Environ Monit Assess (2015) 187: 158 DOI 10.1007/s10661-015-4263-6
River water quality management considering agricultural return flows: application of a nonlinear two-stage stochastic fuzzy programming Ali Tavakoli & Mohammad Reza Nikoo & Reza Kerachian & Maryam Soltani
Received: 31 January 2014 / Accepted: 2 January 2015 / Published online: 5 March 2015 # Springer International Publishing Switzerland 2015
Abstract In this paper, a new fuzzy methodology is developed to optimize water and waste load allocation (WWLA) in rivers under uncertainty. An interactive twostage stochastic fuzzy programming (ITSFP) method is utilized to handle parameter uncertainties, which are expressed as fuzzy boundary intervals. An iterative linear programming (ILP) is also used for solving the nonlinear optimization model. To accurately consider the impacts of the water and waste load allocation strategies on the river water quality, a calibrated QUAL2Kw model is linked with the WWLA optimization model. The soil, water, atmosphere, and plant (SWAP) simulation model is utilized to determine the quantity and quality of each agricultural return flow. To control pollution loads of agricultural networks, it is assumed that a part of each agricultural return flow can A. Tavakoli : M. Soltani School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran A. Tavakoli e-mail:
[email protected] M. Soltani e-mail:
[email protected] M. R. Nikoo School of Engineering, Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran e-mail:
[email protected] R. Kerachian (*) School of Civil Engineering and Center of Excellence for Engineering and Management of Civil Infrastructures, College of Engineering, University of Tehran, Tehran, Iran e-mail:
[email protected]
be diverted to an evaporation pond and also another part of it can be stored in a detention pond. In detention ponds, contaminated water is exposed to solar radiation for disinfecting pathogens. Results of applying the proposed methodology to the Dez River system in the southwestern region of Iran illustrate its effectiveness and applicability for water and waste load allocation in rivers. In the planning phase, this methodology can be used for estimating the capacities of return flow diversion system and evaporation and detention ponds. Keywords QUAL2Kw . SWAP . Uncertainty analysis . Water and waste load allocation . Water quality
Introduction The integration of water and waste load allocation in rivers has been receiving more attention in recent years. Dai and Labadie (2001) introduced an extended version of the MODSIM simulation model (Fredericks and Labadie 1995) called MODSIMQ for water and waste load allocation in river basins. The MODSIMQ incorporates a generalized model for estimating salinity loads in irrigation return flows and predicting mass transport mechanisms in groundwater. They used an iterative procedure based on the Frank-Wolfe nonlinear programming to link the water quantity and quality simulation models. Zhang et al. (2010) proposed a deterministic water quantity–quality simulation model, which considers water production, water supply, pollution loads, and water demands together in a water allocation
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process. According to their approach, a river basin is divided into river reaches and a series of tanks out of river to analyze a water allocation optimization problem considering water quality issues. They applied their model to Jiaojiang River basin in China. Nikoo et al. (2012) developed a methodology for simultaneous water and waste load allocation in rivers using a nonlinear interval parameter optimization model. Their methodology considers the main uncertain parameters and inputs associated with water and waste load allocation problems as interval numbers. Nikoo et al. (2013a) developed a model based on fuzzy transformation method (FTM) for optimal allocation of water and waste load in rivers. The FTM was used to incorporate the existing uncertainties in model inputs. They utilized FTM, as a simulation model, in an optimization framework to construct a fuzzy water and waste load allocation model. In the models developed by Nikoo et al. (2012, 2013a), they assumed that the return flow quality and the ratio of return flow to allocated water are constant for each water user, but in the current paper, they are estimated using the soil, water, atmosphere, and plant (SWAP) model. Recently, linear interval programming has been utilized by Nikoo et al. (2013b, c, d) for water and waste load allocation in river–reservoir systems. In their models, it is assumed that in each month, the qualities of return flows are constant. Tavakoli et al. (2014) developed a water and waste load allocation model for river systems by using factorial interval optimization. In their work, only one water quality variable (i.e., total dissolved solids (TDS)) was considered, and effects of uncertain parameters and their interactions on the optimization model outputs were evaluated by using fractional factorial analysis. They showed that the fractional factorial analysis can be used for reducing the computational time of the optimization model. However, their methodology cannot be applied to largescale problems due to its significant computational cost. In most of water and waste load allocation problems, the available data about the uncertain variables are very limited to be presented as probabilistic distribution, so in this paper, the fuzzy set theory and interval programming are utilized to incorporate the main existing uncertainties in model inputs and parameters. Wang and Huang (2011) developed an interactive two-stage stochastic fuzzy programming (ITSFP) approach for water resources management. They applied their model to a water resource allocation problem without considering the water quality issues. In the current paper, an
Environ Monit Assess (2015) 187: 158
integrated model is proposed for water and agricultural waste load allocation in rivers considering two important water quality indicators (i.e., TDS and total coliform bacteria (TC)). The ITSFP is used for incorporating the main existing uncertainties in the model inputs and parameters. By using QUAL2Kw and SWAP simulation models, the proposed methodology is also able to accurately consider the quantity and quality of river flow and agricultural return flows. In order to control the pollution loads of agricultural dischargers, we here assume that a part of their return flow can be diverted to some evaporation ponds. In order to use solar radiation in wastewater disinfection, we assume that a part of agricultural return flow can be stored in some shallow detention ponds, exposed to solar radiation, and finally, the return flow with a very low concentration of TC is discharged to river. To examine the applicability and efficiency of the proposed methodology, it is applied to the Dez River system in the southwestern part of Iran.
Model framework A flowchart of the proposed methodology is presented in Fig. 1. As illustrated in this figure, the methodology consists of three main steps. At first, the data needed for the simulation and optimization of the system are collected, including the quantity and quality of the agricultural return flows and headwater, monthly water demands, soil profile data in agricultural lands, initial groundwater level, basic weather data, pollution loads reduction costs, crop yield response factors, and area of agricultural lands. Then, the main uncertain inputs of the simulation and optimization models are selected, and their fuzzy boundary intervals are determined. In the second step, in order to consider the impact of water allocation on quantity and quality of agricultural return flows and river flow, the SWAP and QUAL2Kw models are developed. In the third step, to find an initial solution for the main optimization model, an optimization model is developed in which the decision variables represent the water allocation to users without considering the water quality constraints. The optimization model provides short-term policies for water allocation to water users and return flow diversion to both evaporation and detention ponds. In the optimization models developed in this paper, the constraints related to the quality of water at
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Fig. 1 A flowchart of the interactive two-stage stochastic fuzzy optimization model for water and waste load allocation in rivers
checkpoints and the functions estimated by SWAP models are nonlinear. Therefore, in the third step, an iterative linear programming (ILP) method is utilized for solving the nonlinear optimization model. An initial solution is calculated for the ILP using an initial optimization model which only considers the water supply objective function. The ILP uses this initial solution and improves it through several iterations considering both the water quantity and quality constraints. Optimization model stops when the difference between two successive objective function values is less than a predefined tolerance or the number of iterations exceeds a predefined value (Nikoo et al. 2013b, d and Tavakoli et al. 2014). Details of the water and waste load allocation optimization model will be discussed in “The SWAP simulation
model,” “The QUAL2Kw simulation model,” and “Water and waste load allocation optimization model” sections. In the third step, the ITSFP approach is used to incorporate the uncertainties of model inputs and parameters. More detail about this approach will be presented in “Interactive two-stage stochastic fuzzy programming” section. In the following sections, the main components of the methodology are discussed in detail: The SWAP simulation model SWAP simulates transport of water, solutes, and heat in the vadose zone. This program has been developed by Alterra and Wageningen University in Netherlands (2003) and can be used to simulate water and mass
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transport processes at field scale during growing seasons (Van Dam and Feddes 2000). The main model inputs may consist of meteorological data, crop growth, and drainage characteristics. SWAP simulates soil water movement using Richards’ equation and solute transport using advection-dispersion equations: ∂h ∂ ∂h C w ð hÞ ¼ k ðhÞ þ 1 −S ðzÞ ∂t ∂t ∂z
ð1Þ
where Cw is differential water capacity (cm−1), h is soil water pressure head (cm), t is time (day), k is hydraulic conductivity (cm day−1), S is root water extraction rate (day−1), and z is soil depth (cm). SWAP solves the Richards’s equation numerically with an implicit, backward, finite difference scheme. The SWAP model can evaluate the irrigation management schedules with different climatic conditions. Irrigation with fixed date, depth, and quality can be specified as input. In this study, the SWAP model is used for estimating quantity and quality of agricultural return flows. The calibrated SWAP model is used for determining the best metamodel to be incorporated in the optimization model. In order to develop the meta-models, the SWAP model is run many times considering different possible values for allocated water to agricultural water users. The QUAL2Kw simulation model QUAL2Kw is a river and stream water quality simulation model. This one-dimensional model simulates the transport and fate of several constituents (Pelletier et al. 2006). The QUAL2Kw has a general mass balance equation for a constituent concentration ci in the water column of a reach i (the transport and loading terms are omitted from the mass balance equation for bottom algae modeling) as (Kerachian and Karamouz 2005):
reactions and mass transfer mechanisms (mg/L/day), Ei and Ei − 1 are bulk dispersion coefficients between reaches i − 1 and i and i and i + 1, respectively (L/day), ci is concentration of water quality constitute in reach i (mg/L), and t is time (day). A detailed description of the QUAL2Kw and its mathematical basis is provided elsewhere (Pelletier and Chapra 2006). In this paper, the QUAL2Kw is utilized for simulating the river water quality based on different water and waste load allocation scenarios developed by the optimization model. In order to reduce the run-time of proposed optimization model, the QUAL2Kw model is replaced with some meta-models. The meta-model of each water quality indicator represents the relationships between concentrations of water quality indicator at checkpoints based on values of water allocation, return flows, and concentration of water quality indicator in upstream river flow. Interactive two-stage stochastic fuzzy programming In real-world problems, when the subjective judgments of decision makers are influential in decision-making outcomes, the fuzzy set theory can be utilized as an effective tool for incorporating subjective information. Firstly, consider a simple linear programming model with fuzzy parameters as follows: Min ef ¼
n X
e jX j A
ð3Þ
j¼1
Subject to: n X
e i ; i ¼ 1; 2; …; m; ei j X j ≤ C B
ð4Þ
j¼1
Qab;i dci Qi−1 Q E i−1 ¼ ci−1 − i ci − ci þ ðci−1 −ci Þ dt Vi Vi Vi Vi þ
Ei Wi ðciþ1 −ci Þ þ þ Si Vi Vi
X j ≤ 0; j ¼ 1; 2; … n ð2Þ
where Qi is flow at reach i (L/day), Qab,i is abstraction flow at reach i (L/day), Vi is volume of reach i (L), Wi is the external loading of the constituent to reach i (mg/ day), Si is sources and sinks of the constituent due to
ð5Þ
e i ∈fRgm1 , and {R} ei j ∈fRgmn , C where Ãj ∈{R}1×n, B indicate a set of fuzzy parameters involved in the objective function and constraints; Xj ∈{R}1×n and {R} also denote a set of crisp decision variables. A fuzzy set à in X is characterized by its membership function (μÃ), where X represents a space of objects x and μÃ(x) represents the membership degree of x in Ã
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(Zadeh 1965). A fuzzy set à with membership function μà can be presented as follows (Jiménez et al. 2007):
(3), (4), and (5) if it is an optimal solution to the following α-parametric programming problem:
8 0 > > > > < f à ð xÞ μà ðxÞ ¼ 1 > > > > g à ð xÞ : 0
Min ef ¼
∀x∈ð−∞; a1 ; ∀x∈½a1 ; a2 ; ∀x∈½a2 ; a3 ; ∀x∈½a3 ; a4 ; ∀x∈½a4 ; ∞Þ;
n X
ej X j EV A
ð10Þ
j¼1
ð6Þ
Subject to: n X
"
# eBi j eBi j e e ð1−αÞE 1 þ αE2 X j ≤αEC1 i þ ð1−αÞEC2 i ;
ð11Þ
j¼1
where functions fÃ(x) and gÃ(x) represent continuous and monotonically increasing and decreasing functions, respectively. If fà and gà are linear functions, the membership function is trapezoidal and can be denoted by Ã=(a1,a2,a3,a4). If a2 =a3, we obtain a triangular fuzzy number. The α-level set of a fuzzy set à can be defined as Ãα ={x∈R|μà ≥α}, where α∈[0,1]. Since μà is upper semi-continuous, the α-level set of à forms a closed and bound interval Ãα =[f−à 1(α),g−à 1(α)], where f−à 1(α)= inf{x:μÃ(x)≥α} and g−à 1(α)=sup{x:μÃ(x)≥α} (Heilpern 1992). The expected interval of a fuzzy set Ã, denoted by EI(Ã), can be defined as follows (Jiménez et al. 2007): i h EI à ¼ EÃ1 ; E Ã2 Z ¼
1
f Ö1 ðαÞd α ;
0
Z
1
g Ö1 ðαÞd α
:
i ¼ 1; 2; …; m; α∈½0; 1
X j ≤ 0; j ¼ 1; 2; … n
where EV(Ã) represents the expected value of the fuzzy h i e e eB eB C C number Ã; E1 ; E2 and E1 ; E 2 are the expected e respectively, and α is the feasibility e and C, intervals of B degree of a decision vector. As suggested by Wang and Huang (2011), to consider uncertainties expressed as fuzzy boundary intervals, the interactive fuzzy resolution method (Jiménez et al. 2007) can be integrated into the inexact two-stage stochastic programming framework (Huang and Loucks 2000) that leads to an interactive two-stage stochastic fuzzy programming as follows:
ð7Þ
0
ð12Þ
Min ef ¼
n X
e X EV A j j
ð13Þ
j¼1
The expected value of a fuzzy set Ã, denoted by EV (Ã), is the half points its expected interval (Heilpern 1992): EÃ þ E Ã2 EV Ã ¼ 1 : 2
ð8Þ
According to Eqs. (7) and (8), if a fuzzy set à is trapezoidal or triangular, its expected interval and its expected value are calculated as follows (Wang and Huang, 2011): 1 1 EI à ¼ ða1 þ a2 Þ; ða3 þ a4 Þ ; EV à 2 2 1 ¼ ð a1 þ a2 þ a3 þ a4 Þ : ð9Þ 4 Based on Jiménez et al. (2007), vector X0(α)∈Rn is an α-acceptable solution of fuzzy optimization models
Subject to: " # n X eBi j eBi j e e C C ð1−αÞE 1 þ αE 2 X j ≤αE 1 i þ ð1−αÞE 2 i ; j¼1
i ¼ 1; 2; …; m; α∈½0; 1
X j ≤ 0; j ¼ 1; 2; … n
ð14Þ
ð15Þ
where ± represent the lower and upper levels of each e e , and C parameters, X±j is an interval variable, and ñj , B j j are fuzzy boundary intervals. As shown in Fig. 2, the lower and upper bounds of an interval ã± (i.e., ã− and ã+) can be presented as fuzzy sets (Wang and Huang 2011). The optimization models (13), (14), and (15) can be divided into two deterministic submodels which provide the lower and upper bounds of the objective function values. This transformation process is based on an
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h
μ
i X −jopt ; X þjopt ; j ¼ 1; 2; …n
ð22Þ
1
h 0
a~ +
a~ −
interactive algorithm (Wang and Huang 2011). Since the objective is minimizing the total cost, the submodel corresponding to the lower bound of the objective function (f−) can be formulated as follows: −
n X
− e X− EV A j j
ð16Þ
M¼
Subject to: X j¼1
"
−
#
−
−
ð18Þ
where X−j are decision variables. The submodel corresponding to the upper bound of the objective function (f+) can also be formulated as follows: þ
Min ef ¼
n X
þ
e EV A j
X þj
ð19Þ
j¼1
Subject to:
j¼1
ð24Þ
ð17Þ
e C þ ð1−αÞE 2 i ; i ¼ 1; 2; …; m; α∈½0; 1
X −j ≤ 0; j ¼ 1; 2; … n
n X
1−α0
αk ¼ α0 þ 0:1k k ¼ 0; 1; …; ⊂½0; 1 0:1
−
eB eB e C ð1−αÞE1 i j þ αE2 i j X −j ≤ αE1 i
ð23Þ
The ITSFP approach allows decision makers to consider two factors: feasibility of the constraints and satisfaction degree of the objective function. The decision makers have to identify a balanced solution between these two conflicting objectives. The best way to reflect decision makers’ preferences is to express them through natural language, establishing a semantic correspondence for the different degree of feasibility (Zadeh 1975). The discrete values of α can be calculated as follows (Jiménez et al. 2007):
j¼1
n
i
a~ ±
Fig. 2 An interval with fuzzy boundaries (Wang and Huang 2011)
Min ef ¼
f −opt ; f þ opt
"
þ þ# þ eBi j eBi j þ e Ci ð1−αÞE1 þ αE 2 X j ≤ αE 1
ð20Þ
þ
e C þ ð1−αÞE 2 i ; i ¼ 1; 2; …; m; α∈½0; 1
X þj ≤ 0; j ¼ 1; 2; … n
ð21Þ
where X+j are decision variables. The solution of the linear programming (models (3), (4), and (5)) is denoted as follows:
where α0 is the minimum acceptable feasibility degree of decision makers. In the first step of the ITSFP method, the space O1 ={x0(αk),αk ∈M} of the αk-acceptable solution and the corresponding possibility distribution of the e ðαk Þ can be obtained through solving objective values Z models (13), (14), and (15) under each αk. In order to get a decision vector that complies with the expectation of the decision makers, two conflicting factors should be evaluated: the feasibility degree of the constraints and acceptability of the objective function value. After obtaining the results of water and waste load allocation under different e ðαk Þ, the decision makers are asked to provide a goal G Z and its tolerance threshold G. The goal is expressed using e with the following membership function a fuzzy set G (Jiménez et al. 2007): 8 h≥G > :0 h≤G
ð25Þ
In the second step, the satisfaction degree of the fuzzy e is calculated using the following equation progoal G posed by Yager (1979):
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Z
0
e ðαÞ K G~ Z
Water and waste load allocation optimization model
ðhÞμ ~ ðhÞd h Sμ 0 G eZ ðαÞ
Z
¼
ð26Þ
ðhÞd h Sμ 0 eZ ðαÞ
e 0 ðαÞ. where S denotes the support set of the fuzzy set Z In Eq. (26), the denominator is the area under the membership function μ 0 , and the numerator is the possieZ ðαÞ bility of occurrence μ 0 ðhÞ for each h is weighted by eZ ðαÞ e In the third its satisfaction degree μ ~ ðhÞ of the goal G: G
step, in order to identify a balance between feasibility of the constraints and satisfaction degree of the goal in the space of α-acceptable solutions, Jiménez et al. (2007) e and e proposed to build two fuzzy sets, F S, with the 0 following membership functions: μ F~ X ðαk Þ ¼ αk e0 ðαk Þ , respectively. As and μS~ X 0 ðαk Þ ¼ K G~ Z shown in Fig. 3, to have a final solution, the mentioned fuzzy sets are combined to define a fuzzy decision (i.e., e ¼ F∩ e e D S) (Bellman and Zadeh 1970): 0 e ðαk Þ μD~ X 0 ðαk Þ ¼ αk *K ~ Z
ð27Þ
G
where * denotes a minimum t-norm which is used to construct the intersection of two fuzzy sets. Considering the membership function of the fuzzy decision, a solution with the highest membership value is the final decision for the ITSFP problem:
μD~ X
*
0 e ¼ max αk *K ~ Z ðαk Þ αk ∈M
ð28Þ
G
A numerical example of ITSFP can be found in Wang and Huang (2011). ~ S
~ F
~ D
X e Fig. 3 Relation between the fuzzy constraint F , the fuzzy goal e S, e and the fuzzy decision D
This optimization model minimizes the total cost of water and waste load allocation in the river system. Total cost of the system includes the loss caused by deficit irrigation, operating costs of evaporation and detention ponds, as well as the average loss due to losing the diverted agricultural return flows. The loss of diverted agricultural return flow can be estimated considering the average benefit of crops which can be annually produced using a unit volume of water. The water production function proposed by Ghahraman and Sepaskhah (2004) can be utilized for quantifying the loss caused by deficit irrigation: " !# n ðW a Þt Ya ¼ ∏ 1−kyt 1− ð29Þ Y p t¼1 Wp t where, t n Ya Yp kyt (Wa)t (Wp)t
Index of crop growth stage Total number of crop growth stages Actual annual crop yield per unit farm land area (ton) Potential annual crop yield per unit farm land area (ton) Yield response factor in growth stage t Applied water in growth stage t (million cubic meter (MCM)) Potential water demand in growth stage t (MCM)
As in this paper it is assumed that the applied water stress to the crops is less than 30 %, so an additive form of production function can be used instead of the multiplicative production function (Ghahraman and Sepaskhah 2004): " !# n X Ya ðW a Þt ¼AþB 1−kyt 1− ð30Þ Yp Wp t t¼1 where A and B are constants. The potential crop yield is assumed to be equal to crop per drop (CPD) of the crop times its potential water demand. FAO (2003) defined CPD as average agricultural production (crop or benefit) per unit volume of water applied to the unit farm land area. The loss of a diverted return flow can be estimated considering the average benefit of crops which can be
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produced using a unit volume of water. Therefore, the total cost is estimated as follows: Total cost ¼
n X i¼1
xi Y pi 1− 1−k yi 1− Ai Þ þ C di þ C ti ui
Y pi ¼ CPDi ui i ¼ 1; 2; … n
ð32Þ
C di ¼ cdi xdi i ¼ 1; 2; … n
ð33Þ
C ti ¼ cti xti i ¼ 1; 2; … n
ð34Þ
where, Total cost ui xi xdi xti Ai kyi
Monthly total cost of the system (US$) Daily water demands of water user i (MCM/day) Amount of water allocated to water user i (MCM/day) Amount of wastewater diverted to evaporation pond by water user i (MCM/day) Amount of wastewater diverted to detention pond by water user i (MCM/day) Farm land area of water user i (ha) The equivalent yield response factor of crops of water user i
Minimize Z ¼
ð31Þ
cdi/Cdi
Unit/total loss of wastewater diversion of water user i (US$/MCM, US$) cti/Cti Unit/total operational cost of wastewater treatment of water user i (US$/MCM, US$) Ypi Total benefit of crops which can be annually produced by user i CPDi Average benefit which can annually be gained by user i using one unit of water (US$/ha) In the optimization model, the quantities of water allocated to users and return flows diverted to both evaporation and detention ponds are considered as decision variables, and only the existing uncertainties in upstream river flow, water demands, and economic parameters (i.e., CPD, cd, and ct) are incorporated using the ITSFP method. These uncertain variables and parameters are considered as interval numbers in the optimization model, and their ranges of variations have been estimated using available data from the study area. Quantifying the uncertainties of the SWAP parameters needs considerable field data, which are not currently available in the study area. Figure 4 shows a schematic river system, which is used for defining components of the proposed water and waste load allocation optimization model:
2 X xi CPDi ui 1− 1−k yi 1− Ai Þ þ ðcdi xdi Þ þ ðcti xti Þ ui i¼1
ð35Þ
Subject to: Dri ¼ f i ðxi ; GWL; k Þ
i ¼ 1; 2
ð36Þ
2 X i¼1
Cdr1k ¼ p1k x1 ; GWL; k; GWQk ; cupk
ð37Þ
Cdr2k ¼ p2k ðx2 ; GWL; k; GWQk ; c1k Þ
ð38Þ
ðxi Dri Þþ
2 X
xdi ≤ qup ð1−ωu ÞU d Ed
ð39Þ
i¼1
x1 ≤ qup ð1 ωu ÞU d
ð40Þ
x2 ≤ qup ð1 ωu ÞU d ðx1 Dr1 Þ xd1
ð41Þ
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Eva. Pond 1
xd 1
x t1
Upstream
Ud
x1
x t1
Dr2
City ωuUd
Det. Pond 1
1
2
x2
Legend Irrigation Area
xd 2
Dr2
xt 2
Ed
xt 2
City
Det. Pond 2
Detention pond
Eva. Pond 2
Evaporation pond Water quality checkpoint
Fig. 4 A hypothetical river system which is used for illustrating the optimization model formulation
0 ≤ x i ≤ ui
i ¼ 1; 2
ð42Þ
0 ≤ xdi ≤ Dri
i ¼ 1; 2
0 ≤ xti ≤ Dri
i ¼ 1; 2
pik
ð43Þ ð44Þ
GWL k GWQk cupk
c1k ¼ g1k x1 ; xd1 ; cup
k ¼ 1; 2; …; m
ð45Þ
c2k ¼ g2k ðx2 ; xd2 ; c1k Þ
k ¼ 1; 2; …; m
ð46Þ
Ud ωu
cik ≤csk
i ¼ 1; 2
;
k ¼ 1; 2; …; m
ð47Þ
where, Z qup Dri Cdrik
fi
Total cost of the system (US$) Daily upstream river flow (MCM/day) Daily return flow of water user i estimated using SWAP model (MCM/day) Concentration of water quality indicator k in return flow of water user i estimated using the SWAP model (mg/L) A function which provides the quantity of agricultural return flow of water user i. This function can be estimated using the SWAP model
Ed cik csk gik
A function which is estimated using the SWAP model and provides the concentration of water quality indicator k in agricultural return flow of water user i Initial average groundwater level (m) Average hydraulic conductivity (m/day) Initial concentration of groundwater quality indicator k (mg/L) Concentration of water quality indicator k in upstream flow (mg/L) Domestic water demand of the city (MCM/ day) Ratio of generated wastewater to water allocated to the city Environmental water demand (MCM/day) Concentration of water quality indicator k at checkpoint i (mg/L) Standard level for the concentration of water quality indicator k (mg/L) A function which provides the concentration of water quality indicator k at checkpoint i; This function can be estimated using the QUAL2Kw model
Based on the interactive two-stage stochastic fuzzy programming that is illustrated in “Interactive two-stage stochastic fuzzy programming” section, the water and waste load allocation can be extended as follows:
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e ¼ MinZ
Environ Monit Assess (2015) 187: 158
2 X x ∼ i e CPDi ui 1 1−k yi 1 Ai Þ þ ecdi xdi þ ecti xti ui i¼1
ð48Þ
Subject to: Dri ¼ f i x i ; GWL; k
Cdr1k ¼
ð49Þ
i ¼ 1; 2
p1k x 1 ; GWL; k; GWQ; cupk
Cdr2k ¼ pi x 2 ; GWL; k; GWQ; c1k
2 X i¼1
x i
ð50Þ
ð51Þ
2 X Dri þ xdi ≤ αEeq1 up
eq eq x 1 ≤ αE 1 up þ ð1 αÞE 1 up ð1 ωu ÞU d
ð53Þ
eq eq x 2 ≤αE 1 up þ ð1 αÞE 1 up −ð1−ωu ÞU d − x1 −Dr 1 −xd1 ð54Þ
0 ≤ xi ≤ ui
i ¼ 1; 2
ð55Þ
0 ≤ xdi ≤ Dri
i ¼ 1; 2
ð56Þ
0 ≤ xti ≤ Dri
i ¼ 1; 2
ð57Þ
i¼1
þ ð1 αÞEeq1 up ð1 ωu ÞU d E d
Fig. 5 A schematic map of the Dez River and its water users and pollution loads
ð52Þ
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c1k ¼ g1k x1 ; xd1 ; cup
k ¼ 1; 2; …; m
ð58Þ
c2k ¼ g2k ðx2 ; xd2 ; c1k Þ
k ¼ 1; 2; …; m
ð59Þ
cik ≤csk
e ¼ Min Z
i ¼ 1; 2
;
k ¼ 1; 2; …; m
ð60Þ
2 X − Ce PD u þ Δu y 1− 1− 1−k i i yi i i i¼1
Subject to: − 0 ≤ x i ≤ ui þ Δui yi
i ¼ 1; 2
0 ≤ yi ≤ 1 i ¼ 1; 2
ð62Þ
ð63Þ
The constraints (48), (49), (50), (51), (52), (53), and (54) and constraints (56), (57), (58), (59), and (60), should also be considered in models (61), (62), and (63). Here also, models (61), (62), and (63) can be transformed into two deterministic submodels, which are corresponding to the lower and upper bounds of the objective function values. The solution of the water and waste load allocation optimization model are denoted as follows: h i − þ x i ¼ 1; 2 ð64Þ iopt ¼ xiopt ; xiopt h i − þ x diopt ¼ xdiopt ; xdiopt h i − þ x tiopt ¼ xtiopt ; xtiopt
eZ opt
i ¼ 1; 2
i ¼ 1; 2
− þ ¼ eZ opt ;eZ opt i ¼ 1; 2
ð65Þ
In this optimization model, if u±i are considered as uncertain inputs, the existing method for solving inexact linear programming problem cannot be used directly (Huang and Loucks 2000). Therefore, let u±i =u−i +Δuiyi, where Δui =u+i −u−i and yi ∈[0,1] and yi are decision variables. Thus, by introducing decision variables y i , models (13), (14), and (15) can be reformulated as:
x i − ui þ Δui yi
e Ai Þ þ ecdi x þ c x di ti ti
ð61Þ
water quality constraints are nonlinear. This nonlinear optimization model is solved by using the ILP algorithm. To have a good starting point, in the first run, water quality constraints (Eq. (60)) are not considered. More detail can be found in Tavakoli et al. (2014).
Case study In this paper, the efficiency of the proposed methodology is demonstrated by applying it to the Dez River system located in the Khuzestan province in the southwestern part of Iran. This important river supplies domestic water demand of the Dezful City as well as water demands of five major agricultural sectors. In this region, irrigation activities have increased the concentration of TDS and TC, so they usually violate the standards in the downstream part of the river. Figure 5 presents a schematic map of the Dez River system in the study area.
ð66Þ
ð67Þ
In the optimization models (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), (58), (59), and (60), water quality at checkpoint 2 depends on water quality at checkpoint 1, amount of water allocated to water user 2, and amount of diverted return flow to evaporation and detention ponds by water user 2. Therefore, some of
Fig. 6 The lower and upper bounds of daily water demands of agricultural water users in the study area in May
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Table 1 The average characteristics of agricultural return flows in the study area (May) Water user
Discharge (m3/s)
TDS (mg/L)
Total coliform (cfu/100 mL)
DO (mg/L)
1 2
BOD (mg/L)
5.54
1,220
10,000
7.57
25.7
2.53
4,900
9,000
1.6
72
3
9.516
1,860
10,500
7.1
27
4
9.313
2,220
8,000
6.7
27
5
0.8935
5,770
12,000
6.6
23
The annual discharge of the Dez River downstream of the Dez dam is 8 billion m3. Moreover, the monthly environmental water demand is about 240 MCM at the Bande-Ghir region. As the worst water quality condition usually occurs in May, the available water quantity and quality data from this month are used for designing evaporation and detention ponds. The uncertainty of agricultural water demand of a water user in the study area is mainly due to inherent uncertainties in air temperature and rainfall. There is no precise information about water demands of the agricultural water users, so the lower and upper bounds of their water demands are estimated and considered in this study (Fig. 6). Water demand intervals are simply the lower and upper bounds of observed values in available historical records. In this paper, it is assumed that the domestic and environmental water demands are fully supplied. This assumption is applicable for different river systems as supplying domestic and environmental water demands is normally a main priority. Sugarcane is the dominant crop in the study area. Therefore, only this crop is considered in the SWAP model. The farm land area of water users 1 to 5 are respectively 9,478, 4,328, 16,280, 15,933, and 1,528 ha. As the Dez River supplies drinking water in downstream, the TDS, dissolved oxygen (DO),biochemical oxygen
demand (BOD), and TC are considered as river water quality indicators because their monitored values have frequently violated the standards (1,000 mg/L, 2 mg/L, 30 mg/L, and 1,000 cfu/100 mL, respectively). Table 1 presents the main characteristics of agricultural return flows in the study area. As presented in this table, the concentration of TC in agricultural return flow is very high. Agricultural practices such as spreading manure as fertilizer on fields during wet periods and using sewage sludge biosolids can all contribute to TC contamination. In this study, it is assumed that the pollution discharge of each agricultural unit is controlled by diverting a part of its return flow to an evaporation pond, and also another part of it can be stored in a detention pond. The annual evaporation rate is 2,400 mm in the region. For a 4-h detention time, the maximum removal percent of TC is about 93 % in a sunny condition (Dababneh et al. 2012). In addition, the rate of BOD removal in detention ponds is 25–40 % for a 4-h retention time (Vigayan Iyer and Mastorakis 2007). Table 2 presents the fuzzy boundary intervals of the average benefit of crops which can be produced by a water user using one unit of water (CPD), unit cost of wastewater diversion (cd), unit cost of wastewater treatment using a detention pond (ct), and the headwater discharge.
Table 2 The fuzzy boundary intervals of some uncertain inputs of the optimization model Parameter
Value
CPD (US$/ha)
[8, 11, 14]
cd (US$)
[2, 3, 4]
[17, 20, 23] [5, 6, 7]
ct (US$/m3)
[0.55, 0.6, 0.65]
[0.7, 0.75, 0.8]
Headwater discharge (MCM/10-days)
[150, 175, 200]
[225, 250, 275]
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Fig. 7 The concentrations of TC at checkpoint 1, obtained using both the QUAL2Kw model and the meta-model developed for river water quality simulation (cfu/100 mL)
Results and discussion In this study, the QUAL2Kw model was calibrated for simulating the concentration of four water quality indicators of TDS, TC, BOD, and DO in the Dez River. In order to reduce the total run-time of the proposed fuzzy simulation-optimization model, the calibrated QUAL2Kw model is replaced with four meta-models. To develop the meta-models, the calibrated QUAL2Kw model is executed many times considering different values for allocated water to water users. In this paper, a meta-model developed for river water quality simulation can present the concentration of a water quality indicator at a checkpoint located downstream of a reach as a function of allocated water to water user in the reach, quantity and quality of return flow of the water user, as well as the concentration of water quality indicator at a checkpoint located upstream of the reach. It should be noted that the concentration of water quality indicator in headwater is considered to be constant. As an example, the concentrations of TC at checkpoint 1 which are obtained using the QUAL2Kw and the respective metamodel are presented in Fig. 7. This figure clearly illustrates the acceptable accuracy of the meta-model. In this paper, the QUAL2Kw model gives the spatial and temporal variations of the concentration of water quality indicators in the river based on the return flows estimated by SWAP model and the headwater quantity and quality conditions. Therefore, the outputs of SWAP models are used as inputs of QUAL2Kw model. The travel time of pollutants in the river in the study area is less than 1 day. Considering the irrigation frequency of agricultural lands (i.e., one time per 10 days), in each day,
the SWAP model provides the quantity and quality of return flow of an irrigation network due to irrigation of one tenth of the agricultural land irrigating in current day and irrigation of other parts of the agricultural land irrigated in the past 9 days. Therefore, the SWAP model is run for 10 days, and the return flow quantity and quality in the last day are considered in the optimization model. The results of the QUAL2Kw are used for Table 3 The value of SWAP inputs and parameters (Samipour et al. 2010) Variable description
Value
Date of sugarcane seeding
Early October
Date of sugarcane harvest
Beginning October
Irrigation interval (day)
10
Growing period (day)
365
Drain spacing (m)
70
Level of drain bottom (m)
−2
Level of impervious layer (cm)
−800
Horizontal hydraulic conductivity top layer (cm/day) Minimum thickness for runoff (cm)
50 3
Initial ground water level (cm)
−200
Yield response factor
0.5
Maximum rooting depth (cm)
200
Residual water content (cm3/cm3)
0.02
Saturated water content (cm3/cm3)
0.43
Shape parameter alfa of the main drying curve Shape parameter n
0.0227
Saturated vertical hydraulic conductivity (cm/day)
1.548 36
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Fig. 8 The return flow of agricultural water user 3 obtained using both the SWAP model and the SWAP-based meta-model (MCM/10 day)
evaluating the objective function of the optimization model, which is run for the most critical day in terms of river water quality and provides the capacities of water allocation and return flow diversion systems. Samipour et al. (2010) calibrated SWAP parameters in the Dez region, based on the field monitoring data. Therefore, in current study, the SWAP parameters calibrated by Samipour et al. (2010) are used. Table 3 presents the values of SWAP inputs and parameters. In order to estimate the maximum capacities of evaporation ponds and return flow diversion system, it is assumed that ground water level is equal to the level of
underground drainage system when irrigation is started in the 10-day planning period. To estimate the quantity and quality of return flows, five different SWAP models are used for different agricultural water users. The results of SWAP models are used for training five meta-models, which provide the pollution loads. The SWAP-based meta-model of each zone gives the quantity and quality of return flow of the zone based on the volume of allocated water. As an example, in Fig. 8, the return flow of agricultural water user 3, which has been obtained using the SWAP model, is compared with the results of the SWAP-based meta-model.
Table 4 The α-acceptable solution obtained using the proposed methodology Feasibility degree, α
0.4
0.5
0.6
0.7
0.8
0.9
1
x1
[7.1, 13.4]
[7.2, 14.6]
[7.5, 14.7]
[7.6, 15.7]
[7.1, 14.7]
[7.2, 13.7] [7.2, 13.7]
x2
[3.8, 7.9]
[3.2, 7.8]
[3.3, 7.8]
[4.9, 7.8]
[3.8, 7.8]
[3.8, 7.8]
[3.8, 7.8]
x3
[12, 19.7]
[12.7, 21]
[13, 21]
[13.7, 21]
[12, 21]
[12, 19]
[12, 20]
x4
[11.9, 18.9] [12.4, 20.6] [12.7, 20.6] [12.1, 20.7] [11.9, 20.6] [12, 19.6]
[11.9, 20.7]
x5
[1.6, 2.5]
[1.2, 2.7]
[1.2, 2.7]
[2.1, 2.7]
[1.6, 2.7]
[1.7, 2.7]
[1.6, 2.7]
Diverted wastewater of water users xd1 [2.2, 3.1] 1 to 5 (MCM) xd2 [1.3, 2.2]
[1.9, 2.6]
[1.9, 2.7]
[2, 2.8]
[1.9, 2.6]
[2, 2.6]
[2.1, 2.7]
[1.1, 2.3]
[0.9, 2.4]
[1.6, 2.4]
[1.3, 2.3]
[1.3, 2.4]
[1.3, 2.3]
xd3 [1.3, 2.3]
[2.2, 3.2]
[1.4, 3.2]
[1.5, 3.2]
[1.3, 3.2]
[1.3, 3.2]
[1.3, 3.2]
xd4 [1.5, 2.7]
[1.4, 2.7]
[2.1, 2.7]
[1.5, 2.7]
[1.5, 2.7]
[1.5, 2.7]
[1.5, 2.7]
xd5 [0.6, 1.2]
[0.6, 0.7]
[0.5, 0.7]
[0.7, 0.7]
[0.5, 1.2]
[0.4, 1.2]
[0.6, 1.2]
xt1
[1.3, 3.1]
[0.9, 3.6]
[0.7, 3.6]
[1.9, 3.7]
[1.3, 3.6]
[1.3, 3.7]
[1.4, 3.6]
xt2
[0, 0.9]
[0, 0.8]
[0, 0.3]
[0, 1.1]
[0. 0.9]
[0, 1]
[0, 0.9]
xt3
[2, 2.2]
[1.6, 1.9]
[1.4, 1.9]
[1.9,2.5]
[1.9, 2.1]
[2, 2.1]
[2, 2]
xt4
[1.8, 2.3]
[1.3, 2.1]
[1.3, 2]
[2.1, 2.2]
[1.8, 2]
[1.8, 2]
[1.7, 2]
xt5
[0.3, 1]
[0.2, 0.6]
[0.2, 0.6]
[0.2, 0.6]
[0.2, 1]
[0.2, 1.1]
[0.2, 1.1]
Allocated water to water users 1 to 5 (MCM)
Treated wastewater of water users 1 to 5 (MCM)
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Table 5 The possibility distribution of the total system cost (106 $) Feasibility degree, α
ef ðαÞ
0.4
[(4.7, 6.5, 8.2), (8.8, 10.4, 11.9)]
0.5
[(4.1, 5.6, 7.1), (8.4, 9.9, 11.4)]
0.6
[(4.1, 5.6, 7.1), (8.6, 10.1, 11.6)]
0.7
[(5.6, 7.7, 9.8), (10.2, 12, 13.8)]
0.8
[(4.7, 6.4, 8.2), (8.6, 10.1, 11.7)]
0.9
[(4.8, 6.4, 8.2), (8.7, 10.4, 11.9)]
1
[(4.8, 6.5, 8.3), (8.8, 10.5, 12)]
In accordance with average intervals of sugarcane irrigation, the proposed optimization model is developed in the 10-day period during the critical period (i.e., May) for real-time water and waste load allocation. In order to design the required evaporation and detention ponds and determine short-term policies for water allocated to users and return flow diversions, the ITSFP method is used to solve water and waste load allocation optimization model. Tables 4 and 5 present the solution obtained using the proposed methodology under a set of feasibility degrees (α). The solution under each level of α reflects potential system condition variation caused by uncertain inputs of
Table 6 Membership grades of the fuzzy goal and fuzzy decision in the α-acceptable solution Feasibility degree, α
Satisfaction degree of fuzzy goal
Membership grade of fuzzy decision
Mean of membership grade Deviation of membership for fuzzy decision grade for fuzzy decision
Lower bound Upper bound Lower bound Upper bound solution solution solution solution 0.4
0.412
0.367
0.165
0.147
0.156
0.009
0.5
0.267
0.274
0.133
0.137
0.135
0.002
0.6
0.262
0.319
0.157
0.191
0.174
0.017
0.7
0.634
0.667
0.444
0.467
0.455
0.012
0.8
0.413
0.324
0.330
0.259
0.295
0.035
0.9
0.413
0.367
0.372
0.330
0.351
0.021
1
0.413
0.367
0.413
0.367
0.390
0.023
Fig. 9 Mean and deviations of membership grades of the fuzzy decision versus different levels of α
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Fig. 10 Allocated water to water users and their diverted wastewater to evaporation and detention ponds during the 10-day planning period based on α=0.7 (MCM)
a
b
c
d
Fig. 11 Concentration of TDS (a), TC (b), DO (c), and BOD (d) at checkpoint-based optimal water and waste load allocation policies
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± Ce PD c q c i ,e di , e i ,e ti , and ui as well as complexities of their interactions. In this optimization problem, a lower satisfaction degree of the objective function would be related to a higher feasibility of the constraints and vice versa. Therefore, a decision maker should find a balance between feasibility of the constraints and satisfaction degree of the goal. After obtaining the total system costs under different level of α, the decision makers would be asked to provide a fuzzy goal ðGÞ and its tolerance threshold G . In our case, the goals for lower and upper bound total system e costs would be expressed by means of a fuzzy set G
methodology for water and waste load allocation in maintaining the water quality standards. The optimal value of the objective function (i.e., ef ¼ ½ð5:6; 7:7; 9:8Þ; ð10:2; 12; 13:8Þ 106 ) which opt is expressed as an interval with fuzzy lower and upper bounds reflects the potential range of variation of the total cost caused by uncertain inputs and parameters.
with the following membership function (Eq. (27)): 8 if z≥ 9:8 > : 9:8−4:1 0 if z≤ 4:1
In this paper, a new framework was proposed for water and waste load allocations in river systems. The ITSFP method was utilized to handle dual uncertainties expressed as fuzzy boundary intervals that exist in the objective function and the left and right hand sides of constraints. Some meta-models, which have been trained based on the results of SWAP and QUAL2Kw models, were utilized to estimate quantity and quality of irrigation return flows and river water quality at some checkpoints along the river. In this paper, an iterative linear programming was used for solving the nonlinear optimization model. The efficiency and applicability of the methodology were examined using data obtained from the Dez River system and its agricultural networks in the southwestern part of Iran. The optimization model provided the upper and lower bounds of allocated water to water users and diverted return flow to evaporation and detention ponds. These results can be effectively used for sizing water allocation and pollution control systems in the planning phase. The methodology presented in this paper can be easily used for water and waste load allocation in other river systems suffering from pollution loads of agricultural drains. In future works, this methodology can be extended to consider nonpoint pollution sources and incorporate uncertainties of SWAP model parameters.
þ
8 > 13:8−8:4 : 0
if z≥ 13:8 if 8:4≤ z ≤ 13:8
ð69Þ
if z≤ 8:4
Table 6 presents the lower and upper bounds of the e and fuzzy satisfaction degree of the fuzzy goal G decision for different values of α. In order to identify a balance between satisfaction degree of the goal and constraint feasibility and also make a final decision, the mean and deviation of the membership grade for the fuzzy decision are calculated under each α-acceptable solution. Finally, the optimal solution would be the one with the highest mean and lowest deviation of membership grades in the fuzzy decision. Figure 9 presents means and deviations of membership grade for the fuzzy decision under different levels of α. As shown in this figure, the 0.7 feasibility optimal solution with the highest mean of 0.45 and the deviation of 0.01 would be the best choice. Therefore, the optimal values of y1, y2, y3, y4, and y5 are 1, 0.3, 0.9, 0.9, and 0.9, respectively. Figure 10 presents the allocated water to water users and their diverted return flow to evaporation and detention ponds based on feasibility degree (α) of 0.7. These results have been obtained under 0.7 feasibility solution of the optimization model. The concentration of TDS, TC, DO, and BOD at the five water quality checkpoints are presented in Fig. 11. This figure shows the capability of the proposed fuzzy
Summary and conclusion
Acknowledgements The authors would like to acknowledge the financial support of the University of Tehran for this research under grant number 8102060/1/06.
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