Abstract
In this study, the risk analysis is used to determine the return period in which the design cost plus the damage risk cost is minimum. The damage includes the roads and traffic, the lawn areas, and the residential and commercial buildings. The traffic damage is based on two factors, time of delay and social negative impacts. The nonlinear reservoir model is used to convert the rainfall into runoff and the dynamic wave model is utilized to perform the network hydraulic simulation in stormwater management model (SWMM) software. This model is defined as an appropriate model. This model was applied in the risk analysis of a region in Tehran to obtain the optimal return period design. The results indicated that the optimal return period is 10 years. The rational method was also applied to the same region and same return period, but the total design cost of the rational method was greater by 5%. The damage due to the traffic include financial damages due to delays and loss of fuel resources in addition to the dissatisfaction of people due to the psychological burden.
HIGHLIGHT
In this study, risk analysis is used to determine the return period in which the design cost plus the damage risk cost is minimum. The damage included the roads and traffics, the lawn areas, and the residential and commercial buildings. The traffic damage is based on two factors, time of delay and social negative impacts.
INTRODUCTION
Population growth, urban development, agricultural and industrial activities, and the conversion of natural lands to residential areas result in a change in the natural hydrologic cycle of a region (Coffman & Rushton 1999). Also, the increase in impervious areas significantly has altered the pre-urban hydrology and increased the surface runoff in urban environments compared with the undeveloped natural environments. Implementation of storm sewer networks is one of the effective ways to manage water resources. The optimal design of storm sewer networks is one of the most important issues in water engineering as they are very costly. Any attempt to reduce the construction costs of these networks can lead to significant savings. The design of storm sewer networks is always associated with risk. In a proper design, there must be a balance between its cost and the risks that may occur in the future. Therefore, both optimization and risk analysis should be considered in the optimal design. The mathematical formulation of the storm sewer network optimization leads to the conversion of the problem into a constrained nonlinear optimization problem. Therefore, appropriate methods should be used. In recent years, the optimization of storm sewer networks has received marked attention. Hassan et al. (2021) in his paper illustrates the application of a new model combining the Genetic Algorithm with Heuristic Programming (GA-HP) techniques in order to establish the optimal design for sewer networks. The objective is to minimize the construction cost function, which is represented by the depth of excavation and pipe diameter. The proposed GA-HP model has achieved the optimum design task in two stages. Firstly, the GA was applied to obtain the diameters of the pipes needed for the preliminary design of the network. Secondly, HP preliminary designs were used to obtain the optimal slope for those pipes and to determine other characteristics such as the velocity, relative depth of water, excavation depths and total cost of the network. A MATLAB code was used to perform the GA-HP optimization modeling. The results show that the GA-HP model is superior to all previous methods and may be more efficient in the design of large networks. Azargashb Lord et al. (2021) conducted a local rehabilitation in storm water 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 stormwater management model (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). Hussain et al. (2018) in their study, the generation of intensity duration frequency curves (IDF) that integrates climate change effect was conducted for Al-Najaf Governorate in Iraq for the first time. In addition, the effects of land use and climate change on the storm water sewer system of Al-Ameer District was simulated using SWMM. The results indicated that by increasing the sub-catchment area from 50 to 100%, an increment in total surface runoff from 20,380 to 37,350 m3, and total flooding from 10,513 to 26,032 m3 had occurred, respectively. As a response to climate change, changing the return period from 2 to 5 years has increased the total surface runoff from 14,120 to 27,110 m3 (representing 48% of raise), and total flooding increased from 5,914 to 17,591 m3 (accounting for 72% of increment). To conclude, flooding locations and magnitude were identified, while the system failed to discharge surface runoff at critical conditions, whereas the effect of climate change on the storm water drainage system was more adverse than the effect of land use. Zaheri et al. (2019) in their paper, a Cellular Automata-based simulation–optimization approach is proposed for the optimal design of household sewer networks. A two-phase CA is used as the optimization tool, while the EPA's SWMM is used as the simulator. A splitting method is first used to redefine the sewer network design problem in terms of two simpler sub-problems with diameters and nodal elevations of each pipe as decision variables, which are iteratively solved using CA methods in a two-stage manner until convergence is achieved. Results indicate the efficiency and effectiveness of the proposed methods compared to the existing methods. Zeng et al. (2020) designed and implemented a web service framework based on SWMM (WEB-SWMM), which can provide real-time computing services for urban water management. To test the functionality, efficiency, and stability of the WEB-SWMM, WEB-SWMM was applied to an urban area in China. Test results show that WEB-SWMM could provide real-time computing services stably, quickly, and accurately. In general, the implementation of WEB-SWMM enables traditional SWMM to be quickly and efficiently applied to real-time urban storm water management. What is more, the web-based hydrological model framework proposed in this paper is also applicable to most existing hydrological models.
In most of the previous research, the issue of storm sewer network design was raised by minimizing design costs with a specified return period. This approach has the basic disadvantage that the possible damage that will happen in the future is not considered. In fact, one of the important issues in designing storm sewer network is the appropriate return period. The appropriate return period for design is a return period in which the total design costs and probable damage are minimal; Risk analysis is performed to determine the appropriate return period with these goals. Usually the design return period is based on experience. Choosing the Experimental Return Period without performing risk analysis would not be optimal, because in a proper design there must be a good balance between the cost of design and the risks that may occur in the future, and the design must support both aspects. Therefore, the purpose of the present study is to design storm sewer networks using the optimization simulation method in which the design costs are minimal. In this study, the optimal design is determined by minimal costs for different return periods. Risk analysis is then performed to determine the optimal return period in which the total design and damage costs are minimal.
The literature review related to flood damage indicates that all damage has not been considered. Jaf (2015) in her research considered only the damage to the commercial and residential buildings, and Hekmati Far (2006) regarded the damage to the roads only. In the present study, all damage including roads and traffic, lawn areas, and residential and commercial buildings are taken into account. In this regard, the average flood depth was estimated and various damage costs were developed based on real data. These damage costs were used in the risk analysis to estimate the optimal design return period. Two methods of storm sewer system network simulation were used in this research. The first one utilizes the nonlinear reservoir model to convert the rainfall into runoff and the dynamic wave model to perform the network hydraulic simulation. The SWMM software is used to do this task. This model is defined as a appropriate model. The second method of simulation is the rational method. The simulation methods are linked to an optimization solver to obtain the optimal design for every selected return period flood. The decision variables in the optimization program are the diameters and slopes of the pipes.
One of the precise methods to calculate the runoff in storm sewer design is the nonlinear reservoir model used in the SWMM program in which nonlinear differential equations have to be solved. This method is regarded as a appropriate one compared with others. The rational method to calculate the runoff in storm sewer design systems is very simple in concept and application compared with the appropriate one.
There are three common methods to handle the hydraulic analysis of the flow in the conduits; the dynamic wave, the kinematic wave, and the Manning equation methods. Let's call the method in which the runoff is calculated the nonlinear reservoir model and the conduit hydraulic analysis the dynamic wave method as the appropriate method. The rational method leads to greater runoffs and consequently larger conduit sizes which in turn increase the costs. How much is the difference in the design costs of these methods? This is investigated in this research by using a case study in a local region in Tehran city.
The damage costs are required for the risk analysis. Relationships for the damage costs of the land use, infrastructure, and traffic, are provided for the mentioned case study. The genetic algorithm (GA) is utilized as an optimization tool to find the decision variables (pipe diameters, and slopes) so that the summation of the design costs will be minimized. In this regard, the appropriate method and the rational method are compared.
INTRODUCING THE SWMM MODEL
The SWMM software was developed by the United States Environmental Protection Agency in collaboration with Metcalfe & Eddy Engineering and the University of Florida from 1969 to 1971 to simulate the quantitative and qualitative associated aspects with runoff pipes.
The required data are very extensive and include physiographic data of sub-catchments, specifications of system structures, system maintenance information, discharge intensity in the dry period, rainfall information, flood hydrographs, and flow quality in the conduit. The surface water management model is a dynamic model of rainfall–runoff simulation and can simulate the quality and quantity of runoff for urban areas for one single event or a continuous one. It can also be combined with other models. As this model is widely used to design, analyze and estimate the cost of constructing a drainage network system in urban areas, in this study, this hydrological–hydraulic model was selected. Because SWMM is a distributed model it allows a study area to be subdivided into any number of irregularly shaped sub-catchment areas to best capture the effect that spatial variability in topography, drainage pathways, land cover, and soil characteristics have on runoff generation.
The complete form of St. Venant equations is solved in the dynamic wave method and thus produces the most theoretically accurate results. The method accounts for conduit storage, backwater effects, entrance/exit losses, flow reversal, and pressurized flow. The time step should be so small to maintain numerical stability. Numerical instability is characterized by oscillations in the flow and water surface elevation that do not dampen out over time. Stable explicit solutions of the St. Venant equations require smaller time steps than those it takes for a dynamic wave to travel down the length of the conduit (Cunge et al. 1980). In the kinematic wave method, the friction and gravity terms are only considered. SWMM also offers a steady flow analysis option, which assumes that within each computational time step, the flow is uniform and steady. It simply translates inflow hydrographs with no delay or change in shape. The Manning equation is used to relate flow rate to flow area (or depth). Because it ignores the dynamics of free surface wave propagation it is only appropriate for rough preliminary analysis of long-term continuous simulations (Rossman 2016).
Many studies that confirm the accuracy of the dynamic wave model method have been conducted, such as (Helmio 2005), (Zhang 2005), (Anderson et al. 2006), (Kuiry et al. 2010), (Venturelli 2011) and (Akbari & Barati 2012).
METHODOLOGY
The rational method
In the above relation, 
is the peak discharge, 
is the rainfall intensity, 
is the runoff coefficient, 
is the catchment or sub-catchment area, and 
is a constant coefficient depending on the selected units of the equation.
To employ this method in SWMM the followings are implemented (Rossman 2016):
- 1.
Set the sub-catchment's percent imperviousness to 100 C and its percent of imperviousness with no depression storage to 0.
- 2.
Assign the same depression storage depth to both the pervious and impervious areas.
- 3.
Use any values for slope and width, and 0 for both the pervious and impervious Manning's n.
- 4.
Use the Horton infiltration option and let its maximum and minimum infiltration rates be the same.
Setting up a model in this form will produce the same results as if Equation (1) were implemented directly. When the Manning roughness coefficient, n, is 0, SWMM bypasses Equation (3) and simply converts all rainfall excess at each time step into instantaneous runoff (Rossman 2016).
In this method, the peak flow is only determined considering the flow travel time through the conduits to determine the time of concentration. This method is widely used. In 1975, a survey among 37 Canadian municipalities revealed that more than 97% of them used the rational method for the design of the storm sewer systems (Badea & Bacotiu 2002). Many types of research have been done which prove the ability of the rational method (for example Singh & Cruise 1992; Mega et al. 2019; and Shad & Hoveidafard 2015). The Manning equation is used for the hydraulic analysis in network conduits.
A hydrograph is calculated based on the rational method and transferred through the network conduits considering the flow travel time and summed with the other hydrographs obtained for the following sub-catchments.
Optimization model
In this research, the GA is used to minimize the total design cost. GA starts to optimize the problem with an initial population of chromosomes which are randomly generated at the beginning. The chromosomes evolve through successive iterations namely generations in GA (Gen & Cheng 1997). For more details about the GA in water and wastewater engineering, readers are referred to the optimization of water distribution networks of different researchers (Simpson et al. 1994; Dandy et al. 1996; Yaghi et al. 1998; Wu & Simpson 2001).
, 
, and CE are the costs of construction of sewers pipes manholes, and bedding, respectively. 
and 
are the number of pipes and manholes, respectively.
In the above equations, D represents the pipe diameter (m), E is the depth of excavation (m), and 
is the depth of the manhole (m).
It should be noted that the design costs calculated in this study are based on prices in 2020 and the obtained relationships are based on curve fitting based on prices in the same year. For other times and in other years, due to price changes, it is necessary to fit the appropriate curve based on the same year and calculate the appropriate correlation relations of the same year.
The objective function is subjected to the following constraints:
- 1.Flow velocity constraint:(6)
The flow velocity in pipes should be between a minimum and maximum bound to ensure self-cleaning and to prevent scouring. Minimum and maximum flow velocities are chosen as 0.9 and 4.5 m/s, respectively.
- 2.Flow depth constraint:(7)
and 
represent the relative flow depth and maximum relative flow depth in pipes, respectively. 
is selected as 0.9 in this research. - 3.Excavation depth constraint:(8)
is the excavation depth for pipe i, 
and 
are the minimum and maximum excavation depths and they are selected as 1 and 6 meters, respectively.
- 4.The telescopic pattern of pipe diameter constraint:(9)
- 5.Pipe slope constraint:(10)
is the slope of pipe i, and 
is the minimum pipe slope recommended as 0.0005. - 6.
Outlet elevation from each manhole constraint:
The invert of the outlet pipe from each manhole should be equal to or lower than inlet pipes.
The initial population in the GA is randomly produced and then by random-based operators next generations are created. This mechanism often results in infeasible chromosomes which are not accepted for cost evaluation. For handling the aforementioned sewer constraints and avoiding infeasible solutions, several techniques have been proposed for metaheuristics so far. These techniques can be simply classified as the rejecting, repairing, penalizing, and modifying strategies (Gen & Cheng 1997).
, 
are the penalties of the constraints related to the maximum and minimum velocity, respectively, 
, 
, represent the penalties of the constraints related to the maximum and minimum allowable depths of excavation respectively, and 
, is the penalty function for violating the relative depth of the flow. If the values of velocity, excavation depth, and relative depth of the flow are not within the allowable range, heavy penalties will be added to the objective function which in its turn deviates the corresponding objective function from being minimum. The penalty function is zero if the values of the penalty functions are not violated. 
, 
and 
are coefficients correspond to velocity, relative flow depth, and excavation depth, respectively. The values of these parameters are usually chosen as large numbers. The final objective function is rewritten by applying the penalty functions as follows:
The diameter and slope of the pipes are considered as the decision variables that should be determined by the optimization process.
The simulation model (SWMM software) is linked to the GA optimization model. The goal is to determine the storm return period with the optimal corresponding design alternative for which the total design cost plus the probable damage cost is minimum. In this regard, at the first stage, the optimal storm sewer network design for a certain selected rainfall return period is obtained. The required hydrologic data corresponding to the selected rainfall return period and hydraulic data are given to the developed model. The optimization input data such as the list of the commercial pipes, the objective function, and constraints should be given to the program. In addition, the GA parameters include the initial population of chromosomes and the mutation factor. The decision variables (pipe diameters and slopes) are initially selected by the GA. The simulation is then performed by the SWMM program. The GA program calculates the objective based on the results of the simulation. The decision variables are changed by the optimization program until a minimum objective function in which all constraints are satisfied is achieved. The design process is shown in Figure 1.
Storm sewer network simulation-optimization flow chart.
The simulation and optimization models are linked so that the calculations and exchanging of information between them will be performed automatically. The GA in MATLAB is developed to be combined with the simulator model. That is, the algorithm randomly generates the decision variables, which include the diameter and slope of the pipes for a selected storm. These variables are used in the flow simulation and optimization program. The set of decision variables that satisfies all constraints and produces the least value for the objective function (Total design costs) will be selected as the optimal design for the selected storm return period.
The risk analysis is done to find the optimal storm return period. This is performed by repeating the same procedure explained above for other storm return periods. The return period is sought in which the summation of the annual costs of the corresponding optimal design and the risk damage is minimum.
GA optimization parameters
The configuration of the GA was determined using several test implementations with random initial generations on the proposed model to achieve the fastest convergence seeking the optimal solutions. For this purpose, the optimization model was implemented with different values of parameters to find the most appropriate ones. These parameters are listed in Table 1.
Parameters set in GA
| Parameter | Population size | Mutation rates |  |  |  |
|---|---|---|---|---|---|
| Amount | 200 | 0.02 | 1.0E + 9 | 5.0E + 10 | 5.0E + 11 |
| Parameter | Population size | Mutation rates | | | |
|---|---|---|---|---|---|
| Amount | 200 | 0.02 | 1.0E + 9 | 5.0E + 10 | 5.0E + 11 |
Performance of the optimization model
To evaluate the accuracy of the model in this research, a benchmark storm sewer network optimized by many other investigators and depicted in Figure 2 was employed. This network was initially designed and presented by Mays et al. (1976). The network consists of 20 branches, 20 nodes (manholes) and one output.
The developed optimization model is applied to the benchmark network and the results are compared with those of other researchers to ensure the validity of the model.
Figure 3 indicates that the optimal design is achieved within 50 generations and the converged minimal objective function is $ 241,763. The results show the good performance of the optimization method used in this study.
Diagram of benchmark network optimization achievement.
A comparison of the developed optimization model with others is presented in Table 2. The optimal cost of the present study is lower than all other results except that of Afshar (2010) who used the ACO method. This proves that the developed model has a good performance in terms of speed and accuracy.
Benchmark problem result comparison of the present study with others
| Researcher | Optimization method | Optimal cost (dollars) | The optimal cost of the present study (dollars) | Comparison with previous research |
|---|---|---|---|---|
| Mays et al. (1976) | Differential discrete programming | 265,775 | 241,763 | −24,012 |
| Miles & Heaney (1988) | Differential discrete programming | 245,874 | 241,763 | −4,111 |
| Afshar et al. (2006) | Genetic algorithm | 244,747 | 241,763 | −2,984 |
| Afshar (2010) | Ant colony | 241,496 | 241,763 | 267 |
| Afshar (2008) | PSO | 242,162 | 241,763 | −399 |
| Afshar & Rohani (2012) | Cellular machine | 247,412 | 241,763 | −5,649 |
| Mousavi (2016) | Genetic algorithm | 241,956 | 241,763 | −193 |
| Researcher | Optimization method | Optimal cost (dollars) | The optimal cost of the present study (dollars) | Comparison with previous research |
|---|---|---|---|---|
| Mays et al. (1976) | Differential discrete programming | 265,775 | 241,763 | −24,012 |
| Miles & Heaney (1988) | Differential discrete programming | 245,874 | 241,763 | −4,111 |
| Afshar et al. (2006) | Genetic algorithm | 244,747 | 241,763 | −2,984 |
| Afshar (2010) | Ant colony | 241,496 | 241,763 | 267 |
| Afshar (2008) | PSO | 242,162 | 241,763 | −399 |
| Afshar & Rohani (2012) | Cellular machine | 247,412 | 241,763 | −5,649 |
| Mousavi (2016) | Genetic algorithm | 241,956 | 241,763 | −193 |
CASE STUDY
The study area is region 2, district 3 of Tehran with residential land use, roads, highways, and lawn and has a total area of 230 hectares. The region was divided into 17 sub-catchments. The layout of the storm sewer network is shown in Figure 4. The simulation was performed using the SWMM software using the nonlinear reservoir model and the rational method to estimate the surface runoff. The rainfall intensity–duration-return period curves provided by Tehran Surface Water Management Comprehensive Plan (2012) and shown in Figure 5 are used in the analysis to obtain the runoff.
Layout of the storm sewer network of region 2 in Tehran city.
The rainfall–intensity–duration–return period of the station covers the case study region.
The rainfall–intensity–duration–return period of the station covers the case study region.
Model calibration
= Observational data
= Simulated data
And m is the number of data.
According to the results of the calculations of Tehran surface water management plan, the peak flow rate of 50-year precipitation has been achieved by 0.35 m3/s. The same amount of rainfall in the model prepared in SWMM software is 0.38 cubic meters per second for the dynamic wave routing method and 0.45 cubic meters for the rational method. Based on this, the RE index value is equal to 7.89% for the dynamic wave routing method and 22% for the rational method. Also, the RMSE for the dynamic wave routing method is 8.6% and for the rational method is 12.6%. The values obtained for both indicators are acceptable according to the opinions obtained in different sources. The amount of calculated indicators shows that the model has a good performance. For example, the output runoff hydrograph is shown for one of the return periods in Figure 6.
output runoff hydrograph for the case study region.
DAMAGE FUNCTIONS
The damage costs are required for the risk analysis to determine the optimal storm return period. The damage cost is assumed as a function of the average flooding depth in the area out of the sewer system when a runoff greater than the storm sewer runoff design takes place.
The average flood depth for every sub-catchment is determined as follows: Subtracting the inflow hydrograph to the storm sewer network at the manhole obtained by the hydraulic simulator from the runoff hydrograph produced by the sub-catchment to obtain the flooding hydrograph for that sub-catchment. The flooding volume can be easily calculated based on this hydrograph and then dividing the flooding volume by the sub-catchment area results in the average flooding depth for that sub-catchment. This should be repeated for all other sub-catchments. The total average flooding depth is then calculated by taking the weighted average flooding depths of the sub-catchments. In Iran, curves or acceptable patterns are not provided for the depth of damage. Due to the lack of adequate statistics and inaccuracies of flood damage, there is no reliable statistics to estimate these curves according to the damage statistics; Therefore, in the present study, the methods used in previous studies were used with engineering vision and judgment. In the present study, depth-percentage damage charts of residential and commercial buildings provided by Jaf 2015 were used (Figure 7).
Depth-damage percent curve of residential and commercial buildings provided by Jaf (2015).
The depth-damage percent curves of the residential and commercial buildings of this region provided by Jaf (2015) and depicted in Figure 7 that used this region were employed.
In relations (19) and (20), 
is the percentage of damage of residential buildings, 
is the percentage damage of commercial buildings and y is the runoff depth of the sub-catchment in meters.
The damage function of roads
The damage function of lawn areas
The function of damage caused by traffic
According to the statistics of the Deputy Minister of Municipal Transportation and the statistics of the Urban Economic Association of Iran, there is an annual delay of 5.68 million hours due to rainfall in region 2 due to the 1-year return period storm. This amount of delay may result in 6,280 thousand dollars loss of which 3,120 thousand dollars are related to fuel loss, 2,520 thousand dollars are related to waste of time and people's dissatisfaction, and 640 thousand dollars are related to pollution. Traffic damage corresponding to other designs based on storm return periods is depicted in Table 3. It is noted as the sewer system design return period increases, the time delay due to traffic and the corresponding damage decreases. This is also introduced in Figure 8.
Traffic damage for various storm return periods
| Return period | Damage (dollars × 10,000) | Delay time (million hours) | |||
|---|---|---|---|---|---|
| Fuel loss | Waste of time and people's dissatisfaction | Pollution | Total | ||
| 1 | 5.68 | 312 | 252 | 64 | 628 |
| 2 | 4.3 | 236 | 192 | 48 | 476 |
| 5 | 3.2 | 176 | 140 | 36 | 352 |
| 10 | 2.1 | 116 | 92 | 24 | 232 |
| 20 | 1.5 | 84 | 68 | 16 | 168 |
| 25 | 1.3 | 72 | 56 | 12 | 140 |
| 50 | 1 | 56 | 44 | 8 | 108 |
| Return period | Damage (dollars × 10,000) | Delay time (million hours) | |||
|---|---|---|---|---|---|
| Fuel loss | Waste of time and people's dissatisfaction | Pollution | Total | ||
| 1 | 5.68 | 312 | 252 | 64 | 628 |
| 2 | 4.3 | 236 | 192 | 48 | 476 |
| 5 | 3.2 | 176 | 140 | 36 | 352 |
| 10 | 2.1 | 116 | 92 | 24 | 232 |
| 20 | 1.5 | 84 | 68 | 16 | 168 |
| 25 | 1.3 | 72 | 56 | 12 | 140 |
| 50 | 1 | 56 | 44 | 8 | 108 |
Traffic time delay chart for different return periods.
Multiplying the percentage of damage by the value of the construction of each above-mentioned damage yields the damage costs for every flooding event.
RISK ANALYSIS
is the risk cost and 
is the cost of flood damage with frequency x.As was explained in section 8, The average flooding depth of the entire catchment is calculated by taking the weighted average of sub-catchment flooding depths. To clarify the concept more, assuming that a storm sewer system is designed based on a 5 years storm return period and a 25 year return period storm occurs. The system, in this case, cannot pass the flood. At each manhole, the full pipe flood will be passed. The flooding hydrograph at each manhole will be equal to the calculated flood of that sub-catchment minus the full flow passed through the manhole. Dividing the flooding hydrograph flow volume by the sub-catchment area results in the average flooding depth for that sub-catchment. Having done this in the same manner for other sub-catchments, the weighted average flooding depth can be estimated for the entire catchment.
is the design cost, r is the discount rate, T is the pipe design life in years, and 
is the annual rate of repairs and maintenance.In the present study, r, T, and 
are considered equal to 0.08, 30 years, and 0.005, respectively.
OPTIMAL DESIGN RESULTS USING THE APPROPRIATE METHOD
In this section, the results of the optimal design of the storm sewer network are presented using the appropriate method. In this method, the nonlinear reservoir model is used to estimate the surface runoff, and the dynamic wave method was used for flow routing in the conduits. The simulation–optimization developed model is applied for each storm return period to produce the optimal design of each one. The optimal design costs are introduced in column (5) of Table 4. These costs are converted to annual costs given in column (6) using Equation (25). The total damage costs depicted in column (3) are obtained by using Equations (19) to (23). The annual damage costs given in column (4) are calculated according to Equation (24). The total annual cost introduced in column (7) shows that 10 years is the optimal return period in which the summation of the total annual design and damage costs is minimal.
Optimal design results using the appropriate method
| Return period | Frequency | Damage costs (dollars) | Annual damage risk cost (dollars) | Optimal Design cost (dollars) | Annual optimal design cost (dollars) | Total annual cost (dollars) |
|---|---|---|---|---|---|---|
| 1 | 1 | 40,000 | 1,942,080 | 36,000,000 | 3,088,720 | 5,030,800 |
| 2 | 0.5 | 2,092,000 | 1,409,080 | 41,564,000 | 3,566,120 | 4,975,200 |
| 5 | 0.2 | 2,824,000 | 671,680 | 420,80,000 | 3,610,360 | 4,282,040 |
| 10 | 0.1 | 3,772,000 | 341,880 | 44,000,000 | 3,775,120 | 4,117,000 |
| 20 | 0.05 | 4,328,000 | 139,380 | 64,000,000 | 5,491,080 | 5,630,440 |
| 25 | 0.04 | 4,500,000 | 95,240 | 84,000,000 | 7,207,040 | 7,302,280 |
| 50 | 0.02 | 5,024,000 | 0 | 1.2E + 08 | 10,295,760 | 10,295,760 |
| Return period | Frequency | Damage costs (dollars) | Annual damage risk cost (dollars) | Optimal Design cost (dollars) | Annual optimal design cost (dollars) | Total annual cost (dollars) |
|---|---|---|---|---|---|---|
| 1 | 1 | 40,000 | 1,942,080 | 36,000,000 | 3,088,720 | 5,030,800 |
| 2 | 0.5 | 2,092,000 | 1,409,080 | 41,564,000 | 3,566,120 | 4,975,200 |
| 5 | 0.2 | 2,824,000 | 671,680 | 420,80,000 | 3,610,360 | 4,282,040 |
| 10 | 0.1 | 3,772,000 | 341,880 | 44,000,000 | 3,775,120 | 4,117,000 |
| 20 | 0.05 | 4,328,000 | 139,380 | 64,000,000 | 5,491,080 | 5,630,440 |
| 25 | 0.04 | 4,500,000 | 95,240 | 84,000,000 | 7,207,040 | 7,302,280 |
| 50 | 0.02 | 5,024,000 | 0 | 1.2E + 08 | 10,295,760 | 10,295,760 |
Pipe diameters and slopes of the final optimal design with a 10-year storm return period of the appropriate methods are given in Table 5.
Pipe diameters and slopes of the final optimal design with a 10-year return period resulting from the appropriate method
| The appropriate method using the dynamic wave routing | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Pipe | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Diameter(mm) | 450 | 450 | 450 | 525 | 525 | 525 | 300 | 525 | 450 | 450 |
| Slope (%) | 0.497 | 1.281 | 1.045 | 0.359 | 1.188 | 0.681 | 0.82 | 1.176 | 0.732 | 0.171 |
| Pipe | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Diameter(mm) | 525 | 675 | 375 | 750 | 750 | 375 | 900 | 900 | 900 | 900 |
| Slope (%) | 0.258 | 1.007 | 0.797 | 1.494 | 0.288 | 0.694 | 0.942 | 0.312 | 1.461 | 1.068 |
| Pipe | 21 | 22 | 23 | 24 | 25 | |||||
| Diameter(mm) | 1,050 | 1,050 | 450 | 450 | 450 | |||||
| Slope (%) | 1.463 | 1.064 | 0.171 | 0.609 | 0.312 | |||||
| The appropriate method using the dynamic wave routing | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Pipe | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Diameter(mm) | 450 | 450 | 450 | 525 | 525 | 525 | 300 | 525 | 450 | 450 |
| Slope (%) | 0.497 | 1.281 | 1.045 | 0.359 | 1.188 | 0.681 | 0.82 | 1.176 | 0.732 | 0.171 |
| Pipe | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Diameter(mm) | 525 | 675 | 375 | 750 | 750 | 375 | 900 | 900 | 900 | 900 |
| Slope (%) | 0.258 | 1.007 | 0.797 | 1.494 | 0.288 | 0.694 | 0.942 | 0.312 | 1.461 | 1.068 |
| Pipe | 21 | 22 | 23 | 24 | 25 | |||||
| Diameter(mm) | 1,050 | 1,050 | 450 | 450 | 450 | |||||
| Slope (%) | 1.463 | 1.064 | 0.171 | 0.609 | 0.312 | |||||
Each of the land uses plays an important role in the resulting damage. In particular, damage of lawn area and damage caused by traffic, which has not been considered so far by any researcher. Therefore, it can be concluded that considering the damage to the lawn area and traffic which were previously ignored by other researchers, can have a great impact on the design results. The damage to the traffic include wasting fuel and time, pollution generation in addition to the creation of public dissatisfaction due to the psychological burden.
OPTIMAL DESIGN USING THE RATIONAL METHOD
In this method, the rainfall intensity is calculated based on the time of concentration. The time of concentration for each conduit is considered equal to the inlet time plus the flow travel time through upstream conduits. Thus, as the flow is transmitted downstream, the concentration time will be increased and consequently, the rainfall intensity will be decreased. Optimal designs obtained by the rational method for various storm return periods are shown in Table 6.
Optimal design results using the rational method
| Return period | Frequency | Damage costs (dollars) | Annual damage risk cost (dollars) | Optimal Design cost (dollars) | Annual optimal design cost (dollars) | Total annual cost (dollars) |
|---|---|---|---|---|---|---|
| 1 | 1 | 41,600 | 2,019,763 | 37,718,326 | 3,212,268 | 5,232,031 |
| 2 | 0.5 | 2,175,680 | 1,465,443 | 43,547,903 | 3,708,764 | 5,174,207 |
| 5 | 0.2 | 2,936,960 | 698,547 | 44,088,532 | 3,754,774 | 4,453,321 |
| 10 | 0.1 | 3,922,880 | 355,555 | 46,100,176 | 3,926,124 | 4,281,679 |
| 20 | 0.05 | 4,501,120 | 144,955 | 67,054,801 | 5,710,723 | 5,855,678 |
| 25 | 0.04 | 4,680,000 | 99,049 | 88,009,427 | 7,495,321 | 7,594,370 |
| 50 | 0.02 | 5,224,960 | 0 | 125,727,753 | 10,707,590 | 10,707,590 |
| Return period | Frequency | Damage costs (dollars) | Annual damage risk cost (dollars) | Optimal Design cost (dollars) | Annual optimal design cost (dollars) | Total annual cost (dollars) |
|---|---|---|---|---|---|---|
| 1 | 1 | 41,600 | 2,019,763 | 37,718,326 | 3,212,268 | 5,232,031 |
| 2 | 0.5 | 2,175,680 | 1,465,443 | 43,547,903 | 3,708,764 | 5,174,207 |
| 5 | 0.2 | 2,936,960 | 698,547 | 44,088,532 | 3,754,774 | 4,453,321 |
| 10 | 0.1 | 3,922,880 | 355,555 | 46,100,176 | 3,926,124 | 4,281,679 |
| 20 | 0.05 | 4,501,120 | 144,955 | 67,054,801 | 5,710,723 | 5,855,678 |
| 25 | 0.04 | 4,680,000 | 99,049 | 88,009,427 | 7,495,321 | 7,594,370 |
| 50 | 0.02 | 5,224,960 | 0 | 125,727,753 | 10,707,590 | 10,707,590 |
Table 5 indicates that the 10-year return period is the optimal one. The optimal return periods determined by using the appropriate and rational methods are similar. But, the optimal design cost of the rational method is greater by 5 percent. The details of the calculations for the optimal return period of 10 years are shown in Table 7. Tc is the time of concentration and Tf is the flow travel time. The results compared to the appropriate method which is supposed to be the most precise one indicate that diameters of pipes (5, 6, 8, 13, 19, 20, 21, and 16) are increased and the design cost is increased by 5 percent.
Results of the design by the rational method
| Pipe | Length(m) | Increment (ha) | Cumulative area (ha) | Tc (min) | Tf (min) (2 ÷ 9) | Cumulative flow (m3/s) | Pipe dam (mm) | Velocity (m/s) | Design cost (dollars) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| p1 | 200 | 8.35 | 8.35 | 20 | 2.19 | 0.5 | 450 | 1.52 | 46,100,176 |
| p2 | 95 | 3.07 | 11.42 | 22.19 | 1.04 | 0.59 | 450 | 1.52 | |
| p3 | 100 | 4.5 | 15.92 | 23.23 | 0.74 | 0.76 | 450 | 2.26 | |
| p4 | 95 | 6.05 | 21.97 | 23.97 | 0.9 | 1.01 | 525 | 1.75 | |
| p5 | 80 | 7.05 | 29.02 | 24.88 | 0.79 | 1.05 | 600 | 1.68 | |
| p6 | 122 | 6.22 | 35.24 | 25.67 | 1.08 | 1.29 | 600 | 1.89 | |
| p8 | 96 | 4.08 | 39.32 | 26.75 | 0.76 | 1.46 | 600 | 2.1 | |
| p12 | 97 | 5.03 | 44.35 | 27.51 | 0.89 | 1.58 | 675 | 1.82 | |
| p7 | 100 | 4.08 | 4.08 | 20 | 1.19 | 0.12 | 300 | 1.4 | |
| p13 | 90 | 5.8 | 50.15 | 21.19 | 0.97 | 2.09 | 450 | 1.55 | |
| p14 | 150 | 20.88 | 115.38 | 50.55 | 0.77 | 2.62 | 750 | 3.24 | |
| p15 | 86 | 21 | 136.38 | 51.33 | 0.51 | 2.96 | 750 | 2.83 | |
| p9 | 110 | 5.7 | 5.7 | 20 | 1.8 | 0.22 | 450 | 1.02 | |
| p10 | 85 | 4.5 | 10.2 | 21.8 | 1.42 | 0.47 | 450 | 1 | |
| p11 | 56 | 4 | 14.2 | 23.21 | 0.83 | 0.61 | 525 | 1.13 | |
| p17 | 87 | 10.5 | 166.48 | 75.87 | 0.57 | 2.89 | 900 | 2.54 | |
| p18 | 99 | 5.78 | 172.26 | 76.44 | 0.66 | 3.02 | 900 | 2.5 | |
| p19 | 100 | 4.91 | 177.17 | 77.1 | 0.6 | 3.15 | 1,050 | 2.78 | |
| p20 | 102 | 5.54 | 182.71 | 77.7 | 0.61 | 3.3 | 1,050 | 2.77 | |
| p21 | 95 | 2.41 | 185.12 | 78.32 | 0.74 | 3.37 | 1,200 | 2.14 | |
| p16 | 86 | 3.77 | 3.77 | 20 | 1.59 | 0.23 | 450 | 0.9 | |
| p23 | 52 | 5 | 8.77 | 21.59 | 0.96 | 0.5 | 450 | 0.9 | |
| p24 | 46 | 2.06 | 10.83 | 22.56 | 0.68 | 0.55 | 450 | 1.12 | |
| p25 | 52 | 2.05 | 12.88 | 23.24 | 0.96 | 0.6 | 450 | 0.9 | |
| p22 | 63 | 6.51 | 227.88 | 103.26 | 0.33 | 3.67 | 1,050 | 3.2 |
| Pipe | Length(m) | Increment (ha) | Cumulative area (ha) | Tc (min) | Tf (min) (2 ÷ 9) | Cumulative flow (m3/s) | Pipe dam (mm) | Velocity (m/s) | Design cost (dollars) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| p1 | 200 | 8.35 | 8.35 | 20 | 2.19 | 0.5 | 450 | 1.52 | 46,100,176 |
| p2 | 95 | 3.07 | 11.42 | 22.19 | 1.04 | 0.59 | 450 | 1.52 | |
| p3 | 100 | 4.5 | 15.92 | 23.23 | 0.74 | 0.76 | 450 | 2.26 | |
| p4 | 95 | 6.05 | 21.97 | 23.97 | 0.9 | 1.01 | 525 | 1.75 | |
| p5 | 80 | 7.05 | 29.02 | 24.88 | 0.79 | 1.05 | 600 | 1.68 | |
| p6 | 122 | 6.22 | 35.24 | 25.67 | 1.08 | 1.29 | 600 | 1.89 | |
| p8 | 96 | 4.08 | 39.32 | 26.75 | 0.76 | 1.46 | 600 | 2.1 | |
| p12 | 97 | 5.03 | 44.35 | 27.51 | 0.89 | 1.58 | 675 | 1.82 | |
| p7 | 100 | 4.08 | 4.08 | 20 | 1.19 | 0.12 | 300 | 1.4 | |
| p13 | 90 | 5.8 | 50.15 | 21.19 | 0.97 | 2.09 | 450 | 1.55 | |
| p14 | 150 | 20.88 | 115.38 | 50.55 | 0.77 | 2.62 | 750 | 3.24 | |
| p15 | 86 | 21 | 136.38 | 51.33 | 0.51 | 2.96 | 750 | 2.83 | |
| p9 | 110 | 5.7 | 5.7 | 20 | 1.8 | 0.22 | 450 | 1.02 | |
| p10 | 85 | 4.5 | 10.2 | 21.8 | 1.42 | 0.47 | 450 | 1 | |
| p11 | 56 | 4 | 14.2 | 23.21 | 0.83 | 0.61 | 525 | 1.13 | |
| p17 | 87 | 10.5 | 166.48 | 75.87 | 0.57 | 2.89 | 900 | 2.54 | |
| p18 | 99 | 5.78 | 172.26 | 76.44 | 0.66 | 3.02 | 900 | 2.5 | |
| p19 | 100 | 4.91 | 177.17 | 77.1 | 0.6 | 3.15 | 1,050 | 2.78 | |
| p20 | 102 | 5.54 | 182.71 | 77.7 | 0.61 | 3.3 | 1,050 | 2.77 | |
| p21 | 95 | 2.41 | 185.12 | 78.32 | 0.74 | 3.37 | 1,200 | 2.14 | |
| p16 | 86 | 3.77 | 3.77 | 20 | 1.59 | 0.23 | 450 | 0.9 | |
| p23 | 52 | 5 | 8.77 | 21.59 | 0.96 | 0.5 | 450 | 0.9 | |
| p24 | 46 | 2.06 | 10.83 | 22.56 | 0.68 | 0.55 | 450 | 1.12 | |
| p25 | 52 | 2.05 | 12.88 | 23.24 | 0.96 | 0.6 | 450 | 0.9 | |
| p22 | 63 | 6.51 | 227.88 | 103.26 | 0.33 | 3.67 | 1,050 | 3.2 |
CONCLUDING
In this study, using the genetic optimization algorithm, which is integrated with the SWMM simulation model, a method is proposed to reduce the cost of storm sewer network by considering constraints and employing the optimal return period in which total annual design costs and annual damages are minimized. The simulation requires hydrologic and hydraulic calculations. The hydrologic calculations involve the rainfall–runoff relationships and determination of surface runoff in the sub-catchments. The hydraulic calculations deal with flow routing in the storm sewer network. Several methods are available for hydrologic and hydraulic simulations. The nonlinear reservoir model to estimate the surface runoff produced by rainfall over a sub-catchment is the most appropriate and precise one. The rational method is the simplest and has less accuracy compared to the nonlinear reservoir model. Many hydraulic methods are available to simulate the flow in the sewers. The dynamic wave method is the most appropriate and precise one. Then, in order to reduce design costs, the optimal dimensions of pipes were determined by combining the simulation and optimization with one model. In this model, the objective function is based on minimizing the design cost. Then, in order to perform the risk analysis in determining the optimal return period, the damage caused by runoff was calculated. The slope and diameter of the pipes were considered as the decision variable. For each method, risk analysis was performed separately, resulting in the optimal design return period, in which the total design costs and risk of damage are minimized.
In this research, the SWMM program is linked to an optimization program that uses the GA to develop a simulation–optimization program for the design of storm sewer networks. The performance of the developed program in terms of speed and accuracy was checked by solving a benchmark problem. The results indicated good and successful performance.
The developed program was applied for a case study design problem in a district located in Tehran city. The optimal designs for various storm return periods were obtained. To determine the optimal storm return period, risk analysis was utilized. The damage due to storm flooding including residential and commercial buildings, roads, lawn areas, and traffics are considered in a case study of a district in Tehran city. This damage is represented in developed equations in terms of flooding depths. The design return periods for different simulation methods were determined by using risk analysis in which the summation of the design and flooding risk costs is minimal. The GA optimization solver was linked to various storm sewer simulation methods.
The results indicated that the optimal design cost obtained by the rational simulation method is only 5 percent greater than that of the dynamic wave method. Thus, it can be concluded that the rational method is powerful and has an acceptable accuracy in addition to its simplicity and being on the safe side.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST STATEMENT
The authors declare there is no conflict.
%20cropped.png?versionId=7151)