## Abstract

An increase in stormwater frequency following the rapid development of urbanization has drawn attention to the mitigating strategies in recent decades. For the first time, the present study aims to conduct a local rehabilitation in stormwater collecting systems by (i) detecting the critical nodes along with the canal network and (ii) redesigning the critical canal reaches using ant colony optimization (ACO) to create maximum capacity for flood discharge with minimum reconstruction cost while considering the probability of exceedance of the flood as a constraint. Hence, using the SWMM model, the flow in the collection system was simulated, and the inundation points in the study area in the eastern Tehran metropolis were determined. After determining the critical points, the hydraulic stimulation model for the selected canal flows was developed using HEC-RAS software to accurately simulate each critical bridge's flow. Then, the optimal parameters for the canal bed width and canal depth were obtained using ACO and defining a probability objective function using the flood probability exceedance as the redesign constraint. The results from the optimizer were compared with those of the LINGO nonlinear model. Finally, the operational performance of the redesigned system was evaluated using the optimal selected parameters. The results showed that in redesigning the studied canals, the two widening and deepening options are needed to obtain a discharge with sufficient flow capacity in various return periods (RPs). The optimization results for the first to third critical sections for a design discharge with a 100-year RPs showed that the calculated cost was 19.765(*10^{6}), 13.327(*10^{6}), and 43.139(*10^{6}) IR rials (1 USD = 202000 IRR), respectively. For the selected sections, the optimal bed width is 6.97, 8.97, and 10.93 m, and the optimal depth is 3.68, 4.81, and 4.04 m, respectively. The results indicate that the local modification in the eastern flood control canal adequately improved inundation problem reduction in various RPs – i.e., for a 10-year RP, the number of node flooding dropped from 4 to zero, the inundated area from 17% to zero, and the overflow volume from (10–45) to zero. It also reduced overflow volume from (30–65), (43–74), and (70–92) in the status quo to (4–12), (11–27), and (24–36) percent for precipitations with 25, 50 and 100-year RPs, respectively.

## HIGHLIGHTS

Redesigning critical segments in stormwater collection system was conducted.

SWMM model was performed to identify the probable critical points along with the system.

A preprocessing analysis included field visiting, brainstorming sessions and hydraulic modelling with Hec-Ras was conducted.

The optimal redesigning of the critical points was conducted based on the flood exceedance probability via ACO and LINGO.

## INTRODUCTION

The urban flood is a common and widespread natural disaster in which financial and casualties and damages have dramatically increased with no sign of reduction. The number of casualties and the total cost of damages to properties, public infrastructure, and natural resources make it a long-term and frequent natural hazard, which remains to be until it is entirely eradicated (Chen *et al.* 2017). This type of flood is caused by a shortage in drainage canal network capacity in urban areas. In other words, it occurs when the precipitation in urban areas exceeds the runoff drainage system capacity (Tasca *et al.* 2018). Indeed, the development and growth of urbanization lead to an increase in impermeable surfaces, thereby causing a drastic change in the natural basin and creating a need for efficient drainage systems (Bertram *et al.* 2017). For instance, in India, land-use change due to the rising urban development from 16.64 to 44.08% led to the peak flood discharge increasing from 2.61 to 20.9% for precipitations with RPs of 200 to 2 years (Zope *et al.* 2017). In the UAE, with a four-time growth in built environments from 1976 to 2016, the runoff increased from 10 to 15 mm, indicating a 0.15 increase in the runoff coefficient (Shanableh *et al.* 2018). In a region in China, an increase in impermeable surfaces changed the annual runoff from 208% to 413%, with the annual flood volume ranging between 194 and 942 depending on the drainage system performance during the development period (Zhou *et al.* 2019). In another study in China, for a flood with a 30-year RP, the impact of change in precipitation and urbanization by 98.8 and 1.2%, respectively, led to a 1% increase in flood volume. For a flood with a one-year RP, with 42.4 and 57.6% changes in precipitation and urbanization, respectively, the flood volume increased by 24.4% (Bian *et al.* 2020). Research in Beijing also revealed that precipitation with a 100-year RP, the 14.52% increase in land use from 1984 to 1999, and the 9.7% from 1999 to 2009 led to 12.66 mm and 7.7 mm increases surface runoff depth, respectively. Meanwhile, with a 12.96% reduction in 2009–2019, a 12.18 mm reduction was observed in the average depth of surface runoff (Hu *et al.* 2020).

Strategies for reducing the risk of stormwater drainage network failure and inundation damages in urban areas have attracted much attention over the past few decades. These strategies consist of a range of structural, non-structural, and hybrid solutions that have substantially reduced the negative impact of this phenomenon in vulnerable urban areas (Lee & Kim 2018). The surface runoff control measures are recommended, designed, and implemented to achieve specific and limited objectives, such as (i) reducing runoff volume and peak discharge hydrograph at the basin and sub-basin drainage points, (ii) accelerating the recharge of underground waters, and increasing evaporation, and (iii) forecast and warning before floods (Hamel *et al.* 2013). For the first and second objectives, the following solutions can be proposed: implementing different types of low impact development (LID) in urban basins with sufficient execution space and social participation (Hua *et al.* 2020), increasing the capacity of open drainage systems, as well as rehabilitating and renovating stormwater drainage networks under development projects (Matos Silva & Costa 2016); building flood protection facilities such as reservoirs and establishing protection facilities in coastal urban areas such as sea walls, surge fences, levees, anti-backflow regulators, dikes, and building pumping stations (Martinez *et al.* 2021).

Regarding non-structural solutions for flood risk reduction, urban regulations revisions, development of efficient flood forecast-warning systems, and optimal management in the design, operation, and maintenance of floodwater drainage networks can be mentioned. It is necessary to explain that, in some cases, the results of the non-structural methods are either (i) influenced by experts' opinions or (ii) based on simulations using hydrological and hydrodynamic models. The expert opinions and inaccuracies due to the lack of accurate validated and calibrated hydrological and hydraulic data for simulation models lead to uncertainty in non-structural studies' results (Wu *et al.* 2020). Using the combination of structural and non-structural measures can be a valuable solution to prevent, reduce, prepare for, respond to, and recover from flooding hazards. Among such measures is the optimal design of stormwater drainage network capacity and its structures (Farzin & Valikhan Anaraki 2020), optimal operation of floodwater transfer and control structures (Seifert & Moore 2018), design of a real-time automated system for the stormwater drainage pumping station, multi-objective optimization of reservoir management, and operation (Meng *et al.* 2019), and optimization models to determine sustainable development policies and structural measures planning (Nones 2015).

A literature review shows that many structural, non-structural, and hybrid solutions are used depending on the topography of the region and technical, social, economic, and environmental considerations. Moreover, this study investigates the effectiveness of local redesign of urban storm sewer canals as an economical option in rehabilitation projects. A significant fact about canal design parameters such as design discharge, the longitudinal slope of the canal, and roughness coefficients is that these parameters are probabilistic. For example, parameters such as the roughness coefficients and longitudinal bed slope may change over time compared to the design values for reasons such as erosion or deposition. Similarly, the flow discharge may exceed the flow rate considered in the canal design due to unexpected rainfall. This results in uncertainty in determining the flow depth and overflowing or flooding the canal (Sankaran & Manne 2013; Farzin & Valikhan Anaraki 2020). Therefore, considering the uncertainties in the parameters, many studies have been done to design open canals with different objectives and purposes. For instance, Farzin & Valikhan Anaraki (2020) used the metaheuristic algorithms – BA, PSO, and a combination of the two, HBP – to design a canal with minimum construction cost and the flood probability as the constraint, which results in a 32% reduction in the cost. Gupta *et al.* (2018) used the fish swarm optimization algorithm to optimize a trapezoidal canal. The results suggested a reduction in the construction costs compared to the PSO results. Orouji *et al.* (2016) used the shuffled frog leaping algorithm (SFLA), which reduced the canal construction costs compared to other algorithms. A significant point in the above studies is the optimal redesign of the flood control canal capacities in rehabilitation projects. However, since in many dense urban areas, the total renovation of the drainage canal network, widening or deepening, is limited by financial and spatial constraints, identifying the problematic canals and locally redesigning and renovating them is a practical and effective option (Mel *et al.* 2020). Therefore, using optimization tools to locally redesign the canals by considering flood probability as the constraint allows for the maximum flow capacity for flood discharge with the minimum reconstruction cost. In this study, while there are canals in the study area for conveying flood discharge, they are not suited for discharges with other probabilities exceedance due to inadequate capacity. Therefore, the study attempts to increase the discharge flow compared to the design discharge by accounting for the flood probabilities exceedance in the canal parameters and improving the flow capacity in sections of the selected canals from the stormwater drainage network.

For this purpose, after reviewing the studies on the optimal design of canal networks' capacity, the ant colony metaheuristic algorithm was used. It has been used to solve a wide range of single and multi-objective water resources management problems in recent years (Afshar & Moeini 2008). The algorithm was chosen because of its application and success in the optimal multi-purpose design of agricultural water conveying canals (Lord *et al.* 2021), water distribution system design (Hajibandeh & Nazif 2018), irrigation management (Nguyen *et al.* 2016), urban sewer and drainage systems (Moeini & Afshar 2019), optimal operation of reservoirs and surface water management (Afshar *et al.* 2015), groundwater management (Ketabchi & Ataie-Ashtiani 2015), and hybrid use of surface and groundwater resources (Safavi & Enteshari 2016), and environmental issues (Karri *et al.* 2020). According to the above research, the ant colony algorithm has performed successfully as a useful and efficient method in many issues. However, few studies have applied the methods in canal design by accounting for the uncertainty of design parameters and the flood probability as the constraint. Considering the above points, the general purpose of this study is to employ the ACO algorithm in the optimal redesign of the flow capacity and local rehabilitation of flood control canals in dense and old urban areas where other structural and non-structural flood control methods such as LID are not possible. Furthermore, the detailed objectives of the study, according to the flowchart presented in Figure 1, are:

Simulating the flow in the surface runoff drainage network and determining the inundation points in the study area in the old and dense texture of eastern Tehran metropolis using the SWMM sorted by precipitation with specific RPs.

Developing the hydraulic model of flow in the selected canals in the previous step using the HEC-RAS software and hydraulic flow stimulation to simulate the state of flow in each critical bridge accurately.

Developing the ACO by defining a probabilistic objective function with the flood probability exceedance as the constraint in redesigning and determining the optimal canal width and bed depth parameters.

Evaluating the ACO results on the optimal redesign of the selected canals based on LINGO, a classical nonlinear optimization method.

Simulating the area's stormwater drainage network performance after optimally redesigning the selected canals in the fourth step for precipitation with the specific RP and evaluating the network performance after using the selected optimal parameters.

## MATERIALS AND METHODS

### Study area

Tehran's main drainage network has to collect the runoff from the mountains overlooking Tehran and the drainage basins outside and inside the city and transfer them out of the city limits. To analyze Tehran's main drainage system and evaluate its hydraulic capacity, it was divided into four areas. The first area, the eastern flood diversion canal (part of the north, northeast, east, and southeast of Tehran), the second, the western flood diversion canal (part of the north, northwest, west, and southwest of Tehran), the third, the central and southern Tehran; and the fourth, west of the Kan creek (districts 21 and 22 of Tehran municipality) (Movahedinia *et al.* 2019). The study area is the main floodway of the eastern flood diversion canal in the first area. The main eastern flood diversion network's basin (about 207 km) comprises the main canals Velenjak, Maghsoud Beyk, Darabad, Tehranpars, Sorkheh Hesar, 17 Shahrivar, Shahrzad, and eastern flood diversion (Bakhtr, Manouchehri, Abouzar, and Baroutkoubi). These canals and tunnels — except for the 17 Shahrivar tunnel, the Sharzad canal, and its headstreams which transfer the urban runoffs — transfer the runoff from the watersheds overlooking the city. Except for the Sorkheh Hesar canal, which has a west–east direction with a low slope in its first stage, the canals leading to the eastern flood diversion have a north–south direction extending toward the dominant slope of the city bases on the hydraulic analysis report of the current surface water management. The main network's rivers and streams cross through districts 1, 3, 4, 7, 8, 12, 13, 14, 15, and 20 in Tehran. Figure 2 shows the study area.

A review of concentration times within Tehran basins (TehranMunicipality 2012; Movahedinia *et al.* 2019) indicated that, under all circumstances, the sum of total concentration times of the individual basins and the concentration time required for water flow through the flooding channel to reach the outlet was less than 6 h. Thus, the rainfall runoff in the model was duly calculated, and the local rainfall pattern was obtained based on the intensity-duration-frequency (IDF) curves across Tehran, using the alternating block method (depicted in Figure 3). Based on the results of previous studies on precipitation in Tehran, rainfall is most severe towards the middle of the rainy period (October to April), for events taking 3 h and more. This fact was included in the development of the local rainfall pattern for Tehran. The runoff hydraulics were simulated for return periods of 10, 25, 50, and 100 years given the importance of floods in the design of hydraulic structures, particularly the surface water collection channels, and given the urban considerations concerning Tehran District 13 (based on the rainfall data available at weather stations). The Horton equations were employed to determine the infiltration rate.

### SWMM simulation model

^{−1}); E is the evapotranspiration rate (m sec

^{−1}); F (m sec

^{−1}) is the rate of infiltration, and q is water flow in each sub-basin per unit area (m sec

^{−1}) obtained through dividing the flow rate from the Manning equation (Equation (2)) by the surface area of the same sub-basin (As):where W is the width of the sub-area (m); n is Manning's roughness coefficient; (m) is the depth of pothole storage, and S is the slope of the sub-basin. Equation (3) is obtained by combining the above two equations, which is a nonlinear reservoir equation, and by solving that unknown parameter d will be obtained, which is shown in equation three as below:

These equations are solved by numerical analysis in the SWMM model.

### HEC-RAS hydraulic model

In Equation (7), n is the Manning's roughness coefficient, A section area, and R the hydraulic radius.

### Ant colony optimization

#### Ant system (AS)

*m*, first. The second step is to place each ant on an initial decision from which it starts moving. In each iteration of the process, the probability of a path is calculated as follows:where shows the probability for the decision

*j*at the point

*i*to be taken by the ant

*k*in the

*t*th iteration. is the amount of pheromones replaced on decision in the

*t*th iteration. is the heuristic value representing the local cost of choosing option j at point i. The parameters

*α*and

*β*control the relative weight of the pheromone trail and the heuristic value.

*j*at decision point

*i*in the (

*t*+ 1)th iteration. Further, is the pheromone concentration of decision in the

*t*th iteration. The parameter (0) is the pheromone evaporation factor and is obtained from the following equation.where is the number of ants (components of the ACO population), is the cost of the solution made by the

*k*th ant in th iteration, and

*Q*is the pheromone reward factor.

#### The rank-based ant system ()

*et al.*(1999), and develops the idea of elitism to involve a rank-based updating scheme. In the pheromone process, the top ranked solutions receive a pheromone addition rather than the solutions from the entire colony. The decision rule in the algorithm is the same as a rule used in the AS algorithm (Equation (8)). The updating process of the pheromone rule is also the same as a rule used in the AS algorithm, except that is used instead of which is obtained from the following equation (Yin

*et al.*2020): is the number of elitist ants.

For complex problems such as the model presented in the present study (which involves the particular uncertainty of various parameters), using traditional methods to achieve optimal solutions may not be easy because several approximations or simplifications or derived information about model functions are required. Therefore, to solve completely nonlinear, non-convergent, multi-state problems with complex functions, instead of using traditional methods, heuristic methods such as ACO can be used to provide acceptable solutions to the problem. In addition to being easy to understand, this method provides the possibility of linking to existing simulation models, suitable for solving complex mathematical problems and providing several almost optimal solutions, especially for problems with a correlation of decision variables. Examples include scheduling and allocating problems (Reddy & Adarsh 2010; Lord *et al.* 2021).

#### Evaluating the ACO-based model

By comparing the ACO results with LINGO results, the developed model was evaluated and validated. The nonlinear solver relies on successive linear programming for calculating new trails in the LINGO model. Although this method uses an increased number of iterations, a linear approximation is used in each iteration to accelerate the calculations (Lord *et al.* 2021).

The optimization process and the comparison of the evaluation results with nonlinear Lingo are shown in Figure 4.

#### ACO parameters setting

A sensitivity analysis was conducted to optimize the parameters of the ACO to control the algorithm's search behavior, comprising the number of ants, iterations, *α*, , *ρ*, and Qu. In order to accelerate the convergence and not get stuck in the local optimal solutions and reach the optimal global solutions, the value ranges of these parameters in the form of Table 1 are recommended using the values proposed in previous research (Nguyen *et al.* 2016; Lord *et al.* 2021). According to the table, while one of the parameters was tested, others stay constant, and this process will be continued until all the parameters were determined (Yousefikhoshbakht *et al.* 2016; Lord *et al.* 2021).

Parameters . | Values for sensitivity analysis . |
---|---|

Number of ants | 10, 20, 30, 40, 60, 80 |

Number of iteration | 10, 30, 60, 80, 100, 120, 150, 200 |

Alpha (α) | 0.5, 0.6, 0.7, 0.8, 0.9, 1 |

Beta () | 0, 0.01, 0.1, 1 |

Pheromone persistence (ρ) | 0.01, 0.03, 0.04, 0.06, 0.08, 0.09, 0.1 |

Pheromone reward (Qu) | 0.8, 0.9, 1, 1.5 |

Sigma () | 3, 6, 9, 12, 15 |

Parameters . | Values for sensitivity analysis . |
---|---|

Number of ants | 10, 20, 30, 40, 60, 80 |

Number of iteration | 10, 30, 60, 80, 100, 120, 150, 200 |

Alpha (α) | 0.5, 0.6, 0.7, 0.8, 0.9, 1 |

Beta () | 0, 0.01, 0.1, 1 |

Pheromone persistence (ρ) | 0.01, 0.03, 0.04, 0.06, 0.08, 0.09, 0.1 |

Pheromone reward (Qu) | 0.8, 0.9, 1, 1.5 |

Sigma () | 3, 6, 9, 12, 15 |

### Objective function

*et al.*2008). The method is used to estimate the uncertainty of a deterministic model with several parameters with unspecified values (and uncertainty). Moreover, the mean and variance of a random variable that itself is a function of k random variables can be explained as follows (Te Chow

*et al.*1962):

#### Constraints

*Q*is the design flow, the standard deviation of the parameter , and the normal distribution, which to illustrate the above probabilistic constraint can be written as an equality constraint in terms of standard normal variable as follows (Reddy & Adarsh 2010):

#### Variable decision

## RESULTS AND DISCUSSION

### Stormwater collection system operational performance in the status quo

The hydrologic and hydraulic simulation of the status quo in the study area, based on available primary data, data analysis, and field visits, was performed using the SWMM. Since the eastern flood diversion's storm sewer and the main canals leading to it are tasked with collecting and transferring mountain and urban catchment runoffs, the 25-year flood hydrograph was used as the basis for evaluating the area's capacity. Meanwhile, precipitations with 10, 50, and 100-year RPs were also used to assess and control the network's behavior under the current state and determine the public aspects of flood control in urban areas. Then the 6 h design precipitation with the above RPs was calculated and executed. The eastern flood diversion's storm sewer is covered with concrete, and the maximum allowable speed in this canal is 6 m/s. The results from hydraulic calculations in the study area for the present conditions and the 25-year flood showed that the main canal does not have enough capacity for the flow, and water overflew in eight nodes of the eastern flood diversion's main canal. Comparing the records on the inundation locations during floods with the simulation results confirmed these points' risky and problematic nature. The modeling results also showed that out of 207 km of the main network located in the eastern flood diversion catchment, about 54 km have the insufficient hydraulic capacity, mainly in the eastern flood diversion. Analysis of the hydraulic results revealed that the canal's hydraulic power decreased and increased periodically along the way.

After examination, the eastern flood diversion was found to have (critical and non-critical) bottlenecks in many places. This was confirmed by the modeling results as well as the water escapes and land inundation at the eight nodes (bridges or bottlenecks) from its northernmost (Khajeh Abdollah Bridge) to the southernmost (Mahallati Bridge). To explain this better, the hydraulic modeling results of a flood with a 50-year RP at these eight bridges are compared to determine the most critical. These bottlenecks in order of location in the main floodway (from north to south) are bridge one (Khajeh Abdollah) with an 80 m^{3}/s discharge, bridges two and three (Izadi and Janbazan) with a 136 m^{3}/s discharge, bridges four and five (Hoseini and Golestan) with a 162 m^{3}/s discharge, bridges six and seven (Shora and Jafari) with a 167 m^{3}/s discharge, and finally, bridge eight (Mahallati) with a 168 m^{3}/s. The modeling results showed that bridges two to five and seven have enough capacity to allow for the flood discharge with no sign of insufficiency for the size and discharge. The remaining three bridges (bridges one, six, and eight) show a substantial capacity insufficiency. The insufficiency in bridge one is about 75% which is in line with the modeling results, which shows land inundations. This amount was estimated at around 45% for bridge six and 41% for bridge eight. Therefore, hydraulic modeling results using SWMM determined the critical bottlenecks along the eastern flood diversion's floodway, and these three bridges were taken as the basis for subsequent calculations, which are indicated in Figure 5.

### Results of Hec-Ras hydraulic model

To assess more accurately the hydraulic performance of these points, the hydraulic flow simulation for the three selected nodes was carried out and sorted by floods with 10, 25, 50, and 100-year RPs. After presenting the modeling results, by holding brainstorming sessions with urban flood management experts in Tehran municipality, the three critical bottlenecks (bridges one, six, and eight) were approved as renovation and rehabilitation options. The results of the hydraulic flow simulation using SWMM and those of the brainstorming sessions were used as the basis for the field study in this research. Therefore, field visits were planned for each of the bridges one, six, and eight, including field inspections, measurements, and data collection.

Figure 6 illustrates the current state of the three critical bridges and upstream canals. Field inspections revealed that the leading causes of damage to these bridges were (1) inattention to hydraulic criteria in design, execution, and maintenance, (2) improper location of bridges with no regard to the upstream and downstream winding course, (3) incorrect estimation of the bedding depth (in terms of structural and geotechnical criteria) irrespective of erosion and scouring, (4) shortcomings and negligence in maintenance, and (5) blocking a portion of the canal's useful capacity storm canals by bridges. In line with experts' recommendations in brainstorming sessions, to ensure that the selected bridges were critical, the HEC-RAS hydraulic flow simulation model was used to accurately simulate the flow for each critical bridge based on the design discharge with a 50-year RP. Therefore, the field measurement was done, which included precise estimation of the course roughness, surveying the upstream and downstream of the canals, accurate measurement of the bridge sizes, and upstream and downstream reaches of the stormwater canal near the bridges. Cross-sections were determined to develop the critical points model in HEC-RAS based on field observations and the as-built plans of the canals (analyzed and revised in the field visits). Since the flow parameters such as water surface profiles are affected by the flow's upstream and downstream, areas broader than the critical points were modeled. The roughness coefficients in the selected ranges were determined based on the studies under the comprehensive plan of Tehran's surface runoff within the range of the Manning roughness coefficient for the canal with the present condition between 0.015 and 0.021. However, since the erosion in the open canals is an effective factor on the roughness coefficient, and the canals are covered with cement and floodplains with asphalt, the roughness coefficients were on average revised as 0.018 for the channel and 0.025 for the floodplains based on the field observations and experts' opinions. The HEC-RAS model's options for defining the boundary conditions are the specific water level, the depth in the critical model, the normal depth, and the stage-discharge curve. In this model, the normal depth method was used to introduce upstream and downstream boundary conditions. For this purpose, the energy line slope of the canal upstream and downstream should be introduced to the model, and if this parameter is not available, the bed slope of the canal can be used with an adequate estimation. Based on the field survey, the bed slope of the three canals at the upstream and downstream was considered 0.01 in the study range. Moreover, the geometrical properties of the bridge – i.e., the width, shape, and size of the piers, the distance between piers, abutment shape, slab thickness, and the distance between the river bed and the slab – were also measured.

### Optimization results

The ant colony algorithm was executed several times to determine each ACO parameter for objective function (e.g., the number of the ants, the number of iterations, Qu, *ρ*, , *α*, and ) whose results are presented in Table 2. Table 2 also includes the results of 10 random executions for AS and algorithms for each probability value of exceedance, sorted by the three critical points in the study area. In this study, the LINGO software was used as a nonlinear optimization method to obtain a final optimal design for the open canal. According to the results, in each probability exceedance, the execution closer to the optimal solution obtained from LINGO was selected as the suitable optimal solution. Another critical point is that the optimal solutions are closer to the LINGO results than AS's.

. | . | . | Parameters setting . | No. of runs . | . | . | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

. | . | . | . | . | M . | . | Qu . | Sigma . | 1 . | 2 . | 3 . | 4 . | 5 . | 6 . | 7 . | 8 . | 9 . | 10 . | Best . | LINGO . |

First canal | 0.01 | AS | 0.9 | 0.09 | 20 | 0.1 | 1.5 | – | 20.101 | 20.101 | 20.061 | 20.061 | 20.101 | 20.141 | 20.061 | 20.141 | 20.061 | 20.141 | 20.061 | 19.724 |

0.9 | 0.09 | 10 | 0.1 | 1.5 | 6 | 19.781 | 19.733 | 19.765 | 19.733 | 19.765 | 19.765 | 19.733 | 19.781 | 19.733 | 19.765 | 19.765 | ||||

0.013 | AS | 0.7 | 0.08 | 10 | 0.01 | 0.95 | – | 17.303 | 18.296 | 18.296 | 18.296 | 18.303 | 18.303 | 18.303 | 18.296 | 18.303 | 18.303 | 18.296 | 17.618 | |

0.9 | 0.08 | 10 | 0 | 0.95 | 3 | 17.681 | 17.681 | 17.737 | 17.681 | 17.681 | 17.737 | 17.681 | 17.737 | 17.681 | 17.681 | 17.681 | ||||

0.02 | AS | 0.9 | 0.03 | 10 | 0 | 0.8 | – | 15.300 | 15.300 | 15.301 | 15.301 | 15.294 | 15.300 | 15.301 | 15.294 | 15.294 | 15.294 | 15.294 | 15.187 | |

1 | 0.06 | 10 | 0 | 0.95 | 6 | 15.201 | 15.187 | 15.201 | 15.201 | 15.187 | 15.187 | 15.187 | 15.187 | 15.201 | 15.187 | 15.266 | ||||

0.04 | AS | 1 | 0.06 | 10 | 0 | 0.8 | – | 10.532 | 10.532 | 10.529 | 10.529 | 10.532 | 10.532 | 10.529 | 10.529 | 10.529 | 10.532 | 10.529 | 10.385 | |

1 | 0.06 | 10 | 0.1 | 1.5 | 6 | 10.459 | 10.456 | 10.459 | 10.456 | 10.459 | 10.459 | 10.459 | 10.459 | 10.456 | 10.456 | 10.456 | ||||

0.1 | AS | 0.7 | 0.01 | 10 | 0 | 0.8 | – | 6.345 | 6.323 | 6.345 | 6.345 | 6.375 | 6.323 | 6.323 | 6.375 | 6.323 | 6.323 | 6.323 | 6.039 | |

1 | 0.06 | 10 | 0.1 | 1 | 6 | 10.131 | 10.131 | 10.039 | 10.131 | 10.039 | 10.039 | 10.039 | 10.039 | 10.039 | 10.039 | 6.094 | ||||

Second canal | 0.01 | AS | 0.5 | 0.01 | 10 | 0.01 | 0.95 | – | 13.792 | 13.785 | 13.792 | 13.785 | 13.785 | 13.785 | 13.792 | 13.785 | 13.785 | 13.792 | 13.785 | 13.241 |

1 | 0.09 | 10 | 0.1 | 1 | 6 | 13.327 | 13.327 | 13.330 | 13.327 | 13.327 | 13.327 | 13.330 | 13.327 | 13.327 | 13.327 | 13.327 | ||||

0.013 | AS | 0.9 | 0.08 | 10 | 0 | 0.8 | – | 10.989 | 10.894 | 10.989 | 10.989 | 10.989 | 10.984 | 10.989 | 10.989 | 10.989 | 10.984 | 10.894 | 10.268 | |

0.6 | 0.09 | 20 | 0 | 0.95 | 9 | 10.363 | 10.363 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | ||||

0.02 | AS | 1 | 0.09 | 10 | 0 | 1 | – | 8.054 | 8.049 | 8.055 | 8.055 | 8.055 | 8.055 | 8.049 | 8.054 | 8.049 | 8.054 | 8.049 | 7.473 | |

1 | 0.09 | 10 | 0 | 0.95 | 9 | 7.554 | 7.554 | 7.554 | 7.549 | 7.549 | 7.549 | 7.549 | 7.554 | 7.549 | 7.549 | 7.549 | ||||

0.04 | AS | 1 | 0.08 | 10 | 0 | 0.8 | – | 3.803 | 3.796 | 3.803 | 3.796 | 3.796 | 3.796 | 3.796 | 3.806 | 3.803 | 3.803 | 3.796 | 3.483 | |

1 | 0.06 | 10 | 0 | 0.95 | 9 | 3.507 | 3.522 | 3.507 | 3.507 | 3.522 | 3.507 | 3.507 | 3.507 | 3.522 | 3.522 | 3.507 | ||||

0.1 | AS | 0.5 | 0.08 | 10 | 0.1 | 0.8 | – | 1.669 | 1.687 | 1.684 | 1.687 | 1.684 | 1.684 | 1.684 | 1.687 | 1.669 | 1.669 | 1.669 | 1.557 | |

0.9 | 0.09 | 10 | 0.01 | 0.95 | 6 | 1.564 | 1.567 | 1.564 | 1.564 | 1.564 | 1.564 | 1.567 | 1.564 | 1.564 | 1.564 | 1.564 | ||||

Third canal | 0.01 | AS | 0.7 | 0.08 | 10 | 0 | 0.8 | – | 43.479 | 43.469 | 43.479 | 43.469 | 43.458 | 43.479 | 43.479 | 43.469 | 43.458 | 43.469 | 43.458 | 42.770 |

1 | 0.06 | 10 | 0 | 0.95 | 3 | 43.139 | 43.144 | 43.139 | 43.144 | 43.139 | 43.139 | 43.139 | 43.139 | 43.139 | 43.139 | 43.139 | ||||

0.013 | AS | 0.9 | 0.09 | 10 | 0 | 0.8 | 0.5 | 39.172 | 39.193 | 39.193 | 39.183 | 39.193 | 39.172 | 39.172 | 39.172 | 39.172 | 39.172 | 39.172 | 38.570 | |

1 | 0.08 | 20 | 0 | 0.95 | 9 | 38.630 | 38.630 | 38.610 | 38.630 | 38.610 | 38.630 | 38.630 | 38.630 | 38.630 | 38.610 | 38.610 | ||||

0.02 | AS | 0.9 | 0.08 | 10 | 0 | 0.8 | – | 33.152 | 33.152 | 33.154 | 33.152 | 33.154 | 33.152 | 33.158 | 33.152 | 33.158 | 33.152 | 33.152 | 32.660 | |

1 | 0.08 | 10 | 0 | 0.95 | 9 | 33.127 | 33.126 | 33.126 | 33.127 | 33.126 | 33.127 | 33.126 | 33.126 | 33.126 | 33.127 | 33.126 | ||||

0.04 | AS | 0.7 | 0.04 | 10 | 0 | 0.8 | – | 23.947 | 23.950 | 23.950 | 23.946 | 23.946 | 23.952 | 23.952 | 23.952 | 23.946 | 23.946 | 23.946 | 23.448 | |

1 | 0.08 | 10 | 0 | 1 | 6 | 23.606 | 23.606 | 23.606 | 23.606 | 23.607 | 23.606 | 23.606 | 23.607 | 23.606 | 23.606 | 23.606 | ||||

0.1 | AS | 0.9 | 0.01 | 10 | 0 | 0.8 | 0.5 | 16.748 | 16.746 | 16.748 | 16.748 | 16.746 | 16.746 | 16.746 | 16.748 | 16.748 | 16.748 | 16.746 | 16.408 | |

0.01 | 10 | 0 | 6 | 16.425 | 16.425 | 16.419 | 16.425 | 16.419 | 16.419 | 16.419 | 16.419 | 16.419 | 16.419 | 16.419 |

. | . | . | Parameters setting . | No. of runs . | . | . | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

. | . | . | . | . | M . | . | Qu . | Sigma . | 1 . | 2 . | 3 . | 4 . | 5 . | 6 . | 7 . | 8 . | 9 . | 10 . | Best . | LINGO . |

First canal | 0.01 | AS | 0.9 | 0.09 | 20 | 0.1 | 1.5 | – | 20.101 | 20.101 | 20.061 | 20.061 | 20.101 | 20.141 | 20.061 | 20.141 | 20.061 | 20.141 | 20.061 | 19.724 |

0.9 | 0.09 | 10 | 0.1 | 1.5 | 6 | 19.781 | 19.733 | 19.765 | 19.733 | 19.765 | 19.765 | 19.733 | 19.781 | 19.733 | 19.765 | 19.765 | ||||

0.013 | AS | 0.7 | 0.08 | 10 | 0.01 | 0.95 | – | 17.303 | 18.296 | 18.296 | 18.296 | 18.303 | 18.303 | 18.303 | 18.296 | 18.303 | 18.303 | 18.296 | 17.618 | |

0.9 | 0.08 | 10 | 0 | 0.95 | 3 | 17.681 | 17.681 | 17.737 | 17.681 | 17.681 | 17.737 | 17.681 | 17.737 | 17.681 | 17.681 | 17.681 | ||||

0.02 | AS | 0.9 | 0.03 | 10 | 0 | 0.8 | – | 15.300 | 15.300 | 15.301 | 15.301 | 15.294 | 15.300 | 15.301 | 15.294 | 15.294 | 15.294 | 15.294 | 15.187 | |

1 | 0.06 | 10 | 0 | 0.95 | 6 | 15.201 | 15.187 | 15.201 | 15.201 | 15.187 | 15.187 | 15.187 | 15.187 | 15.201 | 15.187 | 15.266 | ||||

0.04 | AS | 1 | 0.06 | 10 | 0 | 0.8 | – | 10.532 | 10.532 | 10.529 | 10.529 | 10.532 | 10.532 | 10.529 | 10.529 | 10.529 | 10.532 | 10.529 | 10.385 | |

1 | 0.06 | 10 | 0.1 | 1.5 | 6 | 10.459 | 10.456 | 10.459 | 10.456 | 10.459 | 10.459 | 10.459 | 10.459 | 10.456 | 10.456 | 10.456 | ||||

0.1 | AS | 0.7 | 0.01 | 10 | 0 | 0.8 | – | 6.345 | 6.323 | 6.345 | 6.345 | 6.375 | 6.323 | 6.323 | 6.375 | 6.323 | 6.323 | 6.323 | 6.039 | |

1 | 0.06 | 10 | 0.1 | 1 | 6 | 10.131 | 10.131 | 10.039 | 10.131 | 10.039 | 10.039 | 10.039 | 10.039 | 10.039 | 10.039 | 6.094 | ||||

Second canal | 0.01 | AS | 0.5 | 0.01 | 10 | 0.01 | 0.95 | – | 13.792 | 13.785 | 13.792 | 13.785 | 13.785 | 13.785 | 13.792 | 13.785 | 13.785 | 13.792 | 13.785 | 13.241 |

1 | 0.09 | 10 | 0.1 | 1 | 6 | 13.327 | 13.327 | 13.330 | 13.327 | 13.327 | 13.327 | 13.330 | 13.327 | 13.327 | 13.327 | 13.327 | ||||

0.013 | AS | 0.9 | 0.08 | 10 | 0 | 0.8 | – | 10.989 | 10.894 | 10.989 | 10.989 | 10.989 | 10.984 | 10.989 | 10.989 | 10.989 | 10.984 | 10.894 | 10.268 | |

0.6 | 0.09 | 20 | 0 | 0.95 | 9 | 10.363 | 10.363 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | 10.362 | ||||

0.02 | AS | 1 | 0.09 | 10 | 0 | 1 | – | 8.054 | 8.049 | 8.055 | 8.055 | 8.055 | 8.055 | 8.049 | 8.054 | 8.049 | 8.054 | 8.049 | 7.473 | |

1 | 0.09 | 10 | 0 | 0.95 | 9 | 7.554 | 7.554 | 7.554 | 7.549 | 7.549 | 7.549 | 7.549 | 7.554 | 7.549 | 7.549 | 7.549 | ||||

0.04 | AS | 1 | 0.08 | 10 | 0 | 0.8 | – | 3.803 | 3.796 | 3.803 | 3.796 | 3.796 | 3.796 | 3.796 | 3.806 | 3.803 | 3.803 | 3.796 | 3.483 | |

1 | 0.06 | 10 | 0 | 0.95 | 9 | 3.507 | 3.522 | 3.507 | 3.507 | 3.522 | 3.507 | 3.507 | 3.507 | 3.522 | 3.522 | 3.507 | ||||

0.1 | AS | 0.5 | 0.08 | 10 | 0.1 | 0.8 | – | 1.669 | 1.687 | 1.684 | 1.687 | 1.684 | 1.684 | 1.684 | 1.687 | 1.669 | 1.669 | 1.669 | 1.557 | |

0.9 | 0.09 | 10 | 0.01 | 0.95 | 6 | 1.564 | 1.567 | 1.564 | 1.564 | 1.564 | 1.564 | 1.567 | 1.564 | 1.564 | 1.564 | 1.564 | ||||

Third canal | 0.01 | AS | 0.7 | 0.08 | 10 | 0 | 0.8 | – | 43.479 | 43.469 | 43.479 | 43.469 | 43.458 | 43.479 | 43.479 | 43.469 | 43.458 | 43.469 | 43.458 | 42.770 |

1 | 0.06 | 10 | 0 | 0.95 | 3 | 43.139 | 43.144 | 43.139 | 43.144 | 43.139 | 43.139 | 43.139 | 43.139 | 43.139 | 43.139 | 43.139 | ||||

0.013 | AS | 0.9 | 0.09 | 10 | 0 | 0.8 | 0.5 | 39.172 | 39.193 | 39.193 | 39.183 | 39.193 | 39.172 | 39.172 | 39.172 | 39.172 | 39.172 | 39.172 | 38.570 | |

1 | 0.08 | 20 | 0 | 0.95 | 9 | 38.630 | 38.630 | 38.610 | 38.630 | 38.610 | 38.630 | 38.630 | 38.630 | 38.630 | 38.610 | 38.610 | ||||

0.02 | AS | 0.9 | 0.08 | 10 | 0 | 0.8 | – | 33.152 | 33.152 | 33.154 | 33.152 | 33.154 | 33.152 | 33.158 | 33.152 | 33.158 | 33.152 | 33.152 | 32.660 | |

1 | 0.08 | 10 | 0 | 0.95 | 9 | 33.127 | 33.126 | 33.126 | 33.127 | 33.126 | 33.127 | 33.126 | 33.126 | 33.126 | 33.127 | 33.126 | ||||

0.04 | AS | 0.7 | 0.04 | 10 | 0 | 0.8 | – | 23.947 | 23.950 | 23.950 | 23.946 | 23.946 | 23.952 | 23.952 | 23.952 | 23.946 | 23.946 | 23.946 | 23.448 | |

1 | 0.08 | 10 | 0 | 1 | 6 | 23.606 | 23.606 | 23.606 | 23.606 | 23.607 | 23.606 | 23.606 | 23.607 | 23.606 | 23.606 | 23.606 | ||||

0.1 | AS | 0.9 | 0.01 | 10 | 0 | 0.8 | 0.5 | 16.748 | 16.746 | 16.748 | 16.748 | 16.746 | 16.746 | 16.746 | 16.748 | 16.748 | 16.748 | 16.746 | 16.408 | |

0.01 | 10 | 0 | 6 | 16.425 | 16.425 | 16.419 | 16.425 | 16.419 | 16.419 | 16.419 | 16.419 | 16.419 | 16.419 | 16.419 |

In Figure 7, the changes in the decision parameters and cost generated by AS and algorithms for various probabilities of exceedance are presented for the Khajeh Abdollah Bridge canal. It is seen that in both methods, as the discharge probability exceedance increases, the cost and the decision parameters decrease. The canal has the highest construction cost with the probability exceedance at 0.01. The canal construction costs using the algorithms and AS is, respectively, at 19.765 (*10^{6}) and 20.061 (*10^{6}) IR rials (1 USD = 202000 IRR), and a comparison shows that the cost calculated by the algorithm was 1.49% lower than that by the AS. By increasing the probability exceedance to 0.1, the construction cost dropped to 6.323 (*10^{6}) and 6.094 (*10^{6}) IR rials for the AS and the algorithm, respectively. The construction cost calculated by the is about 3.76% lower than that by the AS.

In Figure 8, the details of the optimal solution for decision parameters and cost generated by AS and algorithms for various probabilities of exceedance are presented for the Shora Bridge canal. The results showed that the cost and decision parameters decreased by increasing the discharge probability exceedance for both methods. The canal construction cost with an 0.01 probability exceedance is higher than those of other probability exceedances analyzed in this study. The canal construction costs using the algorithms AS and are, respectively, at 13.785 (*10^{6}) and 13.327 (*10^{6}) IR rials which shows that the cost calculated by the algorithm was 3.44% lower than that by the AS. The results show that by increasing the probability exceedance to 0.1, the canal construction cost decreases. The cost estimated by the AS and are at 1.699 (*10^{6}) and 1.564(*10^{6}) IR rials, respectively. The construction cost calculated by the is about 6.71% lower than that by the AS.

In Figure 9, the changes in the decision parameters and cost performed by both mentioned algorithms (AS and ) for various probability exceedance are presented for the Mahallati Bridge canal. As shown in both methods, while discharge probability exceedance increases, the cost, and the decision parameters decrease. According to the results, the canal has the highest construction cost for probability exceedance at 0.01. The canal construction cost estimated by the is at 43.139 (*10^{6}) IR rial, which is 0.74% lower than that by the AS at 43.458 (*10^{6}) IR rials. By increasing the probability exceedance to 0.1, the construction cost dropped to 16.746 (*10^{6}) and 16.419 (*10^{6}) IR rials for the algorithms AS and , respectively. The results showed that the algorithm managed to estimate construction at 1.99% than that by the AS.

### Comparing the results of optimization and LINGO software

Table 3 shows a summary of the canal design results from AS and metaheuristic algorithms for different flow probability exceedances in the three critical canals. Furthermore, the table compares the results from the algorithmic methods with those from the LINGO for evaluation and validation. The comparison results showed that for the three critical canals in the study area, for different probabilities values, the difference between the means of optimal results are 0.54, 0.56, and 0.63 for the and 3.31, 6.8, and 1.77 for the AS, respectively. As shown in the results, the outperformed the AS in finding the optimal solution. It is worth noting the can find optimal results close to those from LINGO within 5 min, which is less than the minimum time recorded for the nonlinear LINGO model. Therefore, it can be concluded that the proposed algorithm is computationally efficient. It should be noted that to solve complicated problems such as the model presented in this study, the use of mathematical models such as LINGO increases the time spent to find the optimal solution, as solving them may need several estimations, simplifications, or the use of derived information. Another result presented in Table 3 is that the variation of the bed width of the canal is lower than that of the canal depth in all three methods and for all three critical canals in the study area. With the increase in flow probability exceedance, the canal depth decreases more than the bed width of the canal.

. | . | AS . | . | LINGO results . | Diff (%) . | Time-consuming . | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

. | Probability exceedance . | Canal bed . | Canal depth . | Objective function . | Canal bed . | Canal depth . | Objective function . | Canal bed . | Canal depth . | Objective function . | AS & LINGO . | & LINGO . | AS . | . | LINGO . |

First canal | 0.01 | 6.95 | 3.72 | 20.061 | 6.97 | 3.68 | 19.765 | 7.00 | 3.67 | 19.724 | 1.71 | 0.21 | 0:04:04 | 0:03:46 | 0:08:45 |

0.013 | 6.95 | 3.50 | 18.296 | 6.95 | 3.50 | 17.681 | 6.98 | 3.41 | 17.618 | 3.85 | 0.36 | 0:03:19 | 0:03:27 | 0:07:16 | |

0.02 | 6.73 | 3.16 | 15.294 | 6.71 | 3.16 | 15.266 | 6.71 | 3.15 | 15.187 | 0.70 | 0.52 | 0:04:23 | 0:04:14 | 0:07:08 | |

0.04 | 6.71 | 2.60 | 10.949 | 6.65 | 2.55 | 10.456 | 6.70 | 2.53 | 10.385 | 5.43 | 0.68 | 0:04:15 | 0:04:12 | 0:08:56 | |

0.1 | 6.45 | 2.00 | 6.323 | 6.33 | 2.00 | 6.094 | 6.30 | 2.00 | 6.039 | 4.70 | 0.91 | 0:04:23 | 0:03:50 | 0:07:48 | |

Second canal | 0.01 | 8.81 | 4.90 | 13.785 | 8.97 | 4.81 | 13.327 | 9 | 4.79 | 13.241 | 4.11 | 0.65 | 0:04:01 | 0:03:09 | 0:07:25 |

0.013 | 8.71 | 4.50 | 10.894 | 8.93 | 4.39 | 10.362 | 8.97 | 4.37 | 10.268 | 6.10 | 0.91 | 0:04:42 | 0:04:39 | 0:07:06 | |

0.02 | 8.66 | 4.10 | 8.049 | 8.90 | 3.99 | 7.549 | 8.94 | 3.97 | 7.473 | 7.71 | 0.14 | 0:04:34 | 0:04:22 | 0:08:02 | |

0.04 | 8.64 | 3.49 | 3.796 | 8.87 | 3.41 | 3.507 | 8.89 | 3.40 | 3.483 | 8.99 | 0.69 | 0:03:15 | 0:03:05 | 0:07:29 | |

0.1 | 8.60 | 3.19 | 1.669 | 8.77 | 3.13 | 1.564 | 8.80 | 3.12 | 1.557 | 7.10 | 0.45 | 0:04:18 | 0:03:50 | 0:08:29 | |

Third canal | 0.01 | 10.90 | 4.06 | 43.458 | 10.93 | 4.04 | 43.139 | 11 | 4.01 | 42.770 | 1.61 | 0.86 | 0:04:36 | 0:04:35 | 0:08:05 |

0.013 | 10.5 | 3.84 | 39.172 | 10.64 | 3.80 | 38.610 | 10.6 | 3.80 | 38.570 | 1.56 | 0.11 | 0:04:14 | 0:04:05 | 0:07:11 | |

0.02 | 10.45 | 3.50 | 33.152 | 10.43 | 3.50 | 33.126 | 10.49 | 3.47 | 32.660 | 1.51 | 1.43 | 0:04:20 | 0:04:17 | 0:08:44 | |

0.04 | 10.38 | 2.98 | 23.946 | 10.39 | 2.96 | 23.606 | 10.41 | 2.95 | 23.448 | 2.12 | 0.67 | 0:04:22 | 0:04:21 | 0:08:08 | |

0.1 | 10.36 | 2.57 | 16.746 | 10.38 | 2.55 | 16.419 | 10.37 | 2.55 | 16.408 | 2.06 | 0.07 | 0:04:38 | 0:04:29 | 0:07:48 |

. | . | AS . | . | LINGO results . | Diff (%) . | Time-consuming . | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

. | Probability exceedance . | Canal bed . | Canal depth . | Objective function . | Canal bed . | Canal depth . | Objective function . | Canal bed . | Canal depth . | Objective function . | AS & LINGO . | & LINGO . | AS . | . | LINGO . |

First canal | 0.01 | 6.95 | 3.72 | 20.061 | 6.97 | 3.68 | 19.765 | 7.00 | 3.67 | 19.724 | 1.71 | 0.21 | 0:04:04 | 0:03:46 | 0:08:45 |

0.013 | 6.95 | 3.50 | 18.296 | 6.95 | 3.50 | 17.681 | 6.98 | 3.41 | 17.618 | 3.85 | 0.36 | 0:03:19 | 0:03:27 | 0:07:16 | |

0.02 | 6.73 | 3.16 | 15.294 | 6.71 | 3.16 | 15.266 | 6.71 | 3.15 | 15.187 | 0.70 | 0.52 | 0:04:23 | 0:04:14 | 0:07:08 | |

0.04 | 6.71 | 2.60 | 10.949 | 6.65 | 2.55 | 10.456 | 6.70 | 2.53 | 10.385 | 5.43 | 0.68 | 0:04:15 | 0:04:12 | 0:08:56 | |

0.1 | 6.45 | 2.00 | 6.323 | 6.33 | 2.00 | 6.094 | 6.30 | 2.00 | 6.039 | 4.70 | 0.91 | 0:04:23 | 0:03:50 | 0:07:48 | |

Second canal | 0.01 | 8.81 | 4.90 | 13.785 | 8.97 | 4.81 | 13.327 | 9 | 4.79 | 13.241 | 4.11 | 0.65 | 0:04:01 | 0:03:09 | 0:07:25 |

0.013 | 8.71 | 4.50 | 10.894 | 8.93 | 4.39 | 10.362 | 8.97 | 4.37 | 10.268 | 6.10 | 0.91 | 0:04:42 | 0:04:39 | 0:07:06 | |

0.02 | 8.66 | 4.10 | 8.049 | 8.90 | 3.99 | 7.549 | 8.94 | 3.97 | 7.473 | 7.71 | 0.14 | 0:04:34 | 0:04:22 | 0:08:02 | |

0.04 | 8.64 | 3.49 | 3.796 | 8.87 | 3.41 | 3.507 | 8.89 | 3.40 | 3.483 | 8.99 | 0.69 | 0:03:15 | 0:03:05 | 0:07:29 | |

0.1 | 8.60 | 3.19 | 1.669 | 8.77 | 3.13 | 1.564 | 8.80 | 3.12 | 1.557 | 7.10 | 0.45 | 0:04:18 | 0:03:50 | 0:08:29 | |

Third canal | 0.01 | 10.90 | 4.06 | 43.458 | 10.93 | 4.04 | 43.139 | 11 | 4.01 | 42.770 | 1.61 | 0.86 | 0:04:36 | 0:04:35 | 0:08:05 |

0.013 | 10.5 | 3.84 | 39.172 | 10.64 | 3.80 | 38.610 | 10.6 | 3.80 | 38.570 | 1.56 | 0.11 | 0:04:14 | 0:04:05 | 0:07:11 | |

0.02 | 10.45 | 3.50 | 33.152 | 10.43 | 3.50 | 33.126 | 10.49 | 3.47 | 32.660 | 1.51 | 1.43 | 0:04:20 | 0:04:17 | 0:08:44 | |

0.04 | 10.38 | 2.98 | 23.946 | 10.39 | 2.96 | 23.606 | 10.41 | 2.95 | 23.448 | 2.12 | 0.67 | 0:04:22 | 0:04:21 | 0:08:08 | |

0.1 | 10.36 | 2.57 | 16.746 | 10.38 | 2.55 | 16.419 | 10.37 | 2.55 | 16.408 | 2.06 | 0.07 | 0:04:38 | 0:04:29 | 0:07:48 |

The chance constraint in the optimal design of open canals enables the user to account for the uncertainty associated with the design parameters. Based on the results obtained, from an implementation perspective, this study's design parameters for the specified probability exceedances can be used to do the optimal design for critical study areas in need of designs with other probability exceedances to allow for discharges over the design value. In canals one to three, for a 0.01 probability exceedance, the acceptable design for bed width is at 6.97, 8.97, and 10.93 m and for canal depth at 3.68, 4.81, and 4.04 m, respectively. The present study is carried out for a certain set of input values and can be easily extended to other combinations of input design parameters for other areas. Many similar studies have been done to optimally design canal parameters that used metaheuristic methods such as GA, PSO (Reddy & Adarsh 2010; Adarsh & Janga Reddy 2015), GSA (Adarsh & Janga Reddy 2015), and HBP (Farzin & Valikhan Anaraki 2020) due to the uncertainties in finding the proper values of design parameters. As mentioned above, it should be noted that the research that used these methods, in theory, can be used for open canals rehabilitation projects. In this research, however, a canals network exists in the study area to allow for the flood discharge, yet it does not suit discharges with other probability exceedances due to insufficient capacity. Therefore, using the studies above, which are intended for renovations, is not economical. In other words, this study attempts to create the conditions that allow for discharges higher than the design value by redesigning the existing channels using the proposed method.

### Hydraulic simulation results using SWMM model after optimization

To that end, the optimal dimensions calculated for the three critical periods substituted the previous ones, a new SWMM model of the study area was developed, and the simulations were performed for precipitations with 10, 25, 50, and 100-year RP In addition to investigating the impact of the local rehabilitation on the floodwater flow in the eastern Tehran floodway, the flood water flow at the critical bridges were also analyzed. In Figure 10, the simulation results illustrate the hydraulic performance of the critical points after rehabilitation and applying the optimal dimensions of canal reaches in the vicinity of these bridges. Furthermore, to compare the impact of local rehabilitation on the critical stormwater canals in eastern Tehran, the network's hydraulic performance before and after applying the optimal dimensions are presented in Table 4. The results indicate that the local rehabilitation of the eastern stormwater canal reduced the inundations in precipitations with various RPs, i.e., for floods with a 10-year RP, the number of node flooding dropped from four to zero, inundated areas from 17% to zero, and the overflow volume from (10–45) to zero. Similar results were obtained by increasing the flood RP for precipitations with 25, 50, and 100-year RPs. Node flooding decreased from 10, 12, and 20 under the present conditions to 1, 2, and 4, respectively. Furthermore, implementing the local rehabilitation in the stormwater canal reduced the overflow volume from (30–65), (43–74), and (70–92) under the present conditions to (4–12), (11–27), and (24–36) for precipitations with 25, 50, and 100-year RPs.

RPs (years) . | Status quo . | RPs (years) . | Renovated project (optimization) . | ||
---|---|---|---|---|---|

T = 10 | Node flooding (No.) | 4 | T = 10 | Node flooding (No.) | 0 |

Flooded area (%) | 17 | Flooded area (%) | 0 | ||

Overflow volume (%) | (10–45) | Overflow volume (%) | 0 | ||

T = 25 | Node flooding (No.) | 10 | T = 25 | Node flooding (No.) | 0 |

Flooded area (%) | 23 | Flooded area (%) | 7 | ||

Overflow volume (%) | (30–65) | Overflow volume (%) | (4–12) | ||

T = 50 | Node flooding (No.) | 12 | T = 50 | Node flooding (No.) | 2 |

Flooded area (%) | 31 | Flooded area (%) | 12 | ||

Overflow volume (%) | (43–74) | Overflow volume (%) | (11–27) | ||

T = 100 | Node flooding (No.) | 20 | T = 100 | Node flooding (No.) | 4 |

Flooded area (%) | 43 | Flooded area (%) | 18 | ||

Overflow volume (%) | (70–92) | Overflow volume (%) | (24–36) |

RPs (years) . | Status quo . | RPs (years) . | Renovated project (optimization) . | ||
---|---|---|---|---|---|

T = 10 | Node flooding (No.) | 4 | T = 10 | Node flooding (No.) | 0 |

Flooded area (%) | 17 | Flooded area (%) | 0 | ||

Overflow volume (%) | (10–45) | Overflow volume (%) | 0 | ||

T = 25 | Node flooding (No.) | 10 | T = 25 | Node flooding (No.) | 0 |

Flooded area (%) | 23 | Flooded area (%) | 7 | ||

Overflow volume (%) | (30–65) | Overflow volume (%) | (4–12) | ||

T = 50 | Node flooding (No.) | 12 | T = 50 | Node flooding (No.) | 2 |

Flooded area (%) | 31 | Flooded area (%) | 12 | ||

Overflow volume (%) | (43–74) | Overflow volume (%) | (11–27) | ||

T = 100 | Node flooding (No.) | 20 | T = 100 | Node flooding (No.) | 4 |

Flooded area (%) | 43 | Flooded area (%) | 18 | ||

Overflow volume (%) | (70–92) | Overflow volume (%) | (24–36) |

## CONCLUSION

The present study attempts to introduce a practical, economical method to carry out the stormwater drainage canal rehabilitation projects in urban areas that can be implemented in the shortest possible time. The solution mentioned above combines structural and non-structural measures that focus only on the localized improvement of critical points and satisfies two conflicting objectives of redesigning stormwater canals while minimizing construction costs. The aspect that distinguished this method from others in rehabilitation projects for urban flood management is the step-by-step rehabilitation of structures with minimum costs. The optimization results for the first to third critical periods for a design discharge with a 100-year RP showed that the calculated cost was 19.765(*106), 13.327(*106), and 43.139(*106) IR rials (1 USD = 202000 IRR), respectively. For the selected periods, the optimal bed width is 6.97, 8.97, and 10.93 m, and the optimal depth is 3.68, 4.81, and 4.04 m, respectively. In addition to considering all urban constraints in the study area, the dimensions also observed the hydraulic constraints while keeping the costs minimum. The design floods drainage simulation by applying the critical reaches redesigned after applying optimal dimensions showed that on average, the rehabilitation reduced inundation throughout the eastern Tehran stormwater canal by 46, 38, 39, and 27.5% for precipitations with 10, 25, 50, and 100-year RPs. Redesigning the entire stormwater canal reaches can reduce the inundation impact to zero; however, it cannot be implemented due to extreme financial constraints.

A significant limitation in this study is the dense texture of neighborhoods and urban areas in the east of the Tehran metropolis, reducing the widening range. Moreover, it was impossible to define a larger depth range to further deepen the reaches because of the high gradient in the stormwater canal parts. In addition to the above limitations, Tehran municipality's severe financial constraint on choosing more reaches limited the rehabilitation options to three after several meetings with the urban flood management department managers. Field studies and accurate measurements at the canals significantly helped reduce the uncertainty stemming from the optimization data, making the research results more applicable.

According to the results, it is suggested that low impact development (LID) methods be used as complementary options along with local rehabilitation to improve the runoff drainage network in terms of both the quantity and the quality of the collected runoff. Since the various LID methods can be implemented in public areas such as parks, gardens, alleys, and streets within the stormwater drainage subnetwork, the inundation is expected to decrease significantly in the eastern Tehran stormwater canal reducing the inflow from the subnetwork. To this end, it is suggested that by using the critical points identification method in this research, critical subnetworks be determined, and LIDs be implemented by prioritizing problematic and critical sub-catchments in each subnetwork.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.