Improving Urban Drainage Systems Resiliency

Water Resources Management https://doi.org/10.1007/s11269-018-2069-3

Improving Urban Drainage Systems Resiliency Against Unexpected Blockages: A Probabilistic Approach J. Yazdi 1 Received: 23 January 2018 / Accepted: 24 July 2018/ # Springer Nature B.V. 2018

Abstract Considering the increase of flood hazards in many large cities, the rehabilitation of hydrourban infrastructures is an important concern for the municipal authorities. A probabilistic approach based on Monte Carlo Simulation (MCS) is presented in this study to improve the resiliency of urban drainage systems when they are subject to unexpected structural blockages. The approach is integrated with SWMM simulation model and an evolutionary search algorithm to find the best set of rehabilitation measures under a significant number of blockage scenarios. Experimental results on the west zone of main drainage system in Tehran city indicate that proposed approach outperforms the conventional hydraulic-based methodology in terms of cost effectiveness and functionality. Results also show that adding the redundancy to the system by bypass lines in bottlenecks is considerably more efficient for flood mitigation and the increase of system resiliency under blockage incidents rather than using conventional methods such as detention ponds and enlargement of the channel sizes. Keywords Resiliency . Rehabilitation . Urban drainage system . Optimization

1 Introduction Urban floods may increase damages to properties and infrastructures, traffic loads, loss of lives, and environmental and health risks. Development of comprehensive surface runoff collective systems is of great importance in urban areas in order to safely transfer stormwater while the economic and administrative constraints is satisfied. Traditional methods of design/ rehabilitation of urban drainage systems (UDSs) are based on providing a sufficient hydraulic capacity for conveyance of design flood discharge. In other words, design or rehabilitation is often carried out with the aim of making a sufficient hydraulic capacity for the system during intensive flood periods. This approach has been addressed and frequently discussed in the literature by applying mathematical simulation models and optimization methods (e.g. see

* J. Yazdi [email protected]

1

Faculty of Civil, Water and Environmental Engineering, Shahid Beheshti University, Tehran, Iran

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Delelegn et al. 2011; Park et al. 2012; Sun et al. 2011; Fu and Butler 2014; Vojinovic et al. 2014; Yazdi and Kim 2015; Garofalo et al. 2017; Xu et al. 2018 among others). The major weakness of these methods is neglecting the impact of unforeseen events which adversely impact on the system capacity. Decreasing the system capacity may be originated from the accidental incidents (such as channel blockage due to sedimentation) or structural failure of the key elements like pumps in the system during the floods. Traditional methods are only based on the hydraulic reliability while in most of the time, structural failure is responsible for the undesirable performance of the system. Therefore, it is important to consider the resiliency of UDSs under unforeseen loads such as blockage when these systems are rehabilitated. Some aspects of the UDS resiliency have been occasionally studied in the literature. For example, Mugume et al. (2015) proposed a novel approach based on resiliency for the analysis of the UDS performance when they are subject to a large number of structural failures. Their results showed that decentralized small detention ponds provide considerable increase in resiliency of the network compared to a large centralized detention during the flood periods. Sweetapple et al. (2018) investigated the effects of two attribute-based interventions (increasing distributed storage and reducing imperviousness) on performance-based resilience measures. They found that these methods provide improvement in performance under system failure in the majority of case studies, but it is also shown that attribute-based intervention development can result in reduced resilience. Yazdi (2018) developed an optimization model to extract cost-effective designs of urban drainage rehabilitation underlying a large set of blockage scenarios. The approach however did not considered the blockage probabilities and bottlenecks priorities. Although the concept of resiliency and its application to UDSs is introduced in the literature, most quantitative studies however tend to focus on investigating the social and administrative aspects and do not consider technical factors such as structural failure or exceptional loads. Considering these concerns, the aim of this paper is presenting a novel resilience-based method of rehabilitation to improve UDS flexibility and resiliency under structural blockage. The developed approach is applied to a part of the main drainage network of Tehran (the capital city of Iran) to study the effectiveness of the approach is examined. Through this resilient framework, the optimal rehabilitation measures are determined using a multiobjective evolutionary algorithm (MOEA) and rainfall-runoff modeling.

2 Methods and Materials 2.1 Resiliency Design or rehabilitation of urban drainage systems based on hydraulic capacity leads to a lack of consideration to other sources of urban flooding such as equipment, pumps, and gates failure, disability of a part of the drainage network and its closure (Mugume et al. 2015). However, urban flooding not only originates from external loads (like heavy rainfalls and urbanization), but they are also related to internal system threats such as malfunction of facilities and blockages. Structural failures can originates from sudden and unexpected collapse like pump failure or chronic and long-term pressures such as pipe obsolescence and sedimentation. These internal factors decrease system service level and may result in severe flooding. Resiliency here is referred to Bthe state of the system that enables limiting failure duration and magnitude to any threat^ (Mugume et al. 2015). The approach proposed in this

Improving Urban Drainage Systems Resiliency Against Unexpected...

research work is optimal increase of redundancy and flexibility on the system with making parallel and axillary tunnels in throats or bottlenecks beside typical actions like putting detention ponds on different parts of the network.

2.2 Blockage Probability Although, the incidence of blockage in sewer and drainage systems is a common problem, little studies have been dealt with this issue. Marlow et al. (2011) reviewed various factors that influence sewer blockages and indicated that blockage rate is influenced by a range of factors, including asset attributes, climatic conditions, water consumption, and soil type. The risk of flooding associated with blockage of culverts/ bridges by natural and anthropogenic debris is often reduced by fitting a trash screen at the structure’s inlet. Wallerstein and Arthur (2012) extended an empirical formula to estimate the probability of delivery of significant debris loads to trash screens by correlating it with pertinent driving variables including channel properties, flows, land use types and the degree of social deprivation. This approach offers a robust means to rank culverts (when there is available data to validate the method) in terms of blockage potential and therefore flood risk. Streftaris et al. (2013) proposed a similar formula, but included the uncertainties of data sets. To allow for randomness in the data set, a Bayesian framework was adopted through which the uncertainty associated with any prediction could be reported using appropriate credible intervals. Using the regression analysis and Bayesian approach or Bayesian network such as the above studies is possible when there is some recorded data about the blockages occurred in the past decades. In many cases like the one in this study, this information is not available or not recorded. In these cases and for urban drainage network rehabilitation under blockage scenarios, the proposed approach here can be used. External loading is assumed here as a fixed design rainfall. This study focuses on a frequent type of internal loads –i.e. blockage and its effect on system performance and the selection of the type of rehabilitation interventions. Many engineering problems are related to the number of events occurred in a specific time interval. These processes are usually described by Poisson distribution. According to a Poisson process, the probability of occurring x number of blockages in a time interval can be estimated as: f ðx; λÞ ¼ pðx; λÞ ¼

λx :e−λ x ¼ 0; 1; 2; … x!

ð1Þ

where the parameter λ is called Bevent rate^ and is defined as the average number of events per interval. Based on this concept, the time interval is deemed here as the return period of the design rainfall. Thus, given the expected number of blockages in this return period (λ), the number of blockages during design return period follows up the above equation. When Poisson process estimated the number of blockages (for example three blockages), engineers are interested to know which three locations on the network are likely to be blocked. To give answer to this question, a likelihood function is proposed in this research for the first time that relates the probability of blockage to the hydraulic parameters of the channels such as size of cross section, flood discharge and the values of sediment loads. The blockage likelihood function can be defined for all elements of the network or just for a few critical elements, based on the expert knowledge. According to the characteristics of

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the studied network, blockage likelihood is defined here only for the network bottlenecks. The proposed formula for the blockage likelihood is as:

Ac LðC Þ ¼ a Ac;m

b

Ls;c þ ð1−aÞ c: Qm;c

!d ð2Þ

where L(C) is the likelihood function of blockage for the element C in the network. This element can typically be a channel, pipe, gate, pump, bridge/culvert or any other component. Ac is the cross section of element C occupied by flood discharge, Ac, m is the maximum cross section of element C; Ls, c is the sediment load carried by flood in the element C and Qm, c is the maximum discharge of element C during flood; a, b and d are constants determined subjectively by engineering judgment. The parameter a is defined as a weighting factor indicating the relative importance of cross-section filling-in (Ac/Ac, m) and non-dimensional sediment load (c. Ls, c/Qm, c) in calculating blockage probability. The parameter c is a constant for scaling the unit of sediment loads and flood discharge. Whatever parameters b and d are larger, the blockage probability of the elements becomes more distinct. The value of both parameters are subjectively assumed as two, in this research. Based on the values of likelihood function, the probability of blockage for each element C, Pb(C) is determined as: Pb ðC Þ ¼

LðC Þ n

∑ LðC i Þ

ð3Þ

i¼1

where n is the total number of critical elements considered in the drainage network. Blockage probability with above equations are employed in MCS accompanied with a simulation and optimization model to find best set of rehabilitation interventions. The proposed methodology is demonstrated in the next sections.

2.3 Simulation Model A simulation model is employed to describe rainfall-runoff process and predict the response of urban drainage network under external (design rainfall) and internal loads (blockages of bridges/culverts). Simulation model has two major modules: rainfall-runoff and flood routing. The former simulates the process of converting design rainfall to the surface runoff and estimates sub-catchments hydrograph while the latter is responsible for flood routing in the channels and conduits to convey surface runoff into the catchment outlet. HEC-HMS (USACE 2010) and EPA SWMM (Rossman 2008) models are used here for rainfall-runoff modeling and flood routing, respectively.

2.4 Optimization Model Optimization model is used to determine the best set of interventions including size of axillary tunnels, detention ponds and control structures such as weirs to improve the performance of UDS considering two or more criteria simultaneously. Here, minimization of the rehabilitation costs and flood volumes are considered as two conflicting


objective functions. The general form of optimization formulation can be written as: i i n h m h K 0 1−minC T ¼ ∑ C i Li f ðH i ; W i Þ þ C i LWi hi þ ∑ C ″j A j H D; j þ ∑ C ‴k Lk Dk i¼1

j¼1

NT

k¼1

ð4Þ

2−minV fT ¼ ∑ V f ;N N ¼1

Subject to : va ≤vmax H i ∈fH 1 ; H 2 ; …; H M g W i ∈fW 1 ; W 2 ; …; W M g LWi ∈fLW1 ; LW2 ; …; LWM g hi ∈fh1 ; h2 ; …; hM g

ð5Þ

where CT and VfT are total rehabilitation costs and total flooding, respectively as the objective functions. Li, Hi and Wi are length, height and width of ith axillary culvert/bridge, respectively; hi and LWi are the height and length of ith side weir, respectively. M is the number of discrete values that Li, Hi, Wi, hi and LWi can take. These parameters (related to the side weir) are a part of decision variables in the optimization problem which will be optimized. Optimization of spillway/weir parameters has been reported by other researchers, too (e.g. see Hosseini et al. 0 2016). Ci and C i are unit cost of culvert/bridge and side weir construction. va is the velocity in ath channel of the network and vmax is the maximum permissible velocity which is considered 6 m/s based on local standards (MGCE 2011a). Vf, N is flooding in Nth node of the network. NT and n are the number of nodes and critical bridge/culverts in the network, respectively. m and K 00 000 are the number of detention pond and axillary channels, respectively. C j and C k are unit cost of jth detention pond and kth axillary channel, respectively. Aj, HD, j, Lk and Dk are the area and depth of jth detention pond and length and diameters of kth axillary channel, respectively. Depth of detention ponds and diameter of axillary channels are also the other part of decision variables. The second objective function does not have an explicit mathematical form of decision variables and for objective function evaluation, running the hydraulic routing model is needed. Hydraulic routing model –i.e. SWMM model, solves the momentum and continuity differential equations and determines depth and velocity of the flood in each channel of the network at different times according to surface runoff hydrographs of sub-catchments which were previously obtained by hydrological model –i.e. HEC-HMS. SWMM simulation model was coupled with optimization algorithm in Matlab environment to jointly generate the values of decision variables and calculate objective functions. Because of the nonlinear and implicit form of objective functions, and on the other hand, discrete nature of the decision variables in the problem studied, the classical optimization technique cannot be used here (Yazdi 2016). For these categories of optimization problems, evolutionary algorithms are best-suited. Therefore, an MOEA, known as Non-dominated Sorting Differential Evolution (NSDE), is used here to solve the above optimization problem and to attain the optimal strategies of rehabilitation. This algorithm combines non-domination and crowding distance (Deb et al. 2002) with DE operators (Yazdi et al. 2016). It exhibited better performance across various benchmarks and engineering optimization problems,

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compared to other well-known MOEAs such as NSGA-II (e.g. see Reddy and Kumar 2006; Moosavian and Lence 2016).

2.5 Resilience-Based Methodology The main steps of proposed probabilistic method are demonstrated here. First, NSDE initializes a population, randomly. Each member of the population is a vector of decision variables representing the type and dimensions of rehabilitation interventions. To evaluate each member, decision variables are defined in the simulation model and then this model is run based on a significant number of blockage scenarios to calculate the second objective function in Eq. (4). For generating the blockage scenarios, based on the return period of design rainfall, expected number of blockages (λ) is determined. Then, the number of blockages, x is determined by sampling from the Poisson distribution. Afterwards, the number of x elements are randomly selected to be (partially) blocked among all critical elements according to their blockage probabilities. This task is carried out using the roulette wheel method in this research, but applying the other sampling methods from discrete distributions is also possible. The cross section area of the selected elements are assumed to be randomly blocked between 50 and 100%. Blockage percentages of less than 50% were neglected due to their lack of considerable effects on flow pattern. Each blockage sample is executed by the simulation model and network flooding is estimated and saved. After a significant number of model realizations, the expected value of network flooding is estimated as the second objective function. The evaluation of the current member is accomplished with calculating the other objective function. Then, next members of population is selected one after the other and evaluated in a similar procedure. Population is sorted based on non-domination and crowding distance, and after mutation and crossover tasks, it is updated. This process continue for a predefined number of iterations as the termination criterion. Non-dominated solutions in the last iteration is considered as the Pareto front solutions representing the tradeoff among the objective functions. Figure 1 illustrates different steps of the proposed methodology.

2.6 Case Study Tehran is the largest and most populous city in Iran. The main drainage network of Tehran has four distinct hydrological catchments. Due to the topology of the region and independent inlets/outlets of four catchments, there is a possibility for each of them to be modeled individually. The west catchment is selected as the study area here. This catchment with an approximate area of 156 km2 consists of 119 km open channels in which 8.8 km of the conduits suffer the lack of enough hydraulic capacity and flooding occurs in some parts of the network underlying the design rainfall with 50-year return period. According to rainfall-runoff modeling, the west area includes 42 sub-catchments and 132 conduits. Figure 2 shows subcatchments and the layout of main drainage network in EPA-SWMM model. Based on the BMaster Plan for Surface Water Management^ studies, one auxiliary tunnel and three detention ponds have been proposed in order to improve the network’s hydraulic capacity shortage (MGCE 2011a) in which their location is plotted in Fig. 2b. The reach of critical bottlenecks in the studied network is also showed in this figure. The bottlenecks include six bridge/culverts with small open area to pass the flow. These bottlenecks are likely to be blocked by sediments and debris loads carried with stormwater in the network and may cause urban flooding in surrounding areas (MGCE 2011a). Therefore, in this study, constructing parallel culverts/


Start

Generating the initial population randomly

Selecting the current member

Defining the current member in simulation model

Determining the number of blockage for the return period of design rainfall

Selecting the blocked elements by sampling from the blockage probabilities

Defining blocked elements in simulation model

Running the model and save the results

No

Is no. of samples enough?

Yes Evaluating objective functions of the current member

No

Are all membe rs selecte d?

Yes Sorting based on non-domination & crowding distance

Eliminating worse members

Is termin ation criterion met?

Yes

No Generating offspring by mutation & crossover

Fig. 1 Proposed methodology to find rehabilitation measures

End

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Fig. 2 (a) Western zone of Tehran main drainage network, (b) location of bottlenecks in hydraulic model

bridges is suggested to mitigate floods raised by structural failure of these critical culverts/ bridges.

2.7 Model Setting Up Design rainfall for Tehran drainage network is selected as a rainfall with 50-year return period and six hours duration. These values are confirmed by municipal authorities during the previous studies (Mahab Ghods Consultant Engineers 2011a, b). Modeling of the study area is based on a similar approach adopted by the Mahab Ghods Consultant Engineers (MGCE 2011a, b), and the same calibrated parameters are used to prepare the rainfall-runoff model, here. Based on the analysis of meteorological data recorded in nine different rainfall gauges in the city of Tehran, the following relation is proposed for the intensity of short-term rainfalls (MGCE 2011b): i ¼ C Alt:RP D−0:645

ð6Þ

where i = the rainfall intensity (mm/h), D = rainfall duration (min) and CAlt. RP = a coefficient related to the return period of design rainfall and mean height of the sub-catchment. The value of CAlt. RP is determined using a lookup table according to the rainfall return period and the average height of sub-catchment. To distribute rainfall depth across 6 h duration, different methods were tested and the Blocal pattern^ was found to be the most critical method in terms of surface runoff amounts. Therefore, this method is selected as the relevant temporal pattern of design rainfall.


For rainfall-runoff modeling by HEC-HMS, loss estimation method, converting the excess rainfall to the surface runoff hydrograph of sub-watersheds and routing the hydrographs in reaches were set to the SCS curve number method, Clark unit hydrograph and Muskingum method, respectively (USACE 2010). Based on the type of soil and land use, 42 subcatchments were considered for hydrological modeling. According to the land uses and soil types, totally 42 sub-catchments were considered in hydrological modeling. After calculating the surface runoff hydrograph of the sub-catchments, these hydrographs were introduced into nodes of drainage network in SWMM hydraulic model of the catchment, in order to routing the flood in the network channels. Instead of simplified methods of flood routing such as Muskingum (Barati 2011) and kinematic wave, full dynamic wave method which is based on numerical solving of Saint-Venant equations, was used in SWMM model to do the task of routing. Before running the coupled SWMM-NSDE algorithm, the parameter of NSDE was tuned and set as: population size = 100, scaling factor β =0.5, crossover probability, Pcr = 0.7 and no. of iterations = 50.

3 Results and Discussion Table 1 indicates the dimensions of bottlenecks in the west drainage network of Tehran city with their blockage probabilities, estimated using Eqs. (2) and (3). For each bottleneck, Ac and Qm, c are taken from the SWMM hydraulic model, Ac, m from the bottlenecks dimensions and Ls, c from the empirical relations between sediment loads and discharges, obtained by the analysis of historical records of sediment loads measured in the hydrometric gauges located at the tributaries of channels. The weighting factor, ‘a’ in Eq. (2) is also considered 0.75, in this research. The reason is that sediment load in different tributaries of the study area was almost the same (Table 1) and thus, the parameter ‘a’ is assumed 0.75 which means allocating the higher weight and higher distribution of the ‘filling ratio of crosssectional area’ for the blockage probability respect to the other factor –i.e. sediment loads. A sensitivity analysis has been done over the values of b and d in Eq. (2) and their values were changed ±50% to see their effects on blockage probability, Pb, and bottleneck priorities. The results have been presented in Table 2. It can be seen that the average change in bottleneck probability is around 14% and − 8% by a relatively large change in the parameter values while the rank of bottlenecks in terms of blockage probability will be still kept. Therefore, the considered value of ‘two’ for these parameters in Eq. (2) is acceptable. Table 1 Dimensions of critical bridge/culverts on the drainage system studied Bridge/Culvert No. Height (m) Width (m) Length (m) Deck thickness (m)

Ac Ac;m

1 2 3 4 5 6

3.3 5 5 5 4.5 5.5

6 7 7 7.5 9.5 5

10 14.5 6 330 22 8

0.4 0.5 0.4 1 0.4 0.4

0.48 0.96 0.93 0.48 0.73 0.25

Ls;c Qm;c

0.9 0.9 0.9 0.8 0.8 0.7

Blockage probability 0.12 0.27 0.27 0.10 0.18 0.05

Yazdi J. Table 2 Sensitivity analysis on values of parameters b and d (Eq. 2) and their effects on Pb Bottleneck no.

1 2 3 4 5 6 Average

Values of parameters b and d 2 (suggested value)

1

Change in Pb (%)

3

Change in Pb (%)

0.12 0.28 0.27 0.10 0.18 0.05

0.14 0.23 0.22 0.14 0.18 0.09

20 −18 −16 30 3 65 14

0.10 0.32 0.30 0.08 0.16 0.04

−14 15 12 −23 −9 −30 −8

Due to the high computational burden of SWMM executions using MCS, the number of 100 samples, randomly generated from the Poisson distribution, are deemed enough for constructing the blockage scenarios. The optimization model is coded and linked with SWMM in MATLAB environment. The running time of the coupled model is nearly 155 h in a computer with Intel® Core™2Duo CPU and 4.00GB RAM, to reach the Pareto optimal solutions. The linkage process is described. Based on the values of decision variables, within the code, geometric data of the studied network in input files of SWMM were modified. Then, executive file of SWMM was run (from the code). Finally, the required data for calculating objective functions was read from the output ASCII file of SWMM. Since, there was no reliable statistics or evidence about the number of blockages on the studied network during last decades, the expected number of blockages during a 50-year period –i.e. the return period of design rainfall, is assumed three different values of λ = 1, 2, 3 and in each scenario, the coupled model is run, separately. Figure 3 shows the Pareto optimal solutions found by SWMM-NSDE algorithm for three considered scenarios. With increasing λ, flooding in the network is expected to rise and Fig. 3 also confirms this, so as the network flooding has higher order of magnitude for larger λ s. Two clusters of solutions are observed in the tradeoff solutions of three scenarios. The reason for this separation is the existence of relatively highcost axillary tunnel (for the fourth bottleneck) in some of the solutions. This option has considerable cost because of the high length of the tunnel compared to similar options (see Table 1). Three solutions are selected from the scenario with largest floods (λ = 3) to analyze the results with more details. Selected solutions are called the low-cost design, median-cost design and highly-cost design presented in Table 3 and depicted with red-color circles in Fig. 3.

Flood Volume (1000 m3/s)

1400 1200

Pareto front, λ=3

1000

Pareto front, λ=2

800

Pareto front, λ=1 Selected solutions

600 400 200 0 0

200000

400000

600000

Costs ($US)

Fig. 3 Pareto front obtained by running NSDE-EPA SWMM framework

800000

1000000

Improving Urban Drainage Systems Resiliency Against Unexpected... Table 3 Selected solutions from the Pareto front Row N.

Strategy no.:

S1 (Low-cost)

S2 (median-cost)

S3 (highly-cost)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Height of Bypath 1 (m) Height of Bypath 2 (m) Height of Bypath 3 (m) Height of Bypath 4 (m) Height of Bypath 5 (m) Height of Bypath 6 (m) Width of Bypath 1 (m) Width of Bypath 2 (m) Width of Bypath 3 (m) Width of Bypath 4 (m) Width of Bypath 5 (m) Width of Bypath 6 (m) Height of Weir 1 (m) Height of Weir 2 (m) Height of Weir 3 (m) Height of Weir 4 (m) Height of Weir 5 (m) Height of Weir 6 (m) Length of Weir 1 (m) Length of Weir 2 (m) Length of Weir 3 (m) Length of Weir 4 (m) Length of Weir 5 (m) Length of Weir 6 (m) Storage Depth of Pond 1 (m) Storage Depth of Pond 2 (m) Storage Depth of Pond 3 (m) Axillary Channel Diameter (m) Total Costs ($US) Total flooding (1000 m3)

– – 4.5 – – – – – 5 – – – – – 2.9 – – – – – 3.6 – – – – – – – 16,611 1103

– 4 4 – 3.5 – – 9 7 – 5 – – 2.9 3.8 – 3.8 – – 4.5 5.4 – 3 – – – – – 118,443 267

5 5 4.5 3.5 3.5 – 10 5 8 5 6 – 1.1 2.6 2.3 4.05 4.7 – 6.9 6.6 4.8 3.6 3 – – – – – 890,940 0

Table 3 shows, only the axillary tunnel for the third bottleneck is emerged in the low-cost design. This bottleneck has the highest blockage probability (see Table 1) and thus, its choice seems to be reasonable compared to other alternatives when there is a limited level of available rehabilitation fund. As the available fund increases, the number of possible interventions also increases, so that in the second selected solution, two other axillary tunnels no. 2 and 5 are also added as the optimal actions. This observation is also consistent with the blockage probabilities whereas the interventions are suggested for the most three likely blocked bottlenecks. The third selected solution –i.e. the highly-cost one includes the parallelization of all bottlenecks except the sixth one which has the lowest blockage probability. The highly-cost design could almost remove flooding of the network resulted from the blockage. It is worth mentioning that the rehabilitation measures suggested by MGCE (including three detention ponds and one axillary channel) have had an expected value of flooding 1125 × 103 m3 with USD 1,585,186 costs. Compared to the optimal solutions, MGCE design has considerably higher order of costs with a little effect on reducing the flooding resulted from bottleneck blockage. An interesting observation is that the suggested measures by the consultant engineers’ proposal (three detention ponds and axillary channel) are not emerged in any optimal solution. From these observations, it can be figured out that conventional methods such as adding storage units and enlargement of system component dimensions are only efficient in the case of insufficient hydraulic capacity and do not have a tangible

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effect on flooding raised by internal loading in the system. Instead, making redundancy and adding elements with parallel functions could efficiently improve system resiliency over the internal loads and failures.

4 Conclusion In this paper, a probabilistic methodology is proposed to improve the resiliency of urban drainage systems underlying unexpected blockage incidents. The probabilistic approach is integrated with coupled SWMM-EPA model and NSDE algorithm to find the best set of rehabilitation measures when there are cost limitations. The results show that increasing the redundancy in the system by adding bypass lines in the bottlenecks could efficiently increase system resiliency against unexpected blockages while the ordinary methods such as detention ponds and enlarging the channel sizes are not effective; although they may have a good performance in the case of existing insufficient hydraulic capacity. It is notable that the proposed probabilistic approach only considered the unexpected blockages, but the other types of internal loads such as component malfunctioning or failure can also be easily included in the analyses by a similar manner. This research only focused on the internal loading while external flooding load is assumed equal to design rainfall. Considering the stochastic nature of external loads and their effects on producing unexpected internal problems such as blockage can be an interesting subject of the research in future. Acknowledgements This research has been supported by the research grant no. 600/1449 funded by Shahid Beheshti University, Tehran, Iran. The author would like to express his gratitude and thanks to Prof. Hewage and Prof. Sadiq, in School of Engineering, University of British Columbia, Kelowna, who read and edited carefully the final draft of the paper. The author also thanks Mr. Saeed Mohammadiun for helps in this research. Their effort is highly acknowledged.

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