Transient Thermal Model

The PV array transient temperature is computed from the steady-state temperatures of previous simulation timesteps using an adaptation of the model of Prilliman¹. The original model acts as a weighted moving average of the steady-state temperature. We recall it here using our own notation and then introduce our modification. In summary, equations (1), (2), and (4) allow us to compute the average transient temperature $T_i$ for timestep $i$, represented by the PVsyst variable TArray.

Prilliman's model

First, the thermal relaxation time $\tau_i \equiv \frac{1}{P_i}$ is evaluated at each timestep $i$ as

\[ \begin{equation} P_i = 0.0046 + 0.00046W_{s,i} - 0.00023m_u - 0.000016W_{s,i}m_u, \end{equation} \]

where

$P_i$ is expressed in $\mathrm{[1/s]}$ and is the inverse of the thermal relaxation time. This bilinear equation was fitted by Prilliman over a set of 3-D finite-element simulations of a PV module cooling down¹.

$W_{s,i}$ is the wind speed in $\left[\mathrm{m/s}\right]$ read from the .MET file for this timestep.

$m_u$ is the PV module unit mass in $\left[\mathrm{kg/m^2}\right]$, obtained from the module area and weight in the .PAN file. This parameter correlates well with the thermal capacity of the PV module, which comes mostly from the glass cover(s). When the weight of the PV module is not defined in the .PAN file, a default value of $m_u = 11.0 ~ \mathrm{kg/m^2}$ is used. Note that for zero wind speed, as $m_u$ increases toward $20 ~\mathrm{kg/m^2}$ and beyond, the relaxation time $1/P_i$ becomes infinite and then negative, which is unphysical. Therefore, we treat any unit mass $\geq 20 ~\mathrm{kg/m^2}$ as invalid and replace it with the default value.

Second, given the previously calculated steady-state temperatures, a first weighted moving-average temperature is calculated for timestep $i$ according to Prilliman's formula:

\[ \begin{equation} T_{\mathsf{Prilliman},i} = \frac{\sum_{k=i-1}^{(t_i - t_{i-1}) \leq (t_i - t_k) \leq \Delta t_c} T_{\mathsf{SS}, k} ~ e^{-P_k(t_i-t_k)}}{\sum_{k=i-1}^{(t_i - t_{i-1}) \leq (t_i - t_k) \leq \Delta t_c} e^{-P_k(t_i-t_k)}}, \end{equation} \]

where

$t_i$ and $t_k$ are the starting times of timesteps $i$ and $k$, respectively.
$T_{\mathsf{SS}, k}$ is the steady-state temperature for timestep $k$, calculated with the previously discussed thermal balance model.
The summation runs over the timesteps strictly prior to the $i$-th timestep, until the time difference $(t_i - t_k)$ becomes larger than the cutoff time $\Delta t_c$.

In the original publication, a cutoff of $20 ~\mathrm{min}$ was proposed, which is relevant for many PV systems. However, for PV modules with thick glass, the thermalization time can approach or exceed this limit. We therefore set $\Delta t_c = 3\times\tau_{\mathsf{max}}$, where $$ \tau_{\mathsf{max}} = 1/(0.0046 - 0.00023m_u). $$ For example, considering a glass/glass PV module with two glass sheets of $3.2 ~\mathrm{mm}$ and an average glass density of $2.5 ~\mathrm{g/cm^3}$, we obtain $m_u = 16 ~\mathrm{kg/m^2}$, leading to $\tau_{\mathsf{max}} = 18 ~\mathrm{min}$, which is dangerously close to the original cutoff and justifies extending it.

PVsyst adaptation of Prilliman's model

An important note on Prilliman's equation (2) is that it does not take into account the steady-state temperature of the present timestep. In the author's words¹, "using the steady-state temperature at the index being considered would include temperature conditions for periods that have not yet occured". However, this implies a different framework from the one used in PVsyst. Prilliman's formula outputs a temperature representative of the start of interval $i$, while PVsyst works with a temperature representative of the average over the time interval $[t_i, t_{i+1}]$. This notably poses a problem when applying Prilliman's formula to hourly data, for which equation (2) returns a temperature delayed by $60 ~\mathrm{min}$ compared to the usual PVsyst temperature.

Therefore, we adapt Prilliman's model to PVsyst's framework as follows. We start by considering that the instantaneous temperature during the interval $[t_i, t_{i+1}]$ follows an exponential decay between the two Prilliman temperatures given for the start and end of the interval, i.e.

\[ \begin{equation} T(t) = \left(T_{\mathsf{Prilliman,i+1}} - T_{\mathsf{Prilliman,i}} \right)\left(1-\exp(-P_it)\right) + T_{\mathsf{Prilliman,i}} \end{equation} \]

The average transient temperature over the interval $[t_i, t_{i+1}]$ can then be calculated analytically by

\[ \begin{equation} \begin{align*} T_i &\equiv \left< T(t) \right>_i = \frac{1}{\Delta t}\int_{t_i}^{t_{i+1}}T(t')\mathrm{d}t' \\ &= \left(T_{\mathsf{Prilliman,i+1}} - T_{\mathsf{Prilliman,i}} \right)\left(\exp(-P_i\Delta t) - 1\right) \left(1+\frac{1}{P_i\Delta t}\right) + T_{\mathsf{Prilliman,i}} \end{align*} \end{equation} \]

where $\Delta t = t_{i+1} - t_i$ is the time interval duration.

This modification of Prilliman's approach has been verified against an exact analytical model and has demonstrated better performance for all the timescales of temperature variation and thermal relaxation time considered here.

Impact of Thermal Inertia on Simulation

The addition of thermal inertia on top of the steady-state model has two main effects on the simulation:

First, the temperature evolution during the day is slightly delayed compared to the irradiance. This is notably visible on clear days, when the peak PV array temperature occurs slightly later than the irradiance maximum.

Second, during cloudy days, the thermal inertia of the module smooths out the temperature during fast irradiance variations.

These behaviors can be seen in the Hourly Graphs, by plotting the variable TArrSS and TArray.

For hourly simulations of conventional PV systems, thermal inertia affects the production E_Grid by about $0.1$-$0.3 ~\%$. In particular settings, it can have a stronger impact. For example, if an inverter is designed to operate close to its overvoltage limit, production can be affected by $1$-$3 ~\%$, as the temperature change induced by thermal inertia mainly impacts the PV array voltage.

Matthew Prilliman, Joshua S. Stein, Daniel Riley, and Govindasamy Tamizhmani. Transient Weighted Moving-Average Model of Photovoltaic Module Back-Surface Temperature. IEEE Journal of Photovoltaics, 10(4):1053–1060, July 2020. URL: https://ieeexplore.ieee.org/document/9095219/ (visited on 2025-04-16), doi:10.1109/JPHOTOV.2020.2992351. ↩↩↩