ing R/S demand from CIS resilience assessment may be misleading since the main purpose of such systems , to meet the current user demand for the R/S they produce , is not considered . For example, focus ing solely on supply leads to investin g limited resources and services in recovering the system that, once recovered, serve s little or no purpose as the users are not ready to consume th e recovered supply capacity. 11 Figure 1: Lack of Resilience of a system as defined in the Re -CoDeS compositional demand 𝐷𝑠𝑦𝑠,𝑅/𝑆(𝑡), supply capacity 𝑆𝑠𝑦𝑠,𝑅/𝑆𝐶(𝑡) and consumption 𝐶𝑠𝑦𝑠,𝑅/𝑆(𝑡) framework (Didier, 2018) iRe-CoDeS e xtension of the Re-CoDeS framework An extension of the Re-CoDeS framework , iRe -CoDeS, to enable disaster resilience quantification for a system -of-interdependent -systems (e.g., a community with interdependent CISs), is presented in this section . Re-CoDeS defines the demand of a component as a set of R/Ss that enable that component to operate . Therefore , whenever this demand is not met by the system , the component ceases to operate and perform its role in the considered system, reflected in the decrease of its supply capacity. This is simulated in iRe -CoDeS by extending the system R/S distribution model to simul ate component interdependencies. Thus, iRe-CoDeS is a disaster resilience quantification framework that views a system -of-interdependent -systems as an 12 assembly of components whose ability to operate is constrained by their interdependenc ies, simulated as a flow of resources and services among components using a demand/supply - based approach. The m ain elements of iRe-CoDeS are presented next , together with the extend ed notions of R/Ss, components, localities , and the demand/supply -based dynamic interdependency model. Resources and Services (R/Ss) iRe-CoDeS refines the Re -CoDeS notion of a R/S by further classifying it into a utilit y or a transfer service. Utilities represent R/Ss supplied by CISs , while t ransfer service s enable the transfer of these utilit ies from a supplier to a user. Components , localities and links As in Re -CoDeS, a component is the lowest level of system discretization considered in the iRe-CoDe S framework. Each component is characterized by its state, comprising its instantaneous demand, supply capacity and consumption values for each R/S, as well as its instantaneous damage and functionality level s. A component can supply and demand none, one or several R/Ss. Some components need utilit y R/Ss to operate. If the transfer of a utility from a supplier to a user requires a distribution network, the demand for the corresponding transfer service is assigned to the user component. Therefore, instantan eous component demand values include the amounts of utility R/Ss a component needs to operate and the amounts of transfer service s needed to deliver utility R/Ss from a supplier to the component. Component instantaneous R/S supply capacity, demand and con sumption are affected by its current damage and functionality level s. Component damage level quantifies the damage state of a component throughout the resilience assessment interval and takes values 13 ranging from 0 (no damage) to 1 (complete damage). As in Re-CoDeS, a component damage level is set immediately after a disaster using that component ’s vulnerability function, and is decreased during r ecovery at that component’s repair rate. C omponent functionality level describes how much of the component’s func tion has been lost due to damage and recovered due to repairs after a disaster. Functionality level of a component varies between 0 (no functionality) to 1 (full functionality). The relation between component ’s damage and its functionality level are defin ed using function s. For example, a component with a binary damage -functionality function has a functionality level of 1 if the damage level is 0, and a functionality level of 0 otherwise. Functionality level of a component with a linear damage -functionality function increases linearly as its damage level decreases due to repairs . Other component behavior types can be constructed in iRe -CoDeS by defining the above -stated functions as needed. The iRe-CoDeS framework presented herein calculates the instantaneous supply capacity of a component as the product of its current functionality level and its supply capacity at full functionality , accounting for the drop in the supply capacity due to component damage . Meanwhile, componen t demand is kept at its pre -disaster level as long as the functionality level is non- zero, otherwise, the demand is set to zero. The relation between the availability of R/Ss a component needs to operate and its ability to supply R/Ss also needs to be defi ned: this is where the main aspect of component interdependency is captured . A simple relation is assumed in the presented iRe -CoDeS framework : a component ceases to operate if any of the R/Ss it needs to operate are not available , reducing its R/S supply capacity to zero. Therefore, for a component to operate at its current (non -zero) functionality level, all its R/S demands must be met in full. More complex interdependency relations can be implemented as needed. 14 A locality is a geographically localized unit that may contain none, one or more components. A link is a component that provides a transfer service for a utility between two localities. Two localities can be linked by none, one or several links. A component in a locality can be an element of a f acility (e.g., a power generator), a facility (e.g., an electric power plant or a building occupied by its inhabitants), or an assembly of facilities and buildings (e.g., a city). The transfer of utilities inside a locality is unconstrained, i.e., the link s inside a locality are abstracted. A neighborhood is an example when such abstraction may be justified if the considered system is an entire city. The scope of a locality depends on the desired level of spatial discretization of the system. At the highest level of discretization, each component is in a separate locality, connected using links. On the other hand, at the lowest level of discretization all components are in a single locality. Links are components and behave in the same way: increase of link damage leads to a decrease of link functionality level that leads to a decrease in transfer service supply capacity of that link. Some links may need utility R/Ss to function. For example, a tunnel needs electric power for lights and ventilation to functio n in a transportation system. In the presented iRe-CoDeS framework, functionality of a link may be pre - conditioned by functionality of a link carrier, e.g., a bridge carrying water pipes and electricity cables . This functional dependency between the carrier and the carried link is modelled using the demand/supply approach, too. Specifically, a bridge provides carrier service to the links on the bridge. When damaged, the functionality level of a bridge is decreased, reducing its supply capacity for carrier services. Implementing the iRe-CoDeS system R/S distribution model Different types of R/Ss are distributed using different methods. In the presented version of iRe-CoDeS, t ransfer services are distributed by employing optimal path algorithms that 15 identify transfer paths between localities t o maximize the ability of the system s to transfer R/Ss with respect to the available transfer service supply capacities of links. Utiliti es are distributed among components following the R/S distribution algorithm presented next, the main element of which is the System Matrix . System Matrix The System Matrix contains the state information for all system components, one row per component. It is used to track the changes of the component states and the dynamics of the system R/S distributions in each time step of the resilience assessment interval. The structure of the System Matrix is used to increase the computational efficiency of interdep endency modelling and LoR evaluation by vectorizing . An illustrative example presented in the Appendix I shows a System Matrix of a simple five -component community (Figure A1 and Table A1). Utility R/S distribution algorithm Once damage and functionality levels of component s are set at the beginning of each time step of the resilience assessment interval, the task of distributing the available transfer services and utility R/Ss among the system components begins, one R/S at a time. When utility R/Ss are distributed, t he components (i.e., the rows of the System Matrix) are ordered in decreasing priority for the utility R/S being distributed first, following the system R/S distribution model (Eq. [3]) . Starting at the top of the R/S distribution priority li st, an attempt is made to satisfy the demand of each component for the distributed R/S using the supply of that R/S from any one or more of the identified suppliers that have a higher priority than the considered component (i.e., are in rows above the comp onent in the System Matrix) . If the supply available to the component is higher or equal than its demand, the component consume s the R/S it needs in full, and contributes to the system -level consumption of the 16 distributed R/S 𝐶𝑠𝑦𝑠,𝑅/𝑆(𝑡) calculated using Eq. [5]. However, if this is not the case, the component does not consume the R/S and ceases to operate, thus, the supply capacity of the component is set to zero, simulating interdependency effects. This process is repeated until the end of the R/S distribution list (i.e., the R/S distribution vector from Appendix I) . The amount of supply available to the component depends on the state of the R/S distribution networks after a disaster. Therefore, the transfer service demand of a component related to the transfer of the considered utility R/Ss is compared to the transfer service supply capacity of the optimal path, consisting of one or several links, connecting the component and the supplier. If the transfer service demand of such a utility R/S transfer is lower than the transfer service supply capacity, the utility R/S is transferred and the component consumes the utility R/S and continues to operate. Otherwise, the supply avail able to the component is set to zero and the component ceases to operate , simulating interdependency effects . The link damage and functionality states are accounted for, for example, to prevent distribution of EP due to a fault in the Electric Power Transm ission System (EPTS) , or to reduce the amount of PW transmitted due to leaks and breaks in the pipes of the Water Supply System (W SS). The topology of the link network as well as the R/S flow laws are taken into account to find paths between R/S suppliers and users. Different measures of path optimality can be used to assess the paths, such as the path length or the amount of transfer losses accumulated along the path in the presented iRe -CoDeS framework . An example sequence of distributions of utility R/Ss in a five -component community (Figure A1) immediately after disaster -induced damage (Table A1) is shown in Figure A2. Demand/supply -based Interdependency modelling Interdependency among components is modeled using component R/S demands (R/Ss needed for its operation) that are supplied by components belonging to various community systems (i.e., CISs) . In the presented iRe-CoDeS framework, it is assumed that a component ceases to 17 operate if its demands f or R/S s required for its operation are not met in full, regardless of its own damage . Consequently, the supply capacities of such components are set to zero . Loss of operation due to interdependency, not due to direct damage, may lead to inoperability o f other components , causing a n interdependency loop among components . Such loops are accounted for in the presented iRe -CoDeS framework by repeatedly distributing R/Ss among components inside a time step of the LoR assessment until the system reaches a stable state , as explained later in this section. T o efficiently allocate computational time for such repeated R/S distributions it is necessary to identify interdependent R/Ss – those R/Ss whose total supply capacity can change during interdependency mode lling. In this way, computational time is saved by not repeating R/S distributions that cannot affect the total supply capacities of other R/Ss. For example, electric power ( EP) is an interdependent R/S, if the suppliers of EP, the electric power plants (EPPs), demand other R/Ss to operate . Hence, when these demands are not met, the EP supply capacity will change, increasing the LoR and potentially causing further interdependency loops. An example of a n independent (i.e., a non -interdependent) R/S is the po table water transfer service , since the pipes supplying this service do not demand other R/Ss to operate. Therefore, the potable water pipe supply capacity cannot change due to interdependency effects. Thus, i ndependent R/Ss are distributed only once in a time step of LoR assessment . Two assumptions are made in the presented iRe-CoDeS framework to efficiently account for interdependency feedback loops in a LoR assessment time step (Hefti, 2019; Henken, 2019) . First, the effect of interdependency loops is considered only in terms of R/S supply capacity , not R/S demand. This ensures that , in a LoR assessment time step, the supply capacity can only decrease while the demand cannot change due to interdependency effects . Therefore, even though the supply capacities of a component are set to zero when its demand for a certain R/S is not met, the R/S demands of the component remain in the System 18 Matrix and in the LoR assessment for that LoR assessment time step . As components can have demands for multiple R/Ss, t his assumption allow s the components inoperable due to the unmet demand for one R/S to consume other R/Ss , potentially reducing the supply available to lower -priority components . In effect, the sensing capabilities of the system are assumed to be slower than the length of the LoR assessment time step. This assumption resolves the causality present in an interdependency loop. Specifically , for a component that needs several different R/Ss to operate, determining which unmet R/S demand triggered the interdependency loop requires an algorithm that does not depend on the order of R/S demand checking . Such an interdependency modelling algorithm is explained later in this section . Second, the interdependency loop is confined to a time step of the community LoR assessment interval . Thu s, inoperability of components due to a n interdependency loop occurring in a time step does not cause inoperability of components in the next time step . Instead, the i nterdependency loop is examined anew in each LoR assessment time step. These two assumptions make it possible to model the interdependencies among components of different CI Ss, as presented by the interdependency modelling algorithm in Figure 2c. At the beginning of each time step of community LoR assessment it is assumed that the reduction of supply capacity and demand of a component is due only to its damage state (i.e., the drop in the functionality level) as it stands in the recovery process, not due to the unmet demand for one or more R/Ss it needs to operate . Therefore , it is assumed that all demands of a component are met and that further R/S distributions can only decrease the supply capacity of the system . Then , the total supply capacit ies of the system for each interdependent R/S are calculated . They serve to identify when the demand/supply -based interdependency modeling algorithm terminates (i.e., when the system reaches a stable state) , by comparing the total supply capacities before and after distributing interdependent R/Ss as described in the next paragraph. 19 The R/S distribution sequence (RSDS) defines the order in which the interdependent R/Ss are distributed. The length of this sequence is equal to 𝑛𝑖𝑅/𝑆, the number of considered interdependent R/Ss. The R/S distribution process starts with the first interdependent R/S, following the order defined in the RSD S. If the demand of a component for that R/S is not met during this distribution process ( e.g., all the available R/S quantities are distributed to higher -priority components), that component ceases op eration and its supply capacity is set to zero, while its R/S demands remain unchanged. The R/S distribution process continues with the next R/S in the RSD S, accounting for the changes in the supply capacities of the components examined previously, until t he last R/S in the RSD S is distributed. If the R/S is a utility, R/S distribution is performed using the System Matrix and the R/S distribution algorithm . If the R/S is a transfer service, the R/S distribution is performed using optimal path algorithms (Figure 2c) . However, the order in which R/S s are distributed, defined in the RSD S, is arbitrary. One pass through the RSD S does not capture all possible permutations of the R/S distribution order. This, in turn, means that some of the interdependency loop triggers may be missed. To account for this, the RSDS is performed 𝑛𝑖𝑅/𝑆 times in a row , defining a R/S distribution vector (RSDV) , where 𝑛𝑖𝑅/𝑆2 is the length of the RSD V using the same RSD S and starting always from the state of the community systems attained at the end of the previous R/S distribution (Hefti, 2019) . It can be shown by induction that such a concatenation of RSD Ss into an 𝑛𝑖𝑅/𝑆2-long sequence accounts for all possible permutations of the R/S distribution order in each time step o f the resilience assessment interval . Once R/Ss are distributed according to the RSDV, the total supply capacities are calculated once again, and they are compare d to the total supply capacities before the R/S distributions defined in the RSDV. When two consecutive RSDV runs result in the same total supply capacities of all interdependent R/Ss, the demand/supply -based interdependency modelling algorithm stops : 20 the system reached a stable state (i.e., a run through an RSDV, where all the possible ways in which R/Ss can affect each other are considered, d oes not change the total supply capacities) . Compared to the original Re -CoDeS framework, iRe -CoDeS requires (𝑛𝑖𝑅/𝑆2−𝑛𝑖𝑅/𝑆) more R/S distributions at each time step of the LoR assessment int erval to simulate the interdependency effects, increasing the computational effort. Figure A2 in Appendix I illustrate s the application of the proposed demand/supply -based interdependency modelling algorithm in a resilience assessment of a simple five -comp onent community . Figure 2: iRe-CoDeS algorithm to quantify community disaster resilience (a ); the “distribute independent R/Ss” step of the iRe -CoDeS algorithm (b); and the demand/supply -based 21 interdependency modelling algorithm ( c) imbedded in the “distribute interdependent R/Ss” step of the iRe -CoDeS community resilience assessment algorithm (a) iRe-CoDeS resilience assessment algorithm The algorithm for quantifying the post -disaster LoR of a system -of-interdependent -systems is shown in Figure 2a. The inputs are system properties (compon ent properties and system topology) and the initial damage level s of the components induced by a disaster . The recovery process is then simulated in a time -stepping loop by decreasing component damage according to it s repair rate . Each time step of the LoR assessment involves calculating the current supply capacity and demand of all components based on their current damage levels, distributing independent R/Ss to calculate their consumption , distributing interdependen t R/Ss and simulating interdependency effects (as explained in the previous section ), and, finally , recovering the components by decreasing their damage level by their repair rate. The distribution of transfer services (i.e., identifying optimal paths between localities) is done before distributing utilities whose distribution require s transfer services in both “distribute independent R/S” and “distribute interdependent R/S” blocks (Figure 2) . This procedure is repeated at each time step until the system reaches a pre-defined recovery target (e.g.: damage levels of all components are zero ; functionality of certain CISs restored to post - disaster target levels ). The LoR for each R/S of the community is computed using Eq . [6]. Case study of a virtual community supplied by interdependent CISs In the following section , disaster resilience of a virtual community supplied by three interdependent CISs is quantified using iRe-CoDeS . The community consists of 20 localities, 32 components in localities , 2 bridges, 27 potable water pipes (PWPs), cooling water pipes (CWPs) and electric power transmission lines ( EPTLs ), and 3600 inhabitants in 9 Building Stock Units ( BSUs ). The topology of the virtual community is shown in Figure 3. 22 Figure 3: Case study virtual community . Considered components in localities are Electric Power Plants (EPPs), Building Stock Units (BSUs), Base Station Controllers (BSCs), Potable Water Facilities (PWFs), Base Transceiver Stations (BTSs) and Cooling Water Facilities (CWFs). Figure 4: Simplified functionality of the CISs The main purpose of the case study is to model the interdependencies between the different CISs : thus, the CISs functionality is deliberately simplified (Figure 4). Physical and technical laws regarding the R/S distribution ( e.g., hydraulic flow laws and electric power transmission rules) are not considered in this case study . The case study utility R/Ss are : the Electric Power ( EP) supplied by the Electric Power Supply System (EPSS); the high-level 23 communication (HLC) supplied by Base Station Controllers (BSCs) followed by the low- level communication (LLC) supplied by Base Transceiver Stations (BTS), both part of the Cellular Communication System ( CCS ); and the Potable Water ( PW) and Cooling Water (CW) supplied by the components of the Water Supply System (WSS) , the Potable Water Facilities (PWFs) and the Cooling Water Facilities ( CWFs ). Table A 2 in Appendix II summarizes pre-disaster component supply capacities, repair rates, and the number s of the considered components in the virtual community , while Table 1 summarizes their R/S demands before the disaster. The case study transfer services are : the CW transfer services (CWTSs) and the PW transfer services (PWTS s), supplied by CWP s and PWPs , respectively ; and the electric power transfer service (EPTS) supplied by EPTLs . For si mplicity, the PWPs, CWPs and EPTLs are assumed have the same topology (i.e. , geo-location and connectivity properties) , as presented in Figure 3. The transfer service supply capacity of these links is assumed to be infinite, hence, when functional (i.e., their functionality level is above 0), these links can transfer any amount of utility R/Ss. Transfer of HLC and LLC is wireless , hence , no physical links are n eeded . Inhabitants of the community populate the building stock units . Each BSU in the virtual community is considered to house 400 people. The portion of the demand imposed on the EPSS and WSS by the inhabitants of the BSUs (Table 1) is, for simplicity, assumed to decrease in proportion to BSU occupancy , i.e. functionality level, after a disaster (Didier et al., 2017) . However, demand for LLC services does not depend on BSU damage and function ality directly , but only on the number of inhabitants remaining in the community after a disaster (Didier et al., 2017) . A time-dependent function describing the change of demand for LLC is given in Eq . [7], representing the immediate post -disaster increase in demand due to emergency calls, and the subsequent return of the demand to the pre -disaster level . Changes in the number and spatial distribution of community inhabitants after a 24 disaster due to relocation into temporary shelters or out -migration is not considered in this case study. 𝐷𝐵𝑆𝑈 ,𝐿𝐿𝐶(𝑡)={𝐷𝐵𝑆𝑈 ,𝐿𝐿𝐶(𝑡<𝑡0) 10∙𝐷𝐵𝑆𝑈 ,𝐿𝐿𝐶(𝑡=𝑡0) 𝐷𝐵𝑆𝑈 ,𝐿𝐿𝐶∙(𝑡−1)∙𝑒−0.3≥𝐷𝐵𝑆𝑈 ,𝐿𝐿𝐶(𝑡<𝑡0)pre-disaster disaster post-disaster ( 7 ) Table 1: Pre-disaster demand for utility R/Ss per each component of the virtual community Component Demand for: EPP BSC BTS CWF PWF BSU Electric Power [MWh] 0.2 0.2 0.1 0.2 0.1 7.7 High Level Communication [E] - - 50 - - - Low Level Communication [E] 0.001 - - 0.001 - 33.3 Cooling Water [Ml/day] 0.05 0.05 - - - - Potable Water [Ml/day] - - - - - 0.086 In this case study, initial damage levels of components are assigned arbitrarily to simulate the effects of a virtual disaster , and are shown in Tables A 3 and A 4. A more realistic simulation of initial damage for the same virtual community exposed to an earthquake is presented in (Blagojević, Kipfer, Didier, & Stojadinović, 2020b, 2 020a) . Damage/functionality functions of the EPPs, BSUs, BSCs and CWF s are linear , while t he BTSs , PWF s, CWPs, PWPs and EPTLs are modelled as binary components . For simplicity, the repair of all components is assumed to start immediately after the disaster (at time t0) and proceed simultaneously at a repair rate specified for each component (Table A 2), resulting in a decrease of component damage levels in each time step of communi ty resilience assessment until component damage is eliminated. The disaster recovery of the community is assumed to 25 end when the damage levels of all of its components reach zer o, i.e., all components are damage -free. Demand/supply -based i nterdependency modelling Table 1 presents the assumed amount of utility R/Ss each component needs to operate and, therefore, highlights the ir interdependencies . For example, a BTS need s 0.1 MWh of EP and 50 E (Erlang, 1925) of HLC to operate. If the transfer of a utility R/S to a component requires a distribution network, then a corresponding transfer service demand for the same amount of R/S is assigned to that component. Thus , the BTS has a transfer service demand for EPTS of 0.1MWh, corresponding to the amount of EP it needs. System R/S distribution model Utility R/Ss are dist ributed according to component priorities : these differ for each utility R/S, but remain constant during the LoR assessment interval . In this case study, distribution of a utility R/S always starts from the components producing that utility R/S. Then, the suppliers of other utility R/Ss have a priority over the BSUs , as shown in Figure A3. Component priorities among the suppliers of R/Ss other than the one being distributed are arbitrarily assumed , but are the same at each time step. Since the transfer of a utility R/S among components in the same locality is unconstrained, optimal paths are defined for each locality pair (e.g., one pair is: locality 103 and locality 105) . Transfer of utility R/S s is assumed to be possible in both directions (e.g., the same optimal path is used to transfer a utility R/S from locality 103 to locality 105 and from locality 105 to locality 103) . Then , if a utility R/S needs to be transferred between a supplier and a user in different localities, the optimal path connecting that locality pair (the locality of the supplier and the user) is used. Optimal paths are calculated for each transfer service separately. A simple optimal path algorithm is employed in this case study. For each 26 locality pair and each transfer service , the optimal path is obtained from a set of pre -defined potential paths, consisting of different links . Due to the simplifying assu mption that all links are binary components and have an infinite transfer service supply capacity, an optimal path is any path from the set of pre -defined paths in which all links are functional (i.e., have a 0 damage level, and hence, a functionality leve l of 1). If no such path exists between two localities for a considered transfer service, the utility R/S requiring that transfer service is not transferred. Consequently, transfer losses are not considered, meaning that the entire amount of utility R/S is either transferred or it is not. The o ptimal path algorithm is ran at each time step of the resilience assessment interval, calculating optimal paths between all locality pairs for each transfer service separately . Since all transfer services in this case study are independent R/Ss and all utility R/Ss are interdependent , the distribution of transfer services, (i.e., the optimal path search) is performed before distributing utility R/Ss (Figure 2) . Once calculated, optimal paths remain constant in a time s tep of the resilience assessment interval. Virtual community r esilience assessment results Disaster resilience of the virtual community is assessed with respect to each considered utility R/S using the presented iRe-CoDeS framework . LoR plots for the considered utility R/Ss are shown in Figure s 5-9, and present the system’s supply capacity, consumption and demand for the considered utility R/S over the resilience assessment interval . The LoR is quantifi ed as the area between the consumption an d demand cur ve (Eq. [6]), using a time step equal to one day. To illustrate the effect s of interdependenc ies between the CISs on community disaster resilience , the results of the analys es with and without interdependency consideration are presente d together. Algorithm presented in Figure 2a is used i n both cases . However, in the independent case, the R/S demands of a component do not have to be met for th at component to operate and have a supply capacity and the operat ion of components is affected only by 27 their damage level . In the interdepende nt case, the unmet demand of a component results in setting its supply capacity to zero, regardless of its damage state. The virtual community recovered to its pre -disaster state in 60 time steps (i.e., 60 days). The n umber of considered interdependent R/Ss is 𝑛𝑖𝑅/𝑆=5 (i.e., all of the utility R/Ss and no transfer service R/Ss) , hence, the RSD S has five elements and the RSDV has 25 elements (i.e., 𝑛𝑖𝑅/𝑆2=25). Therefore, within each time step of the LoR assessment interval , at least one run through the entire RSDV that involve s 25 R/S distributions is performed . For this case study , 2275 utility R/S distributions among 32 components were performed for the entire resilience assessment interval with interdependencies considered . This resilience assessment , implemented in Python, t ook approximately 1.3 seconds on a laptop with 16GB RAM and Intel Core i7 -8750H 2.2Ghz CPU . The dashed blue and green lines in Figures 5—9 represent the supply capacity 𝑆𝑠𝑦𝑠,𝑅/𝑆𝐶 and consumption of the community 𝐶𝑠𝑦𝑠,𝑅/𝑆 , respectively, for the independent case, while the continuous blue and green line represent the supply capacity 𝑆𝑠𝑦𝑠,𝑅/𝑆𝐶∗ and consumption of the community 𝐶𝑠𝑦𝑠,𝑅/𝑆∗ , respecti vely, for the simulations with interdependencies considered. The community ’s demand, 𝐷𝑠𝑦𝑠,𝑅/𝑆 for a R/S is the same for both cases , since it is assumed that the component inoperability caused by interdependency effects affect only the supply capacity and not the demand . The values between the time steps are linearly interpolated . The R/S LoRs of the system are labelled as 𝐿𝑜𝑅𝑠𝑦𝑠,𝑅/𝑆 and 𝐿𝑜𝑅𝑠𝑦𝑠,𝑅/𝑆∗, for the independent and interdependent cases, respectively. Disaster r esilience of the virtual community regarding electric power The considered disaster causes damage to the EPPs, lowering the supply capacity of EP available to the virtual community (Figure 5). However, the evolution of supply capacity available to the community in the independent and interdependent resilience assessments is 28 not the same. In the independent case, the total supply capacity drops from 80MWh to 50MWh due to the damage to the EP Ps. When interdependencies are considered, the supply capacity drops to zero, since the demand of the EP Ps for CW and LLC cannot be met. Thus, the interdependenc ies have an adverse effect on the supply capacity of the EPSS . The demand for EP also decreases due to the damage to the EP users, such as BSUs and BTSs . Even though the supply capacity for EP is higher than the demand , the consumption of EP is al most zero immediately after the disaster in the independent and interdependent resilience assessments. This is due to the damage suffered by the EPTLs that cannot supply enough EPTS to transfer the EP from the EPPs to the users . Therefore, only the users located at the localities that contain EPP s, localities 301 and 305 ( Figure 3), can consume EP immediately after the disaster, since it is assumed that the transfer of R/Ss within a locality is unconstrained. The jumps in the consumption 𝐶𝑠𝑦𝑠,𝐸𝑃 on days 4, 7 and 10 after the disaster , occur when EPTLs connecting the EPPs with the user s in other localities are repaired. In contra st, 𝐶𝑠𝑦𝑠,𝐸𝑃∗ remains zero until day 9, since this is the time needed to repair the suppliers of CW and LLC and the ir distribution networks to meet the demand of EPPs . The system 𝐿𝑜𝑅𝑠𝑦𝑠,𝑅/𝑆=272 .77𝑀𝑊 ℎ in the independent case, while 𝐿𝑜𝑅𝑠𝑦𝑠,𝑅/𝑆∗=456 .86𝑀𝑊 ℎ in the interdependent case, showing a significant effect of CIS interdependencies on the EP resilience of the virtual community . 29 Figure 5: Disaster r esilience of the virtual community regarding EP Disaster r esilience of the virtual community regarding the high -level communication Figure 6 presents the HLC LoR plots. T he disaster causes damage to both BSCs , the suppliers of the HLC . However, immediately after the disaster, HLC supply 𝑆𝑠𝑦𝑠,𝐻𝐿𝐶𝐶 drops from 600E to 350E , while 𝑆𝑠𝑦𝑠,𝐻𝐿𝐶𝐶∗ drops to zer o because the BSCs cannot be supplied with EP and CW . The LoR plot for EP in the interdependent case (Figure 5) show s that the EPSS is not producing any EP until day 10 after the disaster . On day 10, all demand s of the EPP s can be met and they start producing EP and supplying the BSC at locality 105, resulting in an increase in 𝑆𝑠𝑦𝑠,𝐻𝐿𝐶𝐶∗. The other BSC at locality 303 is supplied with enough CW only on day 16, resulting in a nother increase of 𝑆𝑠𝑦𝑠,𝐻𝐿𝐶𝐶∗ (Figure 6), as well as an increase in the consumption of CW (Figure 9). The step -wise increase in the HLC demand , 𝐷𝑠𝑦𝑠,𝐻𝐿𝐶, is due to the binary character of the users of HLC, the BTSs . Since BTSs are binary components , a 30 jump in HLC demand occurs when a BTS is fully repaired (i.e., its damage level is zero) . In addition, BTSs begin to function and produce LLC only when their demand for HLC if fully met. This causes an important drop in HLC consumption from day 22 to day 35, even though the HLC supp ly is available and HLC demand exists. Jumps in HLC consumption occur on day 27 and day 35 in both independent and interdependent resilience assessments when the HLC demand for the BTS s at localities 305 and 307, respectively, is fully met. The system LoRs are 900E and 2650E in the independent and the interdependent cases, respectively, showing a remarkable effect of the unmet BSCs demand for EP and CW in the first 15 days after the disaster . An installation of e mergency EP diesel generators and CW coolers at BS Cs is an effective way to reduce this HLC LoR in the virtual community. Figure 6: Disaster r esilience of the virtual community regarding the HLC Disaster r esilience of the virtual community regarding the low-level communication LLC is provided by 11 BTSs in the virtual community. The LLC LoR plot is shown in Figure 7. Due to the increase in emergency calls after a disaster , LLC demand peak s on the first day 31 after the disaster and decreases to t he pre -disaster level by day 10 . The b inary character of the BTSs results in a stepwise increase of the system’s LLC supply capacity in both independent and interdependen t cases. LLC LoR in the interdependent case is about 10% higher, due mainly to the unmet demands of some BTSs for EP and HLC in the first 15 days following the disaster . Figure 7: Disaster r esilience of the virtual community regarding LLC Disaster r esilience of the virtual community regarding potable water Figure 8 shows the PW resilience of the virtual community . After the disaster, the demand for PW d rops due to the damage suffered by the building stock . The PW demand increases as the BSUs are repaired. The PW supply capacity available to the community drops to zero in both interdependent and interdependent cases because the PWFs are treated as binary components , and all community PWFs suffered a certain amount of damage due to the disaster (Table A 3 in Appendix II). Hence, none of them was operational immediately after the disaster . The 32 stepwise increase of the PW supply capacit y occur s when a PWF is fully repaired. In this case study , the interdependency between PW Fs and the EPSS did not have a large effect on the functionality of the PW Fs. The EPSS delayed the increase of the total PW supply capacity by one time step: in the independent case the first PW F was operational on day 9 , and in the interdependent case it was operational on day 10 . One of the consequences of dynamic interdependency modelling in iRe -CoDeS can be seen on days 25, 40 and 53 , when an unexpected decrease in PW consumption occurs. This is because the PW demand (of the inhabitants returning to the repaired BSUs) increase s faster than the P W supply capacity. On days 25, 40 and 53 the demand of certain BSUs increases to a point where the current PW supply cannot meet their demand in full. This results in redirecting the available PW supply to other BSUs with lower PW demand that can be fully met at that time st ep. As their demand is lower, their consumption is also lower and, thus, there is a drop in the consumption of P W. On days 29, 48 and 56 the PW supply capacity increases, due to the repaired PWFs, leading to an increase in PW consumption. This change in ho w the PW users (i.e., the BSUs) are supplied by the same PW suppliers illustrates how iRe -CoDeS can simulate the change of interdependency relations among components as the community is recovering. 33 Figure 8: Disaster r esilience of the virtual communit y regarding the PW Disaster r esilience of the virtual community regarding cooling water The CW LoR plot is shown in Figure 9. The total CW supply drops from the pre -disaster 0.2Ml/day to 0.055Ml/day in the independent case and to zero in the interdependent case. In the independent case, the total CW supply capacity starts to increase immediately after the disaster, since the CW Fs are recovering. However, in the interdependent case, the total CW supply capacity remains zero until day 10, similarly to other CISs of the virtual community . On day 10, the EPSS recovers, the CW Fs are supplied with EP and start to supply the virtual community with CW again . Thereafter , the total CW supply capacities in both cases are the same. The stepwise increase s in CW consumpt ion are, again, due to the assumption that a user demand has to be fully met for consumption to start . Therefore, jumps in CW consumption occur once the CW supply capacity is large enough to full y meet the demand o f BSCs and EPPs, and once the distribution network (i.e., the water pipes) can transfer the CW from CWFs to BSCs and EPPs . The difference in CW LoRs in independent and 34 interdependent cases is primarily due to the unmet demand of CWFs for EP and LLC in the first 10 days after the disaster. Figure 9: Disaster r esilience of the virtual community regarding the CW R/S Conclusion This paper presents a disaster resilience quantification framework for systems -of- interdependent -systems, the iRe -CoDeS framework. The framework extends the Re -CoDeS disaster resilience quantification framework that measures the resilience of a Civil Infrastructure System (CISs) using the Lack of Resilience (LoR) metric. LoR is a measure of CIS performance, relevant for making rational decisions on how to increase communi ty disaster resilience . It represents the unmet demand of the community for a resource or a service provided by the considered CIS over the resilience assessment interval. The iRe - CoDeS framework discretizes a system (e.g., a community) into components that can belong to various subsystems of the considered system and are located in or between localities. Components can supply the system with one or multiple resources or services and/or demand 35 one or multiple resources or services from the system. Compon ents recover, and thus increase their functionality, due to r epairs that decrease their damage level at a predefined repair rate in each time step of the resilience assessment interval. Interdependencies among components are modeled through the resources a nd services they need to operate. Interdependency relations among components of a system are not pre -defined: instead, they depend on the post-disaster component supplies and demands for various resources and services, on the state of the distribution netw orks, and on the component prioritization during resource and service distribution, all of which can vary during the resilience assessment interval. Additionally , components can be supplied by different suppliers at different times during system recovery. Therefore, the interdependencies modeled in the iRe -CoDeS framework are dynamic, not static, making it possible to consider changes in interdependency relations between components over time. Furthermore, interdependency loops, situations when failure of on e component to function triggers failures of other components, are expected to occur. The presented iRe -CoDeS framework features a demand/supply -based interdependency model and a resource and services distribution algorithm to resolve such interdependency loops and, thus, enables quantification of community disaster resilience considering interdependencies among community components. A case study on a virtual community is used to demonstrate the presented iRe -CoDeS framework. The virtual community consists of the building stock, the electric power supply system, the cellular communication system and the water supply system. After a virtual disaster, d amage levels are assigned to each component of the community, followed by a recovery process that lasts until all components are damage -free. The virtual community supply, demand and consumption of the resources and services, such as the electric power or potable water, over the resilience assessment interval are presented in LoR plots. Two iRe- CoDeS resilience a ssessments are conducted, one with and the other without considering 36 component interdependencies. A comparison of the LoR plots reveals a significant reduction of virtual community disaster resilience due to interdependencies. Furthermore, an analysis of the supply, demand and consumption changes at both the system and the component levels, enabled by the presented iRe -CoDeS framework, reveals the causes of disaster resilience reduction . Such findings point to rational and effective resilience improvement i nterventions, as components and CISs that are significantly impaired by the interdependency effect s are identifi ed. Reducing the dependency of such components on other CISs can effectively increase comm unity disaster resilience . Notably, t he iRe-CoDeS resilience assessment without considering interdependencies reveals the upper limit of such resilience improvement measures. If such upper limit does not meet the community resilience objectives, alternative measures, such as increase s of compone nt robustness and/or system redundancy, should be considered. Acknowledgment The authors would like to thank ETH Zürich for funding the work presented in this publication. Declaration of interest The authors declare that there is no conflict of interest. 37