
Citation: M. Boussaid, A. Belghachi, K. Agroui, N.Djarfour. Mathematical models of photovoltaic modules degradation in desert environment[J]. AIMS Energy, 2019, 7(2): 127-140. doi: 10.3934/energy.2019.2.127
[1] | Mebarek LAHBIB, Mohammed BOUSSAID, Houcine MOUNGAR, Ahmed TAHRI . Analytical assessment of the filed results on the PV system connectors performance under operating temperature. AIMS Energy, 2023, 11(3): 555-575. doi: 10.3934/energy.2023029 |
[2] | Honnurvali Mohamed Shaik, Adnan Kabbani, Abdul Manan Sheikh, Keng Goh, Naren Gupta, Tariq Umar . Measurement and validation of polysilicon photovoltaic module degradation rates over five years of field exposure in Oman. AIMS Energy, 2021, 9(6): 1192-1212. doi: 10.3934/energy.2021055 |
[3] | Miguel de Simón-Martín, Montserrat Díez-Mediavilla, Cristina Alonso-Tristán . Real Energy Payback Time and Carbon Footprint of a GCPVS. AIMS Energy, 2017, 5(1): 77-95. doi: 10.3934/energy.2017.1.77 |
[4] | Eliseo Zarate-Perez, Juan Grados, Santiago Rubiños, Herbert Grados-Espinoza, Jacob Astocondor-Villar . Building-integrated photovoltaic (BIPV) systems: A science mapping approach. AIMS Energy, 2023, 11(6): 1131-1152. doi: 10.3934/energy.2023052 |
[5] | Otto Andersen, Geoffrey Gilpin, Anders S.G. Andrae . Cradle-to-gate life cycle assessment of the dry etching step in the manufacturing of photovoltaic cells. AIMS Energy, 2014, 2(4): 410-423. doi: 10.3934/energy.2014.4.410 |
[6] | Shijian Chen, Yongquan Zhou, Qifang Luo . Hybrid adaptive dwarf mongoose optimization with whale optimization algorithm for extracting photovoltaic parameters. AIMS Energy, 2024, 12(1): 84-118. doi: 10.3934/energy.2024005 |
[7] | Alemayehu T. Eneyaw, Demiss A. Amibe . Annual performance of photovoltaic-thermal system under actual operating condition of Dire Dawa in Ethiopia. AIMS Energy, 2019, 7(5): 539-556. doi: 10.3934/energy.2019.5.539 |
[8] | Abdulrahman Th. Mohammad, Wisam A. M. Al-Shohani . Numerical and experimental investigation for analyzing the temperature influence on the performance of photovoltaic module. AIMS Energy, 2022, 10(5): 1026-1045. doi: 10.3934/energy.2022047 |
[9] | Carmine Cancro, Gabriele Ciniglio, Luigi Mongibello, Antonino Pontecorvo . Methodology for the characterization of the humidity behavior inside CPV modules. AIMS Energy, 2015, 3(4): 666-678. doi: 10.3934/energy.2015.4.666 |
[10] | Amer Braik, Asaad Makhalfih, Ag Sufiyan Abd Hamid, Kamaruzzaman Sopian, Adnan Ibrahim . Impact of photovoltaic grid-tied systems on national grid power factor in Palestine. AIMS Energy, 2022, 10(2): 236-253. doi: 10.3934/energy.2022013 |
Abbreviations: (α, β, γ, η, θ, λ, μ, a, b): Model parameters; R (t): Reliability function; MTBF (τ): the average Lifetime; PV: Photovoltaic; a-Si: Amorphous silicon; pc-Si: Polycrystalline silicon; mc-Si: Single crystalline silicon
Many articles in literature have studied the degradation of photovoltaic modules when exposed to natural environments using accelerated tests to observe degradation in reality [1,2]. A study confirmed that after 20 years of continuous exposure a matrix of 70 polycrystalline silicon photovoltaic modules has undergone an average performance decay of 0.24% per year in a moderate subtropical climate environment [3,4]. Another study stated that after only one year of exposure in a tropical climate environment the electrical powers of two modules of type (a-Si) and (pc-Si) were degraded to 60% and 56% respectively of their initial values [5]. In addition to these results, another study has shown that some photovoltaic modules (mc-Si and pc-Si) had been degraded by 0.22% /year to 2.96% / year for the maximum power [6]. In the long run, the polycrystalline silicon modules have the best reliability with a degradation rate of 0.41% per year in a natural environment [7]. In a tropical environment (Ghana), the exposure of 14 polycrystalline silicon modules during a 19-year period recorded a degradation rate of 21% to 35% of nominal power [8]. The degradation is in the order of 1.2% per year for polycrystalline silicon modules and 0.8% per year for single crystalline silicon modules [9]. An important study that followed the degradation of 204 modules (123 mc-Si and 81 pc-Si) had revealed a degradation variance from 0 to 6% per year for exposure periods of 18 years to 24 years in a subtropical moderate environment [10]. In Saharan environment (southern Algeria for example) the degradation rate of polycrystalline silicon modules was very high ranging from 3.33% / year to 4.64% / year unlike mono-crystalline silicon modules which recorded a rate of 1.22% / year after 28 years of exposure [11,12]. Accelerated tests cannot evaluate totally the effect of natural environment on electrical and optical characteristics of a photovoltaic module [13]. But it is the only method to see the effect of single factor or limited number of climatic factors [13,14]. The return of experimental data within a period of operation in a natural environment allows to predict the lifetime and the degradation over the long term [15]. Our objective in this study is to search in the literature for an adequate model to simulate the reliability of photovoltaic modules (crystalline silicon) exposed in desert environments in order to probably estimate their degradation at any period of their operation. The method consists of using a genetic algorithm (artificial intelligence optimization method) to estimate the unknown parameters of the models and to check the competence of the simulation by comparing with feedbacks of experimental data.
Two kinds of methods in the literature are used to predict the duration of good operation and the reliability of a photovoltaic module exposed in a natural environment, the first that uses the return of experiments, or the second that utilizes accelerated tests [16]. In this study, we use the feedback data that are practically measured in the desert of California and in the Algerian Sahara (Adrar region), extracted from references [16,17,18] to estimate the lifespan of photovoltaic modules (single crystalline silicon) in these environments. In order to calculate the parameters of models we will use a genetic algorithm. The iterative stochastic genetic algorithm uses an initial population to reach an optimal solution of any problem [19]. The initially chosen population has evolved from generation to generation where the most suitable individuals have a great chance of breeding. This mechanism of intelligence is realized by respecting the following steps [20,21]:
1. Creation of an initial population
2. Assessment of individuals in population
3. Selection of adapted individuals
4. Reproduction by crossing and mutation
5. Formation of a new generation
This process is circulated until an optimal solution is obtained. Practically, we represent these steps according to the flowchart below (Figure 1):
Originally, the reliability concerned the high technology systems (nuclear, aerospace...) to guarantee their operational safety. Today, all areas are interested in the study of reliability to make decisions on ratio Cost / gain and to control the failure sources [22,23]. Reliability of a system is a quantity characterizing the safety of operation or measuring the probability of operation of an appliance according to prescribed standards (definition presented in 1962 by the Academy of Sciences). Reliability (or survival function) is expressed by:
R(t)=r−∫t0h(x)dx | (1) |
h(t): Instant failure rate (probability of seeing a failure in a short interval after instant t.
The average time of operation (lifetime) which is the Mean Time Before Failure (MTBF) is given by:
MTBF=∫+∞0R(t)dt | (2) |
According to their instantaneous rates of failure, the parametric reliability models are classified in the literature as follows [24,25,26]:
1. Models of constant rate: Exponential model.
2. Models of monotone rate: Weibul model, gamma model, Gompertz-Makeham model, exponential Weibul model, Mix of exponential models.
3. Models of rates of a bathtub shape: Modified Weibul model, exponential power model, quadratic model, and uniform model.
4. Models of rate in bell form: Generalized Weibul model, normal model, log-normal model, log logistic model, extreme values model.
The characteristics of chosen models are presented in the table 1 below:
Model | Model of two parameters | |
Reliability function | Average lifetime | |
Exponential model | R(t)=e−λtwithλ>0 | MTBF=1λ |
Weibul model | R(t)=e−(tη)β;β,η>0 | MTBF=ηΓ(1+1/β) |
Gamma model | R(t)=1−1Γ(μ)∫θt0xμ−1e−xdx;(μ,θ)>0 | MTBF=μθ |
Exponential power model | R(t)=e1−e(λt)αwithα>0;λ,α>0 | MTBF=∫+∞0e1−e(λt)αdt |
Normal model | R(t)=1−1√2πσ∫+∞0e−(x−μ)22σ2dx | MTBF=μ |
Log-normal model | R(t)=1−1√2πσ∫Int−∞e−(x−μ)22σ2dx | MTBF=eμ+σ22 |
Log logistic model | R(t)=αβαβ+tβorα>0;β>1 | MTBF=∫+∞0αβαβ+tβdt |
Uniform model | R(t)=b−tb−a;fort∈[a,b] | MTBF=∫bab−tb−adt |
Extreme values model | R(t)=e−α(eβt−1)withα>0,β>0 | MTBF=∫+∞0e−α(eβt−1)dt |
Gompertz-Makeham model | R(t)=e−at−bInc(ct−1) | MTBF=∫+∞0e−at−bInc(ct−1)dt |
exponential Weibul model | R(t)=1−{1−e−(tη)β}μorη,β,μ>0 | MTBF=∫+∞01−{1−e−(tη)β}μdt |
Mix of exponential models | R(t)=a1e−tθ1+(1−a1)e−tθ2;θ1,θ2>0;0<a1<1 | MTBF=a1θ1+(1−a1)θ2 |
Modified Weibul model | R(t)=e−(tη)βeμtwith(η;β;μ>0) | MTBF=∫+∞0e−(tη)βeμtdt |
Quadratic model | R(t)=e−(αt+β2t2+γ3t3)α,γ>0;−2√γα≤β≤0 | MTBF=∫+∞0e−(αt+β2t2+γ3t3)dt |
Generalized Weibul model | R(t)=e1−(1+(tη)β)1γwith(η,β,γ)>0 | MTBF=∫+∞0e1−(1+(tη)β)1γdt |
After filtering we present only the cases where the calculated average error is less than 2%. This choice is solely made to limit the size of the study. The other cases are not interesting as our goal is to visualize the most adequate model having the least error.
Estimated parameters and simulated reliability graphs are shown below (Table 2, Figures 2 and 3):
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (71, 1.35, 0.03) | (49, 1.05, 0.01) |
Average lifetime MTBF (years) | 28.75 | 30.47 |
Uncertainty | 0.0039 | 0.0059 |
After 20 years of operation in Adrar region we observe that the Weibul modified model predicts a 30% degradation of starting value of electrical power for this type of photovoltaic modules while the degradation is approximately of 38% in the desert of California.
In the Table 3 and Figures (4, 5) we present the estimated parameters and the simulated reliability:
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, γ) | (38, 2.5, 0.5) | (77, 1.2, 0.4) |
Average lifetime MTBF (years) | 23.45 | 26.09 |
Uncertainty | 0.0184 | 0.0092 |
In this case the generalized Weibul model predicts a degradation of 35% in Adrar region after 20 years of operation while the degradation is approximately of 44% in the desert of California.
The Table 4 and the Figures 6, 7 show the estimated parameters and the simulated reliability of PV module:
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (28, 4.05, 0.66) | (37, 2.65, 0.45) |
Average lifetime MTBF (years) | 22.57 | 23.40 |
Uncertainty | 0.0254 | 0.0028 |
By this model the degradation is about of 38% in Adrar region after 20 years of operation while the degradation is approximately of 46% in the desert of California.
Estimated parameters and simulated reliability graphs are shown below (Table 5, Figures 8 and 9):
In Adrar Sahara | In the desert of California | |
Parameters model (α, β) | (0.015, 0.68) | (0.02, 0.9) |
Average lifetime MTBF (years) | 50.09 | 31.99 |
Uncertainty | 0.0081 | 0.0227 |
By model of extreme values, the degradation is about of 22% in Adrar region after 20 years of operation while it is approximately of 37% in the desert of California.
In this case the estimated parameters and the simulated reliability are shown below (table 6, figure 10 and 11):
In Adrar Sahara | In the desert of California | |
Parameters model (a, b) | (1.6, 77) | (0.4, 48) |
Average lifetime MTBF (years) | 37.70 | 23.80 |
Uncertainty | 0.0074 | 0.0139 |
Finally, the uniform model predicts a degradation of 24% in Adrar region After 20 years of operation while it is approximately of 41% in the desert of California.
These results are summarized in the following Table 7.
Model of Reliability | Average life time (years) | Error Means (%) | |
in Adrar | in California | ||
modified Weibul model | 28.75 | 30.47 | 0.4 |
Uniform model | 37.70 | 23.80 | 1.1 |
Generalized Weibul model | 23.45 | 26.09 | 1.3 |
exponential Weibul model | 22.57 | 23.40 | 1.4 |
Extreme values model | 50.09 | 31.99 | 1.5 |
Average lifetime calculated | 32.51 | 27.15 | |
29.83 ≈ 30years |
The above results indicate that:
1. The experimental data used have guided us to predict the future of solar panels operating in desert environments. We therefore believe that more return data will give us confidence in models and methods.
2. The calculated mean error that present the average relative distance of the graph from the points of comparison is generally small (especially for the modified Weibul model). These reflect the skill of the optimization method used (the genetic algorithm).
3. Extrapolation of curves in longer durations allows informing on the reliability (outside of periods of real measurements).
4. The modified Weibul model is the most adequate of the models tested to simulate the reliability of photovoltaic modules (single crystalline silicon) and to estimate their lifetimes (MTBF) in the desert environments. It predicted a duration of nearly 30 years in the desert of California and of around 29 years for the Adrar area.
5. It should be noted that the degradation of electrical power of photovoltaic modules in Californian desert is significant compared to that of Adrar region in the first step (in the initial period of 30 years).
6. These obtained results are more or less comparable to those stated in references [11,12] (a degradation close to 1.53% /year in this study).
It has been confirmed in this article that the modified Weibul law is the most adequate model compared to other tested models to simulate the reliability function of photovoltaic modules and estimate their lifetime while operating in desert environments (California and Adrar). Using simulation findings, an average lifespan of about 30 years has been predicted for photovoltaic modules exposed in desert regions where the maximum power of the photovoltaic module is degraded to almost 46% of its initial value. The annual rate of degradation is in the order of 1.5% / year. This obtained result is more or less comparable to those presented in the literature. The prediction results must be taken into consideration for any study of construction of solar stations in the Saharan environments.
I thank my fellow researchers in Renewable Energy Research Unit in Saharan areas (URERMS) for all given help.
The authors declare there are no conflicts of interest in this paper.
[1] | Laronde R, Charki A, Bigaud D (2010) Reliability of photovoltaic modules based on climatic measurement data. Int J Metrol Qual Eng 1: 45–49. |
[2] | Boussaid M, Belghachi A, Agroui K (2018) Contribution to the degradation modeling of a polycrystalline photovoltaic cell under the effect of stochastic thermal cycles of a desert environment. Int J Control Energy Electr Eng 6: 66–72. |
[3] | Polverini D, Field M, Dunlop E, et al. (2013) Polycrystalline silicon PV modules performance and degradation over 20 years. Progress Photovoltaic Res 21: 1004–1015. |
[4] | Sample T, Pozza A (2013) 20-Year field exposed polycrystalline silicon PV modules: detailed visual inspection and analysis. European Commission, DG JRC, Institute for Energy and Transport, Ispra (VA), Italy. |
[5] |
Akhmad K, Kitamura A, Yamamoto F, et al. (1997) Outdoor performance of amorphous silicon and polycrystalline silicon PV modules. Sol Energy Mater Sol Cells 46: 209–218. doi: 10.1016/S0927-0248(97)00003-2
![]() |
[6] |
Ndiaye A, Kébé C, Kobi A, et al. (2014) Degradation evaluation of crystalline-silicon photovoltaic modules after a few operation years in a tropical environment. Sol Energy 103: 70–77. doi: 10.1016/j.solener.2014.02.006
![]() |
[7] | Smith K (2016) Degradation effects in photovoltaic modules. ENG470 Engineering thesis, Murdoch University, Western Australia. |
[8] | Quansah D, Adaramola M, Takyi G, et al. (2017) Reliability and degradation of solar PV modules-case study of 19-year-old polycrystalline modules in Ghana. Technologies 5: 22. |
[9] | Köntges M, Kurtz S, Packard C, et al. (2014) Review of failures of photovoltaic modules. IEA PVPST asks 13 External final report IEA-PVPS, ISBN978-3-906042-16-9. |
[10] | Sample T (2011) Failure modes and degradation rates from field-aged crystalline silicon modules. Institute for energy, ISPRA, Italy, NREL, Golden USA. |
[11] |
Kahoul N, Chenni R, Cheghib H, et al. (2017) Evaluating the reliability of crystalline silicon photovoltaic modules in harsh environment. Renewable Energy 109: 66–72. doi: 10.1016/j.renene.2017.02.078
![]() |
[12] | Bandou F, Arab AH, Belkaïd MS, et al. (2015) Evaluation performance of photovoltaic modules after a long time operation in Saharan environment . Int J Hydrogen Energy 40: 13839–13848. |
[13] |
Boussaid M, Belghachi A, Agroui K, et al. (2016) Solar cell degradation under open circuit condition in out-doors-in desert region. Results Phys 6: 837–842. doi: 10.1016/j.rinp.2016.09.013
![]() |
[14] |
Fezzani A, Hadj Mahammed I, Said D, et al. (2017) Degradation and performance evaluation of PV module in desert climate conditions with estimate uncertainty in measuring. Serb J Electr Eng 14: 277–299. doi: 10.2298/SJEE1702277F
![]() |
[15] | Laronde R, Charki A, Bigaud D (2012) Lifetime Estimation of a Photovoltaic Module Subjected to Corrosion Due to Damp Heat Testing. J Sol Energy Eng 135: 010–021. |
[16] | Mani GT, Kuitche J (2013) Accelerated lifetime testing of photovoltaic modules. Photovoltaic Reliability Laboratory, Arizona State University. Available from: www.solarabcs.org/acceleratedtesting. |
[17] | Sadok M, Benyoucef B, Mehdaoui A (2012) Performances et dégradation des modules PV en milieu saharien. Revue des Energies Renouvelables SIENR'12 Ghardaïa 2012: 203–212. |
[18] | Agroui K (2010) Contribution au développement des techniques de contrôle de qualité des modules photovoltaïques de diverses technologies. Thèse, Université de Bechar, Algérie. |
[19] | Melanie M (1999) An introduction to genetic algorithms. |
[20] | Bodenhofer U (2004) Genetic Algorithms: Theory and Applications. Johannes Kepler University in Linz. |
[21] | A Coley D (1998) An introduction to genetic algorithms for scientists and engineers. Singapore: World Scientific. |
[22] | Carrion Garcia A (2015) Reliability modeling and prediction. The Research Center of Dependability and Quality Management, Polytechnic University of Valencia, Spain. |
[23] | Ayyub B, Mccuen R. (1997) Probability, Statistics & Reliability for engineers. New York: CRC Press. |
[24] | Rausand M (2004) System reliability theory: Models, statistical methods and applications. Technometrics 46: 495 |
[25] | Morice E (1966) Quelques modèles mathématiques de durée de vie. Revues de Statistique Appliquée 14: 45–126. |
[26] | Saint Pierre P (2015) Introduction à l'analyse des durées de survie. Université Pierre et Marie Curie, France. |
1. | Abdeldjalil Dahbi, Mohammed Boussaid, Mohammed Haidas, Maamar Dahbi, Rachid Maouedj, Othmane Abdelkhalek, Miloud Benmedjahed, Lalla Moulati Elkaiem, Lahcen Abdellah, 2020, Chapter 54, 978-3-030-37206-4, 506, 10.1007/978-3-030-37207-1_54 | |
2. | Mohammed Boussaid, Abdeldjalil Dahbi, Abdella Lahcen, Lalla Moulaty Elkaiem, 2019, The Interest of Connecting Mini Solar Stations to the Public Electricity Grid in a Desert Environment, 978-1-7281-5152-6, 1, 10.1109/IRSEC48032.2019.9078295 | |
3. | Matheus Rabelo, Muhammad Aleem Zahid, Khushabu Agrawal, KyungSoo Kim, Eun-Chel Cho, Junsin Yi, Analysis of solder joint degradation and output power drop in silicon photovoltaic modules for reliability improvement, 2021, 127, 00262714, 114399, 10.1016/j.microrel.2021.114399 | |
4. | Nabil Kahoul, Hocine Cheghib, Mariano Sidrach-de-Cardona, Zoubida Kherici, Mohamed Younes, 2022, Chapter 44, 978-3-030-76080-9, 363, 10.1007/978-3-030-76081-6_44 | |
5. | Jay Kumar Pandey, 2023, Prediction for Solar Energy Different Climatic Conditions to Harvest Maximum Energy, 979-8-3503-3600-9, 1, 10.1109/ICNWC57852.2023.10127523 | |
6. | Hassan Abdal Haidy Al-Hamzawi, Mohammad Hassan Shojaeefard, Mohammad Mazidi Sharfabadi, Improving the photovoltaic/thermal (PV/T) system by adding the PCM and finned tube heat exchanger, 2023, 13, 2158-3226, 10.1063/5.0179371 | |
7. | Saliou Diallo, Fatim Zahra Melhaoui, Mohamed Rafi, Abdellatif Elassoudi, B. Benhala, A. Raihani, B. Boukili, A. Sallem, M. Qbadou, Understanding Photovoltaic Module Degradation: An Overview of Critical Factors, Models, and Reliability Enhancement Methods, 2023, 469, 2267-1242, 00011, 10.1051/e3sconf/202346900011 | |
8. | Xin Huang, He Wang, Xuefang Jiang, Hong Yang, Performance degradation and reliability evaluation of crystalline silicon photovoltaic modules without and with considering measurement reproducibility: A case study in desert area, 2023, 219, 09601481, 119421, 10.1016/j.renene.2023.119421 | |
9. | Chebabhi Ardjouna, 2023, Mathematic model for photovoltaic module under fault conditions, 979-8-3503-7218-2, 1, 10.1109/ICETS60996.2023.10410696 | |
10. | Abdelkader ELKHARRAZ, Mohammed BOUSSAID, Abdennour El Mohri, Noureddine DJARFOUR, Mourad OTHMANI, A Novel Acceleration Law for Sand Erosion Degradation of Photovoltaic Modules, 2025, 09601481, 122522, 10.1016/j.renene.2025.122522 | |
11. | Abdeldjalil Dahbi, Abderrahmane Khelfaoui, Miloud Benmedjahed, Ahmed Bouraiou, Tidjar Boudjema, Abdeldjalil Slimani, Nouar Aoun, Messaoud Hamouda, 2025, Chapter 80, 978-3-031-80300-0, 766, 10.1007/978-3-031-80301-7_80 |
Model | Model of two parameters | |
Reliability function | Average lifetime | |
Exponential model | R(t)=e−λtwithλ>0 | MTBF=1λ |
Weibul model | R(t)=e−(tη)β;β,η>0 | MTBF=ηΓ(1+1/β) |
Gamma model | R(t)=1−1Γ(μ)∫θt0xμ−1e−xdx;(μ,θ)>0 | MTBF=μθ |
Exponential power model | R(t)=e1−e(λt)αwithα>0;λ,α>0 | MTBF=∫+∞0e1−e(λt)αdt |
Normal model | R(t)=1−1√2πσ∫+∞0e−(x−μ)22σ2dx | MTBF=μ |
Log-normal model | R(t)=1−1√2πσ∫Int−∞e−(x−μ)22σ2dx | MTBF=eμ+σ22 |
Log logistic model | R(t)=αβαβ+tβorα>0;β>1 | MTBF=∫+∞0αβαβ+tβdt |
Uniform model | R(t)=b−tb−a;fort∈[a,b] | MTBF=∫bab−tb−adt |
Extreme values model | R(t)=e−α(eβt−1)withα>0,β>0 | MTBF=∫+∞0e−α(eβt−1)dt |
Gompertz-Makeham model | R(t)=e−at−bInc(ct−1) | MTBF=∫+∞0e−at−bInc(ct−1)dt |
exponential Weibul model | R(t)=1−{1−e−(tη)β}μorη,β,μ>0 | MTBF=∫+∞01−{1−e−(tη)β}μdt |
Mix of exponential models | R(t)=a1e−tθ1+(1−a1)e−tθ2;θ1,θ2>0;0<a1<1 | MTBF=a1θ1+(1−a1)θ2 |
Modified Weibul model | R(t)=e−(tη)βeμtwith(η;β;μ>0) | MTBF=∫+∞0e−(tη)βeμtdt |
Quadratic model | R(t)=e−(αt+β2t2+γ3t3)α,γ>0;−2√γα≤β≤0 | MTBF=∫+∞0e−(αt+β2t2+γ3t3)dt |
Generalized Weibul model | R(t)=e1−(1+(tη)β)1γwith(η,β,γ)>0 | MTBF=∫+∞0e1−(1+(tη)β)1γdt |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (71, 1.35, 0.03) | (49, 1.05, 0.01) |
Average lifetime MTBF (years) | 28.75 | 30.47 |
Uncertainty | 0.0039 | 0.0059 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, γ) | (38, 2.5, 0.5) | (77, 1.2, 0.4) |
Average lifetime MTBF (years) | 23.45 | 26.09 |
Uncertainty | 0.0184 | 0.0092 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (28, 4.05, 0.66) | (37, 2.65, 0.45) |
Average lifetime MTBF (years) | 22.57 | 23.40 |
Uncertainty | 0.0254 | 0.0028 |
In Adrar Sahara | In the desert of California | |
Parameters model (α, β) | (0.015, 0.68) | (0.02, 0.9) |
Average lifetime MTBF (years) | 50.09 | 31.99 |
Uncertainty | 0.0081 | 0.0227 |
In Adrar Sahara | In the desert of California | |
Parameters model (a, b) | (1.6, 77) | (0.4, 48) |
Average lifetime MTBF (years) | 37.70 | 23.80 |
Uncertainty | 0.0074 | 0.0139 |
Model of Reliability | Average life time (years) | Error Means (%) | |
in Adrar | in California | ||
modified Weibul model | 28.75 | 30.47 | 0.4 |
Uniform model | 37.70 | 23.80 | 1.1 |
Generalized Weibul model | 23.45 | 26.09 | 1.3 |
exponential Weibul model | 22.57 | 23.40 | 1.4 |
Extreme values model | 50.09 | 31.99 | 1.5 |
Average lifetime calculated | 32.51 | 27.15 | |
29.83 ≈ 30years |
Model | Model of two parameters | |
Reliability function | Average lifetime | |
Exponential model | R(t)=e−λtwithλ>0 | MTBF=1λ |
Weibul model | R(t)=e−(tη)β;β,η>0 | MTBF=ηΓ(1+1/β) |
Gamma model | R(t)=1−1Γ(μ)∫θt0xμ−1e−xdx;(μ,θ)>0 | MTBF=μθ |
Exponential power model | R(t)=e1−e(λt)αwithα>0;λ,α>0 | MTBF=∫+∞0e1−e(λt)αdt |
Normal model | R(t)=1−1√2πσ∫+∞0e−(x−μ)22σ2dx | MTBF=μ |
Log-normal model | R(t)=1−1√2πσ∫Int−∞e−(x−μ)22σ2dx | MTBF=eμ+σ22 |
Log logistic model | R(t)=αβαβ+tβorα>0;β>1 | MTBF=∫+∞0αβαβ+tβdt |
Uniform model | R(t)=b−tb−a;fort∈[a,b] | MTBF=∫bab−tb−adt |
Extreme values model | R(t)=e−α(eβt−1)withα>0,β>0 | MTBF=∫+∞0e−α(eβt−1)dt |
Gompertz-Makeham model | R(t)=e−at−bInc(ct−1) | MTBF=∫+∞0e−at−bInc(ct−1)dt |
exponential Weibul model | R(t)=1−{1−e−(tη)β}μorη,β,μ>0 | MTBF=∫+∞01−{1−e−(tη)β}μdt |
Mix of exponential models | R(t)=a1e−tθ1+(1−a1)e−tθ2;θ1,θ2>0;0<a1<1 | MTBF=a1θ1+(1−a1)θ2 |
Modified Weibul model | R(t)=e−(tη)βeμtwith(η;β;μ>0) | MTBF=∫+∞0e−(tη)βeμtdt |
Quadratic model | R(t)=e−(αt+β2t2+γ3t3)α,γ>0;−2√γα≤β≤0 | MTBF=∫+∞0e−(αt+β2t2+γ3t3)dt |
Generalized Weibul model | R(t)=e1−(1+(tη)β)1γwith(η,β,γ)>0 | MTBF=∫+∞0e1−(1+(tη)β)1γdt |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (71, 1.35, 0.03) | (49, 1.05, 0.01) |
Average lifetime MTBF (years) | 28.75 | 30.47 |
Uncertainty | 0.0039 | 0.0059 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, γ) | (38, 2.5, 0.5) | (77, 1.2, 0.4) |
Average lifetime MTBF (years) | 23.45 | 26.09 |
Uncertainty | 0.0184 | 0.0092 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (28, 4.05, 0.66) | (37, 2.65, 0.45) |
Average lifetime MTBF (years) | 22.57 | 23.40 |
Uncertainty | 0.0254 | 0.0028 |
In Adrar Sahara | In the desert of California | |
Parameters model (α, β) | (0.015, 0.68) | (0.02, 0.9) |
Average lifetime MTBF (years) | 50.09 | 31.99 |
Uncertainty | 0.0081 | 0.0227 |
In Adrar Sahara | In the desert of California | |
Parameters model (a, b) | (1.6, 77) | (0.4, 48) |
Average lifetime MTBF (years) | 37.70 | 23.80 |
Uncertainty | 0.0074 | 0.0139 |
Model of Reliability | Average life time (years) | Error Means (%) | |
in Adrar | in California | ||
modified Weibul model | 28.75 | 30.47 | 0.4 |
Uniform model | 37.70 | 23.80 | 1.1 |
Generalized Weibul model | 23.45 | 26.09 | 1.3 |
exponential Weibul model | 22.57 | 23.40 | 1.4 |
Extreme values model | 50.09 | 31.99 | 1.5 |
Average lifetime calculated | 32.51 | 27.15 | |
29.83 ≈ 30years |