Use of Nonlinear Membership Functions and the Water Stress Index

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On the use of non-linear membership functions and the water stress index for the environmentally conscious management of urban water systems: Application to the city of Morelia Sergio Armando Medina-González, Maria Guadalupe Rojas-Torres, José María Ponce-Ortega, Antonio Espuna, and Gonzalo Guillén-Gosálbez ACS Sustainable Chem. Eng., Just Accepted Manuscript • DOI: 10.1021/ acssuschemeng.8b00660 • Publication Date (Web): 01 May 2018 Downloaded from http://pubs.acs.org on May 3, 2018

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ACS Sustainable Chemistry & Engineering

On the use of non-linear membership functions and the water stress index for the environmentally conscious management of urban water systems: Application to the city of Morelia Sergio A. Medina-Gonzáleza, Ma. Guadalupe Rojas-Torresb, José M. Ponce-Ortegac, Antonio Espuñaa and Gonzalo Guillén-Gosálbezd* a

Chemical Engineering Department, Universitat Politècnica de Catalunya, EEBE. Av. Eduard Maristany, 16, Edifici I, Planta 6, 08019 Barcelona, Spain.

b

Chemical Engineering Dept., Universidad de La Ciénega del Estado de Michoacán de Ocampo. Avenida Universidad 3000, Sahuayo, Michoacán, México. c

Chemical Engineering Dept., Universidad Michoacana de San Nicolás de Hidalgo, Morelia Michoacán, 58060, México.

d

Centre for Process Systems Engineering (CPSE), Imperial College London, SW7 2AZ, United Kingdom *corresponding author: [email protected]

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Abstract This article proposes a multi-objective optimization strategy based on a fuzzy formulation for the sustainable design and planning of water supply chains in urban areas considering simultaneously economic and environmental objectives. Harvested rainwater and reclaimed water are considered as alternative sources to reduce freshwater consumption while maximizing water revenues and minimizing land usage. As opposed to other works that attempt to minimize water consumption, this work seeks to minimize the water stress index, which quantifies the impact of freshwater consumption considering the specific location where the withdrawals take place. We illustrate the capabilities of this approach through its application to a real case study based on the city of Morelia in Mexico, in which we show that the use of alternative water sources along with an appropriate water distribution plan allows to reduce the pressure over natural reservoirs. Keywords: Fuzzy Programming, Multi-Objective-Optimization, Water Stress Index, Cause-effect relationship

Introduction Water is an essential resource for all anthropogenic activities worldwide, including food production, energy generation and domestic applications, among others. It is well known that industrial, agricultural and domestic activities are the largest contributors towards water consumption at the global level, representing more than 50% of the annual withdrawals1. Accordingly, these massive water requirements are driving the fast 2 ACS Paragon Plus Environment

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depletion of available freshwater, which compromises water availability in the near future. Water scarcity affects in various ways geographic regions across the world due to the uneven spatial distribution of ground water availability as well as the regionspecific climatic conditions. Therefore, sustainable water management strategies that consider the spatial features of water consumption are required to ensure that water is consumed in a sustainable manner, particularly in the industrial and domestic sectors. Water preservation/conservation requires overcoming challenges such as: (i) integration of water reuse/recycle strategies in industrial processes; (ii) development of efficient techniques for the redesign/retrofit of current industries and wastewater treatment for an enhanced operation; and (iii) integration of water efficiency indexes in decision-support tools. The process systems engineering (PSE) community is particularly well positioned to tackle these challenges, as it adopts a holistic systemsbased analysis that seeks integrated solutions where the impact is minimized globally while considering feasibility constraints imposed by universal physical laws and current regulations. Reducing freshwater consumption is a critical problem in sustainability that is often addressed through the implementation of recycle and reuse strategies. The wellknown pinch analysis is a widely used method tackle this problem, in which the combination of different water streams allows to satisfy water quality/quantity requirements while minimizing the amount of fresh water required2-4. Particularly, the integration of mathematical programming and the pinch analysis method was applied to address a variety of water reuse/recycle, regeneration, and wastewater treatment problems5-10, where the main goal was to combine process streams so as to reduce freshwater consumption. 3 ACS Paragon Plus Environment


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Water preservation and its management represents another important problem worldwide that was traditionally addressed through the optimization of water distribution systems considering multiple water sources. In a seminal work, Liu et al.11 applied mathematical programming techniques to optimize the management of water in a Greek island considering alternative sources, such as desalinated seawater, treated wastewater and/or reclaimed rainwater. Later, a similar approach was used to address the optimal design and operation of water networks in an urban area using water storage to maximize demand satisfaction12-14. Multi-objective optimization (MOO) has gained wide popularity in sustainability problems as it allows to simultaneously consider economic and environmental objectives in the design and planning of a wide variety of industrial systems, including water networks15. For instance, Zhang et al.16 proposed a systematic MOO framework, in which the trade-off among the recovered wastewater, regeneration costs and pollutants reduction was investigated in order to assess the potential benefits of reusing wastewater in regional sectors. In another work, the amount of rainwater collected was considered as an alternative water source within a water network17. In this study, three conflicting objectives are considered: one economic and two environmental ones (i.e., freshwater consumption and land use). More recently, the scope of the study was enlarged to optimize dual-purpose power plants and water distribution networks according to economic, environmental and social objectives18. Furthermore, multiobjective models were applied to optimize the use of water in agriculture concerning wheat production19. The overwhelming majority of formulations discussed above that tackle water preservation/conservation problems optimize freshwater consumption rather than the 4 ACS Paragon Plus Environment

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environmental impact associated with the water withdrawals. Nevertheless, the impact of freshwater withdrawals critically depends on the location where they occur. In fact, significant developments in life cycle assessment of water have led to more sophisticated environmental indices that better express the cause-effect relationships between water use and environmental impact. Along these lines, in 2009 the water stress index (WSI) was proposed to describe and model the impact of water consumption considering its local availability20. The authors discussed in their work the many advantages of using the WSI as key indicator to quantify the potential benefits of implementing water reduction strategies. An extensive literature review on water efficiency indexes was carried out in the work by Brown21. This review covers several metrics available for assessing water consumption, including the Falkenmark indicator, social water stress, water resources vulnerability, water supply stress and WSI. It was concluded that the effective management of water resources requires a holistic approach, linking social and economic development with one of the above indexes21. These metrics should attempt to quantify the real impact of water consumption considering the regional aspects of the withdrawals. To the authors’ best knowledge, the aforementioned water efficiency indexes were never included as environmental objective in a multi-objective (MO) water management problem at the urban level like the one addressed herein. Hence, there is significant room for improvement in the way water management is optimized, particularly regarding the selection of appropriate environmental metrics. Another major challenge in MOO models, besides the definition of tailored environmental objective functions, concerns the post-optimal analysis of their solution, which is not unique but rather given by a set of Pareto points. These Pareto solutions need to be analyzed after they are calculated and tools are therefore required for a 5 ACS Paragon Plus Environment


systematic identification of the best option among them. Methods used for this purpose include the Analytical Hierarchical Processes (AHP)22, Weighted Sum Approach (WSA)23, constraint, objective programming, and dictionary ordering24. For more information regarding the concepts and capabilities behind these approaches, readers are referred to the extensive review of Cui et al.24. One of the main disadvantages of the above techniques is that due to the definition of objectives preferences some desirable robust solutions could be wrongly discarded. Alternatively, one can systematically narrow down the number of Pareto solutions following other approaches, including, Pareto filters25-27, ELECTRE methods28, data envelopment analysis29 and bi-level optimization30,31, which can be used as standalone MCDM techniques or combined with traditional MOO methods32. Nevertheless, these alternatives suffer from one of the following limitations: (i) need of a large and accurate list of options (solutions) and, (ii) large computational requirements to produce a solution. In this context, fuzzy programming is a promising alternative to reduce the complexity of MOO models since simultaneously overcomes the above limitations while promoting the generation of well-balanced solutions. Fuzzy programming was first introduced by Zimmermann33 in this specific context and widely used afterwards in a wide variety of problems34-37. This approach defines a singleobjective (SO) function by modelling objectives using membership functions that are finally combined into a single metric. This approach was applied to the selection of manufacturing38 and energy systems in which both energy efficiency and total cost were expressed as fuzzy sets39. Similarly, a MO-fuzzy formulation was used in the planning of heat/cooling networks considering operating costs and energy requirements40. More recently, a nonlinear fuzzy membership function was used to represent imprecise aspiration levels of a set of decision-makers41. In this formulation, a nonlinear objective 6 ACS Paragon Plus Environment

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behavior was assumed without an explicit cause-effect relation between the membership functions and the objective functions. Despite all the studies on fuzzy approaches that address MOO problems, two main challenges remain unsolved. First, the proper definition of membership functions so as to capture the objectives’ behavior and its associated impact (cause-effect). Second, how to properly incorporate the decision-makers’ preferences in the fuzzy model. This work proposes a novel approach for the optimal retrofit and planning of a water distribution network in an urban area based on a MO-fuzzy formulation that makes use of nonlinear membership functions. Three conflicting objectives are considered: , economic profit (Profit), , water consumption (WC) and ,

land usage (LU). The first criterion is commonly optimized in industrial processes and reflects the economic dimension of sustainability. The other two quantify environmental aspects, with the third one measuring as well the level of complexity of the network and the ease of operation. and are formulated assuming linear membership functions following traditional fuzzy methods. For , a nonlinear membership

function is defined that links the WC to the water availability. Hence, one of the main novelties of the work is the adoption of a mathematical approach to capture the causeeffect relationship between water consumption and the associated impact (rather than using WC as proxy of environmental impact). The general formulation of the water network is discussed later in this paper. Additionally, a case study is presented afterwards to illustrate the proposed approach, while the conclusions of the work are drawn in the last part of the article.



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Problem statement We address the re-design and operation of a water network system considering its economic performance and environmental impact (see Fig.1). To derive our mathematical formulation, a standard high-level network is considered encompassing natural water sources that play the role of suppliers (including damns, springs and

deep wells). The supply chain (SC) also includes industrial , agricultural ℎ and

domestic sites acting as water consumers along with potential sites for storage tanks

and artificial ponds (indexed with the subscripts and , for agricultural and domestic

sites, and and , for industrial sites, respectively).

Natural freshwater sources, , can be recharged by direct precipitation, runoff

water and by natural tributaries . Water from natural sources is treated in central

facilities (i.e., mains) and distributed to industrial , agricultural ℎ and/or domestic

sites. Reclaimed water can be directly used to meet the agricultural demands or might be discharged to the environment. Water from external places can be purchased and distributed directly to the final users whenever natural sources cannot satisfy the water demand. Finally, harvested rainwater can be stored in different facilities (storage tanks and artificial ponds) and used as an alternative source to meet the user’s requirements.


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Figure 1. Superstructure for water distribution at macroscopic level.

Given the average capacity of natural water sources, the water demand to be met and supply capacities of alternative water sources as well as purchasing prices and capacity constraints, the goal of the analysis is to identify the best design and planning decisions in terms of maximum economic performance and water perseveration as well as minimum land usage. Further details on how the problem is formally stated can be found in the literature9, 17.

Methodology The proposed approach comprises three main steps as shown in Fig. 2. A MOO model is developed in step 1. Step 2 reformulates the MOO model into a single-objective 9 ACS Paragon Plus Environment


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optimization (SOO) one by using membership functions. Finally, the SOO model is solved in step 3. A detailed description of each step is provided in the ensuing subsections.

Figure 2. Algorithm for the proposed strategy.

Multi-Objective Model The mathematical model presented herein capitalizes on the mixed-integer linear programming (MILP) formulation introduced by Rojas-Torres et. al.17. The model seeks to optimize simultaneously the Profit, WC and LU objectives described next. Due to space limitations, the complete formulation is presented in the Appendix A of the Supporting Information, while the multi-dimensional objective function is described in detail next. Economic Objective 10 ACS Paragon Plus Environment

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The economic objective is calculated from the revenues and expenditures associated with the management of water (Eq. (1)). Revenues are obtained from water sales for

domestic, agricultural, and industrial purposes (). Expenditures account for the

treatment

( ),

distribution

( )

installation/operation costs associated with artificial tanks ( ).

= − − −

and

the

(1)

Environmental Objectives (WC and LU) The was originally used as environmental objective to assess the impact on natural water repositories6,7,42. The calculation is described by Eq. (2).

= "#$% &' + $% &) + $% &* + &

+ "#,ℎ-. + + "(,ℎ-/ ) + "(,ℎ-0 ) .

/

0

The first three terms in the right hand side of Eq. (2) represent the total flow of freshwater from water source (k) to domestic, agricultural and industrial sites, respectively. As commented, freshwater can be purchased from external suppliers and sent to domestic, agricultural and industrial sites, which are considered in the last three terms of Eq. (2). As will be later discussed in the article, this objective does not account for the spatial specificity of the impact, and therefore it is replaced here by the water stress index WSI, which provides a better estimates of the “true” impact of water consumption. The 23 represents the surface occupied by the installation of storage tanks and is determined as follows:

23 = " 4526 + " 4578 + " 459 + " 457: (3) 6

8

9

:

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11

(2)


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Where the first two terms in the right hand side of the equation represent the area occupied by artificial ponds for agricultural/domestic and industrial usage, respectively. Similarly the last two terms model the area occupied by storage tanks for agricultural/domestic and industrial usage, respectively. The overall model M can be described in compact form as follows:

( 0.2), any increment in the water consumption will significantly compromise the water availability for future applications; finally, for large values in WTA (>0.9) the impact becomes irreversible and even when there is still water available in the reservoirs, it is likely that other processes will operate under water limitations. The most appropriate expression to represent this WSI behavior is a sigmoidal function as the shown in Eq. (8), which provides a continuous range between 0.01 and 1 as discussed in the litterature20. Therefore, Eq. (8) can then be used as membership function for quantifying the impact of water consumption:

7 =

1

1 + TU.V∗XYZ [

1 0.01 − 1\

(8)

Eq. (8) describes that. The environmental objective related to water consumption is then calculated via Eq. (9):


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HX^ (=X^ ) = 1 − 7

(9)

Finally, the original MO model is reformulated into model M2 as follows:

(

Use of Nonlinear Membership Functions and the Water Stress Index

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