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Generalized Probabilistic Restoration Prediction
Aman Karamloua, Liyang Mab, Wenjuan Suncand Paolo Bocchinid
aStructural Engineer, DeSimone Consulting Engineers, formerly Research Assistant at Lehigh University
bResearch Assistant, ATLSS Engineering Research Center, Lehigh University
cResearch Associate, ATLSS Engineering Research Center, Lehigh University
dFrank Hook Assistant Professor, Department of Civil and Environmental Engineering, Lehigh University
Abstract: To reduce the negative impacts of indirect losses after an extreme
event, typically caused by long-lasting reduced functionality of critical structures
and infrastructure systems, disaster managers need to plan the disaster relief ac-
tivities considering the post-event resilience as one of the main determining fac-
tors. However, this requires better understanding of the restoration process of the
system and the underlying uncertainties, as well as the impact of each component
on the functionality of the system. In this paper, a technique is presented to sim-
ulate the restoration schedule and recovery curve of damaged infrastructure sys-
tems with multiple components. This is carried out by assembling the restoration
tasks of each damaged component of the system considering the availability of
resources for the restoration activities, and the uncertainties involved in restora-
tion duration. The application of the proposed methodology is presented for two
cases: an electrical distribution substation, and a pair of bridges, all with multi-
ple damaged components. Sample restoration schedules and recovery curves are
presented, and probabilistic restoration functions are derived for every structure.
1 Introduction
Todays communities are composed of several social, economic, and physical units, all function-
ing interdependently to provide the necessary means of life for modern human beings. These
include the supply chain (e.g., food, water, and energy), business sectors and financial institu-
tions, and infrastructure systems (e.g., communication, road, and electric networks). The func-
tionality of these units are typically threatened by several causes, among which natural (e.g.,
earthquake and hurricane) and man-made (e.g., fire spread and terrorist attack) extreme events
are the most critical and devastating. For example, despite the small number of causalities and
human losses, the damage imposed to the socioeconomic elements and physical infrastructures
by the 2011 Christchurch earthquake was both extensive and expansive to the extent that it
completely disrupted the normal life of the residents and forced them to evacuate a large por-
tion of the impacted regions [1]. Similarly, the slow pace and inefficient restoration process
following Hurricane Katrina, caused large-scale migration of the population, which imposed a
permanent loss to the impacted community [2]. Additionally, non-engineering factors such as
demographics, governance, and land-use policies have been found to have a significant impact
on the recovery process. Such observations made researchers and engineers to develop various
methods to improve disaster management activities and mitigate potential losses in order to en-
hance the resilience of communities [3, 4].
Resilience has been defined differently across different fields and domains, such as social, eco-
nomic, and ecology sciences [4]. In engineering fields, Bruneau et al. [5] are considered as the
IASSAR
Safety, Reliability, Risk, Resilience and Sustainability of Structures and Infrastructure
12th Int. Conf. on Structural Safety and Reliability, Vienna, Austria, 6–10 August 2017
Christian Bucher, Bruce R. Ellingwood, Dan M. Frangopol (Editors)
c
2017 TU-Verlag Vienna, ISBN 978-3-903024-28-1
3249
Time
Step-1: Component-
System Definition
Structure-1
Component-11
Component-n1
⋮
⋮
Structure-ns
Component-1ns
Component-nns
⋮
Step-2: Component DSs
Restoration Tasks
Task Duration
Precedence Relations
Resource Requirement
⋮
Task Duration
Precedence Relations
Resource Requirement
Task Duration Sampling
Step-4: Sample
Restoration Schedule
Restoration Tasks
Step-5: Sample
Restoration Curves
⋮
⋮
Structure-1
Structure-ns
𝑄1(𝑡)
Time
𝑄𝑛𝑠(𝑡)
Step-3:
Sampling
Repeat Step-4 and Step-5 for all duration sample sets
Notations: DS=Damage State, ns=Number of Structures in the system,
Component-ij= ith component from jth structure, 𝑄𝑗(𝑡) =Functionality
of jth structure at time t.
Figure 1: The conceptual flowchart of the proposed methodology for restoration and functionality as-
sessment of (single or multiple) structures
pioneers of disaster and infrastructure resilience. They defined community resilience as the abil-
ity of the social units to mitigate hazards, and recover from the damage with a minimal social
disruption. Also, the concept of resilience triangle proposed by Bruneau et al. is one of the most
common tools to quantify resilience, and the underlying idea inspired several researchers to pro-
pose other metrics and methodologies to measure, enhance, and optimize resilience of different
structures and infrastructure systems [6–9]. These include, but are not limited to, healthcare
structures and facilities [10, 11], power grids [12, 13], gas distribution systems [14], bridges
and transportation networks [15–17].
As indicated in the definition of resilience, the assessment of the restoration process of different
units of the community and their functionality immediately after the event and during the recov-
ery phase are among the key steps of resilience assessment. However, evaluating the restoration
process and its characteristics prior to the event is a challenging task. Most of all, this is due
to the considerable amount of uncertainties involved in such process, originated from factors
like decision making, restoration resource availability, weather conditions, etc. Also, the inter-
dependent impact of the functionality and restoration of different structures and infrastructures
makes the simulation of the post-event activities more complicated.
To this respect, this paper presents a technique for post-event restoration and functionality as-
sessment of individual and groups of structures of any type or class. This methodology is an
extension of the technique presented by Karamlou and Bocchini [18, 19] that allows the simu-
lation of the restoration schedule and recovery curve of damaged bridges using a construction
management approach. This paper is organized as follows. Firstly, the proposed framework is
presented conceptually and in general terms. Next, the technique is applied to the restoration
and functionality analysis of two types of structures: a damaged electric power substation, and
a group of two damaged bridges.
2 Conceptual Framework of the Proposed Technique
Karamlou and Bocchini [18, 19] developed a simulation-based framework to compute the sam-
ple construction schedules and restoration functions focusing on the specific case of a single
bridge with multiple damaged components. A more conceptual and generalized presentation of
this technique is provided herein, to make it applicable to other types of structures and to groups
of structures. Figure 1 illustrates the flowchart of the proposed technique.
The first step is the system definition. Here, system refers to a group of structures whose restora-
tion and functionality is being investigated. This requires specifying the components of all stud-
3250
ied structures. To this respect, the components that (1) have a major contribution in delivering
the expected functionality of the structure, (2) are prone to considerable amount of damage
in the case of catastrophic events, and (3) require a significant amount of resources for their
restoration should be considered in the definition of the system. For instance, in the case of
bridges, columns are included in the definition as both field inspections and numerical sim-
ulations have shown that they are among the most vulnerable elements of bridges and their
restoration takes a considerable amount of time and construction resources [18–22]. Similarly,
for electrical distribution substations, circuit breakers and transformers are considered as the
critical components [23].
In the next step (Step-2), the damage and restoration data of the components of each structure
are specified. To this respect, for each component, a set of repair-based damage states is de-
fined. These states are different damage levels to which a particular restoration decision and
repair process is associated. Accordingly, a set of tasks can be compiled for the restoration
of each component at every relevant damage state. The damage state of each component can
be specified through numerical simulations or from inspections typically made after an event.
The duration of each component restoration task is characterized with a probability distribution
function which will be used later to account for the uncertainty in the restoration scheduling of
the (group of) structure(s). Additionally, a selected number of resource items required to execute
each task is specified. These items are typically the equipment and human resources needed to
perform the restoration.
In many cases there is a precedence relationship among the restoration tasks of a component,
or among the components of the studied structures. Such precedence relations are typically due
to common construction logics and approaches [18]. Also, if deemed relevant, precedence rela-
tions can be defined among different structures of the group to model interdependencies in the
restoration of infrastructure systems. For instance, if a bridge needs to be restored before being
able to access and restore a substation, a precedence relation can be defined for the two struc-
tures in the system. However, the discussion of these types of precedence relations to represent
infrastructure interdependencies is beyond the topic of this manuscript.
In Step-3, samples of the duration of the restoration tasks are generated for all components of
the structures. Techniques such as Latin hypercube [24] sampling are preferred for a more com-
putationally efficient simulation.
Each set of samples generated in Step-3 is used to compute one sample restoration schedule for
the investigated structures in Step-4 . This is carried out using a number of optimization-based
scheduling tools developed to reflect some of the uncertainties in scheduling and the flow of in-
formation [18]. The proposed tools are based on the Resource Constrained Project Scheduling
Problem (RCPSP) to compute the sample restoration schedules [25]. RCPSP is an extension of
the classic Critical Path Method that takes into account the resource availability for scheduling,
in addition to precedence relations. The number of available resources for each resource type
used for the restoration tasks is specified given the capabilities of the contractors who are in
charge of the restoration process.
Sample schedules show the start and end time of each restoration task for each component of
the structures. Such information is used to develop a sample restoration function (or recovery
curve) for each structure. The process to compute the restoration function from the restoration
schedule is different for each type of structure. However, general factors that can be consid-
ered for such computations include the disruption caused by and the safety of the crew during
the restoration process, as well as the damage level of the components and the safety of the
structure. This process (Step-4 and Step-5) is repeated for all sets of restoration task duration
samples. The resulting sample restoration functions can be used to compute the time-dependent
probabilistic characteristics of the functionality for each structure during the restoration pro-
3251
𝑇𝑅1𝑇𝑅2
𝐵1
𝐵2
𝐷
1
𝐷2
𝐷3
𝐷4
𝐶
1
𝐶2
𝐶3
𝐶4
𝐷5
𝐷6
𝐷7
𝐷8
𝐷𝐿1
𝐵3
𝐵
4
𝐷9
𝐷
10
𝐷
11
𝐷
12
𝐶5
𝐶6
𝐶7
𝐶8
𝐷
13
𝐷
14
𝐷
15
𝐷
16
𝐷
17
𝐷
18
𝐷
19
𝐷20
𝐶9
𝐶
10
𝐷21
𝐷22
𝐷23
𝐷24
𝐶
11
𝐶
12
𝐶
13
𝐶
14
𝐷25
𝐷26
𝐷27
𝐷28
𝐷29
𝐷30
𝐷31
𝐷32
𝐶
15
𝐶
16
𝐶
17
𝐶
18
𝐷33
𝐷34
𝐷35
𝐷36
𝐷37
𝐷38
𝐷39
𝐷40
𝐶
19
𝐶20
𝐷41 𝐷42
𝐷43 𝐷44
𝐶21
𝐶22
𝐷𝐿2𝐷𝐿3𝐷𝐿4𝐷𝐿5𝐷𝐿6𝐷𝐿7𝐷𝐿8
Figure 2: Layout of a double breaker bus distribution substation
Table 1: Tasks for the restoration of the example distribution substation
ID Description Duration (hours) Resource requirement
Distribution Min Mode Max Manpower
A1Detailed damage assessment Triangular 10 11.5 13 0
A2Repair circuit breaker Triangular 0.5 1.0 1.5 2
A3Repair disconnect switch Triangular 0.5 1.0 1.5 2
A4Repair transformer anchorage Triangular 60.0 84.0 156.0 5
A5Repair transformer radiator Triangular 5.0 8.0 11.0 2
A6Repair transformer bushing Triangular 3.0 4.0 5.0 2
A7Repair bus Uniform 0.0 - 0.4 3
A8Obtain additional repair material from district yard Triangular 1.0 2.0 3.0 1
cess. This includes any statistical moment or dispersion of the functionality distribution of a
structure at each time step during the restoration. Also, the final results can be presented in the
form of probabilistic restoration functions, which will be discussed in the following sections.
3 Application
Two examples are provided in the following to showcase the technique presented in Section 2.
The first is the restoration of a damaged substation, which is a case for a single structure with
multiple damaged components. The second example is the restoration of two bridges, and each
bridge has multiple damaged components.
3.1 Restoration of a Power Substation
In this section the restoration process and recovery curves of a power substation is simulated
using the proposed framework. To this respect, this study uses a double breaker bus substation,
which is among the most common substation layouts around the world [26]. The layout of the
substation along with its critical components considered in this study is shown in Figure 2. In
this figure, TRi,Ci,Di, and Birepresent the ith transformer, circuit breaker, disconnect switch
and bus of the example substation. As it can be observed, the substation has 80 components: 44
disconnect switches, 22 circuit breakers, 4 buses, 2 transformers, and 8 distribution lines.
In the case of the current example, it is assumed that the extreme event has caused the failure
of the anchorage and radiator of TR1and bushing of TR2. Also, bus bars B1and B4, discon-
nect switches D16,D27, and D36, as well as circuit breakers C1,C9,C11 and C21 have been
damaged and require to be restored. Table 1 presents the restoration tasks of the damaged com-
ponents along with their duration properties and resource requirements [26, 27]. Regarding the
3252
0
50
100
010 20 30 40 50 60 70 80 90 100 110 120 130 140 150
𝐴1
𝐴4
𝐴6
𝐴7
𝐴8
𝐴3
𝐴3
𝐴2
𝐴2
𝐴3
𝐴7
𝐴2
𝐴2
𝐴5
𝑄(𝑡)
Time (hours)
(a)
(b)
Figure 3: Substation (a) sample restoration schedule, and (b) sample restoration function
precedence relations among the tasks, it is assumed that task A1is the prerequisite of all other
restoration activities. Also, task A8should be carried out before the restoration of the circuit
breakers. Finally, task A4is the predecessor of tasks A5and A6.
Figure 3(a) shows a sample restoration schedule of the substation considering a random set of
task duration samples. To compute such a schedule, the RCPSP has been solved using the dura-
tion of the restoration tasks and their resource requirements, as well as the precedence relations
as discussed earlier. In this example, the total duration of the substation restoration was con-
sidered as the objective of the RCPSP. In this case, it was assumed that the contractor assigned
10 technicians for this restoration project. Figure 3(a) shows that the process starts with the
simultaneous restoration of the two transformers (TR1and T R2) and bus B1. This is followed
by the restoration of disconnect switches, circuit breakers and bus B4. Also, the total duration
of the restoration is 141 hours for this specific sample.
The total functionality of the substation is assumed to be proportional to the number of func-
tional distribution lines as defined in Equation 1, which is derived using classic system reliability
theory and the concept of minimal path set:
Q(t) = (1/8)
8
∑
i=1
[ΦDLi(t)] = (1/8)
8
∑
i=1"1−
4
∏
j=11−∏
k∈Pi j
xk(t)#(1)
where Q(t)and ΦDLi(t)are the functionality of the substation and the value of the structure
function of the ith distribution line at time t, respectively. ΦDLi(t)assumes the value of one if
power can be transmitted through DLiat time tand zero otherwise. Structure functions can be
written using the the concept of minimal path set as presented in the second part of Equation 1.
In this part, Pi j is the jth path set of DLi, and xk(t)is the value of the state of component kin
the minimal path set Pi j at time t. For the case of the current example, xk(t)is equal to one if
the associated component is not damaged or its necessary restoration process has been com-
pleted. Otherwise, xk(t)is considered to be zero. Based on Figure 3, for each distribution line,
four minimal path sets can be defined. For example, the four minimal path sets for DL6are
3253
Table 2: Minimum path sets for DL6
Pi j Components
P61 {TR2,D26,C13,D25,B3,D29,C15,D30}
P62 {TR2,D27,C14,D28,B4,D32,C16,D31}
P63 {TR1,D6,C3,D5,B1,D41,C21,D42 ,B3,D29,C15,D30}
P64 {TR1,D7,C4,D8,B2,D43,C22,D44 ,B4,D32,C16,D31}
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Time (hours)
P(Q(t) <100%)
Figure 4: Probabilistic restoration functions for the substation
presented in Table 2. The minimal path vectors for the other distribution lines can be written
similarly. Considering the initial damage state of each component and the restoration comple-
tion time of the tasks for each component, Equation 1 can be used to compute the functionality
at each time step. The result is presented in Figure 3(b). It can be observed that after completion
of task A6(t=16 hours) all of the components in P61 are functional and thus DL6is back on
line. Additionally, DL7and DL8are restored at t=16 hours and therefore the functionality of
the system is increased to 37.5%. Similarly, the functionality of the rest of the distribution lines
are restored as their recovery processes are finalized.
One million samples of each task shown in Table 1 has been generated to consider the un-
certainty in the duration of all restoration tasks and the consequent effects on the restoration
schedule and functionality of the substation. The resulting restoration function samples can be
used to compute the probabilistic characteristics of the functionality during the restoration pro-
cess. For example, Figure 4 illustrates the probability of the substation functionality being less
than 100% at each time step. The results show that this damaged substation is most likely to be
fully functional after 38 hours.
3.2 Restoration of a Group of Bridges
In this section the restoration process and recovery curves of a group of two bridges are simu-
lated using the proposed framework. These bridges are a Single Span (SS) Steel Girder bridge,
and a Multi-Span Simply Supported (MSSS) Steel Girder bridge, both among the bridge classes
most typical of the Central and Southern United States [20]. The structural properties of the
bridges have been adapted from other studies and can be found in [20, 22]. Figure 5 shows a
schematic view of these bridges and their components.
The restoration and functionality analyses are performed and the data are briefly provided for a
case study. In this example, it is assumed that both bridges are in the same region and subjected
to a hypothetical seismic event with a high intensity. As a result, for the SS Steel Girder bridge,
the back wall on the left approach has been cracked due to the pounding of the deck. Regarding
the MSSS Steel Girder bridge, it is assumed that there is a considerable amount of residual
displacement in the left rocker bearings and thus the left deck needs to be realigned. Table 3
presents the restoration tasks and some of their properties used for the functionality analysis.
In this table, the tasks which have ID starting with SS and MSSS are associated with the SS
Steel Girder and the MSSS Steel Girder bridge, respectively. Five and four tasks need to be
3254
24.4 m
12.2 m 12.2 m
Left Columns
Right Columns
(b) MSSS Steel Girder bridge
Right
abutment
Left
abutment
Left
foundations Right
foundations
Left fixed
bearings
Left rocker
bearings Mid-fixed
bearings Mid-rocker
bearings Right fixed
bearings
Right rocker
bearings
Left
approach Right
approach
(a) SS Steel Girder bridge
18.3 m
Fixed
bearings Rocker
bearings
Left
abutment
Left
approach
Right
abutment
Right
approach
Figure 5: (a) SS Steel Girder bridge, (b) MSSS Steel Girder bridge
Table 3: Tasks for the restoration of the example SS Steel girder bridge and MSSS Steel girder bridge
ID Description qA(%)Duration (days) Resource requirement
Distribution Min Mode Max Manpower Geo-Machine Crane Mixer
SS1Excavate abutment 50 Uniform 1 - 2 5 2 0 0
SS2Repair abutment cracks 100 Triangular 1 2 3 2 0 0 1
SS3Repair abutment spalls 100 Triangular 1 2 3 2 0 0 1
SS4Abutment backfill 50 Triangular 1 2 4 5 2 0 0
SS5Install joint seal 50 Triangular 1 2 3 3 0 0 1
MSSS1Review shoring plan 100 Triangular 20 30 40 0 0 0 0
MSSS2Install shoring 0 Triangular 1 2 3 5 0 1 0
MSSS3Realign bearings 50 Uniform 1 - 2 5 0 1 0
MSSS4Remove shoring 0 - 1 - 1 5 0 1 0
performed for the restoration of the SS Steel Girder bridge and the MSSS Steel Girder bridge,
respectively. In many cases, it is required to fully or at least partially close the bridge due to
the restoration activities or for the safety of the crew during the construction. To this respect,
the parameter qA reflects the residual functionality of the bridge during the execution of each
task. Also, four different resource types have been considered to carry out the restoration tasks:
manpower, geo-machines, crane, and concrete mixer. It is assumed that the restoration will be
performed by a contractor with 5 crew members, 2 geo-machines, 1 crane, and 1 concrete mixer.
Figure 6(a) shows a restoration schedule sample computed using one set of task duration sam-
ples by solving the RCPSP. To this end, the minimization of the sum of the total duration of the
restoration of the two bridges has been considered as the objective of the RCPSP. In addition
to the schedule, the precedence relations among the tasks of the two bridges are also illustrated
using the arrows. The results show that the process starts simultaneously for both bridges, but
the restoration of the SS Steel Grider bridge finishes first, after 9 days.
Figure 6(b) presents the sample restoration functions of the two bridges computed from the
schedule shown in Figure 6(a) and by taking into account a number of considerations including
bridge structural safety, construction induced traffic disruptions, and temporary repair solutions.
For the case of the SS Steel Girder bridge, it is assumed that the damage to the abutment requires
the reduction of the allowable speed of the vehicles and this is equivalent to the reduction of the
functionality of the bridge to 50%. Therefore, the functionality is restored to its full capacity
after the repair process is finished. Regarding the MSSS Steel girder bridge, the initial damage
3255
𝑆𝑆1
𝑆𝑆2
𝑆𝑆3
𝑆𝑆4
𝑆𝑆5
𝑀𝑆𝑆𝑆1
𝑀𝑆𝑆𝑆2
𝑀𝑆𝑆𝑆3
𝑀𝑆𝑆𝑆4
SS Steel Girder bridge tasks
MSSS Steel Girder bridge tasks
0
50
100
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Time (days)
SS Steel Girder Bridge
MSSS Steel Girder Bridge
𝑄(𝑡)
(a)
(b)
Figure 6: (a) Sample restoration schedule, Sample restoration curve for (b) SS Steel Girder bridge and
MSSS Steel Girder bridge
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
Time (days)
P(Q <100%)
SS Steel Girder bridge
MSSS Steel Girder bridge
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40 45 50
Time (days)
P(Q(t) <100%)
SS Steel Girder bridge
MSSS Steel Girder bridge
Figure 7: Probabilistic restoration functions for bridges
does not require the closure of the bridge. However, the installation and removal of the shorings
(tasks MSSS2and MSSS4) require the bridge to be closed for the safety of the crew. Also, during
the execution of task MSSS3, the functionality is reduced to 50% due to the interruption of half
of the passing traffic [18]. However, since this partial opening is very short (during MSSS2and
MSSS4the functionality is zero), such increase in the functionality has been filtered out using
a window function. Therefore, the functionality remains zero during task MSSS3. More details
about the computation of bridge functionality during restoration can be found in [18].
To study the uncertainty in the restoration scheduling, one million samples of each task shown
in Table 3 have been generated and used to compute sample restoration schedules and their
associated recovery curves. The results have been used to compute the probabilistic restoration
functions for each of the bridges. For instance, Figure 7 shows the probability of the bridge
being at least partially closed (P[Q(t)<100%]) for each of the bridges.
4 Conclusions
This study proposes a generalized simulation-based methodology for post-event restoration
analysis of damaged structures and infrastructures in a probabilistic fashion. The proposed tech-
nique is capable of computing the restoration schedules and functionality recovery curves of
different damaged structures considering the resource availability and the uncertainties in the
restoration process at the component-level.
The application of the methodology is presented for two cases: a damaged substation, and two
damaged bridges. For the case of the substation, the impact of multiple damaged components
(transformers, disconnect switches, buses, etc.) was considered in the functionality analysis.
Regarding the second example, the restoration of a damaged Single Span (SS) and a dam-
aged Multi-Span Simply Supported (MSSS) Steel Girder bridge was investigated. Probabilistic
restoration functions for the example structures were derived using the proposed methodology.
3256
The input data, such as duration distribution and resource requirements of every task for each
damaged component in different damaged structures, can be updated through consulting with
professional engineers and construction managers. Also, field data of restoration schedules of
structures, along with their recovery curves can be used to further adjust the parameters and
assumptions of the proposed technique, and improve the results. Ultimately, this generalized
methodology predicts the probabilistic functionality restoration with time-variant characteris-
tics for individual structures, a group of structures, and networked infrastructure systems. The
functionality probability at different times can help decision-makers to conduct emergency man-
agement activities and adjust mitigation policies with more realistic understanding on the field
performances of critical structures and infrastructure systems.
Acknowledgments
The support provided by the National Science Foundation through award CMMI - 1541177 is
gratefully acknowledged.
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