Reliability Assessment of Solar Power System by Employing Algebra of Logics

Reliability Assessment of Solar Power System by Employing Algebra of Logics

Reliability Assessment of Solar Power System by Employing Algebra of Logics

Pankaj Singh 1 , K. P. Yadav 2, Abhishek Srivastava 3

1 Research Scholars, Department of Electronics Communication Engineering, NIMS University, Jaipur, Rajasthan

2Professor, Mangalmay Institute of Engineering and Technology, Greater Noida, India

3Professor, NIMS University, Jaipur, Rajasthan

Abstract–The author has been used algebra of logics for the formulation and solution of mathematical model of solar power system. Reliability of the whole system has been obtained. Reliability function for the system as a whole has been computed in two different cases e.g., when failures follow Weibull and exponential time distribution. An important reliability parameter, mean time to system failure (M.T.T.F.), has also been obtained to improve practical utility of the model. A numerical computation together with its graphical illustration has been mentioned in the end to highlight important results of the study.

Keyword- Reliability Assessment, Solar Power System, Algebra of Logics, Boolean Algebra etc.

I. INTRODUCTION

In this model, the author has considered a solar power system for its reliability analysis by using algebra of logics. System configuration of considered system has been shown in fig-1. In this system, there are two solar panels in standby redundancy. On failure of solar panel x1 we can follow online

1. Switching device used to online standby solar panel is perfect.

2. Failures are statistically-independent.

SOLAR SYSTEM

Fig-1 (System Configuration)

1. NOTATIONS USED

   solar panel

   x2 with the help of perfect switching device

   x3 .

   Following notations have been used throughout this study:

   x1 , x2	States of solar panels.
   x3	State of switching device.
   x5 , x6 , x7 , x8	States of batteries.
   x4 , x9	States of automatic switching devices.
   x10	State of sine-wave inverter.
   xi	0 in bad state, 1 in good state.
   /	Conjunction/disjunction.
   RS	Reliability of whole system.
   Ri	Reliability corresponding to system state xi .
   RSW t/ RSE t	Reliability functions for whole system, in case failures follow weibull/exponential time distribution.
   M.T.T.F.	Mean time to failure.

   x1 , x2	States of solar panels.
   x3	State of switching device.
   x5 , x6 , x7 , x8	States of batteries.
   x4 , x9	States of automatic switching devices.
   x10	State of sine-wave inverter.
   xi	0 in bad state, 1 in good state.
   /	Conjunction/disjunction.
   RS	Reliability of whole system.
   Ri	Reliability corresponding to system state xi .
   RSW t/ RSE t	Reliability functions for whole system, in case failures follow weibull/exponential time distribution.
   M.T.T.F.	Mean time to failure.

   These solar panels produce the solar energy and it can be

   saved in four batteries

   B1, B2 , B3 and B4 . There are two

   automatic switching devices ASD1 and ASD2 to change the battery links. Batteries supply DC power to the sine-wave

   inverter

   x10

   ,that converts this DC power into AC power.

   Finally, this AC power feed the connected electric equipments. The cables used to connect any two equipments are hundred percent reliable. Note that any one battery is sufficient to feed all the consumer requirements.

   II. ASSUMPTIONS

   The following assumptions have been associated with this model:

   1. Initially, all the components are good and operable.

   2. Reliability of every component of the system is known in advance.

   3. Each component will remain either good or bad.

   4. Transaction of power from one component to any other is

   100% reliable.

   100% reliable.

   100% reliable.

   100% reliable.

2. FORMULATION OF MATHEMATICAL MODEL

   B8 x2

   x3 x8

   (11)

   The possible minimal paths for successful operation of the system, in terms of logical matrix, can be expressed as:

   in equation (3) and using orthogonalisation algorithm, we obtain:

   x1 x4

   x5 x9

   x10 B

   1

   1

   x x x x x

   1 4 6 9 10

   B1 B2

   x1 x4 x7

   x9 x10

   B B B

   x x x x x

   1 2 3

   F x , x

   , x

   1 4 8

   9 10

   B1 B2 B3 B4

   1 2 10 x x x

   x x x

   f x1 , x2 , x10

   2 3 4

   5 9 10

   B

   1

   1

   B2

   B3

   B4 B5

   x2

   x3 x4

   x6 x9

   x10

   B B

   B B

   B B

   x x x x x x

   1 2 3 4 5 6

   2 3 4 7 9 10

   B B B B B B B

   x x x x x x

   1 2 3 4 5 6 7

   2 3 4 8 9 10

   B B B B B B B B

   1 2 3 4 5 6 7 8

   (1)

3. SOLUTION OF THE MODEL We can write the equation (1) again as :

   (12)

   Now we can obtain by using algebra of logics

   Fx1 , x2 , x10 x4

   x9 x10 f x1 , x2 , x10

   (2)

   B1

   x1

   x

   x

   where,

   1 5

   x1 x5

   BB

   x1

   x x

   x x

   1 2 x

   x 1 6

   1 6 1 5

   x1 x7

   x x

   x1

   x5

   x6

   (13)

   f x , x

   , x

   1 8

   1 2 10 x

   x x

   Similarly, we compute the following:

   2 3 5

   x2 x3 x6

   B1 B2 B3 x1 x5 x6 x7

   x

   x

   x

   x

   2 3

   x7

   B B B B

   x

   x x x

   (14)

   x

   x2

   x3 x

   8

   1 2 3 4

   1 5 6

   7 8

   (15)

   Substituting

   (3)

   B1 B2 B3 B4 B5 x1 x2 x3 x5

   (16)

   B x x

   B1 B2 B3 B4 B5 B6 x1 x2 x3 x5 x6

   1 1 5

   (4)

   (17)

   B2 x1 x6

   (5)

   (18)

   B3 x1 x7

   (6)

   and

   B4 x1 B5 x2 B6 x2 B7 x2

   x8

   x3 x3 x3

   x5 x6 x7

   (7)

   (8)

   (9)

   (10)

   (19)

   Making use of equations (4) and (13) through (19), equation

   (12) becomes:

   x1 x5

   x x x

   1 5 6

   x1

   x

   x5 x

   x6 x

   x7

   x x

   f x , x , x

   1 5 6 7 8

   1 2 10

   x x x x

   1 2 3 5

   x1 x2

   x x

   x3 x5

   x x

   x6

   x x

   1 2 3 5 6 7

   x1 x2 x3 x5 x6 x7 x8

   (20)

   Using (20), equation (2) gives:

   F x1 , x2

   , x10

   x1 x4

   x

   1 x4

   x1 x4

   x

   x

   1 x4

   x x

   1 2

   x1 x2

   x5 x5 x5 x5 x3

   x3

   x9 x6 x6 x6 x4

   x4

   x10 x9 x7 x7 x5

   x5

   x10 x9 x8 x9

   x6

   x10 x9 x10

   x9

   x10

   x10

4. SOME PARTICULAR CASES

   (22)

   x

   x

   x

   x

   1 2

   x3 x4

   x5

   x6 x7

   x9 x10

   CASE I: When each component has the reliability R:

   In this case, the reliability of the whole system can be

   x1 x2

   x3 x4

   x5

   x6

   x7

   x8 x9

   x10

   obtained from equation (22),

   (21)

   Since, R.H.S. of equation (21) is disjunction of pair-wise disjoint conjunctions, therefore, the reliability of whole solar system is given by:

   RS Pr Fx1 , x2 , x10 1 =

   R 4R5 2R6 6R7 9R8 5R9 R10

   S

   S

   (23)

   CASE II: When failure rates follow weibull time distribution:

   In this case, the reliability of the whole system is given by:

   23 22

   23 22

   R t exp. a t exp. b t

   SW

   i1

   i

   j 1

   j

   (24)

   where

   Ri = reliability of the component corresponding to system

   state xi

   where is a real positive parameter and

   a1 c 1 5

   a2 c 1 6

   and Qi 1 Ri

   Thus, we may write

   a3 c 1 7

   a4 c 1 8

   a5 c 2 3 5 a6 c 2 3 6 a7 c 2 3 7 a8 c 2 3 8

   a9 c 1 5 6 7 a10 c 1 5 6 8 a11 c 1 5 7 8 a12 c 1 6 7 8

   a13 c 1 2 3 5 6 a14 c 1 2 3 5 7 a c

   CASE III: When failures follow exponential time distribution:

   Exponential time distribution is the particular case of weibull time distribution for 1 and is very useful in

   15 1 2 3 6 7

   a16 c 2 3 5 6 7

   a17 c 1 2 3 5 8

   numerous practical problems. Therefore the reliability

   23

   23

   function for the whole system at any timet, in this case, is given by:

   a c

   R t

   exp. a t

   22

   exp. b t

   18

   a19

   1

   c 1

   2

   2

   3

   3

   6

   7

   8

   8

   SE

   i1

   i

   j 1

   j

   (25)

   a20 c 2 3 5 6 8

   where as and bs have been mentioned earlier.

   i j

   a21 c 2 3 5 7 8

   a22 c 2 3 6 7 8

   Also, in this case, an important reliability parameter

   M.T.T.F , is given by

   a23 c 1 2 3 5 6 7 8

   b1 c 1 5 6

   M .T.T.F. RSE t dt

   0

   b2 c 1 5 7 b3 c 1 6 7 b4 c 1 5 8 b5 c 1 6 8

   1

   1

   23

   i1 ai

   22

   1

   1

   j 1 bj

   (26)

   b6 c 1 7 8

   b c

5. RESULTS AND CONCLUSION

For a numerical computation, let us consider the values:

7 1 2 3 5

b c

(i) i i 1,2, 10 0.001, t 0,1,2 – – – and

8 1 2 3 6

b9 c 2 3 5 6

2 . Using these values in equation (24), we compute the table -1. The corresponding graph has been shown in fig-2.

b10 c 1 2 3 7

(ii)

i 1,2,3 10 0.001, and

t 0,1,2 – – – .

b11

c 2

3

5

7

i

Using these values in equation (25), we compute the table-1

b12 c 2 3 6 7

b13 c 1 2 3 8

and the corresponding graph has been shown in fig-2.

(iii) Putting

b14 c 2 3 5 8

i i 1,2,3 10 0.001,0.002 – – – 0.01 in

b15 c 2 3 6 8

equation (26), we compute table-2 and its graphical

b16

c 2

3

7

8

reorientation has been shown in fig-3.

b17 c 1 5 6 7 8

b18 c 1 2 3 5 6 7 b19 c 1 2 3 5 6 8 b20 c 1 2 3 5 7 8 b21 c 1 2 3 6 7 8 b22 c 2 3 5 6 7 8

Analysis of table -1 and fig-2 reveals that values of reliability function decreases catastrophically, in case, failures follow weibull time distribution but it decreases approximately in constant manner for exponential time distribution. Therefore, reliability function remains better when failures follow exponential time distribution.

A critical examination of table -2 and fig-3 concludes that

M.T.T.F. decreases rapidly for the lower values of failure rate

and c 4 9 10

but it decreases smoothly for higher values of .

where i

is the failure rate of system state

xi ,i 1,2, 10

TABLE-1

Fig-2

TABLE-2

Fig-3

REFERENCES

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