2. Methodology and Architecture
The integrated simulation model has been developed as a Python-based package consisting of modules for simulation of system- and network-level cascading effects resulting from component failures. The overall methodological framework of the integrated simulation platform is illustrated in Fig. 2.1.
Fig. 2.1 InfraRisk integrated simulation platform structure
The platform is based on the widely accepted risk- and resilience analysis frameworks as presented in [Argyroudis2020] and [Balakrishnan2020]. In this framework, the most important component is an interdependent infrastructure model that consists of various infrastructure system submodels of interest. In addition, the major hazards in the region can also be modeled. Further, the vulnerabilities in the network to those hazards are mapped and the direct impacts (physical and functional failures in the infrastructure components) are simulated using the hazard model. For scheduling post-disaster restoration/repair actions, a recovery model is also developed. The restoration actions are prioritized based on specific recovery strategies or optimization methods. The indirect failures in the network are simulated using the interdependent infrastructure model based on the initial failure events and the subsequent repair actions. The component- and system-level operational performance are quantified and tracked using appropriate resilience metrics (Fig. 2.2).
Fig. 2.2 Implementation of the simulation platform to generate performance curves
The basic idea behind the InfraRisk simulation package is to integrate existing infrastructure-specific simulation models through an object-oriented interface so that interdependent infrastructure simulation can be achieved. Interfacing requires identifying and modeling the dependencies among various infrastructure components and time-synchronization among infrastructure simulation models. To address the above challenges, InfraRisk is built using a sequential simulation framework (Figure 2). The advantage of this approach is that it simplifies the efforts for data preparation and enables the complete utilization of component-level modeling features of the domain-specific infrastructure models.
InfraRisk consists of five modules, namely,
integrated infrastructure network simulation
hazard initiation and vulnerability modeling
recovery modeling
simulation of direct and indirect effects
resilience quantification.
In the rest of the section, a detailed discussion on each of the above modules is provided.
2.1. Integrated infrastructure network
This module houses the three infrastructure models to simulate power-, water-, and transport systems. These models are developed using existing Python-based packages. In order to model the power system, pandapower is employed [Thurner2018]. The water distribution system is modeled using wntr package [Klise2018]. The traffic model provides the travel costs for traveling from one point in the network to another and is modeled using the static traffic assignment method [Boyles2020]. All three packages have network-flow optimization models that identify the steady-state resource flows in the respective systems considering the operational constraints. The details of the packages are presented in Table 2.1.
Table 2.1 Infrastructure packages used in the simulation model
Infrastructure |
Package |
Capabilities |
Power |
pandapower |
|
Water |
wntr |
|
Transport |
static traffic assignment package |
|
2.1.1. Power system model
The pandapower package can be used to determine the steady-state optimal power flow for a given set of system conditions. The optimal power flow problem, solved by pandapower, attempts to minimize the total power distribution costs in the system under load flow-, branch-, bus-, and operational power constraints (Equation 1)
where i, j, and k are the power system components, gen is the set of generators, sgen is the set of static generators, load is the set of loads, extgrid is the set of external grid connections, bus is the set of bus bars, trafo is the set of transformers, line is the set of lines, and trafo3w is the set of three winding transformers, \(f_{i}(\cdot)\) is the cost function, \(P_{i}\) is the active power in i, \(Q_{i}\) is the reactive power in i, \(V_{j}\) is the voltage in j and \(L_{k}\) is the loading percentage in k.
2.1.2. Water system model
The wntr package can simulate water flows in water distribution systems using two common approaches, namely, demand-driven analysis (DDA) and pressure-dependent demand analysis (PDA). While DDA assigns pipe flows based on the demands irrespective of the pressure at demand nodes, PDA assumes that the demand is a function of the pressure at which water is supplied. The PDA approach is more suitable for pressure-deficient situations, such as disaster-induced disruptions to water infrastructure. In the case of PDA, the actual node demands is computed as a function of available the water pressure at the nodes as in Equation.
The wntr package can simulate water flows in water distribution systems using two common approaches, namely, demand-driven analysis (DDA) and pressure-dependent demand analysis (PDA). While DDA assigns pipe flows based on the demands irrespective of the pressure at demand nodes, PDA assumes that the demand is a function of the pressure at which water is supplied. The PDA approach is more suitable for pressure-deficient situations, such as disaster-induced disruptions to water infrastructure. In the case of PDA, the actual node demands is computed as a function of available the water pressure at the nodes as in Equation~ref{eq:PDA} cite{Klise2020}.
where is the actual demand at node i at time t, \(D_{i}(t)\) is the desired demand at a node i at t, \(p_{i}(t)\) is the available pressure in node i at t, \(P_f\) is the nominal pressure, and \(P_0\) is the lower pressure threshold, below which no water is consumed. In InfraRisk, the hydraulic simulation is performed using the PDA approach.
2.1.3. Transport system model
The traffic condition in the transport system is modeled using the static traffic assignment method based on the principle of user-equilibrium. Under user-equilibrium, every user tries to minimize their travel costs. The traffic assignment problem considered in InfraRisk package is formulated as follows (Equation~ref{eq:staeqs}) cite{Boyles2020}.
where A is the set of all road links with i and j as the tail and head nodes, \(t_{ij}\) is the travel cost on link \((i,j)\), \(x_{ij}\) is the traffic flow on link \((i,j)\), \(h^{\pi}\) is the flow on path \(\pi \in \Pi\), \(\delta_{ij}^{\pi}\) is an indicator variable that denotes whether \((i,j)\) is part of \(\pi\), \(d^{rs}\) is the total flow between origin-destination pair \((r,s)\).
2.1.4. Modeling interdependencies
The module also consists of an interdependency layer which serves as an interface between infrastructure systems. The interdependency layer stipulates the different pieces of information that can be exchanged among individual infrastructure systems and their respective formats. The interdependency submodule also stores information related to the various component-to-component couplings between infrastructure systems. The module facilitates the communication between infrastructure systems and enables information transfer triggered by dependencies. Currently the following dependencies are considered.
Power-water dependencies, which include dependency of water pumps on electric motors and generators on reservoirs (hydro-power).
Dependencies also exist between road traffic system and the other two infrastructure systems, as the former provides access to the latter. The disruptions to transport infrastructure components and their recovery are key considerations that influence the restoration and recovery of all other infrastructure systems. The module also stores the functional details of all infrastructure components, including their operational status after a disaster.
The interdependency layer communicates with the infrastructure simulators through inbuilt functions (wrappers).
2.2. Hazard initiation and vulnerability modeling
The hazard module generates disaster scenarios and initiates disaster-induced infrastructure failures based on their vulnerability. The hazard initiation and the resulting infrastructure component failures is the first step in the interdependent infrastructure simulation. The probabilistic failure of an infrastructure component is modeled as follows (Equation~ref{eq:failureprob}):
where i is the component, \(p(\cdot)\) is the probability. The probability of failure of a component is computed as the product of the probability of the hazard, the probability of the component being exposed to the hazard if it occurs, and the probability of failure of the component if it is exposed to the event.
In InfraRisk, infrastructure component failures can be induced using five types of hazards.
Point events (e.g., explosions)
Track-based events (e.g., hurricanes and floods)
Random disruption events (e.g., seven random road link failures)
Custom events (e.g., fail five specific pipelines)
Fragility-based events (e.g., earthquakes)
Among these, the first four two of event types are agnostic to infrastructure vulnerability and focus on the proximity of the components to the location of the event. The random events are generated randomly based on user requirements. Custom events can generate disruptions based on the user-defined lists. The fragility-based events are generated based on the component fragility curves and considers the vulnerability characteristics of the components.
2.3. Recovery modeling
The third module, which is the recovery module, determines how the repair actions are sequenced and implemented. The three major factors that influence recovery are the availability of repair crews, repair times of components, and the criteria used for selecting subsequent components to restore. In InfraRisk, the user can specify the number of crews deployed for restoration of the three infrastructure systems, their initial locations in the network, and the repair times of the infrastructure components.
The repair sequence can be derived using two approaches as follows.
2.3.1. Heuristics-based repair sequences
The first approach is to adopt pre-defined repair strategies based on performance- and network-based heuristics. Currently, there are three inbuilt strategies based on the following criteria:
Maximum flow handled: The resource-flow during normal operating conditions could reflect the importance of an infrastructure component to the system. The maximum resource-flow handled by a component, considering the temporal fluctuations, can be used as a performance-based heuristics to prioritize failed components for restoration.
Betweenness centrality: Centrality is a graph-based measure that is used to denote the relative importance of components (nodes and links) in a system. Betweenness centrality is often cited as an effective measure to identify critical infrastructure components cite{Almoghathawi2019}.
Landuse/zone: Certain regions of a network may have consumers with large demands or critical to the functioning of the whole city. Industrial zones and central business districts are critical from both societal and economic perspectives.
While it is comparatively easier to derive repair sequences based on heuristics, they may not guarantee optimal recovery of the system or the network.
2.3.2. Recovery optimization
The second approach is an optimization model leveraging on the concept of model predictive control (MPC) cite{Camacho2007}. In this approach, first, out of n repair steps, the solution considering only k steps (called the prediction horizon) is computed. Next, the first step of the obtained solution is applied to the system and then the process is repeated for the remaining n-1 components until all components are scheduled for repair. In the context of the integrated infrastructure simulation, the optimizer module evaluates repair sequences of the length of the prediction horizon for each infrastructure (assuming that each infrastructure has a separate recovery crew) based on a chosen resilience metric cite{Kottmann2021}. The optimal repair sequence is found by maximizing the resilience metric. At this stage, the optimal repair action in each prediction horizon is computed using a brute-force approach where the resilience metric is evaluated for each of the possible repair sequences. The major limitation of MPC is that it is suitable only for small disruptions involving a few component failures; MPC becomes computationally expensive to derive optimal restoration sequences for larger disruptions due to the large number of repair permutations it has to simulate.
2.4. Simulation of direct and indirect effects
The simulation module implements the integrated infrastructure simulation in two steps, namely, event table generation and interdependent infrastructure simulation.The objective of the event table is to provide a reference object to schedule all the disruptions and repair actions for implementing the interdependent network simulation.
The component failures, repair actions, and the respective time-stamps, are recorded in an event table for later use in the simulation module. The simulation platform uses the event table as a reference to modify the operational status of system components during the simulation, so that the consequences of disaster events and repair actions are reflected in the system performance. The recovery module also stores details including the number of repair crews for every infrastructure system, and their initial locations.
The next step is to simulate the interdependent effects resulting from the component disruptions and the subsequent restoration efforts. One of the main challenge in simulating the interdependent effects using a platform that integrates multiple infrastructure models is the time synchronization.
In order to synchronize the times, the power- and water- system models are run successively for every subsequent time-interval in the event table. The required water system metrics are collected for every one minute of simulation time from the wntr model, whereas power system characteristics at the start of every time interval is recorded from the pandapower model. The power flow characteristics are assumed to remain unchanged unless there is any modification to the power system in the subsequent time-steps in the simulation.
2.5. Resilience quantification
Currently, the model has two measures of performance (MOP), namely, equitable consumer serviceability (ECS) and prioritized consumer serviceability (PCS), to quantify the system- and network steady-state performances. The above MOPs are based on the well-known concepts of satisfied demand cite{Didier2017} and productivity cite{Poulin2021}. Th MOPs are used as the basis for defining the resilience metrics.
Consider an interdependent infrastructure network \(\mathbb{K}\) consisting of a set of infrastructure systems denoted by \(K: K\in \mathbb{K}\). There are N consumers who are connected to \(\mathbb{K}\) and the resource supply from a system K to consumer iin N at time t under normal operating conditions is represented by \(S_{i}^{K} (t)\).
The ECS approach assumes equal importance to all the consumers dependent on the system irrespective of the quantity of resources consumed from the system. For an infrastructure system, the ECS at time t is given by Equation~ref{eq:ecs}.
where \(s_{i}\) is the resource supply at time t under stressed system conditions and \(n_{K}(t)\) is the number of consumers with a non-zero normal demand at time t.
In the case of PCS, the consumers are weighted by the quantity of resources drawn by them. This approach assumes that disruptions to serviceability of large-scale consumers, such as manufacturing sector, have larger effect to the whole region compared to small-scale consumers such as residential buildings. The PCS metric of an infrastructure system at time t is given by the Equation~ref{eq:pcs}.
The normal serviceability component (\(S_{i}^{K}(t)\)) makes both ECS and PCS metrics unaffected by the intrinsic design inefficiencies as well as the temporal fluctuations in demand.
For water distribution systems, pressure-driven approach is chosen as it is reported to be most ideal for the hydraulic simulation under pressure deficient situations. The component resource supply values for water systems are computed as in Equations~ref{eq:water_t_pda}–ref{eq:water_base_pda}.
where \(Q_{i}(t)\) and \(Q_{i}^{0}(t)\) are the water supplied to consumer i during stressed and normal system conditions, respectively.
For power systems, the power supplied to components under normal and stressed system conditions can be calculated using Equations~ref{eq:power_t}–ref{eq:power_base}.
where \(p_{i}(t)\) and \(p_{i}^{0}(t)\) are the power supplied to consumer i under stressed and normal power system conditions.
The ECS and PCS time series can be used to profile the effect of the disruption on any of the infrastructure systems. To quantify the system-level cumulative performance loss, a resilience metric called Equivalent Outage Hours (EOH), based on the well-known concept of resilience triangle’ cite{Bruneau2003}, is introduced. EOH of an infrastructure system due to disaster event `gamma^{K} is calculated as in Equation~ref{eq:system_eoh}.
where \(t_{0}\) is the time of the disaster event in the simulation and \(t_{max}\) is the maximum simulation time (both in seconds). In Equation~ref{eq:system_eoh}, system performance during normal operating conditions is 1 due to the expression of the MOP used (see Equations ~ref{eq:ecs}-~ref{eq:pcs}).
EOH of an infrastructure system can be interpreted as the duration (in hours) of a full infrastructure service outage that would result in an equivalent quantity of reduced consumption of the same service by all consumers during a disaster. The larger the EOH value, the larger the impact on the infrastructure system and thereby on the consumers due to the disruptive event. The EOH metric can effectively capture the response and resilience of the infrastructure system (Equation~ref{eq:system_eoh}), according to the serviceability criteria chosen by the user.
Similar to EOH of a system, the consumer-level EOH can also be quantified, which indicates the equivalent duration of infrastructure service outage experienced by each consumer (Equation~ref{eq:consum_eoh}).
Finally, in order to compute the resilience of the interdependent infrastructure network, a weighted EOH metric is derived (Equation~ref{eq:wEOH}).
By default, equal weights are applied to both water and power systems.
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