Contents:

• Photovoltaic installations
• Calculation solar radiation
• Shadow analysis

Photovoltaic installations

The calculation of the intensity of solar radiation in the plane of the photovoltaic panels is analogous to that used in solar thermal collectors.

The F method of assessing the energy obtained from the installation is applied, irrespective of installation type.

It is assumed that there are three main types of electrical loads in the installation:

• Directly consumed load without a battery, such as fan pumps, etc., operating only when there is sun;
• Directly consumed load and battery, such as a monitoring system, cathodic signaling, etc .;
• The whole load first passes through a battery, then through the inverter to the consumer, for example a refrigerator consumes energy day and night.

In addition to the listed basic types of electrical loads, it is also necessary to know the peak load, regardless of the type of consumer (DC and AC).

Before starting with the sizing of the photovoltaic installation, it is necessary to determine:

• What loads of cargo we have;
• Time of use of the loads within the day;
• Number of days of the week during which the load is used.

For example, the AC load per day is calculated using the following equations, depending on their type:

Directly consumed AC load:

\[AC_{demand,PV}=P_{AC,PV}\cdot h_{AC,PV}, \left[kWh/day\right]\]	(1)

Directly consumed load and rechargeable battery in the installation:

\[AC_{demand,const}=P_{AC,const}\cdot h_{AC,const} \left[kWh/day\right]\]	(2)

The installation stores the energy entirely in accumulators from where it is transformed and consumed:

\[AC_{demand,batt}=P_{AC,batt}\cdot h_{AC,batt}, \left[kWh/day\right]\]	(3)

In the case that we have a combined type of installation / combination of the above types /, the load is calculated according to the following relationship:

\[AC_{demand}=AC_{demand,PV}+AC_{demand,const}+AC_{demand,batt}, \left[kWh/day\right]\]	(4)

In dependencies from (1) to (4) inclusive, AC means AC load. The PV, const, batt indices means directly consumed load, directly consumed load and battery, all the load is consumed by the battery.

`P_{AC,PV}`, `P_{AC,const}`, `P_{AC,bat t}` are the different types of AC loads respectively, [kW];

`h_{AC,PV}`, `h_{AC,const}`, `h_{AC,bat t}` are the times of use of individual types of loads within 24 hours.

The constant current load in the installation is analogous and is calculated according to the following dependencies:

Directly consumed DC load:

\[DC_{demand,PV}=P_{DC,PV}\cdot h_{DC,PV}, \left[kWh/day\right]\]	(5)

Directly consumed load and rechargeable battery in the installation:

\[DC_{demand,const}=P_{DC,const}\cdot h_{DC,const} \left[kWh/day\right]\]	(6)

The installation stores energy entirely in accumulators:

\[DC_{demand,batt}=P_{DC,batt}\cdot h_{DC,batt}, \left[kWh/day\right]\]	(7)

In the case that we have a combined type of installation / combination of the above types /, the load is calculated according to the following relationship:

\[DC_{demand}=DC_{demand,PV}+DC_{demand,const}+DC_{demand,batt}, \left[kWh/day\right]\]	(8)

DC means DC load.

`P_{DC,PV},P_{DC,const} , P_{DC,bat t}` are the different types of DC loads respectively, [kW];

`h_{DC,PV},h_{DC,const} , h_{DC,bat t}` are the times of use of individual types of loads within 24 hours.

The voltage generated by the photovoltaic panels is constant, so it is necessary to calculate the equivalent DC load regardless of the type of installation.

The calculation of the equivalent DC load takes place in the following dependencies:

\[DC_{eq,PV}=DC_{demand,PV}+AC_{demand,PV}/\eta_{inv}/0,01, [kWh/day]\]	(9)

\[DC_{eq,const}=DC_{demand,const}+AC_{demand,const}/\eta_{inv}/0,01, [kWh/day]\]	(10)

\[DC_{eq,batt}=DC_{demand,batt}+AC_{demand,batt}/\eta_{inv}/0,01, [kWh/day]\]	(11)

Where `\eta_{i n v}` is the efficiency of the inverter in the installation.

For the correct sizing of the installation it is necessary to know also the peak AC and equivalent DC load, which are calculated by the following dependencies:

Peak AC load in the installation:

\[Load_{AC,p}=P_{AC,PV}+P_{AC,const}+P_{AC,batt}, \left[kW\right]\]	(12)

Peak DC load in the installation:

\[Load_{DC,p}=P_{DC,PV}+P_{DC,const}+P_{DC,batt}, \left[kW\right]\]	(13)

Equivalent DC peak load in the installation:

\[Load_{DC,eq,p}=Load_{DC,p}+Load_{AC,p}/\eta_{inv}/0,01, \left[kW\right]\]	(14)

The energy produced by the PV modules for one day is calculated according to the following dependence:

\[PV_{output,day,i}=\frac{\eta_{a,i}}{100}\cdot S_{PV,array}\cdot \overline{H_{T,i}}\cdot |0,277777778 \frac{kWh}{MJ}|, \left[MJ/day\right]\]	(15)

Where:

`i` - the corresponding month of the year /i=1..12/;

`\eta_{a,i}` - the efficiency of modules in the installation, taking into account the influence of their temperature and the angle below which solar radiation falls.

`S_{PV,array}` - the area of the PV panels in the installation,[m ²];

`\overline{H_{T,i}}` - solar radiation in the module plane , [kWh/m²ден]

The resulting energy within a month is calculated by the following dependence:

\[PV_{output,month,i}=PV_{output,day,i}\cdot \frac{\eta_{inv,cor}}{100}\cdot Fm_{use,i}\cdot Days_{i}, \left[MJ/day\right]\]	(16)

Where:

`n_{i nv,c o r}` - the efficiency of the inverter in the installation / if we only have a DC load, then the efficiency of the inverter is not involved in the expression /, [%];

`Fm_{use,i}` - the installation's utilization rate during that month;

`Days_{i}` - the number of days in that month.

The energy supplied by the modules in the installation to the direct use load is calculated according to the following dependency:

\[E_{continuous,i}=min\left( PV_{output,day,i} \cdot \left(1-\overline{\phi_{i}}\right),DC_{eq,const}\right), [kWh/day]\]	(17)

The energy supplied by the modules to the dimensional load is calculated by the following dependence:

\[E_{matched,i}=min\left(PV_{output,day,i}-E_{continuous,i}, DC_{eq,PV}\right), \left[kWh/day\right]\]	(18)

The total energy delivered to the load is as follows:

\[E_{delivered,i}=E_{continuous,i}+E_{matched,i}, \left[kWh/day\right]\]	(19)

The power delivered to the battery is:

\[E_{battery,i}=PV_{output,day,i}+E_{delivered,i}, \left[kWh/day\right]\]	(20)

If the installation is connected to the grid, then the amount of energy produced by the modules per month is:

\[PV_{OnGrid,collected,i}=PV_{output,day,i} \cdot Fm_{use,i} \cdot Days_{i} \cdot \frac{\eta_{inv,cor}}{100} , \left[kWh/m\right]\]	(21)

Where `\eta_{i n v,c o r}` is the efficiency of the inverter, taking into account its efficiency, depending on the degree of its load.

The amount of energy exported to the grid is calculated according to the following dependency:

\[PV_{OnGrid,delivered,i}=PV_{OnGrid,collected,i}\cdot\frac{n_{abs}}{100}, \left[kWh/m\right]\]	(22)

Where `\eta_{abs}` is the efficiency of receiving power from the network. When connecting the plant to the central grid, this efficiency is approximately 95 to 96%. If the plant is powering an isolated grid, this performance can reach 100%.

The degree of load of the installation when connected to the grid is calculated by the following dependency:

\[PV_{OnGrid,capacity,i}=\left( \frac{PV_{OnGrid,delivered,i}}{P_{nom,PV} \cdot Fm_{use,i} \cdot Days_{i} \cdot \frac{24}{1000}} \right) \cdot 100, [\%]\]	(23)

The efficiency of the installation's operation when connected to the electricity grid by months is calculated according to the following dependency:

\[PV_{OnGrid,eff,i}=\left( \frac{\frac{PV_{OnGrid,delivered,i}}{S_{PV,array}}}{\overline{H_{T,i}}\cdot 0,277777778 \frac{kWh}{MJ}\cdot Days_{i}}\right)\cdot 100, [\%]\]	(24)

In the case of an isolated installation (not connected to the grid), the energy delivered by the load installation is the following:

\[PV_{OffGrid,delivered,i}=\left( E_{continuous,i}+E_{matched,i}+Load_{met,batt,i}\right)\cdot Fm_{use,i}\cdot Days_{i}, \left[ \frac{kWh}{m}\right]\]	(25)

Where `Load_{met,bat t,i}` is the charge covered by the batteries in the installation per day.

The rate of installation load by months is accordingly:

\[PV_{OffGrid,capacity,i}=\left( \frac{PV_{OffGrid,delivered,i}}{P_{nom,PV} \cdot Fm_{use,i}\cdot Days_{i} \cdot \frac{24}{1000}}\right) \cdot 100, [\%]\]	(26)

The efficiency of the operation of an isolated installation by months is respectively:

\[PV_{OffGrid,eff,i}=\left( \frac{\frac{PV_{OffGrid,delivered,i}}{S_{PV,array}}}{\overline{H_{T,i}} \cdot 0,277777778 \frac{kWh}{MJ}\cdot Days_{i}}\right)\cdot 100, [\%]\]	(27)

Calculation of solar radiation

For this purpose, it is necessary to use climatic data for the daily amount of solar radiation on a horizontal surface for the respective location. This data can be obtained from the embedded database, or the user inserts them on their own. Average monthly solar radiation on an inclined surface is determined by the following dependency:

\[\overline{H}_{T}=\overline{R}\cdot\overline{H}\left[kWh/m^{2} ден\right]\]	(28)

Where:

`\overline{R}` - the correction factor, which is a ratio of full solar radiation of a randomly oriented inclined surface to full solar radiation on a horizontal surface;

`\overline{H}` - average daily solar radiation on a horizontal surface, [kWh/m² ден];

The correction factor R can be calculated using the following dependency:

\[\overline{R}=\left(1-\frac{\overline{H}_{d}}{\overline{H}}\right)\cdot\overline{R}_{b}+\frac{\overline{H}_{d}}{\overline{H}}\cdot\left(\frac{1+\cos\beta}{2}\right)+\rho\cdot\left(\frac{1-\cos\beta}{2}\right)\]	(29)

Where:

`\overline{R_{b}}` - the ratio of the average monthly direct sun radiation to the inclined and the horizontal surface;

`\overline{H_{d}}` - average daily diffuse radiation over a horizontal surface, [kWh/m ден];

`\beta` - angle of inclination of the surface under consideration / angle of inclination of the solar collectors to the horizon/, [^o] ;

`\rho` - the reflection coefficient of the solar radiation from the environment (in particular from the earth's surface). In the absence of data, an average annual value of 0.4 may be taken, or individual values for each month, reflecting the presence of snow cover, vegetation type, etc. may be used.;

The ratio `\frac{\overline{H}_{d}}{\overline{H}}` is determined by the following dependency:

\[\frac{\overline{H}_{d}}{\overline{H}}=1,39-4.03\cdot\overline{K}_{T}+5,53\cdot\overline{K}_T^2-3.11\cdot\overline{K}_T^3\]	(30)

Where `\overline{K_{T}}` is the cloud factor.

The cloud factor can be calculated as the ratio of the average daily solar radiation on a horizontal surface to the average daily solar radiation behind the atmosphere.

\[{H}_{d}=\frac{86400\cdot I_{SC}}{\pi}\cdot\left(1 + 0,033\cdot\cos\left(2\pi\frac{n}{365}\right)\right)\cdot\left(\cos\phi\cdot\cos\delta\cdot\sin\omega_{s} + \omega_{s}\cdot\sin\phi\cdot\sin\delta\right)\]	(31)

Where:

`\delta` - the declination of the sun / determined for the respective day of the month_i/, [^o];

`\omega_s^\prime` - hour sunset angle on a horizontal surface / determined for the respective day of the month day_i/, [^o];

`\omega_s` - hour sunset angle on the inclined surface / set for the respective day of the month day_i/, [^o];

`\phi` - latitude, [^o].

`I_{sc}` = 1367 [W/m²] is the solar constant;

`n` – the number of the day from the beginning of the year / 01.01 is day number 1 of the year, 31.12. is the day number 365 from the beginning of the year /; The variable `day_{i}` can be assumed to be a constant (21st day of month). In the specific case, to improve the accuracy of the model, we work with different values for each month presented in the following array:

\[day_{i}=[17,16,16,15,15,11,17,16,15,15,14,10]\]	(32)

The hour angle of sunset on a horizontal surface is determined by the following dependency:

\[\omega_{s}=\arccos\left(-tg\phi\cdot tg\delta\right)\]	(33)

The hour angle of sunset on the inclined surface is determined by the following dependency:

\[\omega_{s}^\prime=min\left[\omega_{s};\arccos\left(-tg\left(\phi-\beta\right)\cdot tg\delta\right)\right]\]	(34)

The declination of the sun for the respective month is calculated by the following dependency:

\[\delta=23,45\cdot\frac{\pi}{180}\cdot sin\left[2\pi\cdot\left(\frac{284 + n}{36,25}\right)\right] \]	(35)

The ratio of the average monthly solar radiation on the slope and the horizontal surface is calculated by the following dependencъ:

\[\overline{R}_{b}=\frac{\cos\left(\phi-\beta\right)\cdot\cos\delta\cdot\sin\omega_{s}+ \pi/180\cdot\omega\prime_{s}\cdot\sin\left(\phi-\beta\right)\cdot\sin\delta}{\cos\phi\cdot\cos\delta\cdot\sin\omega_{s}+ \pi/180\cdot\omega\prime_{s}\cdot\sin\phi\cdot\sin\delta}\]	(36)

Shadow analysis

If you are near to hills or mountains there may be times when the sun is behind the hills and the solar radiation will be reduced to that coming from the sky or clouds. Our software uses information about the elevation of the terrain with a high resolution(about 10m). This means that for every 10m we have a value for the ground elevation. From these data we have calculated the height of the horizon around requested geographical location from our climate database. These data are then used to calculate the times when the sun is shadowed by hills or mountains. When this happens the solar radiation is then calculated using only the diffuse part of the radiation. The calculations done within system can all make use of this information.

The height of the sun by hours for each month is calculated using the following dependency:

\[\alpha_{s,i,j}=\cos_{ZenAng}(Lat, nDay_{i} , HrAng_{j}) - \arccos(\cos_{ZenAng}(Lat, nDay_{i} , HrAng_{j}))\]	(37)

Where:

`i` - months;

`j` - hours;

`cos_{ZenAng}` - cosine of the zenith angle;

`Lat` - latitude;

`nDay_{i}` - number of days in the month;

`HrAng_{j}` - hour angle of the sun for every hour of the day;

Dependency (37) is calculated 288 times altogether - 12 times per each month i=1..12, 24 times for each hour of the day

Solar radiation by hours, taking into account the shading, is calculated by the following dependency:

\[I_{t,s,i,j}=\alpha_{s,i,j}\cdot Hh_{j}+I_{g,i,j}+I_{b,i,j}+I_{d,i,j}\]	(38)

Където:

`\alpha_{s,i,j}` - the height of the sun by hours for each month;

`Hh` - height of the horizon in degrees from an external database;

`I_{d,i,j}` - Diffuse composition of the solar radiation falling on the surface for the corresponding time;

`I_{g,i,j}` - Solar radiation reflected from the surface of the Earth falling on the surface for the corresponding time;

`I_{b,i,j}` - Direct component of the solar radiation falling on the surface for the respective hour;

Dependency (38) is calculated 288 times altogether - 12 times for each month i=1..12, 24 times for each hour of the day.