Associated model
In order to calculate the degree of correlation data and derive the reliability of RPS, an association model between each element to characterize the relationship is established. The reliability data is calculated based on the relationship among the elements. In order to facilitate the calculation, this section will establish a simplified model of RPS reliability, and describe the formulas and conversion of data used in the calculation of RPS reliability.
The reliability model established shown in Fig. 2, which is used to characterize the relationship of the control process. Matrix A = A1, A2, A3, A4 represents reliability of condition signals feedback for field device. Matrix B = B1, B2, B3, B4, B5 represents reliability of generating control command when received condition signals. Matrix C = C1, C2 = c1, c2, c3, c4, c5, c6 represents reliability of control commands issued. Matrix D = D1, D2 represents reliability of control actions, in which D1 represents of reliability of reactor trip action, and D2 represents of the reliability of ESF action. The relationship of matrix A, B, C, D is shown in Fig. 2.
Contribution factor
In order to calculate reliability of the entire network, it is necessary to define the contribution degree of each node to the next node, for example the reliability of path that through node B5 determined by the reliability of node B5 as well as the reliability of node A1 and A4 (Hou and Chen 1999). The reliability of node B5 is determined by the correlation function. The contribution of A1 and A4 to B5 depends on their importance. If it is assumed that the paths A1 and A4 are equally important, the contribution factor will be 0.5.
Note the contribution of Ai to Bj as Ab
_{
ij
}, Bi to Cj as Bc
_{
ij
}, Ci to Dj as Cd
_{
ij
},thus we establish correlation matrix Ab, Bc, Cd of matrix A, B, C, D. If the reliability of a node is related to n nodes upstream, the reliability contribution of each node upstream to this node is 1/n, thereby the correlation matrix is obtained:
$$ Ab = \left {\begin{array}{*{20}l} { 0,Ab_{12} ,0,0,Ab_{15} } \hfill \\ {0,0,0,Ab_{24} ,0} \hfill \\ {Ab_{31} ,0,0,0,0,} \hfill \\ {0,0,Ab_{43} ,0,Ab_{45} } \hfill \\ \end{array} } \right = \left {\begin{array}{*{20}l} { 0,1,0,0,1/2} \hfill \\ {1/2,0,0,1,0} \hfill \\ {1/2,0,0,0,0,} \hfill \\ {0,0,1,0,1/2} \hfill \\ \end{array} } \right $$
(3)
$$ Bc = \left {\begin{array}{*{20}l} {0,Bc_{12} ,0,0,0,0} \hfill \\ {Bc_{21} ,0,0,0,0,0} \hfill \\ {0,0,0,0,0,Bc_{36} } \hfill \\ {0,0,Bc_{43} ,0,0,0} \hfill \\ {0,0,0,Bc_{44} ,Bc_{45} ,0} \hfill \\ \end{array} } \right = \left {\begin{array}{*{20}l} {0,1,0,0,0,0} \hfill \\ {1,0,0,0,0,0} \hfill \\ {0,0,0,0,0,1} \hfill \\ {0,0,1,0,0,0} \hfill \\ {0,0,0,1,1,0} \hfill \\ \end{array} } \right $$
(4)
$$ Cd = \left {\begin{array}{*{20}l} {Cd_{11} ,0} \hfill \\ {Cd_{21} ,0} \hfill \\ {Cd_{31} ,0} \hfill \\ {0,Cd_{42} } \hfill \\ {0,Cd_{52} } \hfill \\ {0,Cd_{62} } \hfill \\ \end{array} } \right = \left {\begin{array}{*{20}l} {1/3,0} \hfill \\ {1/3,0} \hfill \\ {1/3,0} \hfill \\ {0,1/3} \hfill \\ {0,1/3} \hfill \\ {0,1/3} \hfill \\ \end{array} } \right $$
(5)
It is necessary to be noted that the calculation of contribution factor for a node is mainly concerned with three aspects:

The importance of the transmission path. The paths transmit the signal for safety equipment is more important than for nonsafety equipment.

The importance of the transmitted signal. The signal used for reactor trip is more important than for ESF.

The number of transmission signals.
In this paper, the transmission path and signals are assumed to be the same importance, the contribution factor of nodes are measured by the number of transmission signals.
Numerical relationship
The reliability of the RPS is noted as K. Since node D1 and D2 are output paths to the entire model, the reliability of D1 and D2 represents the reliability of the entire model. The reliability of D1 depends on D1, C1, C2, C3 where the reliability of D1 is (C1*Cd
_{11} + C2*Cd
_{21} + C3*Cd
_{31})*D1, which can be expressed as (C*Cd).*D by matrix. The node reliability considering the contribution of previous node is noted as \( A' \), \( B' \), \( C' \), \( D' \), thus:
$$ B' = (A'*Ab).*B $$
(7)
$$ C' = (B'*Bc).*C $$
(8)
$$ D' = (C'*Cd).*D $$
(9)
Substituting (6), (7), (8), (9) to (10), we get the reliability formula of the entire model:
$$ K = det(D') = det((((((A*Ab).*B)*Bc).*C)*Cd).*D) $$
(11)
Model application
According to the control network model (Fig. 1), the signal flow of reactor trip response and ESF response is sorted out, which is shown in Fig. 2. A1, A2, A3 and A4 represent the uplink paths which feeding control status back. B1, B2, B3, B4 and B5 represent the control signal downlink paths. C1, C2 and C3 represent reactor trip response, while C4, C5 and C6 represent ESF response. D1 represents the reactor trip action, and D2 stands for ESF action (He and Shi 2006).
When the reactor trip condition or ESF condition occurs, device status signal will be feedback via the uplink route A1. Then SCID releases control commands through downlink route B2, which would result in the reactor trip response and ESF response. It controls the related device to generate reactor trip and ESF action.
According to the results calculated in “Establish reliability model” section, we get matrix A, B and C.
Matrix D = D1, D2 represents reactor trip action and ESF action, which is the result of control command issued. Matrix D is set to D = D1, D2 = 1, 1.
According to the formula (11), we get the reliable calculation formula for reactor protection system:
$$ K = det(D') = det((((((A*Ab).*B)*Bc).*C)*Cd).*D). $$
The reliability of RPS is calculated:
$$ K = det\left( {\left {0.33,\,0.10} \right} \right) = 0.215 $$
From the results calculated, we can see that the entire RPS reliability is 0.215. Reactor trip reliability is 0.33, which is higher than the ESF Reliability 0.10. The low reliability of node B5 causes low reliability of ESF, which led to a lower reliability of RPS. In engineering practice, if we want to improve the reliability of RPS, increasing the reliability of the node B5 is particularly important. If we improve the reliability of the node B5 to 0.90 by means, the ESF calculated reliability will be 0.325, compared with 0.10 before optimization significantly improved. Therefore, this method can not only calculate the reliability of RPS but also apply to work in the engineering aspects for fault diagnosis.