic systems Review of Renewable Energy ICRESD-07 Tlemcen 2007 pp C. S. Chin P. Neelakantan H. P. Yoong and K. T. K. Teo Optimization of fuzzy based maximum power point tracking in PV system for rapidly changing solar irradiance Transaction on Solar Energy and Planning ISSN: Vol. 2 June Guohui Zeng Xiubin Zhang Junhao YingChangan Ji A Novel Intelligent Fuzzy Controller for MPPT in Grid-connected Photovoltaic Systems Proc. of the 5th WSEAS/IASME Int. Conf. on Electric Power Systems High Voltages Electric Machines Tenerife Spain December pp M.S. Aït Cheikh C. Larbes G.F. Tchoketch Kebir and A. Zerguerras Maximum power point tracking using a fuzzy logic control scheme Revue des Energies Renouvelables Vol. 10 No pp [8] Pongsakor Takun Somyot Kaitwanidvilai and Chaiyan Jettanasen Maximum Power Point Tracking using Fuzzy Logic Control for Photovoltaic Systems Proceedings of the International MultiConference of Engineers and Computer Scientists 2011 Vol. II IMECS 2011 March Hong Kong. [9] M. El Ouariachi T. Mrabti M.F. Yaden Ka. Kassmi and K. Kassmi Analysis optimization and [13] B.Krishna Kumar Matlab based Modelling of Photovoltaic Panels and their Efficient Utilization for Maximum Power Generation 1st National Conference on Intelligent Electrical Systems NCIES Salem India pp [14] Z. Michalewicz Genetic Algorithms Data Structures Evolution Programs Springer Verlag New York 2nd edition [15] R. Fletcher Practical Methods of Optimization 2nd ed. Wiley New York [16] [17] Ahmed M. Othman Mahdi M. M. El-arini Ahmed Ghitas and Ahmed Fathy Realworld Maximum Power Point Tracking Simulation of PV System Based on Fuzzy Logic Control The third Arab Conference on Astronomy and Geophysics "ACAG-3"NRIAG 8-11 Oct 2012 Helwan Cairo Egypt. AUTHOR PROFILES Mahdi M. El-Arini 1955) has been a professor at Zagazig University Egypt since 1999 where he is now Vice Dean of Faculty of Engineering and Head of Electronics Department in Zagazig University. The main activities of Dr. Mahdi include Voltage and Reactive Power 89

12 Control power system optimal operation and control of power systems contingency evaluation of power systems and system security and system stability. Ahmed M. Othman 1980) He received the B.Sc. and M.Sc degrees from Faculty of Engineering Zagazig University in 2002 and 2007 respectively. He received Ph.D. degree from Electrical Engineering Dept. Helsinki University of Technology TKK) Finland and he is now works as lecturer at the Electrical Power & Machine Dept. Faculty of Engineering Zagazig University Egypt. His main interest research is FACTS devices their control with AI. Ahmed Fathy 1984) He received the B.Sc. and M.Sc degrees from Faculty of Engineering Zagazig University in 2006 and 2010 respectively. Currently he is working towards the Ph.D. degree Electrical Power & Machine Dept. Faculty of Engineering Zagazig University Egypt.His main interest research is optimization processes power system dynamics Photovoltaic system and the application of AI. 90