The temperature of photovoltaic modules is modelled as a dynamic function of ambient temperature, shortwave and longwave irradiance and wind speed, in order to allow for a more accurate characterisation of their efficiency. A simple dynamic thermal model is developed by extending an existing parametric steady-state model using an exponential smoothing kernel to include the effect of the heat capacity of the system. The four parameters of the model are fitted to measured data from three photovoltaic systems in the Allgäu region in Germany using non-linear optimisation. The dynamic model reduces the root-mean-square error between measured and modelled module temperature to 1.58 K on average, compared to 3.03 K for the steady-state model, whereas the maximum instantaneous error is reduced from 20.02 to 6.58 K.

Photovoltaic (PV) systems have become an integral part of electricity grids worldwide, in particular due to a dramatic reduction in costs as well as the drive to mitigate anthropogenic climate change using renewable energy sources. Accurate modelling of PV power production in the field is important for several reasons: (i) forecasts of solar PV power production are becoming indispensable for grid operators, (ii) improvements in performance and efficiency need to be properly characterised under different environmental conditions and (iii) in the meteorological context, it is conceivable that PV power data could be used to gain more information about atmospheric optical properties. Since PV module efficiency is dependent on temperature, an incorrect thermal model will in the end lead to errors in the overall power model, especially in the case of rapidly fluctuating atmospheric conditions such as inhomogeneous cloudiness. Under high irradiance variability, a simplified steady-state description of heat exchange leads to a mismatch between irradiance and module efficiency and thus a bias in the modelled power output. In this work a simple four-parameter model is shown to be sufficient to capture the dynamics of PV module temperature as a function of ambient temperature, shortwave and longwave irradiance and wind speed, and the parameters are fitted to measured data using non-linear optimisation.

Several authors have studied the thermal characteristics of PV systems in some detail (see for instance the reviews in

In order to describe the module temperature dynamically one needs to solve the differential equation governing heat exchange between the module and its environment, which has been studied in detail before. Some examples include

In the present work a simple model built on the works of

The model equations are described in detail in Sect.

From physical considerations the module temperature can be described by the heat balance equation

In this work a simplified parametric model is proposed as follows: the module temperature time series

Although one would expect the effect of thermal emission to be proportional to

The model in Eq. (

The model was validated using data from two different stations and three different PV systems. The first station is a large free-standing solar park made up of 504 modules of 180 Wp each. The solar park is just outside Kempten, Allgäu, close to the Iller river, and a pyranometer measuring station (see Fig.

PV system and measurement station with horizonal and plane-of-array pyranometer along with a small weather station measuring ambient temperature at station 1, situated at 47.683233

At the second station on a farm east of Kempten, two different PV systems were used to validate the model. System 2A is a small system (roughly 6 kWp) with a steep elevation angle of roughly 60

PV system 2A at station 2 (47.653161

PV system 2B at station 2, with a Kipp & Zonen RT1 sensor mounted on the edge of the module in order to measure plane-of-array irradiance and module temperature.

Data frequency, measurement uncertainty and measurement time periods for the three PV systems.

Table

The longwave downward welling irradiance was measured with a frequency of 2 Hz and an uncertainty of 2 % using a secondary standard Kipp & Zonen pyrgeometer, situated on the roof of a high-rise building in Kempten. Although this device is not exactly co-located with the PV systems it still gives a general idea of the sky temperature and improves the model fit, especially in the early morning and late evening. The sky temperature is simply calculated from the irradiance measurements using

The model in Eq. (

Number of days of each type and total number of data points used for the parameter retrieval for each system.

Results for all-sky conditions for both the static and dynamic models.

Histogram of the deviation between modelled and measured module temperature at system 1, for both the dynamic and static models and under all-sky conditions (i.e., all available days).

Comparison of dynamic and static temperature modelling for system 1 on 14 September 2018. Measured module temperature (red) is plotted together with the deviation between modelled and measured module temperature for the dynamic (blue) and static (orange) models, along with ambient temperature (dark red dashed), in units of

The a priori values of the unknown parameters were taken to be

Figure

Comparison of dynamic and static temperature modelling for system 2A on 27 September 2018, see the caption of Fig.

Comparison of dynamic and static temperature modelling for system 1 on 4 October 2018, see the caption of Fig.

The model can also reproduce the thermal behaviour on a clear sky day, as shown in Fig.

In this work a simple four-parameter dynamic thermal model for the temperature of PV systems was proposed, and the model was fitted to data from three different systems using non-linear optimisation. By employing an exponential smoothing kernel it was shown that the time constant (and therefore the heat capacity) of the system can be extracted from data, and the dynamic model could reproduce 1 min instantaneous temperature measurements with an RMSE of between 1.20 and 2.18 K and a maximum absolute deviation of between 5.83 and 7.63 K. Further improvements to this work could be achieved by considering reflection losses as well as losses due to power generation. It could also be conceivable to use the measured PV power to estimate the sky temperature, so that a longwave irradiance measurement is not needed. A comprehensive comparison of the differential equation approach with the method presented here will be carried out in future work.

From the heat balance equation in Eq. (

Assuming that

Data is available as an open-access data set via

The two measurement campaigns were designed and coordinated with contributions from all authors, and the installation and calibration of the various measurement devices was performed by NK, CS, HD, JW and FG. The temperature model was developed by JB, DB, KP, AHC and SM; the software and simulations to implement the model were developed and carried out by JB and DB. JB prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “19th EMS Annual Meeting: European Conference for Applied Meteorology and Climatology 2019”. It is a result of the EMS Annual Meeting: European Conference for Applied Meteorology and Climatology 2019, Lyngby, Denmark, 9–13 September 2019.

This research was carried out under the BMWi project “MetPVNet: Entwicklung innovativer satellitengestützter Methoden zur verbesserten PVErtragsvorhersage auf verschiedenen Zeitskalen für Anwendungen auf Verteilnetzebene”. Thanks go to Philipp Hofbauer and Matthias Struck from egrid applications & consulting GmbH (part of the local grid operator Allgäuer Überlandwerk), for access to the photovoltaic systems in the Allgäu region.

This research has been supported by the Bundesministerium für Wirtschaft und Energie (grant no. 0350009).

This paper was edited by Sven-Erik Gryning and reviewed by David Faiman and one anonymous referee.