149
Original scientific paper
Journal of Microelectronics, 
Electronic Components and Materials
Vol. 54, No. 2(2024), 149 – 161
https://doi.org/10.33180/InfMIDEM2024.207
How to cite:
M. Pirc, “Empirical Light-soaking and Relaxation Model of Perovskite Solar Cells in an Indoor Environment", Inf. Midem-J. Microelectron. 
Electron. Compon. Mater., Vol. 54, No. 2(2024), pp. 149–161
Empirical Light-soaking and Relaxation Model of 
Perovskite Solar Cells in an Indoor Environment 
Matija Pirc
Laboratory of Photovoltaics and Optoelectronics, University of Ljubljana, Faculty of Electrical 
Engineering, Ljubljana, Slovenia
Abstract: Perovskite Solar Cell (PSC) technology is approaching the level of maturity required for some niche applications, primarily 
in indoor environments. However, their metastability, expressed in the form of the light-soaking effect (LSE), makes it difficult to 
accurately estimate their expected real-life performance. This work demonstrates a new approach to LSE modelling, which can be 
used to determine the performance parameters of the PSC based on the history of its irradiance. The model was developed and tested 
on PSC performance data recorded during one month of operation in a realistic uncontrolled indoor environment, two days of which 
were used for the tuning of the model and the rest for its verification. The presented model was compared to two static one-diode 
models, which do not account for the LSE. The energy yield prediction error of the new model was only -0.72 %, the error of the static 
model based on low-light measurements was +6.96 %, and the error of the static model based on measurements under standard test 
conditions (STC) was +7.76 %. EY prediction of the low-light static model can however be arbitrarily improved by cherry picking the 
I-V curve on which to base the model, once the expected result is known. A more meaningful measure of model performance is the 
mean absolute error (MAE) of the predicted power at the maximum power point PMPP . The MAE of PMPP predicted by the new model 
was 16.7% lower than that of the low-light static model and 17.1 % lower than that of the STC static model.
Keywords: perovskite solar cells; light-soaking; indoor photovoltaics; I-V curve; energy yield 
Empirični model svetlobnega prežemanja in 
relaksacije perovskitnih sončnih celic v notranjem 
okolju
Izvleček: Tehnologija perovskitnih sončnih celic (PSC) se približuje stopnji zrelosti, ki je potrebna za vstop na trg nekaterih nišnih 
aplikacij, predvsem v notranjem okolju. Zaradi njihove metastabilnosti, ki se izraža v obliki učinka svetlobnega prežemanja (ang. 
Light-soaking Effect - LSE), težko natančno ocenimo njihovo pričakovano učinkovitost v realnem okolju. To delo prikazuje nov način 
modeliranja  LSE, ki omogoča napovedovanje delovanja PSC na podlagi zgodovine obsevanosti. Model je bil razvit in preizkušen na 
enomesečnih meritvah delovanja PSC v realnih nenadzorovanih notranjih pogojih delovanja. Podatki dveh dni so bili uporabljeni za 
umerjanje modela, preostali podatki pa za preverjanje njegovega delovanja. Predstavljeni model smo primerjali z dvema statičnima 
enodiodnima modeloma, ki ne upoštevata LSE. Napaka napovedi energijskega izplena novega modela je bila le -0,72 %, medtem 
ko je napaka statičnega enodiodnega modela, osnovanega na meritvah pri nizki osvetljenosti, znašala +6,96 % in napaka statičnega 
enodiodnega modela, osnovanega na meritvah pod standardnimi testnimi pogoji (ang. Standard Test Conditions - STC), znašala 
+7,76 %. Napoved energijskega izplena statičnega modela osnovanega na meritvah pri nizki osvetljenosti, je mogoče pri znanem 
želenem rezultatu skoraj poljubno izboljšati z izbiro krivulje I-U, na kateri ta model temelji, zato napaka energijskega izplena ni najboljše 
merilo uspešnosti modela. Boljše merilo je srednja absolutna vrednost napake (ang. Mean Absolute Error - MAE) napovedane moči 
v točki največje moči PMPP . MAE napovedane PMPP novega modela je bila za 16,7 % manjša od MAE napovedi statičnega modela, 
osnovanega na meritvah pri nizki osvetljenosti in 17,1 % manjša od MAE napovedi statičnega modela, osnovanega na STC meritvah.
Ključne besede: perovskitne sončne celice, svetlobno prežemanje, notranja fotovoltaika, I-U karakteristika, energijski izplen
* Corresponding Author’s e-mail: matija.pirc@fe.uni-lj.si

150
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161
1 Introduction
In the field of photovoltaics (PV) perovskite solar cells 
(PSC) have received a lot of attention during the last 
decade due to their high power conversion efficiency 
(PCE) potential [1], [2] and  simple and low cost produc-
tion process [3]–[5]. Despite their great advantages,  
they also have a few weaknesses hindering their wide-
spread adoption, namely the use of toxic materials like 
lead [6]–[8] and their less than desirable long-term 
stability [9], [10]. However, intense research efforts are 
also improving these areas to the point that commer-
cial applications are becoming feasible [11]. One of the 
niches the first commercial PSCs will most likely flour-
ish in is indoor PV [12], [13], due to the mild operating 
environment, and their high and tunable bandgap en-
ergy [14]–[16], which makes them particularly suitable 
for use with artificial light sources that emit light in the 
human visible light spectrum by design. At the same 
time, a vast opportunity for indoor PV is opening with 
the rapid expansion of the Internet of Things (IoT) mar-
ket [17] and PSCs could be the perfect solution.
For any PV use case an estimation of power capacity 
and expected energy yield (EY) is required. The estima-
tion process can be as simple as taking the average irra-
diation and nominal solar cell efficiency to get a rough 
estimation of expected EY or an in-depth analysis tak-
ing into account measured device characteristics, tem-
perature dependence, angular irradiance distribution, 
spectra, etc. [18]–[22]. Such analysis is not only useful 
for adequate sizing of PV installations or PV devices, 
but also for optimizing the structure of PV devices[18], 
[21], real-time monitoring and early fault detection by 
comparing the predicted and real performance of an 
installation, as well as gaining deeper insights into the 
operation of solar cells.
Although methodologies for predicting solar cell per-
formance and calculating EY have evolved significantly 
and are capable of remarkable accuracy in the case of 
established technologies like silicon solar cells, PSCs 
are still a challenge due to their short-term instability, 
widely known as the light-soaking effect (LSE). A sim-
ple approach to sidestepping the challenges of the LSE 
in outdoor environments is to use the performance 
characteristics of a fully light-soaked PSC throughout 
the whole day [22]. This method is quite accurate for 
outside environments on bright sunny days when the 
LSE rapidly reaches saturation. However, in indoor en-
vironments, or even outdoors on particularly cloudy or 
foggy days, the LSE is much slower due to lower irradi-
ance and may not reach saturation at all, yet still signifi-
cantly affects the performance [23], [24]. In these cases, 
a method for predicting the state of the LSE of a PSC 
based on the history of its irradiance could significantly 
improve the accuracy of the PSC energy yield calcula-
tions.
Many mechanisms contribute to the LSE, e.g. light in-
duced ion migration, trap defect passivation, charge 
carrier accumulation, lattice expansion, etc. [25]–[28]. 
The bulk of the research of the LSE focuses on under-
standing the mechanisms behind it and the models 
developed are quite involved and usually require in-
depth knowledge of the specific PSC. For the purpose 
of long-term performance predictions, these models 
are often computationally too intensive and require 
device parameters which are challenging to acquire. 
Long-term performance modelling generally requires 
different, high-level models, which are comparatively 
easy to tune and use on large data sets of environmen-
tal parameters.
In this contribution, we present an empirical model of 
the LSE which was developed by analyzing recorded 
data of PSC performance in a realistic, uncontrolled in-
door environment. The model can be tuned on a few 
days’ worth of measured data and then used to predict 
the I-V curves of the PSC based on the history of its ir-
radiance for any point in time. We also discus the short-
comings of the model and explore the possibilities of 
further research and improvements.
2 Materials and methods
2.1 Measurement Setup
PSC performance and environmental parameters were 
monitored and logged by an in-house designed Indoor 
Monitoring System which is thoroughly described in 
[24], therefore only the relevant details will be repeated 
here. The system maintains four solar cells at the MPP , 
and logs the average values of their performance, irra-
diance, temperature and humidity every 5 minutes. Ad-
ditionally, I-V scans of all the cells are performed every 
half an hour. I-V scans are performed in voltage steps 
of approximately 20 mV every 60 ms, starting at V
OC
 , 
sweeping down to 0 and then back up to V
OC
 . During 
each step, the irradiance is measured and stored with 
the I-V data to facilitate subsequent detection of un-
suitable lighting conditions, e.g. sudden changes in the 
irradiance during an I-V scan. 
The system was designed to mimic a realistic indoor 
PV powered device between I-V scans, which would 
shut down MPP tracking when the energy cost of MPP 
tracking would exceed energy gains. Therefore, when 
the current of the PSC falls below 30 μA, MPP tracking 
and energy harvesting is suspended.

151
The monitoring system was located on a shelf approxi-
mately at the center of the Laboratory of Photovoltaics 
and Optoelectronic office 1 (LPVO-1) in Ljubljana, Slo-
venia, with windows facing north-north-west (336°). 
This means that direct sunlight is possible only for a 
few minutes in the evenings from about a month be-
fore to about a month after the summer solstice, which 
was not the case within the timespan of the measure-
ments used here.
Irradiance was measured with a silicone photodiode 
SFH2440L with an IR-cut filter, to adjust the spectral 
response of the photodiode to be more similar to the 
spectral response of the PSC. Unfortunately, the dif-
ference between both spectral responses is still quite 
large and since the spectrum of light changes consid-
erably when the ratio of natural and artificial lighting 
changes, the irradiance measurement accuracy leaves 
something to be desired. Although the accuracy is suf-
ficient for the LSE modelling and performance predic-
tion, PCE calculations based on these measurements 
are not recommended.
2.2 Perovskite solar cell
The structure of the PSC was glass | ITO | MeO-2PACz 
| perovskite | C
60
 | SnO
2
 | Cu in a p-i-n architecture, 
where the MeO-2PACz monolayer (2-(3,6-Dimeth-
oxy-9H-carbazol-9-yl)ethyl)phosphonic acid ) is a 
hole transport layer (HTL), and the C
60
 and SnO
2
 lay-
ers are electron transport layers (ETL). The perovskite 
absorber is a formamidinium-caesium (FACs) double 
cation perovskite absorber with the chemical formula 
FA
0.83
Cs
0.17
Pb(I
0.83
Br
0.17
)
3
. The back and front contacts are 
connected with 2 self-adhesive copper stripes. The de-
vice is sealed between 2 sheets of glass with two-com-
ponent 5-minute epoxy. The area of the PSC is 1.06 cm
2
 
and under standard test conditions (STC) the short-
circuit current density is 22.3 mA/cm
2
, open-circuit 
voltage 1.13 V, fill factor 78.6 % and PCE 19.9 %. The J-V 
curve under STC and the external quantum efficiency 
(EQE) of the PSC, along with the EQE of the photodiode 
for measuring irradiance are shown in Figs. 1 and 2, re-
spectively.
Figure 2: Spectral response of the PSC and photodiode 
with IR-cut filter for measuring irradiance.
2.3 Data preparation
Because the measurements were performed in an 
uncontrolled environment, not all the measurements 
were valid, e.g. when the irradiance was changing too 
much during the I-V scan. To filter out these scans, the 
standard deviation of the irradiance during each I-V 
scan was calculated and all I-V scans with a standard 
deviation of irradiance larger than 1 % of the average 
irradiance or larger than 0.005 W/m
2
 were ignored. I-V 
scans with an average irradiance lower than 0.15 W/
m2 were also ignored, because the current measure-
ment noise becomes too prevalent at such low irradi-
ance. For easier comparisons with other studies all cell 
parameters and measurements were normalized to the 
area of 1 cm
2
.
3 The Light-soaking Model
The effect of Light-soaking can be readily discerned by 
observing the J-V scans taken throughout a typical day. 
Fig. 3 shows the J-V scans recorded on August 1, 2022 
progressing from light to dark color from the morning 
to the evening. The LSE is most apparent when compar-
ing the J-V scans taken very early in the morning and 
very late in the evening. When the two marked scans 
were acquired, the irradiance was very similar (0.80 and 
0.76 W/m2, respectively), yet the V
OC
 was 90 mV higher 
in the evening (15 %), even though the irradiance was 
slightly lower. 
Even though the LSE is most evident in increased V
OC
, 
the V
OC
 has a very strong dependence on the irradiance 
Figure 1: J-V of the PSC under STC
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

152
G, which makes it impossible to determine the size of 
the LSE from the V
OC
 alone.
Figure 3: J-V scans of the PSC recorded on August 1, 
2022. Line color hue indicates the time of the scan – 
from bright in the morning to dark in the evening. The 
two marked lines show the LSE most prominently.
3.1 One-diode Model Parameters
Power and EY calculations are possible as soon as the 
JV curves for each point in time are determined. Solar 
cell J-V curves are often modelled by a one- or two-
diode model. If the parameters of the model can be 
determined, all other calculations can be performed as 
well. To see if the parameters of such a model could be 
used to predict the influence of the LSE, a mass fit on 
all the J-V scans of the PSC acquired in august 2022 was 
performed using the 2/3 Diode Fit program [29]. With 
some experimentation we determined that a one-di-
ode model is sufficient to achieve a very good fit for all 
the J-V scans and with the low irradiance in the indoor 
environment and therefore small currents, even the 
series resistance of the one-diode model can be safely 
disregarded. The one-diode model used in the study is 
shown in Fig. 4 and it is described by equation (1)
exp1
/
PH S
P
VV
JJ J
nkTq R


 


 
 (1)
where J
PH
 is photo-current density, J
S
 saturation current 
density of the diode, n ideality factor, k Boltzmann con-
stant, T temperature, q electron charge, R
P
 shunt resist-
ance and V voltage of the solar cell.
The photo-current density J
PH
, which, in the case of 
zero series resistance, is equal to the short-circuit cur-
rent density J
SC
, is expected to be directly proportional 
to the irradiance G. This was confirmed by Fig. 5, which 
shows that measurements exhibit linear dependence 
on irradiance with deviations of only a few percent, ex-
cept for a few outliers most likely resulting from local 
shading.
Figure 5: Short-circuit current density J
SC
 of all J-V scans 
vs. irradiance G(blue dots) and a linear fit to the data 
(orange line). The few red dots are considered to be 
outliers. The R
2
 score of the fit without the outliers is 
0.996.
Plotting the time evolution of the fitted parameters of 
the one-diode model unveils a weak correlation be-
Figure 4: One-diode model used to model the PSC.
Figure 6: Time evolution of the ideality factor n (or-
ange squares) and parallel resistance R
P
 normalized to 1 
cm
2
 (blue circles) on August 5, 2022. The gray fill shows 
irradiance.
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

153
tween the ideality factor n and the irradiance G, as well 
as an inverse correlation between the parallel resist-
ance R
P
 and the irradiance G throughout most of the 
day. Fig. 6 shows n, R
P
 and G on August 5, 2022. This 
date was selected because irradiance more or less 
steadily rises during the day, making it easier to per-
ceive the observed correlations. 
Saturation current density J
S
 on the other hand does 
not seem to have an obvious direct correlation to irra-
diance. Therefore Fig. 7 shows its time evolution for all 
days in August. On most days (but not all) the J
S
  starts 
quite high, drops by 2 or 3 orders of magnitude in the 
first 2 hours, and after that more or less steadily falls 
until the evening.
Figure 7: Time evolution of the saturation current den-
sity J
S
 throughout every day of August 2022.
The correlations of n and R
P
 to G are discernible be-
tween approximately 9:00 or 10:00 in the morning and 
18:00 or 19:00 in the evening. Before and after that the 
correlations don’t seem to hold or they are simply lost 
in the noise. When irradiance is low, the noise in the 
measurements makes for a very undefined fit, which 
can result in very large and unrealistic swings in param-
eter values.
To verify the observed correlations, we plotted the 
one-diode parameters of all J-V curves as a function of 
irradiance, shown in Figs. 8 through 10. The majority of 
values of the ideality factor n fall within a narrow range 
between 1.1 and 1.2 with some outliers, especially 
at lower irradiance values, falling outside this range. 
Values that do fall within this range seem to exhibit a 
slight linear dependence on irradiance, confirming the 
previous observation. It is likely that some of the outli-
ers are not just the result of ambiguity when fitting the 
one-diode model to noisy data, but instead the ideality 
factor may also depend on the state of light-soaking. 
However, at this stage we are not yet sure if that is re-
ally the case, and for now we consider data points that 
deviate enough from the main linear trend as outliers. 
The linear fit 
 
0 n
nnkG      (2)
depicted by the orange line in Fig. 8, where G is irradi-
ance, the intercept n
0
 is 1.094 and the linear coefficient 
k
n
 is 0.015 m
2
/W, was performed using the RANSAC lin-
ear regression method (RANdom SAmple Consensus) 
from the python package Scikit-learn [30] with the re-
sidual threshold value of 0.05. The red dots in Fig. 8 are 
the ideality factor values which the RANSAC algorithm 
marked as outliers. Due to the intrinsic randomness of 
the RANSAC algorithm, the fit is slightly different for 
each run. The R
2
 score of the fit without outliers varies 
between 0.5 to 0.7 from run to run. 
Figure 8: Ideality factors n of all the fits to J-V curves vs 
irradiance G (dots) and a linear fit to the data (orange 
line). The red dots are outliers as determined by the 
RANSAC fit method. The R
2
 score of the fit without the 
outliers averages around 0.6.
Parallel resistance values (Fig. 9) mostly fall between 0.8 
and 3 MΩ∙cm
2
. Values larger than that have almost no 
influence on the J-V curve (smaller than the measure-
ment accuracy), and are therefore just an artifact of the 
one-diode model fitting process. The datapoints give 
an impression of negative exponential relation to irra-
diance, which is similar to the inverse correlation ob-
served in the time evolution of parallel resistance and 
irradiance shown in Fig. 6. We modelled the relation 
between parallel resistance RP and irradiance G with 
the equation (3) 
exp
PP MIN PE
K
G
RR R
G

 


   (3)
depicted by the orange line in Fig. 9, where G is irra-
diance, R
PMIN
 is 0.784 MΩ∙cm
2
, R
PE
 is 1.390 MΩ∙cm
2
 and 
G
K
 is 1.440 W/m
2
. The fit was acquired using a similar 
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

154
approach to the RANSAC linear regression method, but 
using our own code built on top of the python lmfit 
package, since the Scikit-learn package does not have 
a built-in function for RANSAC exponential regression. 
In the case of parallel resistance, the outliers are much 
more obvious (out of range of Fig. 9) than in the case 
of the ideality factor, therefore more or less the same 
datapoints are marked as outliers in each run and the 
results of the fit are much more consistent from run to 
run. The R
2
 score of the fit without outliers is 0.637.
Saturation current densities resulting from the fittings 
of the data to the one-diode model, shown in Fig. 10, 
span a large range of values and do not show a direct 
correlation to irradiance as already expected based on 
Fig. 7. Therefore, we assume that the greater part of the 
LSE is encompassed within this parameter.
For additional context, the one-diode parameters of 
the same PSC have been extracted from the I-V curve 
measured after the PSC was fully light-soaked under 
STC. In this case the saturation current density J
S
 is 
1.1∙10
-10
 A/cm
2
, the ideality factor n is 2.31, and paral-
lel resistance R
P
 is infinite (any value above 10 kΩ gives 
good fit results). On the other hand, the model does re-
quire a series resistance of 1.15 Ω. The measured short-
circuit current density is 22.3 mA/cm
2
, which is 13.5 % 
lower than would be expected based on the photo-
current dependence on irradiance shown in Fig. 5. The 
ideality factor is considerably larger than in low light 
conditions, however, based on equation (2) it should 
be about 7 times larger still, showing that the linear fit 
is only valid within the range of low irradiances. 
3.2 Linking one-diode model parameters to a single 
observable of the LSE
The challenge of predicting the size of the saturation 
current lies in its very large range of possible values as 
well as in its close connection to the ideality factor n. A 
small change in the ideality factor n, can be compen-
sated with quite a large change in the saturation cur-
rent density to achieve an almost identical J-V scan fit, 
as shown in Fig. 11. The two fits shown represent the 
measured data practically equally well, but the satura-
tion current density of the second fit is approximately 
5 times larger than that of the first fit while the differ-
ence in the ideality factor is less than 10 %. The parallel 
resistance is the same in both cases. This means that 
predicting the value of the saturation current indepen-
dently from the ideality factor would not be the most 
reliable way of predicting the size of the LSE. Therefore, 
a more robust parameter, combining the saturation 
current density with other parameters, would be a bet-
ter option for predicting the LSE.
Figure 11: Demonstration of strong dependence of 
the J
S
 parameter on the n parameter of the one-diode 
model – under 10 % change in n requires 500 % change 
in J
S
, to get an almost identical fit within the measured 
range of data.
Figure 9: Parallel resistance R
P
 of all the fits to J-V curves 
vs irradiance G (blue dots) and an exponential fit to the 
data (orange line). The R
2
 score of the fit without outli-
ers is 0.637.
Figure 10: Saturation current J
S
 of all the fits to J-V 
curves vs irradiance G.
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

155
Open-circuit voltage is the most obvious parameter 
where the effects of light-soaking are evident. How-
ever, as already mentioned, it also strongly depends on 
the photocurrent, which in turn depends on the irra-
diance. The effect of light-soaking on the V
OC
 could be 
isolated from the direct effect of the irradiance if the I-V 
curves were always measured at the same irradiance. 
Although this was not possible within the experiment, 
such a measurement should only marginally affect 
the shape of the curve through minor changes to the 
ideality factor and the parallel resistance according to 
(2) and (3), provided the scan time at a different irra-
diance was short enough not to meaningfully change 
the amount of light-soaking. On the other hand, once a 
one-diode model fit is obtained, a V
OC
 at a pre-selected 
photo-current can be easily calculated with reasonable 
accuracy. Fig. 12 shows the time evolution of the cal-
culated V
OC
 at a pre-selected photo-current density of 
73.3 μA/cm
2
, which corresponds to 3 W/m
2
 irradiance 
– approximately half of the maximum irradiance meas-
ured in August.
The time evolution of the V
OC
 at the pre-selected cur-
rent indicates that the LSE more or less steadily increas-
es throughout most days, with quite a weak depend-
ence on the irradiance and only shows a decrease when 
the irradiance drops very drastically. Combining these 
observations with the observed speed with which the 
LSE increases under STC or in an outdoor environment 
compared to the indoor environment, leads us to be-
lieve that the speed of the LSE, or at least this param-
eter which indicates the state of the LSE, is proportional 
to the logarithm of the irradiance.
3.3 Discrete linear time-invariant system for LSE 
dependent VOC prediction
Previous studies [26], [27] have shown that at a con-
stant irradiance, the V
OC
 of a PSC follows an increasing 
exponential decay form with an offset V
OCMIN
 (4), remi-
niscent of the voltage of a capacitor while charging to a 
fixed voltage through a resistor, if the offset is ignored. 
 
 1e xp
OC OCMIN OCLS
t
VtVV



 
 


 (4)
The time evolution of the V
OC
 in Fig. 12 looks like it 
could fit the same increasing exponential decay form, 
but with quite a long time constant. In light of this, 
we sought to model the LSE with a linear time-invar-
iant (LTI) system, which takes a logarithm of the time 
resolved irradiance as an input and outputs the time 
resolved V
OC
 at a pre-selected current density. The pre-
dicted V
OC
 calculated for each point in time can then be 
used together with the ideality factor n and parallel re-
sistance R
P
 calculated from the irradiance using (2) and 
(3) to determine the saturation current density J
S
.
The step response of the system shown in (4) has an 
offset V
OCMIN
, which has to be treated separately since a 
system with an offset does not satisfy the homogeneity 
condition for linearity and is therefore not an LTI sys-
tem. The step response of the linear part of the system 
is therefore
 
 1e xp
OC OCLS
t
VtV



 
 


  (5)
where τ is the time constant of the LSE and V
OCLS
 is the 
size of the input step function determined by the irradi-
ance and represents the final increase of the V
OC
 due to 
the LSE.  The V
OC
(t) is assumed to be 0 for t < 0, but is not 
denoted within the equation by multiplication of the 
right-hand side with a unity step function for clarity’s 
sake, as will be the case henceforth.
The LTI system could be modelled by a resistor-capaci-
tor (RC) electrical circuit, and its response calculated by 
any electrical circuit simulator. However, since all the 
Figure 12: Time evolution of V
OC
 at a pre-selected photo-current density of 73.3 μA/cm
2
, equivalent to irradiance of 3 
W/m
2
, during the first 5 and last 5 days of August.
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

156
measured data, including the irradiance, which will be 
the input to the system, is sampled at discrete points 
in time, it is more convenient to construct and run a 
discrete-time LTI system with an equivalent response. 
To do that, we first rewrite (5) as a unit step response 
g(t) and derive the system’s impulse response h(t)
 1e xp
t
gt


 


    (6)

 d
1
exp
d
gt
t
ht
t 




   (7)
Periodically sampling the impulse response with a 
sampling period of T
S
, we get

1
exp
S
S
nT
hn T





   (8)
which can be rewritten in its discrete-time form as 
 

n
hn Ka      (9)
where K equals 1/ τ and a equals exp(-T
S
/ τ). 
The Z-transform of (9)
 

z
Hz K
za


                  (10)
can be used to construct a difference equation of a 
discrete-time system
  
1 yn Kxna yn                    (11)
where x[n] are the consecutive input values of the sys-
tem (logarithm of the irradiance) and y[n] are the con-
secutive output values of the system (increase of the 
V
OC
 due to the LSE).
Before the logarithm function can be applied to the 
irradiance, the irradiance needs to be normalized to a 
reference value. We assume some minimal irradiance is 
required for the processes contributing to the LSE to 
start, although so far, we have not found any reports 
on the matter. Therefore, we define the logarithmic ir-
radiance GL as
10
log
MIN
G
GL
G

                   (12)
The values of the irradiance G are also downward lim-
ited to G
MIN
 to avoid negative and complex values of GL, 
which can appear when negative values of irradiance 
are measured due to noise when the real irradiance 
is below the noise threshold of the instrument. The 
complete model for predicting the V
OC
 is described by 
Pseudo-code 1: 
Pseudo-code 1: V
OC
 prediction model.
INPUT: vector G
OUTPUT: vector V
OC
V
OCLS
[0] = V
OCLS0
for n = 1 to length(G)-1
 G
LIM
 [n]= max(G[n], G
MIN
)
 GL[n] = log
10
(G
LIM
[n]/G
MIN
)
 V
OCLS
[n] = K∙GL[n] + a∙V
OCLS
[n-1]
 V
OC
[n] = V
OCMIN
+V
OCLS
[n]
end for
Before the model can be used for predicting the val-
ues of the V
OC
 at the pre-selected photo-current, the 
constants G
MIN
, V
OCLS0
, K, a, and V
OCMIN
 have to be tuned 
to the specific solar cell. This can be achieved by treat-
ing the constants as parameters of a model, and fitting 
the model to the known values of the V
OC
 at the specific 
times when it was measured. We used the data of the 
first 2 days to tune the model and the rest of the data 
to assess the performance of the model. The values of 
the constants after the tuning process are collected in 
Table 1.
Table 1: Tuned model parameters.
Parameter Value
G
MIN
0.0001 W/m
2
V
OCLS0
0.303 V
V
OCMIN
0.425 V
K 2.949∙10-4 V
a 0.99689
4 Results
Fig. 13 shows the V
OC
 at the pre-selected photo-current 
density of 73.3 μA/cm
2
, corresponding to an irradiance 
of 3 W/m
2
, calculated from the one-diode model fits of 
the measured I-V scans (blue circles) and predicted by 
the new model (orange line). The gray filled area repre-
sents the logarithm of the normalized irradiance, which 
is used as an input to the LTI system predicting the V
OC
. 
Fig. 14 shows the measured (blue circles) and the pre-
dicted (orange dots) P
MPP
 as well as the P
MPP
 predicted 
by  two static models (green Xs and red plusses), which 
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

157
do not take the LSE into account, to provide a basis for 
comparison. Fig. 15 shows the errors of the new mod-
el (orange dots) and the two reference static models 
(green Xs and red plusses). The gray fill in Figs. 14 and 
15 shows irradiance. All three figures show the data for 
the first 5 days and the last 5 days of August 2022.
The V
OC
 predicted by the new model follows the meas-
ured V
OC
 quite well at the beginning of the month, but 
not quite as well at the end of the month. The V
OC
 values 
measured on the 27
th
 and 28
th
 of August (Saturday and 
Sunday) seem to change quite unpredictably, which is 
the result of very low irradiance and therefore a low sig-
nal to noise ratio. The very low irradiance is consistent 
Figure 13: The measured (blue circles) and the predicted (orange line) V
OC
 at the pre-selected photo-current density 
of 73.3 μA/cm
2
, equivalent to an irradiance of 3 W/m
2
, for the first 5 and last 5 days of August. The gray fill shows the 
logarithm of the irradiance normalized to the minimum irradiance, which is used as the input to the prediction LTI 
system.
Figure 14: The measured P
MPP
 (blue circles), the P
MPP
 predicted by the new model (orange line with dots), the P
MPP
 pre-
dicted using the low-light static model (green Xs), and the P
MPP
 predicted using the STC static model (red plusses) for 
the first 5 and the last 5 days of August.
Figure 15: The P
MPP
 prediction errors of the new model (orange dots), the low-light static model (green Xs), and the 
STC static model (red plusses) for the first 5 and the last 5 days of August.
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

158
with the blinds being lowered all the way down. De-
spite this, the predicted V
OC
 values are in the general 
vicinity of the measured values. On the last three days 
of August, the model underestimates the LSE and the 
predicted V
OC
 lags considerably behind the measured 
V
OC
. Such discrepancies between the measurements 
and predictions begin to appear in the second half of 
the month. This behavior is reflected in the predicted 
P
MPP
, which also matches the measured P
MPP
 very well 
at the beginning of the month, and not nearly as well 
towards the end of the month. 
The static models used as a basis for comparison con-
sist of taking a single I-V scan, obtaining a fit to the 
one-diode model and using the obtained parameters 
to calculate the P
MPP
 for every point in time based on 
the irradiance. Energy yield predictions or analysis 
in outdoor environments are often performed us-
ing measured stabilized I-V characteristics of the PSC, 
which means the I-V scan is taken only after the cell 
has been fully light-soaked. Since the fully light-soaked 
state can be reached quite quickly in a bright outdoor 
environment, the error introduced by this approach is 
acceptable in most cases. In the case of low-light envi-
ronments, however, the LSE may not reach saturation 
even after an entire day, therefore an I-V scan taken 
sometime in the middle of the day may be more repre-
sentative. We decided to use both approaches and de-
rive one static model based on an I-V scan performed 
in low-light conditions on the first day of the measure-
ments, when the brightness was the highest (green Xs 
in figures, hereafter referred to as the low-light or LL 
static model) and the other based on the I-V scan of the 
fully light-soaked PSC under STC (red plusses in figures, 
hereafter referred to as the STC static model). The pre-
diction error of the low-light static model is quite well 
balanced between positive and negative values at the 
beginning of the month, confirming a good choice of 
the I-V scan to base the model on. 
It has to be noted that selecting a different I-V scan for 
the low-light static model can result in either better or 
worse predictions. EY calculations in particular can eas-
ily be manipulated by selecting just the right I-V scan to 
get the desired result on a known dataset. Therefore, EY 
calculations and their errors are given for completeness 
and to give an idea of how large errors in the EY can po-
tentially be expected, but are not a reliable measure of 
the quality of the method. Instead, the mean absolute 
error (MAE) and root mean square error (RMSE) of the 
P
MPP
 paint a much clearer picture, although they too de-
pend on the selection of the I-V scan in the case of the 
low-light static model and the selection of the model 
tuning period in the case of the new model. The STC 
static model on the other hand is much less tweakable.
A good selection of the tuning period is essential for the 
performance of the model. Tests showed that the mini-
mum tuning period is 2 days, because this ensures that 
the relaxation of the LSE is well defined in the training 
data. Longer training periods are generally beneficial, 
but not necessarily by much. If the selected training pe-
riod includes representative conditions, increasing the 
training period does not improve the model’s perfor-
mance greatly. However, longer tuning periods usually 
include a wider range of conditions and thus provide a 
more representative dataset and therefore result in a 
better performing model. 
The P
MPP
 predictions of both static models are very 
similar and the low-light static model performs only 
marginally better than the STC static model. However, 
a closer examination of the predictions shows a con-
siderably different picture. Fig. 16 shows the measured 
and predicted J-V (solid lines) and P-V (dotted lines) 
curves, with the MPP marked with Xs on the I-V curves 
and plusses on the P-V curves. The predicted curves of 
the new model and the low-light static model match 
the general shape of the measured data very well. The 
STC static model, on the other hand, predicts a much 
more gradual drop in current density when approach-
ing V
OC
 and considerably higher V
OC
, resulting in a lower 
predicted fill-factor (FF). Although the general shape 
of the predicted J-V curve is a much worse match to 
the measured data than those of the other models, the 
combination of increased V
OC
 and decreased FF coinci-
dentally results in almost the same P
MPP
 as in the case of 
the low-light static model.
Figure 16: Measured (blue dots) and predicted J-V 
curves (solid lines) and P-V curves (dotted lines) of all 
three prediction models: the new model (orange), the 
low-light static model (green), and the STC static model 
(red). The MPP on the J-V curve is marked with Xs, and 
on the P-V curve with plusses.
The measured and predicted EY over the course of the 
month, as well as the MAE and RMSE of all the models 
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

159
are summarized in Table 2. During the first half of the 
month the P
MPP
 predictions of the new model mostly 
outperform the predictions of both static models. 
However, at the end of the month, the errors of the 
static models are sometimes lower than those of the 
new model. Taking the entire prediction period into ac-
count, the MAE of the new model is 16.7 % lower than 
MAE of the low-light static model and 17.1 % lower 
than MAE of the STC static model. The RMSE indicates 
slightly lower improvement of 12.3 % and 15.6 % over 
the low-light and STC static models, respectively. The 
EY prediction errors of both static models are +6.96 % 
and +7.76 %, while the EY prediction error of the new 
model is only -0.72 %, which is extraordinary and prob-
ably due to a bit of luck as well.
Table 2: Performance comparison of the new model 
with the static model.
Parameter
Measure-
ment
static 
model
STC 
static 
model
new 
model
EY
 [mWh/cm
2
]
12.50 13.37 13.47 12.41
EY error [%] - +6.96 +7.76 -0.72
MAE(PMPP) 
[μW/cm
2
]*
- 2.21 2.22 1.84
RMSE(PMPP)
[μW/cm
2
]*
- 3.26 3.39 2.86
* P
MPP
 varies between 0 and 94 μW/cm
2
5 Discussion
The presented model of the LSE in the PSC predicts the 
state of the PSC based on the history of irradiance quite 
well in the first half of the month but starts to deviate 
in the second half. It has to be noted, that the model is 
never completely reset and errors in prediction accu-
mulate with time. Yet, despite this, the error at the end 
of the month still remains strictly within ± 13 μW/cm
2
 
and rarely exceeds ± 5 μW/cm
2
 (14 % and 5 % of the 
observed P
MPP
 range), exhibiting a degree of robustness 
of the model. 
On the other hand, the model cannot predict the small 
variations the measured V
OC
 exhibits (Fig. 13), even at 
the beginning of the month. We believe the same basic 
approach could be used to model those variations as 
well, but with a more sophisticated LTI system. How-
ever, more research is need to isolate individual contri-
butions of external parameters to light-soaking and to 
more carefully identify the system’s response to exter-
nal stimuli.
The experiment that provided the data for this work 
was designed to provide data on PSC performance in 
a realistic, uncontrolled indoor environment and sta-
tistical data on such an environment throughout the 
seasons of the year. As such, it was not optimized to 
gather data required for the LSE modelling. However, 
the recorded data was just accurate enough to facili-
tate the first steps in the LSE modelling in PSCs and 
inspire further research, which will hopefully provide 
higher quality data for a more accurate model as well 
as more reliable and extensive validation. The experi-
ment had several shortcomings if viewed in light of the 
requirements for LSE research that will have to be im-
proved upon in future research:
- Shading in realistic indoor environments can be 
very localized, therefore it is very important to 
place the irradiance sensor as close as possible to 
the device under test (DUT), or better yet, place 
several irradiance sensors on opposite sides of 
the DUT.
- Spectral matching of the irradiance sensor and 
the PSC remains a challenge for now, especially 
if the angular sensitivity of both devices needs to 
match as well. A workaround would be to avoid 
an uncontrolled environment and strictly control 
the spectrum of incident light.
- For the purpose of accurate one-diode model fit-
ting, the current measurement accuracy needs to 
be increased.
- The irradiance within this experiment almost nev-
er exceeded 6 W/m
2
, therefore the model is only 
verified within this low irradiance range. 
In a more specialized experiment, the influence of 
temperature could also be characterized and perhaps 
included in the model. However, within this work, the 
measured temperature was accounted for only as a 
parameter of the one-diode model (1), even though it 
has been established before that it affects the LSE [25] 
as well. With the very limited changes in temperature 
in the indoor environment (26.4 °C to 34.0 °C) we as-
sumed that all other sources of uncertainty overshad-
owed the influence of temperature. 
Focusing on the prediction results for the second half 
of the month, we see several possible reasons for the 
reduced performance of the model. It is possible that 
changes in blinds positions (which were not recorded 
or otherwise logged) changed either the average spec-
trum of light enough to influence the LSE or changed 
the shading conditions, which could have led to a 
smaller indicated irradiance, which could in turn have 
caused the model to underestimate the LSE. PSCs are 
also known for their less than optimal long-term stabil-
ity and it is possible that their performance changed 
enough during the first month to make a noticeable 
M. Pirc; Informacije Midem, Vol. 54, No. 2(2024), 149 – 161

160
difference in how the LSE manifests. However, it is im-
possible to conclusively determine the cause of the 
slightly worse performance at the end of the test pe-
riod from the available data.
6 Conclusions
Within this work a new model for predicting the light-
soaking effect (LSE) of perovskite solar cells (PSC) 
based on the history of their irradiance has been pre-
sented. The model uses the current irradiance to calcu-
late the photo-current I
PH
, the ideality factor n and the 
shunt resistance R
P
 of the one-diode model of the PSC. 
From the irradiance history it calculates the V
OC
 at a pre-
selected current (to isolate the LSE from the influence 
of the photo-current on the V
OC
), which is then used to 
calculate the saturation current of the one-diode mod-
el. The thus calculated one-diode model parameters 
account for the LSE and can be used in further analysis, 
like P
MPP
 or energy yield calculations.
The proposed model was compared to two static 
one-diode models over the course of one month and 
showed a 16.7 % improvement over the low-light static 
model and 17.1 % improvement over the STC static 
model in the mean absolute error (MAE) of the P
MPP
 pre-
diction, achieving a MAE of 1.84 μW compared to the 
2.21 μW and 2.22 μW of the low-light and STC based 
static models, respectively, compared to the daily vari-
ations of P
MPP
 between 0 and approximately 94 μW.
7 Acknowledgments
I would like to thank M. Topič, M. Jošt, Ž. Ajdič, Š. Tomšič, 
M. Jankovec and G. Matič for many fruitful discussions. 
The Research was funded by the Slovenian Research 
and Innovation Agency, program P2-0415. 
8 Conflict of Interest
I have no conflict of interest to declare.
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Arrived: 06. 02. 2024
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