Acta IMEKO, Title


ACTA IMEKO 
ISSN: 2221-870X 
December 2017, Volume 6, Number 4, 121-130

 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 121 

Markov process reliability model for photovoltaic module 
failures 
Loredana Cristaldi, Mohamed Khalil, Marco Faifer 

 Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano 20133, Italy  

 

 

Section: RESEARCH PAPER  

Keywords:  failure analysis; Markov process; photovoltaic module; reliability model  

Citation: Loredana Cristaldi, Mohamed Khalil, Marco Faifer, Markov process reliability model for photovoltaic module failures, Acta IMEKO, vol. 6, no. 4, 
article 19, December 2017, identifier: IMEKO-ACTA-06 (2017)-04-19 

Section Editor: Lorenzo Ciani, University of Florence, Italy 

Received: October 13, 2016; In final form October 6, 2017; Published December, 2017 

Copyright: © 2017 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits 
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited 

Corresponding author: Mohamed Khalil, e-mail: mohamedmahmoud.khalil@polimi.it 

 

1. INTRODUCTION 
Photovoltaic (PV) systems are installed all around the world 

to produce electricity from solar energy. The evaluation of their 
long-term reliability is one of the fundamental values of their 
assets [1] and should be inclusive of both a complete and partial 
outage of PV systems in addition to their operation at a level 
below expectations. In the last years, PV systems have been 
changing from small isolated systems to large grid-connected 
power stations. Therefore, any failure associated with PV 
modules can affect the overall performance of the power 
system and generate safety issues. Consequently, 
comprehensive studies have been carried out to explain the 
failures of PV modules. For instance, Quintana et al. [2] 
reported the different degradation mechanisms of aged PV 
modules in the field, Chattopadhyay et al. [3] presented an 
analysis of degradation data of the Indian survey on PV 
modules, Cristaldi et al. [4] proposed a diagnostic monitoring 
architecture based upon the analysis of the PV module failure 
causes, and a detailed failure analysis of PV module was 
reported by an international energy agency in [5], etc. Based on 
the available studies, seven main failure causes for a PV module 
can be identified in the field: dust, host spot, corrosion, cell 
cracks, semiconductor ageing, and broken interconnects and 
soldier  busses.  PV  encapsulation  failures contribute as well in  

 
 
 

the failure of PV modules. This includes encapsulation 
delamination, discoloration, moisture ingress, and module 
broken glass. The failures of module encapsulation are excluded 
in this analysis in order to have a clear view on the reliability of 
the PV module without the encapsulation. This will allow the 
manufacturers and decision-makers to have the capability to 
enhance the lifetime of PV modules separately. In addition, it 
will provide them with a clear view on the impact of each 
failure mode on PV reliability, and probability of occurrence for 
each failure mode.  

Different techniques from the literature were used to 
estimate the PV module reliability. For instance, Charki et al. [6] 
and Lorande et al. [7] used a Petri network to represent the 
failures of various components in a PV system and estimate its 
availability. Theristis et al. [8] constructed a Markov model for a 
standalone photovoltaic system in order to evaluate Loss of 
Load Probability (LOLP) based upon the failure and repair rate 
of each component in the system. Zini et al. [9] used a fault tree 
technique in order to understand the impact of Balance of 
System (BOS) components on the system overall reliability. 
Reliability block diagrams were considered in [10], [11] to 
measure loss of load probability, availability, and capacity, 
associated with the performance degradation in PV systems. 

ABSTRACT 
This paper presents a Markov process reliability model for a photovoltaic module. This model includes all the possible failures 
associated to the operation of a Photovoltaic (PV) module. Failure modes are identified in detail, afterwards all the possible failure 
states of the PV module during the service are considered. Finally, a complete Markov process is attained to assess the probability of 
each failure mode occurrence and estimate the mean time to failure, probability density function of the time to module failure, hazard 
and survival functions of a PV module. 



 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 122 

Generally, two different reliability scenarios are widely 
encountered during the useful life time and wear out phase of 
systems based on the warranty limits. The first scenario is 
related only to the full performance so any failure within the PV 
modules will have a significant impact on reducing the whole 
system output below the warranty limits. As a result, the system 
states associated with these failures will be treated as a down 
state, and these failures are called reliability failures. The second 
scenario considers the degrade performance as up states, and 
the associated failures are called durability losses or soft failures. 
This is applicable as long as the performance of the PV module 
meets the warranty limits. Both scenarios are based on non-
restorable systems and contain only transitions in the direction 
towards system failures. More details on reliability failures and 
durability losses are available in [12], [13]. In this paper, the 
second scenario is highlighted. For the purpose of abbreviation, 
the term ‘failure’ refers to ‘soft failure’.  

In this work, a complete framework for evaluating the 
reliability of PV modules using a Markov process is presented 
based on the aforementioned failure causes. It is aimed to 
estimate the reliability of the system on the long-term and the 
probability of each failure mode occurrence, in addition to the 
calculation of the probability density function, hazard rate, and 
mean time to failure (MTTF) of a PV module. It is very 
important to mention that the whole framework is distinct from 
the general reliability modelling methodologies in [14], as the 
operational and reliability aspects of photovoltaic modules are 
considered based upon the failures that affect performance 
requirements. Moreover, the Markov model processed, in this 
work, is totally different from the one in [8] whose authors 
provided a component transition diagram to measure the Loss 
of Load Probability (LOLP) based on the failure and repair 
rates of each component in a standalone PV system. In our 
analysis, a Markov model on the different failure modes of PV 
module components is constructed, regardless the PV system 
arrangement, in order to measure the previously mentioned 
reliability indices, where the failure rates of these failure modes 
are the transition rates between the different functional states of 
the PV module component. 

Monitoring activities are essential for repairable systems and 
components with high probability of failures. However, PV 
modules cannot be repaired, and they are replaced once they 
fail. Therefore, it is quite hard to implement the necessary 
monitoring activity, especially if the PV plant is not easily 
accessed by the maintenance crew. The proposed Markov 
model can be used as a statistical tool to measure the failure 
frequency once the probabilities of occurrence are estimated 
[14] in addition to the contribution of each failure mode in the 
failure frequency of the system.   

A Markov Process is a stochastic process that allows to 
model systems characterized with several states and the 
transitions between states. For a random variable X(t) that 
denotes the state of the system at time t and the collection of all 
possible states is called state space denoted by {X= 0,1,..,r}, the 
conditional probability of the state X(s)=i at time s to be at state 
j at time t+s is: 

𝑃{𝑋(𝑡 + 𝑠) = 𝑗|𝑋(𝑠) = 𝑖, 𝑋(𝑢) = 𝑥(𝑢), 0 ≤ 𝑢 ≤ 𝑠}   (1) 

Equation (1) shows the necessity to know the past history of 
the process to know the current state. However, a Markov 

process requires only the knowledge of the current states 
regardless the past history of the system as shown in (9).  This 
feature is an advantage of using a Markov process. 

Through this work, manufacturers and designers will have a 
clear view on the reliability indices of PV modules in order to 
have more reliable PV modules at the end of the design phase. 
In addition, it assists decision makers during the operational 
phase to enhance the maintenance schedules and monitoring 
strategies by prioritizing the failure modules in terms of their 
occurrence and their impact on the functional reliability of PV 
modules. 

This paper is organized as follows: Section 2 presents the 
different failure modes of PV module, Section 3 introduces the 
Markov Process. Section 4 discusses the Markov analysis of 
module failures and Section 5 presents the reliability evaluation 
of PV module. Finally, Section 6 includes conclusions. 

2. PV MODULE FAILURES 

A PV module consists of solar cells interconnected together 
by ribbons, and an encapsulant material to protect the 
components of a PV module from foreign impurities and 
moisture. The failure causes of PV modules are briefly 
described in the following subsections. 

2.1. Dust 
The accumulation of dust on the PV module’s surface area 

can produce spots with varying concentrations. These spots 
vary in shape, location and concentration density. The variation 
in dust accumulation in any place can lead to different 
transmittance of light into the module. This will result in small 
random areas on the PV module with less exposure to solar 
radiation [15], [16].  

Dust deposition depends on its density and size distribution 
[17]; accordingly, the average degradation of a PV module 
performance due to dust fouling is not uniform and highly 
dependent upon the location and weather conditions. For 
instance, Elminir et al. [18] observed a reduction in PV output 
power up to 17.4 % per month in a separate field experiment in 
Egypt during the months of December 2004 to June 2005 while 
Kymakis et al. [19] monitored the performance of a PV park in 
Crete for one year and they found out the soiling losses were 4–
5 % during the winter and 6–7 % during the summer period. 
Cabanillas et al. [20] observed a reduction of modules power up 
to 13 % within 3 months in Mexico. On the other hand, 
experimental investigation on the reduction of the PV output 
efficiency presented in [21] shows that the reduction of 
efficiency reached up to 11.6 % when the dust deposition 
density was fixed at about 8 gm-2. 

Although dust is a detrimental agent whenever solar-energy 
applications are concerned, in literature reviews in [22], [23], all 
the attention was given to estimate the degradation rate of a PV 
module, in short periods up to one year, and no studies are 
available for the failure rate of the dust when the output power 
is below the expectations. Accordingly, the failure rate of dust, 
in the work is calculated based on two important performance 
parameters: the guaranteed output power and the annual 
degradation rate under the effect of dust. 

Currently, many warranties guaranty that the output power 
will be at least 90 % of its initial nominal power in 10-12 years 
and reaches 80 % after 20-25 years. Consequently, if the output 



 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 123 

power is below 80 % of the nominal power, the PV module 
fails [24]. 

Regarding the degradation rate, an interesting six-year 
experimental work by Ryan et al. [25] concluded an annual 
degradation rate of 1.4 %, due to dust, that can be considered 
as an accepted value over long time. 
Generally, PV module power follows a Gaussian distribution 
[26], [27]. Accordingly, the associated probability density 
function is 

𝑃(𝑝) =
1

√2𝜋𝜎
𝑒𝑥𝑝 �–

1
2

( 
𝑝– µ
𝜎

 )� 
  (2) 
 

where p is the power of module, µ is the average power and σ is 
the standard deviation of the PV module’s power. As the 
average power of the PV module decreases linearly versus years 
of operation [28], [29], the average power can be calculated 
from, 

µ(𝑡) = 𝑃0 – 𝐴𝑡 (3) 
where Po is the average power at t=0, A is a parameter that 
reflects the annual reduction in power such that A/Po 
represents the annual degradation rate per year, therefore A/Po 
=0.014. The value of σ  can be related to Po using the concept 
of manufacturing tolerance for the module output power [24], 
so σ =0.0167 Po.  

Taking into account the time variation, the power 
distribution (2) at a defined instant t will fit the following 
distribution: 

𝑃(𝑝, 𝑡) = 1
√2𝜋𝜎

exp �− 1
2
�𝑝−

(𝑝0−𝐴𝐴)
𝜎 

�
2
�   (4) 

By definition, R(t) = 1−F(t) where F(t) is the cumulative 
distribution function. Thus, 

𝑅(𝑡) = 1 − 1
2

 [1 + 𝑒𝑒𝑒 (𝑝−𝑝0+𝐴𝐴
𝜎 √2

)] (5) 

where erf is the gauss error function. Substituting P=0.8 Po , 
A= 0.014 Po, σ =0.0167 Po in (5), gives 
𝑅(𝑡) = 0.5 − 0.5𝑒𝑒𝑒(0.5928𝑡 − 8.4683) (6) 
hence the failure rate function is estimated as follows: 

h(t)= − d
d𝐴
𝑙𝑙𝑅(𝑡) = 0.3344𝑒𝑒𝑝 [−

(0.5928𝐴−8.5746)2]
0.5−0.5𝑒𝑒𝑒 (0.5928𝐴−8.4746)

 (7) 

2.2. Corrosion 
Corrosion of the conductive parts of the cells and the 

interconnections is responsible for the deterioration of the PV 
module and it happens frequently among the PV modules. For 
instance, Gobind H. et al. [31] observed a significant corrosion 
in 31 out of 640 photovoltaic modules at 15 kWp grid-
connected photovoltaic system after three years in service. 
Corrosion results in an increase of ohmic resistance and a 
decrease of the shunt resistance [30]. In general, it contributes 
in 45 % of the PV module's failures [32], [33]. 

2.3. Hot spots 
Shading conditions, mismatching between cell electrical 

characteristics, and bypass diode failure contribute in the 
occurrence of hotspots [34]. In the field, solar cells arrays might 
be subjected to shadows from predictable sources such as 
weather and environmental conditions, and unpredictable 
sources such as bird droppings or fallen leaves. Hot spots in PV 
modules result in contact delamination, melting of 
encapsulation layers and cells damage. 

2.4. Broken interconnect and solar buses failures 
Solar cells consist of two basic elements: front and rear 

contacts, to deliver current to the external circuit. Electrical 
current is carried by bus strips that are soldered to the front and 
back contacts. Interconnect breaks occur as a result of thermal 
expansion and contraction or repeated mechanical stress. Also, 
kinks in interconnects contribute to breaking of interconnects 
[35]. Broken interconnects result in short circuited cells and 
open circuited cells. Moreover, solder bond failures may cause a 
rise of 10 °C in the operating temperature of the cell. This 
increase in temperature will reduce the operational efficiency to 
about 97 % of the healthy conditions [36]. 

2.5. Cells cracks 
Cell cracking is a common problem encountered in 

photovoltaic modules operation. It may develop through 
different stages of the module’s lifetime: during manufacturing, 
the soldering induces high stresses into the solar cells [37], [38], 
and during transport through handling and vibrations [39]. In 
addition, a module in the field experiences mechanical loads 
due to wind (pressure and vibrations) and snow (pressure).  
Cracking of cells occurs at a rate of about 1 % per year. 
Although 1 % failure rate is small, it leads to significant power 
degradation; because it causes around 1-10 % open circuit cell 
failures, a decrease in the filling factor and cells mismatching 
[40], [41]. 

Generally, cell cracks are formed in different lengths and 
orientation. Cracks parallel to busbars have a high probability, 
16 to 25 %, of generating a potentially separated cell area [42]-
[44]. Consequently, the failure rate of cell cracks parallel to bus 
bars are only considered in this work. 

2.6. Semiconductor aging of PV material 
Thermal ageing of semiconductors is a result of exposure to 

UV rays for long intervals of time. High temperature and 
electric fields experienced by the solar cells in a PV module 
result in the transport of atoms and ions that cause lattice 
deformation [45]. The lattice affects the structure of the solar 
cell and changes the electrical properties of the cell. Therefore, 
the old PV module performance is reduced due to 
semiconductor material ageing. Ageing results in an increased 
series resistance or decreased shunt resistance in addition to 
antireflection coating deterioration. It is well known that the 
lifetime of photovoltaic modules follows a Weibull distribution 
with β= 2.6 [46]. Meanwhile, the lifetimes of semiconductor 
devices of special applications are expected to be from 50 to 
100 years [47]. Taking into account the environmental extremes 
in which the PV modules operate, η is assumed to be 50 years 
for β= 2.6. 

3. MARKOV ANALYSIS 
The aforementioned failure causes occur during the 

operation of PV modules and result in poor performance and 
power losses. Therefore, it is crucial to include these failures in 
the reliability model. As already stated in the introduction, 
several reliability techniques were considered in the literature 
and they highlighted the component failures. In this paper, the 
Markov process is selected to carry out the functional reliability 
analysis. Accordingly, the PV module is represented by the 
number of possible states, where each state represents the 
occurrence of one or more failures. 

Markov chains are sequences of random variables in which 
the future variable is determined by the present variable and 



 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 124 

independent on the way in which the present state arose from 
its predecessors. The analysis looks at a sequence of events and 
analyzes the tendency of one event to be followed by another 
[14]. This tendency is the probability evaluation of transition 
from one state to another until the system has reached the final 
state. Thus, a Markov process is defined by a process {p(t), t≥0} 
with state space X= {0, 1, 2, 3….r} and stationary transition 
probabilities, 

𝑃𝑖𝑖 = 𝑃𝑒(𝑝(𝑡) = 𝑗│ 𝑝(0) = 𝑖) for all 𝑖, 𝑗 ∈  𝑋 (8) 
where P(t) is a random variable that denotes the state and 
belongs to state space X. 

In this work, the state space is finite, for instance a system of 
n failures will have 2n states. For a process of probability state 
vector P(t) = [ P0(t), P1(t),….Pr(t)], the distribution  P(t) can be 
estimated from Kolmogorov forward equation as follows, 
d𝑝(𝐴)
d𝐴

= 𝑝(𝑡). 𝐴  (9) 

and 
∑ 𝑝(𝑡)𝑛𝑖=0 = 1  (10) 
where dP(t)/dt is a vector that represents the state probability 
p(t) at time t, and A is the matrix of failure rates between states. 
Considering failure rates from a state i to state j, the failure rates 
can be arranged in the matrix form  
















=

rrr

r

A
λλ

λλ







0

000

 (11) 

A is also known as a generator matrix. Its diagonal elements 
are negative and they represent the rate of staying at the same 
state. It is worth to mention that the sum of the raw entries is 
equal to zero. 

As the number of possible states is finite, (10) is necessary 
since the probabilities of all states at any time t should be one 
and the system can be in one and only one of these states [48]. 
In case of zero repair rates, i.e. Poisson birth-death process, (9) 
can be rewritten as 
𝑑𝑝𝑖(𝐴)
𝑑𝐴

= –  ∑ 𝜆𝑖,𝑖 𝑖≠𝑖 𝑃𝑖(𝑡) + ∑ 𝜆𝑖,𝑖 𝑖≠𝑖 𝑃𝑖(𝑡) (12) 

Through the Markov process, system characteristics can be 
measured like the state visit frequency, system availability, mean 
time to failure, probability distribution of failures and system 
performance. 

Although a Markov process, from the theoretical viewpoint, 
is flexible and versatile, special precautions are necessary to deal 
with the difficulties of practical applications. The main problem 
is that the number of system states and possible transitions 
increases rapidly with the number of events in the system [49] 
and therefore some assumptions are necessary. The usual 
assumptions considered by current standards and references, 
i.e. IEC 61508 [50], IEC-61165 [49], [48], and [14], can be 
summarized as follows: i) failure and repair rate are constant, ii) 
failure and repair events are independent, iii) the transition 
probability from one state to another state occurs within a very 
small time interval, iv) only one event occurs at the same time. 
Furthermore, absorbing states can be used as an assumption 
that helps in numerical mathematical limitations [14], [51]. 
States are absorbing when they are once reached and the system 
will remain there forever. 

The model states are divided into three groups of states: up, 
degrade and down, as shown in Figure 1. This kind of models 

with zero repair rate [49] focuses on reliability evaluation R(t), 
MTTF, and hazard rate. 

The same assumptions and limitations considered in section 
3 are followed up: 

i) Environmental and operational conditions are 
constant, consequently, failure rates of different failure causes 
are constant, with an exception to module aging and dust 
whose failure rates are affected by the ageing and accumulation 
factors respectively. 

ii)  The repair rates, in this model, are equal to zero; 
because system analysts pay a lot of attention to MTTF;, also 
the failure of a PV module means a replacement.  

iii) The degrade state is still an up state. The failures 
occurrence leads to a drop in PV module performance but still 
not below the 80 % warranty. 

Table 1 illustrates the possible states of a PV module 
associated with various failures. Although all failure causes are 
independent, ageing of a PV module is assumed to be a failure 
which affects the operation of all module states. This is a 
reasonable accepted assumption if we consider the replacement 
of the whole PV module as a solution to eliminate that failure. 
Therefore, each state including ageing failure will be treated as 
an absorbing state, as shown in Figure 2. 

From the state transition diagram in Figure 2 and (12), the 
state space equations can be written by inspection as shown 
below: 
𝑃0•(𝑡) = −(𝜆1+ 𝜆2+ 𝜆3 + 𝜆4 +  𝜆5 + 𝜆6) 𝑃0(𝑡) (13a) 
𝑃1•(𝑡) = 𝜆6𝑃0(𝑡) – (𝜆1+ 𝜆2+ 𝜆3 + 𝜆4 +  𝜆5) 𝑃1(𝑡)   (13b) 
𝑃2•(𝑡) = 𝜆5𝑃0(𝑡) – (𝜆1+ 𝜆2+ 𝜆3 + 𝜆4+ 𝜆6 ) 𝑃2(𝑡)  (13c) 

where 𝑃•(𝑡) is the rate of change of state probability with 
respect to time. Absorbing states are of no interest from a 
reliability point of view, since they are reachable from other 
states and the limit of their probabilities with respect to time is 
one:  

lim𝐴→∞ 𝑃absorbing (𝑡) = 1.  
According to [52], the 64 states of the Markov process can 

be reduced into two states: success and failure states, as shown 
in Figure 3, where the probability of failures is the summation 
of all down states. Therefore, the absorbing states probability 
PD(t) represents the probability the system is down,  where PD(t)  
is evaluated from: 

𝑃D(𝑡) =  ∑𝑃absorbing (𝑡) = 1 − ∑𝑃non-absorbing(𝑡) (14) 

Up state
Degraded  

state
Down state

( absorbing state)  
Figure 1. State transition diagram with three states for evaluation of 
reliability for one element. 

 
Figure 3. Equivalent Two-state Markov Process from an Eight-state Markov 
Process [52]. 



 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 125 

4. MARKOV PROCESS OF PV MODULE FAILURES 
The manufacturer warranty period typically exceeds 20 years 

for crystalline silicon modules and 15 years for thin-film 
modules. Unfortunately, there is little field monitored data and 
independent accelerated test data available to support most of 
these warranty claims. The failure rates of PV modules were 
evaluated and listed in Table 2. The failure rates are estimated 

based on the frequency of the PV module failures, noted in 
literature, for certain population over a determined interval. 

4.1. System state probability 
Based on the state space equations (13) and failure rates in 

Table 2, the probabilities of PV module states that are 
associated with one and two simultaneous failure occurrences 
are shown in Figure 4 and Figure 5, respectively.  PV module 
states associated with more than two simultaneous failure 
occurrences are avoided because of their low probabilities of 
occurrence. For example, the state probability of three and four 
simultaneous failure events ranges from 0 to 0.19 and 0 to 
0.025 simultaneously. 

 Figure 4 shows that cell cracks failure event, represented by 
system state P2(t) has the highest probability of occurrence and 
increases steeply. This is due to its high failure rate and its 
associated significant power degradation. On the other hand, 
failures due to dust events of state P16(t) occur, as expected, 
over a large range of operating time because dust is considered 
one of PV environmental extremes which has its impact on the 
operation of the PV system as long as the PV module is in 
service even if its probability of occurrence is small. 

Accordingly, the probability of state P18(t) , in which cell 
cracks and dust failure events might occur simultaneously, is 

Table 1. System states. 

State 
 

A
geing ( λ1) 

D
ust   (λ2) 

H
otspot (λ3) 

Corrosion (λ4) 

Cells cracks 
(λ5) 

Broken 
interconnects 

(λ6) 

System
 state 

0 N N N N N N S 
1 N N N N N Y S 
2 N N N N Y N S 
3 N N N N Y Y S 
4 N N N Y N N S 
5 N N N Y N Y S 
6 N N N Y Y N S 
7 N N N Y Y Y S 
8 N N Y N N N S 
9 N N Y N N Y S 

10 N N Y N Y N S 
11 N N Y N Y Y S 
12 N N Y Y N N S 
13 N N Y Y N Y S 
14 N N Y Y Y N S 
15 N N Y Y Y Y S 
16 N Y N N N N S 
17 N Y N N N Y S 
18 N Y N N Y N S 
19 N Y N N Y Y S 
20 N Y N Y N N S 
21 N Y N Y N Y S 
22 N Y N Y Y N S 
23 N Y N Y Y Y S 
24 N Y Y N N N S 
25 N Y Y N N Y S 
26 N Y Y N Y N S 
27 N Y Y N Y Y S 
28 N Y Y Y N N S 
29 N Y Y Y N Y S 
30 N Y Y Y Y N S 
31 N Y Y Y Y Y F 

 

State 
 

A
geing ( λ1) 

D
ust   (λ2) 

H
otspot (λ3) 

Corrosion (λ4) 

Cells cracks 
(λ5) 

Broken 
interconnects 

(λ6) 

System
 state 

32 Y N N N N N F 
33 Y N N N N Y F 
34 Y N N N Y N F 
35 Y N N N Y Y F 
36 Y N N Y N N F 
37 Y N N Y N Y F 
38 Y N N Y Y N F 
39 Y N N Y Y Y F 
40 Y N Y N N N F 
41 Y N Y N N Y F 
42 Y N Y N Y N F 
43 Y N Y N Y Y F 
44 Y N Y Y N N F 
45 Y N Y Y N Y F 
46 Y N Y Y Y N F 
47 Y N Y Y Y Y F 
48 Y Y N N N N F 
49 Y Y N N N Y F 
50 Y Y N N Y N F 
51 Y Y N N Y Y F 
52 Y Y N Y N N F 
53 Y Y N Y N Y F 
54 Y Y N Y Y N F 
55 Y Y N Y Y Y F 
56 Y Y Y N N N F 
57 Y Y Y N N Y F 
58 Y Y Y N Y N F 
59 Y Y Y N Y Y F 
60 Y Y Y Y N N F 
61 Y Y Y Y N Y F 
62 Y Y Y Y Y N F 
63 Y Y Y Y Y Y F 

 

TABLE 2. PV Module failure rates. 

Failure mode 
 

Parameter 

Ageing of PV module 
β

η
β

λ t=1       

yrs50;6.2 == ηβ   [46-47] 
Dust )(2 th=λ  

Hot spot 1
3 012.0

−= yrλ     [42] 
Corrosion 1

4 023.0
−= yrλ [3] 

Cells cracks 1
5 091.0

−= yrλ          [53] 
Broken interconnect 

& soldier bus failures 
1

6 0031.0
−= yrλ          [54] 



 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 126 

very high as shown in Figure 5.  
Through Figure 4 and Figure 5, it is possible to measure the 

probability of each failure mode occurrence after a certain 
period of operation. For example, Figure 4 shows that the 
probability of occurrence of dust failure and cells cracks are 0 
and 0.4945, respectively after 10 years of operation. These 
probabilities of failure occurrence change after 15 years of 
operation to be 0.1529 and 0.1358 for dust and cells cracks, 
respectively.  

Generally, it can be concluded from Figures 4 and 5 that the 

more operating years, the higher is the probability of 
occurrence for states that have more numbers of failure events. 
The peaks of system states probability curves in Figures 4 and 5 
decrease after a certain time because the probability of the PV 
module being in the absorbing state PD(t) increases by time. 

4.2. Reliability evaluation 
For a system of states X, the up states of the system can be 

grouped in a set B of functioning sets. The failed states, 
therefore are grouped into F=X-B of failed states. 

 
Figure 2. State transition diagram of PV module failure. 



 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 127 

The survivor function R(t) determines the probability that a 
system does not leave the set B of functioning states during the 
time interval. Also it can be evaluated from the subtraction of 
the probability of being into the set F from unity. Thus, the 
survivor function is: 

𝑅(𝑡) = ∑ 𝑃𝑖𝑖∈𝐵 (𝑡) = 1 − 𝑃D (𝑡) (15) 
The Probability Density Function (PDF) indicates the 

distribution of the failure over the entire time range. The larger 
is the value of PD(t), the more failures that occur in the small 
interval time around t. If the time to failure has a Probability 
Density Function, then  

𝑃𝑃𝑃 = − d𝑅
d𝐴

 (16) 

Figure 6 and Figure 7 show the survivor function and the 

Probability Density Function f(t) over the time of the PV 
module failure, respectively. This is calculated by substituting 
the failure rates declared in Table 2 into (13). The reliability of 
the PV module is 0.985 and 0.767 after 10 and 30 years, 
respectively, as shown in Figure 6. 

The MTTF is a basic measure of reliability for non-repairable 
systems that describes the expected time to failure under 
specified experimental conditions. MTTF is given by 

𝑀𝑀𝑀𝑃 = ∫ 𝑅(𝑡)d𝑡
∞
0  (17) 

Meanwhile the hazard rate h(t) is the instantaneous failure 
rate, and is defined by 

ℎ(𝑡) = 𝑒
(𝐴)

𝑅(𝐴)
 (18) 

The hazard function indicates the change in the failure rate 
over the life of a population. Figure 8 shows the hazard 
function of a PV module. By analyzing Figure 8 and (18), it can 
be noticed that the PV module follows up a constant failure 

 
Figure 4. State probability of one failure event. 
 

 

 
Figure 5. State probability of two simultaneous failure events. 

 
Figure 6. PV module Survivor function. 
 

 

Figure 7. PDF of the time to PV module failure. 
 



 

ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 | 128 

rate in the quarter of the first year of the PV module’s 
operation followed by a pronounced wear out behavior. 

It is worth to mentioning that the aforementioned PV 
module reliability results did not include the encapsulate failures 
in order to have a clear view on the reliability of the PV module 
only. The encapsulation of the PV module is subjected to 
failures due to Discoloration and Delamination (D&D), 
moisture ingress and module broken glass. A complete separate 
reliability analysis using a Markov process for failures of a PV 
module encapsulation is conducted in [55].  

The reason behind the very short constant failure rate period 
is the dust failure that will degrade the output power if the 
module is left uncleaned. 

By applying (17) it results that the MTTF of PV modules is 
44.4 years regardless module encapsulation failures. This value 
is relatively higher than the current warranty periods declared 
by manufacturers for PV modules.  

5. CONCLUSION 
It is well known that a high degree of reliability needs to 

identify the possible failure modes of a component or a system. 
A PV module is a critical component in a PV system. Hence, 
the different failure modes of PV modules are discussed in 
detail. A Markov process is used to analyze the probability of 
occurrence for each failure mode in addition to the evaluation 
of survivor function, hazard rate, meantime to failure, and 
probability density function based on these failure modes.  

The Markov process shows that cell cracks have a high 
probability of occurrence. This requires to pay more attention 
to the technology of cell implementation in the module, the 
factory tests, and means of transportation to the site. Dust is 
the dominant failure cause of a PV module along its operating 
life time and results in a very short useful life time phase, three 
months, if the PV module is not cleaned. In addition, both dust 
and cell cracks failures contribute significantly in the probability 
of two simultaneous failure events. For example, the highest 
probability for a PV module to have two simultaneous failure 
events is assigned to both cell crack and dust, P18. This is 

followed by P10, where hotspot and cell cracks occur, and P24, 
where dust and hotspot occur simultaneously.  

The reliability of a PV module is 0.985 and 0.767 after 10 
and 30 years, respectively. Meanwhile, the calculated MTTR is 
44.4 years because a PV module follows up the same aging 
characteristics of semiconductors, when the encapsulation 
failures are excluded from the PV module reliability analysis. 
This highlights the impact of encapsulant failures that reduces 
the lifetime of PV modules to the current values declared by 
manufacturers. 

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    /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke.  Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice