Research Article Monthly Optimal Reservoirs Operation for...

Research ArticleMonthly Optimal Reservoirs Operation for Multicrop DeficitIrrigation under Fuzzy Stochastic Uncertainties

Liudong Zhang12 Ping Guo1 Shiqi Fang1 and Mo Li1

1 Centre for Agricultural Water Research in China China Agricultural University Number 17 Qinghua East RoadHaidian Beijing 100083 China

2 College of Water Conservancy Yunnan Agricultural University Kunming 650201 China

Correspondence should be addressed to Ping Guo guopcaueducn

Received 11 December 2013 Accepted 6 February 2014 Published 18 March 2014

Academic Editor Y P Li

Copyright copy 2014 Liudong Zhang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An uncertain monthly reservoirs operation and multicrop deficit irrigation model was proposed under conjunctive use ofunderground and surface water for water resources optimization management The objective is to maximize the total crop yield ofthe entire irrigation districts Meanwhile ecological water remained for the downstream demand Because of the shortage of waterresources the monthly crop water production function was adopted for multiperiod deficit irrigation management The modelreflects the characteristics of water resources repetitive transformation in typical inland rivers irrigation systemThemodel was usedas an example for water resources optimizationmanagement in Shiyang River Basin China Uncertainties in reservoirmanagementshown as fuzzy probability were treated through chance-constraint parameter for decision makers Necessity of dominance (ND)was used to analyse the advantages of the method The optimization results including reservoirs real-time operation policy deficitirrigation management and the available water resource allocation could be used to provide decision support for local irrigationmanagement Besides the strategies obtained could help with the risk analysis of reservoirs operation stochastically

1 Introduction

Shortage of water resources is a critical constraint for agricul-tural production in arid and semiarid regions Because of thechanging environment agricultural water supply is facing theunpredictable challenge Ashofteh et al estimated that waterdemand volume in the future (2026ndash2039) would increase16 while the average long-term annual volume of runoffwould decrease 07 [1] To meet the challenge reservoirsoperation (RO) is important and particularly difficult [2]

Reservoirs operation plays an important role for arid andsemiarid areas especially when the competition betweenagricultural and ecological water demand is intense Agricul-tural water is critical for food productivity while ecologicalwater is important to regional sustainable developmentReservoirs operation problems concerned with maximiza-tion of irrigation benefits have been emphasized by manyresearchers since the 1970s [3] Typical reservoirs opera-tion method was dynamic programming [3ndash5] Vedula andMujumdar [6] integrated the reservoir release and irrigation

allocation by using dynamic programming but they tookrainfall and potential evapotranspiration as deterministicinputs Furthermore Vedula and Kumar [7] overcome theshortcomings considering the rainfall and inflow as stochas-tic parameters

But when the irrigation water is limited to ecologicalwater the situation is more complicated Unfortunately theprevious studies mainly focused on irrigation water man-agement Mujumdar and Ramesh [8] developed a real-timedynamic programming model for optimal crops water allo-cation and reservoir release but the framework of the modelis deterministic Moradi-Jalal et al [2] considered annualcrops irrigation water for monthly reservoirs operation andoptimal crop pattern But their research did not consider therelationship between water and crops

Optimal water allocation for irrigation district is animportant issue which has received considerable attentionin previous researches Evers et al [9] integrated hydrologicmodel (PRMS) and crop growth simulation model (EPIC)

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 105391 11 pageshttpdxdoiorg1011552014105391

2 Journal of Applied Mathematics

for reservoir management options Dos Santos Teixeira [5]developed a forward dynamic programming (FDP) modelfor real-time reservoirs operation of irrigation districtsGeorgiou and Papamichail [10] aimed to determine optimalirrigation allocation and cropping pattern for multicropsPrasad et al 2012 [11] developed a real-time reservoir releasedecision Among these studies crop yield was critical data fordecision objective There were two ways to determine cropyield (1) estimated by average production per unit area or percube water [12] and (2) calculated through water productionfunction [6 11 13]

However in real-world problem the source of watercould be groundwater and surface (reservoir) water Thenthe problem becomes more complicated to identify the mostefficient strategies for the whole system Reichard [14] applieda stochastic simulation optimization model for Santa Clara-Callcguas Basin surface water-groundwater managementVedula et al [13] optimized multicrop water allocation fora reservoir-canal-aquifer system considering intraseasonalperiod As discussed above previous researchers just con-sidered two issues among reservoirs operation irrigationdistrict ecological water requirement and surface water-groundwater conjunctive use To approach the real situationan integrated water allocation model related to these issueswas considered

Another difficulty in the whole systemwas the uncertain-ties including natural factors and human triggers Uncertain-ties in reservoir and irrigation system caused by river inflowsand crops requirement had been discussed from differentaspects [1 15ndash18] Karamouz and Vasiliadis [15] forecastedthe stream flows by Bayesian stochastic dynamic programing(BSDP) with different conditional probabilities Teegavarapuand Simonovic [16] handled the imprecise loss function forshort time reservoirs operation under fuzzy environmentSeifi and Hipel [19] presented a stochastic model to deal withthe stochasticity of inflows under multiple inflow scenariosChang et al [18] combined grey system with fuzzy stochasticdynamic programming Although uncertainty was consid-ered the reliability of satisfying (or risk of violating) cannotbe reflectedThen Azaiez et al [20] applied chance constraintto explain the potential benefits of groundwater saving

But these methods cannot handle dual uncertainties pre-sented as linkage between fuzzy and probability distributionTo handle this problem Guo and Huang [21] developed atwo-stage fuzzy chance-constraint programming for waterresources management In a similar fashion if adequatesamples of historic reservoir inflow date were obtained thendifferent reservoir inflow levels distributed as randomnessdistribution If the total reservoir release water cannot bemetchance-constraint programming can be applied to obtainthe expected net benefits [22] Nevertheless in the progressof RO alternations were made by decision makers withdifferent goals probabilities of different inflow levels couldnot exactly reflect randomness of inflows alone [23] Thusthe probabilities may own the fuzzy information based onthe decision makerrsquos judgment leading to fuzzy stochasticcharacteristic

In this paper an optimization model for deficient irri-gation water allocation under uncertainties was developed

The model is combined with ground-surface water joint-usemodel monthly reservoirs operation model and multicropirrigation model Fuzzy stochastic characteristic was repre-sented as fuzzy probability reflecting the risk of reservoirrelease water and necessity of dominance (ND) was consid-ered to illustrate it Reservoir water and groundwater wereintegrated as interactive resources for both irrigation districtsand ecological water Irrigation water was calculated throughJensen model under deficit irrigation condition

2 Model Formulation

The available water resources mainly come from the moun-tainous runoff in typical inland area of China According tothe characters of water resources of inland river system asshown in Figure 1 the watershed can be divided into twoparts runoff formation and disappear areas Accumulatedwater flows into mountainous river for water utilization inthe plain region Several reservoirs located mainly in themountain river and some of the water was used for irrigationbut the rest leaked in the piedmont plain as groundwaterrecharge Groundwater was pumped for agricultural irriga-tion once more and the surplus water needs to meet theecological water demand of downstream meanwhile Thewater resources transformation process leads to efficientwater development pattern and the water recycle is improvedHowever excessive development scale and unreasonable allo-cation of water resources can cause the overextraction ofgroundwater and deterioration of the environment

At present maximum utilization of water resources isvery important because of the shortage of water resourcesTo optimize water resources considering exploitation andreplenishment balance of groundwater for upstream anddownstream users the current study was used to simulatemultiperiod reservoirs operation and multicrop irrigationoptimization The objective is to maximize the total cropyields of the entire irrigation districts

This model can be written as follows and the meaningsof parameters are listed in the Nomenclature section All ofvariables are nonnegative

Max119865 =

119870

sum

119896=1

119873

sum

119899=1

[

[

119884119899

max

119879

prod

119905=1

(ET119896119899119905

ET119899119905max

)

120582119899

119905

times 119860119896119899]

]

+

119868

sum

119894=1

119873

sum

119899=1

[

[

119884119899

max

119879

prod

119905=1

(ET119894119899119905

ET119899119905max

)

120582119899

119905

times 119860119894119899]

]

(1)

The expected crop yields are estimated by deficit irrigationcrop water production function The water production func-tion can be used to tackle the irrigation management Wedivide the crop planting stages into monthly periods Thecroprsquos monthly water consumption is the decision variablein the model The deficit irrigation Jensen model [24] isintroduced to reflect crop yields linked to sensitivity of watershortage Sufficient water requirement of crop can lead tomaximum crop yields and water deficit in different stagesmay reduce the actual crop production in different degrees

Journal of Applied Mathematics 3

Solid precipitationLiquid precipitation

in the mountainous region

Mountain river

Water utilizationin the plain region

River waterirrigation area

Well waterirrigation area

Water-resistanthills

Water accumulation

Figure 1 The relationship of water resources transformation in arid inland area

Subject to Avoiding large water deficit during the periodof crop water consumption and not exceeding maximumevapotranspiration constraints

06ET119899119905 max le ET119896119899

119905le ET119899119905 max

06ET119899119905 max le ET119894119899

119905le ET119899119905 max

(2)

Considering the balance equation of the reservoirs located inthe mountain pass of the rivers

119881119896

119905= 119881119896

119905minus1+ 119868119896

119905minus 119877119896

119905minus 119874119896

119905minus 119864119896

119905 (3)

Reservoirs overflow constraint is expressed as follows In eachplanning time period accumulatedwater in themountainousregion flows into the reservoirs Part of the water is utilizedfor the irrigation district If the available storage of reservoirat the end of time period exceeds total effective capacitythe overflow happens The 0-1 type integer variables areintroduced into the model to decide the overflow of thereservoir 1 for overflow from reservoir occurring and 0 fornot occurring119872 is a fixed bigger number

119887119896

119905le119881119896

119905

119862119896

119874119896

119905minus 119887119896

119905times119872 le 0

(4)

Reservoir capacity constraint the available storage of reser-voir is less than the effective capacity of reservoir

119881119896

119905le 119862119896 (5)

In real-world practical problem we allow proper violationsThe constraint is considered as uncertain inputs A risk ofviolating may lead to higher benefit and more reasonablemanagement strategy It is often difficult to define the preciseboundary of the constraint due to the fuzzy decision Withreference to Guo and Huang [21] the constraint is shown bythe following

Pr (119881119896119905le 119862119896) ge 120575119896 (6)

Then we have

1119881119896

119905le 119862119896(119901119896)

(7)

where 119862119896(119901119896)

= 119865minus1

119896(119901119896) is given the cumulative distribution

function of 119862119896 and the probability of violating constraint119896(119901119896) 120575119896 is fuzzy tolerance measures (0 le 120575

119896le 1 119896 =

1 2 119870) and 120575119896 = 1 minus 119901119896

Assume that the fuzzy coefficients are considered trape-zoidal fuzzy sets Thus 1 = (1 1 1) = (08 10 12)indicating that the risk level of the reservoir storage isrepresented as fuzzy uncertainty When the lower bound ofthe coefficient is calculated reservoir storage water was at ahigh risk level

119901119896= (119901119896 1199011198960 119901119896) = (1 minus 120575

119896

1 minus 1205751198960 1 minus 120575

119896)

119901119896= (1 minus 095 1 minus 08 1 minus 06) = (005 02 04)

(8)

It meant that the risk level of the reservoir allowable releasewater distributed as fuzzy stochastic uncertainty Utilizing 120572-cut technique fuzzy parameter under each 120572-cut level canbe included within a closed interval And it is expressed asnecessity of dominance (ND) solutions Consider

((1 minus 120572) 1 + 120572)119881119896

119905le 119862119896((1minus120572)119901

119896

+1205721199011198960)

(9)

(08 + 02120572)119881119896

119905le 119862119896((1minus120572)005+12057202)

(10)

In order to ensure sustainable utilization of reservoir theavailable storage of reservoir at the ending time period isrequested not to be less than the beginning time period

119881119896

119899minus 119881119896

0ge 0 119896 = 1 2 119870 (11)

Water consumption in the irrigation district consists of agri-cultural water and nonagricultural water including domesticand industrial water According to the irrigation district pop-ulation livestock and industrial production nonagriculturalwater including water conveyance loss can be estimated


Multiplying water withdrawal for agriculture by utilizationcoefficient of irrigation water is irrigation water requirement

119877119896

119905= AW119896

119905+ DW119896

119905

IR119896119905= AW119896

119905sdot 120578119896

(12)

By combining the rainfall with water consumption of crops inriver irrigation district during growing stages net irrigationwater requirement is calculated as follows

IR119896119905=

119873

sum

119899=1

(119864119879119896119899

119905minus 119875119896119899

119905) times 119860119896119899 (13)

Subtracting seepage and evaporation loss reservoirs overflowflows into the downstream According to the measuring datathe river water loss is large in river irrigation district

119874119896

119905= RL119896119905+ RO119896119905 (14)

The available water resource is utilized firstly in river irri-gation district and water conveyance loss is unavailablesuch as seepage and evaporation Meanwhile seepage losssupplies groundwater to well irrigation district Recharge ofgroundwater can be written as

GO119905=

119870

sum

119896=1

(120572119864119896

119905+ 120573RL119896

119905+ 120574AW119896

119905(1 minus 120578

119896)) + GI

119905 (15)

Groundwater exploitation is not taken in river irrigationdistrict The recharge of groundwater comes from reservoirseepage river seepage irrigation seepage and the lateralgroundwater inflow frommountains 120572 120573 120574 are groundwaterrecharge coefficients respectively In well irrigation districtgroundwater can be regarded as an entire undergroundreservoir Through conjunctive use of underground andsurface water the underground water reservoir is mainlysupplied from surface water Groundwater quantity balancecan be shown as follows

VG119905= VG

119905minus1+ GO

119905

+ 120596

119870

sum

119896=1

RO119896119905+

119868

sum

119894=1

[1205741015840AW119894119905(1 minus 120578

119894)]

minus

119868

sum

119894=1

119877119894

119905minus GWD

119905minus GWO

119905

(16)

120596 1205741015840 are groundwater recharge coefficients of rivers andagricultural irrigation in well irrigation district

Similarly the groundwater is pumped for agriculturaland nonagricultural purpose in well irrigation district Netirrigation water requirement is calculated as follows

119877119894

119905= AW119894

119905+ DW119894

119905

IR119894119905= AW119894

119905sdot 120578119894

IR119894119905=

119873

sum

119899=1

(ET119894119899119905minus 119875119894119899

119905) times 119860119894119899

(17)

In order to ensure the reasonable exploitation of groundwaterannual exploitation does not exceed allowable amount ofgroundwater

119877119894

119905le 119877max (18)

After the allocation of water resources in plain region the restof the water consists of two parts the surplus water of riversand groundwater Excess underground water overflows fordownstream as springs The surplus water of rivers is writtenas follows

RO1015840119905= (1 minus 120576)

119870

sum

119896=1

RO119896119905 (19)

120576 is river loss coefficient in well irrigation district Thedemand of downstream should be satisfied So the surpluswater should be more than the minimum ecological waterdemand

RO1015840119905+ GWD

119905ge 119882min (20)

The model reflects the characteristics of water resourcesrepetitive transformation in typical inland rivers irrigationsystem Multiperiod reservoirs operation and multicrop irri-gation optimization is reflected under water allocation andconjunctive use of ground and surface water

3 Case Study

Shiyang River Basin (101∘411015840 sim104∘161015840E 36∘291015840 sim39∘271015840N)is located in arid and semiarid region in Gansu provincenorthwest of China which is a typical inland river basin inHexi Corridor with an annual precipitation of 150ndash300mmand potential evaporation of 1200ndash2000mmWater resourcesin the basin are scarce and have been overused so that a seriesof ecological environment problems occurred As shown inFigure 2 the watershed can be divided into three separateriver systems according to hydrogeological units DajingRiver six rivers and Xida River In the six rivers systemthere are twomain basins Wuwei andMinqin basins WuweiBasin was selected as the study area to illustrate the monthlyreservoirs operation for multicrop irrigation optimizationmodel under dual uncertainties

The major water supply of the study area originatesfrom the southern part of the Qilian Mountain and thereare six tributaries of the Shiyang River that is the GulangHuangyang Zamu Jinta Xiying and Donghe rivers Somereservoirs are built to regulate water resources in the moun-tain pass of the rivers except Zamu River There are sixriver irrigation districts corresponding to the six riversrespectively Gulang (GL) Huangyang (HY) Zamu (ZM)Jinta (JT) Xiying (XY) and Donghe (DH) irrigation districtswhich are located in the south of Wuwei Basin The availablewater resource is utilized in river irrigation districts and flowsinto the northern well irrigation districts There are fourwell irrigation districts in the north of the basin Qingyuan(QY) Jingyang (JY) Yongchang (YC) and Qinghe (QH)irrigation districts The surplus water resources flow into thedownstream Minqin Basin


River City nameIrrigation district

Hydrologic stationboundaryIrrigation area

Minqin

Jinchuan

Wuwei

Gulang

Yonchang

Dajing

Rive

r

Huangyan

g Rive

r

Zamu Rive

rJinta RiverXiying iver

Gulang River

Dongda River

Xida Rive

r

GLHY

QYZM

JT

YC JYXY

QHDH

N

(km)0 10 20 30

Figure 2 The study area

The dynamic transformation of water resources in riverbasin complicates the water cycle processThe extensive agri-culture development and improperwater resources allocationlead to environmental deterioration The research of ShiyangRiver Basin water resources transformation was significantto provide decision support for local irrigation managementsand agricultural producers

Because of lower precipitation and higher evaporationrunoff generation is different in Wuwei Basin The basicwater demand of crop growth may not be met Mountainousrunoff is the main source for the crop irrigation Figure 3shows the comparison of precipitation and reference cropevapotranspiration (ET

0) ET0is only related to local meteo-

rological conditions ET0and crop coefficientswill be adapted

to estimate crop water requirement which is the foundationof irrigation water management Long-term average annualprecipitation is 176mm while long-term average annual ET

0

is 1008mm however crop water requirement is larger thanET0As mentioned above mountainous runoff allocation is a

critical factor for entire irrigation districts benefits Runoffstatistical study is used to analyze the data from 1955 to 2009and the six riversmonthly runoff of 2000 (119875 = 50) as shownin Table 1 is selected as the model inflow data

The reservoirs total effective storages in the mountainouspass of rivers are 14520 56440 16260 24000 and 80000times 103m3 corresponding to Gulang Huangyang Jinta Xiy-ing and Donghe The annual losses of the reservoirs areapproximately 900 3300 1000 3300 and 2400 times 103m3respectively In the model the reservoir is not built on Zamuriver when 119896 = 4119881119896

119905and119864119896

119905is equal to zero Beside irrigation

district water withdrawal the mountainous runoff flows into

1 2 3 4 5 6 7 8 9 10 11 1220601001401802200

20406080

100120

1 2 3 4 5 6 7 8 9 10 11 12

Month

Prec

ipita

tion

(mm

)

PrecipitationET0

ET0

(mm

)

Figure 3The comparison of precipitation and ET0of the study area

the downstream river The balance equation is rewritten asfollows

119868119896

119905= 119877119896

119905+ 119874119896

119905 (21)

Irrigation district water withdrawal covers agricultural andnonagricultural purposes and crop irrigation water is themajority of the agricultural water The main planting cropsin these irrigation districts are wheat maize potato flax andmelon Growing periods of main crops last from March toSeptember Irrigated crop area of each irrigation district isshown in Table 2

The present irrigation method is a traditional modelthat manages surface irrigation to meet the high yield cropirrigation water requirement which leads to unreasonablewater allocation between upstream and downstream usersBecause of the scarcity of water resources deficit irrigationis necessary in the study area The Jensen model has been


Table 1 The monthly runoff of the six rivers in 2000 (119875 = 50)

River index Monthly runoff (m3s)Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Gulang 052 088 097 181 309 225 312 506 505 274 183 113Huangyang 146 129 190 284 305 505 246 995 988 297 186 190Zamu 093 073 199 459 805 1460 883 1710 1620 764 316 208Jinta 054 038 061 107 299 1330 784 1060 689 246 090 101Xiying 138 108 257 494 1170 1930 1810 2500 2120 725 327 279Donghe 378 380 447 608 1032 1741 1541 1855 1430 693 503 386

Table 2 The irrigated crop area of each irrigation district

Irrigation district index Irrigated crop area (hm2)Wheat Maize Potato Flax Melon

GL 424241 128871 152388 20695 10347HY 804764 244463 289073 39257 19628ZM 680469 206706 244426 33194 16597JT 471445 143211 169344 22997 11499XY 1352699 410909 485892 65985 32993DH 1230328 373736 441936 60016 30008QY 535788 162756 192456 26136 13068JY 368918 112066 132516 17996 8998YC 503917 153075 181008 24581 12291QH 603739 183397 216864 29451 14725

more widely adopted for deficit irrigation water productionfunctionwhich is a crop production stage water consumptionmodel and can be used to calculate staged irrigation waterCrop yield calculation method by Jensen model has thefollowing expression

119884 = 119884max

119899

prod

119894=1

(ET119894

ET119898119894

)

120582119894

(22)

120582119894is sensitivity index of crop to water deficit ET

119894and

ET119898119894

are staged actual evapotranspiration and maximumevapotranspiration 119884max is maximum crop yield under suf-ficient irrigation water condition and it can be obtained bywater production function in the whole stages The previousresearchers have provided a large number of experimentaldata and fitted water production function to estimate 119884maxwhich usually is shown as the following

119884 = 1198861+ 1198871ET + 119888

1ET2 (23)

1198861 1198871 1198881are empirical coefficients fixed by experimental data

The calculation results of119884max are 809782 1121943 2697829226816 and 305746 kghm2 corresponding to themain crop

For reservoirs operation and irrigation managementcrop sensitivity to water deficit in irrigation interval is moreeffective than in growth stage Kipkorir and Raes [25] haveapplied Jensen model in irrigation interval Tsakiris [26]provides a method of estimating crop sensitivity to waterdeficiency at given time intervals However relevant study

is not reported in the study area The Jensen model is fittedin monthly interval in keeping up with the optimizationmodel as shown in Table 3 Effective rainfall in example yearis 57mm 0mm 62mm 93mm 333mm and 186mmcorresponding to the period from April to September

Annual nonagricultural water consumption in the modelis estimated through the irrigation district population live-stock and industrial production which are shown in Table 4

Parameter determination is important for the optimiza-tion results According to Comprehensive Planning of ShiyangRiver Basin (2007) utilization coefficient of irrigationwater ofriver irrigation area is 065 in Gulang irrigation district 058in Xiying irrigation district and 054 in Huangyang ZamuJinta and Donghe irrigation districts utilization coefficientof irrigation water of well irrigation area is 06

The water of each tributary flows together into ShiyangRiver minus evaporation and leakage loss River flow innorthern Wuwei Basin is low due to severe leakage Accord-ing to the investigation we take 0667 for the value of leakagerate

Complement between the surface water and under-ground water is obvious in the basin

Groundwater is not pumped in southern Wuwei BasinConsidering the underground aquifer as a whole ground-water reservoir sources of its supply are reservoir seepageriver seepage irrigation seepage and the lateral groundwaterinflow Under low rainfall and deep groundwater levelgroundwater is scarcely recharged by the rainfall accordingto Ma et al [27] Rainfall seepage is ignored in the model


Table 3 The fitted value of ET119898(mm) and 120582

Crop index MonthMar Apr May Jun Jul Aug Sep

Wheat ET119898

2354 11929 17603 21921 15323 000 000120582 006 007 015 008 001 000 000

Maize ET119898

000 2782 7076 11255 17192 12906 2303120582 000 004 012 020 031 022 003

Potato ET119898

000 1687 6156 12922 11891 10147 1132120582 000 001 003 007 006 005 000

Flax ET119898

000 1849 10502 15006 14619 7790 000120582 000 012 027 036 035 009 000

Melon ET119898

000 000 5185 6284 12585 6848 2165120582 000 000 006 007 015 008 001

Table 4 Nonagricultural water of irrigation districts (106 m3)

Irrigation district index GL HY ZM JT XY DH QY JY YC QHNonagricultural water 774 930 1671 839 620 153 449 486 707 303

The quantity of groundwater supply is calculated byrecharged coefficients Qu et al [28] estimate reservoir see-page and river seepage recharged coefficients as 085 andirrigation seepage as 08

Calculation results of Yao et al [29] show that thelateral groundwater inflow is 136 times 10

6m3 and the lateralgroundwater outflow is 254 times 106m3

To ensure reasonable water allocation between upstreamand downstream users according to local planning theavailable groundwater mining is controlled below 319 times

106m3 and minimum downstream water requirement is not

less than 218 times 106m3 under local planning requirements

4 Results Analysis

41 Multicrop Deficit Irrigation Management Deficit irriga-tion could result in the lower crop yields and reduce the netbenefits of the irrigation district However it is unavoidablebecause of water resources shortage The relation betweensensitivity of water deficit and crop water use efficiency wasstudied through the use of Jensen model

According to the calculation results of optimal decision-making model deficit irrigation proportions of the five cropsin the whole growth stage in river irrigation district are 061084 071 060 and 068 respectively and in well irrigationdistrict the values are 076 100 100 094 and 060 Itindicates that deficit irrigation should be adopted primarilyin river irrigation district

The optimization model can reach crop water require-ment in each irrigation interval As shown in Figure 4 thepriority deficit irrigation crops are melon and wheat inwell irrigation district ET is monthly optimal crop waterrequirements and ET

119898is monthly potential crop water

requirements The results can provide multiperiod deficitirrigation management for the agricultural producers

0005000

10000150002000025000

3 4 5 6 7 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 5 6 7 8 9Wheat Maize Patato Flax Melon

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m

(a) Optimal crop water requirements in river irrigation

Wheat Maize Patato Flax Melon

0005000

10000150002000025000

3 4 5 56 7 4 6 7 8 9 4 5 6 7 9 4 5 6 7 88 5 6 7 8 9

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m

(b) Optimal crop water requirements in well irrigation

Figure 4 Results of crop deficit irrigation management

42 Monthly Reservoirs Operation In the monthly multicropirrigation system water resource is mainly obtained from themountain river Reservoirs regulation is significant for thewater utilization Reservoirs release and overflow manage-ment can be determined by the model The nonagriculturalwater demand of each irrigation district among time periodsis even but the primary part of agriculture water demanddepends on the crop area and growing periods of crop


01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)

Figure 5 Real-time multireservoirs release

0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)

Figure 6 Real-time reservoirs overflow

Total annual runoff of the six rivers is 11459 times 106m3 andreal-time multireservoirs release can be shown in Figure 5Agricultural water consumption takes most rates in waterconsumption structureThemaximumwater demand occursin June Water withdrawal is closely related to irrigation areaof each reservoir water-supply zone The total annual waterwithdrawal from reservoir is 66574 times 106m3

To avoid the occurrence of shortage and overflow atthe same time period overflow would not happen unlessthe reservoir is full storage Figure 6 shows the real-timereservoirs overflow Overflow occurred in the period fromAugust to October when irrigation water demand decreasedand monthly runoff was still high The annual reservoirsoverflow is 15451 times 106m3 It indicates that mountainousrunoff is utilized by river irrigation district The maximumannual overflow is from Xiying Reservoir with 12424 times

106m3 and the minimum overflow is from HuangyangReservoir with 021 times 106m3

43 Recharge Relationship between Surface Water and Under-ground Water Wuwei Basin is a typical inland river basinconjunctive use of surface water and underground water isnecessary to the arid irrigation area To be more efficientin the groundwater management managers should try torealize recharge relationship between surface water andundergroundwater anddetermine the available undergroundwater

In the model underground water is pumped in thewell irrigation district and supply from reservoir seepage

020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month

Groundwater recharge

Gro

undw

ater

rech

arge

(103m

3)

times103

Figure 7 Monthly groundwater recharge in the basin

05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month

QingyuanJingyang Yongchang

Qinghe

Gro

undw

ater

use

(103m

3)

Figure 8 Monthly groundwater use in well irrigation district

river seepage irrigation seepage and the lateral groundwaterinflow in the southern basin Figure 7 shows the monthlygroundwater recharge of the basin

As a result of conjunctive use of surface water andgroundwater optimization model the groundwater use ineach well irrigation district can be listed in Figure 8 in orderto provide decision support for local irrigation managementThe major groundwater exploitation is from April to AugustGroundwater quantity balance is considered in the modelGroundwater exploitation is less than groundwater rechargeand annual exploitation does not exceed allowable amount ofgroundwater

44 Fuzzy Stochastic Uncertainty In real-world practicalproblem reservoir capacity is not considered as strictlydeterminate numbers and the failure of the constraint isallowed So the chance-constraint programming is consid-ered to analyse the uncertain constraint of the failure of thelimitation of the total reservoir capacity Proper violationunder different risks is studied in the model which providesselectable management strategies for the decision makersNecessity of dominance (ND) solutions under three 120572-cutlevels are calculated

As shown in Figure 9 the lower120572-cut level leads to highertotal crop yield It means that reservoir capacity constraint isloose under lower 120572-cut level and higher violation risk leadsto larger system benefit

In this model the right hand of the reservoir capacityconstraint is allowed to change under the given probabilities


1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075

Figure 9 Total crop yield of the entire irrigation districts underdifferent 120572-cut levels

038

039

040

041

042

0606106206306406506606706806907

Reservoirs overflowReservoirs withdrawal

120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05

Figure 10 Reservoirs withdrawal and overflow under different 120572-cut levels

As shown in Figure 10 water supply from reservoirs is 0668times 109m3 (120572 = 02) and the corresponding figure is 0660times 109m3 (120572 = 075) while water overflow from reservoirsexperiences the opposite change

5 Conclusions

In this study a multicrop irrigation water resources opti-mization model based on Jensen water production functionmodel for real-time reservoirs operation is proposed Theobjective of the model is to achieve the maximum benefit ofirrigation areas considering water stress level of crops andthe dynamic characteristic in the irrigation water resourcesoptimal allocation system The proposed model can pro-vide decision makers with different irrigation water optimalstaged allocation schedules (conjunctive of surface water andgroundwater) of upstream and downstream and agriculturaland nonagricultural water resources and so forth

Considering the uncertainties of irrigation waterresources management system fuzzy chance-constraintprogramming is introduced to the developed modelNecessity of dominance (ND) is adopted to solve fuzzy

chance-constraint program with the membership functionof both fuzzy parameters and probabilities being a triangularfuzzy membership function in this study

The developed model has been applied to Shiyang RiverBasin China to demonstrate the applicability of the devel-oped model The limited water resources supply that ismonthly runoff and effective precipitation the conjunctiveuse of surface water and groundwater and the ecologicalwater demand of downstream are considered As a resultthe irrigation water resources optimal allocation schedulesfor multiple crops of multiple periods of reservoirs operationunder different 120572-cut levels are obtained and analyzed Suchresults are valuable for local irrigation managements andagricultural producers Further study will focus on fieldmeasurement experiments of some parameters for exampleutilization coefficient of irrigation water seepage rechargedcoefficients and river leakage rate to make the developedmodel have more practical value

Nomenclature

119860119896119899 Irrigated crop area

119887119896

119905 0-1 type integer variables to decide the

overflow of the reservoir119862119896 Effective capacity of reservoir

119864119896

119905 Reservoir losses

ET119896119899119905 Actual crop evapotranspiration

ET119899119905max Potential maximum crop

evapotranspiration120582119899

119894 Sensitive index of the crop to water stress

during the stage119865 Expected crop yields of the entire

irrigation districtsGI119905 Lateral groundwater inflow from

mountainsGO119905 Recharge of groundwater from river

irrigation districtGWD

119905 Groundwater supplementary amount fordownstream

GWO119905 Lateral groundwater outflow from wellirrigation district

119894 Well irrigation district index119868119896

119905 Inflow to the reservoir in time period 119905

IR119896119905 Net irrigation water requirement

AW119896119905 Agricultural water requirement in river

irrigation districtAW119894119905 Agricultural water requirement in well

irrigation districtDW119896119905 Nonagricultural water requirement in

river irrigation districtDW119894119905 Agricultural water requirement in well

irrigation district119896 River irrigation district index119899 Crop index119874119896

119905 Reservoir overflow

119875119896119899

119905 Effective precipitation

119877max Annual allowable amount of groundwater


119882min Annual downstream minimum ecologicalwater demand

119877119896

119905 Water consumption in the irrigation

districtRL119896119905 Loss of river

RO119896119905 Remaining flow of the rivers

119905 Time period index119881119896

119905minus1 Reservoir storage at the beginning of time

period 119905119881119896

119905 Reservoir storage at the end of time period

119905

119884119899

max Crop maximum yield of crop n under thecondition of full irrigation method

VG119905 Storage capacity of groundwater

120578119896 Utilization coefficient of irrigation water

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was sponsored by the National Natural Sci-ence Foundation of China (nos 41271536 and 51321001)and National High Technology Research and DevelopmentProgram of China (863 Program) (nO 2013AA102904)

References

[1] P S Ashofteh O B Haddad and M A Marino ldquoClimatechange impact on reservoir performance indexes in agriculturalwater supplyrdquo Journal of Irrigation and Drainage Engineeringvol 139 no 2 pp 85ndash97 2012

[2] M Moradi-Jalal O Bozorg Haddad B W Karney and M AMarino ldquoReservoir operation in assigning optimal multi-cropirrigation areasrdquo Agricultural Water Management vol 90 no1-2 pp 149ndash159 2007

[3] W S Butcher ldquoStochastic dynamic programming for optimumreservoir operation1rdquoWater Resources Bulletin vol 7 no 1 pp115ndash123 1971

[4] J R Stedinger B F Sule and D P Loucks ldquoStochastic dynamicprogramming models for reservoir operation optimizationrdquoWater Resources Research vol 20 no 11 pp 1499ndash1505 1984

[5] A Dos Santos Teixeira and M A Marino ldquoCoupled reservoiroperation-irrigation scheduling by dynamic programmingrdquoJournal of Irrigation and Drainage Engineering vol 128 no 2pp 63ndash73 2002

[6] S Vedula and P P Mujumdar ldquoOptimal reservoir operation forirrigation of multiple cropsrdquo Water Resources Research vol 28no 1 pp 1ndash9 1992

[7] S Vedula and D N Kumar ldquoAn integrated model for optimalreservoir operation for irrigation of multiple cropsrdquo WaterResources Research vol 32 no 4 pp 1101ndash1108 1996

[8] P P Mujumdar and T S V Ramesh ldquoReal-time reservoiroperation for irrigationrdquo Water Resources Research vol 33 no5 pp 1157ndash1164 1997

[9] A Evers R Elliott and E Stevens ldquoIntegrated decision makingfor reservoir irrigation and crop managementrdquo AgriculturalSystems vol 58 no 4 pp 529ndash554 1998

[10] P E Georgiou and D M Papamichail ldquoOptimization model ofan irrigation reservoir for water allocation and crop planningunder various weather conditionsrdquo Irrigation Science vol 26no 6 pp 487ndash504 2008

[11] A S Prasad N Umamahesh and G Viswanath ldquoShort-termreal-time reservoir operation for irrigationrdquo Journal of WaterResources Planning andManagement vol 139 no 2 pp 149ndash1582012

[12] A Afshar M A Marino and A Abrishamchi ldquoReservoirplanning for irrigation districtrdquo Journal of Water ResourcesPlanning and Management vol 117 no 1 pp 74ndash85 1991

[13] SVedula P PMujumdar andGChandra Sekhar ldquoConjunctiveuse modeling for multicrop irrigationrdquo Agricultural WaterManagement vol 73 no 3 pp 193ndash221 2005

[14] E G Reichard ldquoGroundwater-surface water management withstochastic surface water supplies a simulation optimizationapproachrdquo Water Resources Research vol 31 no 11 pp 2845ndash2865 1995

[15] M Karamouz and H V Vasiliadis ldquoBayesian stochastic opti-mization of reservoir operation using uncertain forecastsrdquoWater Resources Research vol 28 no 5 pp 1221ndash1232 1992

[16] R S V Teegavarapu and S P Simonovic ldquoModeling uncertaintyin reservoir loss functions using fuzzy setsrdquo Water ResourcesResearch vol 35 no 9 pp 2815ndash2823 1999

[17] X Li S Guo P Liu and G Chen ldquoDynamic control of floodlimitedwater level for reservoir operation by considering inflowuncertaintyrdquo Journal of Hydrology vol 391 no 1-2 pp 124ndash1322010

[18] F-J Chang S-C Hui and Y-C Chen ldquoReservoir operationusing grey fuzzy stochastic dynamic programmingrdquoHydrologi-cal Processes vol 16 no 12 pp 2395ndash2408 2002

[19] A Seifi and K W Hipel ldquoInterior-point method for reservoiroperation with stochastic inflowsrdquo Journal of Water ResourcesPlanning and Management vol 127 no 1 pp 48ndash57 2001

[20] M N Azaiez M Hariga and I Al-Harkan ldquoA chance-constrained multi-period model for a special multi-reservoirsystemrdquo Computers and Operations Research vol 32 no 5 pp1337ndash1351 2005

[21] P Guo and G H Huang ldquoTwo-stage fuzzy chance-constrainedprogramming application to water resources managementunder dual uncertaintiesrdquo Stochastic Environmental Researchand Risk Assessment vol 23 no 3 pp 349ndash359 2009

[22] A J Askew ldquoChancemdashconstrained dynamic programing andthe optimization of water resource systemsrdquo Water ResourcesResearch vol 10 no 6 pp 1099ndash1106 1974

[23] D Z Fu Y P Li and G H Huang ldquoA factorial-based dynamicanalysis method for reservoir operation under fuzzy-stochasticuncertaintiesrdquoWater Resources Management vol 27 no 13 pp4591ndash4610 2013

[24] M E Jensen ldquoWater consumption by agricultural plantsrdquoWaterDeficits and Plant Growth vol 2 pp 1ndash22 1968

[25] E C Kipkorir and D Raes ldquoTransformation of yield responsefactor into Jensenrsquos sensitivity indexrdquo Irrigation and DrainageSystems vol 16 no 1 pp 47ndash52 2002

[26] G P Tsakiris ldquoA method for applying crop sensitivity factors inirrigation schedulingrdquo Agricultural Water Management vol 5no 4 pp 335ndash343 1982


[27] J Ma Z Ding W M Edmunds J B Gates and T HuangldquoLimits to recharge of groundwater from Tibetan plateau tothe Gobi desert implications for water management in themountain frontrdquo Journal of Hydrology vol 364 no 1-2 pp 128ndash141 2009

[28] Y Qu S Ma and W Qu ldquowater resources transformation anddevelopment and usemodel in the arid area of northwest chinardquoJournal of Desert Research vol 18 no 4 pp 299ndash307 1988

[29] L Yao S Feng X Mao Z Huo S Kang and D A BarryldquoCoupled effects of canal lining andmulti-layered soil structureon canal seepage and soil water dynamicsrdquo Journal ofHydrologyvol 430-431 pp 91ndash102 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


for reservoir management options Dos Santos Teixeira [5]developed a forward dynamic programming (FDP) modelfor real-time reservoirs operation of irrigation districtsGeorgiou and Papamichail [10] aimed to determine optimalirrigation allocation and cropping pattern for multicropsPrasad et al 2012 [11] developed a real-time reservoir releasedecision Among these studies crop yield was critical data fordecision objective There were two ways to determine cropyield (1) estimated by average production per unit area or percube water [12] and (2) calculated through water productionfunction [6 11 13]

However in real-world problem the source of watercould be groundwater and surface (reservoir) water Thenthe problem becomes more complicated to identify the mostefficient strategies for the whole system Reichard [14] applieda stochastic simulation optimization model for Santa Clara-Callcguas Basin surface water-groundwater managementVedula et al [13] optimized multicrop water allocation fora reservoir-canal-aquifer system considering intraseasonalperiod As discussed above previous researchers just con-sidered two issues among reservoirs operation irrigationdistrict ecological water requirement and surface water-groundwater conjunctive use To approach the real situationan integrated water allocation model related to these issueswas considered

Another difficulty in the whole systemwas the uncertain-ties including natural factors and human triggers Uncertain-ties in reservoir and irrigation system caused by river inflowsand crops requirement had been discussed from differentaspects [1 15ndash18] Karamouz and Vasiliadis [15] forecastedthe stream flows by Bayesian stochastic dynamic programing(BSDP) with different conditional probabilities Teegavarapuand Simonovic [16] handled the imprecise loss function forshort time reservoirs operation under fuzzy environmentSeifi and Hipel [19] presented a stochastic model to deal withthe stochasticity of inflows under multiple inflow scenariosChang et al [18] combined grey system with fuzzy stochasticdynamic programming Although uncertainty was consid-ered the reliability of satisfying (or risk of violating) cannotbe reflectedThen Azaiez et al [20] applied chance constraintto explain the potential benefits of groundwater saving

But these methods cannot handle dual uncertainties pre-sented as linkage between fuzzy and probability distributionTo handle this problem Guo and Huang [21] developed atwo-stage fuzzy chance-constraint programming for waterresources management In a similar fashion if adequatesamples of historic reservoir inflow date were obtained thendifferent reservoir inflow levels distributed as randomnessdistribution If the total reservoir release water cannot bemetchance-constraint programming can be applied to obtainthe expected net benefits [22] Nevertheless in the progressof RO alternations were made by decision makers withdifferent goals probabilities of different inflow levels couldnot exactly reflect randomness of inflows alone [23] Thusthe probabilities may own the fuzzy information based onthe decision makerrsquos judgment leading to fuzzy stochasticcharacteristic

In this paper an optimization model for deficient irri-gation water allocation under uncertainties was developed

The model is combined with ground-surface water joint-usemodel monthly reservoirs operation model and multicropirrigation model Fuzzy stochastic characteristic was repre-sented as fuzzy probability reflecting the risk of reservoirrelease water and necessity of dominance (ND) was consid-ered to illustrate it Reservoir water and groundwater wereintegrated as interactive resources for both irrigation districtsand ecological water Irrigation water was calculated throughJensen model under deficit irrigation condition

2 Model Formulation

The available water resources mainly come from the moun-tainous runoff in typical inland area of China According tothe characters of water resources of inland river system asshown in Figure 1 the watershed can be divided into twoparts runoff formation and disappear areas Accumulatedwater flows into mountainous river for water utilization inthe plain region Several reservoirs located mainly in themountain river and some of the water was used for irrigationbut the rest leaked in the piedmont plain as groundwaterrecharge Groundwater was pumped for agricultural irriga-tion once more and the surplus water needs to meet theecological water demand of downstream meanwhile Thewater resources transformation process leads to efficientwater development pattern and the water recycle is improvedHowever excessive development scale and unreasonable allo-cation of water resources can cause the overextraction ofgroundwater and deterioration of the environment

At present maximum utilization of water resources isvery important because of the shortage of water resourcesTo optimize water resources considering exploitation andreplenishment balance of groundwater for upstream anddownstream users the current study was used to simulatemultiperiod reservoirs operation and multicrop irrigationoptimization The objective is to maximize the total cropyields of the entire irrigation districts

This model can be written as follows and the meaningsof parameters are listed in the Nomenclature section All ofvariables are nonnegative

Max119865 =

119870

sum

119896=1

119873

sum

119899=1

[

[

119884119899

max

119879

prod

119905=1

(ET119896119899119905

ET119899119905max

)

120582119899

119905

times 119860119896119899]

]

+

119868

sum

119894=1

119873

sum

119899=1

[

[

119884119899

max

119879

prod

119905=1

(ET119894119899119905

ET119899119905max

)

120582119899

119905

times 119860119894119899]

]

(1)

The expected crop yields are estimated by deficit irrigationcrop water production function The water production func-tion can be used to tackle the irrigation management Wedivide the crop planting stages into monthly periods Thecroprsquos monthly water consumption is the decision variablein the model The deficit irrigation Jensen model [24] isintroduced to reflect crop yields linked to sensitivity of watershortage Sufficient water requirement of crop can lead tomaximum crop yields and water deficit in different stagesmay reduce the actual crop production in different degrees




Mountain river





Water accumulation



06ET119899119905 max le ET119896119899

119905le ET119899119905 max

06ET119899119905 max le ET119894119899

119905le ET119899119905 max

(2)


119881119896

119905= 119881119896

119905minus1+ 119868119896

119905minus 119877119896

119905minus 119874119896

119905minus 119864119896

119905 (3)


119887119896

119905le119881119896

119905

119862119896

119874119896

119905minus 119887119896

119905times119872 le 0

(4)


119881119896

119905le 119862119896 (5)


Pr (119881119896119905le 119862119896) ge 120575119896 (6)

Then we have

1119881119896

119905le 119862119896(119901119896)

(7)

where 119862119896(119901119896)

= 119865minus1



119896le 1 119896 =

1 2 119870) and 120575119896 = 1 minus 119901119896


119901119896= (119901119896 1199011198960 119901119896) = (1 minus 120575

119896

1 minus 1205751198960 1 minus 120575

119896)


(8)


((1 minus 120572) 1 + 120572)119881119896

119905le 119862119896((1minus120572)119901

119896

+1205721199011198960)

(9)

(08 + 02120572)119881119896

119905le 119862119896((1minus120572)005+12057202)

(10)


119881119896

119899minus 119881119896

0ge 0 119896 = 1 2 119870 (11)




119877119896

119905= AW119896

119905+ DW119896

119905

IR119896119905= AW119896

119905sdot 120578119896

(12)


IR119896119905=

119873

sum

119899=1

(119864119879119896119899

119905minus 119875119896119899

119905) times 119860119896119899 (13)


119874119896

119905= RL119896119905+ RO119896119905 (14)


GO119905=

119870

sum

119896=1

(120572119864119896

119905+ 120573RL119896

119905+ 120574AW119896

119905(1 minus 120578

119896)) + GI

119905 (15)


VG119905= VG

119905minus1+ GO

119905

+ 120596

119870

sum

119896=1

RO119896119905+

119868

sum

119894=1

[1205741015840AW119894119905(1 minus 120578

119894)]

minus

119868

sum

119894=1

119877119894

119905minus GWD

119905minus GWO

119905

(16)



119877119894

119905= AW119894

119905+ DW119894

119905

IR119894119905= AW119894

119905sdot 120578119894

IR119894119905=

119873

sum

119899=1

(ET119894119899119905minus 119875119894119899

119905) times 119860119894119899

(17)


119877119894

119905le 119877max (18)


RO1015840119905= (1 minus 120576)

119870

sum

119896=1

RO119896119905 (19)


RO1015840119905+ GWD

119905ge 119882min (20)


3 Case Study






Minqin

Jinchuan

Wuwei

Gulang

Yonchang

Dajing

Rive

r

Huangyan

g Rive

r

Zamu Rive


Gulang River

Dongda River

Xida Rive

r

GLHY

QYZM

JT

YC JYXY

QHDH

N

(km)0 10 20 30







0




119905and119864119896



1 2 3 4 5 6 7 8 9 10 11 1220601001401802200

20406080

100120

1 2 3 4 5 6 7 8 9 10 11 12

Month

Prec

ipita

tion

(mm

)

PrecipitationET0

ET0

(mm

)



119868119896

119905= 119877119896

119905+ 119874119896

119905 (21)











119884 = 119884max

119899

prod

119894=1

(ET119894

ET119898119894

)

120582119894

(22)


119894and

ET119898119894


119884 = 1198861+ 1198871ET + 119888

1ET2 (23)













Wheat ET119898

2354 11929 17603 21921 15323 000 000120582 006 007 015 008 001 000 000

Maize ET119898

000 2782 7076 11255 17192 12906 2303120582 000 004 012 020 031 022 003

Potato ET119898

000 1687 6156 12922 11891 10147 1132120582 000 001 003 007 006 005 000

Flax ET119898

000 1849 10502 15006 14619 7790 000120582 000 012 027 036 035 009 000

Melon ET119898

000 000 5185 6284 12585 6848 2165120582 000 000 006 007 015 008 001









4 Results Analysis






0005000

10000150002000025000


Crop

wat

er

requ

irmen

ts (m

m)

ETET

m



0005000

10000150002000025000

3 4 5 56 7 4 6 7 8 9 4 5 6 7 9 4 5 6 7 88 5 6 7 8 9

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m





01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)


0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)







020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month


Gro

undw

ater

rech

arge

(103m

3)

times103


05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month


Qinghe

Gro

undw

ater

use

(103m

3)








1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Mountain river





Water accumulation



06ET119899119905 max le ET119896119899

119905le ET119899119905 max

06ET119899119905 max le ET119894119899

119905le ET119899119905 max

(2)


119881119896

119905= 119881119896

119905minus1+ 119868119896

119905minus 119877119896

119905minus 119874119896

119905minus 119864119896

119905 (3)


119887119896

119905le119881119896

119905

119862119896

119874119896

119905minus 119887119896

119905times119872 le 0

(4)


119881119896

119905le 119862119896 (5)


Pr (119881119896119905le 119862119896) ge 120575119896 (6)

Then we have

1119881119896

119905le 119862119896(119901119896)

(7)

where 119862119896(119901119896)

= 119865minus1



119896le 1 119896 =

1 2 119870) and 120575119896 = 1 minus 119901119896


119901119896= (119901119896 1199011198960 119901119896) = (1 minus 120575

119896

1 minus 1205751198960 1 minus 120575

119896)


(8)


((1 minus 120572) 1 + 120572)119881119896

119905le 119862119896((1minus120572)119901

119896

+1205721199011198960)

(9)

(08 + 02120572)119881119896

119905le 119862119896((1minus120572)005+12057202)

(10)


119881119896

119899minus 119881119896

0ge 0 119896 = 1 2 119870 (11)




119877119896

119905= AW119896

119905+ DW119896

119905

IR119896119905= AW119896

119905sdot 120578119896

(12)


IR119896119905=

119873

sum

119899=1

(119864119879119896119899

119905minus 119875119896119899

119905) times 119860119896119899 (13)


119874119896

119905= RL119896119905+ RO119896119905 (14)


GO119905=

119870

sum

119896=1

(120572119864119896

119905+ 120573RL119896

119905+ 120574AW119896

119905(1 minus 120578

119896)) + GI

119905 (15)


VG119905= VG

119905minus1+ GO

119905

+ 120596

119870

sum

119896=1

RO119896119905+

119868

sum

119894=1

[1205741015840AW119894119905(1 minus 120578

119894)]

minus

119868

sum

119894=1

119877119894

119905minus GWD

119905minus GWO

119905

(16)



119877119894

119905= AW119894

119905+ DW119894

119905

IR119894119905= AW119894

119905sdot 120578119894

IR119894119905=

119873

sum

119899=1

(ET119894119899119905minus 119875119894119899

119905) times 119860119894119899

(17)


119877119894

119905le 119877max (18)


RO1015840119905= (1 minus 120576)

119870

sum

119896=1

RO119896119905 (19)


RO1015840119905+ GWD

119905ge 119882min (20)


3 Case Study






Minqin

Jinchuan

Wuwei

Gulang

Yonchang

Dajing

Rive

r

Huangyan

g Rive

r

Zamu Rive


Gulang River

Dongda River

Xida Rive

r

GLHY

QYZM

JT

YC JYXY

QHDH

N

(km)0 10 20 30







0




119905and119864119896



1 2 3 4 5 6 7 8 9 10 11 1220601001401802200

20406080

100120

1 2 3 4 5 6 7 8 9 10 11 12

Month

Prec

ipita

tion

(mm

)

PrecipitationET0

ET0

(mm

)



119868119896

119905= 119877119896

119905+ 119874119896

119905 (21)











119884 = 119884max

119899

prod

119894=1

(ET119894

ET119898119894

)

120582119894

(22)


119894and

ET119898119894


119884 = 1198861+ 1198871ET + 119888

1ET2 (23)













Wheat ET119898

2354 11929 17603 21921 15323 000 000120582 006 007 015 008 001 000 000

Maize ET119898

000 2782 7076 11255 17192 12906 2303120582 000 004 012 020 031 022 003

Potato ET119898

000 1687 6156 12922 11891 10147 1132120582 000 001 003 007 006 005 000

Flax ET119898

000 1849 10502 15006 14619 7790 000120582 000 012 027 036 035 009 000

Melon ET119898

000 000 5185 6284 12585 6848 2165120582 000 000 006 007 015 008 001









4 Results Analysis






0005000

10000150002000025000


Crop

wat

er

requ

irmen

ts (m

m)

ETET

m



0005000

10000150002000025000

3 4 5 56 7 4 6 7 8 9 4 5 6 7 9 4 5 6 7 88 5 6 7 8 9

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m





01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)


0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)







020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month


Gro

undw

ater

rech

arge

(103m

3)

times103


05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month


Qinghe

Gro

undw

ater

use

(103m

3)








1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877119896

119905= AW119896

119905+ DW119896

119905

IR119896119905= AW119896

119905sdot 120578119896

(12)


IR119896119905=

119873

sum

119899=1

(119864119879119896119899

119905minus 119875119896119899

119905) times 119860119896119899 (13)


119874119896

119905= RL119896119905+ RO119896119905 (14)


GO119905=

119870

sum

119896=1

(120572119864119896

119905+ 120573RL119896

119905+ 120574AW119896

119905(1 minus 120578

119896)) + GI

119905 (15)


VG119905= VG

119905minus1+ GO

119905

+ 120596

119870

sum

119896=1

RO119896119905+

119868

sum

119894=1

[1205741015840AW119894119905(1 minus 120578

119894)]

minus

119868

sum

119894=1

119877119894

119905minus GWD

119905minus GWO

119905

(16)



119877119894

119905= AW119894

119905+ DW119894

119905

IR119894119905= AW119894

119905sdot 120578119894

IR119894119905=

119873

sum

119899=1

(ET119894119899119905minus 119875119894119899

119905) times 119860119894119899

(17)


119877119894

119905le 119877max (18)


RO1015840119905= (1 minus 120576)

119870

sum

119896=1

RO119896119905 (19)


RO1015840119905+ GWD

119905ge 119882min (20)


3 Case Study






Minqin

Jinchuan

Wuwei

Gulang

Yonchang

Dajing

Rive

r

Huangyan

g Rive

r

Zamu Rive


Gulang River

Dongda River

Xida Rive

r

GLHY

QYZM

JT

YC JYXY

QHDH

N

(km)0 10 20 30







0




119905and119864119896



1 2 3 4 5 6 7 8 9 10 11 1220601001401802200

20406080

100120

1 2 3 4 5 6 7 8 9 10 11 12

Month

Prec

ipita

tion

(mm

)

PrecipitationET0

ET0

(mm

)



119868119896

119905= 119877119896

119905+ 119874119896

119905 (21)











119884 = 119884max

119899

prod

119894=1

(ET119894

ET119898119894

)

120582119894

(22)


119894and

ET119898119894


119884 = 1198861+ 1198871ET + 119888

1ET2 (23)













Wheat ET119898

2354 11929 17603 21921 15323 000 000120582 006 007 015 008 001 000 000

Maize ET119898

000 2782 7076 11255 17192 12906 2303120582 000 004 012 020 031 022 003

Potato ET119898

000 1687 6156 12922 11891 10147 1132120582 000 001 003 007 006 005 000

Flax ET119898

000 1849 10502 15006 14619 7790 000120582 000 012 027 036 035 009 000

Melon ET119898

000 000 5185 6284 12585 6848 2165120582 000 000 006 007 015 008 001









4 Results Analysis






0005000

10000150002000025000


Crop

wat

er

requ

irmen

ts (m

m)

ETET

m



0005000

10000150002000025000

3 4 5 56 7 4 6 7 8 9 4 5 6 7 9 4 5 6 7 88 5 6 7 8 9

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m





01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)


0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)







020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month


Gro

undw

ater

rech

arge

(103m

3)

times103


05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month


Qinghe

Gro

undw

ater

use

(103m

3)








1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Minqin

Jinchuan

Wuwei

Gulang

Yonchang

Dajing

Rive

r

Huangyan

g Rive

r

Zamu Rive


Gulang River

Dongda River

Xida Rive

r

GLHY

QYZM

JT

YC JYXY

QHDH

N

(km)0 10 20 30







0




119905and119864119896



1 2 3 4 5 6 7 8 9 10 11 1220601001401802200

20406080

100120

1 2 3 4 5 6 7 8 9 10 11 12

Month

Prec

ipita

tion

(mm

)

PrecipitationET0

ET0

(mm

)



119868119896

119905= 119877119896

119905+ 119874119896

119905 (21)











119884 = 119884max

119899

prod

119894=1

(ET119894

ET119898119894

)

120582119894

(22)


119894and

ET119898119894


119884 = 1198861+ 1198871ET + 119888

1ET2 (23)













Wheat ET119898

2354 11929 17603 21921 15323 000 000120582 006 007 015 008 001 000 000

Maize ET119898

000 2782 7076 11255 17192 12906 2303120582 000 004 012 020 031 022 003

Potato ET119898

000 1687 6156 12922 11891 10147 1132120582 000 001 003 007 006 005 000

Flax ET119898

000 1849 10502 15006 14619 7790 000120582 000 012 027 036 035 009 000

Melon ET119898

000 000 5185 6284 12585 6848 2165120582 000 000 006 007 015 008 001









4 Results Analysis






0005000

10000150002000025000


Crop

wat

er

requ

irmen

ts (m

m)

ETET

m



0005000

10000150002000025000

3 4 5 56 7 4 6 7 8 9 4 5 6 7 9 4 5 6 7 88 5 6 7 8 9

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m





01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)


0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)







020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month


Gro

undw

ater

rech

arge

(103m

3)

times103


05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month


Qinghe

Gro

undw

ater

use

(103m

3)








1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119884 = 119884max

119899

prod

119894=1

(ET119894

ET119898119894

)

120582119894

(22)


119894and

ET119898119894


119884 = 1198861+ 1198871ET + 119888

1ET2 (23)













Wheat ET119898

2354 11929 17603 21921 15323 000 000120582 006 007 015 008 001 000 000

Maize ET119898

000 2782 7076 11255 17192 12906 2303120582 000 004 012 020 031 022 003

Potato ET119898

000 1687 6156 12922 11891 10147 1132120582 000 001 003 007 006 005 000

Flax ET119898

000 1849 10502 15006 14619 7790 000120582 000 012 027 036 035 009 000

Melon ET119898

000 000 5185 6284 12585 6848 2165120582 000 000 006 007 015 008 001









4 Results Analysis






0005000

10000150002000025000


Crop

wat

er

requ

irmen

ts (m

m)

ETET

m



0005000

10000150002000025000

3 4 5 56 7 4 6 7 8 9 4 5 6 7 9 4 5 6 7 88 5 6 7 8 9

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m





01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)


0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)







020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month


Gro

undw

ater

rech

arge

(103m

3)

times103


05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month


Qinghe

Gro

undw

ater

use

(103m

3)








1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Wheat ET119898

2354 11929 17603 21921 15323 000 000120582 006 007 015 008 001 000 000

Maize ET119898

000 2782 7076 11255 17192 12906 2303120582 000 004 012 020 031 022 003

Potato ET119898

000 1687 6156 12922 11891 10147 1132120582 000 001 003 007 006 005 000

Flax ET119898

000 1849 10502 15006 14619 7790 000120582 000 012 027 036 035 009 000

Melon ET119898

000 000 5185 6284 12585 6848 2165120582 000 000 006 007 015 008 001









4 Results Analysis






0005000

10000150002000025000


Crop

wat

er

requ

irmen

ts (m

m)

ETET

m



0005000

10000150002000025000

3 4 5 56 7 4 6 7 8 9 4 5 6 7 9 4 5 6 7 88 5 6 7 8 9

Crop

wat

er

requ

irmen

ts (m

m)

ETET

m





01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)


0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)







020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month


Gro

undw

ater

rech

arge

(103m

3)

times103


05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month


Qinghe

Gro

undw

ater

use

(103m

3)








1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01000020000300004000050000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs re

leas

e(103m

3)


0100002000030000400005000060000

1 2 3 4 5 6 7 8 9 10 11 12Month

GulangGuangyangZamu

JintaXiyingDonghe

Rese

rvoi

rs o

verfl

ow(103m

3)







020406080

100120

1 2 3 4 5 6 7 8 9 10 11 12Month


Gro

undw

ater

rech

arge

(103m

3)

times103


05000

10000150002000025000

1 2 3 4 5 6 7 8 9 10 11 12Month


Qinghe

Gro

undw

ater

use

(103m

3)








1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1348135013521354135613581360136213641366

Tota

l cro

p yi

eld

(109 kg

)

120572 = 02 120572 = 05 120572 = 075


038

039

040

041

042

0606106206306406506606706806907


120572 = 02 120572 = 075

Rese

rvoi

rs o

verfl

ow (1

09m

3)

Rese

rvoi

rs w

ithdr

awal(109m

3)

120572 = 05



5 Conclusions





Nomenclature


119887119896



119864119896





















119875119896119899





119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877119896






period 119905119881119896


119905

119884119899






Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Monthly Optimal Reservoirs Operation for...

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Transcript of Research Article Monthly Optimal Reservoirs Operation for...