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Research paper

Extended Muskingum method for ood routing

D. Nagesh Kumar a ,*, Falguni Baliarsingh b , K. Srinivasa Raju c

a Department of Civil Engineering, Indian Institute of Science, Bangalore, Indiab Department of Civil Engineering, College of Engineering & Technology, Bhubaneswar, India

c Department of Civil Engineering, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad, India

Received 29 June 2009; revised 7 August 2010; accepted 12 August 2010

Abstract

Routing of oods is essential to control the ood ow at the ood control station such that it is within the specied safe limit. In this paper,the applicability of the extended Muskingum method is examined for routing of oods for a case study of Hirakud reservoir, Mahanadi riverbasin, India. The inows to the ood control station are of two types e one controllable which comprises of reservoir releases for power and spilland the other is uncontrollable which comprises of inow from lower tributaries and intermediate catchment between the reservoir and the oodcontrol station. Muskingum model is improved to incorporate multiple sources of inows and single outow to route the ood in the reach.Instead of time lag and prismoidal ow parameters, suitable coefcients for various types of inows were derived using Linear Programming.Presently, the decisions about operation of gates of Hirakud dam are being taken once in 12 h during oods. However, four time intervals of 24,18, 12 and 6 h are examined to test the sensitivity of the routing time interval on the computed ood ow at the ood control station. It isobserved that mean relative error decreases with decrease in routing interval both for calibration and testing phase. It is concluded that theextended Muskingum method can be explored for similar reservoir congurations such as Hirakud reservoir with suitable modications. 2010 International Association of Hydro-environment Engineering and Research, Asia Pacic Division. Published by Elsevier B.V. All rightsreserved.Keywords: Extended Muskingum; Linear programming; Mahanadi river basin

1. Introduction

Flood routing is an important aspect in reservoir operationfor ood control. This requires suitable ood routing relation-ship explicitly in the formulation of the policy. The releasesfrom reservoir during oods should be so controlled that the

total ow at a downstream station is within the safe limit. Thedownstream station at which the specied maximum ow is tobe restricted is herein after referred as ood control station. Thefactors causing oods at ood control station are the release forpower and spill from reservoir, measured inow to the riverfrom tributaries between the reservoir and the ood controlstation and unmeasured lateral ow from the intermediatecatchment. The last two factors, viz., measured inow fromintermediate catchment and unmeasured lateral ows are not

under human control. Only the release from reservoir can becontrolled considering the safety and other criteria of thereservoir. There would be some time lag in terms of hours oreven days for the release of water from the reservoir to reach theood control station. It is necessary to know the effect of thereleased quantity from the reservoir, at the ood control station

at the time of taking decision about the reservoir releases. Theood routing equation is specically to be developed for thispurpose, and is considered as an important element in reservoiroperation.

Flood routing is necessary for most of the reservoirs ingeneral and very much essential for Hirakud reservoir, Maha-nadi river basin, India in specic. In Mahanadi river basin, threereservoirs were originally proposed for full development of thebasin (Patri, 1993 ). But only one was constructed in the year1956 at Hirakud mainly to mitigate oods. As no additionalreservoirs are constructed till date, the existing reservoir is usedboth for ood control and conservation purposes. Various con-servation purposes of reservoir include irrigation, hydropower,

* Corresponding author. Tel.: 91 80 2293 2666; fax: 91 80 2360 0404.E-mail address: [email protected] (D.N. Kumar).

1570-6443/$- seefront matter 2010International Associationof Hydro-environmentEngineeringandResearch,Asia PacicDivision.PublishedbyElsevierB.V. All rights reserved.doi:10.1016/j.jher.2010.08.003

Available online at www.sciencedirect.com

Journal of Hydro-environment Research 5 (2011) 127 e 135www.elsevier.com/locate/jher


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drinking water supply, low-ow augmentation etc. Averagemonsoon inow into the reservoir, observed in the history, isabout six times more than the live storage of the reservoir. Soood control during high ood is extremely difcult and

sometimes is not possible. To design a suitable ood controlmeasure, it is rst necessary to optimize the resultant ow atNaraz by suitably routing the ood in the reach between thereservoir and Naraz. Efforts are continuing to extract maximumbenets fromthis single reservoir of the basin. It is observed thatthe operating policy in the form of rule curves for the reservoirwas changed six times from 1958 till to date. These changes aredue to the climatic and hydrologic changes in the basin andchanges in objectives of the reservoir. In this regard, sustainableandsystematic ood control policyforHirakud reservoir is to beevolved in a scientic way ( Baliarsingh, 2000 ). The objectivesof the present study are as follows:

1. Exploring the applicability of extended Muskingummethod (with multiple inows e single outow) as oodrouting model which has the ability to account for thelateral inows.

2. Exploring the applicability of Linear Programming fordetermining Muskingum parameters C 0 , C 1 , and C 2 forvarious types of inows.

3. Development of relationship between discharge and otherparameters.

4. Validation of above methodologies to the case study of Hirakud reservoir, Mahanadi river basin, India.

The paper is organized as follows: literature review, des-cription of case study, data for ood routing, results anddiscussion and conclusions.

2. Literature review

The methods of ood routing are broadly classied asempirical, hydraulic, andhydrological ( Fread, 1981 ). A numberof soft computing related techniques were used for ood fore-casting in addition to Muskingum method. A brief literaturereview is presented to provide an overview.

Preliminary concepts and numerous applications of Arti-cial Neural Networks (ANN) to hydrology are available(ASCE, 2000a,b; Fernando and Jayawardena, 1998 ). Chengand Chau (2001), Cheng and Chau (2002) proposed fuzzyiteration methodology and three-person multi-objectiveconict decision model respectively for reservoir ood controloperation for a case study of Fengman Reservoir, China. Chauet al. (2005) employed the Genetic Algorithm based ArticialNeural Network (ANN-GA) and the Adaptive Network basedFuzzy Inference System (ANFIS), for ood forecasting ina reach of the Yangtze River in China. Similar studies arereported by Cheng et al. (2002, 2008a,b) .

Muskingummethod is a hydrologicalood routing technique(Chowet al., 1988 ) which was modied by many researchers.In

the two parameter Muskingum method, there are number of ways for nding the two parameters, K (travel time) and x(weighing factor for prism and wedge storage of routing reach).

These methods were discussed in detail by Singh and McCann(1980) and applied to a set of data to assess their relative ef-cacy. Gill (1978) proposed segmented curve method, in whichleast square method was used to nd out the parameters of

nonlinear form of Muskingum method. Stephenson (1979)demonstrated the way to calculate directly the coefcients of Muskingum method, C 0 , C 1 , and C 2 using Linear Programminginstead of calculating the parameters, K and x.

ODonnell (1985) considered the lateral ow factor inMuskingum two parameter model of single input single output(si-so) nature, which was converted into a three parametermodel. The parameters are K , x, a (a shows the fraction of lateral ow in comparison with inow to the reach). The leastsquare technique is used to nd out these parameters in therouting reach automatically. Khan (1993) extended the si-soood routing model to include lateral ow to form a multiinput single output (mi-so) model with lateral ow.

Tung (1985) developed state variable modeling techniquefor solving the nonlinear form of Muskingum method. Theparameters of the model were found out by four methods of curve tting. Yoon and Padmanabhan (1993) developed a soft-ware, MUPERS, where both linear and nonlinear relationshipswere dealt with. Kshirsagar et al. (1995) found parameters bya constrained, nonlinear (successive quadratic) programming.In this work, the Muskingum equation was used for routing theupstream hydrograph and the intermediate ungauged lateralinow. The lateral inow was calculated by an impulseresponse function approach. Mohan (1997) used genetic algo-rithm for parameter estimation of nonlinear Muskingum

method and compared its performance with the approach byYoon and Padmanabhan (1993) .

Samani and Jebelifard (2003) applied multilinear Musk-ingum method for hydrologic routing through circular conduits.Das (2004) developed a methodology for parameter estimationfor the Muskingum model of stream ow routing. Al-Humondand Esen (2006) presented two approximate methods for esti-mating Muskingum ood routing parameters. Geem (2006)introduced the Broyden e Fletcher e Goldfarb e Shanno (BFGS)technique, which searches the solution area based on gradientsfor estimation of Muskingum parameters.

Choudhury (2007) proposed a multiple inows Muskingummodel. This model appropriately extended the Muskingumphilosophy to multiple inows routing, expressed in a singleinowsingleoutow form. Themodelperformance is comparedwith the nonlinear kinematic wave model. He applied the modelto the ood events in Narmada Basin, India. Das (2007) devel-oped a chance constrained optimization based model, forMuskingum model parameter estimation. Das (2009) developeda methodology for Muskingum models parameter estimationfor reverse stream ow routing for which a fresh calibration wasfound necessary. Chu (2009) applied Fuzzy Inference System(FIS) andMuskingum model in ood routing where rules of FISwere incorporated with the Muskingum formula.

As mentioned in the objectives of the present study,extended Muskingum method with multiple inows e singleoutow as developed by Khan (1993) is employed for a casestudy of Hirakud reservoir, Mahanadi river basin, India. Linear

128 D.N. Kumar et al. / Journal of Hydro-environment Research 5 (2011) 127 e 135


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Programming is adopted for arriving at the values of param-eters C 0 , C 1 , and C 2 instead of time lag and prismoidal owparameters. Brief description of case study is provided in thenext section.

3. Case study

The project considered for the present study, Hirakud reser-voir, is situated in Mahanadi basin, India. The Mahanadi basinlies between 80 300and 86 500East longitudes and 19 200and23 350Northlatitudes. Areaof thisbasin is141,600 sqkmand isbroadly divisible into three distinct zones, the upper plateau, thecentral hill part anked by Eastern ghats, and the delta area.Hirakud dam across Mahanadi river is located in the secondzone.

Mahanadi river originates in Raipur district of MadhyaPradesh and runs for a length of 851 km and joins the Bay of Bengal. After a run of 450 km from its starting point, theHirakud dam was built across the river. Downstream (d/s) of

the dam, the river gets water mainly from two tributaries, Ongand Tel, in addition to free catchment. The river ows down toNaraj, the head of delta and nally joins the Bay of Bengal.The catchment area up to Naraj is 132,200 sq km. On the d/sof Naraj, the river divides into several branches, namely,Birupa, Chitrotpala, Devi, Kushabhadra, Bhargabi, Daya etc.and runs 80 km before discharging into Bay of Bengal. Theschematic diagram of Hirakud project is shown in Fig. 1.

The multipurpose Hirakud reservoir is utilized mainly forthree purposes, ood control, irrigation, and hydropowerproduction in that order of priorities. Hirakud dam is expected

Chipilimapower house

Irrigation release

Inflow to Hirakud reservoir

Hirakud reservoirPower house

Reservoir

Khairimal

River Ong Salebhat

River Suktel Sukma

River Tel Kantamal

Mundali

Naraj

Bay of Bengal

R i v e r

M a h a n a d i

Legend

- Flow measuringstation

- Reservoir- Power house

Fig. 1. Schematic diagram of Hirakud project.



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to control ood in coastal delta area by limiting the ow atNaraj to be within 25,500 cumecs. There are three head regu-lators, which can draw 128.8 cumecs for irrigation purposes.Areas of 160,000 and 110,000 ha are irrigable under the

reservoir during Kharif (June to October) and Rabi (Novemberto February) seasons respectively. The total installed hydro-power capacity of the project is 308 MW, out of which 236 MWcan be produced from seven units of Hirakud hydropowerstation, and 72 MW from three units of Chipilima hydropowerstation, located further d/s of Hirakud dam. The water, used forpower generation at Hirakud, ows from Hirakud hydropowerstation to Chipilima hydropower station through a powerchannel, 22.4 km long. After generating power at Chipilima, thewater ows back into the river.

For this reservoir, ood control is the rst priority. There isno other ood controlling structure downstream of Hirakudreservoir. During monsoon season, the coastal delta part,between Naraj and Bay of Bengal, is severely affected byoods. These ood ows comprise of releases from Hirakudreservoir for power generation and spill and runoff from thedownstream catchment. Naraj where the ow of Mahanadiriver is measured is situated at the head of the delta area. Theow of Mahanadi river at Naraj is considered by the Hirakudauthority as the indicator of occurrence of ood in the coastaldelta area. As Hirakud reservoir is on the upstream side of thedelta area of the basin, it plays an important role in alleviatingthe severity of the ood in this area by suitable regulation of releases from the reservoir. Before making the operationaldecisions, it is necessary to optimize the resultant ow at

Naraz by suitably routing the ood in the reach between thereservoir and Naraz. Next section discusses the data require-ments and analysis for ood routing.

4. Data for ood routing

4.1. Inows into the reach

The 320 km long reach of the river between Hirakudreservoir and Naraj is treated in this study for ood routing.Flood data from 1992 to 1995 are considered for the purpose.

There is one ood each in 1992, 1993, and 1995 and six oodsin 1994. The characteristics of the oods are shown in Table 1(Patri, 1993 ).

The outowfrom the reservoir is the combination of the spill

from the reservoir and the release for power. This outow ismeasured at the dam. Water released for power joins the riverdownstream of the reservoir after generating power at twolocations as shown in Fig. 1 . Both the points are 25km apart andfor simplicity, these two ows are combined into one value.Two tributaries, Ong and Tel, join Mahanadi river on thedownstream of Hirakud reservoir. Flow in tributary Ong ismeasured at Salebhat. The ow in tributary Tel is measured atKantamal. In the case of Suktel, the tributary of Tel, ow ismeasured at Sukma. Suktel joins the tributary Tel d/s of Kant-amal. Location of these tributaries and measuring stations areshown in Fig. 1. The conuence points of these two tributarieswith Mahanadi river are quite close, compared to the length of river reach considered for ood routing purpose. Theows at allthese three stations are therefore considered together, as if theseows are combined together before joining the Mahanadi river.This combined ow at Khairimal is henceforth termed as d/scatchment contribution.

4.2. Lateral ow

The data is tested for the amount of unmeasured lateral owin Hirakud-Naraj routing reach. Duration of each ood event isso chosen that the river level at Naraj is same at beginning andend of the event. As there is no other supporting data, it isassumed that the slope of water surface in this reach is same atthe beginning and at the end of each ood event. Floodsnumbered 4 and 5 occurred consecutively without any timegap, and so did oods numbered 7 and 8. So these two pairs of oods are combined together to form one ood event eachonly for the calculation of lateral ow and are named as oodnumbers 4 and 7 respectively. The volume of inow into thereach is the combination of release for power and spill fromHirakud reservoir and d/s catchment contribution. Thedifference between inow into the reach and outow from thereach, i.e., ow at Naraj, constitutes the lateral ow into the

Table 1Characteristics of oods under consideration for ood routing.

Flood No. Duration of ood event Inuencing variableof ow at Naraj

Maximum ow fromreservoir (10 3 cumecs)

Maximum d/s catchmentcontribution (10 3 cumecs)

Maximum ow atNaraj (10 3 cumecs)

Minimum ow atNaraj (10 3 cumecs)

(1) (2) (3) (4) (5) (6) (7)

1 16/08/92 e 29/08/92 Both RPS a and DC b 15.52 14.11 c 33.98 c 08.012 14/08/93 e 27/08/93 Both RPS and DC 10.16 05.54 22.10 02.203 19/06/94 e 27/06/94 Only RPS 11.18 02.96 17.63 08.644 08/07/94 e 19/07/94 Only RPS 21.22 c 08.04 29.04 03.115 19/07/94 e 28/07/94 Only RPS 17.99 02.52 22.93 08.126 30/07/94 e 10/08/94 Only RPS 11.68 09.74 14.52 07.037 17/08/94 e 26/08/94 Both RPS and DC 11.01 11.25 22.04 11.638 26/08/94 e 12/09/94 Both RPS and DC 16.74 13.54 30.64 12.849 18/07/95 e 30/07/95 Both RPS and DC 13.62 09.85 25.93 01.71

a RPS e Release for power and spill from Hirakud reservoir.b DC - downstream catchment contribution.c Maximum.



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routing reach. These details are shown in Table 2 . I t i sobserved that the unmeasured lateral ow varies widely fromone ood to other. This is not having any specic ratio to thevolume of inow or volume of outow. The ratio of lateral

ow to the volume of inow varies from 1% to 72%, and thatto the volume of outow at Naraj varies from 1% to 42%(columns 6 and 7 of Table 2 ).

4.3. Data sets for calibration and testing

Data of each of the nine oods are plotted individually inFig. 2 showing the inuence of release for power and spillfrom reservoir and d/s catchment contribution on the ow atNaraj. It may be observed from the plots that the oods can beclassied into two categories. In rst category, both outowfrom reservoir and d/s catchment contribution inuence thepattern of ow at Naraj and in second category, only outowfrom reservoir has the inuence. Four oods, numbers 3 to 6fall under second category and remaining ve oods fallunder rst category. This characteristic is denoted in thirdcolumn of Table 1 . Two thirds of the data is used for cali-bration and the remaining is used for testing. The importantfactor in such classication of calibration and testing data setsis that the characteristics of testing data should be denitelypresent in the calibrating set. If the calibrating set is toogeneralized, performance during testing may not be thatencouraging. This aspect is taken care of in the present study.Here, the quantity of ow at various time instances is taken asthe characteristic of ood ow data. The oods of higher

maximum discharge and lower minimum discharge are usedfor calibration, leaving the intermediate cases for testing(Baliarsingh, 2000 ).

Keeping these aspects in consideration, out of ve oods of rst category, ood numbers 1, 8, 9 are used for calibrationand ood numbers 2, 7 for testing. Similarly, out of four oodsin second category, ood numbers 3, 4 are used for calibrationand ood numbers 5, 6 for testing. In total ve oods, i.e.,ood numbers 1, 3, 4, 8, and 9 are used for calibration and fouroods, i.e., ood numbers 2, 5, 6, and 7 for testing.

Presently for Hirakud reservoir, the decisions of gateoperation are being taken once in 12 h during ood. However,

in the present study, it is proposed to evaluate the performanceof gate operation once in 24 h, 18 h, 12 h, and 6 h to nd outthe effect of these variations on the predicted ow at Naraj.

The ood routing equations with ow ordinates at the abovetime intervals are required to be used in the model for corre-sponding gate operation. Accordingly, the ood routingequations with these four routing periods by extended Musk-

ingum method are analyzed in this study.

4.4. Performance measure

In this study, only one output viz., ood ow at the oodcontrol station, Naraj, is predicted. So a single performancemeasure viz., Mean relative error is considered sufcient toevaluate the performance of the various models and is given by

E 1n X

n

l 1

EY l OY l OY l

100 1

where E is mean relative error in %; EY is the estimatedoutow at Naraj; OY is the observed outow at Naraj; n is thenumber of data points in the data set. Section 5 presents resultsand discussion.

5. Results and discussion

The extended Muskingum routing method ( Khan, 1993 )deals with multiple inow e single outow routing process. Inthe present study, the ood hydrograph at Naraj ( Q) dependson two ood hydrographs on the upstream side of the reach;(i) hydrograph of release for power and spill from the reservoir(RPS) and (ii) hydrograph of d/s catchment contribution (DC).Downstream catchment contribution is the summation of owat Kantamal, Sukma, and Salebhat. Downstream catchmentcontribution is assessed by deducting RPS from the total owmeasured at Khairimal.

The relationship used in extended Muskingum method isexpressed as

Q t 1 f RPS t 1 ; DC t 1 ; RPS t ; DC t ; Q t 2

where, Q t is the ow at Naraj (outow from the Hirakud-Narajrouting reach) at beginning of time period t ; RPS t is the releasefor power and spill from Hirakud reservoir at beginning of

time period t ; DCt is the ow from Ong and Tel tributaries asassessed at Khairimal at beginning of time period t . t is thetime period.

Table 2Assessment of unmeasured lateral ow for each ood.

Flood No. Duration of ood Volume of outowfrom the reach(109 cubic meter)

Volume of inowinto the reach(109 cubic meter)

Volume of lateralow into the reach(109 cubic meter)

Lateral ow inpercentage of inow

Lateral ow inpercentage of outow

(1) (2) (3) (4) (5) (3) (4) (6) (5) 100/(3) (7) (5) 100/(4)

1 16/08/92 e 29/08/92 20.55 12.19 08.36 68 402 14/08/93 e 27/08/93 11.84 06.87 04.97 72 423 19/06/94 e 27/06/94 08.49 06.51 01.98 30 234 08/07/94 e 28/07/94 32.47 25.94 06.54 25 20

6 30/07/94 e 10/08/94 11.48 11.31 00.17 01 017 17/08/94 e 12/09/94 44.63 28.46 16.17 57 369 18/07/95 e 30/07/95 10.62 09.21 01.41 15 13



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Khan(1993) developed theextended Muskingummethod formultiple inow e single outow (mi-so) form of ood routing,based on the work of single inow single outow (si-so) formproposed by ODonnell (1985) and it takes care of the ungaugedlateral inow also. The original equation of mi-so form of routing as given by Khan (1993) is

Q t 1 C10 i1t C

20 i

2t / C

m0 i

mt C

11 i

1t 1 C

21 i

2t 1 /

Cm1 imt 1 C2 Q t 3

where, Q t is the outow from the reach at beginning of routinginterval t ; C 0 with superscript 1, 2, . , m denote the C 0 coef-cients associated with inows of tributaries 1, 2, . , mrespectively at the beginning of the routing interval t ; C 1 withsuperscript 1, 2, . , m denote the C 1 coefcients associatedwith inows of tributaries 1, 2, . , m respectively at the end of

the routing interval t ; C 2 is the coefcient associated withoutow from the routing reach at beginning of routing intervalt . it with superscript 1, 2, . , m denote the inow of tributaries

0

10

20

30

40

0 2 4 6 8 10 12 14

Time from beginning of event (Days)

F l o w

( 1 0 c u m e c s

)

0

10

20

30

40

0 2 4 6 8 10 12 14Time from beginning of event (Days)

F l o w

( 1 0 c u m e c s

) Flood 2 (1993)

Flood 1 (1992)

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9


F l o w

( 1 0 c u m e c s

)Flood 3 (1994)

0

10

20

30

40

0 2 4 6 8 10 12


F l o w

( 1 0 c u m e c s ) Flood 4 (1994)

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9


F l o w

( 1 0 c u m e c s

)Flood 5 (1994)

0

10

20

30

40

0 2 4 6 8 10 12Time from beginning of event (Days)

F l o w

( 1 0 c u m e c s

)Flood 6 (1994)

0

10

20

30

40

0 2 4 6 8 10


F l o w

( 1 0

c u m e c s

)

Flood 7 (1994)

0

10

20

30

40

0 2 4 6 8 10 12 14 16


F l o w

( 1 0

c u m e c s

)Flood 8 (1994)

0

10

20

30

40

0 2 4 6 8 10 12


F l o w

( 1 0

c u m e c s

)

Ou tfl ow fr om re se rvo ir d /s catch me nt con tr ib uti on Fl ow a t N ara j

Flood 9 (1995)

Fig. 2. Flow pattern of nine oods used in ood routing.



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1, 2,. , m respectively at beginning of routing interval t . t isthe routing interval.

In the present study, m is 2, one for release for power andspill from Hirakud reservoir and the second for the d/scatchment contribution. Rewriting the equation (3) for thiscase results in

Qt 1

C H 0

RPSt C K

0DC

t C H

1RPS

t 1 C K 1 DC t 1 C2 Q t 4

where superscript H and K of the C 0 and C 1 coefcientsrepresent outow from Hirakud reservoir (RPS) and d/scatchment contribution at Khairimal (DC) respectively.

According to ODonnell (1985) and Khan (1993) , theabove set of C coefcients is valid for any ungauged lateralow into the reach, if it is found by least square technique. Inthis study, Linear Programming (LP) is used for the purpose to

minimize the sum of absolute deviations between observedand calculated ow at Naraj for determining C 0 , C 1 , and C 2 .

a

0

5

10

15

20

25

0 10 20 30 40

Time Step (Periods of 24 hours)

F l o w

( 1 0

3 c u m e c s )

Extended Muskingum

Observed

b

0

5

10

15

20

25

0 10 20 30 40


F l o w

( 1 0

3 c u m e c s )

c

0

510

15

20

25

0 10 20 30 40 50 60 70 80 90


F l o w (

1 0

3 c u m e c s )

d

0

5

10

15

20

25

0 20 40 60 80 100 120 140 160 180Time Ste p (Periods of 6 hours)

F l o w

( 1 0

3 c u m e c s )

Fig. 3. (a)e (d). Performance for different routing intervals of 24, 18, 12, and 6 h in testing phase.

Table 3Coefcients and mean relative error for extended Muskingum method in calibrating phase.

Routing interval (h) Coefcients of extended Muskingum equation Objective function value(outcome of LP optimization)

Mean relativeerror (%)C H 0 C

K 0 C

H 1 C

K 1 C 2

24 0.513 1.138 0.082 0.092 0.509 141.5 21.818 0.431 0.981 0.070 0.322 0.697 188.1 20.212 0.236 0.691 0.164 0.180 0.855 210.0 14.106 0.107 0.656 0.100 0.436 0.958 251.6 7.9



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Minimize Z X5

j 1Xn j 1

t 2pos : err t ; j neg : err t ; j 5

Subject to

Q t ; j pos : err t ; j neg : err t ; j C H 0 RPS t 1 ; j C K 0 DC t 1; j

C H 1

RPS t ; j C K 1

DC t ; j

C2 Q t 1 ; j for j 1 ; 2 ; . ; 5;

t 2 ; 3 ; . ; n j ; n j 1 6

N C H 0 N ; N C K 0 N ;

N C H 1 N ; N C K 1 N ; N C2 N ;

7

pos : err t ; j ! 0; neg : err t ; j ! 0 for j 1 ; 2 ; . ; 5;t 2 ; 3 ; . ; n j ; n j 1 8

In this formulation, t denotes the time period and j is ood

event. As there are ve oods in calibrating set, j ranges from1 to 5. n j is the number of time periods in jth ood event.pos.err and neg.err denote the positive and negative errorvalues respectively. This formulation is valid for all routingintervals, namely 24 h, 18 h, 12 h, or 6 h and even any other.Only n j values are to be changed suitably.

Linear Programming formulation is run for individualrouting interval data. The resulting objective function valuesand values of coefcients C 0 , C 1 , and C 2 are presented inTable 3 . It may be noticed that the C 1 values which correspondto inows at the end of the time interval are all negative except1 out of 8. Thus, the effect of the inows at the end of the timeinterval is actually negative. The C H

0, C K

0values are decreasing

with decrease of routing interval whereas C 2 values areincreasing. In the case of values of C H 1 , C

K 1 , there is no

consistency of decrease or increase. It is observed that theobjective function value of LP increases with decrease of routing interval. The mean relative errors with training data forthe four time intervals are shown in the last column. Thesevalues i.e., 21.8, 20.2, 14.1, 7.9 are decreasing with decrease intime interval. The models, thus obtained, are used for testingto obtain mean relative errors for respective routing intervals.

The total number of testing data sets obtained on combiningthose of the four oods 2, 5, 6 and 7 are 41, 57, 86 and 176 forrouting intervals of 24, 18, 12 and 6 h respectively. Fig. 3ae dpresent the comparison of discharge values predicted byextended Muskingum method and observed values. It is notedfrom Fig. 3ae d that ows predicted by Muskingum method

are less than the observed ows for 15 data sets out of 41(36.6%), 20 data sets out of 57 (35.1%), 38 data sets out of 86(44.2%) and 69 data sets out of 176 (39.2%) for 24, 18, 12,6 hrouting interval respectively.

Table 4 presents number of data sets falling within errorranges of 10%, 10% e 20%, 20%e 30%, 30e 40%, 40e 50%and above 50% for the predicted versus observed ows. It is

observed from Table 4 that for routing interval of 24 h, 18 h,12 h and 6 h, numbers of data sets falling in error range of 0e 20% are 29, 42, 68, 156 (or 70.73%, 73.68%, 79.06%,88.63% of the total numbers of data sets). For routing intervalof 24 h, 18 h, 12 h and 6 h, numbers of data sets falling in errorrange of below 50% are 36, 53, 81, 171 (or 87.80%, 92.98%,94.18%, 97.15% of the total numbers of data sets). Further themean relative errors (in %) of testing data are computed usingequation (1) and these are 23.3, 17.4, 15.9, 9.1 respectively for24, 18, 12, 6 h. It is observed that mean relative error decreasessystematically with decrease in routing interval as observed incalibration phase.

6. Conclusions

The extended Muskingum method is examined in this studyfor its applicability as ood routing method for the case studyof Hirakud reservoir, Mahanadi river basin, India. Nine oodsfrom 1992e 1995 are analyzed for this purpose. LinearProgramming is employed to determine the values of thecoefcients required for the extended Muskingum method. Itis observed that mean relative error decreases systematicallywith decrease in routing interval both for calibration andtesting phase. In addition, it is observed that the ows pre-dicted by Muskingum method are less than the observed owsfor 15 data sets out of 41 (36.6%), 20 data sets out of 57(35.1%), 38 data sets out of 86 (44.2%) and 69 data sets out of 176 (39.2%) for 24, 18, 12 and 6 h routing intervalrespectively.

From these results it can be concluded that the extendedMuskingum method can be explored for similar reservoirconguration such as Hirakud reservoir system with suitablemodications. Limitations, suggested improvements andfuture directions of the present study are

1. In the present study LP is used as the basis for estimatingMuskingum coefcients. Genetic algorithms, fuzzy infer-ence system, Radial basis function and other advancedneuro computing techniques can be explored for thispurpose.

Table 4Number of data sets falling within the ranges of errors (predicted versus observed ows).

Routing interval (h) 0 e 10% 10e 20% 20e 30% 30e 40% 40e 50% Above 50% Total numberof data sets

24 18 11 4 2 1 5 4118 28 14 5 4 2 4 5712 48 20 4 8 1 5 866 127 29 13 1 1 5 176



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2. Floods of 1992 e 1995 only are considered for the presentstudy. More oods may be analyzed as and when sufcientdata becomes available, to make the forecasting morerobust and reliable.

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