Mismatch losses minimization in photovoltaic arrays by arranging modules applying a genetic algorithm

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ScienceDirect Solar Energy 108 (2014) 467–478 www.elsevier.com/locate/solener

Mismatch losses minimization in photovoltaic arrays by arranging modules applying a genetic algorithm Samad Shirzadi ⇑, H. Hizam, Noor Izzri Abdul Wahab Department of Electrical and Electronic Engineering, University Putra Malaysia, 43400 UPM Serdang, Malaysia Centre of Advanced Power and Energy Research, Faculty of Engineering, University Putra Malaysia, 43400 UPM Serdang, Malaysia Received 10 May 2014; received in revised form 20 July 2014; accepted 2 August 2014

Communicated by: Associate Editor Takhir Razykov

Abstract Photovoltaic (PV) arrays consist of series and parallel connections of PV modules. Difference in current–voltage (I–V) characteristics among a batch of modules form an array causes power losses in PV systems referred to as mismatch losses. These power losses are conventionally reduced by module sorting techniques which sort modules based on an I–V parameter such as short circuit current, current at maximum power or maximum power. This work introduces a new method that employs genetic algorithm (GA) to find an arrangement of modules in an array which minimizes mismatch losses more effectively than conventional methods do. Extensive simulations are applied to adapt a GA to the problem of mismatch losses, find the arrangement and demonstrate its superiority over module sorting techniques in terms of mismatch losses decrement and energy yield increment. Instructions for practical application of the suggested method are also provided. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Mismatch losses; Genetic algorithm; Module sorting techniques; Photovoltaic

1. Introduction Based on statistical information acquired from International Energy Agency (IEA) Key World Energy Statistics in 2006, yearly energy consumption of the world counts for around 140,000 TW h and it is predicted to increase up to 60% more than this amount till 2030 (Freris and Infield, 2008). The growing energy demands, finitude of fossil fuels, and environmental issues have motivated human beings to seek for other energy resources. A while after the invention ⇑ Corresponding author at: Department of Electrical and Electronic Engineering, University Putra Malaysia, 43400 UPM Serdang, Malaysia. Tel.: +60 129761733. E-mail addresses: [email protected] (S. Shirzadi), Hashim@ eng.upm.edu.my (H. Hizam), [email protected] (N.I.A. Wahab).

http://dx.doi.org/10.1016/j.solener.2014.08.005 0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.

of the first silicon PV cell at American Bell Laboratories in 1954 (Quaschning, 2009), solar PV energy found its position as a reliable and sustainable energy resource. An outstanding increase in global installed capacity, from 5 GW in 2005 to 40 GW in 2010, stands behind the commercial prosperity of PV industry (Bortnikov et al., 1981). One issue among several issues toward well utilization of PV systems is power losses due to a phenomenon called mismatch which is the subject of this study. As depicted in Fig. 1, statistics available on ISI web of science show that mismatch power losses in PV systems is still a hot topic in academic researches due to considerable number of publication and increasing number of citations it has gotten since 2003. In PV generators, PV arrays convert photo-energy of sun irradiation to electrical energy in DC form. A PV array consists of series and parallel combination of PV modules.

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Citation

No

No

Publication

Year

Year

Fig. 1. Publications on mismatch power losses (left) and citations of mismatch power losses (right).

The same way PV modules are composed of PV cells. This modular nature of PV systems, is advantageous when it helps to wire the system up to desirable level of current, voltage and power. But the fact that PV modules with the same brand and same ratings are not exactly identical, turns the modularity of PV systems to be disadvantageous when it causes sort of power losses known as mismatch losses which are recognized by several research works (Bucciarelli, 1979; Chouder and Silvestre, 2009; Picault et al., 2010). Since PV modules are fabricated in the factory, further investigation and modification in cell level requires damaging the module encapsulation. So investigating mismatch losses mitigation techniques among cells inside modules is of PV module manufacturers’ interest whereas such investigation among modules in arrays is of system operators and installers interest. This paper deals with mismatch losses among modules at array level. Module sorting techniques are regular methods for minimizing mismatch losses in PV arrays in which modules are sorted in arrays by one of their characteristic parameters. This paper elaborates these techniques and reviews the mechanism of mismatch losses to propose a more effective solution. The proposed solution applies a GA to find the optimal arrangement of modules in an array, considering array output power as an objective function to be maximized. The arrangement of modules obtained by the GA is then compared to the arrangements obtained by the sorting techniques in terms of mismatch losses and energy yield. Understanding electrical configuration of PV arrays, difference in PV modules characteristics and mismatch losses mechanism are inevitable steps toward comprehending the problem of mismatch losses in PV arrays and subsequently devising solutions to this problem. Second and third sections elaborate these steps. Section four reviews the conventional treatments for mismatch losses. Section five explains the hypothesis of this study. Sixth section is devoted to the methodology of the work. Sections seven and eight provide results and conclusions respectively.

(BL) and total cross tied (TCT) Ramaprabha and Mathur, 2012. SP configuration which is the most practically used configuration is considered in this study. In this configuration connected modules in series form strings and connected strings in parallel form an array. Obviously all modules per each string work at the same amount of current and all strings in an array work at the same amount of voltage at a time. So it is simply concluded from Kirchhoff laws that the string voltage equals the summation of modules voltage in strings, the array current equals the summation of strings current and array’s voltage equals the voltage of every string. An array of 40 modules and 4 strings with SP configuration is depicted in Fig. 2 as an example. 3. Mismatch losses As previously mentioned, differences in PV modules characteristics together with modularity of PV arrays cause mismatch losses in PV arrays. These differences are discussed in first subsection and second subsection explains how these differences cause mismatch losses. 3.1. Differences in PV modules characteristics A group of modules of the same brand and same nominal ratings are not exactly identical. Their differences are understood by comparing and contrasting their characteristic parameters such as fill factor (FF), maximum power (PMPP), current at maximum power (IMPP), voltage at maximum power (VMPP), short circuit current (ISC) and open circuit voltage (VOC). Difference in module characteristic parameters is called I–V mismatch, since it results in different electrical performance. I–V mismatch comes from either temporary or permanent sources as classified in Fig. 3. Shading or non-uniform illumination might happen by fallen leaves of trees, scattered clouds moving over the PV

2. PV array configurations There are different possible configurations of PV modules in a PV array such as series parallel (SP), bridge link

Fig. 2. SP configuration of an array of 40 modules in 4 strings.

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469

Manufacturing Tolerance Light-induced Degradation Soiling Permanent Factors Discoloration Delaminataion

Sources of I-V mismatch

Cracking Temporary Factors

Shading or non-uniform illumination

Fig. 3. Classification of factors that cause mismatch losses.

array, shadow of an object situated around the PV array or another reason (Sonnenenergie, 2008). This factor can temporarily result in mismatch losses in an array of modules, whereas there are other factors which permanently cause mismatch losses such as manufacturing tolerance, light-induced degradation, discoloration, soiling, delamination and cracking. Referring to datasheet of PV modules are currently available in the market, it is found that despite all advancements of PV modules production technology, there still exist a manufacturing tolerance of ±3% to ±5% in their rating PMPP, IMPP and VMPP. Light-induced degradation, discoloration, soiling and delamination are all matters of aging (Smith et al., 2012). Cracking in cells is a module defect that can happen during shipment, installation or further happen due to hail (Ton et al., 2007) so it also can be considered as a matter of aging. All these permanent factors cause dispersion in PV module characteristic parameters. This variation in module characteristic parameters is the root of mismatch losses (Zilles and Lorenzo, 1992; Reis et al., 2002). This paper focuses on finding a solution for mismatch losses occurs due to all permanent factors. Thus, hereinafter term mismatch losses addresses the mismatch power losses coming from permanent factors and term mismatch refers to permanent I–V mismatch in PV modules. It should be pointed out that aforementioned permanent factors (excluding manufacturing tolerance) have another impact on the PV array performance that is always studied separately entitled “performance degradation” (Smith et al., 2012; Coello, 2011). Performance degradation is not subject of this study. Performance degradation is measurable by comparing the performance parameters of an aged PV system to its fresh condition whereas mismatch losses is measurable by comparing array output power to the summation of individual modules output power. 3.2. Mismatch losses mechanism In practical PV generators, a central Maximum Power Point Tracker (MPPT) system is applied to lead the system to work at its’ possible maximum power. In the presence of such a system all modules are led to work at their IMPP and VMPP which are not exactly identical for all modules as

previously explained. Simply put, under central MPPT, two modules with different IMPP that are connected in series, compromise to work at the lower IMPP and similarly two modules with different VMPP that are connected in parallel work at the lower VMPP. Generalization of these conditions to a large array of PV modules connected in series and parallel, results in the following equation (Chamberlin et al., 1995): ! P mod;i arr P  P MPP iX MPP MML% ¼ 100 ð1Þ mod;i P MPP i where MML% is the percent of mismatch losses, P mod;i MPP is the maximum power produced by ith module if it works independently, and P arr MPP is the output power of the whole array. 4. Conventional solutions According to the mechanism of mismatch losses explained in Section 4, the more variation exists in characteristic parameters of modules forming an array the greater the resulting MML% of that array will be. This fact is the basis of some conventional mismatch mitigation techniques called module sorting techniques which stand for sorting modules in arrays based on one of the modules characteristic parameters such as IMPP, ISC or PMPP (Sonnenenergie, 2008; Webber and Riley, 2013; Bakas et al., 2012) for arrays with SP configuration. Variation of maximum power current of modules ðI mod MPP Þ in strings and variation of maximum power voltage of strings ðV str MPP Þ in the array are in fact sources of mismatch losses (Bucciarelli, 1979; Kaushika and Rai, 2007). Sorting modules by ISC and IMPP reduces mismatch losses by greedily minimizing the variation of I mod MPP in strings (Webber and Riley, 2013; Bakas et al., 2012) but it is further demonstrated that it neglects the variation of V str MPP if it is compared to sorting by PMPP. It is also demonstrated that sorting modules by PMPP returns a balancing outcome str in terms of minimizing the variation of I mod MPP and V MPP , but it does not result in lower mismatch losses in comparison with sorting by ISC or IMPP.

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5. Proposed solution The number of overall possible arrangements of modules in an array is calculated as follows:       m m  ð1  lÞ m  ð2  lÞ X¼   l l l ð2Þ   m  ððn  2Þ  lÞ  ...  8m; n; l 2 N & n > 1 l where X is the number of possible arrangements, l is the number of modules in series per each string, n is the number of parallel strings, and m is the total number of modules in the array. Table 1 shows how dramatically number of overall possible arrangements for an array increases by increasing the number of modules. For example, the second array in Table 1 with 40 modules organised in 4 parallel strings of 10 modules, has 4.7054  1021 possible arrangements and sorting modules by IMPP, ISC and PMPP are only three of them. Since module sorting techniques reduce mismatch losses through minimizing the varstr iation of I mod MPP and V MPP by simply sorting modules and performances of other possible arrangements are unknown, the idea that there might be an arrangement with lower resulting mismatch losses than what is obtained by those few arrangements of module sorting techniques seems worthy of investigation. Finding such an arrangement is the hypothesis of this study. Instead of trying all the possible arrangements to find the best one which is apparently inapplicable due to magnitude of the search space (all possible arrangements), GA which is highly qualified among probabilistic search algorithms is applied to find it. 6. Methodology In order to examine the suggested mismatch mitigation technique and compare it to conventional ones in terms of its capability of mismatch losses reduction and energy yield improvement, some simulations are applied in MATLAB environment. The arrays listed in Table 1 require datasets of modules with dispersed characteristic parameters to represent arrays with mismatch losses. Each module needs to be modeled to be implemented in simulations. A GA must be adapted to the problem of mismatch losses to find the new arrangement of modules in the arrays. Finally the GA based arrangement and three other arrangements need to be applied to each of the arrays

and mismatch losses and energy yield must be calculated for each one respectively. Following subsections cover the methodology of the work step by step. 6.1. PV module modeling This is a common practice in electrical engineering to model an electrical element with some mathematical equations and their equivalent circuit. Without such a model, participation of that element in any credible calculation or simulation would be impossible. One-diode model (Chan and Phang, 1987) is chosen to represent PV modules in different simulations which are carried out in this study. Respective equations are as follows and its equivalent circuit is shown in Fig. 4.     V þ Rs  I V þ Rs  I I ¼ I ph  I 0 exp ð3Þ 1  ns  V t Rp Vt ¼

akT q

ð4Þ

where I and V are the module current and voltage, Rs and Rp are series and parallel resistances, Iph is the photogenerated current, I0 is the diode saturation current, ns is the number of cells in series per each module, a is the diode ideality factor, k represents the Boltzmann constant (1.381  1023J/K), T is the module temperature and q is the electron charge (1.602  1019 C). 6.2. Data sets of modules The stochastic process introduced in Bakas et al. (2012) is used to generate 4 data sets of 18, 40, 65 and 90 PV modules. Studies on the statistical distribution of PV modules characteristic parameters shows that Gaussian distribution explains the distribution of VOC, ISC and PMPP (Reis et al., 2002; Damm et al., 1995). This stochastic process assumes Gaussian distribution for VOC, ISC and PMPP and generalizes it to IMPP and VMPP as well by considering their datasheet values as mean values and 10% of datasheet values as their standard deviations. Datasheet values of CSUN 095-36M mono-crystalline solar module which are listed in Table 2 are used in this study. Next step of the stochastic process is to randomly choose values within the range of ±10% of mean value for I MPP , V MPP , I SC and V OC , and within the range of ±3% of mean value for P MPP . These two allowable selection ranges are in accordance with the manufacturing tolerance reported in the datasheet and standard

Table 1 4 Different arrays and respective number of overall possible arrangements. Array/dataset number

Array dimension

Number of modules

Number of all possible arrangements (size of search space)

1 2 3 4

36 4  10 5  13 5  18

18 40 65 90

1.7153  107 4.7054  1021 8.8090  1041 1.3811  1059

Rs

I +

Iph

D

V

Rp -

Fig. 4. PV module equivalent circuit based on one-diode model.

S. Shirzadi et al. / Solar Energy 108 (2014) 467–478

Sera et al. (2007), Can and Ickilli (2014). Stochastic process that is applied in this study is summarized in Fig. 6. Fig. 7 shows the current voltage curves (I–V curves) at STC for 40 modules that comprise the array with 40 modules. I–V curves show more dispersion in I SC and VOC than what they do at the knee point of the curves. This condition caused by applying the restriction of ±3% of mean value for PMPP which equals the multiplication of IMPP and VMPP. The absolute error between generated characteristic parameters and those acquired from respective module model are calculated by Eq. (5) and their maximum amounts among all the modules are demonstrated for each of the arrays in Table 3 to verify the accuracy of modeling process.

Table 2 Datasheet values of CSUN 095-36M mono-crystalline solar module. Parameter

Value

Unit

PMPP VOC ISC VMPP IMPP Performance deviation of PMPP Performance deviation of VOC, ISC, VMPP and IMPP Voltage temperature coefficients Current temperature coefficients Power temperature coefficients Number of cells in series per module

95 22.5 5.56 18.3 5.21 ±3 ±10 0.307 +0.039 0.423 36

(W) (V) (A) (V) (A) % % %/K %/K %/K Pieces

E x ¼ jxmodeled  xgenerated j Probability Density

0.8

0.4 ±10% of the Mean Value

0.2 0 2.78 3.336 3.892 4.448 5.004 5.56 6.116 6.672 7.228 7.784 8.34

Isc Values (A) Fig. 5. Gaussian distribution for I SC and allowable range for selecting random values which are shown with dashed lines.

deviation of 10% of mean value is chosen by try and error to generate a distinguishable amount of MML% close to what is reported by Picault et al. (2010) which include practical investigations of mismatch losses due to the permanent factors. For instance, the assumed Gaussian distribution for ISC and its relative allowable selection area is shown in Fig. 5. Then desirable number (regarding the number of modules for each array) of meaningful groups of these chosen values (ISC, VOC, IMPP, VMPP and PMPP) are selected to represent the characteristic parameters of PV modules. Finally equivalent circuit parameters (Rs, Rp, Iph, I0 and a) are extracted out of the characteristic parameters for each module based on the modeling approach explained in ,

,

, and

6.3. Sorting modules in arrays As previously stated, sorting modules by ISC, IMPP and PMPP are the conventional ways of mismatch losses mitigation for SP configured arrays. In order to investigate mismatch losses reduction by these techniques MML% should be calculated for each of the arrangements coming from sorting by ISC, IMPP and PMPP respectively. The organization of modules dataset, conveniently allows for

with a mean value equals to their datasheet values and a standard deviation equals to

Choosing random values within the range of ±10% of datasheet value for The extra condition for choosing

and

,

,

, and

is that the multiplication of each selected pairs of VMPP and IMPP must be in the range of ±3% of

datasheet value of Selecting meaningful groups of these generated values (

Extracting equivalent circuits parameters (

ℎ , 0,

,

,

,

and

)

R s , R p , and a) for each group of selected values

Using one-diode model for each module and finding accuracy of PV module modelling

,

,

,

ð5Þ

where E_x is the absolute error, xgenerated is the generated characteristic parameter through stochastic process, and xmodeled is the analogous characteristic parameter acquired through one-diode model. The solution to inaccurate modeling problem is to regenerate the values and recalculate the equivalent circuit parameters regarding the new values. The final outcomes of the stochastic process which is explained above are 4 datasets of modules for 4 mentioned arrays in Table 1. Format of the produced datasets which are called dataset matrixes hereinafter is shown in Table 4. In this format, each row of the dataset matrix includes the characteristic parameters and equivalent circuit parameters of one module. Modules are uniquely labeled by their row numbers and unable to be removed from the array, duplicated or replaced by other modules out of the dataset.

0.6

Assuming Gaussian distribution for 10% of their datasheet values

471

and

and comparing them with the generated values to verify the

Fig. 6. Different steps of the stochastic process applied in this study.

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calculate the array power voltage (P–V) curve out of the array I–V curve. Now finding the array maximum power P MPP is restricted to finding the maximum of array P–V curve (Bishop, 1988). Resulting P–V curves of the second array of 40 modules are drawn in Fig. 8 for 4 different arrangements.

7

Current (A)

6 5 4 3 2

6.5. Adapting a GA to the problem of mismatch losses

1 0

5

10

15

20

25

GAs are inspired by natural genetic evolution of life species. The basic idea of GAs is very simple. Thinking of a herd of prey as population of life species, those that are superior in some important characteristics such as smartness and agility have the chance of survival and breeding. So future generations will inherit better specifications and by passage of time the population will improve and evolve naturally. The same way, it is possible to generate a population of random answers for a specific optimization problem (initialization). Using the optimization criteria (fitness function) as a tool to score these generated answers (evaluation), applying a probabilistic mechanism to select superior pairs of answers (parent selection) and making new pairs of answers out of the selected pairs (crossover) will result in a new population of answers which is averagely better than the previous one. Continuation of this process can provide even better answers and this is how the basic idea of a GA is applied in optimization problems. GAs are useful especially when regular mathematical techniques are not applicable. Applying GAs in a practical problem concerns more details such as determining chromosomes, genes and the dynamic of mutation. Each answer is considered as a chromosome that is a chain of gens. Each gene represents a specification of an answer. Mutation that stands for changing one or more genes within every chromosome of a population of answers, motivates the stabilized population to improve again (Iglesias et al., 2006). Specifications of the problem of finding an optimal arrangement of PV modules in an array is similar to a

Voltage (V) Fig. 7. I–V curves of 40 modules drawn based on one-diode model.

sorting modules by ISC, IMPP and PMPP as well as any other arrangement. 6.4. Calculating I MPP , V MPP and P MPP of the array for different arrangements In order to score the answers by fitness function, calculation of array output power is frequently needed during the algorithm operation. When the dataset of Table 4 is on hand, I–V curve of each module is easy to calculate through Eq. (3). Since Series connection of modules in each string forces them to work at the same amount of current, adding the voltages of modules in each string at the common current intervals provides the strings voltages. Finding voltages of different modules of each string for common current intervals is done by interpolation. So far I–V curves for each string is calculated. As parallel connection of strings in the array forces them to work at the same amount of voltage, adding the strings currents at the common voltage intervals provides the array current. Again strings currents for common voltage intervals are acquired through interpolation. In this way array I–V curve is calculated for every arrangement of modules in the array. Multiplication of current and voltage of the array leads to

Table 3 Maximum absolute error for generated characteristic parameters relative to those acquired from one-diode model among all modules of 4 datasets. Array/dataset number

Array dimension

Number of modules

E_ISC (A)

E_VOC (V)

E_IMPP (A)

E_VMPP (V)

E_PMPP (W)

1 2 3 4

36 4  10 5  13 5  18

18 40 65 90

2.94E05 1.99E05 2.94E05 2.69E05

5.51E06 4.44E06 5.51E06 6.15E06

4.99E04 5.08E04 5.08E04 5.08E04

1.75E03 1.92E03 1.92E03 1.93E03

3.00E04 2.54E04 3.00E04 3.32E04

Table 4 Format of dataset matrixes for each array. Module number

Characteristic parameters

Equivalent circuit parameters

ISC

VOC

IMPP

VMPP

PMPP

Rs

Rp

a

I0

Iph

1 2 3. .. m

5.85 5.35 6.08 .. . 5.94

22.13 23.26 21.08 .. . 23.38

5.35 4.92 5.63 .. . 4.91

17.35 18.77 16.90 .. . 19.71

92.75 92.36 95.13 .. . 96.84

2.35E01 1.49E01 1.70E01 .. . 1.78E05

4.16E+02 4.91E+02 7.17E+02 .. . 2.59E+01

1.49 1.55 1.30 .. . 1.33

7.04E07 5.12E07 1.71E07 .. . 3.30E08

5.85 5.35 6.08 .. . 5.94

S. Shirzadi et al. / Solar Energy 108 (2014) 467–478

473

Fig. 8. P–V curves for 4 different arrangements of 40 modules of the second array.

well-known problem called Traveling Salesman Problem (TSP) that has been solved by GA Liu (1402), Ahmed (2014), Changdar et al. (2014), Wang (2014). Imagining a group of cities with specified distances between them, TSP is to find the shortest path that passes all the cities and returns back to the starting point. Main similarity between finding optimal arrangement of PV modules in arrays and TSP is that both problems deal with searching and finding one specific arrangement of large number of objects with some specific qualities among numerous possible arrangements. Other promising similarity is that PV modules in arrays are as unique as the cities in TSP are. According to Eq. (1) rearranging the PV array affects P mod P arr MPP with i P MPP being constant. In other words minimization of MML% equals the maximization of array output power. Therefore, in order to adapt a GA to the problem of mismatch losses, fitness function is defined as the array output power to be maximized as formulated in Eq. (6). k;i FF ðAk;i Þ ¼ P arr MPP

ð6Þ

where FF is the fitness function, Ak,i is a 1  m vector that k;i shows the arrangement of m modules in the array, P arr is MPP the array output power for k th arrangement at ith generation, k indicates an answer among the population of answers and i indicates one generation of all generations produced during the algorithm operation. In order to consider positions of modules as genes, different arrangements of modules in the array as chromosomes and apply partially mapped crossover (PMX) Naveen Kumar and Rajiv Kumar, 2012, it is needed to assimilate the matrix shape of the PV array to a vector shape of chromosome. To remedy this strings of each PV array are lined up sequentially to form a chain of modules as depicted in Fig. 9 for the array of 40 modules and 4 strings as an example. Positions of modules in the array are kept unchanged before and after this transformation to prevent miscalculation of array output power which is further explained. Since modules are labeled by unique numbers, every random permutation of numbers from 1 to m in Ak,i gives a random arrangement of modules in the array which is

considered a random answer for FF. Mutation concerns swapping the position of two modules in the array. For each of the arrangements that are created at the initialization, during the GA process as random answers or as the k;i final optimal answer P arr MPP is calculated by the method that is explained in Section 6.4. Flowchart of the GA is demonstrated in Fig. 10. Where i counts the GA procedure rounds and indicates one generation among all generations, FF A ðiÞ is the average fitness score of all answers of ith generation, j indicates rounds of GA operation process at which the population of answers stabilizes and stops anymore improvement and FFS(j) is the respective value of fitness function. The FF(Ak,i) in Eq. (6) that calculates the fitness score of answer k of the generation i should not be confused with FFA(i) as the average fitness score of generation i, or FFS(j) as fitness score of the stabilized generation. Maximum fitness score of each generation that is plotted in Fig. 10 is not a necessity to the algorithm flow but an important part of the results. The GA starts with generating random answers to form the initial population. Evaluation of these answers is applied by calculation of fitness function for each of them (FF(Ak,i)). Then parents’ selection is performed by means of a probabilistic procedure called casino roulette wheel which is explained in Iglesias et al. (2006). After that PMX crossover is applied to the selected parents and resulting offsprings are replaced by the current generation. Evaluation, parents selection, crossover and generation replacement are applied sequentially and continuously until stabilization in FF among population of current generation is achieved which means all answers are converged to one answer and stop anymore improvement. At this stage, mutation is applied to all the population of current answers. This mutated population goes through the evaluation, parent selection, crossover, and generation replacement to reach the stability again. This process continues whilst average fitness (FFA), fitness of stabilized generations (FFS) and maximum fitness per each generation are preserved. Finally procedure stops whenever several stabilized populations of answers with same fitness score are observed consecutively and at least one mutation has been

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1

2

Fig. 9. Transformation applied to make vector shape of chromosomes out of matrix shape of PV array.

Initialization: Generating 30 Random Answers

= 1, = 1 and

= +1

(1) = 0

Evaluation by Fitness Function

No Parents Selection

Is Stabilization achieved?

Yes

Crossover ( )= Replacing Offspring with Previous Generation (Selected Parents)

()

No Has Mutation Been Applied so Far?

=2

Yes

Mutation No = +1

( )=

Is ( − 1)? Yes

End

Fig. 10. Flowchart for the GA algorithm.

applied since the beginning of GA process. Fig. 11 shows the evolution of the GA. Dropping points in the left curve of Fig. 11 are the average fitness score breakdowns after every mutation. The introduced GA process that takes a data set of PV modules with determined SP configuration as input and returns an optimal arrangement of the modules in the array as output, is applied to the 4 different datasets that are generated through stochastic process regarding 4 arrays specified in Table 1. Choosing the right size for initial population is important because the bigger the initial population is the more time consuming the simulation program would

be. In order to appropriately size the initial population for each simulation, at first an initial population between 100 and 500 (almost in accordance with what has been experienced for solving TSP by GA (Julstrom, 1996) was selected for each simulation and valid and stable answer (which shows crossing the knee point in the GA evolution curves e.g. Fig. 10 and does not vary for multiple times of simulation repetition) was achieved. Then the initial population was decreased step by step to see if we could get the answer with smaller initial population. Table 5 lists all simulations that are performed for the 4 arrays in this work. Respective results are discussed in the next section.

S. Shirzadi et al. / Solar Energy 108 (2014) 467–478

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Fig. 11. Evolution of GA for the 4  10 array (a) average fitness of each generation and (b) maximum fitness of each generation.

Table 5 Format of data set matrixes for each array. Array number

Number of modules

Array dimension (strings  modules per string)

Initial population

Mutation percentage

Crossover method

1 2 3 4

18 40 65 90

36 4  10 5  13 5  18

20 30 50 70

2 2 2 2

PMX PMX PMX PMX

Table 6 Mismatch losses for each of the arrays. Arrangement

Sorted by I SC Sorted by I MPP Sorted by P MPP Arranged by the GA

MML% Array #1

Array #2

Array #3

Array #4

1.38 1.17 1.57 1.02

1.18 1.13 1.26 0.74

1.27 1.08 1.27 0.74

1.07 1.02 1.20 0.63

7. Results and discussion

In order to have an idea about the advantage of using the new GA based arrangement, the recoverable energy which could be gained by using this arrangement relative to applying the best sorting technique for the presumed arrays is calculated as the differentiation of energy yield under GA based arrangement and under the best sorting technique which is sorting by IMPP. Recoverable energies and recoverable energies percentages for the selected day and also selected month are calculated by Eqs. (7) and (8) respectively. EGA  Es Es   EGA  Es DEr % ¼ 100 Es DEr ¼

In order to have a judgment about effectiveness of suggested arrangement versus those coming from the sorting techniques detailed comparisons in terms of resulting mismatch losses, output power and energy yield of the PV array in the presence of both GA based arrangement and sorting ones are presented hereby. Table 6 shows that the GA based arrangement however decreases the mismatch losses for all arrays but this decrement is sensitive to the size of array. Considering the 4 different arrangements of modules in presumed arrays, energy yield is calculated for June of 2012 and also for 12 of Jun separately based on irradiation and temperature profiles measured at solar site of University Putra Malaysia (UPM) during June of 2012. Measurement time intervals are 1 min and 15 min for the day and the month respectively. These two time periods are chosen due to their relatively short sampling intervals comparing to other available metrological datasets from UPM solar site. Specifications of UPM solar site are brought out in Table 7, and energy yield calculation results are demonstrated in Table 8.

ð7Þ ð8Þ

where DEr is the recoverable energy, %DEr is the percentage of recoverable energy, EGA is the energy yield under GA based arrangement, and Es is the energy yield under the best sorting technique which is sorting by IMPP. Recoverable energies and percentage of recoverable energies for the 4 arrays are reported in Table 9. One interesting comparison is made about variation of str I mod MPP within each string and variation of V MPP within the array. Standard Deviation (SD) is chosen as an index of variation of these two factors. Table 10 depicts the outcome of different arrangements of the second array at STC. As previously mentioned, the more variation exists in series blocks maximum currents and parallel blocks maximum voltages the higher the resulting mismatch losses will be. Variation of I mod MPP in strings that achieved through sorting by I MPP is lower than that of sorting by I SC , but it

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Table 7 Specifications of UPM solar site. Location

Longitude

Latitude

Daily average radiance level (W/m2)

Average monthly ambient temperature (°C)

Daily maximum ambient temperature (°C)

UPM, Selangor, Malaysia

2.99

101.72

253–512

29.60

30.2–36.6

Table 8 Energy yield for each of the arrays during one day and one month. Arrangement

Energy yield (kW h) Array #1

Sorted by I SC Sorted by I MPP Sorted by P MPP Arranged by the GA

Array #2

Array #3

Array #4

Day

Month

Day

Month

Day

Month

Day

Month

14.02 14.05 13.99 14.07

277.12 277.72 276.59 278.13

31.47 31.49 31.45 31.62

622.08 622.43 621.57 624.87

51.02 51.12 51.02 51.29

1008.40 1010.31 1008.37 1013.79

70.87 70.91 70.78 71.19

1400.78 1401.42 1398.88 1407.02

Table 9 Recoverable energy for each of the arrays during one day and one month. Arrangement

Recoverable energy Array #1

Day Month

Array #2

Array #3

Array #4

Wh

%

Wh

%

Wh

%

Wh

%

50 1010

0.36 0.36

141 2792

0.45 0.45

270 5390

0.53 0.53

320 6240

0.45 0.45

Table 10 str Comparison of different arrangements in terms of MML%, variation of I mod MPP and V MPP at STC for the second array of 40 modules. Arrangement

Sorted by ISC Sorted by IMPP Sorted by PMPP Arranged by the GA

MML%

1.18 1.13 1.26 0.74

SD of I mod MPP (A)

SD of V str MPP (V)

1st String

2nd String

3rd String

4th String

0.12 0.09 0.24 0.16

0.11 0.06 0.20 0.23

0.17 0.06 0.21 0.14

0.13 0.11 0.23 0.20

(a)

(b)

(c)

(d)

7.72 9.50 3.30 3.74

Fig. 12. Different arrangements of 40 PV modules in the 4  10 array (a) Sorted by ISC, (b) sorted by IMPP, (c) sorted by PMPP and (d) arranged by the GA.

is vice versa for Vstr MPP variation in the array. It means that a greedier decrement of I mod MPP variation in strings decreases the mismatch losses even for increasing the V str MPP variation. Sorting by P MPP , exhibits an outstanding decrement in mod V str MPP and an increment in I MPP , but despite the relative

balance it shows in terms of current voltage variations, it does not result in lower mismatch losses than those of other sorting techniques. GA based arrangement also exhibits a str balancing results in terms of variations of I mod MPP and V MPP . mod This arrangement returns slightly lower I MPP variation

S. Shirzadi et al. / Solar Energy 108 (2014) 467–478

and higher V str MPP variation than the one of sorting by PMPP and also higher I mod MPP variation relative to arrangements of sorting by ISC and IMPP. The average variations of I mod MPP among 4 strings equals 0.22 A and 0.18 A for sorting by P MPP and GA based arrangement respectively. It seems that GA helps the array to negotiate mismatch losses more effectively by swapping modules among strings during the GA evolution. Ultimate outcome of the GA for an array of m modules is the answer with best fitness score which in fact is a 1  m vector including a permutation of numbers between 1 and m. Since each module is uniquely labeled in the dataset matrixes and regarding the aforementioned reciprocal chromosome to array transformation, this vector equals an arrangement of modules in the array that returns the highest array output power and lowest MML% among all possible arrangements. This arrangement is presented in Fig. 11 along with 3 other arrangements that are resulted from sorting techniques for the second array. Since array has SP configuration, displacement of modules within one string is allowed as well as displacement of strings within the array. This fact helps to see the similarities and differences between the 4 arrangements in Fig. 12. For example, it is noticed that modules number 1, 4, 13, 18, 22, 26 and 40 are all in one string in both sorting by IMPP (Fig. 12.b) which is the best among the sorting techniques and the GA based arrangement (Fig. 12d). This work suggests a new technique for arranging PV modules in arrays. This technique is applicable through some practical steps. First one is to perform I–V measurements for all the modules of the target array to acquire appropriate data which is required for the precise PV modules modeling. These I–V measurements are suggested to be performed under STC, otherwise measured I–V curves have to be translated to STC. Second step is to model the PV modules to come up with a structurally same modules dataset as the one depicted in Table 5. Third step is to apply the GA algorithm explained here to the acquired dataset considering desirable array structure e.g. 4  10 in SP. Forth step is to arrange the modules in the array based on the resulting arrangement of the GA algorithm considering the modules labeling number in the dataset. Mentioned steps form the instructions regarding the new module arrangement technique. Applying and validating this new technique can contribute the future work regarding this study.

8. Conclusion A new method of arranging modules in an array based on GA was demonstrated in this work. This method was compared to conventional techniques of mismatch losses reduction in terms of MML% and energy yield. It seems that a smart arrangement of modules reduces mismatch losses more effectively than what is gained by an objective str minimization of I mod MPP which neglects the V MPP or a

477

str combined minimization of both I mod MPP and V MPP through sorting by P MPP . Results show an energy saving potential for this arrangement technique. Energy yield calculation supports the idea that the suggested arrangement is superior to sorting techniques not only at STC but also under verity of irradiations and temperatures.

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