3.1. Application of Bayesian Network
Khakzad et al. (2013) developed a methodology based on BN for modeling domino effects in process plants, presenting the process units as the nodes of the BN and the possibility of accident propagation among the adjacent nodes as the directed arcs of the BN. In their approach, the conditional probabilities assigned to the nodes were determined using dose-response relationships (probit functions) developed for estimating the damage probability of process units exposed to heat flux (in the event of fire) and shock wave (in the event of explosions) [
8].
Given a primary fire or explosion at a process vessel, among two exposed vessels the one with the highest (conditional) escalation probability is chosen as the secondary unit which would be entailed in the chain of accidents. Considering the possible synergistic effects between the primary and secondary vessels, the sequence and the probabilities of tertiary and higher order vessels getting involved in the chain of accidents can be identified in the same way.
It should be noted that since the explosions are sudden and short-lived, the possibility of having multiple coincident explosions is very low, and thus the principle of synergy rarely applies to the explosions. In the event of fires, however, the possibility of concurrent fires is much higher: industrial fires such as tank fires, pool fires, and jet fires usually take a longer time to burnout (depending on the available fuel) or be suppressed, and thus it is more likely that the heat fluxes of contemporary fires can be superimposed, resulting in a synergistic effect.
The application of the BN-based methodology [
10] to modeling domino effect can be demonstrated using a tank terminal consisting of five fuel storage tanks as shown in
Figure 7a. Considering a primary tank fire at T2, the fire spread between the tanks can be modeled as the BN in
Figure 7b.
The arc from T2 to T3 denotes the possibility of fire spread from T2 to T3 with a certain probability, depending on the intensity of the heat flux T3 receives from T2 and the type of T3 (atmospheric or pressurized) and its dimension. Likewise, if T3 catches fire, the fire may spread to T4, indicated by the arc from T3 to T4.
As previously mentioned, the conditional escalation probabilities of type P(T3 = fire|T2 = fire) can be determined using dose-response relationships (probit functions). In this example, for illustrative purposes, let’s assume that the escalation probabilities can be estimated as
, where Q
ij is the intensity of the heat flux tank Tj receives from fire at tank Ti. The numerator denotes the heat radiation threshold of 15 kW/m
2 required to cause damage to atmospheric storage tanks [
8]. Assuming that the intensity of heat radiation each of the storage tanks may receive from an adjacent tank fire is 40 kW/m
2, the CPT of node T3 given a tank fire at T2 can be presented as
Table 1, where
. Since the tanks are identical and exposed to identical heat radiation intensity (40 kW/m
2), the same CPT can be assigned to all of them. (Except T2, which is the root node and needs a marginal probability table rather than a CPT).
Implementing the BN in the Bayesian network modeling software, GeNIe [
27], the probability of fire spread to each storage tank can be calculated as depicted in
Figure 8. Rank ordering the units based on their marginal probabilities, T1 and T3 can be identified as the secondary units: P(T1 = fire) = P(T3 = fire) = 0.63; T4 as the tertiary unit: P(T4 = fire) = 0.39, and T5 as the quaternary unit: P(T5 = fire) = 0.24. These probabilities make sense as the probability of a lower-order event (e.g., a secondary fire at T3) should be higher than the probability of a higher-order event (e.g., a tertiary fire at T4).
The application of BN to domino effect modeling does not capture the uncertainty arising from different possible accident propagation paths. In the tank terminal in
Figure 7, for instance, if there were two simultaneous primary fires at T2 and T5, two different BNs (
Figure 9) could be developed to model the fire spread.
Having two BNs with the only difference in the direction of the arc between T3 and T4 makes it almost impossible to choose between these two BNs particularly that each BN results in different probabilities for the units. As for developing the CPTs, consider the BN in
Figure 9a as an example: the CPTs of T1 and T3 are the same as
Table 1, but the CPT of T4 is the same as
Table 2 to account for the synergistic effect of T3 and T5 on T4. As such, P(T4 = fire|T3 = safe, T5 = fire) =
while P(T4 = fire|T3 = fire, T5 = fire) =
.
Using the BN in
Figure 9a, the probabilities can be calculated as displayed in
Figure 10a. Rank ordering the units based on their marginal probabilities, T4 is identified as the secondary unit with a probability of P(T4 = fire) = 0.74, and both T1 and T3 as the tertiary units with an identical probability of P(T1 = fire) = P(T3 = fire) = 0.63. This result, however, does not comply with the structure of the BN: Since T4 would catch fire as a secondary unit before T3 catches fire as a tertiary unit, the direction of the arc—which also implies the sequence of the fires—should be from T4 to T3 not the opposite, unlike what is shown in
Figure 10a.
This may make the modeler wonder if the BN in
Figure 9b should have been chosen to model the right order of fires during the domino effect. Using the BN of
Figure 9b, the probabilities can be calculated as in
Figure 10b; as can be seen, again the probabilities of the sequential fires do not comply with the sequence of the fires: structurally and considering the direction of the arcs, T4 seems to catch fire before T3 (note the arc from T4 to T3), however the fire probability at T3 (74%) is higher than that of T4 (63%), which means, T3 has caught fire before T4!
3.2. Application of Dynamic Bayesian Network
As demonstrated in the previous section, the application of conventional BN to domino effect modeling may result (not always) in counterintuitive results (e.g., the probability of a secondary event is less than that of a tertiary event!) This is mainly because the conventional BN is not capable of considering all possible mutual interactions among the involved units, and thus the propagation paths are imposed by the modeler rather than being identified by the model. In other words, the modeler decides about the direction of arcs instead of letting the model identify the most probable path and subsequently align the direction of arcs [
25].
Khakzad [
18] modified the BN methodology previously developed by Khakzad et al. [
10] by using a DBN, where the most likely sequence of events can be identified by the model by considering all the possible interactions among the units. In this regard, the uncertainty on the temporal sequence of T3 and T4 in
Figure 9 can be modeled as the DBN in
Figure 11. The arcs in color red denote the uncertainty as to whether the fire would spread from T3 to T4 or the opposite.
In
Figure 11, the states of T2 and T5 in the first time slice have been instantiated to “fire” as P(T2(0) = fire) = P(T5(0) = fire) = 100%, whereas the states of T1, T3, and T4 have been instantiated to “safe” as P(T1(0) = safe) = P(T3(0) = safe) = P(T4(0) = safe) =100% as the initial evidence (observation) required to quantify the DBN. The CPTs of the nodes of the DBN in
Figure 11 are similar to
Table 1 and
Table 2. For the sake of clarity, the CPT of T3 in the 2nd time slice, i.e., T3(2), following the tank fires at T2 and T5 in the 0th time slice (the beginning of the modeling) is presented in
Table 3.
As can be seen from
Figure 11, the DBN results in identical probabilities of fire spread to T3 and T4 at sequential time steps due to the symmetrical position of T3 and T4 to the primary units T2 and T5. In the 1st time step, the probability of fire spread to T1, T3 and T4 is the same as 0.63 mainly because the DBN has not yet be given sufficient time to take into account all the possible fire interactions [
25]. Given enough time, the probabilities of fire spread to T3 and T4 are seen to be identical and slightly higher than that of T1. As such, either all the three tanks (T1, T3, T4) may catch fire as the secondary units in the 1st time step with a probability of 0.63 (this is the worst-case scenario) or T3 and T4 may catch fire before T1 in the 2nd time step.
The developed DBN can also be used to identify the most critical units contributing to the domino scenarios [
24,
28]. For this purpose, given an equal chance of primary fire for all the units at the 0th time slice (0.1, for illustrative purposes) and considering the mutual interactions among all the units, the (marginal) probability of fire spread to each unit can be calculated as shown in
Figure 12.
As can be seen, T3 has the highest probability of catching fire as time goes by and thus can be identified as the most critical (vulnerable) unit in the terminal. Khakzad and Reniers [
28] demonstrated that the isolation of most vulnerable units from the chain of fires, for instance, via fireproofing or by keeping them empty, would dramatically disrupt the domino effect and thus reduce the vulnerability of the facility more than would the isolation of any other tanks.