How HOMER Calculates the PV Cell Temperature

The photovoltaic (PV) cell temperature is the temperature of the surface of the PV array. During the night, it is the same as the ambient temperature, but in full sun, the cell temperature can exceed the ambient temperature by 30°C or more.

If, in the PV array inputs, you choose to consider the effect of temperature on the PV array, HOMER calculates the cell temperature in each time step and uses it in calculating the power output of the PV array. The following describes how HOMER calculates the cell temperature from the ambient temperature and the radiation striking the array.

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Energy Balance for the PV Array

We start by defining an energy balance for the PV array, using the following equation from Duffie and Beckman (1991):

$$\tau \alpha G_T = \eta_c G_T + U_L (T_c - T_a)$$

where:

• τ = solar transmittance of any cover over the PV array [%]
• α = solar absorptance of the PV array [%]
• G_T = solar radiation striking the PV array [kW/m²]
• η_c = electrical conversion efficiency of the PV array [%]
• U_L = coefficient of heat transfer to the surroundings [kW/m²·°C]
• T_c = PV cell temperature [°C]
• T_a = ambient temperature [°C]

The equation above states that a balance exists between, on one hand, the solar energy absorbed by the PV array, and on the other hand, the electrical output plus the heat transfer to the surroundings.
We can solve that equation for cell temperature to yield:

$$T_c = T_a + \frac{G_T}{U_L} \left( \tau \alpha - \eta_c \right)$$

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Nominal Operating Cell Temperature (NOCT)

It is difficult to measure the value of $\frac{\tau \alpha}{U_L}$ directly, so instead manufacturers report the Nominal Operating Cell Temperature (NOCT), which is defined as the cell temperature that results at an incident radiation of 0.8 kW/m², an ambient temperature of 20°C, and no load operation (meaning $\eta_c = 0$).

Substituting these values and solving for $\frac{\tau \alpha}{U_L}$:

$$\frac{\tau \alpha}{U_L} = \frac{T_{c,NOCT} - T_{a,NOCT}}{G_{T,NOCT}}$$

where:

• T_c,NOCT = nominal operating cell temperature [°C]
• T_a,NOCT = ambient temperature at which the NOCT is defined [20°C]
• G_T,NOCT = solar radiation at which the NOCT is defined [0.8 kW/m²]

If we assume that $\frac{\tau \alpha}{U_L}$ is constant, we can substitute this into the previous equation for cell temperature:

$$T_c = T_a + G_T \left( \frac{T_{c,NOCT} - T_{a,NOCT}}{G_{T,NOCT}} \right) \left( 1 - \frac{\eta_c}{\tau \alpha} \right)$$

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HOMER assumes a value of 0.9 for τα, as suggested by Duffie and Beckman (1991).
Because $\frac{\eta_c}{\tau \alpha}$ is small compared to unity, this assumption does not introduce significant error.

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Maximum Power Point Efficiency

HOMER assumes that the PV array always operates at its maximum power point, as it does when controlled by a maximum power point tracker. That means HOMER assumes the cell efficiency is always equal to the maximum power point efficiency:

$$\eta_c = \eta_{mp}$$

where:

• η_mp = efficiency of the PV array at its maximum power point [%]

Replacing $\eta_c$ with $\eta_{mp}$, we obtain:

$$T_c = T_a + G_T \left( \frac{T_{c,NOCT} - T_{a,NOCT}}{G_{T,NOCT}} \right) \left( 1 - \frac{\eta_{mp}}{\tau \alpha} \right)$$

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Temperature Dependence of Efficiency

However, $\eta_{mp}$ depends on the cell temperature $T_c$.
HOMER assumes that the efficiency varies linearly with temperature according to the following relation:

$$\eta_{mp} = \eta_{mp,STC} \left[ 1 + \alpha_P (T_c - T_{c,STC}) \right]$$

where:

• η_mp,STC = maximum power point efficiency under standard test conditions [%]
• α_P = temperature coefficient of power [%/°C]
• T_c,STC = cell temperature under standard test conditions [25°C]

Because the temperature coefficient of power is typically negative, efficiency decreases with increasing temperature.

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Final Cell Temperature Equation

Substituting the efficiency equation above into the cell temperature equation and solving for $T_c$ yields:

$$T_c = \frac{ T_a + (T_{c,NOCT} - T_{a,NOCT}) \left( \frac{G_T}{G_{T,NOCT}} \right) \left[ 1 - \frac{\eta_{mp,STC}(1 - \alpha_P T_{c,STC})}{\tau \alpha} \right] }{ 1 + \left( \frac{T_{c,NOCT} - T_{a,NOCT}}{G_{T,NOCT}} \right) \frac{G_T \, \eta_{mp,STC} \, \alpha_P}{\tau \alpha} }$$

The temperatures in this equation must be in Kelvin.
HOMER uses this equation to calculate the cell temperature in each time step.