Identifying Critical Isolation Valves in a Water Distribution Network: A Socio-Technical Approach

Isolation valves are critical for reliable functioning of water distri-8 bution networks (WDNs). However, it is challenging for utilities to prioritize 9 valve rehabilitation and replacement given it is often unclear if certain valves

to have an effect on resulting best practices or decisions (Diao et al., 2019).

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Finally, the likelihood of failure of segments has not been included in these 107 studies (Giustolisi, 2020), but considering it can provide utilities with insight 108 on the valves that ought to be repaired sooner.In sum, challenges to prioritiz-   A network model of segments and valves is then generated using the networkx 145 package (Abdel-Mottaleb and Walski, 2020).

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There are still loops in the segment-valve representation, making it compu-147 tationally complex to identify how valves impact segments when any valves are 148 inoperable.This is because there are many potential combinations of valves 149 that can impact isolation of a given segment.Consider the small network shown 150 in Figure 1a, where node 1 (i.e., segment 1) contains the reservoir and node 6 151 (i.e., segment 6) contains a tank.In this figure, if segment 4 must be isolated, 152 at least two operational valves must also exist (one on each path) between 153 segment 2 and segment 4, so that segment 2 is not isolated unnecessarily.The 154 combinations of two operable valves that allow segment 4 to be isolated with-     To mathematically formulate the model, we define S as the set of all seg-291 ments, and s i as the binary decision variable related to segment i ∈ S, where 292 s i = 1 means that segment i must be isolated and s i = 0 means otherwise.
293 Also, we let V be the set of all valves, and v j represent the binary decision 294 variable related to valve j ∈ V , where v j = 1 means that valve is inoperable 295 and v j = 0 means otherwise.Based on the logical network, V i is defined as the 296 set of all the valves affecting the segment i (i.e., must be operational such that 297 segment i is not unnecessarily isolated) such that i=S V i = V .In addition, we -Segment Flow Volume: -Reachability Impact: The constraints of the optimization model are as follows: i∈S j∈Vi Constraints (4) and (5) define the relationship between valves and seg-312 ments such that, if and only if at least one of the valves affecting segment i is 313 inoperable (from the minimum set of valves that must be operable), then seg-314 ment i must be unnecessarily isolated.Note that, since all the objective func- 326 where v i is the binary decision variable showing whether the valve affecting the segment i is inoperable or not.This combined with Constraint (6) results in i∈S j∈Vi Formulation Two In this formulation, the model accounts for the uncertainties 327 that are inherent in two of the objective functions of the study.Uncertainties between zero and one, where γ = 1 means that the decision maker is completely 344 pessimistic and he/she believes that all the coefficients will take their worse 345 values, i.e., a i and b i for all i ∈ S.However, if γ = 0, then the decision maker is 346 completely optimistic meaning that the coefficients will take their best values, 347 i.e., a i and b i for all i ∈ S. If being set to a value in the interval (0, 1), then 348 the decision maker believes that the ratio of parameters that take their worst 349 values is γ.
350 Following these definitions, the objective functions of social vulnerability 351 and segment flow volume impact are defined as follows:

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-Social Vulnerability: -Segment Flow Volume: where r i is a binary variable that shows whether the worst-case is happening for segment i or not.The constraints of the robust optimization model include the constraints of formulation one, i.e., constraints (4)-(8), in addition to the following constraints: Xeon E5-2650 2.2 GHz 12-Core Processors (30MB), 128GB RAM, the RedHat 396 Enterprise Linux 6.8 operating system, and using Gurobi's default setting.Fig. 7: Maps of identified valves using the three formulations, where Formulation 1 is deterministic, Formulation 2 accounts for uncertainty, and Formulation 3 accounts for uncertainty and likelihood of failure Segment-valve represenation and Gomory-Hu tree of a network example Figure 2 representation of the logical network, where valves that were represented by edges are modeled as nodes and segments are also modeled as nodes, and the edges between valves and segments represent logical implications Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.This map has been provided by the authors.
Segment reachability values and the subsequently identi ed critical valves Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.This map has been provided by the authors.
Map of segment ow volume Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.This map has been provided by the authors.Segment age mapped by color, age is used as a surrogate for the likelihood of failure Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.This map has been provided by the authors.
Maps of identi ed valves using the three formulations, where Formulation 1 is deterministic, Formulation 2 accounts for uncertainty, and Formulation 3 accounts for uncertainty and likelihood of failure Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.This map has been provided by the authors.
Maps of valves identi ed by formulations 1 and 3 at different percentile for the frequency of occurrence in the solution set Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.This map has been provided by the authors.

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While many stressors act on water distribution networks (WDNs) (e.g., cli-30 mate change, aging infrastructure, interdependencies), utilities have limited 31 resources to address challenges WDN infrastructure faces.This makes prior-32 itizing WDN components (e.g., pipes, valves, pumps, tanks) to repair, reha-33 bilitate, or replace important.Much research has focused on the prioritization 34 of pipes, based on pipe criticality (Shuang et al., 2014; He and Yuan, 2019; 35 Abdel-Mottaleb et al., 2019).The criticality of pipes is determined using sev-jectives were the minimization of the negative impact on the consumers (e.g., 93 unsupplied water volume) and the minimization of the corresponding number 94 of repair actions.The purpose of their method is real-time decision making 95 after failures assuming valves are operable.96 Furthermore, all of these previous studies do not account for the social 97 vulnerability of communities serviced by the WDN (i.e., customers vulnerable 98 to a disruption in water service).In reality, utilities are often concerned with 99 social indicators to varying degrees and account for it within the score of conse-100 quences of failure in their risk-based asset management.In addition, previous 101 studies do not account for uncertainties associated with both water service 102 needs of customers and customer vulnerability (Shuang et al., 2019), which 103 are inherent in infrastructure networks and the urban environment (Walski, 104 2001; Zischg et al., 2018).Accounting for these uncertainties has been shown 105

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ing isolation valves include the complexity of accounting for multiple possible 110 operable valve scenarios in a network, and accounting for social vulnerability 111 regarding the service population, uncertainties in both hydraulic performance 112 and social vulnerability, and the likelihood of segment failure.These limita-113 tions can be attributed to the methodologies used for analyzing WDN valving.114 To address these limitations, this study develops a methodology and ap-115 plies it to the City of Tampa as a case study.The presented methodology 116 is geared towards long-term planning as opposed to real-time operation, to 117 provide a prioritization of valves to rehabilitate, repair or replace while con-118 sidering the potential inoperability of valves.Graph theory is combined with 119 multi-criteria optimization to identify the pareto-optimal sets of critical valves 6 Noha Abdel-Mottaleb et al.
(i.e., valves which have the most potential adverse impact when inoperable) 121 based on hydraulic performance and social vulnerability of WDN segments.In 122 lieu of a priori knowledge on valve operability, the deterministic model identi-123 fies critical valves and the results are compared with models that account for 124 both uncertainties and likelihood of failure in segments for the City of Tampa.125 2 Methods 126 To identify critical valves, an optimization model is used over the logical impli-127 cations of inoperable valves.The implications of inoperable valves are segment 128 isolation and the potential subsequent loss of water service.Not all segment 129 isolation has the same effect-some segment isolation is more critical than 130 others.To obtain the logical implications, first a segment-valve representation 131 of the network is constructed.Then, the segment-valve representation is sim-132 plified to a gomory-hu tree network model (i.e., an equivalent flow graph), that 133 allows identification of the minimum set of inoperable valves that would cause 134 any given segment isolation.

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The segment-valve representation of the WDN has often been called the math-137 ematical "dual" of the more common pipe-junction representation.In this rep-138 resentation, segments containing pipes are the nodes and isolation valves are 139 the edges, or links, between them.To obtain the segment-valve representa-140 tion, a hydraulic network model based on pipe-junction representation is first 141 created using a hydraulic modeling software (e.g., WaterGEMS).The data 142 structure (e.g., list of lists, dictionary) containing segments and associated 143 isolation valves is exported from the software (e.g., WaterGEMS) to python.
155 out isolating segment 2 correspond to the minimum k-cut between segments 156 2 and 4, where k = 2.A minimum k-cut between two nodes (i.e., segments) 157 means that k edges (i.e., valves) are required to partition the two nodes (i.e., 158 successfully isolate them from each other).If the valves between segments 2 159 and 4, and segments 4 and 5 are both inoperable, then to isolate segment 4, 160 the entire network must also be isolated from the reservoir.This illustrates the 161 combinatorial complexity of assessing the impact of failed valves on segments 162 because of the existence of loops of segments.1632.2 Gomory-Hu Tree of Segment Connectivity to Address Combinatorial 164 Complexity 165 The gomory-hu tree (G-H tree for short) simplifies the task of identifying the 166 paths along which operational valves should exist to minimize unnecessary 167 segment isolation (i.e., the minimum k-cuts between segments).When there 168 are loops of segments, a set of valves instead of a single valve must be op-169 erational to minimize unnecessary segment isolation.The G-H tree contains 8 Noha Abdel-Mottaleb et al. nodes, which represent segments, and edges linking segments that represent 171 the flow paths between each pair of connected segments.Specifically, edges 172 represent the k valves (corresponding to all the minimum k-cut(s) in the seg-173 ment valve network, or the minimum number of valves that must be operable) 174 between two connected nodes for successful isolation.The G-H tree is also 175 called a flow equivalent graph, because each minimum cut(s) represents all of 176 the possible flow paths between two nodes (i.e., segments).The G-H tree of 177 the small example in Figure 1b.In this figure, the edge from segment 2 to seg-178 ment 3 represents the following valves from Figure 1a: segment 2-segment 3, 179 and either segment 2-segment 4, segment 4-segment 5, or segment 5-segment 180 3.The G-H tree of the segment valve representation of the network is obtained 181 using the gomory hu tree function within networkx in python.The valves that 182 are part of the minimum cut(s) are stored in a data structure for the following 183 step.184 (a) segment-valve representation of a small example network (b) Gomory-Hu tree of small example, where edges that represent more than one valve are bolded

Fig. 2 :
Fig.2: representation of the logical network, where valves that were represented by edges are modeled as nodes and segments are also modeled as nodes, and the edges between valves and segments represent logical implications 2.4.1 Formulation280In this study, three level of formulations are used to solve the problem ac-281 counting for different aspects.However, the overall formulation requires the 282 decision-maker to provide the number of critical sets of valves that the model 283 should identify.In other words, for a given user-defined number, the model 284 looks for the worst sets of valves to fail in the network.285 Formulation One The first formulation is the most straightforward one and is 286 based on the two following assumptions.First, we assume that there does not 287 exist any uncertainty in the objective functions over time or in their measure-288 ment.The second assumption is that the likelihood is the same for all valves 289 to be operational (i.e., all segments have an equal likelihood of failure). 290

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and c := (c 1 , c 2 , • • • , c |S| ) 299 as the vectors of non-negative real numbers where a i , b i , and c i are the coef-300 ficients representing the social vulnerability, segment flow volume, and reach-301 ability impact of segment i ∈ S, respectively.302 The optimization model includes three objective functions to be maximized 303 for identifying the worst sets segments to be isolated.Subsequently, from the 304 relationship between segments and valves in the logical network, the worst sets 305 of valves to be inoperable are identified by selecting the sets of valves connected 306 to the identified segments in the logical network.The objective functions of 307 our optimization model are as below: 308

315Proposition 1
tions of our optimization problem are in maximization form, Constraint (5) 316 will be naturally satisfied, and therefore, can be removed from the model.Con-317 straint (6) defines the user-imposed condition on the total number of valves 318 to fail, where k is a positive integer number.Constraints (7) and (8) enforce that the variables representing valves and segments can only take a value of 320 zero or one.321 If each valve in the network only affects one segment, then 322 the formulation can be simplified by replacing the constraints (4)-(7) with the Since, |V i | = 1 by assumption, we can rewrite Constraints (4) and (5) 325 as follows:

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in segment flow volume are assumed to increase (between 5 percent and 95 329 percent of a given flow value, evenly distributed based on quantile) as pipe 330 diameters decrease.Uncertainties in social vulnerability are assumed to in-331 crease as the social vulnerability index increases (also between 5 percent and 332 95 percent of a social vulnerability index value, evenly distributed based on 333 quantile).This assumption is specific to the case study and may differ for other 334 cities. 16 Noha Abdel-Mottaleb et al.In order to mathematically formulate the objectives of the new formulation, 336 we define [a i , a i ] and [b i , b i ] as the intervals of the coefficients representing the 337 social vulnerability and segment flow volume.In these intervals, the lower 338 bounds are the estimated lowest values of the social vulnerability index and 339 segment flow volume, and the upper bounds are the highest values (i.e., the 340 worst-case scenario) that the social vulnerability and segment flow volume 341 impact can be.Furthermore, we define γ as a parameter that represents the 342 level of conservatism of the decision maker.This parameter can take values 343

4674. 2 Fig. 6 :
Fig. 6: Segment age mapped by color, age is used as a surrogate for the likelihood of failure

5124. 3
Differences Between the Formulations 513 Both formulations 1 and 3 identified critical valves in the Northeast (New 514 Tampa) and East Tampa, but the selected individual valves varied.These lo-515 cations are selected by the model because they bridge between the reservoir 516 and the periphery of the WDN.The variation in the individual valves is likely 517 due to the likelihood of segment failure that is accounted for in formulation 518 3.In addition, there are more valves selected by the deterministic model (for-519 mulation 1) in East Tampa.When uncertainties and likelihood of failure are 520 included (i.e., formulation 3) they are not selected, as shown in Figure 8c.521This may be because of more newly installed pipes (as shown in Figure6) 522 and pipe replacement projects (FDOT, 2020; Water, 2018) that contribute 523 to a lower likelihood of failure in East Tampa.Additionally, social vulnera-524 bility is relatively low in East Tampa in comparison with locations such as 525 Ybor or North Tampa; meaning, there is lower uncertainty associated with 526 the social vulnerability of segments in East Tampa, leading the model to se-527 lect less valves in East Tampa when uncertainties are accounted for.Although 528 the models selected many common valves (shown in Figure 8d), the determin-529 istic model selects more valves in proximity to those common valves.This is 530 likely due to the additional constraints (for the likelihood of failure) in for-531 mulation 3 limiting the selected valves in a given location.More importantly, 532the models can vary in the level of criticality assigned to selected valves.For 533 example, formulation 3 selects valves in the Lake Leto area in the 25th per-534 centile (Figure8a), but formulation 1 selects valves from that location at the 535 higher percentiles.In other words, the Lake Leto area is less critical when 536 the likelihood of failure and uncertainty in flow volumes and social vulnera-537 bility are accounted for.This is an example of how including uncertainty and 538 the likelihood of failure in the model provide more complete information for decision-makers.Additionally, including uncertainty and the likelihood of fail-540 ure reduces the number of critical valves identified as the percentile (of the 541 frequency of valves occurring in the solution set) increases.This is because 542 formulation 3 includes additional constraints (i.e., based on their likelihood 543 of failure), thereby eliminating solutions (i.e., sets of valves) that were iden-544 tified by the exact method.The reduced number of critical valves identified 545 at the higher percentiles makes the model results more readily considered by 546 decision-makers.Another advantage of formulation 3 is that it allows valves 547 to be identified as critical (even if at a lower percentile of frequency) where 548 they otherwise may not have been selected at all.For example, if criticality 549 is determined by the likelihood of failure alone, there would be less valves 550 selected in the recently developed Northeast of the city (or the New Tampa 551 location), but because the likelihood of failure is integrated within the model 552 there are still valves selected with high frequency in the Northeast of the city.553This is due to high values of reachability and flow volume of segments in the Implications of the Optimization Models 556 This study applied multi-objective optimization to identify critical isolation 557 valves in a real WDN.The optimization models were solvable within reason-558 able time and provided results that are useful to utilities.In particular the 559 models made it easier to identify valves as critical that may have been missed 560 using single objectives.The optimization model formulations allow decision-561 makers to account for multiple facets of valve criticality.In particular, includ-562 ing social vulnerability in the optimization formulation identifies communities 563 that may suffer unduly from an extended lack of water.If valves in these 564 communities are operable, unsupplied demand to vulnerable customers may 28 Noha Abdel-Mottaleb et al.

)
Formulation Three This formulation is the most realistic model that accounts 357failure within the segments).Though the likelihood of failure is complex and 358 there are models that have been developed accounting for pipe length, age, 359 size, and material among many other things.In this work, we use age as the 360 primary factor because most of the pipes in the City of Tampa are severely 361 aged (Park et al., 2010).362Based on the likelihood of failures in a given segment, we categorize the 363 set of valves into three categories shown by V h , V m , and V l representing 364 the sets of valves affecting segments (i.e., the valves that successfully isolate 365 given segments) with high, medium, and low likelihood of need to be operated, 366 respectively.Following this definition, we simply substitute constraint (6) with 367 the following set of constraints: