Photovoltaic Model
This page documents the functionality of the PV model available in SIMONA.
The initial parts of the model are presented in the paper Agent based approach to model photovoltaic feed-in in distribution network planning. Since then several adaptions has been made that are documented as follows.
The PV Model is part of the SIMONA Simulation framework and represented by an agent.
Parameters
Attributes, Units and Remarks
Please refer to PowerSystemDataModel - PV Model for Attributes and Units used in this Model.
Implemented Behaviour
Output visualization
Calculations
The energy produced by a photovoltaic (pv) unit in a specific time step is based on the diffuse and direct radiation provided by the used weather data. The following steps are done to calculate (= estimate) the power feed by the pv.
To calculate the overall feed in of the pv unit, the sum of the direct radiation, diffuse radiation and reflected radiation is needed. In the following, the formulas to calculate each of these radiations are presented and explained. The sections end with the formula to calculate the corresponding power feed in.
Caution: all angles are given in radian!
The azimuth angle \(\alpha_{E}\) starts at negative values in the East and moves over 0° (South) towards positive values in the West. Source
Declination Angle
The declination angle \(\delta\) (in radian!) is the day angle that represents the position of the earth in relation to the sun. To calculate this angle, we need to calculate the day angle \(J\). The day angle in radian is represented by:
with
n = number of the day in the year (e.g. 1 January = 1, 20 February = 51)
Based on \(J\) the declination angle \(\delta\) (in radian!) can be calculated as follows:
References:
Mousavi Maleki, Hizam, and Gomes [Maleki.2017]
Spencer [Spencer.1971]
Hour Angle
The hour angle is a conceptual description of the rotation of the earth around its polar axis. It starts with a negative value in the morning, arrives at 0° at noon (solar time) and ends with a positive value in the evening. The hour angle (in radian!) is calculated as follows
Since outside German literature the hour angle is defined as negative in the morning, we use the following adaption:
with
ST = local solar time (standard time, in hours)
with
LMT = local mean time (in minutes)
ET = equation of time (in minutes)
with
CET = central eastern time
λ = longitude of the location of the PV panel
with
J = day angle (in radian!)
Note: The used formulas are based on “DIN 5034-2: Tageslicht in Innenräumen, Grundlagen.” and therefore valid especially for Germany and Europe. For international calculations a more general formulation that can be found in Maleki, S.A., Hizam, H., & Gomes, C. (2017). Estimation of Hourly, Daily and Monthly Global Solar Radiation on Inclined Surfaces: Models Re-Visited. might be used.
References:
Watter [Watter.2013]
Mousavi Maleki, Hizam, and Gomes [Maleki.2017]
Wang [Wang.2019]
Sunrise Angle
The hour angles at sunrise and sunset are very useful quantities to know. These two values have the same absolute value, however the sunrise angle (\(\omega_{SR}\)) is positive and the sunset angle (\(\omega_{S}\)) is negative. Both can be calculated from:
with
\(\delta\) = the declination angle
References:
Mousavi Maleki, Hizam, and Gomes [Maleki.2017]
Itacanet [Itaca_Sun]
Solar Altitude Angle
Represents the angle between the horizontal and the line to the sun, that is, the complement of the zenith angle.
with
\(\delta\) = the declination angle
\(\phi\) = observer’s latitude
\(\omega\)= hour angle
References:
Mousavi Maleki, Hizam, and Gomes [Maleki.2017] p. 5
Itacanet [Itaca_Sun]
Zenith Angle
Represents the angle between the vertical and the line to the sun, that is, the angle of incidence of beam radiation on a horizontal surface.
with
\(\alpha_s\) = solar altitude angle
References: See Solar Altitude Angle
Incidence Angle
The angle of incidence is the angle between the Sun’s rays and the PV panel. It can be calculated as follows:
with
\(\alpha_e\) = sun azimuth
\(\alpha_s\) = solar altitude angle
\(\gamma_e\) = slope angle of the surface
\(\delta\) = the declination angle
\(\phi\) = observer’s latitude
\(\omega\) = hour angle
References:
Quaschning [Quaschning.2013]
Mousavi Maleki, Hizam, and Gomes [Maleki.2017] p. 18
Air Mass
Calculating the air mass ratio by dividing the radius of the earth with approx. effective height of the atmosphere (each in kilometer)
References:
Schoenberg [Schoenberg.1929]
Wikipedia [WikiAirMass]
Extraterrestrial Radiation
The extraterrestrial radiation \(I_0\) is calculated by multiplying the eccentricity correction factor
with the solar constant
with
J = day angle
References:
Zheng [Zheng.2017] p. 53, formula 2.3b
Iqbal [Iqbal.1983]
Beam Radiation on Sloped Surface
For our use case, \(\omega_{2}\) is normally set to the hour angle one hour after \(\omega_{1}\). Within one hour distance to sunrise/sunset, we adjust \(\omega_{1}\) and \(\omega_{2}\) accordingly:
with
\(\omega\) = hour angle
\(\omega_{SS}\) = hour angle \(\omega\) at sunset
\(\omega_{SR}\) = hour angle \(\omega\) at sunrise
\(\Delta\omega\) = \(15^\circ \cdot (\frac {\pi}{180^\circ})\) (one hour worth of \(\omega\))
From here on, formulas from given reference below are used:
Please note: \(\frac{1}{180}\pi\) is omitted from these formulas, as we are already working with data in radians.
with
\(\delta\) = the declination angle
\(\phi\) = observer’s latitude
\(\gamma\) = slope angle of the surface
\(\omega_1\) = hour angle \(\omega\)
\(\omega_2\) = hour angle \(\omega\) + 1 hour
\(\alpha_e\) = sun azimuth
\(E_{dir,H}\) = beam radiation (horizontal surface)
Reference:
Duffie [Duffie.2013] p. 88
Diffuse Radiation on Sloped Surface
The diffuse radiation is computed using the Perez model, which divides the radiation in three parts. First, there is an intensified radiation from the direct vicinity of the sun. Furthermore, there is Rayleigh scattering, backscatter (which lead to increased in intensity on the horizon) and isotropic radiation considered.
A cloud index is defined by
Calculating a brightness index
Perez Fij coefficients (Myers 2017):
\(\epsilon\) is sorted into one of eight bins according to its value:
\(\epsilon\) low |
\(\epsilon\) high |
Bin number\(x\) |
---|---|---|
1 |
1.065 |
1 |
1.065 |
1.230 |
2 |
1.230 |
1.500 |
3 |
1.500 |
1.950 |
4 |
1.950 |
2.800 |
5 |
2.800 |
4.500 |
6 |
4.500 |
6.200 |
7 |
6.200 |
\(\infty\) |
8 |
In order to calculate indexes representing the horizon brightness and the brightness in the vicinity of the sun, the following factors are computed.
Horizon brightness index:
Sun ambient brightness index:
Using the factors
and
the diffuse radiation can be calculated:
with
\(\theta_{z}\) = zenith angle
\(\theta_{g}\) = angle of incidence
\(\alpha_{s}\) = solar altitude angle
\(\alpha_{z}\) = sun azimuth
\(\gamma_{e}\) = slope angle of the surface
\(I_{0}\) = Extraterrestrial Radiation
\(m\) = air mass
\(E_{dir,H}\) = direct radiation (horizontal surface)
\(E_{dif,H}\) = diffuse radiation (horizontal surface)
References:
Perez, Seals, Ineichen, Stewart, and Menicucci [Perez.1987]
Perez, Ineichen, Seals, Michalsky, and Stewart [Perez.1990]
Myers [Myers.2017] p. 96f
Reflected Radiation on Sloped Surface
with
\(E_{Ges,H}\) = total horizontal radiation (\(E_{dir,H} + E_{dif,H})\)
\(\gamma_e\) = slope angle of the surface
\(\rho\) = albedo
Reference:
Mousavi Maleki, Hizam, and Gomes [Maleki.2017] p. 19
Output
Received energy is calculated as the sum of all three types of irradiation.
with
\(E_{dir,S}\) = Beam radiation
\(E_{dif,S}\) = Diffuse radiation
\(E_{ref,S}\) = Reflected radiation
A generator correction factor (depending on month surface slope \(\gamma_{e}\)) and a temperature correction factor (depending on month) multiplied on top.
It is checked whether proposed output exceeds maximum (\(p_{max}\)), in which case a warning is logged. If output falls below activation threshold, it is set to 0.