# A New Optimization Approach for Maximizing the Photovoltaic Panel Power Based on Genetic Algorithm and Lagrange Multiplier Algorithm

Authors: Mahdi M. M. El-Arini; Ahmed M. Othman; Ahmed Fathy
Journal: International Journal of Photoenergy
 (2013)

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## Abstract

In 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 to maximize the electrical power of a PV panel. The technique which is based on objective function represents the output power of the PV panel and constraints, equality and inequality. First the dummy variables that have effect on the output power are classified into two categories: dependent and independent. The proposed approach is a multistage one as the genetic algorithm, GA, is used to obtain the best initial population at optimal solution and this initial population is fed to Lagrange multiplier algorithm (LM), then a comparison between the two algorithms, GA and LM, is performed. The proposed technique is applied to solar radiation measured at Helwan city at latitude 29.87°, Egypt. The results showed that the proposed technique is applicable.


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## Body

## 1. Introduction

In the last years, global warming and energy policies have become a hot topic on the international agenda as the 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. PV power generation has an important role to play as 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. Maximum power point tracking (MPPT) algorithms are necessary in PV applications because the MPP of a solar module varies with the irradiation and temperature as shown in Figures1 and 2. So, the use of MPPT is required to obtain the maximum output power from a module [1]. The optimization process can be defined as the finding values of variables that minimize or maximize the objective function and satisfy the constraints. The optimization problems are centered on three main factors which are as follows:(1)
objective function which is to be minimized or maximized;(2)
a set of unknown variables that have effect on the objective function;(3)
A set of constraints that allow the unknown to take on certain values but exclude others.Figure 1
PV module voltage-power at different irradiance levels.Figure 2
PV module voltage-current at different temperature levels.The most popular literature optimization techniques of the PV power are based on the usage of the MPPT. The MPPT control method uses one estimate processes between every two Perturb processes in search for the maximum PV output (EPP) which is proposed in [2].An intelligent approach for MPPT DC-DC Boost converter focused on Perturb and Observe (P&O) algorithm and compared to a designed fuzzy logic controller which is presented in [3]. A comparative study of two types of MPPT which are P&O and incremental conductance method is introduced in [4]. The implementation of fuzzy logic controller based on the change of the PV module power, ΔPpv, and its change with respect to the module voltage, ΔPpv/ΔV is studied in [5], the proposed fuzzy logic determines the size of the voltage then obtains the corresponding Power. The performance of fuzzy logic with various membership functions (MFs) is tested to optimize the MPPT. A novel intelligent fuzzy logic controller for MPPT based on boost converter and single-phase grid-connected inverter is introduced in [6]. An intelligent control method for MPPT of a photovoltaic system under variable temperature and insolation conditions which uses a fuzzy logic controller applied to a DC-DC converter device is proposed in [7]. A fuzzy logic control to control MPPT for a PV system is proposed in [8]. 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 in [9].  MPPT simulation based on using particle swarm optimization (PSO) algorithm is presented in [10]. 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 are presented in [11].This paper presents a new optimization approach to maximize the electrical power of the PV panel. The technique which is based on objective function represents the output power of the PV module and constraints, equality and inequality. First, the dummy variables that have effect on the output power are classified into two categories: dependent and independent. The proposed approach is a multistage one as GA is used to obtain the best initial population at optimal solution, and then the best initial population is fed to Lagrange multiplier algorithm (LM), then a comparison between two algorithms, GA and LM, is performed. The proposed technique is applied to solar radiation measured at Helwan city at latitude 29.87°, Egypt. The results showed that the proposed technique is applicable.


## 2. Mathematical Model of PV Module

The solar flux striking a PV panel can be 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 in the atmosphere, and reflected radiation that has bounced off the ground or other surface in front of the collector as shown in Figure3.Figure 3
Direct, diffuse, and reflected beams that strike the PV panel.The total rate of radiationGC strikes a collector on a clear day as follows [12]:
(1)GC=GBC+GDC+GRC,GC=Ae-km[{(1-cosΣ2)}cosβcos(φS-φC)sinΣ+sinβcosΣ+C(1+cosΣ2)+ρ(sinβ+C)(1-cosΣ2)],m=h2h1=1sinβ,
where M is the air mass, β is the altitude angle, φS is the solar azimuth angle, φC is the PV module azimuth angle, positive for eastsouth and negative for westsouth, Σ is the PV module tilt angle, ρ is reflection factor, and C is sky diffuse factor and is given by the following equation [12]:
(2)C=0.095+0.04sin[360365(d-100)],
where A and k are dependent on the day number and can be obtained by the following equations [1] and d is the day number:
(3)A=1160+75sin[360365(d-275)](W/m2),k=0.174+0.035sin[360365(d-100)].A PV cell can be simulated by a real diode in parallel with an ideal current source,ISC, which depends on striking radiation. The generalized equivalent circuit of the PV cell is given in Figure 4 [13, 14].Figure 4
The equivalent circuit for a PV cell.One can derive the following equations for current and voltage:(4)ISC=I+Id+IP,I=ISC-I0{exp[q(VC+I·RsCnkbTC]-1}-(VC+I·RsCRPC),I0=ISC(eqV/nkbTC-1).The reverse saturation current,I0, is dependent on the temperature and is given by the following equation:
(5)I0(TC)=I0(Tref)*(TCTref)3*exp(qEGnK(1Tref-1TC)).The short circuit current,ISC, depends on the solar radiation and cell temperature as follows:
(6)ISC=[ISCr+Ki(TC-Tref)]*GC,
where I is the cell output current, V is the cell output voltage, and RS is the cell series resistance (Ω).R
P is the cell parallel resistance (Ω), n is the diode ideality factor, kb is the Boltzmann constant (1.38e-23 J/K), TC is the cell temperature (°C), Tref is the reference temperature of the PV cell, I0(Tref) is the cell reverse saturation current at reference temperature, EG is the band gap of semiconductor used in the cell, ISCr is the cell short circuit current at reference temperature and radiation, Ki is the short circuit current temperature coefficient, and GC is the solar radiation strikes a tilted module in W/m2.The cell temperature is calculated as follows:(7)TC=Tambient+(NOCT-20°0.8)*GC,
where Tambient is ambient temperature and NOCT is cell temperature in a module when ambient is 20°C.A PV module consists of series cells,NS, and parallel branches, NP, which are given in Figure 5.Figure 5
The equivalent circuit for a PV module.The PV module’s currentIM under variable operating conditions can be described as follows [15]:
(8)IM=NpISC-NpI0{exp[q(VM/NS+IM·RSM)nkbT]-1}-(VM/NS+IM·RSMRPM),RsM=NSNpRs,RPM=NpNSRP,VM=NSV.


## 3. Proposed Technique

### 3.1. The Proposed Optimization Problem

The mathematical model of any continuous optimization problem can be described as follows:(9)minimizeormaximizef(X,U),
where f(X,U) is an objective function to be minimized or maximized subjected to equality and inequality constraints as follows:(1)
equality constraint(10)qi(X,U)=0,i=1,2,…,l,(2)
parametric inequality constraint(11a)Xjmin≤Xj≤Xjmax,j=1,2,…,f,(11b)Ukmin≤Uk≤Ukmax,k=1,2,…,r,(3)
functional inequality constraint(12)hy(X,U)≤0,y=1,2,…,p,where X is a vector of state, dependant, variables and U is a vector of control (independent) variables. First the dummy variables that have effect on the PV module power are classified into two categories: independent (U) and dependent variables (X)—U=[NS,NP,d,Σ,φC] and X=[β,m,φS,GC,ISC,I0,TC].The proposed objective function is expressed in the following form:(13)maximizePpv(X,U)=f(TC,VM,m,Σ,φC,β,L,ω,GC,I0)=(NsVC)*In+1M=VM*(InM*(1-({*∂(TC)}1(1+ψ(TC)*(1-μ(VM,TC,IM))*∂(TC))-1))+({*∂(TC))-1}({(VM,TC,IM)+γ(VM)}ψ(TC)*ϵ(m,φS,β,Σ,φC)*μ(VM,TC,IM)+γ(VM))×(1+ψ(TC)*(1-μ(VM,TC,IM))*∂(TC))-1)).Subjected to the following proposed parametric constrained:(14)NSmin<NS<NSmax,NPmin<NP<NPmax,dmin<d<dmax⟶1≤d≤365,Σmin<Σ<Σmax⟶0≤Σ≤80,φCmin<φC<φCmax⟶-45≤φC≤45,
the proposed equality constraint is given as
(15)g(U,X)=Voc-184.0293*NsVCT=0.The limits of independent variables are selected according to the following aspects.(1)
WhenΣ=0°, the PV module becomes horizontal and produces power, while for Σ=90, the PV module becomes vertical and produces zero power; so, the selected limits are assumed between 0° and 80°.(2)
The PV module azimuth angle is positive for east of south and becomes negative for west of south; so, the limits are selected as±45°.


### 3.2. Genetic Algorithm

Genetic algorithm, GA, is 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. GA is capable of handling complex and irregular solution spaces and has been applied to various difficult optimization problems [16]. It 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 operatorsthat work on the chromosomesin which the parameters of possible solutions are encoded. In each generation of the GA, the new solutions replace the solutions in the population that is selected for removal. The main elements of GA are 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 Figure 6.Figure 6
A simple GA flow chart.The simplest form of genetic algorithm involves three types of operators: selection, crossover (single point), and mutation.SelectionIt selects chromosomes in the population for reproduction.CrossoverIt randomly chooses a locus and exchanges the subsequences before and after that locus between two chromosomes to create two offsprings.MutationIt randomly flips some of the bits in a chromosome.


### 3.3. The Method of Lagrange Multipliers

Lagrange multipliers, LM, play a vital role in the study of constrained optimization. It can be interpreted as the rate of change in the objective function with respect to changes in the associated constraint function [16]. The main formula of this method can be described as follows:
(16)∇U,Xf(U,X)=-λ∇U,Xg,
where ∇U,Xf=(∂f/∂U,∂f/∂X) and ∇U,Xg=(∂g/∂U,∂g/∂X) are the respective gradients. Constant λ in (16) are called the Lagrange multipliersof the constrained problem.By solving (15) one can get the value of λ and then obtain the value of U vector and X vector at optimal solution and the optimal power extracted from the PV module. The proposed  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 29.87°N and longitude 31.30°E. The station is over a hill top of about 114 m height above sea level.The initial condition of any optimization problem plays an important role in finding the optimal solution so GA program is performed under varying the initial condition of the independent variables until the optimal, maximum, power is obtained then Lagrange multiplier algorithm program is used and compared with GA to investigate the validity and quality of our proposed method.The main steps of the proposed technique can be summarized as follows.(1)
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.(2)
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.(3)
The initial population at the optimal PV output power is fed to the Lagrange multiplier algorithm and comparing the results with those obtained by GA. Figure7 shows these proposed technique steps.Figure 7
The proposed analysis steps.


## 3.1. The Proposed Optimization Problem

The mathematical model of any continuous optimization problem can be described as follows:(9)minimizeormaximizef(X,U),
where f(X,U) is an objective function to be minimized or maximized subjected to equality and inequality constraints as follows:(1)
equality constraint(10)qi(X,U)=0,i=1,2,…,l,(2)
parametric inequality constraint(11a)Xjmin≤Xj≤Xjmax,j=1,2,…,f,(11b)Ukmin≤Uk≤Ukmax,k=1,2,…,r,(3)
functional inequality constraint(12)hy(X,U)≤0,y=1,2,…,p,where X is a vector of state, dependant, variables and U is a vector of control (independent) variables. First the dummy variables that have effect on the PV module power are classified into two categories: independent (U) and dependent variables (X)—U=[NS,NP,d,Σ,φC] and X=[β,m,φS,GC,ISC,I0,TC].The proposed objective function is expressed in the following form:(13)maximizePpv(X,U)=f(TC,VM,m,Σ,φC,β,L,ω,GC,I0)=(NsVC)*In+1M=VM*(InM*(1-({*∂(TC)}1(1+ψ(TC)*(1-μ(VM,TC,IM))*∂(TC))-1))+({*∂(TC))-1}({(VM,TC,IM)+γ(VM)}ψ(TC)*ϵ(m,φS,β,Σ,φC)*μ(VM,TC,IM)+γ(VM))×(1+ψ(TC)*(1-μ(VM,TC,IM))*∂(TC))-1)).Subjected to the following proposed parametric constrained:(14)NSmin<NS<NSmax,NPmin<NP<NPmax,dmin<d<dmax⟶1≤d≤365,Σmin<Σ<Σmax⟶0≤Σ≤80,φCmin<φC<φCmax⟶-45≤φC≤45,
the proposed equality constraint is given as
(15)g(U,X)=Voc-184.0293*NsVCT=0.The limits of independent variables are selected according to the following aspects.(1)
WhenΣ=0°, the PV module becomes horizontal and produces power, while for Σ=90, the PV module becomes vertical and produces zero power; so, the selected limits are assumed between 0° and 80°.(2)
The PV module azimuth angle is positive for east of south and becomes negative for west of south; so, the limits are selected as±45°.


## 3.2. Genetic Algorithm

Genetic algorithm, GA, is 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. GA is capable of handling complex and irregular solution spaces and has been applied to various difficult optimization problems [16]. It 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 operatorsthat work on the chromosomesin which the parameters of possible solutions are encoded. In each generation of the GA, the new solutions replace the solutions in the population that is selected for removal. The main elements of GA are 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 Figure 6.Figure 6
A simple GA flow chart.The simplest form of genetic algorithm involves three types of operators: selection, crossover (single point), and mutation.SelectionIt selects chromosomes in the population for reproduction.CrossoverIt randomly chooses a locus and exchanges the subsequences before and after that locus between two chromosomes to create two offsprings.MutationIt randomly flips some of the bits in a chromosome.


## 3.3. The Method of Lagrange Multipliers

Lagrange multipliers, LM, play a vital role in the study of constrained optimization. It can be interpreted as the rate of change in the objective function with respect to changes in the associated constraint function [16]. The main formula of this method can be described as follows:
(16)∇U,Xf(U,X)=-λ∇U,Xg,
where ∇U,Xf=(∂f/∂U,∂f/∂X) and ∇U,Xg=(∂g/∂U,∂g/∂X) are the respective gradients. Constant λ in (16) are called the Lagrange multipliersof the constrained problem.By solving (15) one can get the value of λ and then obtain the value of U vector and X vector at optimal solution and the optimal power extracted from the PV module. The proposed  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 29.87°N and longitude 31.30°E. The station is over a hill top of about 114 m height above sea level.The initial condition of any optimization problem plays an important role in finding the optimal solution so GA program is performed under varying the initial condition of the independent variables until the optimal, maximum, power is obtained then Lagrange multiplier algorithm program is used and compared with GA to investigate the validity and quality of our proposed method.The main steps of the proposed technique can be summarized as follows.(1)
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.(2)
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.(3)
The initial population at the optimal PV output power is fed to the Lagrange multiplier algorithm and comparing the results with those obtained by GA. Figure7 shows these proposed technique steps.Figure 7
The proposed analysis steps.


## 4. Numerical Analysis

The analysis uses bpsx150 PV module which has 72 cells connected in series and its electrical characteristic is shown in Table1 [17].Table 1
Electrical characteristic of bpsx150 PV module.
Maximum power (Pmax)
150 W
Voltage atPmax(VPmax)
34.5 V
Current atPmax(IPmax)
4.35 A
Warranted minimumPmax
140 W
Short-circuit current (Isc)
4.75 A
Open-circuit voltage (Voc)
43.5 V
Maximum system voltage
600 V
Temperature coefficient ofIsc
(0.065 ± 0.015) %/°C
Temperature coefficient ofVoc
−(160 ± 20) mV/°C
Temperature coefficient of power
−(0.5 ± 0.05) %/°C
NOCT
47 ± 2°CThe bpsx150 PV module is simulated by MATLAB/Simulink toolbox as shown in Figure8. Referring to Figure 8, there are three main inputs which are the module voltage, the solar radiation, and the ambient temperature; these items are labeled in Figure 9 with numbers 1, 2, and 3. Simulink is a software package of MATLAB program for modeling, simulating, and analyzing dynamical systems. It supports linear and nonlinear systems, modeled in continuous time, sampled time, or a hybrid of the two.  For modeling, Simulink provides a graphical user interface, GUI, for building models as block diagrams, using click and drag mouse operations. The detailed configuration of the PV module is given in Figure 9. The analysis of the proposed algorithm is performed on measured solar radiation and ambient temperature as inputs to GA program. These data are measured at Helwan city, for the sunny day of June 10, 2012, and start from 6:10 AM to 5:50 PM. The direct beam solar radiation and the corresponding ambient temperature, Helwan city, at latitude 29.87°, Egypt, are given in Figure 10.Figure 8
Simulink model of the PV module.Figure 9
Detailed configuration of PV module subsystem.Figure 10
Direct beam solar radiation and ambient temperature for Helwan city, Egypt.The initial population of GA program which contains(Σ0,φC0,V0M) plays an important role in finding the optimal power, so they are changed as follows.(1)
The initial tilt angle,Σ0, is changed  at acceptable limits, fixing both φC0 and V0M at acceptable values until the maximum power is obtained as shown in Table 2.Table 2
Effect of changing the initial tilt angle on the module power at 6:10 AM.
Initial solution
Independent variables
Dependant variables
I
M (A)
P
pv (W)
[
Σ
0
°,φc0°,V0M]
Σ
°
φ
C
°
V
M (V)
G
C (Kw/m2)
I
0 (A)
T (°C)
U
0 = [0, 5, 30]
56.5384
44.4320
43.4848
0.0887
4.5125
*
10
-
6
31.8929
0.4231
18.7692
U
0 = [3, 5, 30]
51.2944
44.7359
43.4484
0.0877
4.4992
*
10
-
6
31.8609
0.4186
18.1864
U
0 = [6, 5, 30]
54.6782
44.8905
43.3716
0.0887
4.5131
*
10
-
6
31.8944
0.4233
18.5889
U
0 = [9, 5, 30]
58.7500
45.0000
43.4777
0.0915
4.5519
*
10
-
6
31.9871
0.4365
19.0712
U
0 = [12, 5, 30]
63.5000
45.0000
43.5000
0.0932
4.5768
*
10
-
6
32.0463
0.4448
19.4612
U
0 = [15, 5, 30]
49.6782
44.9171
43.3964
0.0843
4.4509
*
10
-
6
31.7443
0.4021
17.9730
U
0 = [18, 5, 30]
49.2228
44.5000
43.3857
0.0833
4.4368
*
10
-
6
31.7098
0.3972
17.8469
U
0 = [21, 5, 30]
56.2744
44.8397
43.2505
0.0887
4.5122
*
10
-
6
31.8921
0.4230
18.7031
U
0 = [24, 5, 30]
47.3468
42.7325
43.4541
0.0816
4.4142
*
10
-
6
31.6547
0.3894
17.3567
U
0 = [27, 5, 30]
51.5620
44.9079
43.2911
0.0862
4.4777
*
10
-
6
31.8091
0.4112
18.1797
U
0 = [30, 5, 30]
67.2500
44.9020
43.4535
0.0941
4.5888
*
10
-
6
32.0746
0.4489
19.6382
U
0 = [33, 5, 30]
73.7500
45.0000
43.4414
0.0956
4.6110
*
10
-
6
32.1269
0.4563
19.8390
U
0 = [36, 5, 30]
74.7500
45.0000
43.4958
0.0957
4.6129
*
10
-
6
32.1313
0.4569
19.8724
U
0 = [39, 5, 30]
75.7500
45.0000
43.4894
0.0957
4.6126
*
10
-
6
32.1307
0.4568
19.8726
U
0 = [42, 5, 30]
76.0000
45.0000
43.3601
0.0953
4.6063
*
10
-
6
32.1160
0.4547
19.8134
U
0
=
[45, 5, 30]
75.8125
45.0000
43.4999
0.0958
4.6131
*
1
0
-
6
32.1318
0.4570
19.8774
U
0 = [48, 5, 30]
75.7500
45.0000
43.3796
0.0953
4.6069
*
10
-
6
32.1173
0.4549
19.8224
U
0 = [51, 5, 30]
76.0000
45.0000
43.3997
0.0958
4.6130
*
10
-
6
32.1317
0.4569
19.8315
U
0 = [54, 5, 30]
75.7500
45.0000
43.4689
0.0958
4.6131
*
10
-
6
32.1318
0.4569
19.8632
U
0 = [67, 5, 30]
75.8125
45.0000
43.4989
0.0957
4.6129
*
10
-
6
32.1313
0.4569
19.8769
U
0 = [70, 5, 30]
75.8125
45.0000
43.4903
0.0958
4.6131
*
10
-
6
32.1318
0.4570
19.8730
U
0 = [73, 5, 30]
75.8125
45.0000
43.4989
0.0958
4.6131
*
10
-
6
32.1318
0.4570
19.8769
U
0 = [76, 5, 30]
75.7500
45.0000
43.4963
0.0955
4.6099
*
10
-
6
32.1243
0.4559
19.8757
U
0 = [80, 5, 30]
75.7813
45.0000
43.4966
0.0958
4.6130
*
10
-
6
32.1317
0.4569
19.8759The optimal output power for this case is equal to19.8774W at Σ0=45° which is fixed at this value.(2)
Changing the initial PV module azimuth angle,φC0, at acceptable limits with fixed acceptable initial module voltage until the optimal power is obtained as shown in Table 3.Table 3
Effect of changing the initial collector azimuth angle on the module power at 6:10 AM.
Initial solution
Independent variables
Dependant variables
I
M (A)
P
pv (W)
[
Σ
0
°, φc0°, V0M]
Σ
°
φ
C
°
V
M (V)
G
C (Kw/m2)
I
0 (A)
T (°C)
U
0 = [45, −45, 30]
50.6146
44.5382
43.4679
0.0867
4.4841
*
10
-
6
31.8246
0.4134
18.0765
U
0 = [45, −40, 30]
53.9220
44.9146
43.3693
0.0880
4.5023
*
10
-
6
31.8683
0.4196
18.5043
U
0 = [45, −35, 30]
56.1325
44.2593
43.4265
0.0888
4.5140
*
10
-
6
31.8966
0.4236
18.6752
U
0 = [45, −30, 30]
55.7048
44.9776
43.3350
0.0874
4.4944
*
10
-
6
31.8495
0.4170
18.6999
U
0 = [45, −25, 30]
57.7381
44.7576
43.3907
0.0870
4.4883
*
10
-
6
31.8348
0.4149
18.8992
U
0 = [45, −20, 30]
53.4386
44.7737
43.2897
0.0877
4.4994
*
10
-
6
31.8615
0.4187
18.3925
U
0 = [45, −15, 30]
75.7500
44.9988
43.4602
0.0957
4.6130
*
10
-
6
32.1316
0.4569
19.8591
U
0 = [45, −10, 30]
75.7500
44.8301
43.4955
0.0956
4.6102
*
10
-
6
32.1250
0.4560
19.8452
U
0 = [45, −5, 30]
54.9162
44.5165
43.4248
0.0891
4.5189
*
10
-
6
31.9083
0.4253
18.5822
U
0 = [45, 0, 30]
76.0000
44.7835
43.4785
0.0956
4.6104
*
10
-
6
32.1255
0.4561
19.8290
U
0 = [45, 5, 30]
75.7500
45.0000
43.4714
0.0955
4.6100
*
10
-
6
32.1246
0.4559
19.8774
U
0 = [45, 10, 30]
75.7500
45.0000
43.4771
0.0953
4.6062
*
10
-
6
32.1155
0.4547
19.8670
U
0 = [45, 15, 30]
75.7500
45.0000
43.4837
0.0953
4.6069
*
10
-
6
32.1173
0.4549
19.8700
U
0 = [45, 20, 30]
75.7500
45.0000
43.4731
0.0958
4.6131
*
10
-
6
32.1318
0.4570
19.8651
U
0 = [45, 25, 30]
75.7500
45.0000
43.4553
0.0957
4.6122
*
10
-
6
32.1298
0.4567
19.8570
U
0 = [45, 30, 30]
76.0000
45.0000
43.4579
0.0952
4.6052
*
10
-
6
32.1133
0.4543
19.8581
U
0 = [45, 32, 30]
75.7500
45.0000
43.4983
0.0953
4.6068
*
10
-
6
32.1171
0.4549
19.8767
U
0 = [45, 34, 30]
76.0000
45.0000
43.4766
0.0957
4.6124
*
10
-
6
32.1303
0.4567
19.8667
U
0 = [45, 36, 30]
75.7500
45.0000
43.4935
0.0958
4.6130
*
10
-
6
32.1317
0.4569
19.8744
U
0 = [45, 37, 30]
76.0000
45.0000
43.4558
0.0944
4.5943
*
10
-
6
32.0877
0.4507
19.8572
U
0 = [45, 39, 30]
75.7500
45.0000
43.4837
0.0958
4.6130
*
10
-
6
32.1317
0.4569
19.8700
U
0
=
[45, 41, 30]
75.7500
45.0000
43.4997
0.0957
4.6126
*
1
0
-
6
32.1307
0.4568
19.8776
U
0 = [45, 43, 30]
75.7500
45.0000
43.4516
0.0957
4.6118
*
10
-
6
32.1289
0.4565
19.8553
U
0 = [45, 45, 30]
75.7500
45.0000
43.4834
0.0953
4.6063
*
10
-
6
32.1160
0.4547
19.8698The optimal output power for this case is equal to19.8776W at φC0=41 which is fixed at this value.(3)
Changing the initial PV module voltage until the optimal power is obtained as shown in Table4.Table 4
Effect of changing the initial module voltage on the module power at 6:10 AM.
Initial solution
Independent variables
Dependant variables
I
M (A)
P
pv (W)
[
Σ
0
°,φc0°, V0M]
Σ
°
φ
C
°
V
M (V)
G
C (Kw/m2)
I
0 (A)
T (°C)
U
0 = [45, 41, 0]
51.8224
44.5636
43.3348
0.0872
4.4921
*
10
-
6
31.8439
0.4162
18.1819
U
0 = [45, 41, 2]
56.3961
45.0000
43.4018
0.0899
4.5301
*
10
-
6
31.9350
0.4291
18.8057
U
0 = [45, 41, 4]
76.0000
44.9427
43.2811
0.0957
4.6123
*
10
-
6
32.1299
0.4567
19.7672
U
0 = [45, 41, 6]
53.7614
44.8565
43.4569
0.0859
4.4740
*
10
-
6
31.8003
0.4100
18.5142
U
0 = [45, 41, 8]
54.9695
44.5616
43.4382
0.0889
4.5158
*
10
-
6
31.9008
0.4242
18.6007
U
0 = [45, 41, 10]
76.0000
45.0000
43.3597
0.0958
4.6131
*
10
-
6
32.1318
0.4570
19.8132
U
0 = [45, 41, 12]
53.1965
44.7522
43.4134
0.0875
4.4960
*
10
-
6
31.8533
0.4175
18.4128
U
0 = [45, 41, 14]
76.0000
44.9767
43.3995
0.0957
4.6127
*
10
-
6
32.1308
0.4568
19.8273
U
0 = [45, 41, 16]
75.7500
45.0000
43.4834
0.0957
4.6129
*
10
-
6
32.1313
0.4569
19.8698
U
0 = [45, 41, 18]
75.8750
45.0000
43.4408
0.0949
4.6006
*
10
-
6
32.1023
0.4528
19.8504
U
0 = [45, 41, 20]
75.7500
45.0000
43.4871
0.0958
4.6131
*
10
-
6
32.1318
0.4570
19.8716
U
0 = [45, 41, 22]
76.0000
45.0000
43.4579
0.0958
4.6131
*
10
-
6
32.1318
0.4569
19.8581
U
0 = [45, 41, 24]
75.7500
45.0000
43.4499
0.0955
4.6092
*
10
-
6
32.1227
0.4557
19.8545
U
0 = [45, 41, 26]
75.7500
45.0000
43.4741
0.0949
4.6003
*
10
-
6
32.1017
0.4527
19.8728
U
0 = [45, 41, 28]
75.7500
45.0000
43.4839
0.0958
4.6131
*
10
-
6
32.1318
0.4569
19.8701
U
0 = [45, 41, 30]
76.0000
45.0000
43.4818
0.0958
4.6130
*
10
-
6
32.1316
0.4569
19.8776
U
0 = [45, 41, 32]
75.7500
45.0000
43.4565
0.0956
4.6115
*
10
-
6
32.1281
0.4564
19.8575
U
0 = [45, 41, 34]
75.7500
45.0000
43.4324
0.0957
4.6129
*
10
-
6
32.1313
0.4569
19.8466
U
0 = [45, 41, 36]
75.6875
45.0000
43.4969
0.0958
4.6131
*
10
-
6
32.1318
0.4570
19.8760
U
0 = [45, 41, 38]
75.7500
45.0000
43.4403
0.0958
4.6130
*
10
-
6
32.1317
0.4569
19.8501
U
0 = [45, 41, 40]
75.7500
45.0000
43.4393
0.0958
4.6131
*
10
-
6
32.1318
0.4569
19.8497
U
0
=
[45, 41, 42]
76.0000
45.0000
43.5000
0.0957
4.6129
*
1
0
-
6
32.1315
0.4569
19.8778
U
0 = [45, 41, 43]
75.5625
45.0000
43.4655
0.0955
4.6098
*
10
-
6
32.1241
0.4559
19.8615
U
0 = [45, 41, 43.5]
75.5000
45.0000
43.5000
0.0958
4.6130
*
10
-
6
32.1317
0.4564
19.8772The final optimal output power is equal to19.8778W at V0M=42V.Finally, one can derive that the optimal obtained power by GA at 6:10 AM is equal to19.8778W and is occurred at initial population which is equal to (Σ0°,φC0°,V0M)=(45°,41°,42V). The curve shows the change of PV module power with changing the initial population is shown in Figure 11. GA program response is shown in Figure 12 that contains the fitness function, the current best individuals, the average distance between individuals and the number of individuals for each iteration. The constant dependant values at optimal power are δ=17.5165, β=10.7181, m=5.3770, φS=75.8452, and ω=60.Figure 11
The optimal PV module power at changing the initial population in GA at 6:10 AM.The optimal PV module power at changing the initial population in GA at 6:10 AM.
(a)
(b)
(c)
(d)The final population at optimal solution is equal toΣ=75.7500, φC=45.0000, and VM=43.5000.The optimal power and the corresponding tilt angle for 24 hours of the day are shown in Figure13 and Table 5.Table 5
The optimal power and the corresponding tilt angle by GA for June 10, 2012.
Hour
Initial condition at optimal solution[Σ0, φc0, V0M]
Tilt angle at optimal power
Optimal power (W)
6:10 AM
[45, 41, 42]
75.7500
19.8773
7:00 AM
[54, 32, 26]
63.4998
50.0546
8:00 AM
[27, 30, 42]
47.2500
84.8483
9:00 AM
[42, 36, 14]
36.9459
137.1025
10:00 AM
[21, 32, 42]
28.2500
182.8460
11:00 AM
[70, 0, 36]
19.2309
225.0597
12:00 PM
[18, 30, 42]
13.0219
226.6207
1:00 PM
[30, 10, 43.5]
19.3121
217.1806
2:00 PM
[54, 30, 43.5]
28.2298
194.0017
3:00 PM
[3, 5, 32]
36.7493
153.7899
4:00 PM
[73, −40, 43.5]
47.7500
98.8123
5:00 PM
[24, −35, 43.5]
63.5000
56.5923
5:50 PM
[70, −45, 16]
75.8125
20.2942Figure 13
The optimal power and the corresponding tilt angle for June 10, 2012.Once the bpsx150 PV module has a fixed number of series cells,NS=72, and a fixed number of parallel branches, NP=1, then they are not be considered as variables.The initial condition at optimal power obtained from the GA is fed to the Lagrange multiplier algorithm to find the optimal power. The MATLAB function fmincon is used to perform the Lagrange multiplier analysis.The Lagrange multiplier algorithm response is shown in Figure14. A comparison between the GA and Lagrange multiplier algorithm is performed and is shown in Figure 15 and Table 6.Table 6
A comparison between the GA and Lagrange multiplier algorithm.
Hour
Genetic algorithm solution
Lagrange multiplier algorithm results
Error in optimal tilt   angle
Error in optimal  power
Σ
opt
P
opt
Σ
opt
P
opt
6:10 AM
75.7500
19.8773
75.8114
19.8774
−0.0614
-
1
E
-
04
7:00 AM
63.4998
50.0546
63.4993
50.0486
0.0005
0.006
8:00 AM
47.2500
84.8483
47.3668
84.8485
−0.1168
−0.0002
9:00 AM
36.9459
137.1025
36.8401
137.1174
0.1058
−0.0149
10:00 AM
28.2500
182.8460
28.2483
182.8456
0.0017
0.0004
11:00 AM
19.2309
225.0597
19.2598
225.0807
−0.0289
−0.021
12:00 PM
13.0219
226.6207
13.0221
226.6537
−0.0002
−0.033
1:00 PM
19.3121
217.1806
19.2598
217.1956
0.0523
−0.015
2:00 PM
28.2298
194.0017
28.2483
194.0071
−0.0185
−0.0054
3:00 PM
36.7493
153.7899
36.8401
153.8525
−0.0908
−0.0626
4:00 PM
47.7500
98.8123
47.3668
98.8142
0.3832
−0.0019
5:00 PM
63.5000
56.5923
63.4993
56.5563
0.0007
0.036
5:50 PM
75.8125
20.2942
75.8114
20.2752
0.0011
0.019The Lagrange multiplier response.
(a)
(b)
(c)A comparison between the GA and Lagrange multiplier algorithm responses.
(a)
(b)
(c)
(d)A comparison between the GA and Lagrange multiplier algorithm is performed and is shown in Figure14 and Table 6.It is clear that the difference between two proposed solution methods is very small as the initial condition of them is the same; the operating conditions of the PV module do not change. So, one can derive that the error between two methods is acceptable and the proposed method is applicable.The benefits of this proposed approach are as follows.(i)
A new objective function of the PV module output power subjected to constraints is presented.(ii)
A simple optimization methodology to obtain the optimal power extracted from the PV module and corresponding tilt angle based on real data is formulated.(iii)
Two suggested solution techniques are manipulated and the error between them is very small.


## 5. Conclusion

In this work a new optimization approach to maximize the PV module output power is presented. A new proposed objective function of PV module power and constraints is also presented. Genetic algorithm is performed under varying the initial population of independent variables, then Lagrange multiplier algorithm is simulated. The optimal PV module power obtained from both algorithms is compared. The analysis is based on real measured data of solar radiation and ambient temperature at Helwan city at latitude 29.87°, Egypt.  The analysis is done in complete sunny day and the results showed that the error between two proposed solution techniques is acceptable and the proposed constrained optimization method is applicable.



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*Source: 481468-2013-03-25.xml*