Kuroda, Hisao

Maeda, Shigeya

Yoshida, Koshi

English

Hydraulic Engineering Repository

Conference Paper, Published VersionMaeda, Shigeya; Yoshida, Koshi; Kuroda, HisaoGrey Fuzzy Optimization of Total Nitrogen Load Allocationto Nonpoint Sources in WatershedZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/99420Vorgeschlagene Zitierweise/Suggested citation:Maeda, Shigeya; Yoshida, Koshi; Kuroda, Hisao (2014): Grey Fuzzy Optimization of TotalNitrogen Load Allocation to Nonpoint Sources in Watershed. In: Lehfeldt, Rainer; Kopmann,Rebekka (Hg.): ICHE 2014. Proceedings of the 11th International Conference onHydroscience & Engineering. Karlsruhe: Bundesanstalt für Wasserbau. S. 101-108.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. Damit ist sowohl die kommerzielle Nutzung als auch das Teilen, dieWeiterbearbeitung und Speicherung erlaubt. Das Verwenden und das Bearbeiten stehen unter der Bedingung derNamensnennung. Im Einzelfall kann eine restriktivere Lizenz gelten; dann gelten abweichend von den obigenNutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Documents in HENRY are made available under the Creative Commons License CC BY 4.0, if no other license isapplicable. Under CC BY 4.0 commercial use and sharing, remixing, transforming, and building upon the materialof the work is permitted. In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding. 1 INTRODUCTION Restoration of water quality in lakes has been a serious problem in Japan. Kasumigaura Lake, which has the second largest surface water area in Japan, is one of the typical water bodies that need watershed-level water quality management due to complex mechanism of pollutant dynamics in the related water bodies and its basin and competing goals of stakeholders. Generally, mathematical programming is an effective method to describe objectives and constraints with respect to effluent control for the water quality man-agement. Recently, for example, Karmakar and Mujumdar (2006, 2007) have presented grey fuzzy opti-mization models for river water quality management. Maeda et al. (2010a) has developed a robust optimi-zation model for allocating allowable COD (Chemical Oxygen Demand) load among point and nonpoint sources in a river watershed. Li et al. (2014) has presented an inexact chance-constraint quadratic pro-gramming model for managing river water quality. Maeda et al. (2006, 2009, 2010b) have presented line-ar programming models for TN (Total Nitrogen) load allocation to nonpoint sources (NPSs) such as pad-dy field and upland crop field.  In this study, an improvement over our previous model (Maeda et al., 2009) is conducted for control-ling NPS effluent using the two-phase grey fuzzy optimization framework presented by Karmakar and Mujumdar (2007). Uncertainty in TN transport and management goals are quantified with interval num-bers (e.g., Huang et al., 1995), which enhances the applicability of the model solution to the real world. The presented method is numerically demonstrated by computing TN load allocation in the Seimei River basin, included in the Lake Kasumigaura basin, Japan.    Grey Fuzzy Optimization of Total Nitrogen Load Allocation to Nonpoint Sources in Watershed S. Maeda, K. Yoshida & H. Kuroda Ibaraki University, Ibaraki, Japan ABSTRACT: A method for allocating allowable total nitrogen (TN) load among nonpoint sources in a watershed has been developed by adopting the two-phase grey fuzzy optimization approach with the aid of GIS (Geographic Information System). Competing goals of water quality management authorities and TN load dischargers are described with linear imprecise membership functions including interval num-bers. Discharged TN loads from each cell of paddy field, upland crop field, and residential area are as-sumed to be transported under the conventional first-order kinetic removal. Uncertainty of river discharge and self-purification coefficients are also expressed with interval numbers. The method developed is ap-plied to the Seimei River watershed, which is a subwatershed of the Lake Kasumigaura basin in Japan. By solving the linear programming submodels for TN load allocation to cells using the simplex method, the allowable load at each cell is procured, which could be used as a target value in the decision-making re-garding effluent control policies. Keywords: Water quality, Nonpoint source, Watershed, optimization, Fuzzy, uncertainty, GIS, Decision-making ICHE 2014, Hamburg - Lehfeldt & Kopmann (eds) - © 2014 Bundesanstalt für Wasserbau ISBN 978-3-939230-32-81012 FRAMEWORK FOR OPTIMAL ALLOCATION OF TOTAL NITROGEN LOAD 2.1 Pollutant Sources and Management Goals The cell-based optimization model developed by Maeda et al. (2009) is modified by adopting the grey fuzzy optimization framework. As nonpoint sources, only paddy field, upland crop field and residential area in a watershed of a water body are chosen in this study. The watershed is horizontally divided to uni-form cells. A GIS (Geographic Information System) and a DEM (Digital Elevation Model) are employed for delineating the watershed boundary, determining the flow direction for the cells, and identifying the routes of both surface and subsurface flows. Flow direction of those flows are assumed identical for sim-plicity, and determined using the concept of the steepest slope to one of its eight neighboring cells. Sup-pose that TN load issuing from a cell of nonpoint source (called ‘Land Management Unit (LMU)’ hereaf-ter) is transported along the estimated route to the main river in the watershed, and finally delivered to the downstream end of the river. Watershed-level effluent control designed in this study defines the following two competing goals: 1. Prevention of water quality deterioration at the outlet of the watershed, which is generally a goal of wa-ter quality management authorities 2. Profit maximization or cost minimization regarding water-related activities, such as farming, wastewater treatment, etc. in the dischargers’ side 2.2 Goal for Water Quality Management Authorities The allowable discharged load from each LMU, a decision variable in our study, is specified as an inter-val number due to uncertainty in self-purification function, stream discharge, precipitation, etc. The al-lowable load can be expressed as [ , ], , , ; 1, ,± − +≡ = = i i ij j j jL L L j p u c i I  (1) where i = cell number; j = type of nonpoint source cell (p = paddy field, u= upland crop field, c = residen-tial area); jI = number of type-j cells; ijL± = discharged TN load from the cell ji (kg・ day-1); ijL− , ijL+ = lower and upper limits of discharged TN load from the cell ji (kg・ day-1). It should be noted that super-script ±  implies the variable is an interval number, and the subscripts – and + stand for lower and upper limits of the interval number, respectively. Assume that the TN loads discharged from their sources decay subject to the distance-related first-order kinetics. Using the concept of interval number, the amount of TN load that reaches the outlet can be expressed as 1exp( )−+ −=+++  = − +    =∑∑ji i iiwIr r wj j jj ijT L x xQf Lλ λ (2) 1exp( )+− +=−−−  = − +    =∑∑ji i iiwIr r wj j jj ijT L x xQf Lλ λ (3) where [ , ]T T T± − +≡  = TN load that transported to the downstream end of the main river from all the LMUs (kg・ day-1); ,r rrλ λ λ− +± ≡     = watershed-wide self-purification coefficient for surface and sub-surface flows (m-1); irjx  = travel length of TN load discharged with surface and subsurface flows from the LMU ji to the outlet at the riverbank (m); ,w wwλ λ λ− +± ≡     = watershed-wide self-purification coefficient for river flow (m2・ day-1); ,Q QQ − +± ≡     = discharge of river (m3・ day-1); iwjx  = travel length of TN load, generated from LMU ji, from the drainage outlet at the riverbank to the downstream end of the river (m). It is noted that Eqs. (2) and (3) show T +  and T −  are functions of ijL+  and ijL− , respectively. For water quality management authorities, less delivered TN load to the downstream end is preferable. Considering the competing goals of stakeholders in the watershed, the management authorities’ goal can be expressed with the following linear imprecise membership function ( )Tµ ± ±  1021 ( )( ) ( )0 ( )+ −± ±± ± ± ± ±± ±− +< −= ≤ ≤ − >DHD HH DHT TT TT T T TT TT Tµ  (4) where [ , ]D D DT T T± − +≡ = the most desirable total transported TN load for the authorities; [ , ]H H HT T T± − +≡ = acceptable total transported TN load for the management authorities ( )D HT T± ±≤  (see Figure 1). The membership value of ( )Tµ ± ±  approaches 1, which is the most preferable state for the au-thorities, by minimizing T ± .   Figure 1. Linear imprecise membership function for the goal of water quality management authorities.  2.3 Goals for Dischargers Goals of TN dischargers at LMUs of paddy field, upland crop field, and residential area can also be ex-pressed with the imprecise membership functions ( )ijLµ ± ± , defined as 0 ( )( ) ( )1 ( )+ −± ±± ± ± ± ±± ±− + < −= ≤ ≤ − >i ii ii i i ii ii iLj jLj j L Dj j j jD Lj jDj jL LL LL L L LL LL Lµ  (5) where [ , ]i i iL L Lj j jL L L± − +≡  = acceptable discharged TN load for discharger at LMU ji; [ , ]i i iD D Dj j jL L L± − +≡  = the most desirable discharged TN load for discharger at LMU ji (i iD Lj jL L± ±≥ ) (see Figure 2).    Figure 2. Linear imprecise membership function for the goal of discharger at LMU ji.  3 GREY FUZZY OPTIMIZATION MODEL A grey fuzzy optimization model that harmonizes goals mentioned above under constraints can be ex-pressed as the following mathematical programming problem Maximize ξ ±  (6) subject to 1. Achievement level of goal of water quality management authorities 103(height) and ArcGIS 10.2, the flow direction of each cell is estimated by the steepest gradient method with surface elevation. In the Seimei River watershed (25.6km2), 4,496 cells at a resolution of 50m×60m are specified as LMUs using ArcGIS 10.2. The numbers of LMUs of paddy field, upland crop field, and residential area are 1201, 1753, and 1542, respectively (see Figure 4). Values of parameters such as self-purification coefficients, discharge of river flow, and permissible and desirable levels related to goals for the authorities and dischargers, etc. are determined with observed data in 2013 and the literature (Skop and Sørensen, 1998) (see Table 1). Flow length, i.e., length of route where TN load issued from a LMU is transported, is estimated using the obtained flow direction data in the GIS.  The obtained optimal solutions in Cases 1 and 2 are listed in Table 2. The grey degree (Karmakar and Mujumdar, 2007) of decision variable, ( )ijGd L± , is defined as ( )( ) 12+ −±− +−=+i iii ij jjj jL LGd LL L. (13)  Table 1. Parameters used in the application Parameter Value ,r rλ λ− + (m-1) 0.000850, 0.000850 ,w wλ λ− + (m2・day-1) 0.480, 0.480 ,Q Q− + (m3・day-1) 4168.8, 16675.2 ,i iD Dp pL L− + (kg・day-1) 3.38, 4.13 ,i iL Lp pL L− + (kg・day-1) 0.00, 0.452 ,i iD Du uL L− + (kg・day-1) 8.80, 10.8 ,i iL Lu uL L− + (kg・day-1) 0.00, 1.18 ,i iD Dc cL L− + (kg・day-1) 1.03, 1.26 ,i iL Lc cL L− + (kg・day-1) 0.00, 0.138 ,D DT T− + (kg・day-1) 0.00, 72.8 ,H HT T− + (kg・day-1) 3720, 4090  Table 2. Optimal solutions expressed as interval numbers and grey degrees of decision variables in Cases 1 and 2. Case 1: goal of water quality management authorities is considered first; Case 2: goals of TN load dischargers are consid-ered first. Case 1:  Solution ( )ijGd L±  [1.3698, 1.3698]ipL± = (kg・day-1), 1, ,1201i =   0 [3.5647, 3.5647]iuL± = (kg・day-1), 1, ,1753i =   0 [0.4173, 0.4173]icL± = (kg・day-1), 1, ,1542i =   0 [2384, 3403]T ± = (kg・day-1)  [0.0775, 0.4678]ξ ± =   Case 2:  Solution ( )ijGd L±  [1.0920, 1.0965]ipL± = (kg・day-1), 1, ,1201i =   4.11×10-3 [2.8536, 2.8536]iuL± = (kg・day-1), 1, ,1753i =   0 [0.33326, 0.33405]icL± = (kg・day-1), 1, ,1479,1481, 1542i =    [0.33326, 0.33451]= (kg・day-1), 1480i =  2.37×10-3  3.74×10-3  [1906, 2724]T ± = (kg・day-1)  [0.1550, 0.3745]ξ ± =    1055 CONCLUSIONS A GIS-aided optimization method is presented for managing TN load from nonpoint sources in a water-shed. The method includes estimation of flow routes by hydrological analysis with the DEM and the over-lay of maps of land use and flow direction in the GIS. The two-phase grey fuzzy optimization framework is employed to formulate optimal allocation problem of the allowable discharged TN load among cells (LMUs) of paddy field, upland crop field, and residential area. The method developed is applied to the Seimei River watershed, which is a part of the Lake Kasumigaura basin, Japan. It is demonstrated that op-timal solution or allowable TN load for each LMU is numerically procured and the presented method could contribute to more rational effluent control at the watershed level. NOTATION ijL±  discharged TN load from the cell ji  jI  number of type-j cells T ±  total transported TN load at the downstream end rλ ±  watershed-wide self-purification coefficient for surface and subsurface flows wλ ±   watershed-wide self-purification coefficient for river flow Q±  discharge of river irjx   travel length of TN load discharged with surface and subsurface flows from the LMU ji to the outlet at the riverbank iwjx  travel length of TN load, generated from LMU ji, from the drainage outlet at the riverbank to the downstream end of the river DT ±  the most desirable transported TN load for the authorities HT ±   acceptable transported TN load for the authorities ( )Tµ ± ±  linear imprecise membership function on total transported TN load iLjL±  acceptable discharged TN load for discharger at LMU ji iDjL±   the most desirable discharged TN load for discharger at LMU ji ( )Lµ ± ±   linear imprecise membership function on discharged TN load ξ ±    watershed-wide satisfaction level Gd  grey degree ACKNOWLEDGEMENT This study was supported in part by the River Maintenance Fund of River Foundation Grant Number 25-1211-006. 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Grey Fuzzy Optimization of Total Nitrogen Load Allocation to Nonpoint Sources in Watershed

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Grey Fuzzy Optimization of Total Nitrogen Load Allocation to Nonpoint Sources in Watershed

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