Analytical determination of the photovoltaic module maximum power point parameters based on the manufacturer''s datasheet Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

математичне моделювання ф1зичнихi технолог1чних процес1вiтехн1чних систем

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ Ф1ЗИЧНИХ I ТЕХНОЛОГ1ЧНИХ ПРОЦЕС1В I ТЕХН1ЧНИХ СИСТЕМ

УДК 621.383.51

Ye.A. BAGANOV, V.V. KURAK, E.V. ANDRONOVA, V.O. GRAMOV

Kherson National Technical University

ANALYTICAL DETERMINATION OF THE PHOTOVOLTAIC MODULE MAXIMUM POWER POINT PARAMETERS BASED ON THE MANUFACTURER'S

DATASHEET

The analytical approach to predict the maximum power point parameters of photovoltaic module at varying operating conditions such as temperature and solar irradiation level is proposed. Input data for calculations are standard module parameters from manufacturer's datasheet only. Analytical expressions for current, voltage, and power at maximum power point have been obtained via the Lambert W function. Calculation of the maximum power point electrical parameters for several photovoltaic modules realized by proposed approach shows, in general, good accuracy in maximum power point determination, which is comparable with more complicated models, except the case of thin-film modules.

Keywords: Photovoltaic module; Maximum power point; Lambert W function.

е.О. БАГАНОВ, В.В. КУРАК, О.В. АНДРОНОВА, В.О. ГРАМОВ

Херсонський нащональний техшчний ушверситета

АНАЛ1ТИЧНЕ ВИЗНАЧЕННЯ ПАРАМЕТР1В ТОЧКИ МАКСИМАЛЬНО! ПОТУЖНОСТ1 ФОТОЕЛЕКТРИЧНИХ МОДУЛ1В НА ОСНОВ1 КАТАЛОГ1В ВИРОБНИК1В

В робот1 запропоновано аналггачний шдхвд до визначення параметр1в точки максимально! потужносп фотоелектричного модуля при р1зних умовах експлуатаци, таких як температура i штенсивнють сонячного випромiнювання. Вихiдними даними для розрахуншв е стандартнi параметри модуля, що наводяться у каталогах виробник1в. Аналтгичш вирази для струму, напруги та потужносп в точцi максимально! потужносп були отриманi через W-функцiю Ламберта. Проведений за допомогою запропонованого тдходу розрахунок електричних параметрiв в точщ максимально! потужностi для низки фотоелектричних модулiв показав, в цiлому, високу точнють у визначеннi точки максимально! потужносп порiвняно з використанням складшших моделей, за винятком випадку тонкоплiвкових модулiв.

Ключовi слова: фотоелектричнi модуль; точка максимально! потужностц W-функцiя Ламберта.

Е.А. БАГАНОВ, В.В. КУРАК, Е.В. АНДРОНОВА, В.А. ГРАМОВ

Херсонский национальный технический университет

АНАЛИТИЧЕСКОЕ ОПРЕДЕЛЕНИЕ ПАРАМЕТРОВ ТОЧКИ МАКСИМАЛЬНОЙ МОЩНОСТИ ФОТОЭЛЕКТРИЧЕСКИХ МОДУЛЕЙ НА ОСНОВЕ КАТАЛОГОВ ПРОИЗВОДИТЕЛЕЙ

В работе предложен аналитический подход к определению параметров точки максимальной мощности фотоэлектрического модуля при различных условиях эксплуатации, таких как температура и интенсивность солнечного излучения. Исходными данными для расчетов являются стандартные параметры модуля, приводимые в каталогах производителей. Аналитические выражения для тока, напряжения и мощности в точке максимальной мощности были получены через W-функцию Ламберта. Проведенный с помощью предложенного подхода расчет электрических параметров в точке максимальной мощности для ряда фотоэлектрических модулей показал, в целом, хорошую точность в определении точки максимальной мощности по сравнению с использованием более сложных моделей, за исключением случая тонкопленочных модулей.

Ключевые слова: фотоэлектрические модуль; точка максимальной мощности; W-функция Ламберта.

matemathhhe moflehwbahhx 0i3hhhhxi texhonormhhx npouecib i texhivhhx chctem

Formulation of the problem

Mathematical modeling is one of the tools used in the design process for optimization of the technical and economic parameters of electric power supply solar systems and determination of the optimal proportion of photovoltaic power generation in hybrid systems.

The most important component of the mathematical model of a photovoltaic (PV) system is a model of PV module, which, on the one hand, should reflect PV module parameters with sufficient accuracy at variable external conditions, i.e. at different temperatures and levels of solar irradiation and, on the other hand, should not require significant amount of computations. The last requirement is especially important for long-term modeling of photovoltaic system operation and it is satisfied in the best way by analytical solution.

Analysis of last investigations and publications

In most cases, when the energy simulation of photovoltaic system is realized, PV module operation is taken at maximum power condition, assuming that maximum power point (MPP) tracker is used in the system [1-3]. PV module data from manufacturer's datasheet are taken as input.

Based on the outcomes and objectives of model of PV module the approaches to PV module modeling can be divided into three categories: (i) definition of current-voltage (I-V) characteristic equation, (ii) definition of some important points of I-V characteristic, including MPP, and (iii) definition of PV module maximum power only.

The methods of the first category, presented, for example in [4, 5], are widely used now and keep on improving. However, the main disadvantage of such methods for MPP parameters determination is the necessity of the numerical solution of the transcendental equations system that results in significant computation amount for long-term modeling of photovoltaic systems.

In [6] the exact analytical explicit expression of I-V characteristic for single-diode model of PV cell using the Lambert W function was obtained. However, in this case the solving of transcendental equations is also required to determine such MPP parameters as voltage, current, and maximum power.

Model of King [7] represents the second category. This model accurately predicts MPP parameters, but it requires some incoming parameters that are normally not available from the manufacturer's datasheets [4].

The third category is represented, for example, by the model of maximum power definition based on the determination of the fill factor dependence on temperature and solar irradiance [8]. However, the values of short circuit current, open circuit voltage and MPP current and voltage under two levels of solar irradiance and two PV module temperatures are needed to realize this model. In addition, it is not possible to determine operational values of MPP voltage and current, which are required to PV system simulation when DC-DC converter, inverter, and other components of the system are taking into account.

The purpose of the investigation

The purpose of the investigation is to develop non-iterative analytical approach to predict the MPP parameters of PV module under varying operating conditions using only standard parameters from manufacturer's datasheet as input data.

Statement of the basic material of the investigation

A single-diode model of PV cell with ideality factor A corresponds to equivalent circuit presented in

Fig. 1.

R

/4© !sh\

Rsh

U

U I

Ri

Fig. 1. The equivalent circuit of single-diode PV cell model

The load current is defined as

1 - 1 ph Id 1 sh

where Iph is the photocurrent of charge carriers generated by solar irradiation; Id is the forward-biased p-n junction current; Ish is the shunt current of PV cell. Last two currents are determined by the expressions:

mätemäthhhe mo^ehwbähhh obhhhhxi texhohonhhhx npo^cib itexhihhhx chctem

Id = 10

I AVt

ey t

-1

=I oe

- I0 ; Ish =

Ui

R

sh

where Vt = kT/qe is the thermal voltage;

qe = 1,6-10-19 C is the electron charge;

k = 1,3810-23 J/K is the Boltzmann's constant;

T is the absolute temperature of a PV cell;

I0 is the reverse saturation current of the p-n junction;

U1 is the voltage applied to the p-n junction;

Rsh is the shunt resistance of a PV cell.

Then

1 = 1 ph + A) - V

.Ex

R

sh

(i)

The load voltage can be defined as

U = U1 - IRs, (2)

where Rs is the series resistance of a PV cell.

The MPP current will be found under condition of the maximum power transfer to the series-connected load resistance Ri and PV cell series resistance Rs. The correctness of such approach will be discussed below. The power that is transferred to Rs and Ri is

P = U1I =

Aph + 10 -1 oe

.El

Rsh

Ui

Then

dP

dU1

= Aph +1 o -1 oe

av,

1 +

Ui

AVt

2U1

t

R

sh

If this expression is put to zero, then:

Aph +10 2 Um

RshI 0

1 + -

U

1n

AVt

(3)

U

e

0

where U1n is the voltage applied to the p-n junction at MPP.

With the introduction of variable z = U1n / VtA and multiplication of left and right hand parts by e Eq. (3) can be written as

/ph +10 - e2V-A = (1 + z)ed+z)

A0 Rsh 0

The solution of this equation can be expressed using Lambert W function W0(x) [9]:

z = W

1 ph +10

1 -

2VtAz

Rsh (lph +10 )

-1.

(4)

I

0

в1сникхнту№ 4(59), 2016р. математичнемоделюванняф1зичнихi _технолог1чних процес1вiтехн1чних систем

Eq. (4) is transcendental relatively to z, and, hence, to U1n. This equation can be simplified taking into account that the shunt resistance has a significant impact on the I-V characteristic of PV cell at low voltages only [10], when the diode current Id is small, and PV cell efficiency is practically insensitive to variations of Rsh [11]. At the MPP U1n >> AVt and considering the order of Rsh for standard PV module

Uln/

Idn ~ e >> Ishn = Uln/Rsh . Then, as Iph > Idn, and therefore, Iph >> Ishn:

2VtAz 2Ui _ 2Ishn

Rsh ({ph +10 ) Rsh (iph +10 ) {f ph +10 )

that makes possible to consider the expression in the square brackets of Eq. (4) is close to one.

Therefore, to simplify the further model the value of Rsh will be considered infinitely large (Rsh ^ <»). Such consideration is additionally justified for mostly series connection of PV cells in the PV module [8] mainly realized by manufacturers. Also, due to U1n >> AVt, the diode current Id n >> I0 and at MPP the reverse saturation current of p-n junction I0 can be neglected compared to the photocurrent Iph. Then, Eq. (4) can be written as

z = W0

С J ^ 1 ph

v J 0 у

-1,

and, hence,

Uln = VtA

v v

( J Л Л

J ph

1 о у

-1

(6)

Substitution of Eq. (6) in Eqs. (1) and (2) gives the explicit analytical expressions for the load voltage and load current at MPP:

I -1 -1 /• ((7)

In -1 ph 10e ,

un - Uln - InRs. (8)

Maximum power of PV cell at these parameters:

Pn = JnUn .

(9)

Then the PV module parameters can be expressed as:

I = IN ; U = UN ; P = I U N N

nm n p) nm n s' nm n n p s'

(10)

The correctness of the used above approach to MPP parameters determination based on the condition of the maximum power transfer to the series-connected Rs and Ri will be estimated in this part of paper.

In [8] the dependence of fill factor FF of PV module with parasitic resistances on the fill factor FF0 of ideal PV module without resistive effects is given. The dependence can be used for one PV cell:

FF = FF

1 —

v UOC / JSC у

(11)

where UOC and IPVcell are the open circuit voltage and the short circuit current of PV cell respectively.

Considering that the product U1n-In is the maximum power for the ideal PV cell, expression (11) can be rewritten as:

JnUn

InUm

(

JSCUOC 1SCUOC

л

UOC /1 SC у

matemathhhe moflehwbahhx 0i3hhhhxi

texhonormhhx npouecib i texhiwhhx chctem

or taking into account Eq. (7)

U1n - TnRs - U1n

1—

uoc ! tsc /

from which

U

Ur

1 SC

(12)

Verify if obtained expressions for Ui„ and In based on the condition of the maximum power transfer to the series-connected Rs and Ri are identical to Eq. (12). In [12] the approximation of the Lambert W function for large values of the argument is presented:

W (*)-L - L + L2 + L Ï2 -2) + L2 ^ -9L2 + 6)

0W 1 2 L 2L2 6L3

| ¿2 (3L2 - 22L2 + 36L2 -12; + 0

)

12L4

L1

5 A

where L1 = lnx and L2 = lnlnx.

Due to the high value of ratio Iph/I0 and, consequently, a large value of argument x - elph/10 , the approximation can be confined within the first two terms of the series. Then, from Eqs. (6) and (7):

VtA

- w

f T ^

1 ph e ——

V, T0 /

-1 « ln

f T ^

1 ph V T0 v

lnln

( ( ÎT

ln

( T \ 1 ph

V /

ln

1-

1 + ln

ph

V 0 //

ln

i T \ 1 ph

V 10 /

f T ^

1 ph e——

V T0 /

fl ^

-1 - ln

ph

V 10 /

lnln

f T ^

1 ph V T0 /

(13)

1 n - Tph - 10e

W| e1-1

I„

ln| el-lnln| e— 1-1

, T T e V 10 J V 10 'Tph -10e

ph

_T _j IphjI0 _ j - 1 ph T0 f j \ - 1 ph

e—ph

V T0 /

ln

1 --

1

C T ^

ehh

V T0 /

ln

(14)

Open circuit voltage at Rsh^œ [13]:

1

Uoc « VtAln-f^.

(15)

Substitution of Eqs. (13-15) in the Eq. (12) taking into account 1ph~IPVceii gives:

n

+

L

2

\

/

1

0

ln

1 —

1 + ln

i j \W 1 ph

V 0 //

ln

i T \

1 ph

V 17 /

1 —

( T ^

1 ph

ln

At significant levels of photocurrent Iph /10 ^ obtained equation turns into identity. This means that the proposed approach to determination of MPP parameters based on the maximum power transfer to the series-

1

1

0

matemathhhe moflehwbahhx 0i3hhhhxi texhonormhhx npouecib i texhivhhx chctem

connected Rs and Ri correctly describes the influence of Rs on the fill factor and, thus, on the maximum power of PV cell, if the condition Iph /10 — œ takes place, i.e., at a sufficiently high level of solar irradiation.

The basic parameters provided by manufacturers in PV module datasheets are the current (Inm) and the voltage (Unm) at MPP, the open-circuit voltage UOCm, and the short-circuit current IPV cenm, generally, under standard test conditions (STC), as well as the relative temperature coefficients of the short-circuit current ( pIsc ),

open-circuit voltage ( pUoc ) and maximum power ( pP ).

To take into account the MPP dependence on the solar irradiation level and temperature the value of Iph/I0 in Eqs. (6) and (7) can be expressed as a function of short-circuit current and open-circuit voltage. Dependence of Rs on PV module operating conditions has insignificant effect on I-V characteristic [4], so Rs is accepted as independent on temperature and solar irradiation level, i.e. Rs — Rs, ref, where "ref" means reference conditions (STC). Ideality factor is also considered as a constant value [4, 8, 14], A — Aref.

From Eq. (1) at the open-circuit mode, taking into account Rsh —I ph » I o it can be obtained:

U Uoc

UOC , = J _ J eVtA - 0 1 ph 1 0e 0 ,

Rsh

Then:

Uoc

T T V,A I0 - Iphe

j Uoc

1 ph _ VtA

Io ~ e

and taking into account temperature coefficients:

9eUOCm,ref (1+PUOC (t_Tref ))

h

— = e

I

Iph = e NAfkT (16)

0

qeUOCm,ref (1+Puoc jT~Tref ))

J _ J 0 NSArefOT (17)

10 _ 1 phe ,

where T and Tref are the PV module operation temperature and temperature at reference conditions respectively.

The photocurrent dependence on temperature and solar irradiation level can be considered according to the following expression [15]:

-(1 + P/SC (T Tref)) , (18)

ref

Iph - Iph,ref r (1 + pIsc

where Gref and G are the solar radiation intensities at the reference and PV module operation conditions respectively;

Iph, ref is the photocurrent at the reference conditions. Then substitution Eq. (10) into Eq. (5) results in:

kT

qe

( ( qUoCmreet (1+PUOç(T_Tf!+l^ ^

U1n - Aref

Wo

NSArefkT

1

V v ^ J

(19)

After substitution of Eq. (19) into Eq. (7) and taking into account Eqs. (17) and (18) the explicit analytical expression for the load current at MPP can be found:

matemathhhe mo^ehwbahhh obhhhhxi texhonormhhx npouecib i texhivhhx chctem

1n 1ph, ref

G

G,

ref

(l + (T - Tref I

Wr

l - e

m,ref (l+PUoc (t-Tref ^ i NsArefkT _

qeUOCm,ref (l+PuOC (t-Tref ))

(20)

Then using Eqs. (20), (8) and (9) the analytical expressions for the PV cell voltage and power at MPP can be obtained.

To realize the calculations according to the Eqs. (19), (20) and (8) - (10) it is necessary to determine such parameters as Iphref, Rs,ref and Aref at reference conditions. Based on the accepted assumption Rsh ^ ® the PV cell model is reduced to a four-parameter model for which [10]:

1 ph,ref - ISCm,ref ^ Np ;

R

s,ref

NpU nm,ref

N pArefkTref

Aref - _

Ns-1 nm,ref qe x1 SCm,ref 1nm,ref q(2Unm,ref - UOC,ref )

NskTref

I

nm,ref

J -J

SCm,ref nm,ref

+ ln

1 -

I

w

nm,ref

i

SCm,ref

//

PV module temperature can be estimated from the expression [1]:

T - ta + Tnoct 20°C G , A 800

where TA is the ambient temperature;

TNOCT is the PV module temperature at normal operating cell temperature (NOCT) conditions also given in manufacturer's datasheet.

To test the accuracy of the proposed approach the calculations of the maximum power Pnm at different temperatures and solar irradiation levels for several types of silicon PV modules were carried out. The results of calculations marked as "LambF" are shown in Table 1. Operation conditions of PV modules, the experimentally determined power values from a building integrated photovoltaic facility at the National Institute of Standards and Technology (NIST), as well as the calculation results obtained in accordance with the King's (King) and the five-parameter (5-param) models of PV module, presented in the same table, as well as required input data were taken from [4].

In Table 2 the relative deviations of calculated results from experimentally determined maximum power values are shown for corresponding models.

As can be seen from Table 2, the results obtained by means of the proposed approach quite correctly describe the behavior of monocrystalline and polycrystalline silicon PV modules.

As regards the thin film PV modules, in the case of proposed approach the deviation is 5.01% at a solar irradiation intensity of 696 W/m2 and rises to 26.4% when intensity decreases to 189.8 W/m2. Thus, at the intensity of 189.8 W/m2 the accuracy of proposed approach is three times worse than for the King's model and two times worse than the accuracy of 5-parametric model. This result can be connected with incorrect value of Rs in the case of the thin-film PV modules. It was found the Rs value obtained from the Eq. (15) is negative that can be explained by incorrect assumption Rsh ^ ® for thin-film PV modules. The relatively low value of Rsh for thin-film PV cell results in the failure of inequality (5) especially at low solar irradiation level when, according to our estimation, the left side of inequality (5) reaches 0.6. Besides, at low irradiation levels (and, hence, at low Iph values) the Eq. (12) is not held exactly for proposed approach that results in discrepancy between the experimental MPP parameters and calculated ones.

Also the proposed approach was tested by calculation of Inm, Unm and Pnm values for NOCT conditions basing on STC data for PV modules, which datasheets contain parameters at STC and NOCT conditions. In addition, the maximum power temperature coefficient pP was calculated. Comparisons of the calculated MPP

parameters with their datasheets values, as well as corresponding relative deviations of parameters are shown in Table 3.

e

NsArefkT

mätemäthhhe mo^ehwbähhh obhhhhxi texhonormhhx npouecib i texhivhhx chctem

Table 1

Results of the maximum power calculation in comparison with experimental data at variable operating _conditions, W_

Type of PV module G, W/m2 1000.0 882.6 696.0 465.7 189.8

"Source 25.0 39.5 47.0 32.2 36.5

Monocrystalline silicon PV module NIST, [4] 133.4 109.5 80.1 62.7 23.8

King, [4] 133.4 111.4 82.0 61.1 22.5

5-param, [4] 133.4 110.6 82.4 61.0 22.3

LambF 133.4 109.8 83.9 61.2 24.7

Polycrystalline silicon PV module NIST, [4] 125.8 106.8 77.4 56.6 21.2

King, [4] 125.8 109.3 79.1 56.9 18.5

5-param, [4] 125.8 105.6 78.1 55.8 20.6

LambF 125.8 103.1 78.2 56.6 22.6

Thin-film silicon PV module NIST, [4] 104.0 83.7 59.9 40.8 14.4

King, [4] 104.0 87.3 62.3 43.2 15.7

5-param, [4] 104.0 85.5 62.3 44.3 16.3

LambF 104.0 84.1 62.9 46.0 18.2

Table 2

Relative deviations of the calculated results from the experimental values of maximum power, %

Type of PV module G, W/m2 1000.0 882.6 696.0 465.7 189.8

' Source----- 25.0 39.5 47.0 32.2 36.5

Monocrystalline silicon PV module King 0 1.74 2.37 2.55 5.46

5-param 0 1.00 2.87 2.71 6.30

LambF 0 0.27 4.74 2.39 3.78

Polycrystalline silicon PV module King 0 2.34 2.20 0.53 12.74

5-param 0 1.12 0.90 1.41 2.83

LambF 0 3.46 1.03 0.00 6.60

Thin-film silicon PV module King 0 4.30 4.01 5.88 9.03

5-param 0 2.15 4.01 8.58 13.19

LambF 0 0.42 5.01 12.75 26.39

Table 3

Comparison of the calculated MPP parameters with PV module datasheets values at NOCT conditions

Name of PV module Datasheet values Calculated values Deviation, %

1nm, A Unm, V P 1 nm? W PP , n %/K 1nm, A Unm, V P, 1 nm•> W Pp , n %/K ^nm Unm P nm PPn

1 2 3 4 5 6 7 8 9 10 11 12 13

Monocrystalline silicon PV modules

LG300N1K-G4 7.38 29.50 218.00 -0.38 7.46 30.37 226.68 -0.383 1.14 2.94 3.98 0.66

SW 270 mono 7.12 28.30 201.30 -0.45 7.12 28.54 203.24 -0.432 0.01 0.86 0.96 3.92

YL300C-30b 7.30 30.30 220.80 -0.38 7.39 30.31 224.03 -0.376 1.23 0.05 1.46 1.17

Q.PEAK-G3 280 7.03 29.60 208.00 -0.42 7.22 29.39 212.20 -0.409 2.71 0.71 2.02 2.65

Q.PEAK BLK-G3 250 6.60 27.92 184.30 -0.43 6.67 27.97 186.44 -0.419 1.00 0.18 1.16 2.58

JAM6(L) 60-290/PR 7.34 28.89 212.02 -0.41 7.37 29.62 218.30 -0.412 0.42 2.52 2.96 0.55

JC250S-24/Bb 6.57 28.00 184.00 -0.43 6.70 27.88 186.67 -0.445 1.93 0.44 1.45 3,42

LDK-180D-24 3.96 32.80 130.00 -0.47 3.98 33.11 131.74 -0.482 0.49 0.93 1.34 2.51

mätemäthhhe mo^ehwbähhh obhhhhxi texhonormhhx npouecib i texhivhhx chctem

Continuation of the Table 3

1 || 2 3 4 5 || 6 7 8 9 10 11 12 13

Average value 1.11 1.08 1.92 2.18

Polycrystalline silicon PV modules

STC 260P 6.80 27.70 189.00 -0.43 6.92 27.79 192.44 -0.411 1.83 0.33 1.82 4.51

Q.PRO-G4 265 6.75 29.01 195.70 -0.41 6.93 28.50 197.59 -0.423 2.71 1.75 0.97 3.05

TSM-250 PC/PA05 6.70 27.00 181.00 -0.43 6.67 27.95 186.29 -0.446 0.52 3.52 2.92 3.61

SW 235 poly 6.28 27.10 170.40 -0.48 6.31 27.45 173.27 -0.482 0.51 1.29 1.68 0.35

WSP-260P6 6.74 28.21 190.14 -0.43 6.75 29.04 196.14 -0.416 0.19 2.96 3.15 3.36

YL245P-29b 6.54 27.20 177.90 -0.45 6.57 27.71 182.17 -0.449 0.52 1.87 2.40 0.26

INE-255-6PB 6.74 28.10 190.00 -0.41 6.77 28.34 191.98 -0.401 0.50 0.86 1.04 2.31

JT300PMe 6.30 34.30 215.90 -0.43 6.37 34.84 221.91 -0.437 1.11 1.57 2.78 1.53

EX260P-60 7.03 28.00 196.90 -0.44 6.91 28.24 195.19 -0.443 1.68 0.85 0.87 0.16

Average value 1.06 1.67 1.96 2.13

Thin-film silicon PV modules

GS STFS 3.60 30.80 111.00 -0.21 3.49 29.37 102.65 -0.552 2.93 4.63 7.52 162.9 0

U-EA 100 1.53 48.80 74.40 -0.35 1.44 48.53 69.88 -0.591 5.88 0.56 6.07 68.9 3

TW-TF 130 0.80 122.80 99.00 -0.29 0.75 122.71 91.47 -0.483 6.24 0.07 7.61 66.3 9

Average value 5.02 1.75 7.07 99.4 1

As can be seen from Table 3, the proposed approach provides correct description of the MPP behavior for several monocrystalline and polycrystalline PV modules at variable temperature and irradiation conditions. However, in the case of thin-film PV modules the significant discrepancies between the calculated and experimental values take place as before.

Conclusions

1. Analytical expressions for PV module current, voltage and power at MPP based on the Lambert W function have been obtained. Photocurrent, reverse saturation current of p-n-junction, ideality factor and series resistance are used as variables in these expressions.

2. Based on the obtained analytical expressions and PV module data from manufacturer's datasheet the non -iterative approach has been proposed to predict the PV module electrical parameters at MPP at different temperatures and levels of solar irradiation. Calculation of MPP electrical parameters for several PV modules realized by proposed approach shows generally good accuracy in maximum power determination, which is comparable with more complicated models, except the case of thin-film PV modules.

References:

1. Koutroulis E. Methodology for optimal sizing of stand-alone photovoltaic/wind-generator systems using genetic algorithms / Koutroulis E., Kolokotsa D., Potirakis A., Kalaitzakis K. // Sol. Energy. - 2006. -V.80. - P. 1072-1088.

2. Savitha, P.B. Modelling of Photovoltaic Cell/Module under Environmental Disturbances using MATLAB/Simulink / Savitha P.B., Shashikala M.S., Puttabuddhi K.L. // International Journal of Engineering Trends and Technology (IJETT). - 2014. - N9 (1). - P. 48-55.

mätemäthhhe mo^ehwbähhh obhhhhxi texhonormhhx npouecib i texhivhhx chctem

3. Gonzalez A. Optimal sizing of a hybrid grid-connected photovoltaic-wind-biomass power system / Gonzalez A., Riba J.-R., Rius A. // Sustainability. - 2015. - V.7. - P. 12787-12806.

4. De Soto W. Improvement and validation of a model for photovoltaic array performance / De Soto W., Klein S.A., Beckman W.A. // Sol. Energy. - 2006. - V.80. - P. 78-88.

5. Dobos A. An improved coefficient calculator for the California Energy Commission 6 parameter photovoltaic module model / Dobos A. // J. Solar Energy Engr. - 2012. - V.134. - P. 021011-1 -021011-6.

6. Jain A. Exact analytical solutions of the parameters of real solar cells using LambertW-function / Jain A., Kapoor A. // Sol. Energy Mater. Sol. Cells. - 2004. - V.81. - P. 269-277.

7. King D.L. Photovoltaic Array Performance Model / King D.L., Boyson W.E., Kratochvil J.A. // Sandia Report. - Albuquerque, 2004. - 43 p.

8. Zhou W. A novel model for photovoltaic array performance prediction / Zhou W., Yang H., Fang Z. // Appl. Energy. - 2007. - V.84 . - P. 1187-1198.

9. Baganov Ye. Reducing the density of threading dislocations in GaAs epitaxial layers. Efficiency assessment of isovalent Bi doping and Pb doping / Baganov Ye., Krasnov V., Lebed O., Shutov S. // Materials Science-Poland. - 2009. - N2. - P. 355-363.

10. Park J. Simple modeling and simulation of photovoltaic panels using Matlab/Simulink / Park J., Kim H., Cho Y., Shin C. // Advanced Science and Technology Letters. - 2014. - V.73. - P. 147-155.

11. Tsai H.-L. Development of generalized photovoltaic model using MATLAB/Simulink / Tsai H.-L., Tu C.-S., Su Y.-J. // Proc. World Congress on Engineering and Computer Science, San Francisco, USA, 2008. - P. 846-851.

12. Corless R.M. On the Lambert W function / Corless R.M., Gonnet G.H., Hare D.E.G., Jeffrey D.J., Knuth D.E. // Adv. Comput. Math. - 1996. - N5. - P. 329-359.

13. Jadallah A.A. Modeling and simulation of a photovoltaic module in different operating regimes / Jadallah A.A., Mahmood D.Y., Abdulqader Z.A. //Acta Physica Polonica A. - 2015. - N 128 (2-B). - P. B-461 - B-464.

14. Dobos A. P. Procedure for applying IEC-61853 test data to a single diode model / Dobos A. P., MacAlpine S. M. // Proc. 2014 IEEE 40th Photovoltaic Specialist Conference (PVSC), Denver, Colorado, 2014. - P. 2846-2849.

15. Kou Q. A method for estimating the long-term performance of direct-coupled PV pumping systems / Kou Q., Klein S. A., Beckman W. A. // Sol. Energy. - 1998. - V.64 (1-3). - P. 33-40.