It is rather trivial to solve the initial Equations ((3.1)-(3.8)) numerically; however, the calculation of the contribution to the time-dependent probability of station blackout,
3( )
P t , from the minimum possible number of EDG failures that could cause a blackout, two in this model, is needed. While that computation could be carried out analytically, by evaluating all of the integrals that would arise, it is intended to leave those integrals unevaluated. This is to arrive at a result having the form of the nonrecovery integral
found in Rodgers et al. [1], except for the inclusion of common-cause contributions associated with the rate constantλ12. The hope is that the form of this common-cause contribution will suggest how common-cause failures should be incorporated into the nonrecovery integral in the more general setting.
The parameter ε introduced in the state-transition equations will be indispensable to this calculation, which will require the use of an asymptotic expansion process. The solution should be in the form of Equation (3.9).
( )
(0)( )
(1)( )
2 (2)( )
( )( )
0
, 0,..,3
n n
i i i i i
n
P t P t εP t ε P t ∞ε P t i
=
= − − +… =
∑
= (3.9)For all values of n up to the smallest, Pi( )n
( )
t is not identically zero. The anticipated value of this index is n=2, but this needs more proof. The procedure by which the solution will be obtained is to substitute the asymptotic expansions (3.9) into the initial value problem (3.1)-(3.8) and systematically equate the coefficients of terms of like order n in ε, for first n=0, then n=1 and etc. At a given value of n, this will give a determined initial value problem in the four unknowns 0 )( )
P(n t , 1 )
( )
P(n t , 2 )
( )
P(n t , and 3 )
( )
P(n t . The hope is for the initial value problems to be sufficiently simpler than state-transition equations to permit them to be solved analytically, at least up to the value of interest n, assumed as a value of two.
There is an interpretation of the asymptotic solution to order n that can serve as a useful check on the computations done in MATLAB. The asymptotic solution up to order n should agree with the solution that would prevail if a maximum of n failure occurs.
This is where single-point failures, independent and one-way common-cause failures as incorporated in the failure ratesλi are counted as single failures which is with multiplicity of one. By contrast, common-cause doublet failures, as incorporated in the failure rate
λ01, are counted with multiplicity of two.
The results of carrying out the preceding procedure at order zero, equating coefficients ofε0) in the state-transition equations lead to the equations seen in Equations (3.10)-(3.13). The initial conditions for the previous set of equations can be seen in (3.14) through (3.17).
This system can easily be solved. The solution of Equations (3.13) and (3.17) is obvious to see as it is zero throughout, or P3( )0
( )
t ≡0. The solutions from Equations (3.11) and (3.12) can then be readily solved and shown as P1( )0( )
t ≡0and P2( )0( )
t ≡0, respectively. After, that these can be input into Equation (3.10) and the solution can be seen asP0( )0( )
t ≡1. Therefore, for the lowest order in ε, the emergency AC system analyzed performs perfectly. That is in agreement with the interpretation suggested above of the order n as the maximum number of failures allowed for, counting number by multiplicities as described in the first part of Section 3.3. At present that order is zero, meaning zero failures. However, if there are no failures, then the initial conditions shown in Equations (3.5) though (3.8) assert that the system initially is in state 0, both EDGs running, with certainty, probability of one, and if there are no subsequent failures, then that certainty persist for an indefinite time. The results obtained are comforting but usedas mostly a check on our assumptions and analysis. The interpretation suggested for the solution of order n is also checked by the result; although, that is hardly surprising.
Subsection 3.3.b – For the n=1 case for Two-EDG Case
For the lower order case n=0, it is expected that the solution of the differential equation for P3(1) to be the catalyst that permits analytic solution of the remainder of the first-order system of ODEs. If the first order terms in Equation (3.4) were equated with the initial conditions, the result would be Equations (3.18) and (3.19) which is where this methodology would start.
(1)
(0) (0)
3
1 1 0 2
dP
P P
dt = λ +λ (3.18)
( )1
( )
3 0 0
P = (3.19)
Except from the results from the n=0 case, it is known that P1(0)( )t =P2(0)( ) 0t ≡ , so these equations can simplify to Equation (3.20).
(1)
3 0
dP
dt = (3.20)
This equation still using Equation (3.19) as the initial condition and the solution is
) 3
(1 ( ) 0
P t ≡ , is expected. This result already shows that no “single-point failures”, i.e. no failure of a single EDG, where failures are counted with multiplicities as indicated previously, will lead to station blackout, which is a design criterion.
Similarly, the first-order counterpart of Equation (3.3) and of the associated initial conditions result in Equations (3.21) and (3.22).
From the results at order n=0, it is known that P2(0)( ) 0t ≡ andP0(0)( ) 1t ≡ , and from preceding computation we know that P3(1)( ) 0t ≡ . Therefore, Equation (3.21) becomes Equation (3.23) which still uses Equation (3.22) as the initial value.
(1) 2 (1)
1 2 1
dP
dt = −μ P +λ (3.23)
The solution of this linear first order initial value problem, Equation (3.23), is then Equation (3.24).
( )1
( )
(1) 1 1( 0) (1) 1 1(
1)
Similarly, the first-order contribution to P1(1)( )t is seen in Equation (3.25).
( )1
( )
(1) 0 0( 0) (1) 0 0(
0)
The rightmost forms of Equations (3.24) and (3.25) show the fully analytic forms for the probabilities, as a function of time, of the states having a signal failed EDG. The intermediate forms, in terms of the definite integrals, will be most convenient for the purposes of reproducing the lowest nonzero order of P3(n) in the form of the nonrecovery integral in Rodgers et al. [1].The first-order differential Equation (3.1) for P0(1) and associated initial condition are shown in Equation (3.26).
0 0 1 0
(3.24) and (3.25) respectively. Substituting these previously listed equations, (3.24) and (3.25), into Equation (3.26) one can solve for Equations (3.27) and (3.28)
( ) ( )
Algebraically Equation (3.27) simplifies to Equation (3.29).
0 0 1 1
The initial value problem is an evaluation of a definite integral so the solution to Equation (3.29) can be seen in Equation (3.30).
( )1
( )
0 (1) 0 1 (1) 1 0 1This is simplified as shown in Equation (3.31).
( )1
( )
(1) 0 0(
0)
(1) 1 1(
1)
Equation (3.31) can be written in terms of Equations (3.24) and (3.25). This is shown in Equation (3.32).
( )1
( )
( )1( )
( )1( )
0 1 2
P t =−P t −P t (3.32)
With P3( )1
( )
t =0, this implies that the net sum of all the first order probabilities is zero. It would be of some interest to note whether this is true for the probabilities of allThis is not our primary interest that is to obtain the analytic expression for P3( )2 , which will not require knowledge ofP0( )1 to obtain.