A participatory approach based on stochastic optimization for the spatial allocation of Sustainable Urban Drainage Systems for rainwater harvesting.

Journal Pre-proof A participatory approach based on stochastic optimization for the spatial allocation of Sustainable Urban Drainage Systems for rainwater harvesting. María Nariné Torres, John E. Fontecha, Zhenduo Zhu, Jose L. Walteros, Juan Pablo Rodríguez PII:

S1364-8152(18)30799-0

DOI:

https://doi.org/10.1016/j.envsoft.2019.104532

Reference:

ENSO 104532

To appear in:

Environmental Modelling and Software

Received Date: 29 August 2018 Revised Date:

24 September 2019

Accepted Date: 27 September 2019

Please cite this article as: Torres, Marí.Nariné., Fontecha, J.E., Zhu, Z., Walteros, J.L., Pablo Rodríguez, J., A participatory approach based on stochastic optimization for the spatial allocation of Sustainable Urban Drainage Systems for rainwater harvesting., Environmental Modelling and Software (2019), doi: https://doi.org/10.1016/j.envsoft.2019.104532. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

A participatory approach based on stochastic optimization for the spatial allocation of Sustainable Urban Drainage Systems for rainwater harvesting. María Nariné Torresa, b,* John E. Fontechac Zhenduo Zhub Jose L. Walterosc Juan Pablo Rodrígueza Corresponding author a

Centro de Investigaciones en Ingeniería Ambiental - CIIA, Universidad de Los Andes, Bogotá, Colombia

b

Department of Civil, Structural and Environmental Engineering, University at Buffalo

c

Group for Applied Mathematical Modeling and Analytics - GAMMA, Industrial Engineering Department,University at Buffalo

Email address: [email protected] (John E. Fontecha) Email address: [email protected] (Zhenduo Zhu) Email address: [email protected] (Jose L. Walteros) Email address: [email protected] (Juan Pablo Rodríguez) Abstract Rainwater Harvesting (RWH) is the practice of capturing and storing stormwater for later use. In addition to being an alternative source of water for non-potable applications, RWH is also used to effectively reduce stormwater runoff volumes and pollutant loads dropped into sewage systems. RWH is typically carried out by placing a variety of Sustainable Urban Drainage Systems (SUDS) in different locations of the urban landscape. However, because of the staggering number of potential combinations of SUDS typologies and spatial configurations that can be used, identifying a strategy that optimally selects and allocates SUDS to maximize the benefits of RWH is a complex endeavor. One of the difficult challenges that emerges during the optimal design and location of these systems is incorporating the inherent uncertainty of the rainfall. In this paper, we develop a flexible computational framework that couples a Geographic Information System (GIS) with a two-stage stochastic mixed integer linear program (TS-MILP) to select and locate SUDS in order to minimize the use of potable water for irrigation and reduce the water runoff at a minimum cost. This framework incorporates an iterative participatory approach to engage stakeholders in the decision-making process. We tested the proposed methodology on a case study for the central campus at Universidad de Los Andes (Bogotá, Colombia). Our results showed that both the expected value of the total runoff volume as well as the consumption of potable water can be reduced up to 67% and 50%, respectively.

Keywords: Mixed Integer Program , Participatory approach , Rainwater harvesting , Stormwater recycling , Sustainable Urban Drainage Systems , Two-stage stochastic programming

1. Introduction Rainwater Harvesting (RWH) is the practice of capturing and storing stormwater for later use for nonpotable purposes (Jones & Hunt, 2010). Although RWH has been practiced for centuries (AbdelKhaleq & Ahmed, 2007; Chocat & Schilling, 2001; USEPA, 2013), its use has grown remarkably over the last few decades sparked by the benefits that RWH brings for the environment, as well as for the growing number of new potential applications that have stemmed from this practice (Boers & Ben-Asher, 1982). In particular, RWH has recently re-emerged as a solution for regional water scarcity, e.g., drought mitigation and increased demand satisfaction (Campisano et al., 2017), and as a low-cost and environmentally friendly practice to reduce stormwater runoff volumes and its pollutant loads (Aladenola & Adeboye, 2010; USEPA, 2013; Jones & Hunt, 2010). While the earliest RWH techniques were commonly limited to rural environments on suitable natural surfaces, the development of new technologies based on simple inducement and collection methods (Boers & Ben-Asher, 1982) have substantially expanded the possibilities of carrying out RWH in

more complex and dense urban environments. Modern implementations have experimented with novel systems to fulfill the goal of RWH in urban areas, i.e., “the concentration, collection, storage and treatment of rainwater from rooftops, terraces, courtyards, and other impervious building surfaces for on-site use” (Campisano et al., 2017). Recent studies have shown that RWH is a promising practice for surface runoff control (Steffen et al., 2013; Zhang & Hu, 2014; Vaes & Berlamont, 1999; Herrmann & Schmida, 2000) and, more importantly, to reduce the consumption of fresh water, e.g., Aladenola & Adeboye (2010); Rahman et al. (2012); Herrmann & Schmida (2000); Appan (2000); Handia et al. (2003). Indeed, some researchers have theorized that the total consumption of potable water that could be substituted by RWH accounts for up to 80% of the total water consumption of an average household (Ward et al., 2012). Other more conservative reports have shown some variability in the expected reduction, but, nevertheless, a notable reduction in general. For example, for a household in Germany or Australia the reported potable-water-saving potential is up to 60% (Coombes et al., 2000; Herrmann & Schmida, 2000), while in Brazil it ranges from 34% to 92%, depending on the type of RWH used (Ghisi et al., 2006). In addition to its potential to save potable water, RWH is known for its capability of reducing runoff. In fact, the increased attention and expanding global implementation of Sustainable Urban Drainage Systems (SUDS) throughout the last 20 years have boosted the research on the effectiveness of RWH systems to control runoff (Campisano et al., 2017). Recent studies have systematically reported the success of RWH and other SUDS as complementary measures to mitigate the increment of both runoff peaks and runoff volumes caused by urbanization (Burns et al., 2015; Chocat et al., 2007; Kozak et al., 2012). For example, in a study conducted by Selbig & Bannerman (2008), predevelopment hydrological and near-complete build-out conditions were achieved in a basin with SUDS. Other studies have reported SUDS’s high effectiveness in both volume and peak flow attenuation, e.g., Wild & Davis (2009) and Palla & Gnecco (2015). The introduction of SUDS did much more than merely propel RWH; it has also broadened the options of using multiple structures to capture and store stormwater. For example, some green roofs include additional drainage compartments that can be used as rainwater storage (Qin et al., 2016; Vila et al., 2012) allowing excess water to be used for other purposes after appropriate treatment (Woods Ballard et al., 2015). Permeable pavements, detention basins, ponds or swales, have the capacity of temporary storing stormwater that can be later used for irrigation purposes. Such structures often have storage compartments beneath the overlying surfaces (Woods Ballard et al., 2015) or vegetated depressions to accommodate large volumes of water, and thus, attenuate excessive runoff (Lawson et al., 2012). A notable characteristic of most SUDS is that they are diverse and flexible, which gives them the adaptability to be placed in several components of the urban landscape, e.g., sidewalks, highways, roofs, parking lots, green areas. The empiric widespread adoption of SUDS has raised the question of whether there is an efficient methodology to identify an optimal configuration that maximize their benefits. This research question has spurred the development of optimization methodologies that take into consideration physical, social, and economical constraints to spatially allocate SUDS in complex urban environments (Zhang & Chui, 2018). The spatial allocation of SUDS has proven to play an important role when maximizing the benefits of RWH, since particular locations and configurations may have characteristics that favor different objectives. For example, a downstream location will be more suitable for regional control (treating larger runoff volumes), locations with high imperviousness favor infiltration-based SUDS to reduce runoff efficiently and promote groundwater recharge, or hot spots for a particular pollutant may be of special interest in terms of improving water quality in the receiving water bodies (Zhang & Chui, 2018). The problem of selecting and allocating SUDS has been tackled by previous researchers using a wide variety of methodologies. In particular, a notable contribution to the field has been the use of Geographic Information Systems (GIS) to perform spatial feasibility analyses. By means of GIS, researchers have been able to propose a variety of applications regarding SUDS planning. For example, Dearden & Price (2012) and Cooper & Calvert (2011) developed suitability maps for infiltration-based SUDS. Jato-Espino et al. (2016) applied geometric and hydrologic criteria to find

areas to install permeable pavements. Shoemaker et al. (2009) developed a framework and tool, SUSTAIN, to identify feasible locations for 14 types of SUDS, and Tiwari et al. (2018) identified suitable locations for rainwater harvesting structures using remote sensing data. The main contribution of most of these works is to determine feasible locations for certain SUDS. However, no further conclusions have been drawn in terms of the preferred optimal locations and configurations of the selected SUDS to be installed. The first approaches to tackle this problem applied the traditional benefit-cost analysis (Andoh & Declerck, 1997), latter applications used exact optimization methodologies such as linear and dynamic programming (Doyle et al., 1976; Mays & Bedient, 1982; Alaya et al., 2003); while most recent endeavors have focused on the development of metaheuristics, like genetic algorithms, simulated annealing, or tabu search methods. Metaheuristics are generic random search procedures able to identify good solutions of complex optimization problems. The flexibility of metaheuristics allows them to be coupled with other algorithms, like simulations, to efficiently explore a large number of potential solutions, albeit at the expense of not having a guarantee of the quality of the solution found (Glover & Kochenberger, 2006; Maier et al., 2014; Zufferey, 2012). In order to simulate the processes of surface runoff, infiltration, evapotranspiration, and other physical and biological processes, the metaheuristics are generally coupled with a [start]â[end][start] [end][start] [end]calculation engine[start]â[end][start] [end][start] [end] (Lee et al., 2005; Zhang & Chui, 2018). Common calculation engines recently used for this purpose are the Stormwater Management Model (SWMM), e.g., Macro et al. (2018); Cunha et al. (2016); Ghodsi et al. (2016) and the Soil and Water Assessment Tool (SWAT), e.g., Kaini et al. (2012); Zhang et al. (2013), although other works use other non-point source pollution models such as the WQM-TMDL-N and the AGNPS, e.g., Srivastava et al. (2003); Yang & Best (2015). Parallel to the consideration on which optimization techniques are appropriate to tackle the SUDS allocation problem, a compulsory aspect is the integration of stakeholders in the decision-making process (Wynne, 1996; Evans & Plows, 2007). An effective stakeholder engagement results in an increased likelihood of success (Carmona et al., 2013; Voinov et al., 2016), especially when a strong science and policy-making relationship is crucial to effectively address the environmental challenge (Argent et al., 2016). In fact, widely adopted guidance such as CIRIA SUDS Design Manual (WoodsBallard et al., 2007) and EPA guidelines (USEPA, 2017) stress the need to engage stakeholders with the decision-making process from early stages. For this reason, we propose an iterative participatory approach in which the stakeholders provide insights to i) the prioritized order of the objectives, ii) the feasibility of potential solutions, and iii) the satisfactoriness of the proposed result. This work stems from previous contributions in participatory literature, GIS applications, and optimization techniques applied to the spatial allocation of SUDS. We propose the use of a two-stage stochastic mixed integer linear program (TS-MILP) to find the optimal spatial configuration that adapts best to the set of potential outcomes according to their probability of occurrence (one of the two stages evaluates the performance of the set of possible solutions over the different scenarios under consideration) (Delage et al., 2014; Ahmed, 2010; Birge & Louveaux, 2011). The proposed methodology is tailored to identify exact solutions for the complex spatial allocation of SUDS. Our methodology considers/includes: i) the stochasticity of rainfall on the basis of its probability of occurrence (instead of optimizing for the average or the worst-case scenario); ii) the identification of optimal solutions by means of linear programming, which permits the application of efficient optimization methodologies to reduce the solution search space; iii) multiple objectives, allowing the stakeholders to identify an importance hierarchy of preference among them; and iv) adaptability to incorporate a wide variety of stakeholder requirements and preferences. To our best knowledge, previous work on the optimization of the spatial allocation of SUDS has not considered the connections (real and feasible distances) and the harvested water flows from the SUDS to supply water-demanding areas. The closest attempt to achieve this goal is the work carried out by Inamdar et al. (2013), which calculated a radius of influence using spatial data of the offer and demand, but which fails to determine an optimal configuration considering this information.

This paper proposes to calculate the linear distance and associated costs of each connection. The TSMILP herein introduced considers: i) the hydrologic balance of harvested water in SUDS typologies, ii) the infiltration and evapotranspiration processes, iii) the water uptake, and iv) the rainfall uncertainty. Three objective functions are used during the sequential execution of the proposed TSMILP. First, we consider the amount of potable water that is used; second, we account for the total runoff not captured by the system; and third, we account for the total installation cost of the system. Since the proposed model requires the minimization of potentially competing objectives, we use traditional techniques commonly used to solve multicriteria optimization problems. We test the proposed methodology in a real-world case study based on the central campus at Universidad de Los Andes, which consumes around 4,800 m3 of potable water each month. The rest of the paper is organized as follows. Section 2 provides a detailed explanation of the five steps of the proposed methodology, starting from the data processing for the feasible sites identification, to the final visualization of results. Section 3 presents the case study involving the central campus at Universidad de Los Andes (Colombia), while Section 4 addresses our findings, including a closing discussion with stakeholders. Finally, Section 5 concludes the work and outlines future research.

2. Methods The methodology starts with the processing of spatial data with GIS to obtain the parameters required by the TS-MILP. The results from the optimization model are then loaded back onto the GIS platform for visualization. Figure 1 shows the general scheme of the methodology, in which five main steps are identified: (1) feasible sites identification, catchment delineation and stakeholders preferences involvement, (2) water consumption quantification, (3) rainfall scenarios and runoff quantification, (4) optimization, and (5) results visualization. Steps 1, 2 and 3 constitute the appraisal of spatial databases, rainfall data and secondary information to obtain the parameters of the problem. In step 4, the optimization program is parameterized with this information and is later used to calculate the optimal solution, which is finally visualized in the GIS and shared with stakeholders in step 5. This last step provides a space for stakeholders to discuss proposed solutions and give feedback on what can be improved to reach a satisfactory solution. The adaptability of the model plays an important role for incorporating a variety of stakeholder requirements or preferences that may arise after the first proposed solution (see connecting arrow between step 5 and step 1 in Figure 1). The following subsections describe each step of the methodology in detail. Figure 1: General scheme of the methodology. The steps in which the stakeholders are involved are shown in gray. Step 1: feasible sites identification, catchment delineation and stakeholder[start]â[end][start] [end][start] [end]s first inputs. Step 2: water consumption quantification. Step 3: rainfall scenarios and runoff quantification. Step 4: optimization, and Step 5: results visualization. The interaction between steps 1 and 5 is described in subsection 2.7 . Figure 2: Flowchart for the first step of the methodology: Feasible sites identification, watershed delineation, and stakeholders inputs. The stakeholders involvement are shown in gray.

2.1. Feasible sites identification, catchment delineation, and the stakeholders inputs Figure 2 illustrates the flow diagram of the main processes carried out in step 1, the participation of the stakeholders is shown in gray. To begin with, georeferenced information of terrain slope, average water table depth, infiltration rate and buildings are required to create a preliminary map of feasible sites by applying raster calculations. This preliminary selection of feasible sites is shared with stakeholders who are involved in the decision-making process through the development of social cartography exercises (Paulston & Liebman, 1994; Downs & Stea, 1973) from which they can map relevant information about the proposed locations. The information collected from stakeholders is then used to modify the preliminary feasible sites by eliminating conflicting sites and re-evaluating the feasibility of new areas. Figure 3 illustrates this first input of stakeholders in step 1, as well as the subsequent steps with stakeholders interventions (steps 4 and 5, which will be discusses in subsection 2.7).

Figure 3: Graphical overview of stakeholders engagement. Stakeholders involvement is shown in gray. The thick line corresponds to comments concerning the feasible sites selection and parameterization, the thin line to modifications in the optimization model, and dotted line to results presentation. The catchment delineation is obtained by processing a Digital Elevation Model (DEM) and a surface inventory is used to associate a weighted runoff coefficient to each catchment. According to the literature, DEM-based algorithms are not very efficient for catchment delineation in urban areas, e.g., Sanzana et al. (2017); Jankowfsky et al. (2013); therefore, a modification of the methodologies proposed by Parece and Campbell (2015) and Jato-Espino (2016) is applied here (see Figure 2). First, the DEM is corrected by using building polygons in order to create high obstacles that direct the flow (Jato-Espino et al., 2016) and then, the catchment areas are refined manually using layers of storm water pipelines and gullypots (Parece & Campbell, 2015). Finally, an ortophoto of the study area is used to identify the types of surfaces, e.g., grass, roofs, and pavements in the catchments. The resulting surface inventory and the runoff coefficient values reported in the literature, e.g., Butler & Davies (2003); Hindman et al. (2016); ASCE & (WEF); CSWR (2011), are used to calculate a weighted runoff coefficient per catchment.

2.2. Quantification of water consumption Water consumption for irrigation is calculated using historical potable water bills and the irrigation regime implemented in the study area. Since water bills are typically aggregated by buildings (or other billing units) that generally supply multiple green areas and because the irrigation regime varies for different types of vegetation, the plants’ water requirements are used to disaggregate water demand per unit area. These calculations allow the construction of demand maps, which show the spatial distribution of the water demand.

2.3. Rainfall scenarios and runoff quantification Historical hourly rainfall time series are used to set up the rainfall scenarios that will be used in the optimization model. In an effort to identify important characteristics of the long series, an exploratory data analysis is performed. The first step is the separation of the series into precipitation events. For this purpose, the methodology proposed by Brown et al. (1985) is followed, in which a precipitation event is defined to be “a period of one or more consecutive hours of precipitation of at least 0.25 mm followed by one or more hours of no precipitation” (Brown et al., 1985). Following this, a frequency analysis is performed in order to identify the range of durations observed in the rainfall series. For each duration, three intensity categories are defined: low, medium and high. Upper and lower limits are set by dividing the intensities range into three sets of equal length. For notation, each rainfall scenario e corresponds to a designed rainfall level of constant intensity i through a duration d, i.e., = , . To calculate the probabilities of occurrence for each scenario we counted the number of occurrences of the combination , and divided by the total number of rainfall events in the historical series. The runoff volume reaching each feasible site is calculated using the rational method (see Butler & Davies (2003)). To do so, the surface of the study area is classified into k surface types, e.g., grass, roofs, pavements, each holding a runoff coefficient obtained from literature (Butler & Davies, 2003; Hindman et al., 2016; CSWR, 2011; ASCE &, WEF). The runoff volume that reaches a feasible point is essentially the amount of potential harvesting water for surface-storage-based SUDS typologies. However, an additional calculation is required to determine the runoff volume that is available for those typologies in which the storage takes place beneath a porous medium or modular surface. Given that a portion of the water is retained in the media until field capacity is reached (Woods Ballard et al., 2015; Pitt et al., 2008), for each time step ∈ , Equations 1, 2, and 3, obtained from the soil-water balance calculations proposed by Pitt et al. (1999), are applied. The following parameters are required to calculate both the soil water content ( ) and the water available for storage ( ): initial soil moisture ( ), infiltration rate (r), saturated hydraulic conductivity (g), and saturation water content (h).

= =

=

+

+

∀ ∈ | = 1 (1) ∀ ∈ | ≠1

∀ ∈ |

≥ℎ

(2)

(3)

2.4. Optimization model formulation As a result of the steps described above, two subsets containing geographical sites are identified: first, the offering nodes ( ), i.e., spaces available for SUDS to store runoff water, and second, the demanding nodes ( ), i.e., green spaces requiring water for irrigation, both sets belonging to the set of nodes . The set stands for the SUDS typologies, the set represents the time steps considered, and the set ℰ gathers the three precipitation scenarios (low, medium and high). The model is formulated under two assumptions: i) a rainfall event can be split into a wet period and a dry period; ii) runoff occurs only during the wet period, while evapotranspiration and the use of water for irrigation occurs only in the dry period. Figure 4 depicts the scheme of these flows using detention basins as an example. Flows for other SUDS typologies are illustrated in Appendix A. Figure 4: Illustration of the flows occurring during wet and dry periods The parameters of the model, i.e., values that are known before the optimization, are defined as follows. Each demanding node ∈ demands a given amount of water ("# ) that we attempt to satisfy using the runoff water temporarily stored in the SUDS, which must be transported from node n to node m through a pipe that has a unitary installation cost of $%& per meter, or alternatively using potable water from the water utility, which has a cost per cubic meter (θ). For the sake of simplicity, the distance among nodes ('%& ) is calculated as Euclidean distances. It is worth mentioning that connections, and thus values for $%& and '%& , are permitted only under the following circumstances: i) among offering nodes, ii) from offering to demanding nodes, and iii) if the difference in height allows the flow of water by gravity. It is worth mentioning that the model can be easily extended to allow flow against gravity by also considering water pumps installation costs; however, we limit our discussion to the former case. SUDS costs are estimated per unit area for each particular typology (() ); the area to be installed in each offering node is limited by an upper bound, corresponding to the maximum feasible area (*+) ) obtained from the feasibility analysis, and a lower bound (,+) ), depending on the nature of the SUDS typology. The storage capacity of each offering node depends on a generic depth for each SUDS typology (-) ). The potential evapotranspiration and infiltration are .) and /+ , and depend on the SUDS typology and the particular geographical site of the offering node, respectively. The available runoff volume to be harvested (0+ ) is calculated using the rational method, for each offering node, time, and scenario. Other parameters are the available budget (ρ), the probability of occurrence of each scenario , and a binary parameter 1 that takes the value of 1 if the time step is dry (no rainfall) and 0 otherwise. It is worth emphasizing that some of the parameters described above do not exist for all the combinations given by the subindices. For example, $+% will only exists for a given 2 ∈ , 3 ∈ the flow of water is allowed by gravity.

if

Nine sets of decision variables are defined for the model. Let 4+) be a binary variable that takes the value of 1 if a SUDS of typology 5 ∈ is selected to be installed on offering node 2 ∈ and 0 otherwise; 6+) be a continuous variable that represents the area of the typology 5 ∈ to be installed in offering node 2 ∈ ; +% be a binary variable that takes the value of 1 if the connection between offering node 2 ∈ and node 3 ∈ is used and 0 otherwise; and 7+% be the volume sent from offering node 2 ∈ to node 3 ∈ at the time ∈ and scenario ∈ ℰ. We use 8# to denote the volume of potable water from the water utility used in the demanding node ∈ in the time ∈ and scenario ∈ ℰ; + to represent the runoff volume leaving the offering node 2 ∈ (when the maximum storage capacity is exceeded), at the time ∈ and scenario ∈ ℰ; + and 9+ to represent the inventory volume available and the losses (by evapotranspiration and infiltration), respectively, in node 2 ∈ , at time ∈ , and scenario ∈ ℰ. Finally, let :+ be a binary variable that takes the

value of 1 if there is evapotranspiration in node 2 ∈ , time ∈ , and scenario ∈ ℰ and 0 otherwise. A complete list of sets, variables, and parameters is available in Appendix B.

2.5. Objective Functions We consider three competing objective functions in the proposed model: i) the amount of potable water that is being consumed, ii) the runoff that is not captured by the system, and iii) the cost of deploying the system, all of which we attempt to minimize. There are several techniques designed to handle the interaction of multiple objectives for mixedinteger optimization. Perhaps the most commonly used method is known as the weighted-sum approach (Ehrgott, 2005), which consists in aggregating the multiple objectives into a single objective function by introducing a normalization weight associated to each of the objectives. Each objective function is multiplied by its corresponding weight and a single objective function is produced by adding all the resulting weighted terms. The main goal of the weights is twofold. First, since it is likely that the objective functions are defined over a different set of units and vary in scale, the weights are set so that all the objectives are commensurate. Second, depending on the importance that the stakeholders give to the corresponding objectives, the set of weights is also used to account for the relevancy they should carry over the overarching set of decisions. To this end, a larger weight is thus typically given to the objectives of greater importance. Unfortunately, the weighted-sum method has several limitations. In particular, when the scale difference between objectives is of several orders of magnitude, finding the ideal set of weights to properly balance all the objectives is not only challenging but may also drastically affect the results of the optimization process by introducing computational rounding errors. Furthermore, incorporating the relevance of the objective functions along with the scale reduction factors can bring further complications to the task of selecting appropriate weights. In this paper we use a lexicographic multi-objective approach instead, often referred to as the ϵconstraints method (Ehrgott, 2005; Waltz, 1967; Marler & Arora, 2004). This method assumes in general that the stakeholders can provide a lexicographic order, i.e., a given ranking, of the objective functions based on their importance. Following this predefined order, the method optimizes the model sequentially, each time with respect to a new objective function. In order to guarantee that at each iteration, the quality of the solution with respect to the previous objectives is not compromised, the method introduces additional constraints that limit the detriment of the previous objectives to be, at most, a predefined value φ. We note that, as with other multi-objective optimization methods, including traditional meta-heuristic approaches (Deb et al., 2002), the proposed lexicographic method can also be used to identify several optimal non-dominated solutions in the form of a Pareto frontier, for different SUDS configurations, by solving the proposed formulation applying different lexicographic orders and varying the values for φ. One particular case of interest, is when the allowed objective deterioration φ is set to zero. This allows the optimization model to search for alternative solutions within the optimal solution face of the feasible regions. Furthermore, in the specific context of this paper, given the importance of the stakeholders inputs, the flexibility of the proposed approach allows for a guided parameterization of different setups runs to be evaluated and tested. In addition to the weighted-sum and the proposed lexicographic methods, there are other alternatives that can be explored to tackle the multi-objective nature of the problem (Ehrgott, 2005; Marler & Arora, 2004). We decided to use the lexicographic method because it allows us to incorporate the stakeholders requirements within the decision-making process in a more natural way. i.e., it is in general easier for the stakeholders to rank the importance of the objectives rather than to providing a numerical preference. For the specific case of our TS-MILP model, the three objective functions are optimized in the following order (selected by the stakeholders): the first objective function (Equation 4) minimizes potable water consumption, which is the sum of the consumption 8# in each scenario ∈ ℰ, demanding node ∈ , and time step ∈ , multiplied by the probability of occurrence of scenario ∈ ℰ. The second objective function (Equation 5) minimizes the runoff leaving the offering

nodes (regardless of whether a SUDS is installed or not) + in each scenario ∈ ℰ, offering node 2 ∈ , and time step ∈ , multiplied by the probability of occurrence of scenario ∈ ℰ. Finally, the third objective function (Equation 6) minimizes the costs, which includes the cost of the connections $+% +% from offering node 2 ∈ to node 3 ∈ , the installation costs of the SUDS area () 6+) in nodes 2 ∈ using typologies 5 ∈ and the cost of the potable water used ;8# in each scenario ∈ ℰ, demanding node ∈ , and time step ∈ , multiplied by the probability of occurrence of each scenario . min ? min ? min @

∈ℰ ∈ℰ

+∈

@ @

A

#∈

∑

+∈

∑%∈

8#

(4)

+

(5)

∈ |BC D

∈ |BC D

|∃GHI $+% +%

+@

+∈

∑)∈

|∃JHK () 6+)

+?

2.6. Constraints

∈ℰ

@

#∈

A

∈ |BC D

;8#

(6)

We will now discuss the set of constraints that define the solution space of the problem. Expressions (7)–(33) along with the three objectives mentioned above define the proposed mathematical formulation. ∑)∈

|∃JHK

4+) ≤ 1

6+) ≤ *+) 4+) +N

∑)N

(7)

∀2 ∈ M, 5 ∈ |∃*+)

6+) ≥ ,+) 4+)

@

∀2 ∈

∀2 ∈ M, 5 ∈ |∃*+)

|∃JHK () 6+)

+@

+N

∑%N

(8) (9) |∃GHI $+% +%

≤O

(10)

Constraint set (7) guarantees that only one SUDS typology can be installed per offering node, while the sets of constraints (8) and (9) guarantee that the selected area is within the maximum and minimum allotted areas. Constraint (10) represents the budget limitation. The first term corresponds to the cost of installed SUDS typologies, and the second term is the cost of the installed connections. 7+% ≤ P

≤ ∑)N

+%

7+% ≤

+

∀2 ∈ , 3 ∈

+%

|∃JHK 4+)

∀2 ∈ , 3 ∈

∀2 ∈ , 3 ∈

∑%∈

|∃GHI

+,

= 0+, −

+%

, ∈ , ∈ ℰ|∃$+%

≤Q

|∃$+%

(11)

(12)

, ∈ , ∈ ℰ|∃$+% (13)

∀2 ∈ M

(14)

The sets of constraints (11) to (14) relate the variables 7+% , +% and 4+) . First, constraint set (11) ensures that the flow 7+% is zero if variable +% is zero, i.e., if the connection between nodes o and n is not used. Here, the value of M is essentially an upper bound on the flow given by 7+% . Note that if the connection from node 2 ∈ to 3 ∈ does not exist, then 7+% does not exist either. The set of constraints (12) assures that the connection +% can exist only if at least one typology is installed in node o. The set of constraints (13) states that the volume flowing through the connection 7+% has to be less or equal to the inventory + . Finally, the set of constraints (14) limits the number of connections N that can exist from an offering node o. + +

= =

+, +,

+,

∀2 ∈ , ∈ ℰ

+ 0+ − − ∑%∈

+

(15)

∀2 ∈ , ∈ , ∈ ℰ|1 = 0, ≠ 1

7+% + ∑%∈

7%+ − 9+

(16)

∀2 ∈ , ∈ , ∈ ℰ|1 = 1

(17)

The set of constraints (15) to (17) are the inventory constraints. The set of constraints (15) is used for the first time-step, which is defined as a wet period, thus the first balance includes the incoming runoff 0+ and the runoff leaving the node (the runoff that exceed the SUDS capacity). Constraint set (16) is used for subsequent wet periods and includes the condition of the inventory in the previous time step. Constraint set (17) is the inventory for dry periods, which besides including the state of the inventory in the previous time step, adds the inflows and outflows from other nodes (second and third terms on the right-hand of the equation) and the losses of water by evapotranspiration and infiltration (the last term of the equation). To properly estimate these losses (terms 9+ ), the model requires to consider their dependency on the typology of SUDS selected and the inventory of water available during the previous time step at each of the candidate locations. In essence, the losses due to evapotranspiration and infiltration depend on: (1) whether there is a SUDS installed in the location, (2) the evapotranspiration (.) ) and infiltration (/+ ) of the SUDS typology installed, and (3) whether the inventory of water collected by the SUDS has reached the potential evapotranspiration limit over a single time step (i.e., maximum expected evapotranspiration). To incorporate these disjunctive requirements into the proposed formulation, we model the term (9+ ) in a similar fashion as the way semi-continuous variables are modeled in integer programming (see Bertsimas & Weismantel (2005), Guéret et al. (2002), and Nemhauser & Wolsey (1988) for further reference). + +

≥ ∑)∈

|∃JHK 6+)

≤ ∑)∈

|∃JHK 6+)

9+ ≤ P′ Z1 − :+,

9+ ≤ ∑)∈

9+ ≥ ∑)∈

|∃JHK 6+) |∃JHK 6+)

.) + /+ − PT :+

∀2 ∈ , ∈ , ∈ ℰ

.) + /+ − U W X :+ + PT 1 − :+

∀2 ∈ , ∈ , ∈ ℰ (19)

V

∀2 ∈ , ∈ , ∈ ℰ| > 1

[

.) + /+

(20)

∀2 ∈ , ∈ , ∈ ℰ| > 1

.) + /+ − PT :+

(18)

(21)

∀2 ∈ , ∈ , ∈ ℰ| > 1

(22)

Modelling this type of semi-continuous variables is typically done by introducing an auxiliary set of binary variables-here the :+ -which take the value of one if the inventory of water has reached the potential evapotranspiration level and zero, otherwise. Furthermore, auxiliary variables :+ are later used in constraints (20) to (22) to limit the values that the loss terms can take, based on the conditions described above. The sets of constraints (18) to (22) are of similar nature to the constraints that are traditionally used to linearize bi-linear terms (i.e., multiplications of variables, often binary) when solving integer programming models (Bertsimas & Weismantel, 2005). It is important to notice, that despite the fact this approach requires increasing the size of the model by introducing additional sets of variables and constraints, it allow us to model the dependencies of the losses on the type of SUDS selected and the inventory of water stored in the SUDS. +

≤ ∑)∈

|∃JHK -) 6+)

"# = 8# + ∑+∈ 7+%

∀2 ∈ , ∈ , ∈ ℰ ∀ ∈

(23)

, ∈ , ∈ ℰ|1 = 1

(24)

Constraint set (23) guarantees that the inventory does not exceed the storage capacity of the installed SUDS, while the set of constraints (24) guarantees that the demand is satisfied by means of both potable water and inflows from offering nodes. Finally, the sets of constraints (25-33) represent the nature of the decision variables. 4+) ∈ ]0,1^ ∀2 ∈ , 5 ∈ 6+) ≥ 0 +%

| ∃ *+) (26)

∈ ]0,1^ ∀2 ∈ , 3 ∈

7+% ≥ 0 8# ≥ 0 +

∀2 ∈ , 5 ∈

≥0

∀2 ∈ , 3 ∈

∀ ∈

| ∃ *+)

| ∃ $+%

(25)

(27)

, ∈ , ∈ ℰ | ∃ $+% (28)

, ∈ , ∈ ℰ | 1 = 13

∀2 ∈ , ∈ , ∈ ℰ | 1 = 0 (30)

(29)

+

≥0

9+ ≥ 0

∀2 ∈ , ∈ , ∈ ℰ (31)

∀2 ∈ , ∈ , ∈ ℰ (32)

:+ ∈ ]0,1^ ∀2 ∈ , ∈ , ∈ ℰ

(33)

Figure 5 schemes the three steps of the lexicographic multi-objective model. During step 1, the algorithm solves the optimization problem for the first objective function. Let ` ∗ be the corresponding optimal solution. Then, at step 2, the algorithm introduces a new constraint that bounds the detriment of ` ∗ and solves the optimization problem for the second objective. Let `b∗ be the corresponding optimal solution of step 2. Finally, in step three the algorithm adds a constraint that bounds the detriment of `b∗ and solves the problem for the third objective. Figure 5: Scheme of the three steps in the lexicographic multi-objective model

One of the advantages of this type of formulations is that they can be easily parameterized to incorporate the specific needs of a wide variety of scenarios. Moreover, depending on the scale of the problem instance, these can be typically solved without the need of heavily specialized hardware. For the optimization process, any black-box commercial optimizer can be effectively used to obtain optimal solutions.

2.7. Step 5: Visualization of results Because of the spatial nature of the problem, maps are used to communicate results, aiming for an understandable format and graphical language. The maps illustrate the optimal solution, which is defined by the selected offering nodes and the corresponding typology and size, the supplied demanding nodes, and the connection to the offering nodes that supply the harvested water. They are automatically visualized in the GIS by uploading csv files (output of the optimization model containing the characteristics and coordinates of the optimal solution) in ArcPy. The same format and procedure is used to present a sensitivity analysis to assess trade-offs between alternative considerations and/or stakeholders preferences (Mysiak et al., 2005). Understandable and informative maps are expected to encourage stakeholders to contribute with suggestions at each iteration of the framework, while promoting bi-directional social learning (Mostert et al., 2008; Voinov & Bousquet, 2010). For example, modelers learn how to frame the problem while stakeholders develop insights on causal connections (e.g., the soil impermeabilization and its effect in the runoff volume) (Montalto et al., 2013). Figure 3 illustrates the iterative process looping on stakeholders suggestions until reaching the final set of site-typologies spatial configuration. The stop criteria is defined by the question “Is the proposed result satisfactory?”. If “No”, the stakeholders comments and suggestions are retrofitted to the feasible sites selection, the optimization, or the results processing for visualization (see Figure 3). The ability to modify the optimization model is perhaps one of the most valuable aspects of this framework. Here, the stakeholders and modeler communication is crucial to achieve a consensus on when the stop criteria is reached.

3. Case Study The proposed methodology was applied to identify an optimal selection of SUDS as well as their optimal locations in the central campus at Universidad de Los Andes in Bogotá (Colombia), which comprises a constructed area of 6 ha, consisting of 30 facility buildings and consumes around 4,800 m3 of potable water each month. Ninety-seven percent of this demand is supplied by the local water utility while the remaining 3% is obtained from reuse of grey water and harvested rainfall. Irrigation accounts for 13% of the total water, which is supplied by the water utility. The central campus’ gardens are irrigated manually using hosepipes, sprinklers systems or by drip irrigation, and they consume 16.6 m3 per day. The distribution of this demand in the study area is illustrated in Figure 6. Figure 6: Spatial distribution of the demand of water for irrigation in the central campus at Universidad de Los Andes

The available information for the case study includes an ortophoto, raster layers of infiltration rate and water table depth, a buildings vector layer, and a DEM. To consider inter- and intra–annual variability of the groundwater table depth and infiltration rate at city scale, the rasters (50 x 50 m resolution) were obtained by applying Kriging interpolation from 3,384 points from drilling studies undertaken from 1961 to 2011 (Jiménez Ariza et al., 2019). The slope raster was calculated from the 1 m resolution DEM; while the 3 m resolution orthophoto was obtained from a drone flight undertaken by the campus Management Office. Finally, the calculation of the distances from the university buildings was attained using the buildings vector layer. Using the above information (see Figure 2), feasible areas for SUDS siting were identified per typology. Five typologies were used for this study, considering the screening matrices provided by CIRIA, in which the case study is classified as commercial land use, steep slopes (ranging from 0 ∘ to 50 ∘ with a mean of 13 ∘ ), clay soil type (mean infiltration rate 8.5 mm/h), and limited availability of space (see Woods-Ballard et al. (2007); Woods Ballard et al. (2015)): detention basins, retention ponds, rain barrels, permeable pavements, and green roofs. Topographic and environmental constraints obtained from SUDS guidelines (summarized in Table 1) were used and implemented using ArcGIS 10.4.1. The result of this analysis is a map with the feasible sites, which was shared with the campus managers (who played the roles of both of decision makers and stakeholders). Several meetings resulted in a map containing information regarding sites that cannot be modified for SUDS placement and the inclusion of new areas that were not considered previously. Figure 7 shows the map resulting from the social cartography and Figure 8 illustrates the final feasible sites per SUDS typology. A total of 70 feasible sites were identified (1 site for retention ponds, 2 sites for detention basins, 7 for permeable pavements, 12 for green roofs, and 48 for rain barrels), which were considered to supply the demand of 45 demanding nodes. Table 1: Constraints used in the spatial feasibility analysis 1 Parameter

Constraint Retention Detention Green

Permeable

Rain

Ponds

Basins

Max

15

15

30

10

-

Min

-

-

-

0.5

-

Water table

Max

-

0.15

-

-

-

depth (m)

Min

0.6

0.6

-

2

-

Infiltration

Max

-

0.15

-

-

-

rate (mm/h)

Min

-

-

-

7

-

Distance to

Max

-

-

-

-

30

buildings (m)

Min

7.6

6

-

6

-

Slope (%)

Roofs Pavements Barrels

[1]Values obtained from Geosyntec (2010); Jiménez Ariza et al. (2019); RCFC (2010); PADEP (2006); of Watershed Protection (CWP); Middlesex (2003); of Edmonton (2014); Shoemaker et al. (2009); Tech (2013) Figure 7: Identified relevant sites resulting from the social cartography exercise A 2-meter resolution DEM was used to delineate the sub-catchments in the campus, -which was corrected using the buildings[start]â[end][start] [end][start] [end] height. Historical rainfall records were obtained from a nearby rain gauge station located 400 meters away from the campus (see Figure 6), which contained hourly records from the 2000 – 2013 period. The rain series was separated into individual rainfall events following the methodology proposed by Brown et al. (1985) and a frequency analysis was performed for the duration of the rainfall event. Figure 9 shows that 95% of the events have a duration of less than 6 hours, and that precipitation events with a duration of 2 hours are the most frequent, followed by durations of 1 and 3 hours. Precipitation events were categorized per duration; and each category was classified in three ranges of equal length, corresponding to low,

medium, and high intensities. Eighteen scenarios were constructed using 6 durations and three intensities, as shown in Figure 10. Figure 8: Spatial distribution of feasible sites per SUDS typology Figure 9: Frequency and cumulative probability of rainfall events for different durations for the 2000-2013 period. Probabilities of occurrence were calculated using the set of rainfall events ranging from 1 to 6 hours, so that the sum of probabilities from these 18 scenarios is 1. Figure 10 shows that, for all durations, the ”Low” scenario has the highest probability of occurrence and that this probability decreases with the duration of the events (see continuous lines in Figure 10). On the other hand, the total runoff for each scenario in the reference situation (the sum of the runoff reaching each feasible point with no SUDS implemented) increases with duration, thus scenarios with a high probability of occurrence are those with lower runoff (see bars in Figure 10). This configuration of scenarios follows the philosophy of designing for the most frequent events, instead of for the ”worst-case scenario”. Table 2 summarizes the probability of occurrence, duration and intensity range for the 18 scenarios. Synthetic hydrographs were constructed for each scenario by assuming a constant intensity, which corresponds to the upper limit of the range reported in Figure 8. The expected value –calculated by multiplying the mean of each intensity range (see Figure 10) times its corresponding probability of occurrence, and applying the rational method– of the total runoff available for harvesting given these 18 scenarios is 378 m3. Figure 10: Runoff (bars) and probability of occurrence (line series) for different rainfall scenarios . L: low, M: medium, and H: high. 1, 2, 3, 4, 5, 6 corresponds to the duration of the rainfall event. Table 2: Characterization of the 18 scenarios. L: low, M: medium, and H: high Scenario Duration (hr) Intensity range (mm/hr) Probability 1 (1L)

1

0.25 - 5.8

0.215

2 (1M)

1

5.9 - 11.3

0.006

3 (1H)

1

11.4 - 16.8

0.002

4 (2L)

2

0.25 - 4.1

0.322

5 (2M)

2

4.0 - 8.15

0.013

6 (2H)

2

8.16 - 12.1

0.006

7 (3L)

3

0.25 - 4.8

0.198

8 (3M)

3

4.9 - 9.5

0.015

9 (3H)

3

9.6 - 14.2

0.001

10 (4L)

4

0.25 - 3.6

0.097

11 (4M)

4

3.65 - 7.2

0.010

12 (4H)

4

7.3 - 10.8

0.002

13 (5L)

5

0.25 - 2.6

0.060

14 (5M)

5

2.7 - 5

0.010

15 (5H)

5

5.1 - 7.4

0.001

16 (6L)

6

0.25 - 3.2

0.034

17 (6M)

6

3.3 - 6.3

0.0060

18 (6H)

6

6.4 - 9.4

0.0010

The model was implemented in the Python language and solved using the commercial optimizer Gurobi 7.0.2 (Gurobi Optimization, 2017). The experiments were ran in a Dell OptiPlex 7060, equipped with an Intel (R) Core TM i7 – 3.4 GHz processor, 32 GB of RAM, and running Windows 10. The specific parameters found in literature and summarized in tables 3 and 4 were used for this case study. The detriment of the objective values in subsequent steps of the lexicographic multiobjective model was limited to 5%, i.e., e = 0.05. The evaporation rate was calculated based on the mean evaporation rate for Bogotá and the infiltration rate (for retention ponds and detention basins) was extracted using the coordinates of each feasible site from the infiltration rate raster layer. Green roofs parameters are set based on reported values for a lightweight gravel-sized drainage layer and a composite growing media. The budget was set as US $150,000 and only one rainwater-harvesting device was allotted per roof. Also, the unitary cost of each meter of connection was set at US $45 (considering both the costs of a PVC pipe in Bogotá and the installation costs in an urbanized area). Table 3: Parameters used in the case study, per SUDS typology [1] Tipology

Unitary cost Evapotranspiration

Generic

Minimum

Infiltration

area (i b

rate (mm/h)

30.0

-

( )

rate (mm/hr)

depth (m)

Retention ponds

186.6

0.125

1.8

Detention basins

221.2

0.125

1.6

30.0

-

Permeable pavements

190.0

-

3.0

15.0

2.5

Rain barrels

70.0

-

1.1

0.4

0.0

Green roofs

150.0

0.125

0.1

15.0

6.2

$USD/i

b

[1]Values obtained from Geosyntec (2010); Jiménez Ariza et al. (2019); RCFC (2010); PADEP (2006); of Watershed Protection (CWP); Middlesex (2003); of Edmonton (2014); Shoemaker et al. (2009); Tech (2013) Table 4: Parameters used in the case study, for green roofs Parameter Initial moisture of the soil (Field capacity) Infiltration rate Saturated hydraulic conductivity Saturation water content

Value 35% 6.2 mm/h 1250 mm/h 53%

4. Results 4.1. Sensitivity analysis of maximum connection length The allotted length of the connections is an important parameter that influences the amount of runoff water used for irrigation. Since the second objective function of the model is to minimize runoff, the model will allocate as much area for locating SUDS as it is possible to temporarily store runoff water. But, if the connections that permit the flow among the offering and demanding nodes are limiting the supply of stormwater, the demand will be satisfied using potable water. For this reason, to assess the effect of limiting the connection length into the optimal solution, the model was implemented for varying maximum connections[start]â[end][start] [end][start] [end] length ranging from 5 to 30 meters. Connections longer than 30 meters were considered impractical and were not contemplated in this study. Figure 11 shows the optimal spatial configurations for each of the maximum connection length, while Figure 12 describes the percentage of reduction of potable water and runoff volumes, as well as the percentage of budget used.

Figure 11: Spatial configuration of optimal solutions for different maximum length: a) 5 m, b) 10 m, c) 15 m, d) 20 m, e) 25 m and f) 30 m. Figure 12: Comparison among different connections length in the 5-30 meters range: percentage of reduction of potable water and runoff, percentage of budget used, and computational time. As expected, the maximum connection length has an important effect on the resulting optimal configuration. The 5-meters length (shown in Figure 11a) can supply only one demanding site (around 4.2% of the total water demand), in contrast with the 30-meters length (shown in Figure 11f), which supplies 27 demanding sites (60% of the demand). For all cases, the budget used was 87-90% of the total budget and the expected value of runoff reduction is around 62%. Regarding these results, it is worth mentioning that: a) SUDS cannot supply the total demand because demanding sites located in high areas are not connected to any offering node (only connections that flow by gravity are allotted), and because the demand is assumed constant in time while SUDS water availability decreases after the precipitation event (because of water uptake, infiltration and evapotranspiration losses); b) the budget is not completely consumed (even though the runoff reduction does not surpass 50%) because we imposed a limit on the maximum SUDS area available per node, restricting the volume that can be stored (particularly for the rain barrels, which were limited to one rain barrel of 0.4 m3 capacity per roof top); and c) because the second objective function, which minimizes the runoff is being limited by the first objective value (minimizing potable water), the spatial configuration of the SUDS are forcing the model to select those nodes closest to demanding areas (even if these SUDS have the lowest storage capacity). In order to confirm these findings, two analyses were carried out: evaluating the sensitivity of the model to different budgets and different number of rain barrels allowed, and eliminating the first objective function from the analysis (optimizing for runoff reduction and budget only).

4.2. Sensitivity analysis of the budget and the number of rain barrels In the ”best-case” scenario, were there are no budget limitations, the installation costs of all the feasible SUDS representas a total of USD $780,000, which was deemed as too high by the stakeholders. We then performed a sensitivity analysis for different budget levels to evaluate the overall impact that limiting the installation costs has on the expected benefits obtained by the system. To this end, we ran the model for different budgets calculated as a percentage of the largest expected cost of USD $780,000. The percentages used were: $30,000 (4% of the total cost), $45,000 (6%), $60,000 (8%), $90,000 (11%), $120,000 (15%), $150,000 (19%), $180,000 (23%), $210,000 (27%), $240,000 (30%). We performed a second sensitivity analysis with respect to the maximum number of rain barrels that can be installed per rooftop. We find this particular analysis of importance because the number of rain barrels limits the total volume of water that can be stored for irrigation, and since it was previously found that rain barrels are consistently being selected by the optimization model for its closeness to demanding areas (see Figure 8), increasing this number is expected to reduce both the potable water consumption and the runoff. Figure 13 shows the comparison of these reductions for the abovementioned budgets, using a maximum connection length of 30 meters. Figure 13: Reduction percentage of potable water consumption and runoff for different budgets and rain barrels allotted per rooftop. First of all, the results of the model suggest that the difference from allowing 1 to 2 rain barrels resulted in changes in potable water reduction of between 3% and 6% (about 0.3 and 0.6 m3), and runoff reduction of between 11% and 12% (18 and 21 m3). However, the change from allowing 2 to 3 rain barrels resulted in non-significant reductions in potable water (0.00005%-0.1%) and for runoff (0.09-0.2%). This result evidences that using two rain barrels per rooftop increases the storage capacity, but three rain barrels will result in a system that exceeds the offering volume, thus leaving a non-utilized storage capacity that provides no benefits. Additionally, each curve has an inflection point: of USD $50,000 for the potable water reduction curve and of USD $60,000 for the runoff reduction curve. The potable water reduction curve shows no improvement after the inflection point, but the runoff reduction curve keeps increasing for higher budgets, with slopes that tend to zero. For

illustration purposes, we will use the solution obtained with the $60,000 budget and two rain barrels per rooftop. Figure 14 illustrates the expected value of both potable water consumption and runoff reduction for the USD $60,000 budget, 2 rain barrels allotted per rooftop, and maximum length of 30 meters solution. That is, the reduction observed in each scenario multiplied by its probability of occurrence. The figure shows: i) runoff reductions are more sensitive to precipitation scenarios than potable water consumption reductions, ii) ”high precipitation” and higher duration scenarios have the lowest percentage of runoff reductions, while ”low precipitation” and lower duration scenarios (which are also the most probable) have the highest percentage of reductions, and iii) potable water consumption reduction remains almost constant for all durations. All the above affirmations can be explained by the fact that the runoff available is one order of magnitude higher than the potable water. Hence, the required water for irrigation can be guaranteed by the storage capacity of the SUDS selected; the reduction of potable water is no more than 50% because the model assumes a constant demand, while the stored water availability decreases in time, accounting for evapotranspiration, infiltration losses, and drainage rate. Figure 14: Runoff reduction, potable water reduction and probability of occurrence per scenario. L: low, M: medium, and H: high. 1, 2, 3, 4, 5, 6 corresponds to the duration of the rainfall event.

4.3. Comparison with/without potable water

Figure 15 contrasts the spatial configuration obtained for the USD $60,000 budget when potable water is not considered in the optimization model (this is, the new lexicographic model has only the two last objective functions). Table 5 specifies potable water and runoff reduction, as well as the number and areas per typology. As can be observed, when potable water is not considered, bigger SUDS areas are selected for detention basins (49% larger), and one permeable pavement typology is selected. The number of rain barrels remains the same, although the configuration of offering nodes with 1 or 2 rain barrels is variable, no clear pattern is identified. The increase in the SUDS areas is the result of not considering connections, using the surplus budget in constructing larger structures with bigger storage capacity to reduce more runoff. The solution that considers potable water reduces 47% of the runoff, while not considering potable water results in a runoff reduction of 61%. However, if the latter configuration is selected, there is a lost opportunity cost of reducing potable water by 50.6%. An interesting finding is that the spatial distribution of the two solutions (with and without potable water) is similar. It was expected that, when potable water is not considered, the model would select other typologies instead of rain barrels, which is preferred for rainwater harvesting for its closeness with demanding areas and lack of infiltration and evapotranspiration losses. However, obtained results show that even with the sole objective of reducing runoff, rain barrels are preferred, showing that rain barrels are an economic and effective measure to reduce runoff. This result has been found in previous studies such as Jones & Hunt (2010) and Seo et al. (2012); however, other studies that consider water quality have also mentioned that rain barrels may not be cost-effective in sites with substantial areas for infiltration practices (Zhen et al., 2006). Finally, it is worth mentioning that from all the model implementations (changing budgets, connection length and eliminating the potable water reduction), optimal solutions were invariably those that selected all typologies except for green roofs. In spite of the fact that green roofs are the second least expensive typology (see Table 3), three facts are particular about this typology: i) similarly to rain barrels, the catchment area is limited to the roof area, so these two typologies receive a lower runoff volume; ii) green roofs retain a considerable amount of water in the substrate layer, e.g., Ferrans et al. (2018), hence reducing the amount of water available for use in demanding areas; and iii) while green roofs reduce runoff through evapotranspiration, this magnitude is considerably less than the runoff volume losses from a comparably sized infiltration practice (Chui et al., 2016). Figure 15: Spatial configuration of optimal solutions a) considering potable, b) not considering potable water. Table 5: Number of SUDS and total area per typology with and without the potable water reduction objective function

Objective functions Potable water reduction

Considering potable water Not considering potable water 50.6%

-

47%

61%

Runoff reduction Typologies

Number

Area (ib )

Number

141.2

2

Area (ib )

Detention basins

2

210.4

Rain barrels

82

-

82

-

Permeable pavements

0

-

1

75

Connections

-

-

21

-

4.4. Stakeholders comments/suggestions The methodology was designed to aid stakeholders and decision makers in the selection and allocation of SUDS. As shown in Section 2, the stakeholder engagement is proposed as iterative, to guarantee that obtained results are satisfactory. In order to receive their feedback on the areas they consider required further improvement, a meeting with the stakeholders was scheduled to present the obtained results and gave them space to share their insights. Two iterations were required to reach a consensus on a ”satisfactory result” (refer to Figure 3). We implemented their suggestions to refine results accordingly. In this section we will describe stakeholders comments/suggestions and how this new information was included in the second iteration of the method. A topic that was revisited during the meeting was the selection of the SUDS typologies included in the analyses. Decision makers acknowledged that retention ponds and detention basins have never been considered in previous campus projects, since consulting companies previously hired did not explore other SUDS typologies besides green roofs and rain barrels. Campus managers showed special interest in the infiltrating-based structures, considered novel for the university context. Stakeholders have perceptions derived from past experiences that influenced the acceptance of proposed results. They expressed, for example, a reluctance to install rain barrels at ground level. They insisted on locating these structures in the basement of buildings, even if this meant a loss of the hydraulic head and a possible increase of costs (due to pumps requirements). They claimed that rain barrels have large space requirements, are not aesthetic, and generate noise—derived from the pumping through sand filters and activated carbon water treatments. On the other hand, although green roofs were not included in the final solution, the campus managers mentioned they would have discarded these structures from the beginning of the analysis; since they consider they have high maintenance costs and the disadvantage of demanding potable water with added nutrients. These insights are of extreme value for the optimized solution to fulfill decision makers’ expectations. Regarding the assumptions and constraints used to build the model, the stakeholders agreed on considering only connections that flow by gravity, since they prefer to avoid the use of pumps. Additionally, they reinforced the 30 meters connections maximum length assumption, but criticized that feasibility is only determined by the distance between nodes. They suggest considering possible obstacles, such as reinforced walls, pillars, and underneath pipes and wiring. Campus managers also mentioned that in some locations, gardens (or demanding nodes) are not at ground level but on a platform about a meter higher. This information modifies the feasibility of some connections but does not affect the equations governing the optimization model. Comments were also made regarding the plots and information shown in the results. Images similar to those in Figure 11 were shown to explain optimized configurations. Campus managers interpreted the graphs and qualified the proposed solution as feasible and useful, and the budget suggested (USD $60,000) as accessible. However, they mentioned that in order to consider this proposal for future implementation, they require more details from the model, e.g., where the outlet is to be located, where the pipe is to be buried, what the environmental benefits are, etc. They valued the fact that the model is selecting best locations from among a long list of possibilities, but they mentioned the usefulness of ranking few top sites to develop a cost-benefit analysis that allows them to choose

among these reduced prioritized options. Suggestions regarding quantifying the water volumes stored per year were made; claiming that this data will help them evaluate the project financially. Finally, the authors inquired about the perceived value of the tool and the interest of the campus managers in rainwater harvesting projects. Decision makers agreed on the usefulness of the exercise for the optimal selection of SUDS sites; they mentioned that some of the outputs will be considered for future projects since the university campus is largely interested in constructing more and better rainwater harvesting strategies: ”The sustainability is a path we are already building. Our university is evaluating the feasibility of projects that, even if not the most financially favorable, they provide ludic-educative spaces and environmental benefits”. Both potable water savings and runoff reduction are objectives of major importance; however, they mentioned that for the near future, they are focusing their efforts on rainwater harvesting for flushing toilets instead of irrigation purposes, since recent studies based within the campus, have shown that flushing water volumes are considerable higher.

4.5. Implementation of stakeholders comments/suggestions Figure 3 shows the three stages in which stakeholders comments can be addressed. In the first place, there are changes concerning the feasible sites selection and parameterization: i) eliminate green roofs as a SUDS typology option, ii) modify the distance of connecting nodes (to include the pipe required to go around possible obstacles such as reinforced walls), iii) consider the gardens’ platform height above the ground level, iv) increase the cost of rainbarrels to account for pumps for water distribution from the basements. The second category gathers those comments that require a modification in the optimization model equations (examples or these changes are the elimination of one or more objective functions, or restricting the length or number of allotted connections); these type of modifications were performed for the sensitivity analyses and comparisons presented in subsections 4.2 and 4.3. Finally, third category includes comments that modify results’ presentation. For example, i) provide a top-5 ranking sites from which to start the implementations, and ii) report the stored water volume per year. Updated results from the second iteration are shown in Figure 16. Figure 16: Second iteration’s spatial configuration of optimal solution. 30 meters connections alloted, considering potable water and budget $60,000. Top-5 ranking sites are labeled.

Table 6 summarizes the number of SUDS and the area per typology optimized in the second iteration. It is observed that the model prioritize the reduction of potable water use despite the increase on rainbarrels costs (accounting for pumps and pipes to round obstacles). As a result the budget left for the second objective (i.e., minimize runoff volume) is decreased and the infiltration-based SUDS are reduced in area. Also, given that the cost of the pump is the same despite the number of rainbarrels installed (1 or 2), the model is systematically replacing 1 rainbarrel for 2 to reduce the unitary extracosts (see Figures 15 and 16).

The runoff volumes and the harvested water volume to supply the demand were simulated for the period 2010-2017 using the spatial configuration of the optimal solution. Figure 17 shows these volumes per year, the total annual precipitation, and the multi-year mean: 1,720 ik for intercepted runoff volume, and 1,619 ik for harvested water volume to supply the demand. Observe that the runoff volume intercepted (white bars) is bigger than that used to supply the demand (gray bars). Not all the SUDS selected are connected to demanding nodes; specifically, the infiltration-based SUDS are only reducing runoff volume (see Figure 16). Finally, Table 7 provides a top-5 ranking of the nodes-typologies based on a Benefit/Cost ratio (B/C ratio) for i) the demand satisfied with harvested rainwater (ik ) and ii) the runoff volume intercepted (ik ). Since the two criteria gave, as expected, different ranking positions, we organized the B/C ratio following the lexicographic model priorities (potable water first and runoff volume second). These top-5 nodes are shown in Figure 16. A top-20 list of the ranking is available in Appendix C. Observe that the top-20 corresponds to rainbarrels typology only; evidencing again rainbarrels’ efficiency over infiltration-based SUDS for rainwater collecting for reuse. Table 6: Number of SUDS and total area per typology for the second iteration.

Number Area (ib )

Typologies Detention basins Rain barrels Permeable pavements Connections

1

30

90

-

3

45

23

-

Figure 17: Simulated intercepted runoff and harvested runoff volumes to supply the demand (ik ) per year; total annual precipitation (mm) during the period 2010-2017. Table 7: Top 5 suggested SUDS based on B/C ratios. 2RB: 2 rainbarrels Node

Satisfied demand

Runoff intercepted

Satisfied

Runoff

(i k )

demand-based rank

intercepted-based rank

(Typology)

(i k )

76 (2RB)

287.3

6,156

1.5

1

265.3

4

81 (2RB)

236.5

5,044

1.3

2

217.5

16

91 (2RB)

192.7

6,162

1.0

3

265.6

3

88 (2RB)

148.9

5,744

0.8

4

247.6

12

85 (2RB)

140.2

5,364

0.8

5

231.2

15

B/C (ik / Position $USD)

B/C (ik / Position $USD)

5. Discussion and Conclusions This paper developed a methodology that couples GIS feasibility analysis with a stochastic multiobjective model for the selection of sites and SUDS typologies, with the purpose of reducing i) the use of potable water for irrigation, ii) runoff, and iii) the installation costs. We used an iterative participatory approach to engage stakeholders in the decision-making process. This method is especially useful when the data is scarce and detailed hydrologic/hydraulic models are not available, and when a more active participation of the stakeholders is seeked. The methodology was tested using a case study involving the central campus at Universidad de Los Andes to supply the demand of 16.6 m3 per day used in the campus gardens. We found that the expected runoff of the sites selected and the use of potable water can be reduced by 67% and 50%, respectively. Our results, for both potable water and runoff reduction, lay in the ”middle” range of the efficiencies reported in literature for a variety of scales (e.g. individual houses, multi-story residential buildings, neighborhoods, university campuses, and cities (e.g., Steffen et al. (2013); Crowley (2005); Ghisi & de Oliveira (2007); López Zavala et al. (2016)). This, along with the wide range of average annual runoff volume reduction (20% - 100%) (Zhang & Hu, 2014; Gilroy & McCuen, 2009) is explained by the diversity of hydrologic patterns, and the specific characteristics of the site (e.g. roof/lot size ratio, soil permeability, slope) (Karpiscak et al., 1990). The optimization model consistently selected rain barrels over infiltration-based typologies. In spite that rain barrels efficiencies have proven promising in previous studies (Domènech & Saurí, 2011), infiltration-based typologies have been reported to overpass rain barrels performance in terms of runoff peak and volume reduction (Abi Aad et al., 2009). This finding evidences the importance of considering the connections in the optimization model: rain barrels are preferred, in this case, because the rainwater is collected close to the place it is used. Similarly, the effect of the connections between demanding and offering nodes proved decisive when selecting SUDS siting. The location in the watershed (e.g. downstream vs. upstream, or high vs. low imperviousness areas) played an essential role in previous studies (Chang et al., 2009; Di Vittorio &

Ahiablame, 2015; Hopkins et al., 2017); but this effect was undermined in our study when considered where the harvested water is used. As a result, we did not find any pattern that favored the SUDS location besides its closeness to demanding nodes. An advantage of using this iterative participatory method is that it allows stakeholders to suggest changes to the model via the modification of parameters, constraints and objective functions. The proposed framework relies on the use of mixed integer programs as the main optimization engine. However, we note that the proposed model can be replaced by other mechanisms such as metaheuristics to identify similar solutions that also fit the stakeholders needs. A disadvantage of using exact methods such as mixed-integer programs is the difficulty of incorporating the complex hydraulic equations to quantify water flows. Heuristic-based methods, coupled with urban drainage models, are often used to provide a more accurate hydraulic representation; however, note that this implies loosing some desired properties of exact methods such as the solutions’ guaranteed optimality and the adaptability. For example, a common metaheuristic that is used in this context are the genetic algorithms. Some of the changes that may be required by the stakeholders may trigger significant modifications to vital components of the metaheuristic, such as the phenotype description and the crossover and mutation operators. To overcome the lack of precision of other exact methods found in the literature, we strengthened the description of the hydrologic dynamics and thus improve the accuracy of the system representation by introducing an additional set of variables and constraints. We focused our efforts on better accounting for the inventory of water collected by the SUDS and their interaction with the evapotranspiration and infiltration losses. These considerations improve the model accuracy without compromising its solvability. In terms of limitations of our methodology, our methodology may present scalability problems in cases where the number of variables required to model the system becomes too large. In such cases, it is possible for these exact methods to stall before finding optimal solutions. Also, our approach shares the limitations of all hydrological models: a strong dependency on the proper estimation of the parameters, as well as the limitations on the considerations of the hydrological interactions occurring at finer temporal and spatial scales (e.g. soil moisture can change in the order of minutes) (Salvadore et al., 2015; Bach et al., 2014). A consideration that aligns with the above mentioned limitation is the granularity of the time discretization. On the one hand, decreasing the time step length increases the accuracy of the model when considering the temporal variations of the evapotranspiration and infiltration losses’, for they occur in finer intervals of less than an hour (Salvadore et al., 2015). On the other hand, optimizing for a series of events with their corresponding dry periods between rainfalls, or even for a long continuous time series, could increase the computational effort required to solve the resulting problem due to the number of additional variables and constraints that must be considered. Furthermore, introducing a component that accounts for the antecedent dry period can also improve the accuracy of the SUDS hydrological performance evaluation, since this is a relevant parameter for the stormwater retention in soils and SUDS substrate (Berretta et al., 2014). An area of improvement for future work is to consider the variability of water demand in time, which is likely to present inter- and intra- day/week/season fluctuations (Zhou et al., 2002)). This may have an effect on the flows from the offering nodes, which in turn may also affect the expected runoff and the saving of potable water used for irrigation purposes. We note that it is possible to incorporate the stochastic nature of the demand in a similar fashion as the way the rainfall was modeled. This, however, would require a more complex scenario based analysis that can be considered in future efforts.

6. Acknowledgements The authors would like to thank Andrés Felipe Muñoz for designing the social cartography exercises and holding the meetings with campus Management Office. We would also like to acknowledge the help of Engineer Daniel Rodríguez from Bogotá’s water utility in assigning connections costs for the

case study. Finally, we thank the editor and two anonymous reviewers, whose valuable insights improved this manuscript.

Chapter . Appendix A Chapter . Appendix B Table 8: List of sets, parameters and variables of the TS-MILP Sets set of SUDS typologies set of time steps

ℰ

set of scenarios set of nodes set of demanding nodes set of offering nodes

Parameters "#

water demand in node

runoff volume available for storage in node 2 ∈ , scenario

0+

maximum area available of offering node 2 ∈

*+)

θ

cost of the pipe from node 3 ∈

manhattan linear distance from node 3 ∈

'%&

budget

infiltration per hour in node 2 ∈

/+

φ

allowed detriment from previous optimal solution probability of occurrence of scenario

1

∈ℰ

1 if the time step is a dry period, 0 otherwise

Variables

8#

to node i ∈

evapotranspiration per hour from typology 5 ∈

l)

7+%

to node i ∈

unitary cost of the typology 5 ∈

()

+%

for typology 5 ∈

cost of a cubic meter of potable water from the water utility

$%&

6+)

for typology 5 ∈

minimum area available of offering node 2 ∈

,+)

4+)

∈

generic depth of the typology 5 ∈

-)

ρ

∈ ℰ and time

∈

1 if the typology 5 ∈

is selected in offering node 2 ∈ , 0 otherwise

area of the typology 5 ∈

1 if the connection among offering node 2 ∈

volume sent from the offering node 2 ∈

is selected in offering node 2 ∈

and node 3 ∈

to node 3 ∈

exists, 0 otherwise

in the time

volume of potable water from water utility used in demanding node

∈

∈

and scenario ∈ℰ

in the time ∈ and scenario ∈ ℰ

losses volume (evapotranspiration and infiltration) from offering node 2 ∈ in the time ∈ and scenario ∈ ℰ

9+

runoff volume generated in offering node 2 ∈

+

inventory volume available in offering node 2 ∈

+

:+

in the time

in the time

∈ ∈

and scenario and scenario

∈ℰ ∈ℰ

auxiliary variable used in the linearization of evapotranspiration and infiltration losses for offering node 2 ∈ , time ∈ and scenario ∈ ℰ

Chapter . Appendix C Table 9: Top-20 nodes-SUDS based on B/C ratio Node (Typology)

Satisfied demand (i k )

Runoff intercepted (i k )

Satisfied

Runoff

demand-based

intercepted-based

B/C (ik /$USD)

Position

1.5

1

B/C (ik /$USD)

Position

265.3

4

76 (2RB)

287.3

6,156

81 (2RB)

236.5

5,044

1.3

2

217.5

16

91 (2RB)

192.7

6,162

1.0

3

265.6

3

88 (2RB)

148.9

5,744

0.8

4

247.6

12

85 (2RB)

140.2

5,364

0.8

4

231.2

15

73 (2RB)

131.4

6,053

0.7

5

260.9

10

83 (2RB)

113.9

6,104

0.6

6

263.1

7

71 (2RB)

96.4

5,724

0.5

7

246.8

13

115 (2RB)

96.4

6,585

0.5

7

283.9

1

92 (2RB)

69.2

5,711

0.4

8

246.2

14

80 (2RB)

52.6

6,067

0.3

9

261.5

9

70 (2RB)

42.0

6,142

0.2

10

264.8

5

82 (2RB)

26.3

6,175

0.1

11

266.2

2

74 (2RB)

17.5

6,116

0.1

11

263.6

6

114 (2RB)

823.4

2,047

0.1

11

2.5

20

113 (2RB)

0

6,585.5

0.0

11

283.1

1

75 (2RB)

0

6,585.5

0.0

11

283.1

1

79 (2RB)

0

6,585.5

0.0

11

283.1

1

86 (2RB)

0

6,585.5

0.0

11

283.1

1

78 (2RB)

0

6,585.5

0.0

11

283.1

1

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Highlights • • • • •

A flexible computational framework to select and allocate SUDS was developed. The objectives are minimizing the use of potable water for irrigation and reducing the water runoff at a minimum cost A Geographic Information System (GIS) is coupled with a two-stage stochastic mixed integer linear program (TS-MILP). The proposed methods can be easily adapted to incorporate a wide variety of stakeholder requirements or preferences. An iterative participatory approach is used to include stakeholders in the decision-making process.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: