This research is being carried out to study how best management practices (BMPs) can mitigate the negative effects of urban floods during extreme rainfall events. Strategically placing BMPs throughout open areas and rooftops in urban areas serves multiple purposes of storage of rainwater, removal of pollutants from surface runoff and sustainable utilisation of land. This situation is framed as a multiobjective optimisation problem to analyse the trade-offs between multiple goals of runoff reduction, construction cost and pollutant load reduction. Output includes a wide range of choices to choose from for decision makers. Proposed methodology is demonstrated with a case study of Greater Hyderabad Municipal Corporation (GHMC), India. A historical extreme rainfall event of 237.5 mm which occurred in 2016 and extreme rainfall event of 1,740.62 mm corresponding to representative concentration pathway (RCP) 2.6 were considered for analysis. Two multiobjective optimisation algorithms, namely non-dominated sorting genetic algorithm-III (NSGA-III) and constrained two-archive evolutionary algorithm (C-TAEA) are used to solve the BMP placement problem, following which the resulting Pareto-fronts are ensembled. K-Medoids-based cluster analysis is performed on the resulting ensembled Pareto-front. The proposed ensembled approach identified ten possible BMP configurations, with costs ranging from Rs. to surface runoff reduction ranging from to and pollutant load removal ranging from tonnes. Use of BMPs in future events has the potential to reduce surface runoff from , while simultaneously removing tonnes of pollutants for cost ranging from The proposed framework forms an effective and novel way to characterise and solve BMP optimisation problems in context of climate change, presenting a view of the urban flooding scenario today, and the likely course of events in the future.

  • The multiobjective BMP problem is solved using NSGA-III and C-TAEA.

  • Cluster analysis is performed using K-Medoids.

  • Accounting for climate change through RCP 2.6.

Urban floods, caused by extreme rainfall or peak discharge events, have been extensively modelled and studied in multiple case studies from India (Bisht et al. 2016; Vemula et al. 2019) and globally (Einfalt et al. 2009; Audisio & Turconi 2011). These floods cause massive damage even after relatively short durations of rain (Deekshith 2020). This problem is likely to be compounded in the near future because of rapid urbanisation of cities (Franco et al. 2017) coupled with climate change, which increases the frequency of extreme rainfall events. Being able to mitigate the urban flooding problem even partially would lead to significant savings, in terms of cost and improvements in the quality of life, making this problem worth investigating.

The use of best management practices (BMPs) has been frequently discussed in literature and was found to be a very effective strategy in managing storm water runoff, reducing flash floods and conserving water (Li et al. 2019a). While there is ample evidence attesting to the effectiveness of BMPs, the best or most optimal way of placing BMPs is not obvious. Adding to the complexity of the decision-making process, there are multiple types of BMPs that can be built – bioretention, rain barrels, infiltration trenches, vegetated rooftops or permeable pavements, each of which has restrictions on where it can be placed. Over the years, numerous researchers studied optimisation of BMP placement to achieve hydrological as well as cost perspectives, which are described below.

Early literature in the field indicates that researchers use optimisation algorithms such as scatter search, integer programming, or binary linear integer programming (BLIP) (Zhen et al. 2004; Loáiciga et al. 2015) for solving single objective BMP optimisation problems. Reddy & Kumar (2021) reviewed swarm intelligence-based algorithms in water resources engineering, for example, flood management practices, design of detention systems and BMPs.

Srivastava et al. (2003) used the annualised agricultural non-point source (AnnAGNPS) watershed model in conjunction with a genetic algorithm (GA) to optimise placement of agricultural BMPs in Mahantango Creek, Pennsylvania, USA. Veith et al. (2003) experimented with five different heuristic search algorithms to identify the cost-effective placement strategy for BMPs for a watershed in the Ridge & Valley region, Virginia, USA where GA was found suitable. Hsieh & Yang (2007) employed discrete differential dynamic programming to minimise BMP costs in Fei-Tsui Reservoir Watershed Area, Taiwan. Gitau et al. (2006) used GA as well as the soil and water assessment tool (SWAT) to minimise costs of phosphorus removal below a certain user-defined threshold in Town Brook Watershed, New York. Artita et al. (2013) employed SWAT and species conserving GA to perform BMP optimisation with minimum cost to a case study of Silver Creek Watershed, Southern Illinois, USA. Shamsudin et al. (2014) optimised placement of detention pond BMPs using analytical probabilistic models (APMs) and particle swarm optimisation (PSO) for two case studies in Malaysia. PSO was found to be better compared to the APM method. Loáiciga et al. (2015) applied linear programming and BLIP for optimising placement of BMPs for City of Los Angeles, USA. More recent approaches employed multiobjective optimisation algorithms, especially certain evolutionary algorithms (Reddy & Kumar 2021). Unlike single objective algorithms which often converge to one optimal solution, a multiobjective algorithm identifies a Pareto-front consisting of a set of solutions.

Rodriguez et al. (2011) analysed trade-offs between minimisation of pollution and net cost for Lincoln Lake Watershed. They used multiobjective GA to generate Pareto-front. Zare et al. (2012) modelled the BMP placement problem with three objectives to a case study of Tehran, Iran. Non-dominated sorting genetic algorithm-II (NSGA-II) generated 200 Pareto-optimal solutions. These were then clustered using K-Medoids to reduce the number of choices to 10. Karamouz & Nazif (2013) used NSGA-II in conjunction with the storm water management model (SWMM) to Tehran, Iran. They used global climate model (GCM)-HadCM3 to simulate climate change scenarios. Chen et al. (2015) applied NSGA-II to Daning River watershed in the Three Gorges Dam area. They also compared NSGA-II with GA, and found that NSGA-II Pareto-front was more narrowly concentrated, leading to better solutions. Aminjavaheri & Nazif (2018) used GA, NSGA-II in conjunction with SWMM and other supporting methods for a case study of Tehran, Iran. The objective was to determine the optimal BMP placement in data-poor catchments.

A few studies were also performed using dual drainage models (DDMs), namely PCSWMM, Info SWMM and SWMM-CADDIES to study the interaction between the surface and sewer runoff and their use in connection with BMPs in urban areas. He (2015) studied the DDM and its capacity in John Street Watershed, Champaign, Illinois, USA under minor and major drains and compared it with SWMM. Results showed that DDM worked satisfactorily under major high intensity storms. Nanía et al. (2019) studied the efficacy of implementing green infrastructure such as rain barrels, pervious pavements and green roofs along with a DDM in the Metropolitan Area of Chicago, Illinois, USA. They concluded that rain barrels have limited effectivity of <0.5% in minimising surface runoff, while both green roofs and pervious pavements produce runoff reductions of 7.47% and 5.66%, respectively. Yin et al. (2020) analysed the use of low impact development (LID) facilities in Zhuhai, Guangdong province, China, using SWMM and cellular automata dual-drainage simulation (CADDIES) 2D model and concluded that LIDs are effective at reducing surface runoff during small to medium-sized rainfall events. Panos et al. (2021) studied the dual drainage system of Denver, Colorado, USA. They assessed how the storm network would perform under various rainfall conditions and the changes in land use. They observed that adding distributed bioretention units throughout the study area helped in increasing the capacity of storm water network.

While many variants of conventional GA and NSGA-II have been applied to the BMP optimisation problem, it was observed that the usage of non-dominated sorting genetic algorithm-III (NSGA-III) and constrained two-archive evolutionary algorithm (C-TAEA) has been limited or non-existent for optimising the placement of BMPs. In addition, from a Pareto-front with thousands of possible BMP placement options, selecting suitable option(s) becomes very challenging. This can be facilitated with clustering algorithms (Zio & Bazzo 2011). Accordingly, the following aspects are discussed as a part of the manuscript:

  • 1.

    Optimal BMP placement situation is explored as a multiobjective optimisation problem involving runoff reduction, construction cost and pollutant load reduction.

  • 2.

    Two multiobjective optimisation algorithms, NSGA-III and C-TAEA, are used for identification of Pareto-optimal solutions.

  • 3.

    A non-dominated sort is applied on individual solutions to arrive at an ensembled Pareto-set. Solutions of this set are clustered to identify and group similar solutions, using the K-Medoids algorithm.

This framework is applied to a case study of Greater Hyderabad Municipal Corporation (GHMC), India. The following sections present an overview of the case study and data collection, methodology/methods employed, followed by results and conclusions.

GHMC has an area of 625 km2 and partitioned into 16 zones based on storm water network. The study area falls in the catchment of Musi River, whose total drainage area is 11,000 km2. GHMC has an annual average rainfall of nearly 840 mm (Agilan & Umamahesh 2015). Figure 1 presents topography of the region under study. Elevation of the study area varies from 462 to 635 m above mean sea level. Due to its undulating terrains, water tends to collect in the low-lying areas (Vemula et al. 2019). The yellow points in Figure 1 represent flood nodes or critical points that are the most susceptible to floods during extreme events. Hence, there is a need for runoff reduction that can be facilitated by implementing BMPs.

Figure 1

GHMC basin topography. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wst.2021.283.

Figure 1

GHMC basin topography. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wst.2021.283.

Close modal

Figure 2 depicts the land use-land cover (LULC) and drainage network of the GHMC. Figure 2 shows that most of the area is covered under buildings (red colour) or roads (grey colour), representing a typical urban area. The black arrows represent the storm water drainage network of GHMC.

Figure 2

GHMC land use and drainage network.

Figure 2

GHMC land use and drainage network.

Close modal

The present study uses data collected during a nine-day historic extreme rainfall event during 20–28 September, 2016. Rainfall and runoffs for this historic event are presented in Figure 3. Highest amount of rainfall was recorded on 23 September 2016. GHMC experienced a weighted average rainfall of 237.5 mm during this extreme event. Data on storm drainage network, soil, LULC, runoff, inflow, and water quality at Hussain Sagar (Madhuri et al. 2021) was collected.

Figure 3

Rainfall and runoff in the GHMC area during the historic extreme event of 20–28 September, 2016.

Figure 3

Rainfall and runoff in the GHMC area during the historic extreme event of 20–28 September, 2016.

Close modal

In addition, the present work attempts to account for climate change by considering a future extreme rainfall event of 1,740.62 mm, corresponding to the representative concentration pathway (RCP) 2.6 scenario based on GCM, Geophysical Fluid Dynamics Laboratory – Climate Model version 3 (Vemula et al. 2019; Swathi et al. 2020; Madhuri et al. 2021). Here, 1,740.62 mm rainfall refers to an extreme event of magnitude 580.52 mm, 579.67 mm and 580.44 mm for three consecutive days for 2040.

Hydrologic engineering center-hydraulic modelling system (HEC-HMS) was employed to simulate surface runoff of a catchment (Sarminingsih et al. 2019). The runoffs generated at outlet for historic and future event are 190.45 mm and 1,027.28 mm respectively, which were calculated based on calibrated parameters of HEC-HMS.

The US Environmental Protection Agency (EPA)-SUSTAIN siting tool was used to identify potential BMP sites (EPA-SUSTAIN 2014; Gao et al. 2015). The SUSTAIN tool takes multiple topographical features including drainage area, slope, imperviousness, soil type, and land use for identifying possible locations to place BMPs (see Table S1 in Supplementary Material for details about feasibility constraints). The BMPs processed by the siting tool can be grouped into two broad categories – non-rooftop BMPs (grassed swales, sand filter, bioretention, vegetated filter strip, porous pavement, wet pond and constructed wetland) and rooftop BMPs (infiltration trench, infiltration basin and rain barrel). Rooftop BMPs processed by the siting tool do not overlap with each other; however, a single location of non-rooftop area can be potentially suitable for many of these BMPs, and in such cases, any type of BMP is selected and can be placed in that particular location. On the other hand, rain barrels can be placed on rooftop areas of size less than 50 m2, while infiltration trenches are used for rooftop areas between 50 m2 and 200 m2, and infiltration basins are used for rooftops greater than 200 m2 (APWALTA 2002; HMDA 2009; HMWSSB 2012). In total, 545,895 possible BMP sites were identified in this process using the BMP siting tool, EPA-SUSTAIN. Figure 4 presents spatial locations of BMPs and shows is seen that rooftop areas on buildings are covered with infiltration basins (light blue), infiltration trenches (pink) and rain barrels (purple).

Figure 4

Distribution of identified BMPs in the GHMC.

Figure 4

Distribution of identified BMPs in the GHMC.

Close modal

More details about the results of HEC-HMS, EPA-SUSTAIN and BMP placement are available from the corresponding author on request. The next section presents methodology.

Methodology used for optimising BMP placement (after identifying a set of all BMPs) is presented in Figure 5. Details about mathematical model formulation including objective function, decision variables and constraints, description about employed optimisation algorithms and clustering are explained in this section.

Figure 5

Methodology for optimising BMP placement.

Figure 5

Methodology for optimising BMP placement.

Close modal

Objective functions and decision variables

Each BMP has the capacity to reduce ‘a certain amount of’ runoff (, in ), while reducing some pollutant load (, in ), and this will require ‘a certain amount of’ capital expense to construct. For a single BMP , these objectives can be expressed as:
formula
(1)
formula
(2)
formula
(3)
where are area, runoff reduction efficiency, depth and pollutant reduction efficiency, respectively, of the BMP; × is the runoff at top of ith BMP. The value of k depends on the market costs of construction in the area. The values of for each type of BMP are reported in Table 1 (Battiata et al. (2010) for vegetated filter strip; Rao & Surinaidu (2012) for bioretention; and HMWSSB (2012) for the remaining BMPs) Table 1 shows that of each type of BMP varies from 10% to 80%, while varies from 0% to 75%.
Table 1

Pollutant and runoff reduction efficiencies of BMPs

S. no. (1)BMP type (2)Notation (3) (4) (5)
Grassed swales GS 35 60 
Sand filter surface SF 45 50 
Bioretention BR 60 80 
Vegetated filter strips VF 50 57 
Porous pavement PP 25 75 
Wet ponds WP 75 10 
Constructed wetland CW 75 10 
Infiltration basin IB 15 70 
Infiltration trench IT 25 70 
10 Rain barrel RB 70 
S. no. (1)BMP type (2)Notation (3) (4) (5)
Grassed swales GS 35 60 
Sand filter surface SF 45 50 
Bioretention BR 60 80 
Vegetated filter strips VF 50 57 
Porous pavement PP 25 75 
Wet ponds WP 75 10 
Constructed wetland CW 75 10 
Infiltration basin IB 15 70 
Infiltration trench IT 25 70 
10 Rain barrel RB 70 
It is noted that all three objectives are linearly proportional to area, and can be rewritten as:
formula
(4)
where α, β and γ are fixed values for each BMP. Now that this relationship has been established for a single BMP, the next step is to define the optimisation problem formally. Given a set of potential BMP sites (obtained through the EPA-SUSTAIN tool), Pareto-optimal configurations to place BMPs need to be identified. A BMP configuration is a subset of all possible BMP sites, chosen to be constructed. For each BMP site, none of the area, a fraction of the area, or all of the area(s) can be used to construct a BMP. If multiple BMPs () have same values of the tuple, such that (α, β, γ) = , they all can be treated equivalently.
formula
(5)
where is area of the BMP, . BMPs are grouped according to Equation (5) to form buckets – all BMPs that have the same values of [α, β, γ] will form a bucket with total area , which is defined as:
formula
(6)
After this process, K buckets are formed. The total runoff reduction, total monetary cost () and total pollutant load reduction () of a BMP configuration are defined in terms of these K buckets , which have areas . Here, the subscript T means ‘total’. In each bucket, all of the area, a fractional of it, or none of it can be selected; and the proportion of selected area is denoted by .
formula
(7)

Constraints

As noted in the previous section, BMP sites processed by the siting tool might contain overlapping BMPs, which is something that the optimisation approaches consider as a constraint.

A multipolygon, M, is defined as a set of polygons. It is noticed that each individual BMP site is a polygon. There are seven types of non-rooftop BMPs, forming seven multipolygons, denoted as M1, M2…M7. For each candidate solution , the area used by each type of BMP is calculated, and is denoted by Z1, Z2Z7. The multipolygon union operation, Union (Mi, Mj), takes multiple multipolygons and combines the areas of both to form a multipolygon. To ensure that one piece of land is not used more than once, the following constraints are added:
formula
(8)
formula
(9)
formula
(10)

Now that the multiobjective optimisation decision variables and constraints have been setup, the next section describes the optimisation algorithms.

Optimisation algorithms

NSGA-III (Jain & Deb 2014) was proposed for optimisation of many objective functions by building (up) on the NSGA-II background (Deb et al. 2002). Main steps in the optimisation process are:

  • Evaluate objective values of current population, Pi, and perform a non-dominated sort to rank the members by the level of domination in objective space. Population points for the next generation (Pi+1) are created through selection, crossover and mutation.

  • Create H reference lines passing through the origin distributed throughout M-dimensional objective space, for an optimisation problem with M objectives. NSGA-III uses normalised objective space with reference directions/points.

  • Calculate the perpendicular distance of each point of the population from the reference lines. Points are assigned to the ‘closest’ reference line (the least perpendicular distance).

  • After this assignment, a niche preservation operation is performed, in which points associated with each vertex are added to the next generation, Pi+1 (Deb & Jain 2014).

C-TAEA (Li et al. 2019b) solves constrained multiobjective problems while attempting to balance conflicting requirements of convergence, diversity and feasibility.

  • The algorithm does this by maintaining two separate archives (or lists) called convergence-oriented archive (CA) and diversity-oriented archive (DA), both of which contain P points. The CA stores Pareto-front at the end of each generation, while the DA is used to explore relatively less explored areas of the Pareto-front. It requires only one hyperparameter, i.e. set of reference directions (consisting of H elements). The algorithm uses a population size (P) such that P=H.

  • During this reproduction step, C-TAEA updates CA and DA separately, and select parents from the previous CA and DA using a modified tournament selection.

Hypervolume (Fonseca et al. 2006) and running metric (Blank & Deb 2020a) are used to assess the performance of optimisation algorithms. Hypervolume is a commonly used metric for tracking the progression of multiobjective optimisation problems. Hypervolume works by measuring the volume bound by the Pareto-front and the nadir point after every generation. Nadir point is defined as the point in objective space, constructed by choosing the worst possible objective values for each objective. Running metric, which is based on inverse generational distance (IGD), evaluates the difference in objective space over generations.

Clustering Pareto-optimal solutions

The K-Medoids clustering algorithm (Park & Jun 2009) is used to cluster the Pareto-front into C clusters. The C cluster centres formed with this are guaranteed to be a part of the original input data, as opposed to K-Means, for which no such guarantee exists. A suitable value of C can be determined by experimenting with a wide range of values and analysing trends in error as the number of clusters changes.

This section contains discussion on the results related to bucketing, optimisation and clustering.

Bucketing

The 545,895 BMPs (see the data collection section) are bucketed to form 91 non-rooftop buckets and 23 rooftop buckets for the historic event (columns 3 and 4 of Table 2), based on Equations (1)–(4) which also depend on rainfall and runoff, whereas the number of buckets for future events are 21 and 3, respectively (columns 5 and 6 of Table 2). Each bucket corresponds to one decision variable in the optimisation, representing the proportion of area in the bucket to be chosen. All optimisation programs for running these experiments were written in Python3 with PyMoo framework, which has been extensively used and calibrated on optimisation of benchmark problems (Blank & Deb 2020b). GeoPandas is used for preprocessing.

Table 2

Number of rooftop and non-rooftop BMPs

BMP type (1)Number of BMPs (area in km2) (2)Number of buckets in historic event (3)Number of BMPs in each bucket along with area in km2 (in parenthesis) for historic event (4)Number of buckets in future event (5)Number of BMPs in each bucket along with area in km2 (in parenthesis) for future event (6)
GS 2,713 (0.88) 14 15 (0.004), 19 (0.009), 19 (0.006), 45 (0.018), 48 (0.016), 55 (0.023), 86 (0.023), 115 (0.053), 118 (0.042), 127 (0.037), 405 (0.15), 496 (0.147), 523 (0.142), 642 (0.219) 100 (0.041), 216 (0.08), 2,397 (0.769) 
SF 4,243 (2.65) 14 16 (0.004), 36 (0.038), 40 (0.03), 56 (0.037), 89 (0.102), 117 (0.099), 162 (0.173), 203 (0.1), 212 (0.165), 223 (0.219), 504 (0.27), 819 (0.394), 823 (0.475), 943 (0.542) 206 (0.201), 390 (0.342), 3,647 (2.105) 
BR 3,257 (2.72) 14 2 (0.002), 33 (0.03), 39 (0.047), 53 (0.047), 80 (0.104), 90 (0.101),143 (0.096), 162 (0.157), 189 (0.205), 229 (0.248), 383 (0.279), 564 (0.424),586 (0.456), 704 (0.531) 170 (0.204), 393 (0.407), 2,694 (2.116) 
VF 3,332 (1.08) 14 15 (0.004), 21 (0.006), 25 (0.01), 48 (0.016), 51 (0.021), 73 (0.03), 111 (0.03), 148 (0.069), 166 (0.063), 169 (0.049), 461 (0.169), 604 (0.161), 665 (0.194), 775 (0.254) 124 (0.051), 274 (0.102), 2,934 (0.922) 
PP 1,289 (0.71) 13 5 (0.003), 10 (0.009), 11 (0.012), 23 (0.009), 36 (0.03), 36 (0.03), 45 (0.036), 65 (0.022), 100 (0.079), 204 (0.093), 222 (0.108), 241 (0.128), 291 (0.146) 59 (0.04), 145 (0.116), 1,085 (0.55) 
WP 206 (0.1) 11 1 (0), 2 (0.001), 5 (0.01), 7 (0.003), 13 (0.009), 14 (0.012), 17 (0.006), 27 (0.007), 34 (0.018), 38 (0.017), 48 (0.018) 12 (0.012),19 (0.006), 175 (0.082) 
CW 206 (0.1) 11 1 (0.0005), 2 (0.001), 5 (0.01), 7 (0.003), 13 (0.009), 14 (0.012), 17 (0.006), 27 (0.007), 34 (0.018), 38 (0.017), 48 (0.018) 12 (0.012), 19 (0.006), 175 (0.082) 
All non-rooftop BMPs 15,246 (8.24) 91  21 15,246 (8.24) 
IB 93,449 (40.28) 11 792 (0.357), 1,073 (0.387), 1,104 (0.454), 2,817 (1.038),5,338 (2.051), 5,361 (2.731), 8,824 (5.171), 9,463 (3.832), 13,921 (6.383), 21,603 (8.648), 23,153 (9.231) 93,449 (40.284) 
IT 379,410 (43.235) 11 3,261 (0.354), 3,496 (0.406), 6,981 (0.797), 21,393 (2.485), 21,850 (2.283), 23,441 (2.403), 25,827 (3.115), 34,664 (3.953), 54,374 (5.924), 69,804 (8.127), 114,319 (13.387) 379,410 (43.235) 
RB 57,790 (2.094) 57,790 (2.094) 57,790 (2.094) 
All rooftop BMPs 530,649 (85.61) 23 - 3 530,649 (85.61) 
All BMPs 545,895 (93.85) 114 - 24 545,895 (93.85) 
BMP type (1)Number of BMPs (area in km2) (2)Number of buckets in historic event (3)Number of BMPs in each bucket along with area in km2 (in parenthesis) for historic event (4)Number of buckets in future event (5)Number of BMPs in each bucket along with area in km2 (in parenthesis) for future event (6)
GS 2,713 (0.88) 14 15 (0.004), 19 (0.009), 19 (0.006), 45 (0.018), 48 (0.016), 55 (0.023), 86 (0.023), 115 (0.053), 118 (0.042), 127 (0.037), 405 (0.15), 496 (0.147), 523 (0.142), 642 (0.219) 100 (0.041), 216 (0.08), 2,397 (0.769) 
SF 4,243 (2.65) 14 16 (0.004), 36 (0.038), 40 (0.03), 56 (0.037), 89 (0.102), 117 (0.099), 162 (0.173), 203 (0.1), 212 (0.165), 223 (0.219), 504 (0.27), 819 (0.394), 823 (0.475), 943 (0.542) 206 (0.201), 390 (0.342), 3,647 (2.105) 
BR 3,257 (2.72) 14 2 (0.002), 33 (0.03), 39 (0.047), 53 (0.047), 80 (0.104), 90 (0.101),143 (0.096), 162 (0.157), 189 (0.205), 229 (0.248), 383 (0.279), 564 (0.424),586 (0.456), 704 (0.531) 170 (0.204), 393 (0.407), 2,694 (2.116) 
VF 3,332 (1.08) 14 15 (0.004), 21 (0.006), 25 (0.01), 48 (0.016), 51 (0.021), 73 (0.03), 111 (0.03), 148 (0.069), 166 (0.063), 169 (0.049), 461 (0.169), 604 (0.161), 665 (0.194), 775 (0.254) 124 (0.051), 274 (0.102), 2,934 (0.922) 
PP 1,289 (0.71) 13 5 (0.003), 10 (0.009), 11 (0.012), 23 (0.009), 36 (0.03), 36 (0.03), 45 (0.036), 65 (0.022), 100 (0.079), 204 (0.093), 222 (0.108), 241 (0.128), 291 (0.146) 59 (0.04), 145 (0.116), 1,085 (0.55) 
WP 206 (0.1) 11 1 (0), 2 (0.001), 5 (0.01), 7 (0.003), 13 (0.009), 14 (0.012), 17 (0.006), 27 (0.007), 34 (0.018), 38 (0.017), 48 (0.018) 12 (0.012),19 (0.006), 175 (0.082) 
CW 206 (0.1) 11 1 (0.0005), 2 (0.001), 5 (0.01), 7 (0.003), 13 (0.009), 14 (0.012), 17 (0.006), 27 (0.007), 34 (0.018), 38 (0.017), 48 (0.018) 12 (0.012), 19 (0.006), 175 (0.082) 
All non-rooftop BMPs 15,246 (8.24) 91  21 15,246 (8.24) 
IB 93,449 (40.28) 11 792 (0.357), 1,073 (0.387), 1,104 (0.454), 2,817 (1.038),5,338 (2.051), 5,361 (2.731), 8,824 (5.171), 9,463 (3.832), 13,921 (6.383), 21,603 (8.648), 23,153 (9.231) 93,449 (40.284) 
IT 379,410 (43.235) 11 3,261 (0.354), 3,496 (0.406), 6,981 (0.797), 21,393 (2.485), 21,850 (2.283), 23,441 (2.403), 25,827 (3.115), 34,664 (3.953), 54,374 (5.924), 69,804 (8.127), 114,319 (13.387) 379,410 (43.235) 
RB 57,790 (2.094) 57,790 (2.094) 57,790 (2.094) 
All rooftop BMPs 530,649 (85.61) 23 - 3 530,649 (85.61) 
All BMPs 545,895 (93.85) 114 - 24 545,895 (93.85) 

It is noticed from Equations (1)–(3), which define the relation between objectives and decision variables, that volume reduction, cost and pollutant load reduction are all linearly proportional to the area of BMP. The proportionality constant between these values and areas are represented by (α, β, γ) respectively, as defined in Equation (4), and provides an insight into how BMPs were bucketed (or grouped) based on their similarities. Alternatively, (α, β, γ) can be thought of as runoff reduction per unit area, cost required per unit area and pollutant reduction per unit area for each bucket of BMPs. For example, column 3 of Table 2 shows that infiltration trenches are divided into 11 buckets for the historic event. This means there are 11 possible tuples of values (α, β, γ), one for each bucket, and every infiltration trench corresponds exactly to one of them (Equation (4)). For example, one such tuple of (α, β, γ) for infiltration trenches is (0.1662 mg/m2, Rs. 180.60/m2, 1,288.98 mg/m2), and there are a total of 114,319 BMPs (column 4 of Table 2) in this bucket. Similarly, other buckets would have different values of (α, β, γ), and once the buckets are similarly created for all BMPs, Equation (5) is used to calculate the objective values.

It is noticed that the number of buckets in the future scenario is less than that in the historic event. For the historic event, data from a number of rain gauges throughout GHMC was collected, leading to spatial variations in rainfall. These spatial variations lead to more buckets in the historic event than in the future event. Rainfall computation for the future event is based on one grid according to GFDL-CM3-RCP 2.6. However, the area of GHMC is smaller than the chosen grid area, and hence the rainfall corresponding to the entire GHMC is uniform throughout (17.36°N, 78.47°E) with no spatial variation.

Optimisation algorithms

Extensive hyperparameter tuning was conducted to narrow down the final values of hyperparameters used in this study (Deb & Jain 2014; Li et al. 2019b). In addition to parameters like population size, number of generations and mutation probabilities, both these algorithms also require reference directions that are created using Das & Dennis (1998) and the Riesz s-energy approaches (Blank et al. 2021). The number of reference directions should be such that HP, where P is the population size (Deb & Jain 2014). Both NSGA-III and C-TAEA use tournament selection. Table 3 summarises the search space of the hyperparameters and the final hyperparameters arrived at for both these algorithms.

Table 3

Hyperparameter search space and parameters chosen

Algorithm (1)Parameter (2)Search space (3)Final parameter chosen (4)
NSGA-III Population 250, 500, 1,000 1,000 
Generations 250–2,000 500 
Reference directions Das–Dennis, Riesz s-energy Das–Dennis method (p = 43) to create 990 reference directions 
Mutation probability 0.1, 0.2, 0.3 0.1 
C-TAEA Population 250, 500, 1,000 1,000 
Generations 250–2,000 500 
Reference directions Das–Dennis, Riesz s-energy Das–Dennis method (p = 43) to create 990 reference directions 
Mutation probability 0.1, 0.2, 0.3 0.1 
Algorithm (1)Parameter (2)Search space (3)Final parameter chosen (4)
NSGA-III Population 250, 500, 1,000 1,000 
Generations 250–2,000 500 
Reference directions Das–Dennis, Riesz s-energy Das–Dennis method (p = 43) to create 990 reference directions 
Mutation probability 0.1, 0.2, 0.3 0.1 
C-TAEA Population 250, 500, 1,000 1,000 
Generations 250–2,000 500 
Reference directions Das–Dennis, Riesz s-energy Das–Dennis method (p = 43) to create 990 reference directions 
Mutation probability 0.1, 0.2, 0.3 0.1 

Results of optimisation algorithms related to historic and climate change aspects are presented in the next sections.

Historic rainfall event

NSGA-III and C-TAEA are used on the historic rainfall event to obtain one Pareto-front from each of the algorithms. Results of both algorithms are combined by performing a non-dominated sort on the union of all points in the Pareto-fronts. Figure 6(a) presents ensembled Pareto-front and Figure 6(b) presents hypervolume of the two optimisation approaches. Figure 6(a) shows that a majority of points, i.e. 646 were contributed by C-TAEA (646 points), as opposed to 193 points contributed by NSGA-III. It is observed from the hypervolume plot of Figure 6(b) that there is a significant improvement in hypervolume in the first 100 generations, followed by a plateau. This suggests that 500 generations are enough for the optimisation process to stabilise because there is diminishing improvement between the 100th and 500th generations. This is further validated by attempting to run the algorithms for 2,000 generations, for which similar trends were observed.

Figure 6

(a) Ensembled Pareto-front for historic rainfall scenario. Purple points were identified by C-TAEA, and yellow points were identified by NSGA-III. (b) Plots of hypervolume of C-TAEA (red) and NSGA-III (green). Dotted red and green lines represent performances of the other sets of parameters, experimented to arrive at the best set of parameters (solid lines). (c, d) Running metric plot for first 70 generations of optimisation on historic rainfall event with (c) NSGA-III algorithm; (d) C-TAEA algorithm.

Figure 6

(a) Ensembled Pareto-front for historic rainfall scenario. Purple points were identified by C-TAEA, and yellow points were identified by NSGA-III. (b) Plots of hypervolume of C-TAEA (red) and NSGA-III (green). Dotted red and green lines represent performances of the other sets of parameters, experimented to arrive at the best set of parameters (solid lines). (c, d) Running metric plot for first 70 generations of optimisation on historic rainfall event with (c) NSGA-III algorithm; (d) C-TAEA algorithm.

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The running metric (Blank & Deb 2020a, 2020b) is used to track the progress of the optimisation algorithm. Two plots of the running metric for NSGA-III and C-TAEA are shown in Figure 6(c) and 6(d), respectively. The x-axis shows is the number of generations completed from the beginning of optimisation run, and the y-axis denotes the change in objective space. Here, the running metric is calculated and plotted after every ten generations. To plot the running metric in the 10th generation, the algorithm calculates the IGD of the first nine generations using the non-dominated points identified in the 10th generation as the reference Pareto-set for the calculation of IGD. A similar procedure is used for 20th, 30th, 40th and 50th generation, each of which is represented by different coloured lines. Figure 6(c) and 6(d) show that all lines begin with high slopes, but then start to plateau, becoming more and more parallel with reference to the x-axis. This is in agreement with the behaviour observed from the hypervolume plots. This indicates significantly faster improvement in the first few generations, followed by slower improvements.

Table 4 presents a comparision of optimisation approaches for historic event.

  • Columns 3–4, 5–6 and 7–8 depict the ranges of V, P, C in the points of the Pareto-front, respectively.

  • Column 9 denotes the number of Pareto-optimal points identified by each algorithm. There are a total of 1,186 points identified across both algorithms.

  • Column 10 provides information on how many points were contributed by each algorithm to the ensemble Pareto-front. For example, NSGA-III identified 196 Pareto-optimal points, of which 193 became a part of the ensembled Pareto-front. The other three points were dominated by the points from C-TAEA and were removed. Out of 839 points in the ensembled Pareto-front, were contributed by NSGA-III. C-TAEA achieves the highest performance (in terms of hypervolume and running metric) suggesting that this algorithm might be well suited to the BMP optimisation problem.

Table 4

Optimisation results for historic rainfall

Algorithm (1)Hypervolume (1028 mg m3 Rs.) (2)Min runoff red (106 m3) (3)Max runoff red (107 m3) (4)Min pollutant red (tonnes) (5)Max pollutant (tonnes) (6)Min cost (108 Rs.) (7)Max cost (1010 Rs.) (8)Pareto-front points (9)Contribution (10)
NSGA-III 3.23 0.71 1.41 5.37 90.6 8.25 1.68 196 193 (23.01%) 
C-TAEA 3.55 1.26 1.47 7.46 100.6 16.87 2.19 990 646 (76.99%) 
Ensemble 3.58 0.71 1.47 5.37 100.6 8.25 2.19 1,186 839 (100%) 
Algorithm (1)Hypervolume (1028 mg m3 Rs.) (2)Min runoff red (106 m3) (3)Max runoff red (107 m3) (4)Min pollutant red (tonnes) (5)Max pollutant (tonnes) (6)Min cost (108 Rs.) (7)Max cost (1010 Rs.) (8)Pareto-front points (9)Contribution (10)
NSGA-III 3.23 0.71 1.41 5.37 90.6 8.25 1.68 196 193 (23.01%) 
C-TAEA 3.55 1.26 1.47 7.46 100.6 16.87 2.19 990 646 (76.99%) 
Ensemble 3.58 0.71 1.47 5.37 100.6 8.25 2.19 1,186 839 (100%) 

As discussed in the methodology section, the next step is to cluster the identified Pareto-optimal points. The K-Medoids algorithm is used for this. The medoids (or centroids) of each cluster can then be used to represent the cluster.

To decide the optimal number of clusters needed, the Euclidean error function (E) was plotted against the number of clusters (C) as shown in Figure 7. Both error and slope (rate of change of error function with reference to number of clusters) are plotted. It is noticed that there is a significant reduction in error as the number of clusters increases from 0 to 10. A similar trend is also observed for the slope. This suggests that ten clusters would be a suitable number for analysis, and this is the value of C used for both historic and future rainfall events. The ten points, one from each cluster, can be presented to a decision maker, in lieu of 839 choices identified in the Pareto-front. Figure 8 presents the results of clustering. Each cluster is represented in a different colour and the centroid is marked with an arrow.

Figure 7

Plot of Euclidean error function as the number of clusters increase.

Figure 7

Plot of Euclidean error function as the number of clusters increase.

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Figure 8

Clustered Pareto-front for historic event.

Figure 8

Clustered Pareto-front for historic event.

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The values of objective functions and the configuration of BMPs required to achieve each of these points, i.e. 1–10 shown in Figure 8, are depicted in Table 5. Table 5 presents cluster centres and their physical meanings. Column 1 gives the point or option number, which corresponds to the point numbers labelled in the Pareto-front of Figure 8. These ten points are the cluster centres. Columns 2–4 provide the total runoff reduction, total pollutant load reduction and the cost of construction, respectively for each option. Columns 5–14 provide information on the area occupied by BMPs of each type. Column 15 provides total area of BMPs for that option. Column 16 presents source of the algorithm based on which option/Pareto-point is generated. Table 5 provides the link between mathematical optimisation and real-world application by summarising how much area of each type of BMP must be used to achieve the given values of objectives (runoff reduction, pollutant reduction and cost).

Table 5

Cluster centres and their physical meaning (historic event)

Areas occupied by BMPs (km2)
Option/point (1)V (2)P (tonnes) (3)C Rs (4)VF (5)WP (6)BR (7)CW (8)PP (9)GS (10)SF (11)RB (12)IB (13)IT (14)ALL (15)Algorithm (16)
0.37 25.5 0.43 0.03 0.009 0.026 0.008 0.023 0.03 0.027 0.011 3.91 17.92 22.00 C-TAEA 
0.87 59.0 0.95 0.01 0.002 0.005 0.001 0.005 0.01 0.002 0.003 15.62 36.76 52.41 NSGA-III 
1.17 77.8 1.28 0.01 0.002 0.009 0.001 0.004 0.01 0.002 1.985 30.49 37.98 70.49 NSGA-III 
1.29 85.4 1.42 0.02 0.006 0.021 0.005 0.024 0.03 0.006 0.067 38.04 39.42 77.63 C-TAEA 
1.36 88.9 1.50 0.04 0.007 0.028 0.007 0.024 0.02 0.015 0.018 38.62 42.77 81.55 C-TAEA 
1.38 90.9 1.58 0.23 0.008 0.164 0.007 0.019 0.02 0.015 0.102 39.64 43.08 83.29 C-TAEA 
1.40 92.7 1.66 0.04 0.010 0.691 0.013 0.026 0.03 0.017 1.173 39.02 43.05 84.07 C-TAEA 
1.43 94.8 1.76 0.23 0.013 0.887 0.011 0.022 0.14 0.011 1.820 39.62 43.13 85.87 C-TAEA 
1.44 97.3 1.90 0.23 0.020 1.674 0.012 0.038 0.14 0.013 1.139 40.14 43.13 86.53 C-TAEA 
10 1.45 99.8 2.08 0.29 0.026 2.120 0.006 0.032 0.15 0.275 1.238 40.04 43.11 87.29 C-TAEA 
Areas occupied by BMPs (km2)
Option/point (1)V (2)P (tonnes) (3)C Rs (4)VF (5)WP (6)BR (7)CW (8)PP (9)GS (10)SF (11)RB (12)IB (13)IT (14)ALL (15)Algorithm (16)
0.37 25.5 0.43 0.03 0.009 0.026 0.008 0.023 0.03 0.027 0.011 3.91 17.92 22.00 C-TAEA 
0.87 59.0 0.95 0.01 0.002 0.005 0.001 0.005 0.01 0.002 0.003 15.62 36.76 52.41 NSGA-III 
1.17 77.8 1.28 0.01 0.002 0.009 0.001 0.004 0.01 0.002 1.985 30.49 37.98 70.49 NSGA-III 
1.29 85.4 1.42 0.02 0.006 0.021 0.005 0.024 0.03 0.006 0.067 38.04 39.42 77.63 C-TAEA 
1.36 88.9 1.50 0.04 0.007 0.028 0.007 0.024 0.02 0.015 0.018 38.62 42.77 81.55 C-TAEA 
1.38 90.9 1.58 0.23 0.008 0.164 0.007 0.019 0.02 0.015 0.102 39.64 43.08 83.29 C-TAEA 
1.40 92.7 1.66 0.04 0.010 0.691 0.013 0.026 0.03 0.017 1.173 39.02 43.05 84.07 C-TAEA 
1.43 94.8 1.76 0.23 0.013 0.887 0.011 0.022 0.14 0.011 1.820 39.62 43.13 85.87 C-TAEA 
1.44 97.3 1.90 0.23 0.020 1.674 0.012 0.038 0.14 0.013 1.139 40.14 43.13 86.53 C-TAEA 
10 1.45 99.8 2.08 0.29 0.026 2.120 0.006 0.032 0.15 0.275 1.238 40.04 43.11 87.29 C-TAEA 

From Table 5, it can be inferred that:

  • 1.

    Costs are ranging from Rs. to surface runoff reduction ranging from to and pollutant load removal ranging from tonnes.

  • 2.

    The area occupied by rooftop BMPs, especially infiltration trench and infiltration basin, is more compared to non-roof top BMPs, mainly constructed wetlands and wet ponds. This may be due to a larger percentage of buildings compared to the open space, satisfying the conditions provided in EPA-SUSTAIN (2014).

  • 3.

    A stabilisation is observed from option 5 where runoff reduction is 1.36 × 107 m3 for a cost of Rs.1.5 × 1010. There is minimal increase in runoff reduction even though there is an increase in the cost.

  • 4.

    In option 2 and option 3, the difference in cost is Rs. 0.33 × 1010 and the difference in runoff reduction is 0.3 × 107 m3, respectively. However, between option 8 and 9, the difference in cost is Rs. 0.14 × 1010 and the difference in runoff reduction is around 0.01 × 107 m3. This suggests that the cost-effectivity of runoff reduction (runoff reduction/cost) is better for the smaller configurations. This is because, during optimisation, all best-performing BMPs are taken first and worst-performing BMPs are considered later.

  • 5.

    It is noticed that for option 1, which is the least expensive option costing Rs. 4.30 , the runoff reduction volume of and pollutant reduction of 25.5 tonnes are observed. This corresponds to a per-unit cost of = Rs. 1,162/m3 for runoff reduction. On the other hand, for option 10 (costing Rs. ), the runoff reduction volume of and pollutant reduction of 99.8 tonnes are observed. This corresponds to a per-unit cost of Rs. 1,434.48/m3. This increase in per-unit runoff reduction cost from option 1 to option 10 provides more evidence to the fact that the optimisation algorithm picks less expensive variants of BMPs before picking more expensive variants. For non-roof top BMPs, the area occupied by bioretention, is considerably more compared to other non-rooftop BMPs. This may be due to terrain characteristics, slope, elevation and distance from streams. This BMP has an added advantage of aesthetic look with flora and fauna.

Spatial visualisation of BMPs for the selected zone and selected option (see Table 5 and Figure 8) are presented in Supplementary Information S2.

Future rainfall event

The discussion and results presented so far depict the efficacy of BMPs for the historic event. In this section, the optimisation approach was performed in the context of a future rainfall event to see the effectiveness of BMPs. Following an approach similar to the one used in the historic event, 24 buckets are formed for this future scenario (column 5 of Table 2). Performing optimisation using NSGA-III and C-TAEA led to the identification of 852 points. Ensembling led to the formation of a Pareto-front with 689 points, as summarised in Table 6.

Table 6

Optimisation results for future rainfall event (RCP 2.6)

Algorithm (1)Hypervolume (1028 mg m3 Rs.) (2)Min runoff red (106 m3) (3)Max runoff red (107 m3) (4)Min pollutant red (tonnes) (5)Max pollutant (tonnes) (6)Min cost (108 Rs.) (7)Max cost (1010 Rs.) (8)Pareto-front points (9)Contribution (10)
NSGA-III 7.86 6.02 1.08 1.63 303 1.16 2.10 129 124 (17.99%) 
C-TAEA 7.92 7.09 1.09 151 307 1.70 2.18 723 565 (82.01%) 
Ensemble 7.96 6.02 1.09 1.63 307 1.16 2.18 852 689 
Algorithm (1)Hypervolume (1028 mg m3 Rs.) (2)Min runoff red (106 m3) (3)Max runoff red (107 m3) (4)Min pollutant red (tonnes) (5)Max pollutant (tonnes) (6)Min cost (108 Rs.) (7)Max cost (1010 Rs.) (8)Pareto-front points (9)Contribution (10)
NSGA-III 7.86 6.02 1.08 1.63 303 1.16 2.10 129 124 (17.99%) 
C-TAEA 7.92 7.09 1.09 151 307 1.70 2.18 723 565 (82.01%) 
Ensemble 7.96 6.02 1.09 1.63 307 1.16 2.18 852 689 

Figure 9(a) and 9(b) show the Pareto-front and the hypervolume plots for the future scenario optimisation. It is noted from the hypervolume plot that NSGA-III outperformed C-TAEA for the first 100 generations, before the latter was overtaken by the former. Plots of running metric are not presented here due to the similarity in trends to the historic event rainfall. Figure 10 presents the ensembled and clustered Pareto-front. Each cluster is marked in a different colour, and each cluster centre is labelled with an arrow and cluster number. Objective function values and configuration of BMPs required to achieve each of these points, i.e. 1–10, are shown in Figure 10 (see Table 7).

Table 7

Cluster centres and their physical meaning (future event)

Option/point (1)Objectives
Areas of BMPs used (km2)
Runoff red (108 m3) (2)P (tonnes) (3)Cost (1010 Rs.) (4)VF (5)WP (6)BR (7)CW (8)PP (9)GS (10)SF (11)RB (12)IB (13)IT (14)ALL (15)Algorithm (16)
0.13 42.4 0.20 0.003 0.006 0.011 0.003 0.003 0.005 0.001 0.046 0.01 10.40 10.49 C-TAEA 
0.49 125.2 0.72 0.000 0.001 0.002 0.005 0.001 0.001 0.001 2.042 17.49 20.44 39.98 NSGA-III 
0.70 198.4 1.06 0.010 0.007 0.012 0.004 0.015 0.020 0.003 0.936 19.29 37.36 57.66 C-TAEA 
0.90 238.8 1.34 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.103 37.27 36.67 74.05 NSGA-III 
1.03 272.0 1.54 0.015 0.004 0.028 0.004 0.004 0.022 0.003 1.124 39.95 43.13 84.29 C-TAEA 
1.05 282.2 1.68 0.013 0.004 0.626 0.009 0.003 0.081 0.003 2.087 40.28 43.23 86.34 C-TAEA 
1.06 287.9 1.80 0.120 0.007 1.140 0.008 0.002 0.195 0.005 2.078 40.28 43.23 87.07 C-TAEA 
1.07 293.9 1.89 0.021 0.008 1.812 0.013 0.028 0.030 0.030 2.085 40.28 43.23 87.53 C-TAEA 
1.07 299.5 1.99 0.110 0.010 2.262 0.006 0.009 0.080 0.027 1.929 40.28 43.22 87.93 C-TAEA 
10 1.08 305.4 2.10 0.404 0.002 2.522 0.009 0.025 0.102 0.085 1.551 40.28 43.23 88.22 C-TAEA 
Option/point (1)Objectives
Areas of BMPs used (km2)
Runoff red (108 m3) (2)P (tonnes) (3)Cost (1010 Rs.) (4)VF (5)WP (6)BR (7)CW (8)PP (9)GS (10)SF (11)RB (12)IB (13)IT (14)ALL (15)Algorithm (16)
0.13 42.4 0.20 0.003 0.006 0.011 0.003 0.003 0.005 0.001 0.046 0.01 10.40 10.49 C-TAEA 
0.49 125.2 0.72 0.000 0.001 0.002 0.005 0.001 0.001 0.001 2.042 17.49 20.44 39.98 NSGA-III 
0.70 198.4 1.06 0.010 0.007 0.012 0.004 0.015 0.020 0.003 0.936 19.29 37.36 57.66 C-TAEA 
0.90 238.8 1.34 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.103 37.27 36.67 74.05 NSGA-III 
1.03 272.0 1.54 0.015 0.004 0.028 0.004 0.004 0.022 0.003 1.124 39.95 43.13 84.29 C-TAEA 
1.05 282.2 1.68 0.013 0.004 0.626 0.009 0.003 0.081 0.003 2.087 40.28 43.23 86.34 C-TAEA 
1.06 287.9 1.80 0.120 0.007 1.140 0.008 0.002 0.195 0.005 2.078 40.28 43.23 87.07 C-TAEA 
1.07 293.9 1.89 0.021 0.008 1.812 0.013 0.028 0.030 0.030 2.085 40.28 43.23 87.53 C-TAEA 
1.07 299.5 1.99 0.110 0.010 2.262 0.006 0.009 0.080 0.027 1.929 40.28 43.22 87.93 C-TAEA 
10 1.08 305.4 2.10 0.404 0.002 2.522 0.009 0.025 0.102 0.085 1.551 40.28 43.23 88.22 C-TAEA 
Figure 9

(a) Ensembled Pareto-front for the future rainfall scenario. Purple points were identified by C-TAEA and yellow points were identified by NSGA-III. (b) Plot of hypervolume of C-TAEA (red) and NSGA-III (green). Dotted red and green lines represent performances by other sets of parameters experimented to reach the best set of parameters (solid lines).

Figure 9

(a) Ensembled Pareto-front for the future rainfall scenario. Purple points were identified by C-TAEA and yellow points were identified by NSGA-III. (b) Plot of hypervolume of C-TAEA (red) and NSGA-III (green). Dotted red and green lines represent performances by other sets of parameters experimented to reach the best set of parameters (solid lines).

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Figure 10

Ensembled and clustered Pareto-front.

Figure 10

Ensembled and clustered Pareto-front.

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Table 7 presents cluster centres and their physical meaning for future event. It is noted from Table 7 that the area occupied by rooftop BMPs is relatively more than the area occupied by non-rooftop BMPs. Pollutant load reduction continues to increase even though runoff reduction value remains almost constant, as observed in options 5–10. The difference in cost between options 2 and 3 is Rs. 0.34 × 1010, while the difference in runoff reduction is 0.21 × 108 m3. The cost difference between implementing options 7 and 8 is Rs. 0.09 × 1010, while the change in runoff reduction values is 0.1 × 108 m3.

Spatial visualisation of BMPs for the selected zone and selected option (see Table 7 and Figure 10) are presented in Supplementary Information S2.

In summary, it is noted from the 2016 historic extreme rainfall event that implementing option 10 (at a cost of Rs. ) would have reduced surface runoff by , while also removing 99.8 tonnes of pollutant load. For a similar cost, it is noticed that almost of runoff (∼13 times increase) is reduced and 305 tonnes of pollutant (∼3 times increase) are removed, indicating how investments in BMPs today would potentially have a multiplier effect in the face of higher rainfalls caused due to climate change. This denotes the necessity of the optimisation approach proposed because even relatively small percentage savings translate into huge sums for large projects of this type.

In this work, a multiobjective optimisation framework for the BMP placement problem was explored in a case study of GHMC for both historic and future rainfall events. NSGA-III and C-TAEA were evaluated and compared by analysing hypervolumes, running metrics and values of objectives. It was found that C-TAEA, designed specifically for constrained optimisation problems, is an effective approach, comparable to widely used algorithms such as NSGA-III. Results of both algorithms are ensembled to form a combined Pareto-front. This Pareto-front is then clustered through a K-Medoids algorithm to reduce the dimensionality of various choices that the decision maker will have to choose from. The optimisation process ensures that every incremental investment of capital has the maximal impact possible, by identifying configurations that show the most runoff reduction or pollutant load reduction for every available capital investment.

Results obtained favours placement of rooftop BMPs (infiltration trench, infiltration basin and rain barrels) on buildings throughout the city for runoff reduction. This situation is expected in the context of rapid urbanisation. However, this consequently means that the success of BMP placement, to a large extent, is dependent on the participation of residents, which is a major challenge. On the other hand, the main limitation of non-rooftop BMPs is the open land availability.

Authors thank officials of GHMC and other agencies for their help. Special acknowledgements to P. Sumasri, Deputy Executive Engineer at GHMC for provding valuable suggestions while preparing the revised manuscript.

Data cannot be made publicly available; readers should contact the corresponding author for details.

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