Critical infrastructure network modelling for flood risk analyses: Approach and proof of concept in Accra, Ghana

In flood risk analysis, it is state‐of‐the‐art to determine the direct consequences of flooding for assets and people. Flooding also disrupts critical infrastructure (CI) networks, which are vital in modern society. Cascading effects in a CI network can exceed the hydrological catchment boundaries. The effects of directly impacted CI cascade to other infrastructures, which are thus indirectly affected by a flood. A robust modelling approach of CI networks is a basis for including these effects in flood risk analysis. One challenge is to balance the simplicity of the modelling approach, the reproduction of a CI network's complexity and the decisions made based on potential model outputs. In this article, a topology‐based modelling approach of CI networks for catchment‐wide flood risk analyses is proposed. The basic model elements are points, connectors and polygons, which are utilised to represent a multisectoral and layered CI network. The newly defined approach is implemented as CI network module to the state‐of‐the‐art flood risk analysis framework ProMaIDes. It analyses the CI's direct and cascading impacts as well as the indirect disruption of CI services triggered by flooding scenarios. It quantifies the consequences by determining the number of disrupted CI users or the disruption time. A proof of concept in Accra, Ghana demonstrates the method's capabilities.


| INTRODUCTION
An extreme natural event becomes a catastrophe or hazard through contact with society. Ever-expanding cities and their growing critical infrastructure (CI) supply networks affect the potential risk through extreme events by increased exposure as well as insufficient preparation and adaptation (Ferguson, 2021). The impacts on society, for example emitted by flooding, are transmitted through expanding and more closely webbed CI networks (Ferguson, 2021). More understanding and appreciation are needed of network analytics in order to improve flood risk analyses and subsequently flood management. Flood consequences, as one element of flood risk analyses, can be differentiated according to their tangibility as well as their cause of disruption (Patt & Jüpner, 2020). Table 1 shows one possible way of categorizing flood consequences based on Merz (2006). Direct flood consequences are caused by flood waters, whereas indirect consequences are not caused by contact with water or are outside the area and time period of a flood. Tangible consequences can be measured in monetary terms. Intangible consequences are not effectively represented as a monetary value (e.g., loss of life). The field of direct, tangible flood consequences has been researched in detail for several years (e.g., Delalay et al., 2020;Huizinga et al., 2017;Wagenaar et al., 2018). The analysis of direct, intangible flood consequences is migrating from state-ofthe-science to state-of-the-art approaches as an integral part of a flood risk analysis (Bachmann & Schüttrumpf, 2014;Jonkman, 2007;Jüpner et al., 2018;Kreibich et al., 2017;Merz et al., 2018;Wagenaar et al., 2019). The analysis of indirect and intangible consequences of flooding is receiving very little acknowledgement in current practise (Dassanayake et al., 2022). The presented categorisation is widely used in FRM practises but does rarely include CI.
CI provide services and goods that are essential and vital for societal functions, health, safety, security and the economic and social well-being of people (Federal Office of Civil Protection and Disaster Assistance-BBK, 2022; Burzel et al., 2014). CI are the technical structures relevant for supplying those services and are organised in sectors such as energy, water, nutrition, information and communication technology (ICT), health, transportation and more (Federal Office of Civil Protection and Disaster Assistance-BBK, 2022). CI networks are formed by considering the interdependencies of these individual CI sectors (CIPedia contributors, 2023). Fekete (2019) extensively defines the cascading effect as transmission through the CI network of a direct CI disruption caused by natural or manmade hazards. Indirect consequences occur in the form of CI service disruptions and are amplified by cascading effects. Table 1 is complemented by two types of flood consequences considering the role of CI.
During Hurricane Sandy, for example, direct and also indirect disruption of the energy, water and transport sectors could be observed (The New York Times, 2012). The 2021 flood events in Western Europe affected energy and water supplies as well as ICT and other CI sectors for days, weeks and months (Fekete & Sandholz, 2021;WDR, 2021). In both cases, cascading effects led to consequences beyond the directly impacted area. Kuhlicke et al. (2021) stated in the aftermath of these events that approaches for quantifying the cascading effects of CI networks during hazard events are required. The structured integration of CI services and cascading effects into flood risk analyses is still rare. Vorogushyn et al. (2018) call for the integration of indirect flood consequences beyond the flooded area due to the failure of CIs into the flood risk analysis. de Bruijn et al. (2019) and Pant et al. (2018) point out that more consideration should be given to CI in the research of flood resilience. Ouyang (2014) provides a set of categories for CI network modelling techniques which contains among others: empirical methods, agent-based and system dynamics-based methods, network-based methods and high-level architecture. For this article, a small range of the modelling approaches is introduced. Based on this, the relevant factors for the presented CI network modelling approach are chosen. Ani et al. (2019) defined empirical methods as established techniques to account for effects on CI. This method type derives network information from expert knowledge and gathers recurring patterns of CI cascades (Murdock et al., 2018). The empirical methods can be combined with quantitative tools depending on the amount and type of information gathered (Murdock et al., 2018).
Agent-based and system dynamics-based methods are designed to react to external or internal impulses according to Brown (2007) and Rinaldi et al. (2001). For agentbased methods, the smallest decision-making units or individuals are considered and form a bottom-up hierarchy of the model dynamic. By contrast, system dynamicsbased approaches depict the adaptivity of networks as a whole and change the metrics of the entire system to form a top-down method (Ouyang, 2014). Network-based methods utilise network elements such as points and connectors, also referred to as nodes and edges. These elements are associated with states of operation, for example, functional or disrupted, which determine the network elements' operability. The boundary of CI analyses ranges from local to regional or even international, as shown by Verschuur et al. (2022). The network-based methods can be further classified in flow-based modelling and topology-based modelling approaches. Flow-based methods focus on goods and services delivered by CIs, whereas topology-based methods focus primarily on the topological presence of a CI, as Mühlhofer et al. (2023) apply in a general natural hazard modelling approach for cascading effects. Flood risk analyses using these methods consider, for example, the effect of flooding on roads and the potential damage due to loss of transport connectivity (Scoccimarro et al., 2020) as well as the resulting economic damage . Emanuelsson et al. (2014) describe a model focussing on the water sector and its incoming dependencies from the energy sector. It combines a probabilistic flood assessment with an analysis of potential cascading effects on other compounds in the water sector. For a more detailed explanation of the available modelling approaches, refer to Ouyang (2014).
The methods outlined above can be combined if necessary. Pant et al. (2018) use a combination of the flowbased method and the topology-based method to model flood inundation impacts on a CI network. The energy sector's electricity grid is modelled in a hierarchical order depending on the relevance and functionality in the supply grid. The direct impact on the electricity infrastructure and the indirect impact caused by electricity outages are one output of the model and account for the cascading effect. "Electricity grid customers disrupted" is used as a key metric for the disruptiveness of a cascade and provides an approach to quantifying the CI's cascades (Murdock et al., 2018;Pant et al., 2018).
Another method is defined by Ouyang (2014) as a high-level architecture method which is characterized by the ability to combine multiple complex sectors in one model. Arosio et al. (2020) and Pant et al. (2017) provide examples of this modelling approach.
The objective of this article is to define a new CI network modelling approach, focusing on theory, practise and proof of concept. In the introduction to this article, available approaches were briefly outlined. Based on these, the new CI network modelling approach and its basic elements are defined. Additionally, element properties, network characteristics and the calculation procedure are explained. A proof of concept based on a case study outlines the applicability and potential benefits of the presented approach. The CI network modelling approach is used to quantify CI consequences and to highlight affected areas and cascading effects for flooding events in Accra (Ghana). Finally, the results and limitations of the approach and the added value of integrating CI disruptions in flood risk analyses are discussed and concluded.

| CI NETWORK MODELLING APPROACH
One purpose of this modelling approach is to supply another metric and a robust modelling approach for decision-making in flood risk management by adding another consequence type to flood risk analysis. Another purpose is to highlight vulnerable points' dependencies in the CI network for which mitigation measures are relevant and show the spatial extent CI disruptions can potentially have. The intention is not to represent the CI network and its service precisely, but to provide a rough overview for a specific topic to the overarching perspective of flood risk managers.
The functions represented by this modelling approach focus on the basic service that CI supplies to civil society, based on the exposure of individual CI elements. A combination of the number of users disrupted and the duration of disruption is chosen as a metric for this modelling approach, as suggested in the empirical approach by Murdock et al. (2018).
The structure of the new modelling approach presented here combines the network topology-based approach comparable to Pant et al. (2018) and the highlevel architecture modelling approach for flood risk analysis. Figure 1 shows how a model schema based on the combined methods is shaped. The hierarchical representation of one sector is extended by including schemas for two additional sectors whose CI elements are represented as triangles and diamonds. In addition to connections within the sector marked by unicoloured arrows and defined as sectoral connectors, this schema also contains striped arrows, highlighting intersectoral dependencies. This enables the method to consider cascading effects within (sectoral) and across (intersectoral) CI sectors and merges them into a system considering interdependency.

| Definitions and model elements
In this article, the defined modelling approach is referred to as the CI network module. At the beginning of the explanation of the CI network module, definitions of the model elements and their attributes are outlined. The CI network module describes network models utilising three types of CI elements: point, polygon and connector elements. Table 2 summarises the properties necessary to define the three element types. Every point element is assigned to a CI sector and level, in addition to the obvious properties (index, x and y coordinates, name). The level indicates the relevance within one sector. The higher a point element's level, the more important it is for the sector's functionality. Additionally, it is a requirement to have data on the threshold for every point element, which defines the water depth causing a disruption. Exceedance of the threshold value at a point element results in a disruption and is cascaded through the connector elements to other point and polygon elements. Once disrupted, the recovery time indicates the time of disruption for an inundated point element. The point elements and the polygon elements are binary, which means they are either functional or disrupted.
Connector elements define other CI elements' directional dependencies. Connector elements indicate that a sink point element relies on the functionality of a source point element. Additionally, connector elements are classified as either sectoral or intersectoral connectors.
Polygon elements define the spatial extent that their source point elements are associated with, or at least partially associated with. In addition to the basic attributes (polygon index, sector index, name and coordinates) the polygon is defined by the number of end-users or consumers it supplies with a service. Polygons are always F I G U R E 1 A schematic network high-level architecture modelling approach including three CI elements from three sectors. The point elements are arranged on sector levels and form a hierarchy for sector 1 (points) and sector 2 (triangles). Sector 3 (diamonds) is not organised hierarchically.

T A B L E 2
Attributes of input elements of the CI network module: points, polygons, and connectors.

Elements Input attributes Description
Point (Nodes, Vertices) Point-, sector index Unique identification for point; Sector specific identification.
Sector level Quantitative representation in the hierarchy per sector.  (Rinaldi et al., 2001). These are: the energy sector, the ICT sector, the water sector and the waste management sector. The primary CI sectors are chosen as such due to their physical attributes and their standardised operating principles, comparable to Rinaldi et al. (2001). Primary sector structures are connected to structures from the secondary sectors. Secondary sectors are not internally driven by physical interdependencies but by logistic, cyber and geographical interdependencies. Secondary sectors include, for example, the emergency and health sector, the state and public sector and the transport and logistics sector.

| CI network characteristics
After the setup of the model, the network characteristics and attributes of individual CI elements are derived. They show that networks are more than the sum of their individual components (Ferguson, 2021). The network characteristics allow highlighting of the role of individual CI elements with outstanding importance for network reliability, without taking the hazards into account. Table 3 lists the network characteristics derived from the CI network's point and polygon elements. Two characteristics are commonly used in network models, as shown by Arosio et al. (2020), to characterize point elements: the hub value H, summarising the number of outgoing connectors per point element, and the authority value A, describing the incoming connectors per point element. Point elements with a high H indicate a high potential to affect other elements of the CI network in the event of a disruption; complementary to that, it signals which point in the network is predestined for resistance enhancing measures, such as mobile flood protection. An extraordinarily high value of A supports the identification of vulnerable elements within a network that would benefit from back-up options, such as emergency generators.
Two characteristics are newly introduced to summarise the behaviour of cascading effects within a CI network: the cascade vulnerability value V describes the vulnerability of end-user elements to be disrupted by cascading effects; the cascade potential value P focuses on the potential of a point element to cause cascading effects.
A (potential) cascade is technically defined as the way back following the connectors from an end-user element (point or polygon) to a starting point (see dotted lines in Figure 2). A point with A = 0 can be a starting point of a cascade; an element with H = 0 is an end-user element (see Table 3). The numbers of elements per cascade are defined as m.
The cascade vulnerability value V is determined per end-user element. For the determination of V all cascades are checked (see Figure 2). Per cascade j a cascade weight w j,i is determined at each affected point i of the cascade. w j,i depends on the cascade weight w j,i-1 of the element before. It is equal to w j,i-1 (the cascade weight of the element i-1 before) divided by the number of incoming connections of the same sector n sec,i-1 at the element i-1: Equation (1) considers redundant connections. The cascade weight w j,0 of the end-user element is always defined as 1, required for the determination of the flowing cascade weights w j,i . However, it is not used further in the calculation of V. Per cascade j the sum over all cascade weights w j,i of each affected point i is applied: If multiple cascades j = 1…n of the same sector are connected to the end-user element the minimum value is relevant: Network characteristics derived from the network configuration.

Characteristics Description
Derived After an internal sectoral check, an intersectoral check is executed. If multiple cascades j = 1…l of the different sectors are connected to the end-user element the maximum value is relevant: The cascade potential value P is derived for every point i in the CI network, excluding end-user point elements. It counts the potentially disrupted end-user elements in case of failure. P is calculated by the sum at each point i over all w j,i derived from all potential cascades o independent from the sector:

| Calculation procedure
The calculation procedure of the CI network module is explained to give an understanding of the CI network's interaction with the simulated flood events and also to show the granularity of the network model. The point elements of the CI network are superposed with the inundated area, in general derived from a 2D hydrodynamic model. The geographic location of the point elements determines the water depth derived from hydrodynamic 2D raster cells. Two calculation modes are available: (1) a steady state calculation of the CI network based on the maximum water depth in the connected hydrodynamic 2D element, and (2) an unsteady state calculation, connecting the water depth for every time step of the hydrodynamic to the point elements. In both cases the maximum (steady state) or the water depth per time step (unsteady state) is compared to the thresholds of every point element. Exceedance of the threshold results in a direct failure state of the point element. The connector elements propagate the failure to the sink point or polygon elements, resulting in a cascading failure without any delay. This indirect failure state can be within the same sector (sectoral failure) or an intersectoral failure or disruption (see Figure 3). After the specific recovery time of the directly affected point element has passed, functionality-also from the cascading effects-is regained. It is assumed that during the recovery period of a directly affected point element the water depth itself has no further influence on the recovery time. Several values to quantify the damage to a CI network are conceivable: the number of affected points per sector, the number of people disrupted from CI services per sector P dis,Sec or the disruption time per sector t Sec . A further summarising quantification value is a combination of t Sec [s] and P dis,Sec [P] as suggested by Murdock et al. (2018). This quantification is introduced as the population time T Pop,Sec [PÁs] and calculated per CI sector: The derivation of the cascade vulnerability value V for two polygon end-user elements and the determination of cascade potential values P for the points A, B, and C. The dotted pathways indicate a cascade chain and are connected to their V on the right.
T Pop,Sec is quantified within the steady state mode. The final flood risk R CI [PÁd/a] is calculated as the sum over all hydraulic boundary scenarios n over the product of the damage value, for example, T Pop,Sec,k [PÁs], multiplied by the probability of reoccurrence per hydraulic scenario p hyd,k [1/a]: Figure 3 shows the calculation procedure of the steady state mode for a schematic CI network which is confronted with a flood event, marked with blue wave hatching. The inundation affects the transformer station directly (CI point, sector: energy). The maximum water depth of 1.5 m exceeds the defined water depth threshold of 0.5 m. Starting from the transformer, the failure is propagated to the hospital (sector: health) and the water supply, disrupting CI services for end-users. This is represented by the end-user polygon element I (sector: health; intersectoral disruption) and the end-user polygon element III (sector: water; intersectoral disruption). The disruption of all CI services is determined by the recovery time of 2 d of the directly affected transformer station, as indicated by the stopwatch. The ICT point element is not affected by the disruption due to redundancy of this connector with another point element from the energy sector. The population time per sector is therefore concluded as shown in Figure 3.
Within the steady state calculation mode, the direct and cascading failures and the time of disruption of the CI services are calculated at once for the whole flooding event. This is mainly required for the damage and risk calculation. In the unsteady state mode, the direct and cascading failures are calculated per time step. These results are suggested for a visualisation of the cascading failure and recovery over time.

| IMPLEMENTATION AND APPLICATION OF MODELLING APPROACH: INPUT AND MODEL SETUP
After the technical definition of the CI network module, an explanation is provided on how to theoretically utilise the model approach for an application. The CI network module is integrated in the publicly available PROMAIDES software package which consists of range of features for a holistic flood risk analysis (Bachmann, 2012). Plugins for a user-friendly model setup and results visualisation have been developed for QGIS, the open geographical information system (GIS) that is used to prepare the data input and output for the CI network module (Schotten & Bachmann, 2022). Figure 4 gives an overview of the required input data and the model setup. The first step highlighted in Figure 4 is to identify the objective of the CI network model for a potential application area. The intended analysis is often driven by key events of previous flood events, which created awareness of potential cascading effects (de Bruijn et al., 2019). In the next step, it is important to define the boundary for the CI network in relation to the extent of the hydrological and hydrodynamic boundaries of the flood risk analysis. The impacts cascaded through the CI network are expected to exceed the area impacted by the hydraulic event or the river catchment area; this needs consideration when gathering the input data.
Ideally, the input data is compiled from data directly received from the CI operators. However, due to a lack of availability or accessibility other sources must be used. In Figure 4, the second step highlights the necessity of collecting point and raster information for the model setup and lists potential data sources. For the identification of point elements in the specific sectors, web mapping services or OpenStreetMap contributors (2021) can provide a basis. To determine the number of end-users, a range of datasets can be used to supply population density datasets ( Figure 4, step 2) such as the high-resolution data from Meta-Data for Good (2022). The point elements and their attributes are defined based on the previous input ( Figure 4, step 3).
In addition to the input data explained above, data on recovery time is needed, though not easily gathered and almost impossible to obtain from public sources. Thus, it is complemented by findings from empirical studies, as described by Murdock et al. (2018) and Koks et al. (2021). For all types of data need presented in this modelling approach, collaborative CI stakeholder engagement tools are recommended in order to derive more information on the CI network sectors, elements and characteristics (Burzel et al., 2014;de Bruijn et al., 2016de Bruijn et al., , 2018Murdock et al., 2018).
The data on CI end-users is also not publicly and consistently available for all elements and sectors. To compensate for the lack of data, an alternative is used to determine an estimate for this information. The alternative method is created assuming that point elements deliver their service only to the population closest to them (Pant et al., 2018). Thus, Voronoi polygons are created for all point elements with the same sector and level. To avoid an overestimation of the Voronoi polygons, it must be ensured that there is a row of data surrounding the boundary of the area of interest. The polygon elements are used to derive the number of endusers from raster data (Figure 4, step 4). This method also helps the connectors to be assigned in the model itself. It connects source and sink by checking which sink elements are in a source element's closest-distance Voronoi polygon (Figure 4, step 5). Figure 4, step 6 concludes the technical workflow of the setup of the CI network model by merging the individual elements.

| PROOF OF CONCEPT IN ODAW CATCHMENT, ACCRA, GHANA
In the following section, the CI network modelling approach that has been developed is tested as a proof of concept in a real-world application. The study area is the catchment of the Odaw River in Accra, the capital of Ghana, which frequently experiences flooding in its urban environment (Almoradie et al., 2020;Ntajal et al., 2022). The catchment covers about 271 km 2 and is located close to the sea (see Figure 5). The CI network model is part of an existing modelling chain that includes a hydraulic model and flood consequence models for economic damage based on Huizinga et al. (2017) and affected and endangered persons based on Jonkman (2007), for an integrated flood risk analysis . For the hydraulic model, a DEM with a 30 m resolution was used and the floodplains were represented by 414,000 25 Â 25 m 2 raster cells. Synthetic block rain events of 24 h with three different return periods (HQ10, HQ100 and HQ1000) are applied, uniformly distributed over the whole area (Krvavica & Rubini c, 2020).

| Setup of CI network model for Accra
The CI network model itself includes 419 point elements, 472 polygon elements and 1124 connector elements (see Figure 5). Six different CI sectors are included: energy, water, ICT, health, emergency services, and transportation. They are represented by electricity substations, telecommunication towers, water supply facilities, hospitals, fire and police services and logistic point structures such as airports and highway traffic lights. The CI network model's input data exceeds the hydrological catchment boundary of the Odaw River. Thus, modelling of cascading effects beyond the hydrological catchment boundaries is possible.
The location and the CI sector index of the point elements are based on OpenStreetMap contributors (2021), complemented with data from web mapping services. Every point element is accompanied by a polygon to mark the area of influence on other sink point elements for the definition of connectors. For CI end-user polygons, the polygon is also used to derive the number of end-users from a population density dataset (Meta-Data for Good, 2022). Due to a lack of more specific datasets from CI operators, Voronoi polygons are used to mark the areas of closest distance to the source point element, as described previously. In the transportation sector, point elements are end-user elements given the attribute of end-users derived from historic traffic and passenger numbers.
A stakeholder workshop with local CI operators was held to validate the present input and to add additional relevant point elements. The workshop was also used to confirm the CI element properties such as recovery time and water depth threshold with experts from the field of CI operation and crisis response (see Table 4) as well as to identify the sectors' dependencies that should be highlighted with connector elements. The workshop discussion was accompanied by a schematic representation of the network shown in Figure 6 (Deltares, 2018). It features all integrated CI sectors of the CI network model as well as the civil population. Figure 6 also shows which sectors have dependencies and are linked through connector elements.
Stakeholder engagement methods like the CIrcle method provide a good discussion platform for CI stakeholders to build a community and enable discussion about the identification of CI sectors, their interdependencies and missing information for the CI network modelling and risk management objectives (Burzel et al., 2014;de Bruijn et al., 2016de Bruijn et al., , 2018Murdock et al., 2018).

| Specific setup of water sector
During the workshop interaction, specific sectors were discussed and differentiated more extensively. This proof of concept is used to demonstrate that the findings of these discussions on the water sector can be accommodated within the modelling approach of the CI network module. The representation of the energy, ICT, emergency services, and transportation sectors is derived from the standard setup approach described previously. Representing the fresh water supply sector and its physical system is a stress test for the CI network modelling method.
The water sector is represented by different types of supply elements with specific dependencies, as shown in Figure 7. Forming the backbone of the supply are water treatment and desalination plants. Booster stations within the area ensure there is sufficient pressure in the water pipes from treatment and desalination plants to the household level (Adank et al., 2011). They form the highest level in the hierarchy of the CI network models' water supply sector. Connected to the supply of the booster stations are the district offices and the water tank hydrants. The water tank hydrants fill up the water tank trucks that deliver water to households. The percentage of the population connected to the pipe-based distribution system and the water tank truck-based distribution can be seen in Figure 7. These percentages are multiplied by the total number of potential end-users in the Voronoi polygons of end-user polygons: end-user suppliers are the district representations, water tank truck hydrants and the smallscale independent providers (SSIP) and community managed small town water systems (CMWS). The SSIP and the CMWS operate outside of the normal water supply network and do not depend on the booster stations.
The water tank trucks and their supply through hydrants are assumed to be able to switch their operation mode to an emergency state and raise their capacities. This emergency mode only activates once a connected structure is disrupted. The emergency mode is not operable if higher level point elements like the booster stations are affected. Due to a lack of available data, the modelling of the water supply structure ignores the supply via other private water companies besides the SSIP and the CWSM, as well as freshwater consumption via vendors and kiosks. No data was available on the role of Ghana Water Company district representations, so it was assumed to have a technical function within the supply system. Figure 8, left-hand image shows a filtered version of the CI network model that highlights the water supply structures mentioned above, the connected energy sector point elements and the Odaw River system and its Overview of CI element properties used for CI network model in Accra. catchment. After superposing the CI network model with hydraulic modelling results, the disruption of the system is derived. Figure 8, right-hand image shows the polygons affected. The comparison of both images in Figure 8 shows that the area of impacted CI users exceeds the catchment area.

| Network characteristics
An analysis of the CI network characteristics gives an overview of the potential disruption of the CI network if triggered by an external impact. To get an understanding of the network, network characteristics are briefly F I G U R E 7 Water sector represented in the critical infrastructure network model of the Odaw Catchment, Accra, Ghana (Adank et al., 2011). analysed. Figure 9 highlights the point elements in the CI network with cascade potential values P (see section 2.2) bigger than 10. These point elements are all associated with the included primary CI sectors: energy, ICT and water. The point element with P = 95 is an electricity station located about 200 m from the Odaw River. It gives an indication for further investigation and potential mitigation measures before the combination with the modelled inundations. The outstanding network characteristics give additional information about vulnerable point elements and potential cascades. Table 5 summarises the minimum and maximum values of the introduced network characteristics and provides an understanding of the representation of the sectors in the CI network. Extensive analysis of the network characteristics is mainly of interest for mitigation measure planning and is outside the scope of the work presented in this article.

| RESULTS
Different types of output can be generated from the CI network model. One type of output is the spatial datasets resulting from the steady state calculation mode (see section 2.3). From the unsteady state calculation mode, spatial data is available for every time step. Figure 10 shows the catchment of the Odaw, highlighting a different aspect of the CI network model in each quadrant. The quadrants show how to superpose the results of a hydraulic model run with the input (b, d) and output (a, c) of the CI network model. In part (d) of the image, the F I G U R E 9 Point elements in the critical infrastructure network model of Accra with cascade potential values P > 10 featuring the energy, information and communication technology and water sectors. 1 km radius equals a cascade potential value P = 50.

T A B L E 5
Minimum and maximum value of all introduced network characteristics per CI sector. hydraulic model and the main channel of the investigated Odaw River are shown superposed with energy sectors' point elements and the associated polygon elements. Part (b) connects the energy sector's points with the ICT sector's points. The polygons show the area supplied with mobile network connectivity. Part (a) highlights the effect of flooding on the structures shown in part (b). Energy and ICT sector points that are directly or indirectly disrupted are marked with a red circle and the polygons from the ICT sector are hatched in black. Part (c) shows the equivalent of the health and energy sector.

Sector
Information for decision makers is presentable in maps to provide accessibility to the spatial extent of consequences for the CI network, comparable to the quadrants of Figure 10. Specific filters in geoinformation systems enable the visualisation of the output data for the sectors of interest. The full spectrum of generated data superposed is not helpful for decision makers in flood risk management or CI operations.
Another type of information output is the summary of the consequences per flood event and, ultimately, the total flood risks in the flood risk matrix (see Table 6). The flood risk matrix shows the annuality of the three flood events and their derived probability of reoccurrence p hyd . In addition to the established categories of consequences C, such as economic damage or consequences for people, it also summarises the population time T Pop (see Equation (6)) per sector. The resulting risk R CI sec (see Equation (7)) is calculated per category and is listed in the last column. The last line shows the overall CI risk sum R CI for the complete CI network. With the CI consequences, a new dimension is added to the flood risk matrix.

| DISCUSSION AND OUTLOOK
The technical approach to modelling CI networks presented here is required for flood risk management procedures, as proposed by Burzel et al. (2014) and . The challenge of this type of analysis is more organisational than technical, however. Responsibilities for flood risk adaptation of CI are split across several stakeholders, for example, CI operators from a wide range of sectors or public authorities. To tackle this challenge, the consistent participation of CI stakeholders is highly recommended. The CI network modelling approach represents the real CI network based on logical and physical dependencies of the CI elements. Empirical approaches were used to derive additional information about the CI network as well as its elements and attributes. The modelling approach itself was tested and verified by simple, synthetic models representing different aspects of network characteristics (e.g., linear, split or circular networks). Within the modelling approach, several technical assumptions are made: the state of a CI element is defined as functional or disrupted without any intermediate states; the recovery time is started once the water depth threshold is crossed even though the CI element might still be inundated. These technical assumptions will not always reflect real conditions and must be communicated. However, for flood mitigation planning on a regional scale these assumptions are rated as suitable, but for a detailed network performance analysis they may be too limited.
The main purpose of the presented proof of concept in Accra is to demonstrate the approach itself and its capabilities. The model's uncertainties arise mainly from two factors. First, uncertainties are caused by the raw input data itself (e.g., localisation, extent of the structure itself and dependencies of the CI elements, recovery times, water depth threshold values, and number of endusers per polygon). To rate the introduced level of uncertainty, a sensitivity analysis concerning these parameters is recommended. The second cause of uncertainty is the transformation of the raw data into model input data, for example, the use of Voronoi polygons as a nearestneighbour assumption for the connection of CI elements. Both types of uncertainties could be significantly reduced if more intense use could be made of direct and sensitive network information (often available from the CI providers). New potential procedures enabling this option would be, for example, the anonymisation of CI end-user data preventing privacy issues or the derivation of data from information shared during CI operator participations.
Validation of the model with data from a real flood event is a priority in the future. In the presented proof of concept, only plausibility checks of data and model results with local stakeholders were performed. However, the challenge remains to acquire reliable validation data, especially in environments with scarce data, such as Accra. Complex datasets for flood events, related cascading effects, CI failures and disruption of CI services are currently not available due to a lack of awareness and inconclusive responsibilities. In the future, it is recommended to authorities to collect and share this information or at least encourage local news agencies to do so. However, it must be stated that the physical and logical character of the approach reduces the requirement for validation.

| CONCLUSION
The CI network modelling module is presented in this study as a new CI network modelling approach for the quantification of multisectoral CI service disruption on a regional scale by flood events. It combines high-level architecture with a network topology-based approach. Multisectoral CI networks are modelled with three types of CI elements: points, polygons and connectors. The relation to CI end-users is established in the model by sectoral and intersectoral dependencies.

T A B L E 6
Flood risk matrix combining classic flood damage in bright gray and CI network consequences in white measured by population time. In addition to established CI network characteristics like the hub value H and authority value A, two new key figures have been introduced. The cascade vulnerability value V shows the vulnerability of a CI end-user polygon. The cascade potential value P helps to identify point elements with high potential for disruption within a cascade, leading to cascading failures.
When virtually confronted with flooding scenarios, the CI network model shows direct disruptions and also indirect disruptions within a sector and on an intersectoral level through cascading effects. It summarises CI service disruption quantitatively by the number of disrupted CI end-users, the time of their disruption or the product of the two, introduced as the population time T Pop . Results for each time step of the CI networks' interaction with the flood model can help the cascading behaviour to be understood. The CI network model is adaptable to the precision of the available data coverage and specific stakeholder interests. The simple structure of three CI elements is applicable from a simple design with few elements to a complex multisectoral multilayer model. It is suited for applications driven by publicly available data sources, by specific and complex datasets or by qualitative information from CI operators and other stakeholders. This modelling approach balances the level of detail the CI model replicates real CI networks and the detail necessary for an improved flood risk management. The approach provides a basis for the interaction of flood risk managers and a multisectoral group of CI operators on a regional scale.
The CI network modelling approach is proven in a proof of concept for the Odaw catchment in Accra, using data from publicly available data sources and complemented and validated through individual and participatory CI stakeholder engagement. A flood risk matrix is derived in which CI disruptions are quantified in addition to usual flood consequences. Additionally, it is proven that CI disruptions can exceed the catchment area of the Odaw River. All in all, a range of possibilities for flood risk analysis has been defined through the CI network module. Opportunities are enabled to integrate CI networks and their disruptions into flood risk management procedures.