Multi-Objective tool to optimize the Water Resources Management using Genetic Algorithm and the Pareto Optimality Concept Issam Nouiri
Water Resources Management An International Journal - Published for the European Water Resources Association (EWRA) ISSN 0920-4741 Water Resour Manage DOI 10.1007/s11269-014-0643-x
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Author's personal copy Water Resour Manage DOI 10.1007/s11269-014-0643-x
Multi-Objective tool to optimize the Water Resources Management using Genetic Algorithm and the Pareto Optimality Concept Issam Nouiri
Received: 5 January 2013 / Accepted: 23 April 2014 # Springer Science+Business Media Dordrecht 2014
Abstract This paper examines the development of a multi-objective tool, called “ALL_WATER”, in optimizing Water Resources Management. The objectives of satisfying demand and reducing costs were taken into consideration while at the same time respecting water salinity requirements and hydraulic constraints. A Multi-Objective Genetic Algorithm (MOGA) and the PARETO optimality concept were used to resolve the formulated problem. The tool developed was used to help optimize the daily management schedule of a real case study in Tunisia. The hydraulic system is made up of three surface water sources, one demand site, two transfer links and three supply links. Within a short computation time, a PARETO front was identified made up of a set of 72 optimal solutions. The modeling approach and the decision-making flexibility, both shown in the case study, prove that the developed tool is able to efficiently identify a set of optimal solutions on a PARETO front. The developed tool will be able to be used for a large variety of water management problems. Keywords ALL_WATER . Water . Management . Optimization . Genetic Algorithm . Tunisia
1 Introduction Identifying and evaluating water resources under Climate Change (CC) conditions is a continual goal which presents a great challenge for managers. Improving the effectiveness of Water Resources Management (WRM) despite its increasing complexity is another serious problem facing managers and decision makers (DM). Managing water efficiently is a difficult task due to several factors: the complexity of natural and man-made hydraulic systems (higher number of sources, demand sites and transfer and supply links); the length and time intervals of the management period; and the variability of management priorities according to the time and place (Chang 2008; Giupponi 2007; Ren et al. 2013). Managers have to find the optimal solution that satisfies demand requirements in terms of volume and quality, while trying to minimize costs throughout the management area over a given period as well (Zhang et al.
I. Nouiri (*) National Institute of Agronomy of Tunisia, 43 Avenue Charles Nicolle 1082 le Mahrajène, Tunis, Tunisia e-mail:
[email protected]
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2012). Environmental and socio-economic aspects can be additional management objectives (Giupponi 2007). Optimal solutions must therefore satisfy a large number of constraints. In order to address the aspects of multiple criteria and higher dimensionality of the WRM problem, researchers have proposed specialized optimization approaches and tools. Koutsoyiannis et al. (2003) developed a Decision Support System (DSS) for optimal management of a surface water resource system. Mysiak et al. (2005) proposed a DSS for WRM integrating hydrological models with multiple-criteria evaluation procedures. Rees et al. (2006) developed a software system to assist practitioners in balancing natural water availability with the requirements of water users. Giupponi (2007) developed a DSS for integrated water resource management (IWRM) to help the DM “make right priorities” and define and evaluate “numerous alternatives.” Khare et al. (2007) proposed a simple economic-engineering optimization model to explore the potential of conjunctive use of surface and groundwater resources. Letcher et al. (2007) provided a generalized conceptual framework taking into consideration water allocation, agricultural production and water use decisions and their interaction with the stream system. The method of Li et al. (2007) was developed for WRM under uncertain conditions to help water resource managers identify desired system designs at times of water shortage or flooding. Dvarioniene and Stasiskiene (2007) presented a structured Integrated WRM model for managing water within complex production systems in industrial companies. To overcome the limitations of cost-benefit analysis, Prato and Herath (2007) used a multiple-criteria decision analysis to evaluate and rank alternative integrated catchment management. Moradi-Jalal et al. (2007) developed a mathematical model for optimal multicrop irrigation in areas associated with reservoir operation policies within a reservoir-irrigation system. Cai (2007) proposed a methodology for integrated water resource-economic modelling. Sechi and Sulis (2007) inserted quality constraint on an existing water management optimization model. Ioris et al. (2008) proposed formulating and applying a framework of catchment-level WRM indicators designed to integrate environmental, economic and social aspects of sustainability. Van Cauwenbergh et al. (2008) developed a DSS to rank different sustainable planning and management alternatives. Ayvaz (2009) studied the problem of groundwater management to develop sustainable strategies. Liu et al. (2010) addressed the IWRM using an optimization-based approach. Sedki and Ouazar (2011) explored optimal pumping schedules that meet current and future water demands while minimizing the risk of adverse environmental impacts. Gaivoronski et al. (2011) addressed the problem of water management and obtained a “robust” decision policy. Yazdi and Salehi Neyshabouri (2012) presented an algorithm for the optimal design of structural and nonstructural flood mitigation measures. For inter-basin water transfer system decision-making, Zhang et al. (2012) developed a multi-party, multi-objective decision/bargaining model based on the “satisfaction principle”. The main objectives taken into consideration by the works cited above are satisfying demand (Letcher et al. 2007; Van Cauwenbergh et al. 2008; Sedki and Ouazar 2011; Zhang et al. 2012), maximizing the net benefit of water use (Cai 2007; Khare et al. 2007; Li et al. 2007; Moradi-Jalal et al. 2007; Prato and Herath 2007), and minimizing water costs and use (Dvarioniene and Stasiskiene 2007). Liu et al. (2010) used the minimization of capital and operating costs as a single objective. Gaivoronski et al. (2011) used a coupled cost-risk objective function to minimize the risk of inappropriate management decisions. Socioeconomic and environmental aspects were taken into consideration in the works of Prato and Herath (2007), Van Cauwenbergh et al. (2008), Ayvaz (2009) and Sedki and Ouazar (2011). Minimizing investment costs of flood mitigation measures and potential damage to the flood plain were the objectives considered in Yazdi and Salehi Neyshabouri (2012). In order to resolve their formulated problems and identify solutions, authors used mathematical optimization methods, such as linear programming (Cai 2007; Khare et al. 2007; Moradi-Jalal
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et al. 2007), stochastic integer programming (Li et al. 2007), dynamic programming (Letcher et al. 2007) and mixed integer linear programming (Liu et al. 2011). Ayvaz (2009) used the Harmony search (HS) optimization algorithm for solving the problem of groundwater management. Sedki and Ouazar (2011) used a genetic algorithm (GA) to resolve the optimization problem. Yazdi and Salehi Neyshabouri (2012) used a Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to resolve the formulated multi-objective problem. In order to obtain an ideal multi-party decision, bargaining was first broken down into two stages by Zhang et al. (2012), and then decision alternatives were chosen using fuzzy pattern recognition. The sum of weighted objective functions is usually used to transform multi-objective problems into single-objective ones. This approach poses a problem in comparing importance and choosing subjective values of objective weights, depending on management priorities and on the manager’s or DM’s point of view. In order to enrich the spectrum of available tools, a multi-objective tool for the WRM optimization has been proposed. This research is a contribution to provide managers and DMs with the possibility of computing a compromise solution to opposite objectives while at the same time respecting physical and socio-economic constraints of the WRM problems. The proposed tool can be used i) for large types of problems (irrigation and/or drinking water, surface and/or groundwater, fresh, salty and/or waste water); ii) on different scales (from farm to regional levels and in one basin or inter basins); and iii) for private or public management problems. Even though the model definition cannot exhaust all of the possibilities in water schemes, it is able to cover most of them. A constrained multi-objective formulation has been proposed for WRM problems. As recommended by Zhang et al. (2012), the conflictive demand satisfaction and operation cost reduction objectives have been taken into consideration, as well as salinity and hydraulic constraints. The problem resolution was performed by Multi objective Genetic Algorithm (MOGA). Collette and Siarry (2003) classified GA as global search heuristics, frequently used to solve complex problems which are difficult to solve using conventional techniques (Goldberg 1991; Hrstka and Kucerova 2004). Similar to Darwin’s theory of evolution, GA evolves populations of individuals (solutions) using genetic operators: selection, crossover and mutation. An evolutionary strategy is generally associated with these operators in order to allow one generation to progress to the next. This iterative process is halted when evolution stagnates or when the maximum number of generation is reached (Back et al. 2000; Goldberg 1991). In GA, solutions are represented as chromosomes formed by a set of genes. Each gene represents a bit that must be filled by a decision variable value in its definition interval. Initially, genes and chromosomes are randomly created in the research area. Next, an intelligent research process will combine selected solutions according to their performance (crossover) and randomly modify some bit values (mutation) to improve population performance. MOGA presents the same basic schema as the single objective GA: variable coding, selection, crossover and mutation. However, evolution strategies and solution comparisons are different. In the problem resolution process, the Pareto optimality concept is used (Goldberg 1991). For a minimization problem of “i” objectives, solution “x” dominates solution “y” within research area “E” if and only if “fi (x)” is less than or equal to “fi (y)” for all objectives "i" and it exists at least “i” as “fi (x)” is less than “fi (y).” If a solution is not dominated by any other solution, it is called “non-dominated.” The set of non-dominated solutions of “E” form the Pareto front. In order to evaluate the performance and usefulness of the developed tool for managers and DMs, it was tested on a real case study in Tunisia of a water treatment plant (WTP) producing drinking water for Tunis, a city with about 2 million inhabitants. The WTP uses surface water from two reservoirs and an artificial channel. The daily question facing the WTP manager is how
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much water to take from each of the three sources in order to satisfy the water demand while reducing the unit cost of the water produced and supplying water with acceptable salinity.
2 General Conceptual model In the general conceptual model proposed (Fig. 1), sources are represented by rectangles with indexes “se”. Demand sites “D” are represented by circles with index “d”. Connections between the model nodes (sources and demand sites) are ensured by transfer and supply links represented in the diagram in Fig. 1 by dotted and continuous lines, respectively. It can be assumed that within any given water management area, multiple sources with different water storage capacities and salinities can be used to satisfy water requirements (quantities and salinities) of demand sites. For groundwater sources, the storage capacity corresponds to the maximum water volume the aquifer can hold. For surface water sources (reservoirs), the storage capacity is the maximum water volume that can be stored. This capacity is usually defined by the land topography and spillway level as well as by a security volume for flood control. In this research, water transfer between sources was considered possible. Each transfer link from source “j” to source “se” had a maximum transfer capacity of “FmaxS (j, se)” (dotted lines). This parameter is defined by the physical pipe characteristics and/or channels used (geometry, roughness, length), the land topography (elevations of the hydraulic system nodes) as well as the characteristics of pumping and booster stations, if there are any. At the same time, the pumping defines the unit cost of water transfer: “CUS (j, se)” between sources. In order to consider water transfer between management areas, each of the water sources “se” could be supplied by an external water source “sext” with an inflow “I (se, sext, t)” at every time interval “t”. The proposed conceptual model also took into consideration that each source was subject to water loss “L (se, t)” at every time interval “t” due to infiltration and evaporation. It was also assumed that each of the demand sites must be supplied by at least one water source through a supply link (solid lines) that could be a pipe or a natural or artificial channel. As for transfer, each of the supply links from source “se” to demand site “d” needs to be characterized by its maximum supply capacity “FmaxD (se, d)” and its unit supply cost: “CUD (se, d)”. The following connectivity diagram (Fig. 1) represents the general conceptual model proposed: To model each water source “se”, maximum and minimum storage capacities must be known, “Vmax (se)” and “Vmin (se)”, respectively. In order to compute the water balance for each water source, time variable inflow “I (se, sext, t)” and water loss “L (se, t)” are required. The salinity of the water supplied to sources “se” from an external source “sext” over time: “Qapp (se, sext, t)” and the unit cost of the transfer from “sext” to “se”: “CUI (se, sext)” are required to compute the salinity and the unit cost of the mixed water in sources “se.” Initial Fig. 1 Connectivity between water sources and demand sites within a management area
I(j,t)
I(se+1,t)
D(d) FmaxD(j, d)
FmaxD(se+1, d) FmaxD(se, d)
S(j)
L(j,t)
S(se+1)
I(se, t)
FmaxS(j, se)
S(se) L(se,t)
FmaxD(se, d+1) D(d+1)
L(se+1,t)
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source conditions must be specified in order to begin system modeling, in particular, the initial available volume “Vini (se)” and initial water salinity “Qini (se)” for each source “se.” For demand sites “d”, it is necessary to determine the required water volume “D (d, t)” and the maximum acceptable salinity “Qty (d, t)” at each time interval within the management period. The decision variables to be identified by the proposed tool, and which constitute the model outputs are: the water supplied by each water source “se” to each demand site “d” at any time interval “t”: “FD (se, d, t)” and the water transfers between sources “se” and “j”: “FS (se, j, t)”, over the management period “0-Tmax.”
3 Optimization model 3.1 WRM problem formulation In this contribution, the WRM has been formulated as a constrained multi-objective problem. The first objective is to satisfy water needs of demand sites “d” at every time interval “t” of the management period “0-Tmax.” This objective can be expressed by minimizing the following unmet demand function “fD.” ! T max NX Dmax NSE . max X X ð1Þ fD ¼ F D ðse; d; tÞ − Dðd; t Þ D d; t se¼1 t¼Δt d¼1 Where “Δt” is the simulation time interval. It depends on the dynamic of the managed system and the management objectives. It can be day, week, month, year or any time step in between; “NDmax” is the number of demand sites; “NSEmax” is the number of water sources in the management area. In this contribution, the unmet demand function has been assumed linearly penalized. The effect of the non linear formulation of “fD” as well as scaling of the objective function values on computation efficiency need to be explored to study the opportunities to improve the tool’s capabilities. With the present formulation, “fD” can fluctuate between zero and positive values depending on the number of demand sites and time intervals. In the best possible situation all demands will be met and “fD” will equal zero. The second objective proposed is to minimize the unit cost of water over the management area throughout the study period: “fC”. The sum of the costs of water supplied to demand sites, those transferred between sources and the cost of the inflows to sources was computed. The total cost was then divided by the total volume of water supplied, transferred and the inflows in order to compute the objective function as formulated in equation (2): " T max NSE Dmax max NX X X fC ¼
t¼Δt
se¼1
F D ðse; d; tÞ C UD ðse; d Þ þ
NSE max NSE max X X se¼1
d¼1 T max NSE Dmax max NX X X t¼Δt
se¼1
d¼1
F S ðse; j; t Þ C US ðse; jÞ þ
j¼1
F D ðse; d; t Þ þ
se¼1
NSE max NSE max X X se¼1
ðseÞ NSE max Nsext X X
j¼1
F S ðse; j; t Þ þ
# I ðse; sext; tÞ C UI ðse; sextÞ
sext¼1
ðseÞ NSE max Nsext X X
I ðse; sext; t Þ
se¼1
sext¼1
ð2Þ Where “Nsext (se)” is the number of inflows to source “se.” The values of “fC” are scaled between the upper and lower unit costs of water characterizing sources, inflows and transmission links in the management area. As for the first objective function, an exploration of the effect of other formulations of “fC” on computation efficiency and tool robustness has been planned for the second step of this research.
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It is important to emphasize that water treatment processes before using surface or waste water may be included in the unit cost of water. This unit cost may also incorporate pumping costs required to transfer or supply water, to abstract groundwater or to desalinize salty or sea water. When pumping makes up the greater part of the unit water cost for all sources, as is the case for groundwater abstraction and desalinization, it is possible to use unit energy consumption (KWh.m−3) instead of monetary unit cost. Doing so will circumvent the effect of spatialtemporal pricing variability and make it possible to compare both system performances over time and with different hydraulic systems within different management areas. In order to meet the requirements for water salinity at the demand sites at each time interval of the management period, the constraints expressed by equation (3) must be complied with: Qðd; tÞ ≤ Qtyðd; tÞ
∀d and ∀ t
ð3Þ
Where “Q (d, t)” is water salinity at demand site “d” at time interval “t”, due to the mixture of inflow water to demand site “d.” In order to simplify the problem, it has been assumed that water mixes completely and instantaneously in the source and demand sites. The upper and lower values for acceptable water storage volumes in sources “se” constitute hydraulic constraints, expressed by equation (4): V min ðseÞ ≤ V ðse; t Þ ≤ V max ðseÞ
∀se and ∀t
ð4Þ
Where “V (se, t)” is the water storage volume in source “se” at time interval “t”. Maximum flow capacities of transmission links from sources to demand sites and to other sources are considered hydraulic constraints, expressed by equations (5) and (6), respectively: F D ðse; d; tÞ ≤ F max Dðse; d Þ ∀se; ∀d and ∀t
ð5Þ
F S ðse; j; tÞ ≤ F max S ðse; jÞ ∀se; ∀j and ∀t
ð6Þ
At the time t =0 of the management period, sources need to be characterized by their initial water storage volumes and salinities. Equations (7) and (8) express these initial conditions: V ðse; 0Þ ¼ V ini ðseÞ ∀se
ð7Þ
Qðse; 0Þ ¼ Qini ðseÞ∀se
ð8Þ
Where “V (se, 0)” is the water volume in source “se” at time interval “0”; “Q (se, 0)” is the water salinity in source “se” at time interval “0”. To check if salinity and volume constraints are respected, mixed water salinity in sources “se” and demand sites “d” as well as water volumes in the sources must be computed. Equation (9) is used to calculate the resulting water salinity in demand site “d” at time interval “t”: NSE max X
Qðd; tÞ ¼
se¼1
F D ðse; d; tÞ Qðse; tÞ
NSE max X
∀d and ∀t
ð9Þ
F D ðse; d; tÞ
se¼1
Where “Q (se, t)” is the water salinity in source “se” at time interval “t.” The value of this parameter is the result of the mixture of initial water in source “se” with water coming from
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other sources within the management area and inflow water from outside of it. The updated salinity in source “se” at time interval “t” is computed by equation (10): V ðse; t−ΔtÞ Qðse; t−ΔtÞ þ
NSE max X
Nsext ðseÞ X
F S ð j; se; t−ΔtÞ Qð j; t−ΔtÞþ
j¼1
Qðse; tÞ ¼
I ðse; sext; t−ΔtÞ Qðsext; tÞ
sext¼1
V ðse; t−ΔtÞ þ
NSE max X
Nsext ðseÞ X
F s ð j; se; t−Δt Þþ
j¼1
I ðse; sext; t−ΔtÞ
sext¼1
ð10Þ Where “Q (sext, t)” is the water salinity of inflow “sext” at time interval “t.” Equation (11) details how the water balance in sources “se” is computed: V ðse; t Þ ¼ V ðse; t−Δt Þ þ
Nsext ðseÞ X
I ðse; sext; t−ΔtÞ þ
NSE max X
sext¼1
−
NSE max X j¼1
F S ðse; j; t−ΔtÞ−Lðse; t−Δt Þ−
F S ð j; se; t−ΔtÞ
ð11Þ
j¼1 NX Dmax
F D ðse; d; t−Δt Þ
d¼1
3.2 Resolution methodology Synthesizing numerous comparative studies on MOGA (Lis and Eiben 1997; Zitzler and Thiele 1998; Knowles and Corne 1999; Esquivel et al. 1999; Deb et al. 2000 and Leiva et al. 2000) made it possible to develop a Multi-Sexual Genetic Algorithm (MSGA) to resolve the WRM problem formulated in this research. This algorithm is elitist and characterized by weak parameter numbers and good distribution of optimal solutions on the PARETO front. The Pareto optimality concept is used at each iteration of the MOGA to compare solutions, according to their objective functions, and to select the non-dominated ones. The main computational steps of the MSGA developed are described below: 1. Randomly create an initial population with “Tpop” solutions. The MSGA randomly assigns a sex to each solution. The sex number is equal to the number of objectives, two in this case. For this problem, real coded solutions were used. 2. Evaluate solutions “s” by computing the objective functions “fD (s)” and “fC (s)” and verifying the constraints. Each solution is evaluated by two “fitness” functions integrating one objective function and its associated constraints. For this tool, “fitness 1” represents “fD (s)” and violations of acceptable volumes in sources and water salinity requirements in demand sites as presented in equation (12). fitness1ðsÞ ¼ f D ðsÞ 1 þ Viol max V max ðseÞ þ Viol max V min ðseÞ þ Viol max Qmax ðd Þ Se ¼ 1; …; NSEmax :and d ¼ 1; …; N Dmax ð12Þ Where:
&
“ViolmaxVmax (se)” is the maximum violation of the maximum acceptable volume of “se” expressed by equation (13): V ðse; t Þ−V max ðseÞ t ¼ Δt; …; T max ð13Þ Viol max V max ðseÞ ¼ Max 0; V max ðseÞ
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&
“ViolmaxVmin (se)” is the maximum violation of the minimum acceptable volume of “se” expressed by equation (14): V min ðseÞ−V ðse; tÞ ð14Þ Viol max V min ðseÞ ¼ Max 0; t ¼ Δt; …; T max V min ðseÞ
&
“ViolmaxQmax (d, t)” is the maximum violation of the acceptable water salinity in “d” expressed by equation (15): Qðd; t Þ−Qmax ðd; tÞ Violmax Qmax ðd Þ ¼ Max 0; t ¼ Δt; …; T max ð15Þ Qmax ðd; tÞ The violation functions used in equations (13), (14) and (15) give results between 0, if there is no violation, and 1, when the violation is the maximum. The linear penalization used of the unmet demand leads to a distinction between solutions with small differences in the unmet demand function as well as in one of the penalty terms. Another option to highlight the effects of constraint violations is to multiply the non-zero penalty terms. A sensitivity analysis makes it possible to compare different formulation options and choose the best one in such a problem. This is a research perspective that could be addressed to improve the computation efficiency of the developed tool. The objective function “fC (s)” is represented by “fitness 2” in the MSGA, expressed by equation (16): fitness2 ¼ f C ðsÞ
ð16Þ
3. Identify non-dominated solutions by the PARETO optimality concept. 4. Create a set of identified non-dominated solutions; make an archive of size “Tarch”. 5. Evolution: A new population is created using the solutions of the previous population.
6. 7. 8. 9. 10. 11.
5.1. Elitism: “Pe” percent of the “Tpop” solutions of the new population are copied from the archive of non-dominated solutions. 5.2. Selection: Two solutions are selected from the previous population according to their fitness functions to participate in the crossover. 5.3. Crossover: The selected solutions participate in an arithmetic crossover to produce two new solutions with a probability “Pc”. 5.4. Mutation: Some “bits” of each created solution can be randomly changed in the research area with a probability “Pm”. Evaluate created solutions as in step (2). Identify non-dominated solutions in the current population as in step (3). Insert non-dominated solutions of the current population into the archive. Update the archive using the PARETO optimality concept. If “Tarch” is not reached then return to step 5. If iteration number is less than the maximum “Gmax”, then repeat steps 5 to 11.
The exposed resolution methodology was coded using the Visual Basic 6 programming language. A multi-objective optimization tool was built with a user-friendly interface. As all
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kinds of water resources (conventional and non-conventional, fresh and salty) can be modeled, the tool developed was called “ALL_WATER.” 3.3 Case study The developed tool was used to aid in management decisions in a real drinking water system in Tunisia. The case study is made up of three water sources (S1, S2 and S3) and a single demand site (D1). S1 is an artificial channel with permanent flow, supplied by pumping from the main Tunisian river (Medjerda). S2 and S3 are reservoirs “Mornaguia” and “Ghedir El Golla,” respectively. These sources are located near the WTP and are used as a back-up in case of a water shortage from S1, maintenance, or when water from S1 cannot be used because of poor quality due to floods and uncommonly high levels of pollution. They are also used to regulate water salinity in case of high levels of salinity in S1. Sources S2 and S3 can be supplied by both S1 and runoff from their respective catchments. Demand site D1 is a WTP which supplies drinking water to the city of Tunis. Surface water coming from S1, S2 and S3 is treated in D1 using pre-chlorination, decantation and filtration processes. Clean water is then chlorinated and supplied by gravity to about 50 tanks connected to the drinking water system. In Fig. 2, (a) presents a satellite image of the north of Tunisia with the location of the WTP, hereafter called “GEG”. Image (b) shows the locations of the water sources and the demand site D1. From S1 (blue line in the upper right hand corner), two pumped pipes (red lines) are used to supply reservoirs S2 and S3. Source S3 is able to supply D1 by gravity through pressurized pipe (blue line) while S1 and S2 need pumping to supply D1 (blue lines). According to the supply history and weather forecasts, managers of the drinking water supply systems forecasted their daily demand and asked the WTP manager to provide the required volume. Therefore, the daily management of the WTP had to be adjusted in order to produce the necessary water volume while minimizing pumping costs and ensuring acceptable salinity. Figure 3 presents the conceptual model of the real case study. S1 is considered an unlimited water source because of its upstream storage capacity. For this reason, a large hypothetic volume was adopted (Table 1). S1 is also characterized by a high concentration of salt (2.0 g L−1). Reservoirs S2 and S3 have a total storage capacity of 20 million m3. Their salt concentrations are acceptable for drinking (1.0 g L−1). Water losses from S2 and S3 are mainly caused by infiltration and evaporation. By using current and forecasted climatic conditions along with the results of experimental evaporation measurements, a manager is able to estimate the amount of evaporation from the surface area of the body of water. S2 has a water loss of between 10,000 and 13,500 m3 Day−1. Water loss from S3 fluctuates between 2,100 and 3,500 m3 Day−1. The maximum weekly water loss over the studied period is estimated at 0.63 and 0.49 % of the storage capacities of S2 and S3, respectively. Maximum transfer and supply capacities and unit costs of water transmission links between sources and the demand site due to energy consumed by pumping are summarized in Table 2. Within the management period, daily consumption of D1 fluctuates between 356,000 and 402,000 m3 Day−1. Average demand over the study period is estimated at 374,570 m3 Day−1 (4.335 m3 s−1). Fig. 4 shows the daily pattern of D1water demand. The maximum acceptable water salinity in “D1”: “Qty (D1)” is constant at 1.50 g L−1. Given the characteristics of the current water management problem, it could be stated that the best source in terms of storage capacity is S1. However, its use is limited due to the low supply capacity of the transmission link to D1. Use of S1 is also at a disadvantage because its salinity level (2.0 g L−1) is greater than the acceptable value. S2 and S3, which have lower
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a
b
Fig. 2 Google satellite images of the North of Tunisia (a) and the study area (b)
water salinities (1.0 g L−1), have to be used in order to improve the quality of the drinking water produced. Transmission links from S1 have the highest unit cost (20 10−3 USD m−3). Supply of D1 from S3 is done by gravity. Therefore, this supply link needs to be used first if the objective of reducing cost is a priority. However, this supply link also has a lower maximum supply capacity and therefore cannot satisfy the entire water demand alone. The data presented show that S2 should be used secondly because of its good quality and low cost, as compared to S1. Thus, using S1 to supply D1 will only be necessary if the water volumes of S2 and S3 and/or their supply capacities are insufficient. It is important to emphasize that the average daily pumping cost during the study period is estimated at 0, 4994 and 7491 USD when using only S3, S2 or S1, respectively.
Author's personal copy Fig. 3 Conceptual model of the real case study
FS(1, 2, t)
FD(2, 1, t)
S2
L2(t)
FD(1, 1, t)
S1 FS(1, 3, t)
D1
D(1, t)
FD(3, 1, t)
S3
L3(t)
This WRM problem consists of defining water transfer from S1 to S2 and S3 and the contribution of S1, S2 and S3 to satisfy the demand of D1, in order to reduce the average unit cost of water and be consistent with acceptable volumes and salinity. While the case study appears to be simple, 35 decision variables (5 flows x 7 days) must be defined in order to optimize its management for 1 week.
4 Results and discussions “ALL_WATER” software was used to optimize water management of the case study presented. The input data about demand sites, sources and transmission links were saved in separate text files read by the developed tool. In order to choose the MOGA and optimization parameters, sensitivity analysis was done. Initial tests converged to a population size of Tpop =50, a maximum number of iterations Gmax =1500, a crossover probability of pc =0.6, a mutation probability of pm =0.1 and a percentage of elitism of Pe =40 %. After running for 6 minutes, the tool evaluated 75,000 possible solutions (50 × 1500) and identified the 72 optimal solutions presented in the PARETO front in Fig. 5. The graph in Fig. 5 expresses the relationship between satisfying water demand and the associated penalties: fitness 1 and the unit cost of water: fitness 2. The manager may choose the optimal solution from the PARETO front, according to his priorities. This relationship demonstrates that, in order to guarantee demand will be met and all constraints taken into consideration, the optimal unit cost of water is the highest, estimated at 10.9 10−3 USD m−3: solution A. From the same relationship, the manager can apply solution B, with the minimum unit cost of water (9.59 10−3 USD m−3). This solution does not ensure that the entire water demand will be met. Another possibility is to apply an intermediate solution as a compromise between the two extremes, A and B. For example, solution C ensures a unit cost of water of 9.89 10−3 USD m−3 and acceptable water satisfaction and constraint violations. For the 7-day management period, implementing optimal solutions “A”, “B” and “C” will require a total cost of 28,538; 25,108 and 25,894 USD, respectively.
Table 1 Characteristic volumes of the water sources (106 m3) and initial salinities (g L−1)
Water source
Vmax
Vmin
Vini
Qini
S1 S2
1000 15
1 5
1000 15
2.0 1.0
S3
5
1
5
1.0
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Table 2 Water transfer and supply link characteristics
Transmission links
Maximum supply (m3 Day−1)
Unit cost of pumping (10−3 USD m−3)
S1 – S2 S1 – S3
300,000 300,000
20 20
S1 – D1
300,000
20
S2 – D1
336,960
13
S3 – D1
172,800
0
The PARETO graph in Fig. 5 emphasizes the asymptotic tendency of the relationship between the two objective functions optimized for high values of “fitness 1” and “fitness 2.” Results demonstrate that the water unit cost will not increase significantly (3.1 %) if the manager chooses solution “C” instead of “B” and the demand satisfaction will be highly improved (0.25). However, choosing solution “A” instead of “C” will generate an increase of 13.7 % of the water unit cost and a weak improvement in demand satisfaction (0.04). Thus, this developed tool can be a support system for managers in making optimal decisions. This compensates for the difficulty of redoing the same computation for every change in management priorities, as with single objective approaches. Table 3 presents the relative differences “RD =100 * (VS – VD)/VD” (%) between volumes that are demanded (VD) and supplied (VS) for the optimal solutions A, B and C presented above. Table 3 demonstrates that with solution A, the water supplied to D1 is less than the demand volume for 3 days. The minimum value of RD is estimated at −1.46 %. Then for 4 days, the water supplied to D1 is greater than the demand volume with a maximum RD of 1.47 %. It is important to emphasize that acceptable flow meter accuracy in hydraulic systems is usually around ±2 %. Therefore, solution A could be considered acceptable in terms of satisfying demand. Solution B provides a lower unit cost of water but does not guarantee that water demand will be met, especially on days 1 and 3, where the RDs are −30.73 and −28.77 %, respectively. In addition, the demand may not be met on days 2, 5 and 7. Only on days 4 and 6 will satisfying water demand be considered guaranteed with solution B. As for the intermediate solution (C), the minimum value of RD on day 3 is observed at −5.23 %. In terms of quality, the three optimal solutions yielded water salinities less than the maximum acceptable concentration (1.50 g L−1). Table 4 clearly illustrates the developed tool’s performance in identifying solutions that respect quality constraints.
Fig. 4 Daily pattern of D1 water demand Water demand (m3/Day)
410000 400000 390000 380000 370000 360000 350000 340000 330000 1
2
3
4 Day
5
6
7
Author's personal copy Fig. 5 PARETO front identified by “ALL_WATER” in the real case study
A
C B
In terms of source water volumes, solutions A, B and C all allow acceptable volumes in S2 and S3, as shown in Table 5. In analyzing source contribution to satisfying demand, small percentages are observed for source S1 for the three optimal solutions A, B and C. Figure 6 presents the percentage of contribution of S1, S2 and S3 in satisfying demand for solution A. For solution A, Fig. 6 demonstrates that S2 and S3 are used to satisfy the majority of the water demand, especially on days 5 and 6, when water demand is the lowest. For this solution, S1 appears to be at a disadvantage due to its salinity and water unit cost. Maximum S1 contribution is estimated at 29 % on the third day. The intermediate solution (C) reflects a comparative contribution of sources as in A. Figure 7 shows that, even given a compromise between the two objectives, S2 and S3 can satisfy the greater part of the water demand. S1 is used to satisfy less than 26 % of the water demand. For solution B, since more importance is accorded to the objective of reducing unit water cost, Fig. 8 shows that S2 and S3 can satisfy more than 80 % of the total demand of D1 on some days. On day 3, their combined contribution is near 95 %. It should also be pointed out here that, when satisfying demand is not the priority, the optimization tool appears to disregard S1 due to its high water unit cost. Figure 8 also demonstrates that S3 is the main source used to satisfy water demand. Its contribution is always greater than 43 % and reaches 62 % on days 1 and 3. As predicted in the case study, the developed tool was able to compute an optimal solution principally using sources S2 and S3 to satisfy the water demand in D1 and reduce the unit cost
Table 3 Relative difference for optimal solutions A, B and C
Day
RD (A)
RD (B)
RD (C)
1
+0.93
−30.73
+0.59
2
+0.64
−6.97
−0.08
3
−1.40
−28.77
−5.23
4
−1.39
−0.75
−2.09
5
−1.46
−9.14
−3.75
6
+1.47
+1.45
+1.48
7
+0.09
−2.89
+5.91
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Table 4 Water salinity supplied to D1 for optimal solutions A, B and C
Solution/Day
1
2
3
4
5
6
7
A
1.15
1.12
1.30
1.24
1.16
1.13
1.20
B C
1.21 1.15
1.12 1.11
1.07 1.27
1.20 1.20
1.14 1.14
1.13 1.13
1.17 1.16
of water. The optimal solutions identified also complied with the constraints taken into consideration. The PARETO front designed by the optimal solutions provides an interesting relationship between the objectives taken into consideration. It can help managers make the right decisions given different situations of the hydraulic system and different priorities. This opportunity cannot be offered by single-objective approaches. Given the detailed performances of the optimal solutions “A”, “B” and “C”, the manager of the real system studied would be encouraged to apply solution “C” in order to produce the required drinking water for the city of Tunis. This solution ensures acceptable water demand satisfaction and requires a low total cost. In addition, it respects the hydraulic system and quality constraints. Although ALL_WATER efficiently and effectively computed the real case study, some future enhancements could be considered. The tool could be enriched by adding additional objectives. An environmental objective to minimize drawdown in aquifers would allow a wider spectrum of applications. The optimal solutions would thus be displayed as an optimal surface in three dimensions. However, this would make it more complicated for managers to choose which optimal solutions to apply and would be a disadvantage to using the tool. One solution to this problem would be to use the weighted sum approach in which only two “fitness” functions are maintained that integrate all objectives and constraints. This issue should be addressed in a future research project. Nonlinear formulation of the objective functions and penalties could be explored to improve the computational efficiency of ALL_WATER. Using local search techniques to refine optimal solutions on the PARETO front could improve solution performance. Given the current structure of the tool developed, different design options for the hydraulic system can be assessed in terms of satisfying demand and reducing costs and then the best option can be chosen. The current version of the tool is not capable of optimizing the system design for WRM.
Table 5 Patterns of water volume (Mm3) in sources S2 and S3 for solutions A, B and C
Day
Solution A
Solution B
Solution C
V (S2)
V (S3)
V (S2)
V (S3)
V (S2)
V (S3)
1
14.852
4.866
14.965
4.856
14.844
4.862
2
14.672
4.772
14.844
4.730
14.683
4.748
3
14.591
4.642
14.775
4.582
14.605
4.600
4
14.525
4.522
14.701
4.435
14.530
4.455
5 6
14.397 14.283
4.417 4.304
14.600 14.470
4.302 4.166
14.408 14.256
4.316 4.180
7
14.155
4.183
14.341
4.030
14.128
3.989
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Percentage in demand satisfaction
100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 1
2
3
4 Day S1
S2
5
6
7
S3
Pourcentage in demand satisfaction
Fig. 6 Contribution percentage of sources S1, S2 and S3 for satisfying demand of D1 using solution A
100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 1
2
3
S1
4 Day
5
S2
6
7
S3
Fig. 7 Contribution percentage of sources S1, S2 and S3 for satisfying demand of D1 using solution C
Pourcentage in demand satisfaction
100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 1
2
3
4
5
6
7
Day S1
S2
S3
Fig. 8 Contribution percentage of sources S1, S2 and S3 for satisfying demand of D1 using solution B
Author's personal copy I. Nouiri
5 Conclusions This paper presented the development and testing of a Multi-objective tool for WRM optimization. The proposed multi-objective formulation of the problem integrates demand satisfaction, reduction of the unit cost of water and compliance with salinity and hydraulic constraints. A MOGA and the PARETO optimality concept were used to resolve the formulated problem and to compute its decision variables. Testing “ALL_WATER” to optimize water management of a real case study demonstrated the efficiency of the developed methodology. A set of optimal solutions, forming a PARETO front, were identified after a single run and short computation time. The manager must simply choose among the optimal solutions identified according to his management priorities without any additional computation necessary. The developed tool is efficient and flexible for WRM optimization for large hydraulic systems and all types of water resources. In addition, any water demand category can be included in terms of volume and salinity. Water management can also be optimized for single and multiple time periods. Acknowledgments The authors would like to thank the National Institute of Agronomy of Tunisia (INAT) and the Tunisian National Drinking Water Utility for their support of this work.
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