Sustainable urban drainage systems are multi-functional nature-based solutions that can facilitate flood management in urban catchments while improving stormwater runoff quality. Traditionally, the evaluation of the performance of sustainable drainage infrastructure has been limited to a narrow set of design objectives to simplify their implementation and decision-making process. In this study, the spatial design of sustainable urban drainage systems is optimized considering five objective functions, including minimization of flood volume, flood duration, average peak runoff, total suspended solids, and capital cost. This allows selecting an ensemble of admissible portfolios that best trade-off capital costs and the other important urban drainage services. The impact of the average surface slope of the urban catchment on the optimal design solutions is discussed in terms of spatial distribution of sustainable drainage types. Results show that different subcatchment slopes result in non-uniform distributional designs of sustainable urban drainage systems, with higher capital costs and larger surface areas of green assets associated with steeper slopes. This has two implications. First, urban areas with different surface slopes should not have a one-size-fits-all design policy. Second, spatial equality must be taken into account when applying optimization models to urban subcatchments with different surface slopes to avoid unequal distribution of environmental and human health co-benefits associated with green drainage infrastructure.

Global climate change, rapid expansion of cities, and the aging of existing urban drainage infrastructure raise new challenges for urban flood management (Raei et al.

The design of sustainable urban drainage systems is a daunting task due to their inherent hydrological and hydraulic complexity together with the conflicting stakeholder interests that often characterize urban planning (Horgan and Dimitrijević

Despite the extensive literature on the subject, most of the simulation-optimization studies address one to three design goals, which are insufficient to comprehensively assess the co-benefits of sustainable drainage infrastructure. Moreover, there is still a paucity of insight into the effect of the average surface slope on the spatial distribution of sustainable drainage system components when this is determined using optimization models. The importance of this lies in the fact that subcatchment slopes can affect the pattern of stormwater detention and infiltration resulting in a biased distribution of floods in cities with various topographic features. Accordingly, when using an optimization model on this subject, the search algorithm may find a sustainable drainage system cost-effective where specific drainage system components are allocated to particular subregions. Although the optimization solution may be efficient in terms of flood management, it can raise concerns about social justice and spatial equality, one of the pillars of the sustainable development goals, in urban drainage system design (Zheng et al.

This study shows how the average surface slope of urban catchments can impact equality in the spatial distribution of sustainable drainage components in urban areas if an optimization model is used to support design decisions. To this end, we apply a many-objective optimization approach to a synthetic case study under different slope scenarios. We introduce parallel axis plots laid alongside system design maps as a summary graphical representation of optimization results for stakeholder deliberations. Results show that urban areas with varying slopes within the same catchment should not have a one-size-fits-all sustainable drainage design. At the same time, care should be taken in ensuring that differences in average surface slope do not result in an unequal distribution of co-benefits associated with green drainage infrastructure.

The simulation of an urban drainage system requires a rainfall-runoff and hydraulic routing model. In this study, the simulations were carried out using the Storm Water Management Model (SWMM) developed by the U.S. Environmental Protection Agency (Rossman _{f} is the friction slope, and _{L} is the local energy loss per unit length of conduit.

The sustainable drainage assets considered in this study include permeable pavements, infiltration trenches, bio-retention cells, rain gardens, rain barrels, and green roofs. Since each of these assets has different performance characteristics, their efficient combination can help achieve an effective design for a specific urban drainage system (Leng et al.

A 29-ha synthetic urban drainage system case study with 8 subcatchments, 13 junctions, and 13 conduits, was selected to demonstrate the design formulation described above and investigate the relationship between average surface slope and drainage element performance (Fig.

Schematic map of the synthetic case study

A synthetic 100-year, 2-h hyetograph with 5-min increments was defined using the Alternating Block Method as an extreme rainfall event. The impervious surfaces were assumed to be composed of rooftops and driveways with equal ratios of surface areas. Two land-use classifications were defined, including residential and undeveloped areas, and the Event Mean Concentration method was applied to estimate wash-off load of total suspended solids. To maximize efficiency of the sustainable drainage system, the decision variables consider combinations of two sustainable drainage types and their surface areas, represented by four integer values in each subcatchment. The surface area of the sustainable drainage components was parameterized as a percentage of the impervious surfaces in each subcatchment. The maximum allowable surface area was set to 15% of the impermeable area of each subcatchment. The area of the subcatchments, land coverage and slope scenarios are summarized in Table

Subcatchment settings for the case study

Subcatchment | Surface area (ac) | Coverage | Average surface slope (%) | |||
---|---|---|---|---|---|---|

Residential | Undeveloped | Scenario 1 | Scenario 2 | Scenario 3 | ||

S1 | 10 | 100% | – | 0.01 | 3 | 6 |

S2 | 10 | 100% | – | 0.01 | 3 | 6 |

S3 | 5 | 100% | – | 0.01 | 3 | 6 |

S4 | 5 | 100% | – | 0.01 | 3 | 6 |

S5 | 15 | 75% | 25% | 0.01 | 3 | 6 |

S6 | 12 | 100% | – | 0.01 | 3 | 6 |

S7 | 4 | 100% | – | 0.01 | 3 | 6 |

S8 | 10 | 50% | 50% | 0.01 | 3 | 6 |

We link the Controlled NSGA-II (CNSGA-II) (Deb and Goel

As mentioned, several simultaneous benefits may be sought in sustainable urban drainage infrastructure design related to efficiency of a drainage system in reducing flood damages and improving its environmental performance (CRC for Water Sensitive Cities

In this paper, the following five objective functions are considered:
_{Cost} is capital cost, _{FloodV} is total flood volume, _{FloodD} is flood duration, _{PeakR} is peak runoff, and _{TSS} is total suspended solids (TSS).

The capital cost was calculated for the urban catchment as follows:
_{s} is the number of subcatchments, _{ij} and _{ij} are the surface area and capital cost of each drainage component, respectively. The capital costs for the drainage assets were extracted from databases published by Herrera Environmental Consultants (

The total flood volume is defined as:
_{i} is flood volume at the

The average manhole flood duration in the urban catchment is defined as:
_{i} is flood duration at the _{f} represents the number of flooded nodes.

Peak runoff is defined as:
_{i} is the peak runoff in each subcatchment.

Finally, the overall total suspended solids (TSS) load was extracted from the numerical results.

To represent the locations of sustainable drainage components, the crossover, mutation, and reproduction operators in the genetic algorithm were adapted to produce integer-valued individuals. Moreover, as an optimization constraint, solutions with two identical sustainable drainage types in each subcatchment were flagged as infeasible solutions. This, however, does not prevent the model from finding solutions with just one type of sustainable drainage systems or even a no-intervention option in a subcatchment, as these can be obtained by selecting the no-intervention option or zero surface area for sustainable drainage assets. The population size of 200 was selected based on a crowding spread study. A function tolerance of 10^{−3} for 100 consecutive iterations was used as the stopping criterion, which resulted in around 22,000 function evaluations before the optimization stopped.

Many-objective optimization allows analysts and their stakeholder clients to identify Pareto-optimal engineered water system designs and their performance trade-offs considering multiple metrics of performance. The term “many-objective” (Fleming et al.

To illustrate here how a range of high-value designs can be extracted from the Pareto-optimal solution set provided by the many-objective optimization, we look at three example design solutions that correspond to alternative sets of stakeholder priorities. The first set of priorities selects the least-cost drainage system design that fits within a prescribed range of acceptable flood volume and flood duration. Such a design might be sought if priority is given to reducing flood damages and securing normal transportation traffic near flooded manholes. The second design selects the least-cost option amongst designs that fit within a prescribed range of flood volume and average peak runoff. Finally, a third design option is chosen which corresponds to the least-cost option that meets a given constraint on the total suspended solids.

Figure

Five-dimensional representation of the Pareto-front for the selected case study considering three surface slope scenarios including (

Although these plots are accurate and complete, they do not lend themselves easily to urban stakeholder learning and design deliberation. To enable this, we use parallel axis plots (Inselberg

Many-objective optimization of sustainable drainage infrastructure in a flat urban catchment prioritizing flood attenuation;

Many-objective optimization of sustainable drainage infrastructure in an urban catchment with an average surface slope of 3% prioritizing flood attenuation;

Many-objective optimization of sustainable drainage infrastructure in an urban catchment with an average surface slope of 6% prioritizing flood attenuation;

The results show the value of applying a many-objective optimization approach when there are multiple design goals that facilitate the necessary functionality of sustainable urban drainage systems. For example, in Fig. ^{3} to 582,000 m^{3} in regions with steeper surface slopes while the mean peak runoff and total suspended solids are reduced by 57% and 70%, respectively. The results also imply that the average surface slope can bias the search algorithm in favor of specific types of sustainable urban drainage components. For instance, larger surface areas of rain gardens are found to be preferable in steeper slope scenarios compared to small slopes. However, no significant change was observed in surface areas of green roofs in response to changes in the surface slope, whereas the optimization suggests the use of rain barrels only for steeper surface slopes. Here, the number of barrels can be obtained based on the surface area values of interventions allocated to each subcatchment. For example, in Figs.

Using the same procedure described above, six portfolios were extracted from the set of Pareto-optimal solutions according to the second and third set of preferences. Figure

Bar chart representation of the Pareto-optimal sustainable urban drainage infrastructure for each catchment surface slope scenario according to;

Figure

Sunburst diagram summarizing surface areas of the selected sustainable urban drainage system designs for each surface slope scenario and design preference. The figure shows the impact of average surface slope on sustainable urban drainage design obtainable from an optimization model

Urban drainage system design is a complex problem, which necessitates several performance criteria to facilitate sustainability and resilience of cities against floods. Large cities are usually characterized by spatial variations of surface slopes, affecting infiltration and detention patterns of stormwater runoff. Surface slope is an important topographic factor that can influence the efficiency of sustainable urban drainage components. This work has demonstrated the use of a many-objective optimization approach for selecting portfolios of drainage infrastructure within an urban catchment with three average surface slope scenarios. The Storm Water Management Model (SWMM) was linked to an evolutionary optimization algorithm (CNSGA-II) to search for Pareto-optimal configurations of sustainable drainage assets in several urban subcatchments interconnected by a conventional drainage network. For each subcatchment, the algorithm selects a combination of two types of drainage assets from amongst seven different options and determines the efficient surface areas of each component type by five design objectives, i.e., minimizing capital cost, flood volume, flood duration, average peak runoff, and total suspended solids. To demonstrate the selection of particular drainage designs corresponding to different trade-offs between the design objectives, the solution space was narrowed down by filtering specific optimization objectives according to stakeholder preferences and/or environmental constraints. Different visualization techniques were employed to analyze the results, including a novel plot where a system design schematic is placed alongside a parallel axis trade-off plot. This optimization approach was applied to urban catchments with three different slope scenarios to investigate how surface slope impacts the design of sustainable urban drainage systems. It was found that variations of surface slopes in an urban area play an important role in controlling the optimal distribution of sustainable drainage components, suggesting higher investment in green infrastructure in subcatchments with steeper surface slopes. However, since sustainable drainage assets provide a set of co-benefits for both the environment and human health, an unbalanced distribution of sustainable drainage assets in large urban areas may raise equality concerns.

The application of optimization models to large urban drainage networks is hindered by their extensive computational requirements as the optimization time increases exponentially with the number of decision variables. Future work should develop strategies for application of similar optimization approaches to larger drainage networks and consider equality and equity metrics to ensure fairness in the distribution of green infrastructure benefits.

This work was jointly supported by Thames Water Utilities Ltd. and The University of Manchester.

This work was jointly supported by Thames Water Utilities Ltd. and The University of Manchester.

Not applicable.

The results of this work were obtained using a custom-made Matlab code and the software SWMM developed by the US Environmental Protection Agency.

The authors declare that they have no conflict of interest in this work.

Not applicable.

All of the authors consent to participate in this research work.

All of the authors consent to publish this work.

Flow cross-sectional area

Surface area of sustainable drainage assets

Capital cost

_{Cost}

Capital cost objective

_{FloodV}

Flood volume objective

_{FloodD}

Flood duration objective

Flood duration

Flood volume

Objective function vector

Gravity acceleration

_{L}

Local energy loss per unit length of conduit

Number of sustainable drainage assets

Hydraulic head

Peak runoff

Discharge

Number of manholes

_{f}

Number of flooded nodes

_{s}

Number of subcatchments

_{f}

Friction slope

Time

Distance

Vector of decision variables

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.