Array Thermal losses

Thermal Model

The objective is to evaluate the array (or cell) temperature during the simulation. The cell temperature is a basic input parameter of the one-diode model. In other words, during the simulation:

We start from the ambient temperature TAmb, defined in the climatic data.
We evaluate the array (or cells) temperature at sun (named TArray), using the thermal model described below,
This cell's temperature is a basic input value for the determination of the PV array electrical behaviour, using the one-diode model.

NB: In PVsyst we consider the Array temperature as equivalent (equal) to the Cell temperature. We may indeed neglect the temperature drop of heat fluxes in the glass and backsheet, as the thermal resistance of the glass is much lower than the thermal resistance of the interface glass-ambient (convection).

Thermal Model: The cell temperature is evaluated by an energy balance, accounting for all incoming and outcoming energy fluxes in the array.

The cells absorb the energy provided by the sun (irradiance), minus the reflected part on the glass. At the thermal equilibrium, this energy flux should be compensated by the array cooling thermal losses, which are mainly convective, and the extracted electrical energy.

\(Ginc · Alpha · (1 - Effic) = U · (Tcell - Tamb)\)

where:

G_inc is the incoming irradiance on the module or PV array. During the simulation, this value is the effective irradiance on the PVarray, i.e. GlobEff.
Alpha is the absorption coefficient of solar irradiance, i.e. (1 - reflection). The usual value chosen for the Absorption coefficient Alpha is 0.9. It may possibly be modified in the PV module definition dialog, but this is not recommended. The measured U-value is closely related to the choice of Alpha.
Effic is the PV efficiency (related to the module area), i.e. the electrical energy removed from the module. When possible, the PV efficiency is calculated according to the operating conditions of the module. Otherwise it is taken as 19%. See below.
Tamb is the ambient temperature, according to the weather data,
U is the "heat loss factor", expressed in [W/m²·k]. This is a heat transfer coefficient, determining the heat flux as proportional to the temperature difference between two media. This U-value is quite equivalent to the Heat transfer factor [W/m²·k], used in building physics for the characterization of walls or windows.

The left side of this equation describes the different energy fluxes specific to a PV array, the right side defines the necessary heat transfer for ensuring the thermal equilibrium.

From this equation, we can easily extract the cell temperature:

\(Tcell = Tamb + 1 / U · ( Alpha · Ginc · (1 - Effic) )\)

U-value

The heat loss factor is the main input parameter used during the simulation for the evaluation of the PV array behavior due to the temperature with respect to running at 25°C (may be a loss, when TArray greater than 25°C, or a gain below).

However this parameter may be varying with the wind speed: as the heat loss is mainly convective, it is sensitive to the air circulation on the array. Therefore this parameter can be split into a constant component Uc and a factor proportional to the wind velocity Uv :

\(U = Uc + Uv · WindVel\)

(Uc in [W/m²·k], Uv in [W/m²·k / m/s], WindVel = wind velocity in [m/s]).

Using the wind dependency Uv is not easy. This requires wind velocity in the climatic data, and a good knowing of the Uv parameter. PVsyst doesn't provide a Uv value as default in the present time, as we don't have sufficient measured information.

NB: According to the meteorological standards, the wind velocity should be measured on a mast at a 10 meter height, in a free environment. This is rarely the case when measuring a wind velocity for the monitoring of a PV system. The real value at the collectors level may be lower by a factor of -35 to -50%. Therefore the wind parameter Uv-value should be adapted to the way of recording the wind velocity.

U-value determination

The U-factor depends on the mounting mode of the modules (sheds, roofing, façade, trackers, floating system, etc...). When Uv is not defined explicitly, the U-value includes an "average wind velocity" contribution. In the PVsyst default values, the wind contribution is supposed to correspond to an average wind speed of about 3 m/s.

For free circulation all around the tables (i.e sheds arrangement), this coefficient refers to both faces, i.e. twice the area of the module. If the back of the modules is more or less thermally insulated, this should be lowered, theoretically up to half the value when the back side is fully insulated (i.e.doesn't participate anymore to thermal convection and radiation transfer).

The determination of the parameters Uc and Uv is indeed a difficult question. We have some measured data for usual free mounted arrays, but there is a severe lack of information when the modules are semi-integrated.

We don't have any simple way for the evaluation of the U-values in a general case. The the only reliable way of determining this parameter is to measure it on-site.

Default and proposed values

In the absence of reliable measured data, PVsyst proposes default values without wind dependency (i.e. assuming an average wind velocity):

For free-standing (open-rack) systems, i.e. with air circulation all around the collectors), according to our measurements on several installations:
- Uc = 29 W/m²·k, Uv = 0 W/m²·k / m/s
Therefore for fully insulated backside (no heat exchange at the backside, only one side contribution to the convecting heat exchange), the U value should be divided by 2:
- Uc = 15 W/m²·k, Uv = 0 W/m²·k / m/s
For intermediary cases (semi-integration, air duct below the collectors), the value should be taken between these 2 limits, but preferably lower than 22 W/m²·k as the air heat removing is often not very efficient. The default value proposed by PVsyst for any new project is
- Uc = 20 W/m²·k, Uv = 0 W/m²·k / m/s
- We have chosen this value as general default, because we consider that it is more representative of usual rooftop systems, managed by "less professional" people who will not necessarily modify the PVsyst default. For big systems, we suppose that trained engineers will indeed adjust this parameter (for example at 29 W/m² for row-like big power plants).
For domes, a manufacturer has measured the U-value on several installations (height about 40 to 70 cm above the ground):
- Uc = 27 W/m²·k, Uv = 0 W/m²·k / m/s

Now if reliable hourly wind velocity data are present in the data, we can't propose a "universal" Uv value. This would be dependent on many parameters, and especially the way of measuring this wind velocity. PVUSA proposes the following thermal correlation, widely used for the free-standing (open-rack) situations when wind speed data are available from official climatic data:

Uc = 25 W/m²·k, Uv = 1.2 W/m²·k / m/s

This corresponds to our default 29 W/m2, when the average wind velocity is 3.3 m/sec (rather usual in continental - not coast situations).

Thermal behaviour of an array on a tilted roof

When installing a PV array on a tilted roof, with an air-duct between the roof and the modules, the air circulation is driven by the temperature difference between the incoming air (ambient) and the outcoming air; i.e. a rather weak motor (low air speed).

As the thermal capacity of the air is low, the air will be heaten up when passing under the very first modules, so that there will not be a significant heat exchange in the upper modules, they will be in the "fully insulated" situation. In this situation, the array temperature is very difficult to evaluate, and may be strongly inhomogeneous. Developing an accurate model would require a detailed description of the air duct thickness, its length, etc. This is out of the scope of PVsyst.

PVsyst doesn't treat this inhomogeneity in the present time, it considers an average temperature.

Thermal model calculation

The expression : \(Tcell = Tamb + 1 / U · ( Alpha · Ginc · (1 - Effic) )\) includes an a priori unknown quantity, the efficiency of the PV array. When evaluating the cell temperature:

If we are not within a simulation, PVsyst supposes an array efficiency of 20%, reasonable with modern PV modules.
Within the simulation, we avail of the ambient temperature and the irradiance (GlobInc). We may calculate the TArray with an initial efficiency of 20%, calculate the array electrical behaviour using the one-diode model, and reintroduce the calculated efficiency in this expression (one iteration is largely sufficient).

NOCT Values

Some practicians - and most of PV module's catalogues - usually specify the NOCT coefficient ("Nominal Operating Cell Temperature"), which you are advised to completely forget... Please have a look at this topic !