International Journal of Photoenergy

Volume 2008, Article ID 356261, 7 pages

http://dx.doi.org/10.1155/2008/356261

## RMS Current of a Photovoltaic Generator in Grid-Connected PV Systems: Definition and Application

IDEA Group, Department of Electronic Engineering, University of Jaén, 23071 Jaén, Spain

Received 18 September 2007; Accepted 19 February 2008

Academic Editor: Ugo Mazzucato

Copyright © 2008 P. J. Pérez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper includes a definition of a new and original concept in the photovoltaic field, *RMS current* of a photovoltaic generator for grid-connected systems. The *RMS current* is very useful for calculating energy losses in cables used in a PV generator. As well, a *current factor* has been defined in order to simplify *RMS current* calculation. This factor provides an immediate (quick and easy) calculation method for the *RMS current* that does not depend on the case particular conditions (orientation, location, etc.). *RMS current* and *current factor* values have been calculated for different locations and modules.

#### 1. Introduction

The *RMS current* of a photovoltaic
generator of a grid-connected photovoltaic system (GCPVS) is a new
concept that allows an easy calculation of certain matters related to
photovoltaic systems, like the energy losses in photovoltaic generator cables.
This new concept implies a new approach of photovoltaic systems engineering, an
approach that allows the use of new photovoltaic system design methods, more
similar to those used in other engineering fields.

The current generated by a photovoltaic cell, module,
or generator depends on its technological and/or inherent characteristics and
on the temperature and radiation exposure. Such a dependence on radiation and
temperature implies a remarkable random character in the yearly current, which
means that it can be studied as a stochastic process. However, in necessary
calculations for the design and evaluation of photovoltaic systems, experts can
use the radiation and temperature data of typical meteorological year [1], as they
are very close to the real values of the system. Therefore, we can *state that
a photovoltaic cell, module, or generator current in GCPVS is periodical, and the
period is one year*.

Considering all these facts, we can define a cell *RMS current* [] by the expression (1) where is the instant current of the cell maximum
power point and *T* is one year: A module *RMS current* []
can be defined in the same way that in the case of a cell, but using the module
instant current in the maximum power point []: If
we suppose all the parallel cell branches of the module provide the same
current, the module *RMS current* will
be the cell *RMS current* multiplied by
the amount of parallel cells [] in the module: Likewise, supposing all the modules are identical and
show the same tilt, we could define the generator *RMS current* as the module *RMS
current* multiplied by the number of parallel modules () in the generator:

#### 2. Methods

The calculation procedure of the RMS current is the following one.

Initially, the value of the cell current at the maximum
power point for ten minute intervals [2] is calculated from the
following values: daily mean monthly irradiation and temperature, and cell
parameters measured at standard test conditions. The applied model is based on
the following procedure.(1)Calculation
of the direct and diffuse daily mean monthly irradiation through those
expressions proposed by Liu and Jordan
[3] and
correlations obtained by Page [4].(2)Irradiance
calculation from daily irradiation according to the method proposed by Whillier [5].(3)Calculation of irradiances
within generator following the model proposed by Perez et al.
[6] for
the diffuse component considering the transmittance losses due to Fresnel
reflection losses, dirt and low irradiance levels.(4)Calculation
of the ambient air temperature, supposing it can be modelled according to two
half-waves of two cosine functions [7].(5)Calculation of short-circuit
current; fill factor and current at cell maximum power point [8]. Taking current
values at cell maximum power point as starting point, the cell *RMS current* in GCPVS can be calculated through the following approximation: where (this value equals the ten-minute interval
measured in days), (number of days for each month) and are the value of at the medium point of the ten-minute
interval expressed in Amps.

#### 3. Results and Discussion

##### 3.1. RMS Current of a Cell Different Locations and Orientations

This paragraph includes those values obtained through
the previously described calculation method for the *RMS current* of a photovoltaic cell, located in different European locations
and with several cell orientations. For that purpose, four European locations
have been chosen: *Bergen (Norway),
Bonn (Germany),
Paris (France), and Madrid (Spain),* located in northern, central, and southern Europe,
respectively. Main reasons for this choice are the availability of the
radiation and temperature data and the aim of studying the behaviour of the *RMS current* in several European
latitudes.

The studied cell shows the following values at
standard test conditions (Cell
temperature = 25^{°}C, Air Mass = 1.5, Irradiance = 1000 W/m25)(i)Short-circuit current () = 3.26 A.(ii)Open-circuit voltage () = 0.6 V.(iii)Maximum power point current () = 3.05 A.(iv)Maximum power point voltage () = 0.48 V.(v)Maximum power point () = 1.46 W.(vi)Normal operating cell temperature (NOCT) = 47^{°}C. Daily mean monthly maximum, minimum temperature, and irradiation values of studied
locations match the last ten years mean values provided by the Langley Nasa Research Center (http://eosweb.larc.nasa.gov/see).

*RMS current* values () for cell orientations between 90^{°} East and 90^{°}
West, and cell tilts between 0^{°} and 90^{°}, in 10^{°} steps, have been calculated and
a total of 190 values have been obtained for each location. *RMS current* values obtained for studied locations
have been represented in Figure 1.

Table 1 shows the mean value [] of
the 190 *RMS current* values obtained
for each studied location, standard deviation [], and coefficient
of variation []: Obviously, values obtained show that *RMS current* values depend largely on
cell orientation and latitude. This dependency is very similar to the case of
irradiation. This way, the *RMS current* in the south of Europe is 70% greater than in the North and values dispersion
due to cell orientation is really high, with standard deviations until 113 mA
and coefficients of variation of 13%.

##### 3.2. Definition of Current Factor

Previous values show that *RMS current* values are strongly dependent on different factors.
This characteristic makes difficult its application in quick methods for
designing or analysing photovoltaic systems operation, as the calculation
procedure is long and
tedious and it should be made for each location, orientation, and module type.

To facilitate *RMS current* calculation and encourage its use within photovoltaic field, a new
parameter has been introduced, the so-called *current factor*. This factor allows
an easy and simple calculation method for the *RMS current* and offers a great advantage: it can be applied to a
wide range of orientations, latitudes, and temperatures. In this sense, it allows
a simple and direct calculation for virtually all the interest cases related to
grid-connected PV systems.

The *current factor* [*F _{I}*] of a photovoltaic cell is defined as the ratio between
its

*RMS current*and the yearly mean daily irradiation at cell surface, both parameters normalised to the standard conditions values; If irradiance remained 1000 W/m

^{2}in a constant and continuous way during the whole year, (which we call at standard test conditions),

*RMS current*value would match current value at maximum power point and the yearly mean daily radiation would be 24 kWh/m

^{2}. Thus,

*current factor*is defined as the ratio between cell

*RMS current*[], divided by current at maximum power point at standard test conditions [], and the yearly mean daily irradiation at cell surface [], divided by reference yearly mean daily irradiation [ kWh/m

^{2}], this way, a dimensionless and standardised (equivalent to the unit at standard test conditions) factor is obtained.

If we considered all parallel cell strings of a module
and all parallel module strings of a generator to provide the
same current, the *current factor* value
would be the same for the cell, the module, and the generator.

##### 3.3. Current Factor for Different Locations and Orientations

Within this section, the *current factor* value is calculated for a photovoltaic cell placed
in different European locations, differently positioned, and with different
cell orientations. Studied locations and cell are the same rather than those
studied in the previous section.

Figure 2 represents those values obtained for *current factor* and Table 2 shows most
remarkable values.

Considering those values obtained for *current factor* (*F _{I}*),
we can remark the following.(a)

*Current Factor*mean values range between 1.58 for the case of Spain and 1.63 for Norway, increasing according to latitude. Although

*RMS current*values in southern Europe are up to 70% higher than in northern Europe,

*current factor*value is lower in the South, but only 3% lower than in North.(b)Dispersion of values due to cell orientation is low, with standard deviations around 0.04 and coefficients of variation under 3%. From the above data, we can conclude that

*current factor*values are very homogeneous for the whole Europe and much less dependent on cell tilt and orientation than in

*RMS current*case. We can observe a wide area with very homogeneous values in central orientations and tilts, that is, from 70

^{°}East to 70

^{°}West, and tilts from 0

^{°}to 70

^{°}.

Another
interesting matter is that *current factor* mean values are very similar to the values obtained for this factor for
those orientations maximising the annual irradiation received by the cell, as
shown in the Table 3.

##### 3.4. Current Factor for Different Photovoltaic Modules

With the aim of providing *RMS current* values for other locations and cell types as well as
checking the low dependency of *current factor* with respect to location and module type, this section includes values obtained
through the previously described calculation method for the *current factor* of different types of
photovoltaic modules situated in different locations.

For that purpose, ten locations have been chosen:
Camberra (AUS), Airport Darwin (AUS), Tokio (JAP), Yakushima (JAP), Kushiro
(JAP), Los Angeles (USA); New York (USA); Atlantic City (USA); Munich (GER);
and Berlin (GER). The main reason for this choice is studying the behaviour of *RMS current* in those countries with the
highest PV power installed until 2004.

The calculated values correspond to five types of photovoltaic modules available in market with different characteristics. Mainly, they show different short-circuit current and different manufacturing techniques (monocrystalline and multicrystalline silicon). Table 4 includes values of modules main electrical parameters, whose data are provided by manufacturers in modules specification sheets at standard test conditions.

Figure 3 shows the dependency of *RMS current* values with respect to the location and the module
type. Figure 4 shows that mean value obtained for *current factor* is very similar in all those different modules and
locations that have been studied.

##### 3.5. Values Obtained for RMS Current and Current Factor in Real System

Within this apart, *RMS current* and *current factor* values calculated from the theoretical model applied
in this paper are compared to those values obtained from monitored values in *UNIVER Project*
[8] System 1
and the “*Pérgola Fotovoltaica*” [9].

The *UNIVER Project* (THERMIE Program:
SE/00383/95/ES/UK) is made up of four grid-connected PV systems and a
total power of around 200 kWp at standard test conditions. System 1 is
located at University of Jaén Campus parking
and shows 38^{°} West orientation and a 7^{°} tilt. The generator real power is 62 kWp at standard test conditions, which means a generator real current at
maximum power point of 135 A.

The “*Pérgola
Fotovoltaica*” is a 2 kWp grid-connected PV system integrated at
terrace of University of Jaén Escuela Politécnica Superior. The generator
is made up of 23 photovoltaic modules Isofotón I-88 with a 15.45 A current at maximum
power point. This generator is divided into four series-connected subgenerators;
three of which are made up of 6 series-connected modules; meanwhile the fourth
one only has five modules. Subgenerators orientations are 6^{°}, 21^{°}, 36^{°}, and 51^{°} East, respectively, and all of them show the
same tilt, 15^{°}. However, in order to simplify the analysis of *RMS current*, we are supposing that all the modules show a 30^{°} East
orientation and the generator current at maximum power point at standard test
conditions is 15.45 A.

To compare *RMS current* and *current factor* values calculated by the theoretical model to those values obtained from
monitored data, the following procedure has been followed.(1)Calculating (theoretical) *RMS current* and (theoretical) *current
factor* values by the theoretical model applied in previous sections, from
irradiation monthly mean daily values obtained from irradiance monitored data.(2)Calculating (real) *RMS
current* from instant current monitored values in studied systems
generators.(3)Calculating (real) *current
factor* value from irradiation yearly mean daily value obtained from
monitored data and (real) *RMS current* value. Table 5 includes both theoretical and real *RMS current* and *current factor* values corresponding to “*Pérgola Fotovoltaica”* during 1997 and 1998, and to *UNIVER Project* system 1 during 2000 and
2001. These data show how real
values for *current factor* match
theoretical ones, with these ones being slightly lower than real values.

##### 3.6. Approximate Calculation Procedure of the RMS Current

From (7), we can demonstrate that the *RMS current* of a cell, module, or
generator in GCPVS equals the
product obtained by multiplying the *current factor* by the current at maximum power point at standard test conditions
and by yearly mean daily radiation divided by the reference radiation [ kWh/m^{2}]: Starting from the previous data, we can consider that
value of the *current factor* that will
be more usual in a typical GCPVS is 1.59. Thus, we obtain a new equation that
provides, on the one hand, an immediate (quick and easy) calculation of *RMS current* and, on the other hand, an
approximate value (small error) of *RMS current* which can be useful in
the field of engineering,

##### 3.7. RMS Current Application

One of the *RMS current* applications is the calculation of the power or energy losses taking place
in generator cables of a GCPVS. The dissipated mean power equals the value obtained by
multiplying the cables resistance by the square value of generator *RMS current*: From this point, we can easily estimate the yearly energy losses in generator cables: Cables resistance is defined as the product obtained by multiplying material
resistivity and length divided into cross-sectional area. In the case of copper
conductors, the following value is obtained: When operating with both previous equations, the following expression (13) is
obtained for the yearly energy losses in generator cables. This expression
shows how energy losses in generator cables are in direct proportion to cable
length and *RMS current* as well as in
inverse proportion to the cross-section area: If we use the approximate calculation (9) of the *RMS current*, we get an immediate approximation of the energy losses in generator
cables in a typical GCPVS: The energy
lost in a generator with different cross-sections and currents will be obtained
by adding up the energy losses in different cables, considering for each cable,
its length, its cross-sections, and its RMS current. This is the case of a
generator made up of several parallel branches using low cross-section cables
for connecting modules and larger cross-section cables for connecting the
generator to the inverter.

#### 4. Conclusions

The *RMS current* of a photovoltaic generator is a new concept that allows an easy calculation of
certain matters related to photovoltaic systems, as the energy losses in
photovoltaic generator cables. The *RMS current* value is very dependent on the used cell type and the radiation, so its value
has to be calculated for each generator, as it changes according to the module
type, orientation, and location.

The *current factor* solves out to a great extent the strong *RMS
current* dependency with respect to orientation and location. And, therefore, the *current
factor* allows a simple, easy, and quick calculation of the *RMS current* of
a cell, module, or generator

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