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(k1,k2,…,km) -out-of-n system and its reliability
Accepted Manuscript (k1 , k2 , . . . , km )-out-of-n system and its reliability Serkan Eryilmaz
PII: DOI: Reference:
S0377-0427(18)30462-X https://doi.org/10.1016/j.cam.2018.07.036 CAM 11822
To appear in:
Journal of Computational and Applied Mathematics
Received date : 6 April 2018 Revised date : 16 June 2018 Please cite this article as: S. Eryilmaz, (k1 , k2 , . . . , km )-out-of-n system and its reliability, Journal of Computational and Applied Mathematics (2018), https://doi.org/10.1016/j.cam.2018.07.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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(k1 ; k2 ; :::; km )-out-of-n system and its reliability by Serkan ERYILMAZ*
Abstract
This paper is concerned with a system consisting of multiple types of components and having (k1 ; k2 ; :::; km )-out-of-n structure. The (k1 ; k2 ; :::; km )-out-of-n system is a system consisting of ni components of type i, i = 1; 2; :::; m, and functions if at least k1 components of type 1, k2 components of type 2, ..., km components of type m work, n =
Pn
i=1
ni : The exact and approximate expressions are obtained for the
survival function of the system under concern. The weighted-(k1 ; k2 ; :::; km )-out-of-n system is also de…ned and studied. This weighted model is applied to evaluate the wind power system that consists of two wind plants. Keywords. Dependence; k-out-of-n system; Reliability approximation; Wind power system *
Atilim University, Department of Industrial Engineering, 06836, Incek, Ankara, Turkey, e-mail:
[email protected], Fax: +90 312 586 80 91
1
1
Introduction
Most of the complex engineering systems consist of components which perform di¤erent types of tasks. The components that have di¤erent functions within the system are expected to have di¤erent reliability characteristics. Thus, most of the systems are composed with multiple types of components which are classi…ed not only with respect to their roles within the system but also with respect to their failure time distributions. For example, a certain production system may consist of machines performing di¤erent types of tasks, e.g. cutting, packing. The cutting and packing machines are expected to have di¤erent failure time distributions. Recently, systems with multiple types of components have attracted great deal of attention in reliability literature. Aboalkhair et al. (2014) studied the problem of nonparametric predictive inference for reliability of a k-out-of-m:G system with multiple component types. Feng et al. (2016) studied imprecise system reliability and component importance for a system consisting of multiple types of independent components. Patelli et al. (2017) proposed a simulation method for reliability of a system with multiple types of independent components using survival signature. Eryilmaz et al. (2018) obtained expressions for marginal and joint reliability importances for a coherent system that consists of multiple types of dependent components. Eryilmaz et al. (2018) studied mean residual life of coherent systems consisting of multiple types of dependent components. 2
The ordinary k-out-of-n system consists of n components which perform same tasks, and operates if at least k components work properly (Kong and Ye (2016), Amari et al. (2018), Cui et al. (2018)). In a more general setting, we might have multiple types of components having di¤erent functions, and di¤erent numbers of components of each type may be required for the proper operation of the whole system. Such a situation can be modeled by (k1 ; k2 ; :::; km )-out-of-n system. The (k1 ; k2 ; :::; km )-out-of-n system is a system consisting of ni components of type i, i = 1; 2; :::; m, and functions if at least k1 components of type 1, k2 components of type 2, ..., km components of type m work, n =
Pn
i=1
ni . The (k1 ; k2 ; :::; km )-out-of-
n system is useful for modeling various real life complex systems. For example, an airplane consists of various subsystems such as hydraulic system, radar system, and communication system. Active redundancy is applied to most of these subsystems. For example, hydraulic system redundancy may be achieved by multiple pressure sources which help to ensure that the entire hydraulic system is not lost in the event of a single component failure. Thus, for a total of m subsystems of an airplane, at least ki components of type i, i = 1; 2; :::; m are required for proper operation of the airplane. Because all components are harmonically integrated to constitute the whole system, dependence is inevitable between various subsystems. Thus assuming a statistical dependence not only between the components of each subsystem but also among subsystems is very realistic and more applicable.
3
In this paper, we study the (k1 ; k2 ; :::; km )-out-of-n system assuming that the random failure times of components of the same type are exchangeable and dependent, and that the random failure times of components of di¤erent type are dependent. In particular, we have two levels of dependence. The 1st level (or local) dependence de…nes the dependence between the components of same type, and the 2nd level (or global) of dependence is a dependence among di¤erent types of components. We obtain exact and approximate expressions for the survival function of the (k1 ; k2 ; :::; km )-out-of-n system under such a dependence model. The weighted(k1 ; k2 ; :::; km )-out-of-n system is also de…ned and studied. The results are illustrated and an application concerned with wind power systems is also presented. The paper is organized as follows. In Section 2, we de…ne the (k1 ; k2 ; :::; km )-outof-n system model and study its survival function. Section 3 involves extension of the model to a system with weighted components. In Section 4, some numerical results are presented on survival function of the (k1 ; k2 ; :::; km )-out-of-n system. Section 5 contains an application of the weighted-(k1 ; k2 ; :::; km )-out-of-n system to evaluate the capacity of the wind power system that consists of two wind plants.
2
The (k1 ; k2 ; :::; km )-out-of-n system
We …rst …x the notation that will be used throughout the paper. n: The total number of components,
4
ni : The number of components of type i; i = 1; 2; :::; m; ki : The minimum number of working components of type i for the operation of the system, (i)
Tj : The lifetime of component j of type i; j = 1; 2; :::; ni ; (i)
(i)
(i)
(i)
Tr:ni : The rth smallest among T1 ; T2 :::; Tni ; Tk1 ;:::;km :n : The lifetime of the (k1 ; k2 ; :::; km )-out-of-n system, (i)
Sni (t): The total number of working components of type i at time t: Recall that the (k1 ; k2 ; :::; km )-out-of-n system is a system consisting of ni components of type i, i = 1; 2; :::; m, and functions if at least k1 components of type 1, k2 components of type 2, ..., km components of type m work. Thus the lifetime of the (k1 ; k2 ; :::; km )-out-of-n system is represented as (1) (2) (m) k1 +1:n1 ; Tn k2 +1:n2 ; :::; Tn km +1:nm );
Tk1 ;:::;km :n = min(Tn for ki
(1)
ni ; i = 1; 2; :::; m:
Any coherent system that consists of multiple types of dependent components can be written as a (generalized) mixture of (k1 ; k2 ; :::; km )-out-of-n systems. Indeed, if T denotes the lifetime of the coherent system, then
P fT > tg =
n1 X n2 X
k1 =0 k2 =0
nm X
cn1 ;n2 ;:::;nm (k1 ; k2 ; :::; km )
k =0
m n o (1) (2) (m) P min(Tn1 k1 +1:n1 ; Tn2 k2 +1:n2 ; :::; Tnm km +1:nm ) > t ;
5
(2)
where the coe¢ cients cn1 ;n2 ;:::;nm (k1 ; k2 ; :::; km ) depend on the structure of the coherent system (Eryilmaz (2016)). Another motivation for studying the (k1 ; k2 ; :::; km )-outof-n system lies in the representation given by (2). It is assumed that the random failure times of components of the same type are exchangeable and dependent, and that the random failure times of components of di¤erent type are dependent. That is, there are two levels of dependence: The 1st level (or local) dependence de…nes the dependence between the components of same type, and the 2nd level (or global) of dependence is a dependence among di¤erent types of components. More explicitly, for …xed i, the vector of failure (i)
(i)
(i)
times (T1 ; T2 :::; Tni ) is assumed to have exchangeable distribution, and the vec(1)
(1)
(1)
(2)
(2)
(2)
(m)
tors (T1 ; T2 :::; Tn1 ); (T1 ; T2 :::; Tn2 ); :::; (T1
(m)
; T2
(m)
:::; Tnm ) are dependent. Be-
low, we give an example for illustrating the dependence model. Example 1 Let n o (1) (1) (m) (m) (1) (m) (m) P T1 > t1 ; :::; Tn(1) > t ; :::; T > t ; :::; T > t 1 1 n1 nm nm 1 " # n1 nm X X (1) (m) ; = 1+ 1 tj tj + ::: + m (j)
for ti
0, i = 1; :::; nj ; j = 1; :::; m;
(3)
j=1
j=1
i
> 0;
> 0. That is, the joint distribution of
the components is modeled by multivariate Pareto distribution which is a useful model (i)
(i)
(i)
in reliability. Then, the joint survival function corresponding to (T1 ; T2 :::; Tni ) is " # ni n o X (i) (i) (i) P T1 > t1 ; :::; Tn(i)i > t(i) = 1+ i tj : (4) ni j=1
6
Obviously, the joint survival function given by (4) is exchangeable.
The second level or global dependence creates a dependence among (1) (2) (m) k1 +1:n1 ; Tn k2 +1:n2 ; :::; Tn km +1:nm .
The originality of the model lies in consider-
Tn
ing such a dependence. We …rst obtain an exact expression for the survival function of the system. Using the representation given by (1), n o (1) (2) (m) P fTk1 ;:::;km :n > tg = P Tn k1 +1:n1 > t; Tn k2 +1:n2 > t; :::; Tn km +1:nm > t = P Sn(1) (t) 1
=
n1 X
k1 ; Sn(2) (t) 2
nm X
r1 =k1
k2 ; :::; Sn(m) (t) m
km
P Sn(1) (t) = r1 ; Sn(2) (t) = r2 ; :::; Sn(m) (t) = rm(5): m 1 2
rm =km (1)
(2)
(m)
From Theorem 1 of Eryilmaz (2017), the joint distribution of Sn1 (t); Sn2 (t):::; Snm (t) can be written as P Sn(1) (t) = r1 ; :::; Sn(m) (t) = rm m 1 =
n1 r1 nm Xrm n1 nm X n 1 r1 n m rm ::: ( 1)j1 +:::+jm ::: r1 rm j =0 j1 jm jm =0 1 n o (1) (m) P min(T1:r1 +j1 ; :::; T1:rm +jm ) > t
(6)
Using (6) in (5), one obtains P fTk1 ;:::;km :n > tg =
n1 X
r1 =k1 nX 1 r1
nm X
n1 nm ::: r1 rm
rm =km
nm Xrm
( 1)j1 +:::+jm
n1
r1
j1 jm =0 n o (1) (m) P min(T1:r1 +j1 ; :::; T1:rm +jm ) > t : j1 =0
7
:::
nm
rm jm
Letting r1 + j1 = i1 ; ::::; rm + jm = im ; the survival function can be written as P fTk1 ;:::;km :n > tg =
n1 X
nm X
i1 =k1
im =km
o n (m) (1) (i1 ; :::; im )P min(T1:i1 ; :::; T1:im ) > t ;
(7)
(i1 ; :::; im ) =
i1 X
r1 =k1 n a;b;c
2.1
=
im X
i1 r1 +:::+im rm
( 1)
m Y j=1
rm =km
rj ; ij
nj r j ; nj
ij
;
n! : a!b!c!
An approximation
The complexity of (7) may prevent its direct use especially for large values of m and ni s. Therefore, a simpler formula for computing P fTk1 ;:::;km :n > tg is desirable. In this section, we present an approximation using
P
(m \
i=1
Ai
)
'
m Y1
P fAi Ai+1 g
i=1 m Y1 i=2
(8)
P fAi g
(see, e.g. Block et al. (1991)). Clearly, P fTk1 ;:::;km :n > tg = P where Ai
n (i) Tni
ki +1:ni
(m \
i=1
Ai
)
;
o > t ; i = 1; 2; :::; m: Thus, for approximating the survival
function we need to get expressions for the following probabilities: n (i) P fAi g = P Tni 8
ki +1:ni
o >t ;
and n (i) P fAi Ai+1 g = P Tni
(i+1) t; Tni+1 ki+1 +1:ni+1
>
ki +1:ni
o >t :
Because the components of same type are exchangeable, n (i) P fAi g = P Tni
ki +1:ni
ni o X >t = P Sn(i)i (t) = r ; r=ki
n o (i) where P Sni (t) = r can be computed from P
Sn(i)i (t)
=r =
n i r X
ni r
ni
( 1)j
r j
j=0
o n (i) (i) P T1 > t; :::; Tr+j > t :
Thus n (i) P fAi g = P Tni = =
ni n i r X X
ki +1:ni
( 1)j
r=ki j=0 ni X s=ki
o >t ni r
ni
r j
n o (i) (s)P T1:s > t ;
o n (i) P T1:r+j > t
where (s) =
s X
( 1)s
r
r; s
r=ki
ni r; ni
s
:
On the other hand, n (i) P fAi Ai+1 g = P Tni =
ki +1:ni
ni+1 ni X X
r1 =ki r2 =ki+1
>
(i+1) t; Tni+1 ki+1 +1:ni+1
o >t
n o (i) (i+1) P Sni (t) = r1 ; Sni+1 (t) = r2
9
By using (6), we obtain n o (i) (i+1) P Sni (t) = r1 ; Sni+1 (t) = r2 =
ni r1
nX i r1 ni+1 Xr2
ni+1 r2
j1 =0
ni
( 1)j1 +j2
r1
ni+1 r2 j2
j1
j2 =0
o n (i+1) (i) P min(T1:r1 +j1 ; T1:r2 +j2 ) > t : Thus n (i) P fAi Ai+1 g = P Tni =
(i+1)
ki +1:ni
> t; Tni+1
ni+1 ni r1 ni+1 r2 ni X X X X
r1 =ki r2 =ki+1 j1 =0
ki+1 +1:ni+1
( 1)j1 +j2
j2 =0
o >t
ni r1
ni+1 r2
ni
r1 j1
ni+1 r2 j2
n o (i) (i+1) P min(T1:r1 +j1 ; T1:r2 +j2 ) > t
=
ni+1 ni X X
i1 =ki i2 =ki+1
n o (i) (i+1) (i1 ; i2 )P min(T1:i1 ; T1:i2 ) > t ;
where (i1 ; i2 ) =
i1 i2 X X
( 1)i1
r1 +i2 r2
r1 =ki r2 =ki+1
r1 ; i1
ni r 1 ; ni
i1
r2 ; i2
ni+1 r2 ; ni+1
i2
:
Thus the survival function of the system can be approximated by
P fTk1 ;:::;km :n > tg '
m Y1
ni+1 ni X X
i=1 i1 =ki i2 =ki+1
n o (i) (i+1) (i1 ; i2 )P min(T1:i1 ; T1:i2 ) > t
ni m Y1 X i=2 s=ki
n o (i) (s)P T1:s > t
:
(9)
The approximation formula given by (9) is simpler in terms of computational complexity when compared with the exact expression given by (7). 10
3
Extension to a system with weighted components
In a system with weighted components, each component contribute di¤erently to the performance of the whole system. See, e.g. Li and Zuo (2008), Armutkar and Kamalja (2014, 2017), Rahmani et al. (2016), Zhu and Boushaba (2017), Meshkat and Mahmoudi (2017). In this section, we de…ne weighted-(k1 ; k2 ; :::; km )-out-of-n system. The components of same type are assumed to have the same weight/contribution. The weight of all components of type i is denoted by ! i ; i = 1; :::; m. The entire system is able to perform its intended task i¤ the total weight of functioning components of type 1 is at least k1 ; the total weight of functioning components of type 2 is at least k2 ; :::; and the total weight of functioning components of type m is km : The total weight of functioning components of type i at time t is de…ned by Wi (t) =
ni X
(i)
! i I(Tj > t) = ! i Sn(i)i (t);
j=1
(i)
(i)
(i)
(i)
where I(Tj > t) = 1 if Tj > t; and I(Tj > t) = 0 if Tj
t:
Let Tk!1 ;:::;km :n denote the lifetime of a weighted-(k1 ; k2 ; :::; km )-out-of-n system. Then P Tk!1 ;:::;km :n > t = P fW1 (t)
k1 ; W2 (t)
11
k2 ; :::; Wm (t)
km g :
By conditioning on the number of working components of each type, we obtain P
Tk!1 ;:::;km :n
>t
n1 X
=
r1 :! 1 r1 k1 nX 1 r1
nm X
n1 nm ::: r1 rm
rm :! m rm km nm Xrm
n1
( 1)j1 +:::+jm
r1
j1 jm =0 n o (m) (1) P min(T1:r1 +j1 ; :::; T1:rm +jm ) > t : j1 =0
:::
nm
rm jm (10)
The lifetime of weighted-(k1 ; k2 ; :::; km )-out-of-n system can be represented as Tk!1 ;:::;km :n = min(T1 ; T2 ; :::; Tm );
(11)
where T1 =
(1) Tr:n 1
T2 =
(2) Tr:n 1
.. .
n (1) i¤ W1 (Tr 1:n1 )
n (2) i¤ W2 (Tr 1:n2 )
(m) i¤ Tm = Tr:n m
k1 and
(1) W1 (Tr:n ) 1
k2 and
(2) W2 (Tr:n ) 2
o
< k1 ; o
< k2 ;
o (m) km and Wm (Tr:n ) < k m : m
n (m) Wm (Tr 1:nm )
The representation given by (11) is useful to simulate the lifetime of the system. An approximation for P Tk!1 ;:::;km :n > t can be obtained using (8) with P fAi g = P fWi (t) =
ni n i r X X
ki g = P ! i Sn(i)i (t) ( 1)j
r:! i r ki j=0
12
ni r
ni
r j
ki n o (i) P T1:r+j > t
and n P fAi Ai+1 g = P ! i Sn(i)i (t) =
ni X
ki ; ! i+1 Sn(i+1) (t) i+1 ni+1
nX i r1 ni+1 Xr2
X
r1 :! i r1 ki r2 :! i+1 r2 ki+1 j1 =0
ni
r1 j1
ni+1 r2 j2
ki+1
o
( 1)j1 +j2
j2 =0
ni r1
ni+1 r2
n o (i+1) (i) P min(T1:r1 +j1 ; T1:r2 +j2 ) > t :
P fAi g represents the probability that the total weight of functioning components of type i is at least ki . Thus the survival function of the system can be approximated by P Tk!1 ;:::;km :n > t
'
m Y1
ni X
ni+1
nX i r1 ni+1 Xr2
X
i=1 r1 :! i r1 ki r2 :! i+1 r2 ki+1 j1 =0 m Y1
j2 =0
n o (i) (i+1) ( 1)j1 +j2 Cni ;ni+1 (r1 ; r2 ; j1 ; j2 )P min(T1:r1 +j1 ; T1:r2 +j2 ) > t
n ni i r X X
( 1)j
ni r
ni r j
i=2 r:! i r ki j=0
Cni ;ni+1 (r1 ; r2 ; j1 ; j2 ) =
4
ni r1
ni+1 r2
ni
r1 j1
n o (i) P T1:r+j > t
ni+1 r2 : j2
Numerical illustrations
To study the performance of the approximation given by (8), we consider a system consisting of three di¤erent types of components, i.e. m = 3. Assume that the joint survival function of components’lifetimes is modeled by (2) with and
1
= 1;
2
= 2;
3
=3
= 3: In Tables 1, we compute exact (when possible) and approximate values of
R((k1 ; :::; km ); (n1 ; :::; nm ) : G) = P fTk1 ;:::;km :n > tg for di¤erent choices of the number 13
of components of each type. For larger values of ni s, the combinatorial terms involved in (7) prevent the use of the equation and hence for n1 = 20; n2 = 15; n3 = 10, we compare approximation with the simulated value. The approximation performs quite well for high reliability values in all cases. However, as it is clear from Table 1, it might be poor with an increase in t for some values of ni s. Overall the approximation given by (8) can be used for highly reliable structures and its potential use, in general, depends on the values of ni s. n1
n2
n3
k1
10 15 10 5
Exact
Approximation
k2
k3
t
3
4
0.05 0.9127 0.9104 0.1
0.6282 0.6053
0.25 0.1804 0.1450 0.5 20 15 10 5
3
4
0.0419 0.0284
0.05 0.9173 0.9125 0.1
0.6375 0.6384
0.25 0.1875 0.1869 0.5 20 15 10 5
3
2
0.0452 0.0431
0.05 0.9900 0.9858 0.1
0.8780 0.8770
0.25 0.4089 0.4045 0.5 14
0.1284 0.1211
Table 1. Exact and approximate values of R((k1 ; k2 ; k3 ); (n1 ; n2 ; n3 ) : G)
5
An application
A wind plant typically consists of wind turbines which convert the kinetic energy in the wind into electrical energy. If the wind speed is below the cut-in wind speed (vci ), then no electrical power is produced since in this case the power in the wind is not enough. If the wind speed is between the cut-in wind speed and rated wind speed (vr ), then the wind turbine generates power and there is a cubic relationship between the wind speed and the generated power. If the wind speed is above the rated wind speed and below the cut-out wind speed (vco ), then the wind turbine generates power at a constant rate Pr . If the wind speed is above the cut-out wind speed, the turbine is stopped since it is in a danger of mechanical failure. The values of vci ; vco ; vr and Pr are determined by the wind turbine model. The power generated by a single wind turbine as a function of wind speed random variable V is given by (see, e.g.Louie Sloughter (2014)).
g(V ) =
8 > > > > > > > > > > > > > > <
0;
if V < vci or V
(V 3 v 3 )
Pr (v3 v3ci) ; > r ci > > > > > > > > > > > > > : Pr ;
15
if vci
V < vr
if vr
V < vco
vco
(12)
Consider two wind plants that are located in two di¤erent regions having similar wind regimes. If these plants are not so far from each other, then there should be some kind of dependence between the wind speeds which are respectively denoted by V1 and V2 . The dependence between wind speeds ensures dependence between the lifetimes of wind turbines located in the two plants. We are interested in the aggregate power produced by these two plants. Assume that wind plant i consists of ni wind turbines of type i, i = 1; 2: Each wind plant consists of same type of wind turbines, and di¤erent wind plants have di¤erent types of wind turbines. The weight of a turbine is measured by its generation capacity. If ! i denotes the power generated by the wind turbine of type i, then using (12) (1)
!1 =
Zvr
Pr(1)
(1)
(1)
(v 3
(vci )3 )
(1) ((vr )3
vci
h1 (v)dv (1) (vci )3 )
+ Pr(1) P vr(1)
(1) V1 < vco ;
+ Pr(2) P vr(2)
(2) V2 < vco ;
(2)
!2 =
Zvr
Pr(2)
(2)
(2)
(vci )3 )
(v 3 (2) ((vr )3
vci
h2 (v)dv (2) (vci )3 )
(13)
(i)
(i)
where hi is the truncated probability density function of the wind speed Vi on (vci ; vr ); i = 1; 2, i.e. hi (v) =
where Fi (v) = P fVi
vg :
8 > > < > > :
(i)
(i)
fi (v) ; Fi (vr ) Fi (vci )
if v 2 (vci ; vr )
0
otherwise
16
;
The aggregate power produced by wind plant i at time t is represented as Wi (t) =
ni X
(i)
! i I(Tj > t);
j=1
(i)
where Tj
denotes the time until failure of wind turbine j in the ith plant, i = 1; 2:
Our main interest is P fW1 (t)
k1 ; W2 (t)
k2 g which corresponds to the probability
that the total aggregate power produced by wind plant 1 is at least k1 and the total aggregate power produced by wind plant 2 is at least k2 .
For given wind turbine
models and wind speed distributions, this joint probability can be calculated using equation (10). Weibull distribution has been found to be suitable for modeling wind speed data. Let Fi (v) = 1 for v
e
v ai
ci
;
0, where ci and ai represent respectively the shape and scale parameters,
i = 1; 2: The characteristics of turbines that are used in plants are given in Table 2. Wind turbine of Type 1 Wind turbine of Type 2 Pr
800kW
1000kW
vci
3m/s
3.5m/s
vr
15m/s
15.5m/s
vco
25m/s
25m/s
Table 2. Characteristics of wind turbines
17
For a1 = 2; c1 = 10; a2 = 2:5; c2 = 9; using (13) and the data in Table 4, we obtain ! 1 = 248.5145 kW and ! 2 = 275.3492 kW. Assuming (2) for the joint distribution of components’lifetimes with P fW1 (t)
k1 ; W2 (t)
1
= 1;
2
= 1:2;
= 2, and using (10), we can calculate
k2 g. Table 3 includes some numerical results for selected
values of n1 ; n2 ; k1 ; k2 ; and t. If each wind plant consists of ten turbines, then the probability that each plant produces at least 2000 kW power with probability 0.6920 for t = 0:05: n1
n2
k1
k2
P fW1 (0:05)
k1 ; W2 (0:05)
k2 g
10 10 2000 2000 0.6920 2000 2500 0.3387 1000 2000 0.8734 15 10 2000 2000 0.8721 2000 2500 0.3904 1000 2000 0.8742 3000 2000 0.7382 Table 3. Capacity exceedance probabilities
6
Summary and conclusions
In this paper, we have studied a new system model called (k1 ; k2 ; :::; km )-out-of-n: According to this new model, the system consists of m types of components and 18
functions if at least ki out of ni components of type i function for all i = 1; 2; :::; m. The survival function of the system has been evaluated for the very general case when the components of the same type exchangeable and dependent, and that the components of di¤erent type are dependent. The approximation is quite useful especially when the system consists of many di¤erent types of components. The extension of the model to a system with weighted components is also presented with the model called weighted-(k1 ; k2 ; :::; km )-out-of-n system. As stated and illustrated in the paper, many real world systems can be modeled by the help of the weighted-(k1 ; k2 ; :::; km )-out-of-n system.
Acknowledgments
The author thanks two reviewers for their supportive and constructive comments and suggestions.
References Aboalkhair, A.M., Coolen, F.P.A., MacPhee, I.M. (2014) Nonparametric predictive inference for reliability of a k-out-of-m:G system with multiple component types. Reliability Engineering & System Safety, 131, 298-304. Amari, S.V., Wang, C., Xing, L. and Mohammad, R. (2018) An e¢ cient phased-
19
mission reliability model considering dynamic k-out-of-n subsystem redundancy, IISE Transactions, in press. Armutkar K.P. and Kamalja K.K. (2014) Reliability and importance measures of weighted-k-out-of-n system. Int J Reliab Qual Saf Eng, 21. Armutkar K.P. and Kamalja K.K. (2017) E¢ cient algorithm for reliability and importance measures of linear weighted-(n,f,k) and systems. Computers & Industrial Engineering, 107, 85-99. Block H.W., Costigan, T. Sampson, A.R. (1991) Second order Bonferroni-type, product-type and setwise probability inequalities. In: Stochastic Orders and Decision Under Risk. In: IMS Lecture Notes-Monograph Series, 19:74–94. Cui, L., Gao, H. and Mo, Y. (2018) Reliability for k-out-of-n:F balanced systems with m sectors. IISE Transactions, in press. Eryilmaz, S. (2016) Reliability of systems with multiple types of dependent components. IEEE Transactions on Reliability, 65, 1022-1029. Eryilmaz, S. (2017) The concept of weak exchangeability and its applications. Metrika, 80, 259-271. Eryilmaz, S., Coolen, F.P.A. and Coolen-Maturi, T. (2018) Marginal and joint
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reliability importance based on survival signature. Reliability Engineering & System Safety, 172, 118-128. Eryilmaz, S., Coolen, F.P.A., Coolen-Maturi, T. (2018) Mean residual life of coherent systems consisting of multiple types of dependent components. Naval Research Logistics, 65, 86-97. Feng, G., Patelli, E., Beer, M., Coolen, F.P.A. (2016) Imprecise system reliability and component importance based on survival signature. Reliability Engineering & System Safety, 150, 116-125. Kong, Y. and Ye, Z. (2016) Interval estimation for k-out-of-n load-sharing systems. IISE Transactions, 49, 344-353. Li, W. and Zuo, M.J. (2008) Reliability evaluation of multi-state weighted kout-of-n systems. Reliability Engineering & System Safety, 93, 160–167. Louie H, Sloughter J.M. (2014) Probabilistic modeling and statistical characteristics of aggregate wind power. Large Scale Renewable Power Generation, Green Energy and Technology; 19-51, Springer Science+Business Media Singapore. Meshkat R.S., Mahmoudi E. (2017) Joint reliability and weighted importance measures of a k-out-of-n system with random weights for components. Journal of Computational and Applied Mathematics, 326, 273-283.
21
Patelli, E., Feng, G., Coolen, F.P.A. and Coolen-Maturi, T. (2017) Simulation methods for system reliability using the survival signature. Reliability Engineering & System Safety, 167, 327-337. Rahmani, R-A., Izadi,M., Khaledi, B-E. (2016) Importance of components in k-out-of-n system with components having random weights. Journal of Computational and Applied Mathematics, 296, 1-9. Zhu, X. and Boushaba, M. (2017) A linear weighted (n, f, k) system for nonhomogeneous Markov-dependent components. IISE Transactions, 49, 722-736.
22
PII: DOI: Reference:
S0377-0427(18)30462-X https://doi.org/10.1016/j.cam.2018.07.036 CAM 11822
To appear in:
Journal of Computational and Applied Mathematics
Received date : 6 April 2018 Revised date : 16 June 2018 Please cite this article as: S. Eryilmaz, (k1 , k2 , . . . , km )-out-of-n system and its reliability, Journal of Computational and Applied Mathematics (2018), https://doi.org/10.1016/j.cam.2018.07.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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(k1 ; k2 ; :::; km )-out-of-n system and its reliability by Serkan ERYILMAZ*
Abstract
This paper is concerned with a system consisting of multiple types of components and having (k1 ; k2 ; :::; km )-out-of-n structure. The (k1 ; k2 ; :::; km )-out-of-n system is a system consisting of ni components of type i, i = 1; 2; :::; m, and functions if at least k1 components of type 1, k2 components of type 2, ..., km components of type m work, n =
Pn
i=1
ni : The exact and approximate expressions are obtained for the
survival function of the system under concern. The weighted-(k1 ; k2 ; :::; km )-out-of-n system is also de…ned and studied. This weighted model is applied to evaluate the wind power system that consists of two wind plants. Keywords. Dependence; k-out-of-n system; Reliability approximation; Wind power system *
Atilim University, Department of Industrial Engineering, 06836, Incek, Ankara, Turkey, e-mail:
[email protected], Fax: +90 312 586 80 91
1
1
Introduction
Most of the complex engineering systems consist of components which perform di¤erent types of tasks. The components that have di¤erent functions within the system are expected to have di¤erent reliability characteristics. Thus, most of the systems are composed with multiple types of components which are classi…ed not only with respect to their roles within the system but also with respect to their failure time distributions. For example, a certain production system may consist of machines performing di¤erent types of tasks, e.g. cutting, packing. The cutting and packing machines are expected to have di¤erent failure time distributions. Recently, systems with multiple types of components have attracted great deal of attention in reliability literature. Aboalkhair et al. (2014) studied the problem of nonparametric predictive inference for reliability of a k-out-of-m:G system with multiple component types. Feng et al. (2016) studied imprecise system reliability and component importance for a system consisting of multiple types of independent components. Patelli et al. (2017) proposed a simulation method for reliability of a system with multiple types of independent components using survival signature. Eryilmaz et al. (2018) obtained expressions for marginal and joint reliability importances for a coherent system that consists of multiple types of dependent components. Eryilmaz et al. (2018) studied mean residual life of coherent systems consisting of multiple types of dependent components. 2
The ordinary k-out-of-n system consists of n components which perform same tasks, and operates if at least k components work properly (Kong and Ye (2016), Amari et al. (2018), Cui et al. (2018)). In a more general setting, we might have multiple types of components having di¤erent functions, and di¤erent numbers of components of each type may be required for the proper operation of the whole system. Such a situation can be modeled by (k1 ; k2 ; :::; km )-out-of-n system. The (k1 ; k2 ; :::; km )-out-of-n system is a system consisting of ni components of type i, i = 1; 2; :::; m, and functions if at least k1 components of type 1, k2 components of type 2, ..., km components of type m work, n =
Pn
i=1
ni . The (k1 ; k2 ; :::; km )-out-of-
n system is useful for modeling various real life complex systems. For example, an airplane consists of various subsystems such as hydraulic system, radar system, and communication system. Active redundancy is applied to most of these subsystems. For example, hydraulic system redundancy may be achieved by multiple pressure sources which help to ensure that the entire hydraulic system is not lost in the event of a single component failure. Thus, for a total of m subsystems of an airplane, at least ki components of type i, i = 1; 2; :::; m are required for proper operation of the airplane. Because all components are harmonically integrated to constitute the whole system, dependence is inevitable between various subsystems. Thus assuming a statistical dependence not only between the components of each subsystem but also among subsystems is very realistic and more applicable.
3
In this paper, we study the (k1 ; k2 ; :::; km )-out-of-n system assuming that the random failure times of components of the same type are exchangeable and dependent, and that the random failure times of components of di¤erent type are dependent. In particular, we have two levels of dependence. The 1st level (or local) dependence de…nes the dependence between the components of same type, and the 2nd level (or global) of dependence is a dependence among di¤erent types of components. We obtain exact and approximate expressions for the survival function of the (k1 ; k2 ; :::; km )-out-of-n system under such a dependence model. The weighted(k1 ; k2 ; :::; km )-out-of-n system is also de…ned and studied. The results are illustrated and an application concerned with wind power systems is also presented. The paper is organized as follows. In Section 2, we de…ne the (k1 ; k2 ; :::; km )-outof-n system model and study its survival function. Section 3 involves extension of the model to a system with weighted components. In Section 4, some numerical results are presented on survival function of the (k1 ; k2 ; :::; km )-out-of-n system. Section 5 contains an application of the weighted-(k1 ; k2 ; :::; km )-out-of-n system to evaluate the capacity of the wind power system that consists of two wind plants.
2
The (k1 ; k2 ; :::; km )-out-of-n system
We …rst …x the notation that will be used throughout the paper. n: The total number of components,
4
ni : The number of components of type i; i = 1; 2; :::; m; ki : The minimum number of working components of type i for the operation of the system, (i)
Tj : The lifetime of component j of type i; j = 1; 2; :::; ni ; (i)
(i)
(i)
(i)
Tr:ni : The rth smallest among T1 ; T2 :::; Tni ; Tk1 ;:::;km :n : The lifetime of the (k1 ; k2 ; :::; km )-out-of-n system, (i)
Sni (t): The total number of working components of type i at time t: Recall that the (k1 ; k2 ; :::; km )-out-of-n system is a system consisting of ni components of type i, i = 1; 2; :::; m, and functions if at least k1 components of type 1, k2 components of type 2, ..., km components of type m work. Thus the lifetime of the (k1 ; k2 ; :::; km )-out-of-n system is represented as (1) (2) (m) k1 +1:n1 ; Tn k2 +1:n2 ; :::; Tn km +1:nm );
Tk1 ;:::;km :n = min(Tn for ki
(1)
ni ; i = 1; 2; :::; m:
Any coherent system that consists of multiple types of dependent components can be written as a (generalized) mixture of (k1 ; k2 ; :::; km )-out-of-n systems. Indeed, if T denotes the lifetime of the coherent system, then
P fT > tg =
n1 X n2 X
k1 =0 k2 =0
nm X
cn1 ;n2 ;:::;nm (k1 ; k2 ; :::; km )
k =0
m n o (1) (2) (m) P min(Tn1 k1 +1:n1 ; Tn2 k2 +1:n2 ; :::; Tnm km +1:nm ) > t ;
5
(2)
where the coe¢ cients cn1 ;n2 ;:::;nm (k1 ; k2 ; :::; km ) depend on the structure of the coherent system (Eryilmaz (2016)). Another motivation for studying the (k1 ; k2 ; :::; km )-outof-n system lies in the representation given by (2). It is assumed that the random failure times of components of the same type are exchangeable and dependent, and that the random failure times of components of di¤erent type are dependent. That is, there are two levels of dependence: The 1st level (or local) dependence de…nes the dependence between the components of same type, and the 2nd level (or global) of dependence is a dependence among di¤erent types of components. More explicitly, for …xed i, the vector of failure (i)
(i)
(i)
times (T1 ; T2 :::; Tni ) is assumed to have exchangeable distribution, and the vec(1)
(1)
(1)
(2)
(2)
(2)
(m)
tors (T1 ; T2 :::; Tn1 ); (T1 ; T2 :::; Tn2 ); :::; (T1
(m)
; T2
(m)
:::; Tnm ) are dependent. Be-
low, we give an example for illustrating the dependence model. Example 1 Let n o (1) (1) (m) (m) (1) (m) (m) P T1 > t1 ; :::; Tn(1) > t ; :::; T > t ; :::; T > t 1 1 n1 nm nm 1 " # n1 nm X X (1) (m) ; = 1+ 1 tj tj + ::: + m (j)
for ti
0, i = 1; :::; nj ; j = 1; :::; m;
(3)
j=1
j=1
i
> 0;
> 0. That is, the joint distribution of
the components is modeled by multivariate Pareto distribution which is a useful model (i)
(i)
(i)
in reliability. Then, the joint survival function corresponding to (T1 ; T2 :::; Tni ) is " # ni n o X (i) (i) (i) P T1 > t1 ; :::; Tn(i)i > t(i) = 1+ i tj : (4) ni j=1
6
Obviously, the joint survival function given by (4) is exchangeable.
The second level or global dependence creates a dependence among (1) (2) (m) k1 +1:n1 ; Tn k2 +1:n2 ; :::; Tn km +1:nm .
The originality of the model lies in consider-
Tn
ing such a dependence. We …rst obtain an exact expression for the survival function of the system. Using the representation given by (1), n o (1) (2) (m) P fTk1 ;:::;km :n > tg = P Tn k1 +1:n1 > t; Tn k2 +1:n2 > t; :::; Tn km +1:nm > t = P Sn(1) (t) 1
=
n1 X
k1 ; Sn(2) (t) 2
nm X
r1 =k1
k2 ; :::; Sn(m) (t) m
km
P Sn(1) (t) = r1 ; Sn(2) (t) = r2 ; :::; Sn(m) (t) = rm(5): m 1 2
rm =km (1)
(2)
(m)
From Theorem 1 of Eryilmaz (2017), the joint distribution of Sn1 (t); Sn2 (t):::; Snm (t) can be written as P Sn(1) (t) = r1 ; :::; Sn(m) (t) = rm m 1 =
n1 r1 nm Xrm n1 nm X n 1 r1 n m rm ::: ( 1)j1 +:::+jm ::: r1 rm j =0 j1 jm jm =0 1 n o (1) (m) P min(T1:r1 +j1 ; :::; T1:rm +jm ) > t
(6)
Using (6) in (5), one obtains P fTk1 ;:::;km :n > tg =
n1 X
r1 =k1 nX 1 r1
nm X
n1 nm ::: r1 rm
rm =km
nm Xrm
( 1)j1 +:::+jm
n1
r1
j1 jm =0 n o (1) (m) P min(T1:r1 +j1 ; :::; T1:rm +jm ) > t : j1 =0
7
:::
nm
rm jm
Letting r1 + j1 = i1 ; ::::; rm + jm = im ; the survival function can be written as P fTk1 ;:::;km :n > tg =
n1 X
nm X
i1 =k1
im =km
o n (m) (1) (i1 ; :::; im )P min(T1:i1 ; :::; T1:im ) > t ;
(7)
(i1 ; :::; im ) =
i1 X
r1 =k1 n a;b;c
2.1
=
im X
i1 r1 +:::+im rm
( 1)
m Y j=1
rm =km
rj ; ij
nj r j ; nj
ij
;
n! : a!b!c!
An approximation
The complexity of (7) may prevent its direct use especially for large values of m and ni s. Therefore, a simpler formula for computing P fTk1 ;:::;km :n > tg is desirable. In this section, we present an approximation using
P
(m \
i=1
Ai
)
'
m Y1
P fAi Ai+1 g
i=1 m Y1 i=2
(8)
P fAi g
(see, e.g. Block et al. (1991)). Clearly, P fTk1 ;:::;km :n > tg = P where Ai
n (i) Tni
ki +1:ni
(m \
i=1
Ai
)
;
o > t ; i = 1; 2; :::; m: Thus, for approximating the survival
function we need to get expressions for the following probabilities: n (i) P fAi g = P Tni 8
ki +1:ni
o >t ;
and n (i) P fAi Ai+1 g = P Tni
(i+1) t; Tni+1 ki+1 +1:ni+1
>
ki +1:ni
o >t :
Because the components of same type are exchangeable, n (i) P fAi g = P Tni
ki +1:ni
ni o X >t = P Sn(i)i (t) = r ; r=ki
n o (i) where P Sni (t) = r can be computed from P
Sn(i)i (t)
=r =
n i r X
ni r
ni
( 1)j
r j
j=0
o n (i) (i) P T1 > t; :::; Tr+j > t :
Thus n (i) P fAi g = P Tni = =
ni n i r X X
ki +1:ni
( 1)j
r=ki j=0 ni X s=ki
o >t ni r
ni
r j
n o (i) (s)P T1:s > t ;
o n (i) P T1:r+j > t
where (s) =
s X
( 1)s
r
r; s
r=ki
ni r; ni
s
:
On the other hand, n (i) P fAi Ai+1 g = P Tni =
ki +1:ni
ni+1 ni X X
r1 =ki r2 =ki+1
>
(i+1) t; Tni+1 ki+1 +1:ni+1
o >t
n o (i) (i+1) P Sni (t) = r1 ; Sni+1 (t) = r2
9
By using (6), we obtain n o (i) (i+1) P Sni (t) = r1 ; Sni+1 (t) = r2 =
ni r1
nX i r1 ni+1 Xr2
ni+1 r2
j1 =0
ni
( 1)j1 +j2
r1
ni+1 r2 j2
j1
j2 =0
o n (i+1) (i) P min(T1:r1 +j1 ; T1:r2 +j2 ) > t : Thus n (i) P fAi Ai+1 g = P Tni =
(i+1)
ki +1:ni
> t; Tni+1
ni+1 ni r1 ni+1 r2 ni X X X X
r1 =ki r2 =ki+1 j1 =0
ki+1 +1:ni+1
( 1)j1 +j2
j2 =0
o >t
ni r1
ni+1 r2
ni
r1 j1
ni+1 r2 j2
n o (i) (i+1) P min(T1:r1 +j1 ; T1:r2 +j2 ) > t
=
ni+1 ni X X
i1 =ki i2 =ki+1
n o (i) (i+1) (i1 ; i2 )P min(T1:i1 ; T1:i2 ) > t ;
where (i1 ; i2 ) =
i1 i2 X X
( 1)i1
r1 +i2 r2
r1 =ki r2 =ki+1
r1 ; i1
ni r 1 ; ni
i1
r2 ; i2
ni+1 r2 ; ni+1
i2
:
Thus the survival function of the system can be approximated by
P fTk1 ;:::;km :n > tg '
m Y1
ni+1 ni X X
i=1 i1 =ki i2 =ki+1
n o (i) (i+1) (i1 ; i2 )P min(T1:i1 ; T1:i2 ) > t
ni m Y1 X i=2 s=ki
n o (i) (s)P T1:s > t
:
(9)
The approximation formula given by (9) is simpler in terms of computational complexity when compared with the exact expression given by (7). 10
3
Extension to a system with weighted components
In a system with weighted components, each component contribute di¤erently to the performance of the whole system. See, e.g. Li and Zuo (2008), Armutkar and Kamalja (2014, 2017), Rahmani et al. (2016), Zhu and Boushaba (2017), Meshkat and Mahmoudi (2017). In this section, we de…ne weighted-(k1 ; k2 ; :::; km )-out-of-n system. The components of same type are assumed to have the same weight/contribution. The weight of all components of type i is denoted by ! i ; i = 1; :::; m. The entire system is able to perform its intended task i¤ the total weight of functioning components of type 1 is at least k1 ; the total weight of functioning components of type 2 is at least k2 ; :::; and the total weight of functioning components of type m is km : The total weight of functioning components of type i at time t is de…ned by Wi (t) =
ni X
(i)
! i I(Tj > t) = ! i Sn(i)i (t);
j=1
(i)
(i)
(i)
(i)
where I(Tj > t) = 1 if Tj > t; and I(Tj > t) = 0 if Tj
t:
Let Tk!1 ;:::;km :n denote the lifetime of a weighted-(k1 ; k2 ; :::; km )-out-of-n system. Then P Tk!1 ;:::;km :n > t = P fW1 (t)
k1 ; W2 (t)
11
k2 ; :::; Wm (t)
km g :
By conditioning on the number of working components of each type, we obtain P
Tk!1 ;:::;km :n
>t
n1 X
=
r1 :! 1 r1 k1 nX 1 r1
nm X
n1 nm ::: r1 rm
rm :! m rm km nm Xrm
n1
( 1)j1 +:::+jm
r1
j1 jm =0 n o (m) (1) P min(T1:r1 +j1 ; :::; T1:rm +jm ) > t : j1 =0
:::
nm
rm jm (10)
The lifetime of weighted-(k1 ; k2 ; :::; km )-out-of-n system can be represented as Tk!1 ;:::;km :n = min(T1 ; T2 ; :::; Tm );
(11)
where T1 =
(1) Tr:n 1
T2 =
(2) Tr:n 1
.. .
n (1) i¤ W1 (Tr 1:n1 )
n (2) i¤ W2 (Tr 1:n2 )
(m) i¤ Tm = Tr:n m
k1 and
(1) W1 (Tr:n ) 1
k2 and
(2) W2 (Tr:n ) 2
o
< k1 ; o
< k2 ;
o (m) km and Wm (Tr:n ) < k m : m
n (m) Wm (Tr 1:nm )
The representation given by (11) is useful to simulate the lifetime of the system. An approximation for P Tk!1 ;:::;km :n > t can be obtained using (8) with P fAi g = P fWi (t) =
ni n i r X X
ki g = P ! i Sn(i)i (t) ( 1)j
r:! i r ki j=0
12
ni r
ni
r j
ki n o (i) P T1:r+j > t
and n P fAi Ai+1 g = P ! i Sn(i)i (t) =
ni X
ki ; ! i+1 Sn(i+1) (t) i+1 ni+1
nX i r1 ni+1 Xr2
X
r1 :! i r1 ki r2 :! i+1 r2 ki+1 j1 =0
ni
r1 j1
ni+1 r2 j2
ki+1
o
( 1)j1 +j2
j2 =0
ni r1
ni+1 r2
n o (i+1) (i) P min(T1:r1 +j1 ; T1:r2 +j2 ) > t :
P fAi g represents the probability that the total weight of functioning components of type i is at least ki . Thus the survival function of the system can be approximated by P Tk!1 ;:::;km :n > t
'
m Y1
ni X
ni+1
nX i r1 ni+1 Xr2
X
i=1 r1 :! i r1 ki r2 :! i+1 r2 ki+1 j1 =0 m Y1
j2 =0
n o (i) (i+1) ( 1)j1 +j2 Cni ;ni+1 (r1 ; r2 ; j1 ; j2 )P min(T1:r1 +j1 ; T1:r2 +j2 ) > t
n ni i r X X
( 1)j
ni r
ni r j
i=2 r:! i r ki j=0
Cni ;ni+1 (r1 ; r2 ; j1 ; j2 ) =
4
ni r1
ni+1 r2
ni
r1 j1
n o (i) P T1:r+j > t
ni+1 r2 : j2
Numerical illustrations
To study the performance of the approximation given by (8), we consider a system consisting of three di¤erent types of components, i.e. m = 3. Assume that the joint survival function of components’lifetimes is modeled by (2) with and
1
= 1;
2
= 2;
3
=3
= 3: In Tables 1, we compute exact (when possible) and approximate values of
R((k1 ; :::; km ); (n1 ; :::; nm ) : G) = P fTk1 ;:::;km :n > tg for di¤erent choices of the number 13
of components of each type. For larger values of ni s, the combinatorial terms involved in (7) prevent the use of the equation and hence for n1 = 20; n2 = 15; n3 = 10, we compare approximation with the simulated value. The approximation performs quite well for high reliability values in all cases. However, as it is clear from Table 1, it might be poor with an increase in t for some values of ni s. Overall the approximation given by (8) can be used for highly reliable structures and its potential use, in general, depends on the values of ni s. n1
n2
n3
k1
10 15 10 5
Exact
Approximation
k2
k3
t
3
4
0.05 0.9127 0.9104 0.1
0.6282 0.6053
0.25 0.1804 0.1450 0.5 20 15 10 5
3
4
0.0419 0.0284
0.05 0.9173 0.9125 0.1
0.6375 0.6384
0.25 0.1875 0.1869 0.5 20 15 10 5
3
2
0.0452 0.0431
0.05 0.9900 0.9858 0.1
0.8780 0.8770
0.25 0.4089 0.4045 0.5 14
0.1284 0.1211
Table 1. Exact and approximate values of R((k1 ; k2 ; k3 ); (n1 ; n2 ; n3 ) : G)
5
An application
A wind plant typically consists of wind turbines which convert the kinetic energy in the wind into electrical energy. If the wind speed is below the cut-in wind speed (vci ), then no electrical power is produced since in this case the power in the wind is not enough. If the wind speed is between the cut-in wind speed and rated wind speed (vr ), then the wind turbine generates power and there is a cubic relationship between the wind speed and the generated power. If the wind speed is above the rated wind speed and below the cut-out wind speed (vco ), then the wind turbine generates power at a constant rate Pr . If the wind speed is above the cut-out wind speed, the turbine is stopped since it is in a danger of mechanical failure. The values of vci ; vco ; vr and Pr are determined by the wind turbine model. The power generated by a single wind turbine as a function of wind speed random variable V is given by (see, e.g.Louie Sloughter (2014)).
g(V ) =
8 > > > > > > > > > > > > > > <
0;
if V < vci or V
(V 3 v 3 )
Pr (v3 v3ci) ; > r ci > > > > > > > > > > > > > : Pr ;
15
if vci
V < vr
if vr
V < vco
vco
(12)
Consider two wind plants that are located in two di¤erent regions having similar wind regimes. If these plants are not so far from each other, then there should be some kind of dependence between the wind speeds which are respectively denoted by V1 and V2 . The dependence between wind speeds ensures dependence between the lifetimes of wind turbines located in the two plants. We are interested in the aggregate power produced by these two plants. Assume that wind plant i consists of ni wind turbines of type i, i = 1; 2: Each wind plant consists of same type of wind turbines, and di¤erent wind plants have di¤erent types of wind turbines. The weight of a turbine is measured by its generation capacity. If ! i denotes the power generated by the wind turbine of type i, then using (12) (1)
!1 =
Zvr
Pr(1)
(1)
(1)
(v 3
(vci )3 )
(1) ((vr )3
vci
h1 (v)dv (1) (vci )3 )
+ Pr(1) P vr(1)
(1) V1 < vco ;
+ Pr(2) P vr(2)
(2) V2 < vco ;
(2)
!2 =
Zvr
Pr(2)
(2)
(2)
(vci )3 )
(v 3 (2) ((vr )3
vci
h2 (v)dv (2) (vci )3 )
(13)
(i)
(i)
where hi is the truncated probability density function of the wind speed Vi on (vci ; vr ); i = 1; 2, i.e. hi (v) =
where Fi (v) = P fVi
vg :
8 > > < > > :
(i)
(i)
fi (v) ; Fi (vr ) Fi (vci )
if v 2 (vci ; vr )
0
otherwise
16
;
The aggregate power produced by wind plant i at time t is represented as Wi (t) =
ni X
(i)
! i I(Tj > t);
j=1
(i)
where Tj
denotes the time until failure of wind turbine j in the ith plant, i = 1; 2:
Our main interest is P fW1 (t)
k1 ; W2 (t)
k2 g which corresponds to the probability
that the total aggregate power produced by wind plant 1 is at least k1 and the total aggregate power produced by wind plant 2 is at least k2 .
For given wind turbine
models and wind speed distributions, this joint probability can be calculated using equation (10). Weibull distribution has been found to be suitable for modeling wind speed data. Let Fi (v) = 1 for v
e
v ai
ci
;
0, where ci and ai represent respectively the shape and scale parameters,
i = 1; 2: The characteristics of turbines that are used in plants are given in Table 2. Wind turbine of Type 1 Wind turbine of Type 2 Pr
800kW
1000kW
vci
3m/s
3.5m/s
vr
15m/s
15.5m/s
vco
25m/s
25m/s
Table 2. Characteristics of wind turbines
17
For a1 = 2; c1 = 10; a2 = 2:5; c2 = 9; using (13) and the data in Table 4, we obtain ! 1 = 248.5145 kW and ! 2 = 275.3492 kW. Assuming (2) for the joint distribution of components’lifetimes with P fW1 (t)
k1 ; W2 (t)
1
= 1;
2
= 1:2;
= 2, and using (10), we can calculate
k2 g. Table 3 includes some numerical results for selected
values of n1 ; n2 ; k1 ; k2 ; and t. If each wind plant consists of ten turbines, then the probability that each plant produces at least 2000 kW power with probability 0.6920 for t = 0:05: n1
n2
k1
k2
P fW1 (0:05)
k1 ; W2 (0:05)
k2 g
10 10 2000 2000 0.6920 2000 2500 0.3387 1000 2000 0.8734 15 10 2000 2000 0.8721 2000 2500 0.3904 1000 2000 0.8742 3000 2000 0.7382 Table 3. Capacity exceedance probabilities
6
Summary and conclusions
In this paper, we have studied a new system model called (k1 ; k2 ; :::; km )-out-of-n: According to this new model, the system consists of m types of components and 18
functions if at least ki out of ni components of type i function for all i = 1; 2; :::; m. The survival function of the system has been evaluated for the very general case when the components of the same type exchangeable and dependent, and that the components of di¤erent type are dependent. The approximation is quite useful especially when the system consists of many di¤erent types of components. The extension of the model to a system with weighted components is also presented with the model called weighted-(k1 ; k2 ; :::; km )-out-of-n system. As stated and illustrated in the paper, many real world systems can be modeled by the help of the weighted-(k1 ; k2 ; :::; km )-out-of-n system.
Acknowledgments
The author thanks two reviewers for their supportive and constructive comments and suggestions.
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