Impact of dust and temperature on photovoltaic panel performance: A model-based approach to determine optimal cleaning frequency

Abstract

Enhancing the reliability of photovoltaic (PV) systems is of paramount importance, given their expanding role in sustainable energy production, carbon emissions reduction, and supporting industrial growth. However, PV panels commonly encounter issues that significantly impact their performance. Specifically, the accumulation of dust and the rise in internal temperature lead to a drop in energy production efficiency. The primary issue addressed in this paper is using mathematical modeling to determine the optimal cleaning frequency. This paper first focuses on stochastic modeling for dust accumulation and temperature changes in PV panels, considering varying environmental conditions and proposing a model-based approach to determine the optimal cleaning frequency. Dust accumulation is described using a Non-homogeneous compound Poisson process (NHCPP), while temperature evolution is modeled using Markov chains. Within this framework, we consider the impact of wind speed and rainfall on dust accumulation and temperature. These factors, treated as covariates, are modeled using a two-dimensional time-continuous Markov chain with a finite state space. A Condition-based cleaning policy is proposed and assessed based on the degradation model. Optimal preventive cleaning thresholds and cleaning frequency (periodic and non-periodic) are determined to minimize the long-term average maintenance cost. The gain achieved by non-periodic inspections compared to periodic inspections ranges from 3.83% to 9.37%. Numerical experiments demonstrate the performance of the proposed cleaning policy, highlighting its potential to improve PV system efficiency and reliability.

1. Introduction

In recent decades, photovoltaic (PV) technology has become a pivotal renewable energy source, experiencing substantial growth [1]. Currently, solar PV systems account for 60% of the total increase in renewable energy capacity. As the installation of PV systems expands, ensuring their reliability and availability has become increasingly critical [2], [3]. PV panels typically have a lifespan of 20-25 years, but they are susceptible to various forms of environmental degradation during their operation [4] [5]. Key factors contributing to this degradation include dust accumulation [6], [7], [8], [9], [10], high temperatures [11], [12], climate change [13], humidity, hail, and snow. If not promptly addressed, these factors can cause significant energy losses. The degree and severity of pollution on the surface of solar modules vary according to the environment in which the PV module is located [14]. For instance, in some desert regions, energy loss due to accumulated dust can range from 20% to 70% [15], [16].

A comprehensive review of existing literature reveals that dust and temperature are consistently significant factors in PV panel degradation [1], [17]. Regular cleaning using the appropriate methods based on the type of dust can mitigate performance loss in PV modules [18]. Managing the operational characteristics of PV systems, such as reliability, availability, maintainability, safety, efficiency, and energy production forecasting, is essential but challenging [19]. Existing research primarily quantifies efficiency losses through time-consuming experiments [20], [21], [11]. However, there is a notable gap in the literature regarding comprehensive degradation modeling and maintenance policy development for PV panels using mathematical models.

A stochastic process is a mathematical model used to describe the uncertain or random evolution of a phenomenon over time. It is commonly used in degradation modeling [22], [69]. In this framework, the most commonly used stochastic processes include the Wiener process [23], gamma process [24], Poisson process [25], [26], and Markov chains [27]. Dust accumulation and temperature changes on PV panels are considered random phenomena. Dust accumulation is affected by various factors such as wind speed, rainfall, location, and human activities, while temperature changes are influenced by solar radiation, weather conditions, time of day, and location. These factors introduce variability and uncertainty into the process, making it difficult to determine the exact amount of dust and temperature changes at any given time. The NHCPP is an effective tool for modeling events such as dust accumulation, which occur at non-constant rates over time and are influenced by factors such as environmental conditions or human activities. Moreover, Markov chains can also be used in environmental temperature modeling since the future ambient temperature trend does not depend on its past, as evidenced by [28], [29].

Environmental conditions are important to consider when modeling the degradation of PV panels for efficient maintenance planning. Given the unpredictability of environmental conditions, they can be categorized as random variables, referred to as covariates [30], [31]. Most condition-based maintenance (CBM) literature ignores the effect of environmental factors on maintenance decision-making for PV systems. Factors such as wind, rain, surface coating, inclination angle, humidity, and wind speed affect the degradation of the system [32], [33], [34]. This article focuses on the impact of wind and rain on accumulated dust and panel temperature. Taking these factors into account can improve maintenance planning.

PV panel maintenance is crucial due to their high cost. At the PV system level, [35] presented a review of four types of maintenance strategies: preventive, predictive, corrective, and urgent. Among maintenance actions, cleaning and cooling of the panel surface is the most common [36]. Many papers have reviewed maintenance policies for multi-component systems [37], [38], [39], especially for PV systems [40], [41].

A CBM policy involves gathering and analyzing real-time data to make maintenance decisions based on the existing state of a component or system, as referenced in various studies [27], [30], [42], [43]. Given that the degradation and failure of industrial systems often become evident only through inspections, planning inspections and maintenance actions based on the system's current state is more appropriate. This paper comprehensively models the degradation of PV panels by considering the effects of dust and temperature and the influence of wind and rain. It also determines the optimal cleaning frequency to enhance energy efficiency, aiming to minimize the long-term average cost rate of the maintenance policy, including non-periodic inspections. Table 1 provides a comprehensive summary of numerous studies conducted on maintenance policies for PV systems.

Table 1.

Literature Comparison Table.

Reference	degradation cause	Maintenance Policy	Methods(data collection)	Limitation
[44]	dust	cleaning	physical experiment	time-consuming
[45]	soiling. dust, temperature	regular cleaning; implementing hydrophobic coatings	experimental testing	long-term degradation of the coating not addressed; additional costs and large-scale feasibility not discussed
[46]	temperature	cooling	experimental data collection	lack of consideration of environmental factors on the effectiveness of the cooling techniques
[47]	System runtime	preventive replacement; minimal Repair	simulation	assumption of perfect maintenance; waste generated from replacing PV panels; lack of consideration of environmental factors
[48]	high temperature, low irradiance, bad weather, natural degradation of materials	corrective (replacement) maintenance, preventive maintenance	failure rates from literature; faults from industrial maintenance reports	limited Sample Siz; lack of consideration of environmental factors; limited to systems from 50 kW to 1.8 MW
[49]	overloads due to lack of insulation; storm or lightning; failure of the ventilation	corrective maintenance	data from Solarig company	limited generalizability; Dependence on maintenance personnel expertise
[50]	dust, shading, corrosion	offline reconfiguration options of the PV modules	simulation	high cost requirements; difficult to scale for large systems; limited Utility of Real-Time Measurement
[51]	shadow occurs and bird dropping; ambient high temperature	cloud computing	Geographic Information System (GIS)	high cost; technological constraints
[52]	System runtime	replacement	distributed e-maintenance platform based on AR	complexity of the equipment; limited generalizability
[53]	soiling; low irradiance; material degradation	Predictive Maintenance	Artificial neural network PV health monitoring system	susceptibility to false positives; need for delayed alarm triggering
[54]	Pollution and dust	scheduling cleaning	remote monitoring system for online tracking	High Design Requirements; ease of access and security
This article	dust, temperature	Condition-based Maintenance(cleaning); non-periodic inspection	mathematical model; simulation	real-time monitoring data integration

The primary motivation of this paper is to address the deficiencies in existing research by developing a comprehensive model that integrates environmental factors (dust accumulation, temperature variation, wind speed and rainfall) into the maintenance planning of PV systems. The need for a robust model that can accurately predict and manage the degradation of PV panels under various environmental conditions is crucial for optimizing their performance and longevity.

The objectives of this paper are to construct a PV degradation model and propose a CBM policy that takes into account environmental conditions. The ultimate goal of this paper is to determine the optimal cleaning frequency to improve energy efficiency and minimize the long-term average cost rate of the maintenance (cleaning) policy.

The scope of this paper encompasses the development of a stochastic process model to incorporate the impacts of dust and temperature on PV panel degradation. It also includes the influence of environmental covariates such as wind speed and rainfall, aiming to create a more dynamic and effective maintenance policy for PV farms.

The contribution of this paper is the development of a novel stochastic process model that integrates the effects of dust and temperature on PV panel degradation while incorporating wind speed and rainfall as covariates. This approach provides a comprehensive understanding of how environmental factors influence degradation and proposes a condition-based maintenance (CBM) policy to minimize the long-term average cost through non-periodic inspections, thereby enhancing the efficiency and reliability of PV systems. The gain achieved by non-periodic inspections compared to periodic inspections ranges from 3.83% to 9.37%. The study provides guidelines for PV maintenance tailored to various environmental conditions.

2. Degradation modeling

2.1. System description

We consider a PV panel to be a single-component system, starting its operation in a new state. The PV panel experiences two phenomena that decrease power production efficiency: dust accumulation and an increase in inner temperature. These two factors are influenced by the surrounding environment, particularly wind speed and rainfall. For this study, we do not consider the impact of shadows from chimneys or other similar structures. When the efficiency of a PV panel drops to a certain level, referred to as “failure” in this paper, it should be properly cleaned. A failure threshold $L_c$ is defined to describe the state of failure, and the total decrease in efficiency at time t is represented by $Y(t)=D(t)+W(t)$, where $D(t)$ and $W(t)$ represent the decrease in efficiency due to dust accumulation and PV temperature, respectively. The independence and additivity of the efficiency reduction due to dust and temperature are supported by the findings of [55], [56]. Take the desert region in [9] for example, the outdoor temperatures typically range from 29 °C to 41 °C in summer and 7 °C to 20 °C in winter. Additionally, the dust accumulation density was observed to be 10.254 $g/m^2$.

2.2. Degradation model for dust accumulation

Dust accumulation on the surface of PV panels creates a physical barrier between the incoming sunlight and the semiconductor materials within the panels, diminishing the amount of sunlight that reaches the panels and consequently lowering power generation efficiency. Let us denote stochastic process $\{X(t),t\ge 0\}$ as the dust accumulation on the PV panel, with $X(0)=0$. Its increments over any time interval are non-negative, positive, and stochastically independent. Dust accumulation can be modeled as a discrete-time jump process, where successive jumps are influenced by factors such as wind speed, human activities, nearby construction, and other environmental events. By considering dust accumulation as a jump process, we capture the intermittent nature of dust deposition and the random fluctuations in its arrival rate.

2.2.1. Model for effect of dust accumulation

The dust accumulation process is considered incremental, with increases at random times. Let $T_j$ be the dust increase times. Let's consider that $\lambda (t)$ is the arrival rates of ${(T_j)}_{j\ge 0}$. The number $N(t)$ of dust accumulation increases up to time t can be written as:

$$N(t)=\sum _{i=1}^{\infty }{\mathbf{1}}_{\{T_i\le t\}}$$ (1)

and $\{N(t),t\ge 0\}$ is a non-homogeneous Poisson process (NHPP). Let $X_j$ be the increase of dust thickness at time $T_j$. It follows an exponential distribution with parameter μ. Therefore, the total dust accumulated at time t is defined by:

$$X(t)=\sum _{j=1}^{N(t)}{X}_j,t\ge 0.$$ (2)

Hence, the dust accumulation process on the PV panel is described by an NHCPP with the arrival rate function $\lambda (t)$ (more details in subsection 2.4.2) and jump sizes exponentially distributed with parameter μ. NHCPP allows the modeling of non-homogeneous intensity functions, which can be useful in situations where the dust accumulation rate is not constant over time. It assumes that the increments of the process (i.e., the amount of dust accumulated in a given time interval) are non-negative, positive, and stochastically independent. This implies that the amount of dust accumulated at any moment depends on the underlying random process and is not influenced by previous or future increments.

Considering results obtained in [57], the relationship between the reduction of PV panel power generation $D(t)$ and the amount of dust accumulation $X(t)$ is proposed as follows:

$$D(t)=\frac{-0.521}{1+{(\frac{X(t)}{12.653})}^{1.74}}+0.532.$$ (3)

2.2.2. Derivation for PDF of $D(t)$

Let $p(n,t)$ be the probability of having n jumps in the time interval $(0,t]$. It is given by:

$$p(n,t)=\mathbb{P}(N(t)=n)=\frac{{(\mathrm{\Lambda }(t))}^n}{n!}{e}^{-\mathrm{\Lambda }(t)},t\ge 0,n=0,1,\cdots$$ (4)

where

$$\mathrm{\Lambda }(t)=E[N(t)]=\underset{0}{\overset{t}{\int }}\lambda (x)dx.$$ (5)

The cumulative distribution function (CDF) of $X(t)$ is derived as follows.

$$F(x,t)=\mathbb{P}(X(t)\le x)=\sum _{n=0}^{\infty }\mathbb{P}(\sum _{j=1}^{N(t)}{X}_j\le x|N(t)=n)\cdot \mathbb{P}(N(t)=n)=\sum _{n=0}^{\infty }\mathbb{P}(\sum _{j=1}^n{X}_j\le x)\cdot p(n,t).$$ (6)

As $X_j\sim \exp (\mu )$, the total accumulated dust after n increases, denoted by $S_n={\sum }_{j=1}^n{X}_j$, follows a gamma distribution with shape parameter n and scale parameter μ. The corresponding probability distribution function is:

$$f_{S_n}(x)=\frac{{\mu }^n{x}^{n-1}{e}^{-\mu x}}{\gamma (n)},x\ge 0,$$ (7)

for $n=1,2,\cdots$, where γ is the Euler gamma function, and $\gamma (n)=(n-1)!$. The Probability density function (PDF) of $X(t)$ can be easily derived as follows:

$$f_{X(t)}(x)=\sum _{n=0}^{\infty }{f}_{S_n}(x)\cdot p(n,t).$$ (8)

Substituting equation (3) in equation (8) and by the transformation method, the PDF of $D(t)$ denoted by $f_{D(t)}$ is obtained as follows:

$$f_{D(t)}(y)=\sum _{n=0}^{\infty }{f}_{S_n}\left(12.653{(\frac{0.521}{0.532-y}-1)}^{\frac{1}{1.74}}\right)\cdot p(n,t)\cdot \frac{12.653\times 0.521}{1.74}{(\frac{0.521}{0.532-y}-1)}^{\frac{-0.74}{1.74}}\cdot {(0.532-y)}^{-2}.$$ (9)

2.3. Model for the effect of PV temperature

The surface temperature of a PV panel can significantly impact its efficiency. Under high temperatures, the electrical conductivity of PV materials weakens. This reduction in conductivity diminishes the current flow inside the PV panel, thereby decreasing the generation of electrical energy. Additionally, high temperatures can increase the internal resistance of the PV panel, further reducing the efficiency of electrical energy output.

The following subsections delve into three crucial aspects of PV temperature modeling. Firstly, the correlation between PV temperature and ambient temperature is investigated, aiming to model PV surface temperature indirectly. Secondly, a comprehensive model of ambient temperature is constructed that considers the impact of wind and rain conditions. Finally, by discerning the linear correlation between ambient temperature and PV panel surface temperature, along with the ambient temperature model, a model is developed demonstrating how PV panel temperature influences their efficiency.

2.3.1. Correlation of PV temperature and ambient temperature

While dust accumulation on the panel surface may affect its performance and efficiency, its effect on the panel surface temperature is typically negligible. The primary factor influencing the panel temperature is the ambient temperature. When the ambient temperature is high, the panel temperature also increases, and vice versa. A linear correlation between solar cell temperature and air temperature is proposed in [58], with the linear correlation parameter found to be 0.88. Our hypothesis and model aim to simplify the analysis by primarily considering ambient temperature, particularly under conditions where solar radiation intensity is either constant or varies minimally.

2.3.2. Model for ambient temperature

Let $T_{\text{ambient}}(t)$ describe the actual ambient temperature at any given point in time t. The evolution of the ambient temperature state is modeled with a non-homogeneous Markov chain $\{Z_{temp}^t,t\ge 0\}$. The relationship between $T_{\text{ambient}}(t)$ and $Z_{temp}^t$ be understood as follows: $T_{\text{ambient}}(t)$ provides a precise measurement of temperature at time t, while $Z_{temp}^t$ essentially categorizes these continuous temperature measurements into discrete states or intervals. Instead of dealing with diverse possible temperatures, $Z_{temp}^t$ handles a limited number of temperature states, where each state represents a range of temperatures. For example, all temperatures $T_{\text{ambient}}(t)$ between 20 °C and 25 °C might be categorized into one state.

The state space of $\{Z_{temp}^t,t\ge 0\}$ is given by $S_{temp}=\{1,2,\cdots ,M\}$, and the transition matrix conditionally depends on wind speed and rainfall states. Strong rainfall and wind can cool down the ambient temperature. Thus, in the Markov chain model of temperature, the transition matrix of temperature states is influenced by the states of wind and rain. More details about the influences of wind and rain on ambient temperature changes can be found in subsection 2.4.3.

2.3.3. Model for the effect of PV temperature

We can examine the average effect of temperature on PV panels over time as temperature consistently changes. For instance, during summer when the temperature is higher and there is no wind or rain, we need to clean PV panels more often. Let $W_k$ represent the mean efficiency reduction due to temperature on the day k of the PV panel power generation. Inspired by [59], $W_k$ is modeled as:

$$W_k=\frac{1}{t_k}\underset{0}{\overset{t_k}{\int }}0.4\text{\%}{(T_{panel}(u)-25)}^{+}du=\frac{1}{t_k}\underset{0}{\overset{t_k}{\int }}0.4\text{\%}{(0.88⁎T_{ambient}(u)-25)}^{+}du,$$ (10)

where $t_k$ represents the daylight duration of day k. We consider e.g. 0 as the sunrise time and $t_k$ as the sunset time for the day k. In equation (10), ${(g(u))}^{+}$ denotes the positive part of $g(u)$. Given the rapid variation in temperature over a whole day, we utilize the average temperature to represent the temperature on a given day. Specifically, only PV panel surface temperatures above 25 degrees Celsius affect PV panel efficiency.

For simplicity's sake, we consider hereafter that the ambient temperature is constant throughout the day and equal to its mean value. Hence for day k we replace $T_{ambient}(t)$ by

$$\overline{T_{ambient}^{(k)}}=\frac{1}{t_k}\underset{0}{\overset{t_k}{\int }}{T}_{ambient}(u)du.$$ (11)

As a consequence of integrating Eq. (10) and Eq. (11), $W_k$ can be expressed as:

$$W_k=0.4\text{\%}{(\overline{T_{panel}^{(k)}}-25)}^{+}=0.4\text{\%}⁎{(0.88⁎\overline{T_{ambient}^{(k)}}-25)}^{+}.$$ (12)

Various bins of ambient temperature can be defined, with each bin representing a specific range of temperatures. For each bin, the mean daily temperature is considered as a random variable $Z_{temp}^k$ with a given distribution. For illustration purposes, the distribution of temperature in each bin is assumed to be uniform. Each bin defines a specific temperature state, and the state's transition is modeled as a Markov chain.

2.3.4. Derivation for CDF of $W_k$

For the temperature state $Z_{temp}^k=i$, let $f_i(\cdot )$ denote the corresponding uniform distribution. The CDF of $W_k$ is denoted by $G(k,x)$ and can be obtained as follows for $x>25$:

$$G(k,x)=\mathbb{P}(W_k\le x)=\sum _{i=1}^M\mathbb{P}(W_k\le x|Z_{temp}^k=i)\cdot \mathbb{P}(Z_{temp}^k=i)=\sum _{i=1}^M(\underset{25}{\overset{x}{\int }}{f}_i(u)du)\cdot \mathbb{P}(Z_{temp}^k=i)\cdot$$ (13)

Over a longer period, the Markov process gradually converges to a steady state and is independent of the initial state. Considering the stationary state,

$$W=\underset{k\to \infty }{\lim }{W}_k,$$ (14)

$$\underset{k\to \infty }{\lim }G(k,x)=\mathbb{P}(W\le x)=\sum _{i=1}^M(\underset{25}{\overset{x}{\int }}{f}_i(u)du)\cdot \mathbb{P}(Z_{temp}=i),$$ (15)

where $\mathbb{P}(Z_{temp}^t=i)={\sum }_{j=1}^m\mathbb{P}(Z_{temp}^t=i|Z_t=j)\cdot \mathbb{P}(Z_t=j)$. Please note that $Z_t$ represents the state of covariates. This will be further explained in subsection 2.4.3.

2.4. Environmental condition modeling

2.4.1. Model for wind and rain as covariates

We partition the wind speed and rainfall domain into m subsets. Let $Z_t$ be the random variable associated with the subset of the operating environment (wind speed and rainfall) for the panel at t. To model the covariates, which are time-related random variables representing environmental conditions, we propose a time-continuous Markov chain $\{Z_t,t\ge 0\}$.

The state space of the covariates is constructed and denoted as $S=\{1,2,\cdots ,m\},m=s^2$. It describes the different states of the couple wind speed and rainfall, where there are s different wind speed values and s different rainfall values. For instance, $Z_t=j$ represents the event that $j=(w,r)$ i.e. that the wind speed value is w and the rainfall value is r, $1\le w\le s,1\le r\le s$.

2.4.2. Environmental influence on dust accumulation

The failure of the system is attributed to the accumulation of dust, which is influenced by environmental factors such as wind speed and rainfall. Dust deposition decreases when the wind speed increases, resulting in a higher power output and vice versa [60]. To characterize the impact of the environment on the degradation process caused by dust accumulation, we develop a model inspired by the Cox proportional hazards model, as discussed in [61]. Specifically, the dust accumulation process is described by the NHCPP with the arrival rate function $\lambda (t)=h(t,Z_t),(t>0,Z_t\in S)$ depending on the covariates as follows:

$$\lambda (t)=\alpha e^{-{\sum }_{j=1}^m{\beta }_j{\mathbb{I}}_{Z_t=j}},$$ (16)

where α is the so-called baseline hazard rate and β_j is the regression coefficient associated to the covariate state j. The generator matrix Q associated to $\{Z_t,t\ge 0\}$ is as follows:

$$Q=\left(\begin{array}{cccc}\hfill q_{11}\hfill & \hfill q_{12}\hfill & \hfill \cdots \hfill & \hfill q_{1m}\hfill \\ \hfill q_{21}\hfill & \hfill q_{22}\hfill & \hfill \cdots \hfill & \hfill q_{2m}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill q_{m1}\hfill & \hfill q_{m2}\hfill & \hfill \cdots \hfill & \hfill q_{mm}\hfill \end{array}\right),$$ (17)

where $q_{ij}$ represents the transition rate from state i to j. The rate at which the process leaves state i, denoted as $q_i$, can be calculated using $q_i=-q_{ii}={\sum }_{j=1,i\ne j}^m{q}_{ij},q_{ij}>0$. Now, consider $t_{ij}$, which is the duration the process stays in state i before transitioning to state j. The distribution of $t_{ij}$ follows an exponential pattern with a rate of $q_{ii}$. The transition matrix $P_t$ is calculated using the formula:

$$P_t(i,j)=e^{tQ}(i,j)=I+\sum _{k=1}^{\infty }{Q}^k{t}^k/k!,$$ (18)

here, $P_t(i,j)$ is defined as:

$$P_t(i,j)=\mathbb{P}\{Z_{t+s}=j|Z_s=i\},s>0.$$ (19)

In Fig. 1, an example of the efficiency reduction due to dust and covariates is illustrated.

Figure 1.

Efficiency reduction process due to dust: α = 6,β_j = (0.1,0.2,0.3,0.4),μ = 2,m = 4.

2.4.3. Environmental influence on temperature

The temperature of PV panels is significantly influenced by environmental temperature, wind, and rain. As the surrounding temperature rises, the temperature of the PV panels typically increases correspondingly. Additionally, when wind blows over the surface of the panels, it enhances thermal transfer between the panels and the ambient air, facilitating conduction and convective heat dissipation. This airflow aids in reducing the temperature of the panels [62], [63]. Moreover, when rainwater covers the surface of the panels, evaporation occurs, absorbing heat from the panels and contributing to cooling [64]. A multivariate Markov chain is employed to model the interactions and dependencies among environmental temperature, wind, and rain, defining the states of panel surface temperature as different combinations of these variables. Let $P_{temp}^{(w,r)}$ be the transition matrix associated with the ambient temperature model when the wind speed is in state w and rainfall is in state r. The bidimensional state space of the covariates (wind and rain) is constructed and transformed into a one-dimensional space denoted as $S=\{1,2,\cdots ,m\}$, with $m=s^2$. The state transition matrices of the Markov chain of ambient temperature differ for different covariate states.

2.4.4. Example

As an example, consider three possible temperature states: cold ($T_{ambient}(t)\le 25$ °C), moderate ($25<T_{ambient}(t)\le 35$ °C), and hot ($35<T_{ambient}(t)\le 50$ °C), along with two possible rain states: no rain (N) and rain (R), and two possible wind states: calm (C) and windy (W). Then, the covariate states can be expressed as a $4\times 4$ matrix with $S=\{\{\text{“}N\text{”},\text{“}C\text{”}\},\{\text{“}N\text{”},\text{“}W\text{”}\},\{\text{“}R\text{”},\text{“}C\text{”}\},\{\text{“}R\text{”},\text{“}W\text{”}\}\}$. This can be represented by a state space $S=\{1,2,3,4\}$.

To estimate the transition matrix for rain and wind conditions $P_t$, we use data from [68]. Firstly, determine the sequence of states over time, count the transitions between each state, and calculate the probabilities of transitioning from one state to another. The transition matrix for rain and wind conditions $P_t$ can be obtained as:

$$P_t=\left(\begin{array}{cccc}\hfill 0.6\hfill & \hfill 0.3\hfill & \hfill 0.1\hfill & \hfill 0\hfill \\ \hfill 0.3\hfill & \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0.2\hfill \\ \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0.5\hfill & \hfill 0.2\hfill \\ \hfill 0\hfill & \hfill 0.4\hfill & \hfill 0.4\hfill & \hfill 0.2\hfill \end{array}\right),$$ (20)

For covariates initially in the state $Z_0=1$, we represent the probability distribution as ${\pi }^n=({\pi }_1^n,{\pi }_2^n,{\pi }_3^n,{\pi }_4^n)$, where ${\pi }_i^n=\mathbb{P}(Z_t=i|Z_0=1),(i=1,2,3,4)$. This distribution can be obtained by ${\pi }^n=(1,0,0,0)P_t^n$. The stationary distribution π is given by $\pi =(0.375,0.325,0.175,0.125)$.

Conditional transition matrices (the time horizon is day) for the temperature states, given rain and wind conditions, are defined as follows:

$$P_{temp}^{(1)}=\left(\begin{array}{ccc}\hfill 0.2\hfill & \hfill 0.5\hfill & \hfill 0.3\hfill \\ \hfill 0.3\hfill & \hfill 0.4\hfill & \hfill 0.3\hfill \\ \hfill 0.4\hfill & \hfill 0.4\hfill & \hfill 0.2\hfill \end{array}\right),P_{temp}^{(2)}=\left(\begin{array}{ccc}\hfill 0.5\hfill & \hfill 0.4\hfill & \hfill 0.1\hfill \\ \hfill 0.3\hfill & \hfill 0.5\hfill & \hfill 0.2\hfill \\ \hfill 0.4\hfill & \hfill 0.4\hfill & \hfill 0.2\hfill \end{array}\right),$$

$$P_{temp}^{(3)}=\left(\begin{array}{ccc}\hfill 0.3\hfill & \hfill 0.4\hfill & \hfill 0.3\hfill \\ \hfill 0.4\hfill & \hfill 0.3\hfill & \hfill 0.3\hfill \\ \hfill 0.5\hfill & \hfill 0.3\hfill & \hfill 0.2\hfill \end{array}\right),P_{temp}^{(4)}=\left(\begin{array}{ccc}\hfill 0.4\hfill & \hfill 0.4\hfill & \hfill 0.2\hfill \\ \hfill 0.4\hfill & \hfill 0.3\hfill & \hfill 0.3\hfill \\ \hfill 0.5\hfill & \hfill 0.3\hfill & \hfill 0.2\hfill \end{array}\right).$$

In Fig. 2, the multivariate Markov chain model illustrating the variation of temperature, rain, and wind is presented. This model captures the effects of the rain and wind conditions on the temperature and allows us to make predictions about future temperature behavior given the current weather conditions. In this paper, the effect of temperature on PV panels is not continuously increasing and is not permanent but temporary. As shown in the first subplot, the efficiency reduction evolution due to temperature is not increasing but fluctuating.

Figure 2.

Multivariate Markov chain model.

2.5. Degradation process increments

Let $D(t,Z_t)$ and $W(t,Z_t)$ be the efficiency loss process of the PV panel power generation due to dust accumulation and PV temperature, respectively, given that the covariate state $Z_t$. An example trajectory of the total efficiency reduction process is shown in Fig. 3.

Figure 3.

A trajectory of total degradation process with the covariates evolution.

Let $Y(t,Z_t)$ be the cumulative decrease in efficiency until time t, given the covariate state $Z_t$. This represents the combined impact on efficiency loss, resulting from both dust accumulation and PV temperature:

$$Y(t,Z_t)=D(t,Z_t)+W(t,Z_t),t\ge 0.$$ (21)

The formal conditional CDF of the total degradation process for a covariate state can be expressed:

$$F_{Y(t,Z_t)}(x)=\mathbb{P}(D(t,Z_t)+W(t,Z_t)\le x)=\underset{0}{\overset{x}{\int }}\underset{0}{\overset{y}{\int }}{f}_{D(t,Z_t)}(y-u)g_{W(t,Z_t)}(u)dudy,$$ (22)

where $f_{D(t,Z_t)}(x)$ and $g_{W(t,Z_t)}(x)$ are respectively the PDF of the $D(t,Z_t)$ and $W(t,Z_t)$ for a given $Z_t$ (see Appendix B).

Let be ${\tau }_i$ the ith jump time of the Markov chain $\{Z_t,t\ge 0\}$ after time t, Let be $F_{Y(t+\mathrm{\Delta }t,Z_t))-Y(t,Z_t))}^{Z_t}$ the CDF of increment $Y(t+\mathrm{\Delta }t,Z_{t+\mathrm{\Delta }t}))-Y(t,Z_t))$. The calculation is developed in (see Appendix C).

$$F_{Y(t+\mathrm{\Delta }t,Z_{t+\mathrm{\Delta }t})-Y(t,Z_t)}^{Z_t}=\sum _{k=0}^{\infty }\sum _{Z_1}\cdots \sum _{Z_k}\mathbb{P}(Y(t+\mathrm{\Delta }t,Z_{t+\mathrm{\Delta }t})-Y(t,Z_t)<x|Z_t=1,E_k^{\mathrm{\Delta }t},Z_{{\tau }_1}=z_1,\cdots ,Z_{{\tau }_k}=z_k)\times \mathbb{P}(Z_{{\tau }_1}=Z_1|Z_t=1)\cdots \mathbb{P}(Z_{{\tau }_1}=z_k|Z_{{\tau }_{k-1}}=z_{k-1})\mathbb{P}(E_k^{\mathrm{\Delta }t}),$$ (23)

where $E_k^{\mathrm{\Delta }t}$ represents the occurrence of k transitions in the covariate over the time interval Δt.

3. Maintenance policy

3.1. Maintenance assumption

• •Dust and high temperatures can reduce a PV panel's efficiency over time. Online monitoring can track gradual performance loss, but sudden failures may remain undetected. Inspection technology typically involves physical examination and diagnostic tools, providing detailed information on panel conditions such as visible damage, electrical output, and thermal characteristics. Inspection is necessary to ensure timely maintenance, costing $C_i$ per unit.
• •When the efficiency loss crosses the failure threshold $L_c$, the panel enters a “failure” state, necessitating corrective action costing $C_c$. The panel is not physically “failed”, but its power generation is too inefficient for continuous operation. We won't replace the “failed” panels but implement cleaning operations.
• •When the efficiency loss exceeds the over-degraded threshold $L_p$ but remains below the failure threshold $L_c$, the system is in a “worn-out” state, necessitating preventive action costing $C_p$.
• •Since corrective maintenance cannot be planned, it is more complex to implement and thus more costly than preventive maintenance. Thus, it is assumed that $C_p<C_c$. The corrective cost, $C_c$, includes all costs associated with system failure, excluding the unavailability cost. The unavailability cost, denoted as $C_d$ per time unit, is incurred when the system fails without maintenance.
• •Since we regard efficiency loss due to dust accumulation as panel degradation, a cleaning policy improving power generation efficiency is considered a maintenance policy.
• •Corrective and preventive actions both involve cleaning the PV panels. The only difference is the amount of dust present at the time of implementation.
• •Inspection and maintenance actions are instantaneous. After maintenance, the panel is as good as new, and maintenance can only be performed during inspections.
• •Maintenance actions do not affect the covariate state.

3.2. Inspection policy

In this subsection, we explore two different inspection policies to reveal the system state: periodic and non-periodic inspection policies. During the inspection and cleaning process, incidents such as impacts on the PV panels, accidental stepping on them, or other types of damage are not considered in this paper.

3.2.1. Periodic inspection policy

Under the periodic inspection policy, the system undergoes regular inspections at predefined intervals. These intervals are denoted as ${(t_n)}_{n\in \mathbb{N}}$, where $t_n=n\tau$, $\tau \in {\mathbb{N}}^{⁎}$, $n\in \mathbb{N}$. Here, τ plays a crucial role as a variable in formulating the maintenance policy.

3.2.2. Non-periodic inspection policy

Non-periodic inspections are based on specific conditions or events, rather than a fixed schedule. The amount of dust on the PV panel can vary due to factors such as weather and season [65], [66]. Cleaning on a fixed schedule can result in unnecessary or delayed cleaning. Inspired by the inspection rules of [67], [30], we suppose that the system undergoes non-periodic inspections at given times ${(t_n)}_{n\in \mathbb{N}}$. At each inspection time, two decisions have to be made:

• •whether the panels should be cleaned preventively or correctively, or no maintenance actions are needed.
• •defines when to implement the next inspection.

The inspection scheduling function determines when and how inspections should be scheduled based on these conditions or events. The following inspection scheduling function is used in the non-periodic inspection policy:

$$m(x)=1+\max \{A-(A/B)⁎x;0\},$$ (24)

where $m(\cdot )$ represents a function that decreases from $[0,L_p)$ to $[m_{min},m_{max}]$ with $m_{min}=m(L_p)>0$ and $m_{max}=m(0)=1+A>m_{min}>0$. Parameters A and B are decision variables and determine the shape of the function. Fig. 4 gives an example of the inspection evolution process of the maintained system.

Figure 4.

Inspection evolution of the maintained system: L_c = 0.6;L_p = 0.5;A = 1.4,B = 2;⁎:inspection.

The length of inspection interval is related to the degree of degradation, and the next inspection time is then:

$$t_n=t_{n-1}+m(Y(t_{n-1})),$$ (25)

where $Y(t_{n-1})$ describes the system degradation level after the maintenance implementation at time $t_{n-1}$.

3.3. Evaluation of the maintenance policy

The cost of maintenance is determined by the inspection time function, denoted as $m(\cdot )$, and the preventive threshold, represented by $L_p$.

• •
  When $Y(t_n,Z(t_n))\ge L_c$ and $Y(t_{n-1},Z(t_{n-1}))<L_c$, and the first failure time since the last corrective maintenance occurs at time $t_f$ between $t_{n-1}$ and $t_n$, corrective maintenance is implemented with maintenance cost $C_c$, inspection cost $C_i$ and unavailability cost $C_d\cdot d(t)$. $d(t)$ is the time that the system stayed in a failure state in $(0,t)$.

  $$\begin{array}{cc}\hfill & C_{\text{corrective}}(t_f)=C_c+C_i+d(t)\cdot C_d,\hfill \\ \hfill & d(t)=\underset{0}{\overset{t}{\int }}{\mathbf{1}}_{\{Y(s,Z(s))\ge L\}}ds,\hfill \\ \hfill & t_f=\underset{t}{\inf }\{Y(t_n,Z(t_n))\ge L_c,t\ge 0\}.\hfill \end{array}$$ (26)
• •
  When $L_p\le Y(t_n,Z(t_n))<L_c$, preventive maintenance is implemented with action cost $C_p$ and inspection cost $C_i$.

  $$C_{\text{preventive}}=C_p+C_i.$$ (27)
• •
  When $0<Y(t_n,Z(t_n))<L_p$, no maintenance action is implemented and only inspection cost $C_i$ is incurred.

  $$C_{no}=C_i.$$ (28)

3.3.1. The cumulative maintenance cost $C(t)$

Let's define $N_i(t)$, $N_p(t)$, and $N_c(t)$ as the cumulative counts of inspections, preventive actions, and corrective actions, respectively, up to time t. Similarly, let $d(t)$ represent the accumulated downtime up to the same point. With this setup, the total maintenance cost $C(t)$ up to time t is calculated as follows:

$$\begin{array}{cc}\hfill C(t)& =N_i(t)\cdot C_i+N_p(t)\cdot C_p+N_c(t)\cdot C_c+d(t)\cdot C_d.\hfill \end{array}$$ (29)

The expected average cost per time unit, calculated across a long period, can be established:

$$\mathbb{E}{C}_{\infty }=\underset{t\to \infty }{\lim }\frac{\mathbb{E}(C(t))}{t}.$$ (30)

In this maintenance model, decisions are made based on the information from scheduled inspections. The state of the system at any given inspection point is influenced only by its state during the last inspection. This means that the degree of deterioration and the state of any covariates at time $t_n$ are determined exclusively by their levels at $t_{n-1}$.

3.3.2. Maintenance cost between two inspections

Calculating the expected maintenance cost in between two inspection instances $t_n$ and $t_{n-1}$ within a time interval T, which is the time between those two inspections, and then extrapolating that to a long time horizon, can also be a useful approach to analyzing and predicting long-term maintenance costs. The total cost during interval T can be expressed as $C(T)$:

$$\begin{array}{cc}\hfill C(T)& =N_i(T)\cdot C_i+N_p(T)\cdot C_p+N_c(T)\cdot C_c+d(T)\cdot C_d,\hfill \end{array}$$ (31)

where $N_p(T),N_c(T)$ represent the cumulative counts of preventive and corrective actions, respectively, taken during interval T. $d(T)$ the downtime during interval T. The expected average cost per unit of time, specifically between two inspections, would be the expected cost during the time interval T divided by the expected value of $\mathbb{E}(C(T))$:

$$\mathbb{E}{C}_T=\frac{\mathbb{E}(C(T))}{E(T)}.$$ (32)

3.4. Optimization of the maintenance policy

For periodic inspection policy, we aim to establish a decision-making rule that reduces the expected average maintenance costs over an extended period. Under the maintenance policy being considered, the optimal values for the preventive maintenance threshold $L_p^{⁎}$, and inspection interval ${\tau }^{⁎}$, are determined in the following way:

$$(L_p^{⁎},{\tau }^{⁎})=arg\underset{(L_p,\tau )}{\min }\mathbb{E}{C}_{\infty }(Z).$$ (33)

For non-periodic inspection policy, the optimization goal is to determine the optimal value of $L_p^{⁎}$ and the optimal inspection parameters $A^{⁎}$ and $B^{⁎}$ that minimize the expected average maintenance cost over a long period.

$$(L_p^{⁎},A^{⁎},B^{⁎})=arg\underset{(L_p,A,B)}{\min }\mathbb{E}{C}_{\infty }(Z).$$ (34)

Taking the periodic checking policy as an example, the optimization process can be shown in the following pseudo-code of Algorithm 1.

Algorithm 1.

Pseudo-code.

4. Results and discussion

A numerical implementation is presented to illustrate the performance of the proposed model. To begin, the main goal is to examine the impact of the unit cost of the different maintenance operations on the preventive maintenance threshold $L_p$ and the optimal inspection function. In the simulation program, the initial parameters set are as follows: $L_c=0.65,\mu =1,\alpha =2,{\beta }_j=(0.2,1,2,4)$. For temperature parameters setting, we consider the example in subsector 2.4.4, $Z_{temp}$ is a 3-state Markov chain. When $Z_{temp}^t=1$, $T_{ambient}(t)\sim U(0,25),g_1(\cdot )=\frac{1}{25}$. When $Z_{temp}^t=2$, $T_{ambient}(t)\sim U(25,35),g_2(\cdot )=\frac{1}{10}$. When $Z_{temp}^t=3$, $T_{ambient}(t)\sim U(35,50),g_3(\cdot )=\frac{1}{15}$. Conditional transition matrices for the temperature states, given rain and wind conditions, are defined as the example in subsector 2.4.4.

We need to identify all the covariate states. As the example in subsector 2.4.4, the covariate space has initial state $Z_0=1$ with state space $S=\{1,2,3,4\}$. Here the transition rate matrix is set as

$$Q=\left(\begin{array}{cccc}\hfill -3/8\hfill & \hfill 1/8\hfill & \hfill 1/8\hfill & \hfill 1/8\hfill \\ \hfill 1/8\hfill & \hfill -3/8\hfill & \hfill 1/8\hfill & \hfill 1/8\hfill \\ \hfill 1/8\hfill & \hfill 1/8\hfill & \hfill -3/8\hfill & \hfill 1/8\hfill \\ \hfill 1/8\hfill & \hfill 1/8\hfill & \hfill 1/8\hfill & \hfill -3/8\hfill \end{array}\right),$$

and hence the stationary distribution $\pi =({\pi }_1,{\pi }_2,{\pi }_3,{\pi }_4)=(0.25,0.25,0.25,0.25)$. The different values of the covariate represent different states of the wind and rain. The higher the wind speed, the lower the rate of dust accumulation. Small rain brings dust to the PV panel, while strong rain can remove the dust on the panel's surface. In effect, these environmental conditions can be considered as the working conditions that affect the system state (its failure rate).

4.1. Illustrative examples: cost calculation by Eq. (30) and Eq. (31)

As an illustrative example, we assume that $\tau =1,C_i=20,C_p=60,C_c=100,C_d=250$ to optimize preventive maintenance threshold $L{p}^{⁎}$ in periodic inspection maintenance policy. It is seen from Fig. 5 and Fig. 6 that under both cases, the calculations from the simulations demonstrate convergence when the number of simulations reaches 40000, suggesting that the results obtained using the two methods are in agreement.

Figure 5.

Optimal expected cost evolution (Eq. (30)).

Figure 6.

Optimal expected cost evolution (Eq. (31)).

4.2. Influence of unit maintenance costs

Typically, PV panels are very expensive. When a PV system is not working, it means it is not functioning as intended, which can have a significant impact on operations and production loss. Here, we consider $C_d=250$. Four cases regarding unit maintenance costs are presented:

• •case I (baseline costs)): $C_i=20,C_p=60,C_c=100,C_d=250$;
• •case II (high preventive costs): $C_i=20,C_p=100,C_c=100,C_d=250$;
• •case III (high inspection costs): $C_i=100,C_p=60,C_c=100,C_d=250$;
• •case IV (low unavailability costs): $C_i=20,C_p=60,C_c=100,C_d=100$.

To analyze the impact of maintenance unit cost on maintenance decisions, we analyze the following points:

• •Periodic inspection policy: determining $L_p^{⁎}$ and ${\tau }^{⁎}$, to minimize the expected long-term average maintenance cost, considering that covariate Z behaves as a general Markov chain.
• •Non-periodic inspection policy: determining $L_p^{⁎}$, $A^{⁎}$, and $B^{⁎}$, to minimize the expected long-term average maintenance cost, considering that covariate Z behaves as a general Markov chain.
• •Evaluating the differences in results between the periodic and non-periodic inspection policy under the aforementioned conditions.

To summarize the simulation results, the $L_p^{⁎}$ and τ⁎ under periodic inspection policy are summarized in the Table 2, Table 4 and Fig. 7. $L_p^{⁎}$ and $A^{⁎}$ and $B^{⁎}$ under non-periodic inspection policy are summarized in the Table 3, Table 5. The analysis is summarized as follows:

• •In Case III, the high costs associated with inspections make it the most costly case.
• •Case IV is characterized by low unavailability costs and is always the most economical option, closely related to the baseline case.
• •For Case II, the peak value of the optimal preventive maintenance threshold arises from the high costs linked to preventive actions.
• •The optimal inspection intervals are extended in Cases III and IV. This is attributed to the high costs of inspections in Case III and the low unavailability costs in Case IV.

Table 2.

Periodic inspection policy (Eq. (30)).

Case	$(L_p^{⁎}(\text{\%}),\tau ⁎(month),E{C}_{\infty }^{⁎})$
I	(29, 1.4, 59.28)
II	(36, 1.5, 75.58)
III	(17, 1.7, 106.10)
IV	(28, 1.9, 53.49)

Table 4.

Periodic inspection policy (Eq. (31)).

Case	$(L_p^{⁎}(\text{\%}),\tau ⁎(month),E{C}_{\infty }^{⁎})$
I	(29, 1.4, 59.24)
II	(37, 1.5, 77.25)
III	(17, 1.9, 105.2)
IV	(26, 2, 53.47)

Figure 7.

Expected average cost as a function of L_p and τ (periodic inspection policy).

Table 3.

Non-periodic inspection policy (Eq. (30)).

Case	$(L_p^{⁎}(\text{\%}),A^{⁎},B^{⁎},E{C}_{\infty }^{⁎})$
I	(39, 0.5, 0.44, 51.61)
II	(41, 0.6, 0.7, 64.79)
III	(31, 0.9, 0.57, 95.22)
IV	(38, 0.9, 0.33, 47.56)

Table 5.

Non-periodic inspection policy (Eq. (31)).

Case	$(L_p^{⁎}(\text{\%}),A^{⁎},B^{⁎},E{C}_{\infty }^{⁎})$
I	(35, 0.5, 0.4, 52.24)
II	(42, 0.6, 0.29, 64.90)
III	(30, 0.9, 0.16, 96.09)
IV	(34, 1, 0.09, 48.61)

4.3. Comparison of dynamic and constant environments

Environmental factors can affect maintenance decision-making. To understand this impact, we can modify environmental variables and assess their effects on maintenance policies. This helps optimize maintenance policies for different environmental scenarios. According to Eq. (16), the arrival rate $\lambda (t)$ of the NHCPP depends on the parameter β and parameter β is the regression parameter of covariate state $Z_t$. The parameter β serves as a measure of the impact that a dynamic environment has on a system's degradation. As β represents the effect of environmental dynamics, it follows that the expected average maintenance cost in the most challenging environmental conditions would exceed that in more favorable settings. In this purpose, we set β =[0.2, 1, 2, 4] for four covariate states, that is, Z=1: β =0.2; Z=2: β =1; Z=3: β =2; Z=4: β =4.

In order to analyze the impact of maintenance unit cost on maintenance decisions, we analyze the following points:

• •Determining the minimum maintenance cost in the case of the covariate Z as a general Markov chain.
• •Calculating the minimum maintenance cost in the case of the covariate Z set at a constant value of $Z=i$, where i ranges from 1 to 4.
• •
  Computing a weighted average of the minimum costs for each $Z=i$ (with i being 1, 2, 3, or 4), using the steady-state probabilities as weighting factors:

  $$\bar{C}=\sum _{i=1}^{4}E{C}_{\infty }^{⁎}(i){\pi }_i,E{C}_{\infty }^{⁎}(i)=\min E{C}_{\infty }(Z=i).$$ (35)

Considering the effect of wind and rain on dust washing and temperature reduction, according to the failure rate function (16), higher β means slower dust accumulation, lower PV temperature and time to failure is longer, then longer inspection interval τ (see Table 6, Table 7, fixed $L_p=40\text{\%}$). The results show that when the system stays in the riskiest environment, marked as $Z=1$, it's recommended to schedule maintenance inspections more frequently than in lower-risk situations $(Z=2,3,4)$. In any case, the non-periodic inspection policy is more economical and efficient compared with the periodic inspection policy.

Table 6.

Periodic inspection policy.

Case	I ($\tau ⁎,E{C}_{\infty }^{⁎}$)	II ($\tau ⁎,E{C}_{\infty }^{⁎}$)	III ($\tau ⁎,E{C}_{\infty }^{⁎}$)	IV ($\tau ⁎,E{C}_{\infty }^{⁎}$)
Z	(1.0, 66.11)	(1.4, 79.34)	(2.2, 116.23)	(2.0, 55.99)
Z=1	(0.9, 67.39)	(1.3, 79.85)	(2.1, 117.96)	(1.5, 57.74)
Z=2	(1.0, 66.59)	(1.3, 79.4)	(2.2, 117.04)	(1.7, 56.28)
Z=3	(1.0, 65.43)	(1.4, 78.63)	(2.4, 115.24)	(2.0, 55.26)
Z=4	(1.3, 64.75)	(1.5, 77.56)	(2.4, 114.02)	(2.1, 54.24)
$\bar{C}$	60.04	78.86	116.07	55.88

Table 7.

Non-periodic inspection policy.

Case	I ($A^{⁎},B^{⁎},E{C}_{\infty }^{⁎}$)	II ($A^{⁎},B^{⁎},E{C}_{\infty }^{⁎}$)	III ($A^{⁎},B^{⁎},E{C}_{\infty }^{⁎}$)	IV ($A^{⁎},B^{⁎},E{C}_{\infty }^{⁎}$)
Z	(0.5, 0.26, 57.97)	(0.7, 0.08, 71.87)	(1.0, 0.07, 107.98)	(0.9, 0.2, 52.26)
Z=1	(0.4, 0.16, 58.81)	(0.5, 0.26, 73.13)	(0.9, 0.15, 108.99)	(0.8, 0.29, 53.62)
Z=2	(0.5, 0.15, 57.98)	(0.6, 0.28, 72.94)	(1.0, 0.15, 108.66)	(0.9, 0.11, 52.27)
Z=3	(0.5, 0.42, 57.41)	(0.6, 0.19, 70.37)	(1.0, 0.23, 107.05)	(1.0, 0.22, 52.04)
Z=4	(0.6, 0.14, 56.76)	( 0.7, 0.31, 69.45)	(1.1, 0.63, 106.69)	(1.1, 0.05, 50.79)
$\bar{C}$	57.74	71.47	107.85	51.93
Gain earned by non-periodic inspection (in percentage of periodic inspection)	3.83%	9.37%	7.08%	7.07%

5. Conclusion

In conclusion, this paper presents a mathematical model for optimizing maintenance policies for PV systems. The model focuses on the impact of environmental factors such as dust accumulation, increased surface temperature, wind speed, and rainfall on the efficiency of PV panels. The proposed maintenance policies combine preventive and corrective cleaning to minimize long-term maintenance costs. The model offers practical tools for renewable energy management by optimizing cleaning schedules and reducing maintenance costs.

In this work, we establish a theoretical framework for analyzing the cleaning costs of a single photovoltaic (PV) panel. This framework assumes that the cleaning price per panel remains constant regardless of the total number of panels and that the inspection cost is independent of panel quantity. This foundation serves as a theoretical foundation for future research, where we will extend the model to encompass a solar farm, a system comprised of numerous PV panels. While our research provides a solid theoretical foundation for single-panel maintenance, it is not without limitations. Future research plans to extend this methodology to solar farms as multi-component systems and extend this work by exploring additional environmental variables, deploying the model across diverse geographical locations, or integrating advanced predictive analytics for more adaptive maintenance planning. Moreover, the impact of the PV tilt angle on dirt intensity and automated PV cleaning will be a focal point of future investigations.

Furthermore, we aim to investigate the economic trade-offs between different maintenance strategies under varying environmental conditions. Another area of interest is the incorporation of real-time monitoring data to enhance the precision of maintenance schedules. Additionally, we plan to study the effects of panel aging and degradation over time to refine our model's predictive accuracy. Integrating machine learning techniques to predict cleaning needs based on historical performance data and environmental patterns is also a promising direction. Future research will consider integrating solar radiation as a variable in the temperature model to enhance its accuracy. Moreover, the factor that rising dust blocks part of the sun's rays and absorbs some of the atmospheric heat, reducing the temperature, will be considered. Finally, collaboration with industry partners to validate and refine our model through field trials and practical implementations will be crucial for practical applicability.

CRediT authorship contribution statement

Yaxin Shen: Writing – original draft, Validation, Software, Methodology, Conceptualization. Mitra Fouladirad: Writing – review & editing, Supervision, Methodology, Conceptualization. Antoine Grall: Writing – review & editing, Supervision, Methodology, Conceptualization.

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Yaxin SHEN reports financial support was provided by China Scholarship Council. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Covariate states between time interval Δt

Define $R_t=\inf \{Z_{t+\mathrm{\Delta }t}\ne Z_t\},\mathrm{\Delta }t>0,t>0$. Let k present transition times of covariate ${\tau }_1,{\tau }_2,\cdots ,{\tau }_k$ and the corresponding covariate states are $Z_1,Z_2,\cdots ,Z_k$ in time interval Δt in $(0,\mathrm{\Delta }t)$. Based on this definition, we have ${\tau }_2={\tau }_1+R_{{\tau }_1},\cdots ,{\tau }_k={\tau }_{k-1}+R_{{\tau }_{k-1}}$, and $R_{{\tau }_i}\simeq exp(q_{Z_{{\tau }_i}})$ according to the property of homogeneous Markov chain and denote $f_{R_{{\tau }_i}}$ as its probability density function (PDF). In the time span Δt, the event where a specific covariate undergoes exactly k changes is denoted as $E_k^{\mathrm{\Delta }t}$, and it can be described by:

$$E_k^{\mathrm{\Delta }t}=R_0+\sum _{i=1,}^k{R}_{{\tau }_i}\le \mathrm{\Delta }t,R_0+\sum _{i=1,}^{k+1}{R}_{{\tau }_i}>\mathrm{\Delta }t.$$ (A.1)

The conditional probability that k state transitions occur in Δt given k transitions is defined as

$$\mathbb{P}(E_k^{\mathrm{\Delta }t}|Z_{{\tau }_1}=Z_1,\cdots ,Z_{{\tau }_k}=Z_k)=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}\mathbb{P}(R_{{\tau }_k}<\mathrm{\Delta }t-r|r=x,Z_{{\tau }_1}=Z_1,\cdots ,Z_{{\tau }_k}=Z_k)\cdot f_r(x)dx=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}\underset{0}{\overset{\mathrm{\Delta }t-r}{\int }}{f}_{R_{{\tau }_k}}(y)f_r(x)dydx,$$ (A.2)

where $r=R_0+{\sum }_{i=1}^{k-1}{R}_{{\tau }_i}$. $f_{R_{{\tau }_i}}$ is the PDF of $R_{{\tau }_i}$. Given that $R_{{\tau }_i}$ follows an exponential distribution, $\exp (q_{Z_{{\tau }_i}})$, we can specify its PDF $f_{R_{{\tau }_i}}$ based on the properties of the exponential distribution. Thus, we have

$$f_{R_{{\tau }_i}}(x)=q_{Z_{{\tau }_i}}{e}^{-q_{Z_{{\tau }_i}}x}\text{ for }x\ge 0,\text{ and }0\text{ otherwise.}$$

This represents the form of the PDF for $R_{{\tau }_i}$. Note that $q_{Z_{{\tau }_i}}$ is an element of the transition rate matrix Q, determined by the given state $Z_{{\tau }_i}$.

The likelihood of precisely k transitions happening within the time interval Δt is expressed by a certain probability.

$$\mathbb{P}(E_k^{\mathrm{\Delta }t})=\underset{Z_1\ne Z_0}{\sum _{Z_1=1,}^m}\underset{Z_2\ne Z_1}{\sum _{Z_2=1,}^m}\cdots \underset{Z_k\ne Z_{k-1}}{\sum _{Z_k=1,}^m}\mathbb{P}(E_k^{\mathrm{\Delta }t}|Z_{{\tau }_1}=Z_1,\cdots ,Z_{{\tau }_k}=Z_k)\cdot \mathbb{P}(Z_{{\tau }_1}=Z_1,\cdots ,Z_{{\tau }_k}=Z_k).$$ (A.3)

Appendix B. Conditional PDF of the $D(t|Z_t)$ and $W(t|Z_t)$

PDF of $D(t)$ and CDF of $W(t)$ have been derived in Equ. (9) and Equ. (15) separately. Then the conditional PDF of the $D(t|Z_t)$ and $W(t|Z_t)$ conditionally to covariate $Z_t$ are respectively $f_{D(t|Z_t)}(x)$ and $g_{W(t|Z_t)}(x)$ as:

$$f_{D(t|Z_t)}(x)=\mathbb{P}(E_k^t)\cdot f_{D(t)}(x),$$ (B.1)

where $E_k^t$ means that there are k transitions of covariate states in time interval t.

$$\mathbb{P}(W(t|Z_t)\le x)=\mathbb{P}(E_k^t)\cdot \mathbb{P}(W_t\le x)$$ (B.2)

$$g_{W(t|Z_t)}(x)=d\mathbb{P}(W(t|Z_t)\le x)/dx.$$ (B.3)

Appendix C. $Y(t|Z_t)$ with covariate during time interval Δt

In the following appendix, we present the derivation of the degradation increment distribution over the time interval Δt. Initially, we construct the distribution function for efficiency reductions given precisely k covariate changes within Δt, as detailed in the appendix A. Therefore, we have the degradation increment distribution due to dust with covariate during time interval Δt:

$$F_{Y(t+\mathrm{\Delta }t)-Y(t)}^{Z_t}(x)=\sum _{k=0}^{\infty }\underset{Z_1\ne Z_0}{\sum _{Z_1=1,}^m}\cdots \underset{Z_k\ne Z_{k-1}}{\sum _{Z_k=1,}^m}\mathbb{P}(Y(t+\mathrm{\Delta }t)-Y(t)<x|Z_t=1,E_k^{\mathrm{\Delta }t},Z_{{\tau }_1}=z_1,\cdots ,Z_{{\tau }_k}=z_k)\cdot \mathbb{P}(Z_{{\tau }_1}=z_1|Z_t=1)\cdots \mathbb{P}(Z_{{\tau }_k}=z_k|Z_{{\tau }_{k-1}}=z_{k-1})\cdot \mathbb{P}(E_k^{\mathrm{\Delta }t}).$$ (C.1)

Fig. C.8 provides an example comparing the histogram of simulated degradation with the PDF of degradation over the same time interval, as numerically calculated by the given Equ. (C.1).

Figure C.8.

Comparison of PDF and histogram of the degradation increment between Δt = 0.8.