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Basic Modelling of a Residential Photovoltaic

System

Alexandre Oliveira de Almeida

Don´

at B´

anki Faculty

´

Obuda University

Budapest, Hungary

aoalmeida@stud.uni-obuda.hu

Abstract

This paper provides a brief description of the workings of a residential solar system with the aim to create basic mathematical

models of said system and compare the results with a real-life example, to analyze how accurate are the models in comparison

to reality, using knowledge acquired during the Systems Engineering class of the Mechatronics Engineering undergraduate course

at ´

Obuda University.

Index Terms

Mathematical model, photovoltaic energy, residential photovoltaic system.

I. INTRODUCTION

With the rising concerns about climate change and carbon emissions, the search for alternate, non-polluting energy sources

has seen an increase, and, along with a worldwide rise in energy costs, many people are looking for an alternative system to

power their homes. The solar photovoltaic (PV) system is the most popular among residential users, mainly due to its easiness

of installation and use, and cost effectiveness. A report produced in 2014 by the International Energy Agency predicted that

the sun could be the largest electricity source by 2050 [1], with solar photovoltaic systems accounting up to sixteen percent

of the world’s electricty demand in the same timeframe.

The most basic PV system comprises of an array photovoltaic cells (solar panels), which generate energy through the

photovoltaic effect, and one or more inverters. Solar panels produce energy in direct current (DC) form, and in order for it

to be used it needs to be converted to alternating current (AC), hence the need of a PV inverter. The inverter will basically

convert the DC energy produced by the solar panels to AC, so that it can either be used by the household or fed back to the

grid, offsetting energy costs in the electricity bill, for example. The array of PV cells may be divided onto ”strings”, which

allow the inverter to better optimize the energy output from the panels. Other components may be added to the system, such as

the string-box, for example, used as a security device, composed of surge protectors and circuit breakers that offer short-circuit

and overcurrent protections. If an anomaly is detected by the string-box, the energy ﬂow is stopped, preventing the anomaly

from damaging the inverter and other devices connected to the energy grid.

II. SY STE M ANA LYSIS

As stated in the previous section, a basic photovoltaic system has two main components: the photovoltaic cells and the

inverter. In order to analyze the system from an engineering point-of-view, the PV system will be simpliﬁed: we will ignore

the inverter and focus only on the solar panels, to create mathematical models that can estimate daily and monthly energy

production of the system. The inverter is left out of the analysis to simplify the modelling, as most solar inverters already have

a high efﬁciency (over 98%), which won’t impact the analysis of the system.

Fig. 1. Simple diagram of a PV system. Adapted from [2]

For comparing the accuracy of the models, we will utilize a real world PV system installed at the author’s household in Brazil.

It has an array of 23 solar panels, distributed along two strings, connected to two solar power inverters, which characterizes a

multiple input, multiple output system (MIMO). To simplify the modelling, the system used for analysis will consider a single

solar panel array and string consisting of 23 solar panels, characterizing a multiple input, single output system (MISO).

(a) Aerial view of the household (b) Overview of the electrical system

(c) Connections of the ﬁrst inverter (d) String box of the ﬁrst inverter

(e) Second inverter (f) String box of the second inverter

Fig. 2. Electrical installation of the PV system

A. Characteristics of the Real System

The real world system used for comparison is composed of an array of 23 solar panels, 20 with a peak power of 360W and

three with a peak power of 400W, distributed along two strings, connected to two inverters. The solar panels are inclined at

a 19° angle, 19° East from the geographic North coordinate (azimuth). The efﬁciency of the solar panels at Standard Testing

Conditions (STC) are 18.51% and 19.90%, respectively.

By analysing the datasheet of both types of solar panels used in the system, it’s very important to note that the peak power

previously mentioned is measured at STC, which speciﬁes an irradiance of 1000 W/m² and cell temperature at 25°C. The peak

power is slightly reduced at Nominal Operating Cell Temperature (NOCT), as this condition considers an irradiance of 800

W/m², cell temperature at 45°C, ambient temperature at 20°C and wind speed at 1 m/s. The peak power values for NOCT are

268W for the 360W modules and 297W for the 400W ones.

TABLE I

SOL AR PAN EL S PARAMETERS

Module 360W (STC) 360W (NOCT) 400W (STC) 400W (NOCT)

Max Power – Pmax (W) 360 268 400 297

Open-circuit Voltage – VOC (V) 47.4 43.8 48.7 46.48

Short-circuit Current – ISC (A) 9.7 7.8 10.79 8.35

Voltage at Max. Power – Vmp (V) 38.5 36 40.70 37.73

Current at Max. Power – Imp (A) 9.36 7.46 9.84 7.88

Module Efﬁciency – η(%) 18.55 — 19.90 —

With the peak power values at NOCT, it is also possible to calculate the efﬁciency at the same conditions, using simple

rules of three, as seen in (1) and (2).

η360W=268W·18.51%

360W= 13.78% (1)

η400W=297W·19.90%

400W= 14.78% (2)

B. Mathematical Models

As this paper focuses on creating basic models to help estimate daily and monthly energy production of a PV system, the

ambient temperature, wind speed and cell temperature variables included in NOCT conditions will be skipped. Although some

previous Physics knowledge may be required to create the mathematical models, the analyzed system can be considered as a

gray box type, as we have some knowledge of the inner workings of the system. As previously mentioned, the analyzed system

consists of a solar array of 23 solar panels. It is worth to simplify it even more, considering the solar array of 23 panels as a

single entity; in other words, the system used will be a single input, single output system (SISO).

One frequently asked question by owners of PV systems is ”How much energy will the system produce in a month?”. In

order produce an answer to this question, we propose the creation of a very simple mathematical model that can estimate the

monthly production of a photovoltaic system. This model will be expressed by (3). The dimensional analysis can be seen in

(4).

Emonth =H·A·n·ηavg (3)

where: H =the average solar irradiation of the month

A=the solar panel surface area,

n=the number of solar panels,

ηavg =the average efﬁciency of the solar panels.

kW h =kW h

m2

·m2

·n1

1

·ηavg

1

1(4)

The average efﬁciency of the solar panels can be calculated using a weighted arithmetic mean, as there are two types of

modules being used in the system. The result is 18.73% for STC and 13.91% for NOCT and can be veriﬁed in (5) and (6).

ηavg =20 ·18.55% + 3 ·19.90%

23 = 18.73% (5)

ηavg =20 ·13.78% + 3 ·14.78%

23 = 13.91% (6)

At ﬁrst sight, it may seem that the model is missing an important piece of information: the peak power output of the

solar panels. Honsberg and Bowden [3] have shown that the efﬁciency is a function of the maximum power of the solar cell

multiplied by its ﬁll factor, where the ﬁll factor is deﬁned as the ratio between the maximum power output of the solar panel

and the product of its open-circuit voltage and short-circuit current. Therefore the peak power output parameter can be replaced

by the efﬁciency of the solar cells.

The ﬁnal missing piece of the model is the solar irradiation parameter (H), which can be obtained through the PVGIS

application [5] or through the CRESESB website [6]. We will utilize both to compare how accurate they are in relation to the

real world PV system.

To obtain the solar irradiation from PVGIS, we need to set a few parameters in the application. The location of the house

is selected in the map. Then we have to choose the solar radiation database from two options: the National Solar Radiation

Database (NSRDB), by the National Renewable Energy Laboratory, or the SARAH database, provided by the EUMETSAT

Climate Monitoring Satellite Application Facility (CM SAF). Other than operational differences, the NSRDB provides data up

until 2015, whereas the SARAH database provides data until 2016. Both databases will be used.

After selecting the database, we have to select the date range of the irradiation data and deﬁne the data we want to receive.

The date range will be the latest year where data is available (2015 for NSRDB, 2016 for SARAH), and we will check ”Global

horizontal irradiation” and ”Global irradiation at angle”, with a value of 19° for the latter setting.

Fig. 3. Conﬁguration settings of the PVGIS application

For the CRESESB website, a similar approach is needed. The coordinates of the location are inserted into the website and it

gives us three databases to choose from. The selected database will be ”Belo Horizonte”, and the options ”Plano Horizontal”

(horizontal plane) and ” ˆ

Angulo igual a latitude” (plane with angle corresponding to latitude) will be checked.

Fig. 4. Data output from CRESESB

With the data in our hands, it is now possible to create the graphs for the mathematical model.

(a) Daily average (b) Monthly sum of daily average

Fig. 5. Solar irradiation obtained from the databases

(a) Daily production average (b) Monthly production

Fig. 6. Comparison of estimated and real energy production

Fig. 7. Comparison of STC and NOCT efﬁciencies for estimating monthly production

It is important to note that the different databases have different solar irradiation values for every month. By comparing

with data obtained from the inverters of the PV system during the months of September to November of 2021, we can see

that only in September the production output surpassed the estimates, whereas it was lower in the other months. Finally, the

difference between the NOCT and STC efﬁciencies when applied in the mathematical model can be seen in Fig. 7.

The resolution for the year 2021 is not ideal, so we need to adjust some parameters of the model. The PV system had 20

360W and three 400W solar panels installed in September 2021; the 400W solar panels were installed during the month of

September 2021. Therefore, to facilitate the analysis of the accuracy of the mathematical model, the amount of solar panels

will change from 23 to 20, and the average NOCT efﬁciency will be set to that of the 360W solar panels, 13.91%, as deﬁned

in (1).

(a) Daily production average (b) Monthly production

Fig. 8. Comparison of estimated and real energy production from August 2020 to August 2021

TABLE II

VARIA NC E PE RC EN TAGE O F RE AL WO RL D SY ST EM DATA CO MPA RE D TO MATH EM ATIC AL M OD EL S

Month PVGIS-SARAH PVGIS-NSRDB CRESESB

Aug ’20 3.4940 -4.1263 7.3079

Sep ’20 18.1239 12.05983 17.0403

Oct ’20 -7.4364 -11.9295 -3.2554

Nov ’20 5.1980 -1.6368 3.3809

Dec ’20 -0.3427 -4.8985 1.4228

Jan ’21 13.5045 8.4057 18.5277

Feb ’21 -25.6387 -24.4442 -24.4927

Mar ’21 7.7725 11.6612 15.8998

Apr ’21 10.8877 12.6584 14.0330

May ’21 2.8660 -2.4273 10.8621

Jun ’21 3.2593 -2.9969 4.8557

Jul ’21 7.9951 3.7680 17.0490

Aug ’21 5.7278 -2.0570 9.6240

We can calculate the average yearly solar irradiation by performing a sum on the monthly average values. It is also possible

to obtain the average ”real” irradiation by manipulating (3), as seen in (7).

H=Emonth

A·n·ηavg

(7)

TABLE III

AVERA GE YE AR LY SO LA R IR RAD IAT IO N (IN KWH/M² )

PVGIS-SARAH PVGIS-NSRDB CRESESB Real system

2181.3300 2269.0200 2108.7000 2060.0210

The variance of the ”real” irradiation compared to the data obtained from the PVGIS and CRESESB databases can be

calculated with the average yearly solar irradiation, as seen in Tab. IV.

TABLE IV

VARIA NC E PE RC EN TAGE O F RE AL S OLA R IR RA DI ATIO N CO MPA RE D TO VALU E OB TAI NED F RO M DATABAS ES

PVGIS-SARAH PVGIS-NSRDB CRESESB

-5.5612 -9.2110 -2.3085

C. Model Review

In the ﬁrst revision of this paper, the databases utilized were a bit outdated, as the solar irradiation data they provided

were from 2015 and 2016. After some research, it was possible to obtain more recent data, from the Instituto Nacional de

Meteorologia (INMET) database [7]. The data utilized from INMET dates from 2020, thus it was possible to analyze the

production data in 2020 from the PV system. The previous modiﬁcations made to the base system are still in place.

However, the data provided by INMET does not consider the inclination angle of 19° of the solar panels. It is safe to assume

that the solar irradiation data considers a horizontal plane. Therefore, to use apply these new values in the model, a conversion

is needed. Luckily, Honsberg and Bowden [8] provide some equations to ﬁnd the solar irradiation in an inclined plane, as seen

in (8), (9) and (10).

α= 90 −ϕ+δ(8)

where: α=the elevation angle of the Sun,

ϕ=the latitude,

δ=the declination angle.

δ= 23.45◦sin 360

365(284 + d)(9)

where: d =the day of the year.

Hmodule =Hhorizontal ·sin (α+β)

sin α(10)

It is needed, however, to modify (8) to consider the Southern Hemisphere, as seen in 11.

α= 90 + ϕ−δ(11)

where: ϕ=the latitude, relative to the North (negative values for Southern Hemisphere).

With this modiﬁcation in place, we need to calculate the average irradiation for every month by using a simple sum. After

calculating the average monthly irradiation, it is now possible to apply these values to the model, as seen in Fig. 9.

(a) Daily production average (b) Monthly production average

Fig. 9. Comparison of estimated and real energy production during the year of 2020.

III. CONCLUSION

The mathematical model is not always accurate, as shown by the data previously presented. In the best case scenario for

the ﬁrst revision of this paper, the model produced a result 2.31% higher than the real data, and in the worst case it is off by

9.21%. Depending on the database chosen, the model can be considered accurate. There is also the weather factor, as weather

is, most of the times, unpredictable, as seen in the off curve point in February 2021 in Fig. 8 (a), where it was very cloudy

and rainy month that could not be predicted by the aforementioned databases.

In relation to the most recent data obtained from INMET, the resolution is much better than the previous databases used in

the model. There are also off the curve points, such as the production in February 2020, which was a very rainy month as

well, as seen in Fig. 9 (a). The differences can also happen due to the location from which the solar irradiation was measured,

as the solar irradiation may be higher or lower depending on where it is measured, mainly due to cloud coverage, which the

data provided does not consider.

In the future, the model could be reworked to include the efﬁciency lost due to the rise in temperature of the solar cells.

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