Probabilistic Methods for Modeling Interdependent Complex Systems
Assistant Professor School of Civil and Environmental Engineering
Georgia Institute of Technology
Iris Tien, Ph.D.
Food Energy Water Nexus Workshop | NSF and AIChE Panel 3: Systems Approaches
October 8, 2015
Iris Tien Probabilistic Methods for Modeling Interdependent Complex Systems
Systems modeling
• Complex systems, comprised of many interconnected components
• Interdependencies between systems The Interdependent Network Design Problem for Optimal Infrastructure System Restoration 9
a. Representation of the gas network in Shelby County, with service areas.
b. Representation of the water network in Shelby County, with service areas.
c. Representation of the power network in Shelby County, with service areas.
d. Representation of the subareas (interconnected areas), shared by water and gas networks in Shelby County.
Fig. 2. Graphical representations of the gas, water and power networks at a transmission level in Shelby County, TN, and
the geographically interdependent areas between the gas and water networks.
These test network descriptions are taken from Hernandez-Fajardo & Dueñas-Osorio (2010), Hernandez-Fajardo & Dueñas-Osorio (2011), and Song & Ok (2010) where a more in-depth analysis as to why and how these networks are interconnected is provided. Second, this example takes under consideration the geographical interdependence between the water and the gas networks. Given that both networks are underground, there is a shared area preparation cost related to the reconstruction process of each component, that is, there is a saving potential by repairing co-located components from the water and the gas networks simultaneously.
Even though the sim+INDP allows including more interdependencies, such as physical interdependence between the gas and the power networks, the authors did not include them in order to keep the example realistic yet simple to understand and study.
Notice that even though some previous works on interdependent infrastructure recovery (Lee II et al., 2007; Cavdaroglu et al., 2011; Nurre et al., 2012) have mentioned the existence and importance of geographical correlation, most have focused on the failure probability correlations between components, but not in the savings in time, money, and efforts that could be made if assigning simultaneous recovery jobs for co-located components. Given that this example considers geographical interdependence, it is important to define a set of geographical spaces . In particular, is defined as the set of areas resulting from intersecting the service areas of the gas and water networks, which are the ones under geographical interdependence in this study.
Figure 2 shows the gas, water, and power networks, as well as the intersection areas that constitute . Note that these intersection areas are complementary and mutually exclusive. As expected, the cost of preparing a given subarea is positively correlated to its own size.
Regarding the disaster scenario, Adachi & Ellingwood (2009) presented a realistic earthquake scenario for Shelby County, with epicenter at and ( from Memphis center), and an approximate average magnitude of . For such an epicenter, this study includes magnitudes within a range of to . Within this range of magnitudes, the number of Monte-Carlo replications is limited to when results show steady behaviors. For this example, the authors assume that there is only one limited resource used for the reconstruction process (constraints (6)), denoted by , and that the amount used recovering each component is exactly one unit of that resource. The INDP formulation easily allows considering a more realistic set of constraints, like having a limiting budget for the recovery of each component. Nevertheless, the conclusions that such analysis could provide would be too specific for our case study, and would hardly be generalizable for other systems. On the other hand, by assuming that each component uses a similar amount of resources for its recovery, the results will be driven solely by the impact of such recovery in the performance of the system. Note that under the assumptions proposed for this case study, the limited resource would be equivalent to the maximum amount of components (nodes and arcs) to be repaired per iteration of the iINDP. We use values of from 3 to 12, in order to analyze the sensitivity of the framework with respect to this parameter. In particular, the authors chose as the lower bound of , such that it is always possible to reconstruct at least one component from each of the three networks. Likewise, the authors chose as the upper bound of in this example, given that for greater values constraints (6) would not highly affect the reconstruction strategy. Figures 4-7 show the results associated to the evolution of the costs involved in the recovery process; as expected, note how the costs depend on the magnitude of the earthquake. The cost
3
9
1
2
10
11
6
15
8
5 17
7
4
12
16
13 14
9
8
76
5
4
3
21
0
15
14
13
12
11
10
312
11
0 6 12 18 24 303km
Gas service areasGas distribution stationsGas pipelines
98 7
65
43
2 1
494847464544
4342 41
4039
38
37 36
353433
32
31 30292827 26
25 242322
2120 19 18
1716
15 14
1312
11 103937
343311
0 6 12 18 24 303km
Water service areasWater distribution stationsWater pipelines
539
0 6 12 18 24 303km
SubareasGas distribution stationsWater distribution stationsGas pipelinesWater pipelines
The Interdependent Network Design Problem for Optimal Infrastructure System Restoration 9
a. Representation of the gas network in Shelby County, with service areas.
b. Representation of the water network in Shelby County, with service areas.
c. Representation of the power network in Shelby County, with service areas.
d. Representation of the subareas (interconnected areas), shared by water and gas networks in Shelby County.
Fig. 2. Graphical representations of the gas, water and power networks at a transmission level in Shelby County, TN, and
the geographically interdependent areas between the gas and water networks.
These test network descriptions are taken from Hernandez-Fajardo & Dueñas-Osorio (2010), Hernandez-Fajardo & Dueñas-Osorio (2011), and Song & Ok (2010) where a more in-depth analysis as to why and how these networks are interconnected is provided. Second, this example takes under consideration the geographical interdependence between the water and the gas networks. Given that both networks are underground, there is a shared area preparation cost related to the reconstruction process of each component, that is, there is a saving potential by repairing co-located components from the water and the gas networks simultaneously.
Even though the sim+INDP allows including more interdependencies, such as physical interdependence between the gas and the power networks, the authors did not include them in order to keep the example realistic yet simple to understand and study.
Notice that even though some previous works on interdependent infrastructure recovery (Lee II et al., 2007; Cavdaroglu et al., 2011; Nurre et al., 2012) have mentioned the existence and importance of geographical correlation, most have focused on the failure probability correlations between components, but not in the savings in time, money, and efforts that could be made if assigning simultaneous recovery jobs for co-located components. Given that this example considers geographical interdependence, it is important to define a set of geographical spaces . In particular, is defined as the set of areas resulting from intersecting the service areas of the gas and water networks, which are the ones under geographical interdependence in this study.
Figure 2 shows the gas, water, and power networks, as well as the intersection areas that constitute . Note that these intersection areas are complementary and mutually exclusive. As expected, the cost of preparing a given subarea is positively correlated to its own size.
Regarding the disaster scenario, Adachi & Ellingwood (2009) presented a realistic earthquake scenario for Shelby County, with epicenter at and ( from Memphis center), and an approximate average magnitude of . For such an epicenter, this study includes magnitudes within a range of to . Within this range of magnitudes, the number of Monte-Carlo replications is limited to when results show steady behaviors. For this example, the authors assume that there is only one limited resource used for the reconstruction process (constraints (6)), denoted by , and that the amount used recovering each component is exactly one unit of that resource. The INDP formulation easily allows considering a more realistic set of constraints, like having a limiting budget for the recovery of each component. Nevertheless, the conclusions that such analysis could provide would be too specific for our case study, and would hardly be generalizable for other systems. On the other hand, by assuming that each component uses a similar amount of resources for its recovery, the results will be driven solely by the impact of such recovery in the performance of the system. Note that under the assumptions proposed for this case study, the limited resource would be equivalent to the maximum amount of components (nodes and arcs) to be repaired per iteration of the iINDP. We use values of from 3 to 12, in order to analyze the sensitivity of the framework with respect to this parameter. In particular, the authors chose as the lower bound of , such that it is always possible to reconstruct at least one component from each of the three networks. Likewise, the authors chose as the upper bound of in this example, given that for greater values constraints (6) would not highly affect the reconstruction strategy. Figures 4-7 show the results associated to the evolution of the costs involved in the recovery process; as expected, note how the costs depend on the magnitude of the earthquake. The cost
3
9
1
2
10
11
6
15
8
5 17
7
4
12
16
13 14
9
8
76
5
4
3
21
0
15
14
13
12
11
10
312
11
0 6 12 18 24 303km
Gas service areasGas distribution stationsGas pipelines
98 7
65
43
2 1
494847464544
4342 41
4039
38
37 36
353433
32
31 30292827 26
25 242322
2120 19 18
1716
15 14
1312
11 103937
343311
0 6 12 18 24 303km
Water service areasWater distribution stationsWater pipelines
539
0 6 12 18 24 303km
SubareasGas distribution stationsWater distribution stationsGas pipelinesWater pipelines
The Interdependent Network Design Problem for Optimal Infrastructure System Restoration 9
a. Representation of the gas network in Shelby County, with service areas.
b. Representation of the water network in Shelby County, with service areas.
c. Representation of the power network in Shelby County, with service areas.
d. Representation of the subareas (interconnected areas), shared by water and gas networks in Shelby County.
Fig. 2. Graphical representations of the gas, water and power networks at a transmission level in Shelby County, TN, and
the geographically interdependent areas between the gas and water networks.
These test network descriptions are taken from Hernandez-Fajardo & Dueñas-Osorio (2010), Hernandez-Fajardo & Dueñas-Osorio (2011), and Song & Ok (2010) where a more in-depth analysis as to why and how these networks are interconnected is provided. Second, this example takes under consideration the geographical interdependence between the water and the gas networks. Given that both networks are underground, there is a shared area preparation cost related to the reconstruction process of each component, that is, there is a saving potential by repairing co-located components from the water and the gas networks simultaneously.
Even though the sim+INDP allows including more interdependencies, such as physical interdependence between the gas and the power networks, the authors did not include them in order to keep the example realistic yet simple to understand and study.
Notice that even though some previous works on interdependent infrastructure recovery (Lee II et al., 2007; Cavdaroglu et al., 2011; Nurre et al., 2012) have mentioned the existence and importance of geographical correlation, most have focused on the failure probability correlations between components, but not in the savings in time, money, and efforts that could be made if assigning simultaneous recovery jobs for co-located components. Given that this example considers geographical interdependence, it is important to define a set of geographical spaces . In particular, is defined as the set of areas resulting from intersecting the service areas of the gas and water networks, which are the ones under geographical interdependence in this study.
Figure 2 shows the gas, water, and power networks, as well as the intersection areas that constitute . Note that these intersection areas are complementary and mutually exclusive. As expected, the cost of preparing a given subarea is positively correlated to its own size.
Regarding the disaster scenario, Adachi & Ellingwood (2009) presented a realistic earthquake scenario for Shelby County, with epicenter at and ( from Memphis center), and an approximate average magnitude of . For such an epicenter, this study includes magnitudes within a range of to . Within this range of magnitudes, the number of Monte-Carlo replications is limited to when results show steady behaviors. For this example, the authors assume that there is only one limited resource used for the reconstruction process (constraints (6)), denoted by , and that the amount used recovering each component is exactly one unit of that resource. The INDP formulation easily allows considering a more realistic set of constraints, like having a limiting budget for the recovery of each component. Nevertheless, the conclusions that such analysis could provide would be too specific for our case study, and would hardly be generalizable for other systems. On the other hand, by assuming that each component uses a similar amount of resources for its recovery, the results will be driven solely by the impact of such recovery in the performance of the system. Note that under the assumptions proposed for this case study, the limited resource would be equivalent to the maximum amount of components (nodes and arcs) to be repaired per iteration of the iINDP. We use values of from 3 to 12, in order to analyze the sensitivity of the framework with respect to this parameter. In particular, the authors chose as the lower bound of , such that it is always possible to reconstruct at least one component from each of the three networks. Likewise, the authors chose as the upper bound of in this example, given that for greater values constraints (6) would not highly affect the reconstruction strategy. Figures 4-7 show the results associated to the evolution of the costs involved in the recovery process; as expected, note how the costs depend on the magnitude of the earthquake. The cost
3
9
1
2
10
11
6
15
8
5 17
7
4
12
16
13 14
9
8
76
5
4
3
21
0
15
14
13
12
11
10
312
11
0 6 12 18 24 303km
Gas service areasGas distribution stationsGas pipelines
98 7
65
43
2 1
494847464544
4342 41
4039
38
37 36
353433
32
31 30292827 26
25 242322
2120 19 18
1716
15 14
1312
11 103937
343311
0 6 12 18 24 303km
Water service areasWater distribution stationsWater pipelines
539
0 6 12 18 24 303km
SubareasGas distribution stationsWater distribution stationsGas pipelinesWater pipelines
From: Gonzalez et al, “The interdependent network design problem for optimal infrastructure system restoration,” in review
e.g., water, power, and gas infrastructures
Iris Tien Probabilistic Methods for Modeling Interdependent Complex Systems
Systems modeling
• Support decision making in system design, management, and rehabilitation
• Achieve efficient resources management and improved system performance (reliability, resilience)
• Challenges – Uncertain information – Evolving information – Large, complex
systems
Iris Tien Probabilistic Methods for Modeling Interdependent Complex Systems
One method: Bayesian networks (BNs)
• DAG: nodes represent RVs, links dependencies between variables
• Advantages – Uncertain information: probabilistic model to
support decision making under uncertainty
– Evolving information: evidence entered into BN propagates through the network, allows for updating as new information becomes available
• Traditionally limited by the size and complexity of system that can be tractably modeled as a BN à algorithms to enable larger systems to be modeled as BNs
X5
X4 X3
X2 X1
p(x) = p(xi | Pa(xi ))i=1
n
∏
Iris Tien Probabilistic Methods for Modeling Interdependent Complex Systems
0.100 0.101 0.106 0.107
0.108 0.163
0.384
0.0
0.2
0.4
0.6
0.8
1.0
10-1
5
25-3
0
37-3
9
43-4
5
55-5
7 4-
9
16-1
8
31-3
3 1-
3
49-5
1
19-2
1
40-4
2
22-2
4
34-3
6
46-4
8
52-5
4
58-5
9
P(c
omp
fail
| sys
fail)
component #s
0.260 0.262 0.277 0.279
0.281 0.425
1.000
0.0
0.2
0.4
0.6
0.8
1.0
10-1
5
25-3
0
37-3
9
43-4
5
55-5
7 4-
9
16-1
8
31-3
3 1-
3
49-5
1
19-2
1
40-4
2
22-2
4
34-3
6
46-4
8
52-5
4
58-5
9
P(s
ys fa
il | c
omp
fail)
component #s
• Prior probabilities of failure for all components = 0.1
Forward inference
Backward inference
Example
2
1
3
5
4
6
8
7
9
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15
Substation 1
32
31
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35
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45
Substation 3
17
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Substation 2
47
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53
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Substation 4
I
III
II
58
59
O
I. Tien and A. Der Kiureghian, “Algorithms for Bayesian Network Modeling and Reliability Assessment of Infrastructure Systems: Part II – Heuristic Augmentations to Improve Efficiency,” Reliability Engineering and System Safety, in review
Iris Tien Probabilistic Methods for Modeling Interdependent Complex Systems
Modeling dynamic systems with data
• Graphical dynamic Bayesian network model
y1
z0
w0
z1 … …
f0
ν1
yk+1
wk-1
zk
wk
zk+1
fk
νk+1
yk
νk
wn-1
zn
yn
νn
fk-1 fn-1
probabilistic input external
force
measurements measurement
noise
system state
I. Tien, M. Pozzi, and A. Der Kiureghian, “Probabilistic Framework for Assessing Maximum Structural Response Based on Sensor Measurements,” Structural Safety, in review
Iris Tien Probabilistic Methods for Modeling Interdependent Complex Systems
Conclusions for FEW Nexus
• Complex systems modeling • Capturing interdependencies, tradeoffs among
the three systems
• Collecting / validating / integrating data • Metrics for system performance and
sustainability
contact: [email protected]
