J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 Journal of the Nigerian Society of Physical Sciences Performance Analysis of Photovoltaic Systems Using (RAMD) Analysis Anas Sani Maihullaa,∗, Ibrahim Yusufb aDepartment of Mathematics, Sokoto State University, Sokoto bDepartment of Mathematical Sciences, Bayero University, Kano Abstract The primary aim of this present study is to examine how reliability, availability, maintainability, and dependability (RAMD) are used to describe the criticality of each sub-assembly in grid- connected photovoltaic systems. A transition diagram of all subsystems is produced for this analysis, and Chapman-Kolmogorov differential equations for each variable of each subsystem are constructed using the Markov birth-death process. Both random failure and repair time variables have an exponential distribution and are statistically independent. A sufficient repair facility is still available with the device. The numerical results for reliability, maintainability, dependability, and steady-state availability for various photovoltaic device components have been obtained. Other metrics, such as mean time to failure (MTTF), mean time to repair (MTTR), and dependability ratio, which aid in device performance prediction, have also been measured. According to numerical analysis. it is hypothesized that subsystem S4, i.e. the inverter, is the most critical and highly sensitive portion that requires special attention in order to improve the efficiency of the PV device plant. The findings of this research are very useful for photovoltaic system designers and maintenance engineers. DOI:10.46481/jnsps.2021.194 Keywords: Photovoltaic system, Mean Time to Repair, Mean Time to Failure. Article History : Received: 05 April 2021 Received in revised form: 26 June 2021 Accepted for publication: 30 June 2021 Published: 29 August 2021 c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: W. A. Yahya 1. Introduction The dependability of photovoltaic systems and their compo- nents/elements such as solar cells, PV modules, electrical stor- age systems, inverters, regulators, etc is a critical problem in production efficiency and financially competitive photovoltaic installations. A product’s reliability can be described as the likelihood of system completing its tasks within a given time frame under stipulated conditions. The consistency of a prod- uct is calculated on this basis, with the reliability principle be- ∗Corresponding author tel. no: +234(0) Email address: anasmaihulla@gmail.com (Anas Sani Maihulla ) ing used in nearly all fields of engineering, including preven- tive maintenance of structures and their components. PV sys- tems have become a common solution for residential houses and other autonomous applications due to numerous reward schemes and local market conditions in many European coun- tries, as well as around the world. RAMD is regarded as one of the most important fields for increasing profitability. RAMD modeling can help to improve safety and environmental efficiency, both of which are essential factors in every industry. Therefore, it becomes important to evaluate each and every part or subsystem of the system in or- der to execute safety and environmental performance. Many re- 172 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 173 searchers have presented appreciable works in this regard. Re- searchers such as; Huffman [1], Hamdy et al. [2] and Graaff et al. [3] investigated on significant issues concerning the re- liability, and safety of PV systems, RAM study of large-scale grid-connected solar-PV systems is conducted using a range of reliability approaches. Zhang et al. [4] and Hu et al. [5] conduct reliability block diagram (RBD) and fault tree analysis (FTA). In the sense of an FTA, the physical structure is translated into a logical diagram, with each block representing a device part. The failure rate is the only thing that describes each block. The overall system’s reliability is calculated by the failure rates of each sub-assembly, so any failure is critical. Failure rates are frequently assumed to be constant. Gahlot et al. [6] study stochastic analysis of a two units’ complex repairable system with switch and human failure using copula approach. Also re- cently, Chiacchio [7] investigated Dynamic performance evalu- ation of photovoltaic power plant by stochastic hybrid fault tree automaton model. RAMD technique was used by Saini et al. [8] and Goyal et al. [9] to define the most vulnerable aspect of serial processes such as evaporation systems in the sugar in- dustry and water treatment plants. This study deconstructs the efficiency indices of the power generation system using STP. The power system was studied using simple probability theory principles and the Markovian birth-death process. As a Markov process proceeds from one stage to the next. Van Casteren et al. [10] used the Weibull Markovian approach to conduct con- sistency checks on electrical power systems. Eti et al. [11] ex- plored a basic issue in order to maximize resource distribution in a thermal power plant. The operational availability assess- ment was carried out in order to improve the system’s effec- tiveness. Ebeling [12] proposed many methods for testing the reliability and maintainability of devices with differing failure and repair rates. S. Gupta et al. [13] Studied Simulation mod- eling and analysis of complex system of thermal power plant. As a case study Tsarouhas et al. [14] investigated the relia- bility, availability, and maintainability of a strudel production line using the best suited allocation of failure and repair rates. Carazas et al. [15] proposed a framework for evaluating gas tur- bine power plant efficiency indicators. The study of reliability assessment of system having two subsystem in series config- uration attended by human operator using Gumbel-Hougaard family copula was carried out by Singh et al. [16]. Singh [17] analyzed the performance of a system with imperfect switch consisting of two subsystem arranged in series through cop- ula approach. Yusuf et al. [18] focus on reliability analysis of computer system configured as series-parallel system hav- ing three dissimilar subsystem via copula approach. Raghav et al. [19] dealt with reliability prediction and performance eval- uation of distributed system consisting with similarity software and server architecture using joint probability distribution via Copula approach. To address the issues raised in the previous literature on the reliability of grid-connected PV systems, this paper provides a full comprehensive RAM analysis for all sub-assemblies of grid-connected solar PV systems with a low reliability grid, taking into account failure details and repair interval (period of identification and replacement of the PV system). Further- more, the aim of this paper is to describe the criticality of each sub-assembly of grid-connected PV systems in terms of relia- bility. The scope of this paper has also been broadened to es- tablish the best probability density function for the failure rate of each solar-PV device subassembly. The rest of this paper is structured as follows; Section 2 captures materials and meth- ods. Section 3 provides RAMD indices for PV system. Section 4 focuses on discussion of the results and the paper is concluded in Section 5. 2. Materials and Methods The RAMD analysis was used in this study. RAMD is one of the most significant fields for rising profitability. RAMD modeling aids in the improvement of safety and environmental performance, both of which are critical in any industry. In this study the RAMD analysis has been performed on PV system plant. PV system mainly consists of five components namely PV modules, controller, batteries, inverter and Distribution Board. All components are arranged in series configuration. The brief description of the subsystems is given below: 1. Subsystem R (Solar module): There are two units of so- lar panel which is connected to the following unit in par- allel. One is operational while the other one is on standby mode. Failure of the two units leads to system failure. 2. Subsystem S (Charge controller): There is one unit of charge controller which is connected to the following unit in series. Failure of this unit leads to system failure. 3. Subsystem T (Battery): This subsystem has two units of batteries which is connected in parallel to another subsys- tem. One is operation while the other one is in standby mode. This unit’s failure causes complete system failure. 4. Subsystem U (Inverter): It consists of one unit of in- verter. This unit’s failure causes complete system failure as it is connected to the following unit in series. 5. Subsystem V (Distribution Board): It consists of one unit of Distribution Board. This unit’s failure causes com- plete system failure as it is connected to the following unit in series. The RAMD analysis was focused on fail- ure rates, and the model data in this article was estab- lished theoretically. 2.1. Assumptions 1. Each subsystem’s failure and repair rates are distributed exponentially. 2. The Failure and repair rates are statistically independent of one another. 3. There are no concurrent faults in the subsystem. 4. There are plenty of maintenance and replacement options. Repairmen are still present in the factory, and the restored machine performs as well as new. 173 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 174 Table 1. Failure and repair rates of component of the photovoltaic system Subsystem Failure Rate (ω) Repair Rate (β) S 1 ω1= 0.001 β1 = 0.5 S 2 ω2= 0.002 β2 = 0.7 S 3 ω3= 0.003 β3 = 0.9 S 4 ω4= 0.004 β4 = 1.1 S 5 ω5= 0.005 β5 = 1.3 2.2. Notation R, S , T , U, and V Represent states under which the subsys- tem is operating at maximum ability. r, s, t, u, and v reflect the conditions in which a subsystem has broken. 3. RAMD indices for PV system 3.1. RAMD indices for PV modules Figure 1. Transition diagram PV modules d dt P0 (t) = −ω1 P0 + β1 P1 (1) d dt P1 (t) = − (β1 + ω1) P1 + ω1 P0 + β1 P2 (2) d dt P2 (t) = −β1 P2 + ω1 P1 (3) Under steady state, equations (1) - (3) yield P1 = β1 ω1 P2 (4) Substituting equation (4) into (1) under steady state we have: P2 = ω21 β21 P0 (5) Substituting (5) into (3) under steady state we have P1 = ω1 β1 P0 (6) Using normalization condition P0+ P1+ P2= 1 It follows that: P0 = β21 β21+ω 2 1 +β1ω1 (7) 3.2. RAMD indices for subsystem R Figure 2. Transition diagram for subsystem R d dt P0 (t) = −ω2 P0+ β2 P1 (8) d dt P1 (t) = −β2 P1+ω2 P0 (9) Under steady state equations (8) and (9) reduces to: P1 = ω2 β2 P0 (10) Using normalization condition P0+ P1= 1 It follows that P0= β2 β2+ω2 (11) 174 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 175 Table 2. Variation of Reliability of subsystems with time Time (in days) Rs1 (t) Rs2 (t) Rs3 (t) Rs4 (t) Rs5 (t) Rsys (t) 0 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 30 0.92774 0.91393 0.90032 0.88692 0.87372 0.59155 60 0.86071 0.83527 0.81058 0.78663 0.76338 0.34994 90 0.79852 0.76338 0.72979 0.69768 0.66698 0.20701 120 0.74082 0.69768 0.65705 0.61878 0.58275 0.12246 150 0.68729 0.63763 0.59156 0.54881 0.50916 0.07244 180 0.63763 0.58275 0.53259 0.48675 0.44486 0.04285 210 0.59156 0.53259 0.47951 0.43171 0.38868 0.02535 240 0.54881 0.48675 0.43171 0.38290 0.33960 0.01500 270 0.50916 0.44486 0.38868 0.33960 0.29671 0.00887 300 0.47237 0.40657 0.34994 0.30120 0.25924 0.00525 330 0.43823 0.37158 0.31506 0.26714 0.22650 0.00310 360 0.40657 0.33960 0.28365 0.23693 0.19790 0.00184 Table 3. Variation of maintainability of subsystem with time. Time (in days) Ms1(t) Ms2(t) Ms3(t) Ms4(t) Ms5(t) MS ys(t) 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 30 0.77687 0.95021 0.98889 0.99752 0.99945 0.72778 60 0.95021 0.99752 0.99988 0.99999 1.00000 0.94773 90 0.98890 0.99988 1.00000 1.00000 1.00000 0.98876 120 0.99752 0.99998 1.00000 1.00000 1.00000 0.99750 150 0.99945 0.99999 1.00000 1.00000 1.00000 0.99944 180 0.99988 1.00000 1.00000 1.00000 1.00000 0.99988 210 0.99997 1.00000 1.00000 1.00000 1.00000 0.99997 240 0.99998 1.00000 1.00000 1.00000 1.00000 0.99998 270 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 300 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 330 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 360 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 Figure 3. Transition diagram for subsystem R 3.3. RAMD indices for subsystem T d dt P0 (t) = −ω3 P0 + β3 P1 (12) d dt P1 (t) = − (β3 + ω3) P1 + ω3 P0 + β3 P2 (13) d dt P2 (t) = −β3 P2 + ω3 P1 (14) Under steady state, equation (1) - (3) yield P1 = β3 ω3 P2 (15) Substituting 4 into 1 under steady state we have: P2 = ω23 β23 P0 (16) Substituting 5 into 3 under steady state we have P1= ω3 β3 P0 (17) Using normalization condition P0+ P1+ P2= 1 It follows that: P0= β23 β23+ω 2 3 +β3ω3 (18) 175 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 176 Table 4. RAMD indices for the photovoltaic system RAMD indices ofSubsystems PV modulesS 1 Charge crS 2 Battery bankS 3 InverterS 4 Distributionboard PV System Reliability e−0.0025t e−0.003t e−0.003t e−0.004t e−0.0045t e−0.0175t Maintainability 1 − e−0.05t 1 − e−0.1t 1 − e−0.15t 1 − e−0.2t 1 − e−0.25t 1 − e−0.75t Availability 0.76923 0.97087 0.97668 0.98039 0.98232 0.70246 MTBF 400 333.33 285.71 250 222.22 1491.27 MTTR 20 10 6.67 5 4 45.67 Dependability 0.95729 0.97308 0.97867 0.98153 0.98328 0.87985 Dependability ratio 20 33.33 42.86 50 55.56 Table 5. Variation in system reliability as a result of PV Module failure rate variation PV System PV System PV modules PV modules Time (in days) ω1 = 0.0011 ω1=0.0012 ω1 = 0.0011 ω1=0.0012 0 1.00000 1.00000 1.00000 1.00000 30 0.61693 0.62438 0.96754 0.96464 60 0.38060 0.38985 0.93613 0.93053 90 0.23480 0.24341 0.90574 0.89763 120 0.14486 0.15198 0.87634 0.86589 150 0.08937 0.09489 0.84790 0.83527 180 0.05513 0.05925 0.82037 0.80574 210 0.03401 0.03700 0.79374 0.77724 240 0.02098 0.02310 0.76797 0.74976 270 0.01295 0.01442 0.74304 0.72325 300 0.00799 0.00901 0.71892 0.69768 330 0.00493 0.00562 0.69559 0.67301 360 0.00304 0.00351 0.67301 0.64921 3.4. RAMD indices for subsystem U d dt P0 (t) = −ω4 P0+ β4 P1 (19) d dt P1 (t) = −β4 P1+ω4 P0 (20) Under steady state equations (8) and (9) reduces to: P1 = ω4 β4 P0 (21) Using normalization condition P0+ P1= 1 It follows that P0= β4 β4+ω4 (22) 3.5. RAMD indices for subsystem V d dt P0 (t) = −ω5 P0+ β5 P1 (23) d dt P1 (t) = −β5 P1+ω5 P0 (24) Under steady state equations (8) and (9) reduces to: P1 = ω5 β5 P0 (25) Using normalization condition P0+ P1= 1 It follows that P0= β5 β5+ω5 (26) Using the following equations R (t) = ∫ ∞ t f (x) d x Reliability function (27) Availability = Li f e time T otal time = Li f e time Li f e time + Repair time = MT T F MT T F + MT T R (28) 176 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 177 Table 6. Variation in system reliability as a result of charge controller failure rate variation PV System PV System charge controller charge controller Time (in days) ω2 = 0.0013 ω2=0.0014 ω2 = 0.0013 ω2=0.0014 0 1.00000 1.00000 1.00000 1.00000 30 0.62251 0.62064 0.96175 0.95887 60 0.38752 0.38520 0.92496 0.91943 90 0.24123 0.23907 0.88959 0.88161 120 0.15017 0.14838 0.85556 0.84535 150 0.09348 0.09209 0.82283 0.81058 180 0.05820 0.05715 0.79136 0.77724 210 0.03623 0.03547 0.76109 0.74528 240 0.02255 0.02202 0.73198 0.71462 270 0.01404 0.01367 0.70398 0.68523 300 0.00877 0.00848 0.67706 0.65705 330 0.00544 0.00526 0.65116 0.63002 360 0.00339 0.00327 0.62625 0.60411 Table 7. Variation in reliability of system due to variation in failure rate of Battery bank PV System PV System Battery bank Battery bank Time (in days) ω3 = 0.0015 ω3=0.0016 ω3 = 0.0015 ω3=0.0016 0 1.00000 1.00000 1.00000 1.00000 30 0.62414 0.62625 0.95600 0.95313 60 0.36455 0.39219 0.91393 0.90846 90 0.24783 0.24561 0.87372 0.86589 120 0.15567 0.15382 0.83527 0.82531 150 0.09778 0.09633 0.79852 0.78663 180 0.06142 0.06033 0.76338 0.74976 210 0.03858 0.03778 0.72979 0.71462 240 0.02423 0.02366 0.69768 0.68113 270 0.01522 0.01482 0.66698 0.64921 300 0.00956 0.00928 0.63763 0.61878 330 0.00606 0.00581 0.60957 0.58978 360 0.00377 0.00364 0.58275 0.56214 Table 8. Variation in reliability of system due to variation in failure rate of inverter PV System PV System Inverter Inverter Time (in days) ω4 = 0.0017 ω4=0.0018 ω4 = 0.0017 ω4=0.0018 0 1.00000 1.00000 1.00000 1.00000 30 0.60230 0.63192 0.95028 0.94743 60 0.36277 0.39932 0.90303 0.89763 90 0.21850 0.25233 0.85813 0.85044 120 0.13160 0.15945 0.81546 0.80574 150 0.07926 0.10076 0.77492 0.76338 180 0.04774 0.06367 0.73639 0.72325 210 0.02875 0.04024 0.69977 0.68523 240 0.01732 0.02543 0.66498 0.64921 270 0.01043 0.01607 0.63192 0.61508 300 0.00628 0.01562 0.60050 0.58275 330 0.00378 0.00642 0.57064 0.55211 360 0.00228 0.00405 0.54227 0.52310 M (t) = 1 − e(− −t MT T R ) Maintainability function (29) MT BF = ∫ ∞ 0 R (t) dt = ∫ ∞ 0 e−θt = 1 θ (30) MTBF = Mean Time between Failure MT T R = 1 β Mean Time to Repair (31) 177 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 178 Table 9. Variation in system reliability due to variation in delivery board failure rate PV System PV System distribution board distribution board Time (in days) ω5 = 0.0019 ω5=0.002 ω5 = 0.0019 ω5=0.002 0 1.00000 1.00000 1.00000 1.00000 30 0.63954 0.63763 0.94460 0.94176 60 0.40902 0.40657 0.89226 0.88692 90 0.26158 0.25924 0.84282 0.83527 120 0.16729 0.16530 0.79612 0.78663 150 0.19700 0.10540 0.75201 0.74082 180 0.06843 0.06721 0.71035 0.69768 210 0.04376 0.04285 0.67100 0.65705 240 0.02799 0.02732 0.63381 0.61878 270 0.01790 0.01742 0.59870 0.58275 300 0.01145 0.01111 0.56553 0.54881 330 0.00732 0.00708 0.53420 0.51685 360 0.00468 0.00452 0.50459 0.48675 Figure 4. Transition diagram for subsystem U β= Repair rate,ω = Failure rate d = ξ ω = MT BF MT T R Dmin = 1 − ( 1 d − 1 ) ( e− ln d d−1 − e− d ln d d−1 ) 4. RAMD Analysis System Reliability Rsys (t) = Rs1 (t)×Rs2 (t)×Rs3 (t)×Rs4 (t)×Rs5 (t) = e−(ω1 +ω2 +ω3 + ω4 )t (32) System Availability Asys = As1 × As2 × As3 × As4 × As5 (33) Figure 5. Transition diagram for subsystem V System Maintainability Msys(t) = MS 1(t) . Ms2(t). Ms3(t) . Ms4(t). Ms5(t) = ( 1 − e−β1 (t) ) × ( 1 − e−β2 (t) ) × ( 1 − e−β3 (t) ) × ( 1 − e−β4 (t) ) × ( 1 − e−β5 (t) ) = 1 − e−(β1 +β2 +β3 +β4 +β5 )t (34) System dependability Dmin(sys) = Dmin(s1)×Dmin(s2)×Dmin(s3)×Dmin(s4)×Dmin(s5)(35) System dependability Dmin(sys) = Dmin(s1) × Dmin(s2) × Dmin(s3) × Dmin(s4) Dmin = 1 − ( 1 d − 1 ) ( e− ln d d−1 − e− d ln d d−1 ) d = β ω = MT BF MT T R d1 = β1 ω1 = 0.05 0.0025 = 20 178 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 179 d2 = β2 ω2 = 0.1 0.003 = 33.33 d3 = β3 ω3 = 0.15 0.0035 = 42.86 d4 = β4 ω4 = 0.2 0.004 = 50 d5 = β5 ω5 = 0.2 0.004 = 55.56 Dmin(s1) = 1 − ( 1 20 − 1 ) ( e− ln 20 20−1 − e− 20 ln 20 20−1 ) Dmin(s1) = 0.95729 Dmin(s2) = 1 − ( 1 33.33 − 1 ) ( e− ln 33.33 33.33−1 − e− 33.33 ln 33.33 33.33−1 ) Dmin(s2) = 0.97308 Dmin(s3) = 1 − ( 1 42.86 − 1 ) ( e− ln 42.86 42.86−1 − e− 42.86 ln 42.86 42.86−1 ) Dmin(s3) = 0.97867 Dmin(s4) = 1 − ( 1 50 − 1 ) ( e− ln 50 50−1 − e− 50 ln 50 50−1 ) Dmin(s4) = 0.98153 Dmin(s5) = 1 − ( 1 55.56 − 1 ) ( e− ln 55.56 55.56−1 − e− 55.56 ln 55.56 55.56−1 ) Dmin(s5) = 0.98328 Dmin(sys) = 0.95729 × 0.97308 × 0.97867 ×0.98153 × 0.98328 Dmin(sys) = 0.87985 System availability Asys =  β21 β21 + ω 2 1 + ω1 β1  × ( β2 β2 +ω2 ) ×  β23 β23 + ω 2 3 + ω3 β3  × ( β4 β4 +ω4 ) × ( β5 β5 +ω5 ) Asys = ( (0.05)2 (0.05)2 + (0.025)2 + (0.05) × (0.0025) ) × ( 0.1 0.10 + 0.0030 ) × ( (0.05)2 (0.05)2 + (0.0035)2 + (0.15)(0.0035) ) × ( 0.20 0.20 + 0.0040 ) ( 0.25 0.25 + 0.0045 ) Asys = 0.76923 × 0.97087 × 0.97668 × 0.98039 × 0.98232 Asys = 0.70246 5. Discussion To reflect the effect of system factors, we established for- mulations for availability, reliability, maintainability, and de- pendability for each subsystem of the model under considera- tion. Table 2 shows that after 300 days of operation, the PV device plant’s reliability stays 0.00525, however the distribu- tion board’s reliability is significantly poor (0.25924) across all subsystems. This sensitivity analysis reveals that the distribu- tion board’s reliability of the system requires special attention and close monitoring. As a result, machine designers must de- velop a maintenance plan. Tables 5- 9 showed how the relia- bility behavior of different subsystems improved over time and with varying failure rates. 6. Conclusion In this paper, an analytical analysis for a case was performed to obtain the reliability metrics of the various subsystems and procedure as presented in Table 1. Tables 2 and 3 demonstrate the findings for the reliability and maintenance habits of various subsystems, respectively. Table 4 summarizes the remaining RAMD phases. Based on this study, it is hypothesized that sub- system S4, i.e. the inverter, is the most critical and highly sen- sitive portion that requires special attention in order to improve the efficiency of the PV device plant. Hence it is concluded that machine designers must develop a maintenance plan for the in- verter to avoid total breakdown of the whole system and some techniques can be opted to enhance the system reliability. This research will show the method’s utility and could also be used by process engineers to learn how to apply RAMD concepts to process design. Real data acquisition in working environments will be the focus of future study. References [1] D. L. Huffman & F. Antelme, “Availability analysis of a solar power sys- tem with graceful degradation”, In: Proceedings of the Reliability and Maintainability Symposium, Fort Worth, TX (2009). [2] M.A. Hamdy, M. E. Beshir & S. E. Elmasry, “Reliability analysis of pho- tovoltaic systems”, Appl. Energy 33 (1989) 253. [3] D. De Graaff, R. Lacerda & Z. Campeau, “Degradation mechanisms in Si module technologies observed in the field; their analysis and statis- tics”, In: Presentation at PV Module Reliability Workshop; NREL. Den- ver, Golden, USA (2011). [4] P. Zhang, W. Li, S. Li, Y. Wang & W. Xiao, “Reliability assessment of photovoltaic power systems: Review of current status and future perspec- tives” Appl. Energy 104 (2013) 822. [5] R. Hu, J. Mi, T. Hu, M. Fu & P. Yang, “Reliability research for PV system using BDD- based fault tree analysis. In Proceedings of the 2013 Interna- tional Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering (QR2MSE)”, Chengdu, China (2013) 359. [6] M. Gahlot, H.I. Agaya, V.V. Singh & I. Abdullahi, “Stochastic analysis of a two units’ complex repairable system with switch and human failure using copula approach”, Life circle Reliability and safety Engineering 9 (2020) 1. [7] F. Chiacchio, F. Famoso, D. D’Urso, S. Brusca, J.I. Aizpurua & L. Cedola, “Dynamic performance evaluation of photovoltaic power plant by stochastic hybrid fault tree automaton model”, Energies 11 (2018) 1 https://doi.org/10.3390/en11020306 [8] Saini, M. & A. Kumar, “Performance analysis of evaporation system in sugar industry using RAMD analysis”, J Braz. Soc. Mech. Sci. Eng. 41 (2019) 175. https://doi.org/10.1007/s40430-019-1681-3 179 Maihulla & Yusuf / J. Nig. Soc. Phys. Sci. 3 (2021) 172–180 180 [9] D. Goyal, A. Kumar, M. Saini & H. Joshi, “Reliability, maintainability and sensitivity analysis of physical processing unit of sewage treatment plant” SN Appl. Sci. 1 (2019) 1507 https://doi.org/10.1007/s42452-019- 1544-7 [10] J. F. L. Van Casteren, M. H. J. Bollen & M. E. Schmieg, “Reliability assessment in electrical power systems: the weibull-markov stochastic model”, IEEE Transactions on Industry Appl. 36 (2000) 911 [11] M. C. Eti, S. O. T. Ogaji & S. D. Probert, “Reliability of the Afam electric power generating station, Nigeria”, Applied Energy 77 (2004) 309. [12] C. Ebeling, An introduction to reliability and maintainability engineer- ing, 10th edn. Tata McGraw-Hill, New Delhi, Published by McGraw-Hill (2008). [13] S. Gupta & P.C. Tewari, “Simulation modeling and analysis of complex system of thermal power plant”, J. of Industrial Eng. and Management 2 (2009) 387. [14] P. Tsarouhas, T. Varzakas & I. Arvanitoyannis, “Reliability and maintain- ability analysis of strudel production line with experimental data: a case study” J. Food Eng. 91 (2009) 250. [15] F.J.G. Carazas & G.F.M. Souza, “Availability analysis of gas turbines used in power plants”, Int. J. of Thermodynamics 12 (2009) 28. [16] A. K. Lado & V. V. Singh, “Cost assessment of complex repairable system consisting two subsystems in series configuration using Gumbel Hougaard family copula”, International Journal of Quality & Reliability Management 36 (2019) 1683. https://doi.org/10.1108/IJQRM-12-2018- 0322 . [17] V. V. Singh, P. K. Poonia & A. H. Adbullahi, “Performance analysis of a complex repairable system with two subsystems in series configura- tion with an imperfect switch”, J. Math. Comput. Sci., 10 (2020) 359. DOI: 10.28919/jmcs/4399 [18] I. Yusuf, A. L. Ismail & V. V. Singh et al., “Performance Analysis of Multi-computer System Consisting of Three Subsystems in Series Con- figuration Using Copula Repair Policy”, SN Comp. Sci. 1 (2020) 241. https://doi.org/10.1007/s42979-020-00258-0. [19] D. Raghav, D. Rawal, I. Yusuf, R. H. Kankarofi & V. Singh, “Reliability Prediction of Distributed System with Homogeneity in Software and Server using Joint Probability Distribution via Cop- ula Approach”, Reliability: Theory & Applications, 16 (2021) 217.https://doi.org/10.24412/1932-2321-2021-161-217-230 180