PART V. Practices and Emerging Applications
4.3 Multi-state System Reliability Indices Evaluation
Generating Function
Ushakov [25] introduced the UGF and formulated its principles of application [26]. The most systematical description of mathematical aspects of the method can be found in [6], where the method is referred to as a generalized generating sequences approach. A brief overview of the method with respect to its applications for MSS reliability assessment was presented by Levitin et al.[4]. The method was first applied to the real system reliability assessment and optimization in [27].
The UGF approach is based on definition of a u-function of discrete random variables
Multi-state System Reliability Analysis and Optimization 65 and composition operators over u-functions.
The u-function of a variable X is defined as a polynomial
u(z)= K k=1
pkzXk (4.6) where the variable X hasK possible values and pk is the probability that X is equal to Xk. The composition operators over u-functions of different random variables are defined in order to determine the probabilistic distribution for some functions of these variables.
The UGF extends the widely known ordinary moment generating function [6]. The essential difference between the ordinary generating func- tion and a UGF is that the latter allows one to evaluate an OPD for a wide range of systems char- acterized by different topology, different natures of interaction among system elements, and the different physical nature of elements’ performance measures. This can be done by introducing dif- ferent composition operators over UGF (the only composition operator used with an ordinary mo- ment generating function is the product of poly- nomials). The UGF composition operators will be defined in the following sections.
In our case, the UGF, represented by polyno- mialU (z)can define an MSS OPD,i.e.it represents all the possible states of the system (or element) by relating the probabilities of each state pk to performance Gk of the MSS in that state in the following form:
U (t, z)= K k=1
pk(t)zGk (4.7) Having an MSS OPD in the form of Equa- tion 4.6, one can obtain the system availability for the arbitrarytandWusing the following operator δA:
A(t, F, W )=δA[U (t, z), F, W]
=δA
K k=1
pk(t)zGk, F, W
= K k=1
pk(t)α[F (Gk, W )] (4.8)
where
α(x)=
1, x≥0
0, x <0 (4.9) A multi-state stationary (steady state) availabil- ity was introduced asPr{G(t)≥W}after enough time had passed for this probability to become constant. In the steady state, the distribution of state probabilities is
pk= lim
t→∞Pr{G(t)=Gk} G(t)∈ {G1, . . . , GK} (4.10) The MSS stationary availability may be defined according to Equation 4.1 when the demand is constant or according to Equation 4.2 in the case of variable demand. Thus, for the given MSS OPD represented by polynomial U (z), the MSS availability can be calculated as
EA(W,q)= M m=1
qmδA(U (z), F, Wm) (4.11) The expected system output performance value during the operating time (Figure 4.1) defined by Equation 4.4 can be obtained for givenU (z)using the followingδGoperator:
EG=δG(U (z))=δG
K k=1
pkzGk
= K k=1
pkGk
(4.12) In order to obtain the expected unsupplied demand EU for the given U (z) and variable demand according to Equation 4.5, the following δUoperator should be used:
EU(W,q)= M m=1
qmδU(U (z), F, Wm) (4.13) where
EU(Wm)=δU(U (z), F, Wm)
=δU
K k=1
pkzGk, F, Wm
= K k=1
pkmax(−F (Gk, Wm),0) (4.14)
66 System Reliability and Optimization
Example 1. Consider, for example, two power system generators with nominal capacity 100 MW as two different MSSs. In the first generator some types of failure require the capacity to be reduced to 60 MW and some types lead to a complete generator outage. In the second generator some types of failure require the capacity to be reduced to 80 MW, some types lead to capacity reduction to 40 MW, and some types lead to a complete generator outage. So, there are three possible relative capacity levels that characterize the performance of the first generator:
G11=0.0 G12= 60
100=0.6 G13=100 100=1.0 and four relative capacity levels that characterize the performance of the second one:
G21=0.0 G22= 40 100=0.4 G23= 80
100=0.8 G24=100 100=1.0 The corresponding steady-state probabilities are the following:
p11=0.1 p12=0.6 p31=0.3 for the first generator and
p21=0.05 p22=0.25 p23=0.3 p24=0.4 for the second generator.
Now we find reliability indices for both MSSs forW=0.5(required capacity level is 50 MW).
1. The MSS u-functions for the first and the second generator respectively, according to Equation 4.7, are as follows:
UMSS1 (z)=p11zG11+p21zG12+p13zG13
=0.1+0.6z0.6+0.3z1.0
UMSS2 (z)=p12zG21+p22zG22+p23zG23+p42zG24
=0.05+0.25z0.4+0.3z0.8+0.4z1.0 2. The MSS stationary availability (Equation 4.8)
is
E1A(W )=A1(0.5)
=
G1k−W≥0
pk=0.6+0.3=0.9 E2A(W )=A2(0.5)
=
G2k−W≥0
pk=0.3+0.4=0.7 3. The expected MSS performance (Equa-
tion 4.12) is EG1=
3 k=1
p1kG1k
=0.1×0+0.6×0.6+0.3×1.0=0.66 which means 66% of the nominal generating capacity for the first generator, and
EG2= 4 k=1
p2kG2k
=0.05×0+0.25×0.4+0.3×0.8 +0.4×1.0=0.74
which means 74% of the nominal generating capacity for the second generator.
4. The expected unsupplied demand (Equa- tion 4.14) is
EU1(W )=
Gk−W <0
pk(W−Gk)
=0.1×(0.5−0.0)=0.05 EU2(W )=
Gk−W <0
pk(W−Gk)
=0.05×(0.5−0.0)
+0.25×(0.5−0.4)=0.05 In this case, EU may be interpreted as expected electric power unsupplied to consumers.
The absolute value of this unsupplied demand is 5 MW for both generators. Multiplying this index by the considered system operating time, one can obtain the expected unsupplied energy.
Note that since the reliability indices obtained have different nature, they cannot be used interchangeably. In Example 1, for instance, the first generator performs better than the second
Multi-state System Reliability Analysis and Optimization 67 one when availability is considered (E1A(0.5) >
EA2(0.5)), the second generator performs better than the first one when expected productivity is considered (EG2> EG1), and both generators have the same unsupplied demand (E1U(0.5)= EU2(0.5)).