water

Article Optimal Allocation of Water Resources from the “Wide-Mild Water Shortage” Perspective

Huaxiang He 1,2 , Mingwan Yin 2,*, Aiqi Chen 3, Junqiu Liu 2 , Xinmin Xie 2 and Zhaohui Yang 2

1 State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100048, ; [email protected] 2 State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China; [email protected] (J.L.); [email protected] (X.X.); [email protected] (Z.Y.) 3 College of Land Science and Technology, China Agricultural University, Beijing 100193, China; [email protected] * Correspondence: [email protected]; Tel.: +86-10-68785708

 Received: 14 July 2018; Accepted: 17 September 2018; Published: 20 September 2018 

Abstract: A major objective of the optimization of water resources allocation is to ensure the supply an adequate amount of water to users at the right time and maximize the utilization of water resources. However, in case of insufficient water supply, water shortage is likely to occur intensively for specific water users or in specific periods, referred to as a “concentrated water shortage”. The risk of a concentrated water shortage should be shared across a wider range of users and periods, so that it would have a less severe impact on each calculation unit in each period, which we refer to as the “wide-mild water shortage”. In this study, the nonlinear weight of the water supply objective function can be converted into a piecewise linear weight based on the law of diminishing marginal utility, making it possible to reduce or even eliminate the concentrated water shortage and thus making the allocation of water resources more reasonable. The case study in the basin in shows that the improved method results in a significant increase in water shortage units but a significant reduction in water shortage range. As a consequence, water shortage is more uniformly distributed from April to June, which contributes to solving the concentrated water shortage problem in May. However, it should be noted that to what extent the wide-mild water shortage can be realized depends not only on the marginal utility of water demand, but also on the available water supply and the regulative capacity of water supply projects. In spite of this, the improved method enables water to be supplied more suitably for users at the appropriate time, which contributes to improving the utilization of water resources and helping decision-makers better address the problem of concentrated water shortage.

Keywords: water resources allocation; wide-mild water shortage; marginal utility; piecewise linear function

1. Introduction Rapid industrialization and urbanization in China have led to a growing demand for water resources for domestic, agricultural, industrial, and ecological purposes in recent years, making the optimal allocation of water resources an important, challenging task. With better understanding of real-world problems, advancing data availability and reliability, researchers are committed to developing large-scale and complex water resources allocation models, and to developing effective algorithms for solving the allocation models [1]. Commonly used methods for the optimization of

Water 2018, 10, 1289; doi:10.3390/w10101289 www.mdpi.com/journal/water Water 2018, 10, 1289 2 of 17 medium- and long-term water resource allocation include dynamic programming, linear programming, and nonlinear programming [2,3]. However, in order to solve an optimization problem using dynamic programming, the principles of optimality and non-aftereffect must both be satisfied [4]. Dynamic programming may also suffer from the curse of dimensionality in solving complex water resources allocation problems in which the storage demands and complexity costs grow exponentially with the dimension of the state space. Nonlinear programming is useful in solving optimization problems where some of the constraints or the objective function are nonlinear [5] and difficult to be linearized [6], or a large error may arise from linearization. However, it is important to note that nonlinear programming may not always converge on the global optimal solution. Linear programming has the advantages of easy modeling, easy availability of necessary parameters, and convergence to the global optimal solution. Obviously, linear programming is preferred to solve medium- and long-term water resource allocation problems [7–15]. The linear programming method has been used to solve the allocation of water resources in the Mahanadi River in northeastern India, in which the seasonal unit-price of surface water or groundwater can be taken as the objective function weights [16]. When the available water is insufficient to meet all the agricultural water demand, the optimization result may cause the occurrence of concentrated water shortages in a certain period due to the lower unit-price in agriculture than that in other users. Even if more water has been supplied for agriculture in the early period, zero full water costing [17] will be presented resulting in no crop harvest due to a seriously concentrated water shortage. A water economy optimization model has been developed by Mirchi in South Florida to quantitatively weigh water management priorities [18]. A characteristic of the penalty function, which is only determined by the relative magnitude by investigation and analysis, is that the function itself has no practical significance. An interval multi-objective programming model has been established to optimize the irrigation water allocation in in China [19]. The interval function, which was solved by linear programming, was used as the weight of the objective function. The max–min operator [20] is used to solve the maximum and minimum of the interval function by generalization as a linear or nonlinear function. If the interval function is generalized as a nonlinear function, the solution method is the same as the nonlinear programming. If the interval function is generalized as a linear function, the assumption is that the marginal utility function is linear and inconsistent with the actual curve. Furthermore, economic benefit/utility [21–23], ecological flow [24], soil moisture uniformity and yield [25], maximization of economic, ecological, and social benefits [26], and investment, operation, and environmental costs [27] can also be taken as the objective function weights and solved by linear programming. Nevertheless, it is noteworthy that all the above weights used to maximize the total water supply benefit or minimize the loss resulting from water shortage are derived under the assumption that the benefit yield of every unit water supply is the same, regardless of the satisfaction level of the water demand. Obviously, this assumption does not hold given the law of diminishing marginal utility. As a consequence, what appears to be theoretically optimal may not be feasible in practical settings. Brown et al. [28,29] showed that, in some cases, the marginal utility of insufficient instream flows was considerably higher than that of the water consumption of users. For these seasons, it is necessary to take into account the law of diminishing marginal utility in water resources allocation. Suppose there are n agricultural plots along the water channel in an irrigation district, which are denoted as i = 1, 2, ... , n from the channel head to its terminal end, and the effective utilization coefficient of irrigated water follows the order of η1 > η2 > ...... > ηn due to the impact of transport distance and water loss. All plots are assumed to have the same soil properties and are planted with the same crop species. The water demand benefit per unit area, as well as the benefit per unit water use λi, is the same, and thus the average benefit per unit water use at different periods is also the same. Thus, the total water demand of the irrigation district is the sum of the water demand of each plot n (Wreal_d = ∑ Wi,net/ηi). If the objective is to maximize the total irrigation benefit and water supply is i=1 allocated according to λi, the programming is a linear problem. Thus, if the available water supply is higher than or equal to the total water demand Wreal_d, the water demands of all plots can be well Water 2018, 10, 1289 3 of 17 satisfied; whereas, in the case of insufficient water supply, the water demands of plots can be satisfied in order from the channel head to the terminal end until the supply of water is exhausted. In this circumstance, water shortage is likely to occur in plots at the terminal end at any time, which is referred to as a “concentrated water shortage” in this study due to the optimization rules. It is important to note that if crop fails due to an extremely low supply of water in a specific period, then the actual benefit of previous periods is equal to 0. This problem is rarely taken into account in most previous Water 2018, 10, x FOR PEER REVIEW 3 of 17 models. For large-scale water resources systems responsible for supplying water from multiple sources to multiplebe userswell satisfied; in multiple whereas, periods in the case via of multiple insufficient water water supply supply, channels, the water demands the allocation of plots can scheme be based on the benefitsatisfied per in unit order water from usethe channel rather head than to on the the terminal marginal end until utility the may supply easily of water result is exhausted. in the occurrence In this circumstance, water shortage is likely to occur in plots at the terminal end at any time, which of concentrated water shortage. is referred to as a “concentrated water shortage” in this study due to the optimization rules. It is Generally,important there to note are that two if crop kinds fails ofdue concepts to an extremely about low water supply shortages, of water in a whichspecific period, are “concentrated then water shortage”the actual and benefit “wide-mild of previous water periods shortage”. is equal to 0. Concentrated This problem is waterrarely taken shortage into account refers in to most dense water shortage occurringprevious models. in specific For large-scale water users water orresources periods, systems which responsible is not affected for supplying by the water precipitation from and water demand.multiple Wide-mild sources to multiple water shortage, users in multiple the antonym periods ofvia concentrated multiple water water supply shortage, channels, refersthe to the allocation scheme based on the benefit per unit water use rather than on the marginal utility may occurrenceeasily of relatively result in the uniform occurrence water of concentrated shortage inwater a specific shortage. period or a specific domain. Similar to the “strategic” andGenerally, “tactical” there water are two resources kinds of concepts allocation abou problemst water shortages, mentioned which by are Turgeon “concentrated [30], water concentrated water shortageshortage” is “strategic”and “wide-mild and water wide-mild shortage”. water Concen shortagetrated iswater “tactical”. shortage refers to dense water Theshortage optimization occurring resultsin specific based water users on or concentrated periods, which is water not affected shortage by the precipitation are often and unrealistic, water demand. Wide-mild water shortage, the antonym of concentrated water shortage, refers to the and sometimes very bad, which will have a direct impact on people’s trust and application. In order to occurrence of relatively uniform water shortage in a specific period or a specific domain. Similar to overcomethe this “strategic” problem, and the “tactical” constraints water of resources minimum allocation water problems supplyare mentioned added toby theTurgeon water [30], demand of each calculationconcentrated unit water in each shortage period. is “strategic” and wide-mild water shortage is “tactical”. However,The given optimization the balance results conditionsbased on concentrated between water water shortage demand are andoften waterunrealistic, supply and in each calculationsometimes unit, it very is very bad, difficult which will to have define a direct the impact constraints on people’s of the trust minimum and application. water In supply order to in a long overcome this problem, the constraints of minimum water supply are added to the water demand of series of inflows.each calculation Too lowunit in constraints each period. may not solve the problem of concentrated water shortage; while constraintsHowever, that given are too the high balance may conditions be infeasible between for water optimization. demand and Accordingly, water supply thein each objective of this studycalculation is to solve unit, the it nonlinearis very difficult water to define resources the constraints allocation of the problem minimum using water the supply linear in programminga long based onseries the rule of inflows. of diminishing Too low constraints marginal may utility, not so andlve the this problem study of may concentrated provide awater simple shortage; but effective while constraints that are too high may be infeasible for optimization. Accordingly, the objective of way for the optimization of water resources allocation. this study is to solve the nonlinear water resources allocation problem using the linear programming based on the rule of diminishing marginal utility, and this study may provide a simple but effective 2. Diminishingway for Marginalthe optimization Utility of water in Water resources Resources allocation. Utilization The concept that the marginal utility of each homogenous unit decreases as the supply of units 2. Diminishing Marginal Utility in Water Resources Utilization increases (and vice versa), provided that other conditions are the same, is referred to as the “law of The concept that the marginal utility of each homogenous unit decreases as the supply of units diminishing marginal utility”. This law has been widely recognized as the explanation of numerous increases (and vice versa), provided that other conditions are the same, is referred to as the “law of economicdiminishing phenomena marginal in the utility”. field ofThis economics, law has been and widely exists recognized in reality as the including explanation its of application numerous to the water resourceseconomic field. phenomena For example,in the field Figureof economics,1 shows and exists the agricultural in reality including water its supplyapplication benefit to the and its marginalwater utility. resources In water field. deficient For example, northeast Figure China,1 shows deficit the agricultural irrigation wate makesr supply itpossible benefit and to its save more water formarginal other purposes utility. In water or to deficient irrigate northeast more land. China, On deficit the irrigation contrary, makes excessive it possible irrigation to save more may cause water for other purposes or to irrigate more land. On the contrary, excessive irrigation may cause waterloggingwaterlogging and consequently and consequent thely the reduction reduction of ofcrop crop yield. It Itis isevident evident that thatthe law the of lawdiminishing of diminishing marginalmarginal utility should utility should be considered be considered in orderin order to to ensureensure the the allocation allocation of water of water resources resources to be more to be more realistic andrealistic effective. and effective.

Figure 1.FigureThe 1. benefit The benefit and and marginal marginal utility utility curves ofof agricultural agricultural water water supply. supply.

Water 2018, 10, 1289 4 of 17

3. Water Resources Optimal Allocation from the Perspective of “Wide-Mild Water Shortage”

3.1. The Concept of Wide-Mild Water Shortage The law of diminishing marginal utility is explicitly neglected in current methods for the optimization of water resources allocation based on benefit per unit water use, which can easily cause the occurrence of concentrated water shortage. As a consequence, (a) the theoretically optimal scheme may not be optimal in practice; (b) a concentrated water shortage that occurs in a specific period may result in a significant benefit/loss not only in this period, but also in earlier and later periods; and (c) given the complex interactions among different water users, the sudden occurrence of a concentrated water shortage for one water user can also bring about a significant benefit/loss of other water users, such as hydroelectric power generation. Thus, in case of insufficient water supply, the risk of concentrated water shortage should be shared across calculation units, periods, and users. In doing so, more calculation units, periods, and users may suffer from water shortage, but the water shortage would have less severe impacts on each calculation unit in each period, which we refer to as the concept of “wide-mild water shortage”. The wide-mild water shortage is expected to be achieved by optimizing water resource allocation based on the law of diminishing marginal utility. Theoretically, such a problem is a nonlinear problem that can be solved by nonlinear programming. In fact, the optimization of water resources allocation is very complex, and linear programming, instead of the nonlinear programming, is often used in this context. In this study, the nonlinear function is approximated by a piecewise linear function, in which the total water demand in each period is divided into piecewise water demands with different marginal utilities. The demands of different calculation units, periods, and users are optimized in the water resources system, and the rest are the same as that in the optimization of water resources allocation based on benefit per unit water use.

3.2. Piecewise Linear Function For a complex water resources system, it is extremely difficult, if not impossible, to establish an accurate piecewise linear function specifically for each water demand in each unit, period, and user, and to accurately determine their marginal utilities and, very often, it is not necessary to do so. Although the allocation of water resources based on the benefit per unit water use may occasionally lead to concentrated water shortage that is unacceptable and, as stated before, the accuracy requirement can be well satisfied. In order to determine the piecewise linear function and marginal utilities, it is important to (a) determine the rank of the average benefit for each water demand, which is the same as the optimization based on the benefit per unit water use; (b) determine the marginal utilities for different satisfaction levels of water demands and then divide them into piecewise segments according to the percentage of water demands. However, the number of piecewise segments should not be too large; (c) examine whether there is an overlap of marginal utilities among different water demands. A sufficiently high marginal utility should be set for those periods during which water shortage is not allowed, and the precision can be set to a level that would have no impact on optimization; (d) examine whether there is a difference in benefit of the same water demand in different calculation units. If a significant difference is found, they should be treated differently; otherwise they can be treated in the same way; and (e) analyze the relationship of the same water demand at different periods in a specific calculation unit. Piecewise segments can be obtained according to the percentage of water demands; however, adjustment can be made if necessary. Much effort has been made to better understand the water use benefit function, such as the Cobb–Douglas production function [31] and piecewise function [32,33], which are not described in detail herein. Water 2018, 10, 1289 5 of 17

3.3. Methodology The objective function for the optimization of water resources allocation based on the average benefit per unit water use (hereafter referred to as the “original method”) can be described as follows.

KN TN JN RN TN HN Max Obj = Max[ ∑ ∑ ∑ BLk,t,j × WLk,t,j + ∑ ∑ ∑ BRr,t,h × WRr,t,h] (1) k=1 t=1 j=1 r=1 t=1 h=1

The major constraints concerning the water supply are as follows,

WLk,t,j ≤ DWLk,t,j (2)

WRr,t,h ≤ DWRr,t,h (3)

IKJN WLk,t,j = ∑ XWLk,t,j,i (4) i=1 IRHN WRr,t,h = ∑ XWRr,t,h,i (5) i=1 where Obj is the objective function; k is the sequence number of calculation units; KN is the total number of calculation units; t is the sequence number of periods in a year; TN is the total number of periods in a year and TN = 12 when the calculation is performed on a monthly basis; j is the sequence number of socioeconomic water use types, JN is the total number of socioeconomic water use types; r is the sequence number of rivers/lakes; RN is the total number of rivers/lakes; h is the sequence number of river/lake ecological flow types; HN is the total number of river/lake ecological flow types; i is the sequence number of water sources; IKJN is the total number of water sources for the water use type j in the calculation unit k; IRHN is the total number of water sources for the water use type h of river/lake r; DWLk,t,j is the water demand j in the calculation unit k; XWLk,t,j,i is the water supply from water source i to DWLk,t,j; WLk,t,j is the total water supply from each water source to DWLk,t,j; DWRr,t,h is the ecological water demand h of river/lake r; XWRr,t,h,i is the total water supply from water source i to DWRh,t,r; WRr,t,h is the total water supply from each water source to DWRr,t,h; BLk,t,j is the average benefit of WLk,t,j per unit water supply; and BRr,t,h is the average benefit of WRr,t,h per unit water supply, respectively. The improved objective function based on the law of diminishing marginal utility (hereafter referred to as the “improved method”) is:

KN TN JN SJN RN TN HN SHN Max Obj = Max[ ∑ ∑ ∑ ∑ BLSk,t,j,s × WLSk,t,j,s + ∑ ∑ ∑ ∑ BRSr,t,h,s × WRSr,t,h,s] (6) k=1 t=1 j=1 s=1 r=1 t=1 h=1 s=1

The following water demand constraints are considered.

SKJN DWLk,t,j = ∑ DWLSk,t,j,s (7) S

SRHN DWRr,t,j = ∑ DWRSr,t,h,s (8) S Constraints in Equations (2)–(5) can be converted into Equations (9)–(12), respectively:

WLSk,t,j,s ≤ DWLSk,t,j,s (9)

WRSr,t,h,s ≤ DWRSr,t,h,s (10) Water 2018, 10, x FOR PEER REVIEW 6 of 17

≤ WRSrths,, , DWRS rths ,, , (10)

Water 2018, 10, 1289 IKJSN 6 of 17 = WLSkt,, js , XWLS kt ,, jsi , , (11) i=1 IKJSN WLS = IRHSNXWLS (11) k,t,j,s = ∑ k,t,j,s,i WRSrths,, ,i=1 XWR rthsi ,, , , (12) i=1 IRHSN where s is the piecewise number ofWRS a waterr,t,h,s =demand,∑ XWR SKJNr, t,ish, s,thei total piecewise number of water(12) demand j in the calculation unit k, SRHN is thei=1 total piecewise number of water demand h of where s is the piecewise number of a water demand, SKJN is the total piecewise number of water river/lake r, DWLSkt,, js , is the socioeconomic water demand of the segment s for the water demand demand j in the calculation unit k, SRHN is the total piecewise number of water demand h of river/lake h in the calculation unit k, DWRSrths,, , is the ecological water demand of the segment s for the water r, DWLSk,t,j,s is the socioeconomic water demand of the segment s for the water demand h in the calculationdemand j unitof river/lakek, DWRSr ,tr,h ,sinis the ecologicalcalculation water unit demand t, BLS ofrt,, the js , segmentis the marginals for the waterutility demand of the jsocioeconomicof river/lake r waterin the calculationdemand of unitthe segmentt, BLSr,t, j,s isfor the the marginal water demand utility of j thein the socioeconomic calculation unit water k, demand of the segment s for the water demand j in the calculation unit k, WLSk,t,j,s is the water supply WLSkt,, js , is the water supply corresponding to the socioeconomic water demand, BRSrths,, , is the corresponding to the socioeconomic water demand, BRSr,t,h,s is the marginal utility of the ecological marginal utility of the ecological water demand of the segment s for river/lake r in the calculation water demand of the segment s for river/lake r in the calculation period t, and WRSr,t,h,s is the water supplyperiod correspondingt, and WRSrths,, to , the is the ecological water supply water corresponding demand. to the ecological water demand. TheThe diminishingdiminishing marginalmarginal utility utility can can be be described described as: as: >>>∀ BLSkt,, j ,1 BLS kt ,, j ,2...... BLS kt ,, jSKJN , k , t , j (13) BLSk,t,j,1 > BLSk,t,j,2 > ...... > BLSk,t,j,SKJN∀k, t, j (13) >>>∀ BRSrth,, ,1 BRS rth ,, ,2...... BRS rt ,, jSRHN , r , t , h (14) BRSr,t,h,1 > BRSr,t,h,2 > ...... > BRSr,t,j,SRHN∀r, t, h (14) Accordingly,Accordingly, thethe agriculturalagricultural waterwater demanddemand processprocess isis shownshown inin FigureFigure2 .2. The The left left figure figure is is the the piecewisepiecewise ofof waterwater demand.demand. TheThe blue,blue, red,red, green,green, andand graygray areasareas standstand forfor fourfour piecewisepiecewise segments.segments. TheThe totaltotal areaarea ofof blue,blue, red,red, andand greengreen isis thethe water water demand. demand. TheThe greygrey areaarea representsrepresents excessexcess waterwater demand.demand. TheThe rightright figurefigure isis marginalmarginal utilityutility curvecurve (black(black curve).curve). TheThe blackblack curvecurve shouldshould bebe divideddivided intointo piecewisepiecewise segmentssegments ofof thethe correspondingcorresponding waterwater supply,supply, shownshown asas thethe blackblack bar.bar. TheThe blue,blue, red,red, green,green, andand graygray barsbars representrepresent thethe averageaverage valuevalue ofof thethe blackblack barbar withinwithin thethe percentagepercentage ofof waterwater demands:demands: thethe closercloser toto thethe bottom,bottom, thethe greatergreater thethe marginalmarginal utility.utility. TheThe greygrey areaarea ofof thethe waterwater supplysupply exceedsexceeds thethe maximummaximum waterwater demanddemand so so that that its its marginal marginal utility utility is is negative. negative.

Figure 2. A schematic of the piecewise waterwater supplysupply andandthe thecorresponding corresponding marginal marginal utility utility functions. functions. 4. Case Study

4.1. Study Area The Nen River basin is located in the northeast region of China and flows through Province, and Provinces, China with a drainage area of 298,500 km2. The average Water 2018, 10, x FOR PEER REVIEW 7 of 17

4. Case Study

4.1. Study Area Water 2018The, 10Nen, 1289 River basin is located in the northeast region of China and flows through Heilongjiang7 of 17 Province, Inner Mongolia and Jilin Provinces, China with a drainage area of 298,500 km2. The average annualannual precipitationprecipitation isis 455455 mm,mm, thethe averageaverage annualannual andand monthly precipitation ofof different frequencyfrequency areare shownshown inin FigureFigure3 .3. InIn 2013,2013, thethe basinbasin hashas aa totaltotal population population ofof 16.56 16.56 million million with with an an urban urban populationpopulation ofof 8.228.22 million,million, aa grossgross domesticdomestic productproduct (GDP)(GDP) ofof 861.6861.6 billionbillion RMB,RMB, anan industrialindustrial addedadded valuevalue ofof 396.6396.6 billion billion RMB, RMB, a a total total grain grain yield yield of of 36.74 36.74 billion billion kilograms, kilograms, and and a livestocka livestock population population of 25.2of 25.2 million. million. It has It has an effectivean effective irrigation irrigation area area of 2.5 of million2.5 million hectares, hectares, a forest a forest and fruitand growingfruit growing area ofarea 21.4 of thousand21.4 thousand hectares, hectares, and a and grassland a grassland area of area 4.5 of thousand 4.5 thousand hectares. hectares. By the By year the 2013, year the2013, basin the 3 hasbasin 429 has water 429 water supply supply projects projects with awith total a total capacity capacity of 16.0 of 16.0 billion billion m3 andm and an effectivean effective capacity capacity of 3 10.4of 10.4 billion billion m3 m. There. There are are three three main main water water resource resource zones zones called called NEJ, NEJ, JQ, JQ, and and BST BST fromfrom upstreamupstream 3 toto downstream.downstream. The The total total water water supply supply is is37.7 37.7 billion billion m , m with3, with the thesurface surface wate water,r, groundwater, groundwater, and andrecycled recycled water water accounting accounting for 61.4 for%, 61.4%, 38.5%, 38.5%, and 0.1%, and 0.1%, respectively. respectively. The main The maincrops cropscultivated cultivated in the inNen the River Nen basin River include basin includerice, corn, rice, wheat, corn, and wheat, soybean. and Sowing soybean. often Sowing begins often in early begins or middle in early April, or middleand the April,water anddemand the waterreaches demand a peak reaches40–80 days a peak after 40–80 seeding days (approximately after seeding from (approximately May to August). from MayThe agricultural to August). Thewater agricultural demand accounts water demand for appr accountsoximately for 74% approximately of the total 74%water of thedemand total waterin the basin. However, the runoff peak occurs from July to September, indicating that the precipitation, demand in the basin. However, the runoff peak occurs from July to September, indicating that the runoff, and agricultural water demand are not temporally concurrent (see Figure 4). The most precipitation, runoff, and agricultural water demand are not temporally concurrent (see Figure4). pronounced imbalance between water supply and demand is observed in May, resulting in the The most pronounced imbalance between water supply and demand is observed in May, resulting in highest probability of severe water shortage. the highest probability of severe water shortage. In this study, the water related information including water resources, economy and ecosystem In this study, the water related information including water resources, economy and ecosystem of of the Nen River basin are simplified into nodes (water resources projects, control sections, and the Nen River basin are simplified into nodes (water resources projects, control sections, and calculation calculation units) and lines (channels, rivers, etc.) [34], and the resulted water resources allocation units) and lines (channels, rivers, etc.) [34], and the resulted water resources allocation network chart is network chart is shown in Appendix Figure A1. The system consists of 47 calculation units, 33 shown in AppendixA Figure A1. The system consists of 47 calculation units, 33 reservoirs, 48 river and reservoirs, 48 river and channel nodes, 161 water supply channels, water release channels, and river channel nodes, 161 water supply channels, water release channels, and river channels. The water is channels. The water is supplied mainly for urban domestic use, rural domestic use, irrigation, supplied mainly for urban domestic use, rural domestic use, irrigation, industrial use, urban ecological industrial use, urban ecological environment, river, and lake ecological environment, etc. The water environment, river, and lake ecological environment, etc. The water sources include surface water, sources include surface water, groundwater, and recycled water from urban sewage. The calculation groundwater, and recycled water from urban sewage. The calculation is done on a monthly basis. is done on a monthly basis. The monthly runoff data for the period 1956–2013 and the monthly water The monthly runoff data for the period 1956–2013 and the monthly water demand data of the current demand data of the current year (2013) and the planning year (2020) are collected and analyzed. year (2013) and the planning year (2020) are collected and analyzed.

FigureFigure 3.3. TheThe averageaverage annualannual andand monthlymonthly precipitationprecipitation inin thethe NenNen RiverRiver basin.basin.

Water 2018,, 10,, 1289x FOR PEER REVIEW 88 ofof 1717

Figure 4. The intra-annual variation in the precipitationprecipitation and agricultural water demand in the Nen River basin. 4.2. Parameter Determination 4.2. Parameter Determination The model inputs include water demands, water sources, water resources projects, operational The model inputs include water demands, water sources, water resources projects, operational constraints, water resources allocation system, benefits of various water demands, etc. The runoff, constraints, water resources allocation system, benefits of various water demands, etc. The runoff, allowable exploitation quantity of groundwater, and parameters of water resources projects (mainly allowable exploitation quantity of groundwater, and parameters of water resources projects (mainly including water storage, division, pumping, and transfer projects), and channel parameters can be including water storage, division, pumping, and transfer projects), and channel parameters can be obtained by survey (see AppendixA Table A1). However, it is difficult to obtain the benefit information obtained by survey (see Appendix Table A1). However, it is difficult to obtain the benefit information of various water demands. In this study, the average benefit of each water demand can be obtained by of various water demands. In this study, the average benefit of each water demand can be obtained survey and statistics, and then the marginal utilities for different satisfaction levels of water demand are by survey and statistics, and then the marginal utilities for different satisfaction levels of water obtained by analysis of typical examples and expert discussion. The applicability of each typical benefit demand are obtained by analysis of typical examples and expert discussion. The applicability of each function for the water demand in different calculation units is rated by experts [35]. The benefits typical benefit function for the water demand in different calculation units is rated by experts [35]. of urban, river, and lake ecological environment are not available in statistics. Thus, the relative The benefits of urban, river, and lake ecological environment are not available in statistics. Thus, the importance or benefit per unit water use for different water users is rated by experts, and then the relative importance or benefit per unit water use for different water users is rated by experts, and average benefits and marginal utilities are determined. According to the precision requirements and then the average benefits and marginal utilities are determined. According to the precision the difficulty in determining the marginal utilities, the water demand for each water user is divided requirements and the difficulty in determining the marginal utilities, the water demand for each into three segments, and thus only three marginal utility values need to be determined in this study. water user is divided into three segments, and thus only three marginal utility values need to be 5.determined Results and in this Discussion study.

5. ResultsIn this and study, Discussion we only discuss the water supply and shortage quantity obtained by the original and improved methods in the planning year of 2020. In this study, we only discuss the water supply and shortage quantity obtained by the original 5.1.and Waterimproved Supply methods and Shortage in the planning year of 2020. Before model improvement, the water supply of NEJ, JQ, and BST (see AppendixA Table A2) 5.1. Water Supply and Shortage were 0.93 billion cubic meters, 6.62 billion cubic meters, and 24.18 billion cubic meters, respectively. AfterBefore model model improvement, improvement, their respectivethe water supply water supplies of NEJ, JQ, were and 0.93 BST billion (see Appendix cubic meters, Table 6.61 A2) billion were cubic0.93 billion meters, cubic and meters, 24.17 billion 6.62 billion cubic meters,cubic meters, with little and change.24.18 billion cubic meters, respectively. After modelBefore improvement, model improvement, their respective water water shortages supplies of NEJ,were JQ,0.93 and billion BST cubic regions meters, totaled 6.61 0 billion cubic meters, 0.22and billion24.17 billion cubic meters,cubic meters, and 0.33 with billion little cubic change. meters, respectively. After model improvement, theyBefore totaled model 0 billion improvement, cubic meters, water 0.23 billion shortages cubic of meters, NEJ, JQ, and and 0.33 BST billion regions cubic totaled meters, 0 billion respectively, cubic withmeters, little 0.22 change. billion Among cubic all meters, domains, and agricultural 0.33 billion water cubic shortage meters, is still respectively. the highest. After model improvement,Obviously, they water totaled supply 0 willbillion not cubic change meters, due to 0.23 the modelbillion improvement.cubic meters, and 0.33 billion cubic meters, respectively, with little change. Among all domains, agricultural water shortage is still the highest. Obviously, water supply will not change due to the model improvement.

Water 2018, 10, 1289 9 of 17

5.2. Changes in the Units with Water Shortage The water resource allocation in 47 calculation units of four primary water users during a 58-year period (1956–2013) in the Nen River basin was optimized (a total of 130,848 sets of data), and Table1 shows the changes in total number of units with water shortage before and after model improvement under different precipitation frequency conditions. Clearly, the number of units with water shortage increases in the improved model, all of which are related to agricultural water shortage.

Table 1. Changes in the units with agricultural water shortage before and after model improvement.

Precipitation Frequency Model Numerator * Denominator # Ratio (%) Before improvement 492 0.66 p = 50% 74448 After improvement 853 1.15 Before improvement 205 0.70 p = 75% 29328 After improvement 285 0.97 Before improvement 90 0.33 p = 90% 27072 After improvement 192 0.71 Before improvement 787 0.60 Total 130848 After improvement 1330 1.02 * Numerator = Number of years in a given precipitation frequency× Number of calculation units with water shortage in the specific years × Number of users × Number of months in one year. # Denominator = Total number of years (mentioned above) × Number of calculation units × Number of users × Number of months in one year.

5.3. Comparison of Water Shortage and Process The water demand for irrigation reaches a peak in the 40-day ponding period from early April to middle/late May in the Nen River basin; the water consumption for irrigation accounts for approximately 1/6 of the total annual water consumption. However, precipitation is low in April on average, accounting for only 3.3% of the total annual precipitation. The most pronounced imbalance between water supply and demand is observed in April, resulting in the highest probability of severe water shortage. Figure5 shows changes in water shortage and the difference in the value of water shortage of nine units from April to June before and after model improvement. The differences in water storage quantity in May obtained by original and improved methods (red bars) vary substantially and are predominantly negative; whereas that in April (green bars) and June (blue bars) are mostly positive. In addition, the water shortage quantity increases significantly in June under a precipitation frequency of 75% and 90%. It can be concluded that the improved method results in wide-mild water shortage from April to June, which contributes to solving the concentrated water shortage problem and the imbalance between inflow and water supply from April to June. Figures6 and7 show the comparison of monthly water shortage for two typical units (YH and WYESYH, the most water shortage areas in Nen River basin) obtained by original and improved methods at a precipitation frequency of 75% and 90%, respectively. Clearly, the improved method results in a wide-mild distribution of the water shortage for YH and WYESYH in May at a 75% precipitation frequency, indicating a significant improvement of the concentrated water shortage. The similar results are also obtained at a 90% precipitation frequency, and the water shortage becomes less severe in June and July. However, a more obvious decrease in the peak of water shortage is obtained at a precipitation frequency of 75% than 90%. Water 2018, 10, 1289 10 of 17 Water 2018, 10, x FOR PEER REVIEW 10 of 17

(a) p = 50%

(b) p = 75%

Figure 5. Cont.

Water 2018, 10, x FOR PEER REVIEW 11 of 17

Water 2018, 10, 1289 11 of 17 Water 2018, 10, x FOR PEER REVIEW 11 of 17

(c) p = 90%

Figure 5. The difference in water shortage quantity before and after improvement under different precipitation frequencies.

Figures 6 and 7 show the comparison of monthly water shortage for two typical units (YH and WYESYH, the most water shortage areas in Nen River basin) obtained by original and improved methods at a precipitation frequency of 75% and 90%, respectively. Clearly, the improved method results in a wide-mild distribution of the water shortage for YH and WYESYH in May at a 75% precipitation frequency, indicating a significant(c) p improvement= 90% of the concentrated water shortage. The similar results are also obtained at a 90% precipitation frequency, and the water shortage

Figure 5. The difference in water shortage quantity before and after improvement under different becomesFigure less 5. severeThe difference in June and in water July. shortageHowever, quantity a more before obvious and decrease after improvement in the peak under of water different shortage precipitationprecipitation frequencies. frequencies. is obtained at a precipitation frequency of 75% than 90%. Figures 6 and 7 show the comparison of monthly water shortage for two typical units (YH and WYESYH, the most water shortage areas in Nen River basin) obtained by original and improved methods at a precipitation frequency of 75% and 90%, respectively. Clearly, the improved method results in a wide-mild distribution of the water shortage for YH and WYESYH in May at a 75% precipitation frequency, indicating a significant improvement of the concentrated water shortage. The similar results are also obtained at a 90% precipitation frequency, and the water shortage becomes less severe in June and July. However, a more obvious decrease in the peak of water shortage is obtained at a precipitation frequency of 75% than 90%.

(a) YH (b) WYESYH Water 2018, 10, x FOR PEER REVIEW 12 of 17 FigureFigure 6. 6. ChangesChanges in in agricultural agricultural water water shortage shortage at at a a precipitation precipitation frequency frequency of 75%.

(a) YH (b) WYESYH

Figure 6. Changes in agricultural water shortage at a precipitation frequency of 75%.

(a) YH (b) WYESYH

FigureFigure 7. 7. ChangesChanges in in agricultural agricultural water water shortage shortage at at a a precipitation precipitation frequency of 90%.

5.4. Comparison of Water Shortage Range Figure 8 shows the comparison of the average annual water shortage range, minimum water shortage range (except 0), and maximum water shortage range obtained by the original and improved methods. It shows that despite the significant increase in the number of water shortage units, the water shortage range is reduced, which may give the false impression that the original method could obtain more satisfactory results. However, the high probability of concentrated water shortage in a specific period (i.e., the peak water demand period in irrigation) is largely ignored in the original operation, which may consequently lead to benefit loss. After the improvement, the problem of large water shortage range can be well-solved, making the optimization operation more acceptable.

Figure 8. Comparison of water shortage range of the Nen River basin.

Figures 9 and 10 show changes in water shortage range for the two units (YH and WYESYH) according to the monthly average water shortage and maximum water shortage. Obviously, after improvement, the water shortage range is obviously reduced, indicating that the water demand can be better satisfied by the improved method proposed in this study.

Water 2018, 10, x FOR PEER REVIEW 12 of 17

Water 2018, 10, 1289 12 of 17

(a) YH (b) WYESYH

5.4. Comparison of WaterFigure Shortage 7. Changes Range in agricultural water shortage at a precipitation frequency of 90%.

Figure5.4.8 shows Comparison the of comparison Water Shortage ofRange the average annual water shortage range, minimum water shortage range (except 0), and maximum water shortage range obtained by the original and improved Figure 8 shows the comparison of the average annual water shortage range, minimum water methods. Itshortage shows range that despite(except 0), the and significant maximum water increase shorta inge therange number obtainedof by waterthe original shortage and improved units, the water shortage rangemethods. is reduced, It shows whichthat despite may the give significant the false incr impressionease in the number that the of water original shortage method units, could the obtain more satisfactorywater shortage results. range However, is reduced, the which high may probability give the false of impression concentrated that the wateroriginal shortage method could in a specific period (i.e.,obtain the peak more watersatisfactory demand results. period However, in the irrigation) high probability is largely of concentrated ignored inwater the shortage original in operation,a specific period (i.e., the peak water demand period in irrigation) is largely ignored in the original which mayoperation, consequently which may lead consequently to benefit lead loss. to benefit After loss. the After improvement, the improvement, the the problem problem of of large large water shortage rangewater canshortage be well-solved, range can be well-solved, making the making optimization the optimization operation operation more more acceptable. acceptable.

Figure 8. Comparison of water shortage range of the Nen River basin. Figure 8. Comparison of water shortage range of the Nen River basin.

Figures9 andFigures 10 show 9 and changes 10 show inchanges water in shortage water shor rangetage range for the for two the unitstwo units (YH (YH and and WYESYH) WYESYH) according to the monthlyaccording average to the water monthly shortage average andwater maximum shortage and water maximum shortage. water shortage. Obviously, Obviously, after improvement after , the water shortageimprovement, range the is water obviously shortage reduced, range is obviousl indicatingy reduced, that theindicating water th demandat the water can demand be better can satisfied be better satisfied by the improved method proposed in this study. by the improvedWater 2018, method 10, x FOR PEER proposed REVIEW in this study. 13 of 17

Figure 9.FigureThe 9. monthly The monthly average average water water shorta shortagege range range of two of typical two typical units. units.

Figure 10. The maximum water shortage range of two typical units.

6. Conclusions In this study, the “wide-mild water shortage” perspective is proposed based on the law of diminishing marginal utility in order to solve the problem of concentrated water shortage. We argue that the risk of concentrated water shortage should be shared across a wider range of users and periods, so that it would have less severe impacts on each individual user in each period. It is necessary to redefine the weight of variables in the water supply objective function. Assuming that the weight is a continuous nonlinear function, and can be converted into a piecewise linear weight, it is possible to reduce or even eliminate the concentrated water shortage, thus making the allocation of water resources more reasonable.

Water 2018, 10, x FOR PEER REVIEW 13 of 17

Water 2018, 10, 1289 13 of 17 Figure 9. The monthly average water shortage range of two typical units.

FigureFigure 10. The 10. The maximum maximum water water shortageshortage range range of oftwo two typical typical units. units.

6. Conclusions6. Conclusions In thisIn study, this study, the “wide-mildthe “wide-mild water water shortage” shortage” perspective perspective is proposed is proposed based based on the on law the of law of diminishingdiminishing marginal marginal utility utility in order in order to to solve solve the the problem of of concentrated concentrated water water shortage. shortage. We argue We argue that thethat risk the of concentratedrisk of concentrated water wa shortageter shortage should should be sharedbe shared across across a wider a wider range range of of users users and and periods, periods, so that it would have less severe impacts on each individual user in each period. It is so that it would have less severe impacts on each individual user in each period. It is necessary to necessary to redefine the weight of variables in the water supply objective function. Assuming that redefinethe the weight weight is a ofcontinuous variables nonlinear in the water function, supply and can objective be converted function. into a Assumingpiecewise linear that weight, the weight is a continuousit is possible nonlinear to reduce function, or even eliminate and can the be conc convertedentrated intowater a shortage, piecewise thus linear making weight, the allocation it is possible to reduceof water or even resources eliminate more reasonable. the concentrated water shortage, thus making the allocation of water resources more reasonable. The case study was carried out in the Nen River Basin, northeast China and shows that the improved method results in a significant increase in water shortage units, but a significant reduction in the water shortage range. As a consequence, the water shortage is more uniformly distributed from April to June, which contributes to solving the concentrated water shortage problem and the mismatch between inflow and water supply from April to June. However, it should be noted that the extent to which the wide-mild water shortage can be realized depends, not only on the marginal utility of water demand, but also on the available water supply and the capacity of water supply projects. In spite of this, the improved method enables the water to be supplied more uniformly at the appropriate time, which contributes to improving the allocation efficacy of water resources and helps decision-makers better deal with the problem of concentrated water shortages.

Author Contributions: M.Y. designed the study. H.H. wrote the manuscript. A.C. and J.L. performed the data analysis. X.X. and Z.Y. reviewed and approved the manuscript. Funding: The study was financially supported by the National Science Foundation for Distinguished Young Scholars of China, grant No. 51709274, the Special Funds for Scientific Research of Public Welfare in the Ministry of Water Resources, grant No. 201501013, the Independent Research Fund of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Grant No. 2016TS03, as well as The National Science and Technology Major Project of Water Pollution Control and Prevention of China, grant No. 2012ZX07201-006 and 2008ZX07207-006. Conflicts of Interest: The authors declare no conflicts of interest. Water 2018, 10, 1289 14 of 17

Appendix A

Table A1. Engineering parameters.

Flood Control Storage Capacity Dead Capacity Name Catchment Area km2 Capacity Million m3 Million m3 Million m3 BR1 66,382 8610 6456.3 487.5 BR2 3745 995 740 310 BR3 683 117 74 7 BR4 1660 281 162 14 BR5 15,112 260 177 23 BR6 2241 298 49 5 BR7 13,500 150 84 14 BR8 / 450 315 110 BR9 7780 1253 1067 34 BR10 548 235 207 63.5 BR11 342 51 40.6 4 BR12 60 209 390.3 34.5 BR13 35 110 107 10 BR14 5300 405 405 105 PR1 8250 1685 1483 320 PR2 10,720 989 859 242 PR3 24,384 / 350 50 PR4 32,229 3331 3507 1007 PR5 19,487 3113 2554 556 PR6 16,137 3508 2926 248 PR7 1990 187 143 32 PR8 2050 754 754 95 PR9 2072 307 132 45 PR10 438 70 31 1 PR11 853 96 43.64 6.33 PR12 1773.6 240 220 104 PR13 2444 450 380 132 PR14 8200 3100 3100 1455 PR15 12,426 1640 1486 198 PR16 1790 538.2 352 19.4 PR17 4206 76 45 8 PR18 7610 574 498 31 PR19 9050 360 132 86 ”/” indicates that the information is not known.

Table A2. Basic information about water resource zones

Water Resource Zone Catchment Area km2 Water Use (in 2013) Billion m3 Sub Units NEJ 67,775 0.36 GGHRUP/GH/GGHR TONEJR NMH/NMEH/NER JQ 99,678 4.82 TOTH/ALH/YH/YLH/TH TO JQ CEH/THE/HLH/JQ TO BST/BST BST 131,049 7.43 TO SHJ/WYESYH/AZXH/ZLXH Water 2018, 10, x FOR PEER REVIEW 15 of 17

PR16 1790 538.2 352 19.4 PR17 4206 76 45 8 PR18 7610 574 498 31 PR19 9050 360 132 86 ”/” indicates that the information is not known.

Table A2. Basic information about water resource zones

Water Resource Catchment Water Use (in Sub Units Zone Area km2 2013) Billion m3 NEJ 67,775 0.36 GGHRUP/GH/GGHR TONEJR NMH/NMEH/NER JQ 99,678 4.82 TOTH/ALH/YH/YLH/TH TO JQ Water 2018, 10, 1289 CEH/THE/HLH/JQ TO BST/BST TO 15 of 17 BST 131,049 7.43 SHJ/WYESYH/AZXH/ZLXH

FigureFigure A1. A1.Water Water resources resources allocation allocation network chart chart of of Nen Nen River River basin. basin.

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