Ability Assesment of A Solar PV System Integrated with Greenhouse

Ability Assesment of A Solar PV System Integrated with Greenhouse

Ganesh Kumar Thakur1, Pawan Sharma 2 , Bandana Priya3

1. Assistant Professor,Department of Mathematics, Krishna Engineering College, Ghaziabad, India,

2. Assistant Professor, Department of Mathematics, Krishna Engineering College, Ghaziabad, India,

3. Assistant Professor, Department of Mathematics, G.L.Bajaj Institute of Technology& management, Greater Noida, India

Abstract

In this paper, the author deals with the ability assessment of a solar photovoltaic (PV) system integrated with greenhouse. The power produced from the solar PV system has been used to operate the required heating/cooling equipments inside the greenhouse. The block diagram of solar PV system has been shown in fig 1-(a). The authors have been used supplementary variables technique to mathematical formulation of the model. The difference-differential equations of various flow states are then solved subjected to Laplace transform. The reliability function, availability function and M.T.T.F have obtained. Steady-state behavior of the system and a particular case (when repairs follow exponential time distribution) have also been computed to improve practical utility of the model. A numerical example together with its graphical illustration has also appended in the end to highlight important results of the study.

Keywords : Reliability assessment, Markovian analysis, Laplace transform, MTTF, Steady state behavior

1. INTRODUCTION

   The whole system under consideration consists of four main subsystems A, B, C, and E, connected in series. Subsystem A is a solar panel and produced DC power from sunlight. Subsystem B is a charge controller and it controls the charging of batteries. Subsystem C is a battery bank and stores the DC power

   produced by the solar panel. Here, in this model, the subsystem C has two units, namely

   C1 and C2 in

   standby redundancy. Originally, one battery bank

   C1 works and on failure of C1 we can online standby

   battery bank C2 by the help of imperfect switching device D. In last, the subsystem E is an inverter and it converts 3.0KVA DC power to 220V/50Hz AC power. The flow of states for this system has been shown in fig-1(b).

   C1

   Inverter E

   Charge controller

   B

   Solar panel A

   Bat	tery Bank C

   D

   Switching

   Device Supply to

   C2 Greenhouse

   Fig-1 (a): Block Diagram of Solar PV System

   A

   PEV (r,t)

   EV

   EV (r)

   A (x)

   PB ( y,t)

   PA (x,t)

   B

   P0 (t)

   B ( y)

   EV

   (1 D )C

   E

   E (m)

   A

   PE (m,t)

   P R (z,t)

   A

   1

   P (x,t)

   C

   C (z)

   C

   0

   PC1 (t)

   A (x)

   B

   B

   D D

   (n)

   E (m)

   E

   B ( y)

   C

   PC1 ( y,t)

   D

   1

   C

   P (n,t)

   E

   1

   C

   P (m,t)

   C

   PW (t)

   States:

   Good

   Failed

   Fig-1 (b): Flow of states

2. ASSUMPTIONS

   The following assumptions have been associated with this model:

   (i.) Initially at t=0, all the subsystems and the system as a whole is operable.

   (ii.) Repair to subsystem C has given only if its both units are failed. In this case, the system has to wait for repair otherwise repair facilities are always available.

   (iii.) The whole system can also fail due to environmental reasons.

   (iv.) Failures are S-independent and nothing can fail from a failed state. (v.) Repairs are perfect i.e., after repair components work as new.

   (vi.) All failures follow exponential time distribution whereas all repairs follow general time distribution.

3. NOMANCLATURE

   P t/ PC1 t

   Pr {At time t, system is operable while unit C1 is working/failed}.

   0 0

   P j,t / PC1 j,t

   Pr {At time t, system is failed due to failure of ith subsystem and elapsed

   i i

   C

   PW t

   C

   P R z,t

   PEV r,t

   Ps Si x

   repair time lies in the interval j, j while unit C1 is working /failed}. Pr {At time t, system is failed due to failure of subsystem C and is waiting for repair}.

   Pr {At time t, system is ready for repair of subsystem C and elapsed repair time lies in the interval z, z }.

   Pr {At time t, system is failed due to environmental reasons and elapsed repair time lies in the interval r, r }.

   Laplace transform of function Pt .

   i (x) exp . i (x)dx, i and x.

   M.T.T.F. Mean time to failure.

   i Failure rate of ith

   subsystem.

   EV

   Failure rate due to environmental reasons.

   i j

   First order probability that ith subsystem will be repaired in the time interval

   j, j , conditioned that it was not repaired up to the time j.

4. FORMULATION OF MATHEMATICAL MODEL

   Probability considerations and limiting procedure yield the following set of difference-differential equations governing the behaviour of considered system, which is continuous in time and discrete in space:

   d

   dt

   A B

   E EV

   1 D

   C P0 t

   PA

   0

   x, t

   A x dx

   • PB y, tB ydy PE m, tE mdm

     C

     C

     0

   • PEV

   0

   r, t EV

   0

   r dr

   0

   P R z,t

   zdz

   (1)

   x

   y

   t A

   t B

   x PA

   y PB

   x, t 0

   y, t 0

   (2)

   (3)

   mP m,t 0

   (4)

   m t

   E E

   d

   C1

   C1

   dt

   A B E C

   D EV P0

   t C 1

   D P0 t

   • PA

     0

     x,t

     A x dx

     (5)

     B

     B

     PC1 y, t

     0

     ydy

     PC1 m, t

     E

     E

     0

     mdm

     PC1 n, t

     D

     D

     0

     ndn

     xPC1 x,t 0

     (6)

     x t

     A

     A

     C1 y,t 0

     (7)

     y t B

     y PB

     1

     mPC m,t 0

     (8)

     m t

     E E

     nPC1 n,t 0

     (9)

     n t

     D D

     PW t

     PC1 t

     (10)

     t

     C C 0

     zPR z,t 0

     (11)

     z t

     C C

     rP

     r,t 0

     (12)

     r t

     EV EV

     Boundary conditions are :

     PA 0,t A P0 t

     (13)

     PB 0,t B P0 t` (14)

     PE 0,t E P0 t

     (15)

     A

     0

     A

     PC1 0,t PC1 t

     (16)

     PC1 0,t

     PC1 t

     (17)

     B B

     PC1 0,t

     0

     PC1 t

     (18)

     1. D 0

        PC1 0,t PC1 t

        (19)

     2. E 0

     C

     PR 0,t PW t

     (20)

     C

     P 0,t

     P t PC1 t

     (21)

     EV EV 0 0

     Initial conditions are:

     P0 0 1, otherwise all state probabilities at t=0 are zero. (22)

5. SOLUTION OF THE MODEL

   Taking Laplace transforms of equations (1) through (21) subjected to initial conditions (22) and then on solving them one by one, we obtain the following Laplace transforms of various transition-state (depicted in fig-1b) probabilities:

   P0 s

   1

   Bs

   (23)

   P A s

   A

   Bs

   DA s

   (24)

   PB s

   PE s

   B

   Bs

   E

   Bs

   DB s

   DE s

   (25)

   (26)

   PC1 s C 1 D

   (27)

   0 AsBs

   PC1 s AC 1 D D

   s

   (28)

   A AsBs A

   PC1 s BC 1 D D

   s

   (29)

   B AsBs B

   PC1 s EC 1 D D

   s

   (30)

   E AsBs E

   PE1 s CD 1 D D

   s

   (31)

   D

   W

   AsBs D

   2 1

   (32)

   s AsBs

   PC s C D

   R

   2 1

   (33)

   s AsBs

   PC s C

   D DC s

   P s EV

   C 1 D

   D s

   (34)

   EV Bs 1 As EV

   D j 1 S i j for

   all

   i and j

   (35)

   where, i j

   As C EV s1 ADA s B DB s E DE s D DD s

   (36)

   and Bs s1 ADA s B DB s E DE s EV C 1 D

   2 1 S C s 1

   (37)

   C D EV 1 C

   D S EV s

   s As

   It is worth noticing that

   As

   Sum of equations (23) through (34) =

   1 (38)

   s

6. STEADY-STATE BEHAVIOUR OF THE SYSTEM

   By using Abels lemma, viz.,

   Lim P t Lim s

   Ps P (say), provided the limit on L.H.S. exists,

   t s0

   one can obtain the steady-state probabilities from equations (23) through (34):

7. PARTICULAR CASE

   When all repairs follow exponential time distribution

   In this case, we have obtained the following Laplace transforms of various flow state probabilities from

   equations (23) through (34) by putting

   S i s

   i

   s i

   , for all i:

   P0 s P A s P B s

   P s

   1

   Cs

   A

   Cs.

   B

   Cs.

   E .

   1

   s A

   1

   s B

   1

   (39)

   (40)

   (41)

   (42)

   E Cs

   s E

   PC1 s C (1 D )

   (43)

   0 EsC(s)

   PC1 s AC (1 D ) 1

   (44)

   A EsC(s) s

   A

   PC1 s B C (1 D ) 1

   (45)

   B EsC(s) s

   B

   PC1 s E C (1 D ) 1

   (46)

   E EsC(s) s

   E

   PC1 s C D (1 D ) 1

   (47)

   D

   W

   EsC(s) s D

   2 (1 )

   (48)

   PC s

   C D

   (s )EsC(s)

   R

   2 (1 ) 1

   (49)

   PC s

   C D

   (s )E s C(s) s C

   EV

   C (1 D ) 1

   (50)

   and P EV

   s Cs 1

   E(s)

   . s

   EV

   A B

   E D

   (51)

   A

   where, E s

   C EV

   s1 s

   s B

   s E

   s D

   and

   Cs

   s1

   A

   s A

   • B

     s B

   • E

     s

     EV

     E

   • C

     (1 D )

     2 (1 )

     (1 )

     (52)

     C D C

     1 C D EV

     EV

     (s )E(s)

     s C

     EV

     E(s)

     s

8. RELIABILITY AND M.T.T.F. OF THE SYSTEM

   Laplace transform of systems reliability is given by:

   Rs

   1

   s A B E EV

   C 1 D

   R(t) L1 R(s)

   exp. A B E EV

   Also, M.T.T.F. = lim R(s)

   s0

   C 1 D t

   (53)

   A

   B

   • E

     1

   • EV

   • C

     1 D

     (54)

9. AVAILABILITY EVALUATION

   We have

   1

   P s 1

   C (1 D )

   A

   up s

   • B

   • E

   • EV

   • C

     1 D

     s A

   • B

   • C

   • D

   • E

   • EV

   On taking inverse Laplace transform, we obtain

   Pup (t) 1 Aexp . A B E EV

   C 1 D t

   where,

   Aexp . A B C D E EV t

   A C (1 D )

   D (1 C )

   (55)

   (56)

   It is important to note that

   Pup (0) 1

   Also, Pdown(t) 1 Pup (t)

   (57)

10. NUMERICAL COMPUTATION

    For a numerical computation, let us consider the values

    A 0.001,

    B 0.02,

    C 0.003, D 0.4,

    E 0.04,

    EV

    0.005 an t = 0,1,2,—–.

    By using these values in equations (53), (54) and (55), one can compute the table- 1, 2, 3 and 4. The corresponding graphs have been shown in fig-2, 3, 4 and 5, respectively.

11. RESULTS AND DISCUSSION

    In this paper, the authors have considered a solar PV system to evaluate its reliability measures. Supplementary variables technique has been used to formulate a mathematical model for the system. The so obtained difference-differential equations have been solved by using Laplace transform. Reliability, availability and M.T.T.F. of the system have computed. Steady-state behaviour and a particular case (when all repairs follow exponential time distribution) have also mentioned to make the model more compatible. A numerical example has also appended in last to highlight important results of the study. By using this numerical example tables- (1) through (4) have been computed and the corresponding graphs have been shown by the figs- (2) through (5), respectively.

12. REFERENCES

    1. Adachi, K. and M. Kodama (1980); Availability analysis of two unit warm standby system with inspection time, Microelectron Reliab., 20, 449-455.

    2. Agnihotri, R.K. and S.K. Satsangi (1996); Two unit identical system with priority based on repair and inspection, Microelectron Reliab., 36, 279-282.

    3. Dhillon, B.S. and N-Yang (1992); Reliability and availability analysis of warm standby with common cause failure and human error, Microelectron Reliab., 32, 561-576.

    4. Goel, L.R., P. Srivastava and R. Gupta (1992); Two unit cold standby system with correlated failures and repairs, Int. Jr. of System Science, 23(3), 379-391.

    5. Goel, L.R. and P. Srivastava (1991); Profit analysis of a two unit redundant system with provision for rest and correlated failures and repairs, Microelectron Reliab., 31(5), 827-833.

    6. Gopalan, M.N., R. RadhaKrishna and A. Vijay Kumar (1984); Cost benefit analysis of a two unit cold standby system subject to slow switch, Microelectron Reliab., 24, 1019-1021.

    7. Kumar, A. and R. Lal (1979); Stochastic behaviour of a two unit standby system with constant failure and intermittently repair facility Int. Jr. of System Science, 10(6), 589-603.

    8. Murari, K. and Vibha Goel (1984) Comparison of two-unit cold standby reliability models with three types of repair facilities Microelectron Reliab., 24(1), 35-39.

    9. Osaki, S. (1972); Reliability of a two-unit standby redundant system with preventive maintenance, IEEE Trans. Reliab., R-21, 24-29.

    10. Singh, S.K. and A.K. Mishra (1994); Profitevaluating of a two unit cold standby redundant system with two operating systems; Microelectron Reliab., 34(4), 747-750.

1.2

1

R(t) —-->

0.8

0.6

0.4

0.2

0

R(t)

1 2 3 4 5 6 7 8 9 10 11

t —-->

Table-1 Fig-2

1.2

1

Pup —-->

0.8

0.6

0.4

0.2

0

Pup(t)

1 2 3 4 5 6 7 8 9 10 11

t —-->

Table-2 Fig-3

Table-3 Table-4

15.4

15.2

M.T.T.F. —-->

15

14.8

14.6

14.4

14.2

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

D ——>

M.T.T.F.

Fig-4

15.4

15.2

M.T.T.F. —-->

15

14.8

14.6

14.4

14.2

14

0 0 0 0.01 0.01 0.01

EV —-->

M.T.T.F.

Fig-5