An analysis of MLR and NLP for use in river flood routing and comparison
with the Muskingum method
Mohammad ZareManfred Koch
Dept. of Geotechnology and Geohydraulics,University of Kassel, Kassel, Germany
Contents
Introduction
Literature review
Study methods
Study area and flood events used
Results and discussion
Conclusions 1
Introduction
occurrence of floods has resulted in tremendouseconomic damages and life losses.
the correct prediction of the rise and fall of a flood(flood routing) is important.
The fundamental differential equation to describeone-dimensional unsteady river flow is the Saint-Venant equation.
2
Introduction
Because of the nonlinearity of the convectiveacceleration term in the Saint-Venant equation, inits most complete form it can only be solvednumerically.
Nowadays, dynamic wave, diffusion wave andkinematic wave method are widely used inpractice.
3
Introduction
The numerical implementation of the kinematicwave approximation is usually the Muskingum orthe Muskingum-Cunge method.
Although Muskingum method is not a very accurateMethod, this routing method is alive and well andby no means exhausted.
4
Introduction
the determination of the routing coefficients in theMuskingum model is solved by trial and errormethod and graphical method.
In this study, two new parameter estimationtechniques, namely, nonlinear programming (NLP)and multiple linear regression (MLR), will beapplied to the routing of three flood events in a riversection.
5
Literature Review
6
Year Researcher Description
1951 Hayami Firstly, presented Diffusive wave theory
1955 to
1989
Various researcher
There have been a lot of investigations since thento what extent the various simplifications in theSaint-Venant Eqs. are valid for routing in aparticular channel
1978 Gillused linear least squares to determine the twounknown parameters in the prism/wedge storageterm which is the basis of the Muskingum method
Literature Review
7
Year Researcher Description
1997 Mohanapplied a genetic algorithm to estimate theparameters in a nonlinear Muskingum model
2004 Das
estimated parameters for Muskingum modelsusing a Lagrange multiplier formulation totransform the constrained parameter optimizationproblem into an unconstrained one
2009Oladghafari& Fakheri
determined the routing parameters of theMuskingum model for three flood events (whichare also at the focus of the present study) in areach of the Mehranrood river in northwestern Iranby the classical (graphical) procedure
Study methodsKinematic/diffusion wave / Muskingum wave routing method
One of the most widely used methods for river flood routing is the Muskingum method.
Eq.1
Using the concept of a wedge-/prism storage for a stream reach, whereby the total actual storage is written as a weighted average of the prism-storage Sprism=KQ and the wedge storage Swedge=KX(I-Q)
Eq.2 8
dS/dt = I(t) – Q(t)
S=K[XQ+(1-X)(I-Q]
Study methodsKinematic/diffusion wave / Muskingum wave routing method
where K is a reservoir constant, (about equal to thetravel-time of a flood wave through the stream reach),and X a weighting factor, both of which a usuallydetermined in an iterative manner from observed input-and output hydrographs, the discrete Muskingum-equations are directly obtained from the time-discretization of Eq. 1
Eq.3
where j = (1,…,m) indicates the time step and C1, C2 andC3 are the routing coefficients, which include the twoconstants K and X , as well as the time step ∆t 9
jjjj QCICICQ 32111 ++= ++
Study methodsGeneral formulation of a constrained nonlinear programming problem (NLP)
The main purpose of nonlinear programming(NLP) is to find the optimum value of a functionalvariation, while respecting certain constraints.
The NLP-problem is generally formulated as
Eq.4
10
E
hhhn
n
x
xxxts
xf
⊂Ω∈
=== 0)(,...,0)(,0)(:.
)(min
21
Study methodsGeneral formulation of a constrained nonlinear programming problem (NLP)
There are lot of methods for solving the NLPproblems.
One of these methods that is used by WinQSB-software is penalty function method
Penalty function method is transformedconstrained problem into an unconstrained one.
11
Study methodsGeneral formulation of a constrained nonlinear programming problem (NLP)
In penalty function method the constrained problemchanges to
Eq.6
for each stage k, where x(k) is the solution at thatstage; µ(k) is the penalty parameter; and E(x(k)) is thesum of all constraint-violations to the power of ρ
Iterations k=1,2..kmax continue until µ(k)*E(x(k))ρ <δ,then x(k+1) is the optimal solution, otherwiseµ(k+1) = β*µ(k), with β a constant
12
[ ] max,....,2,1))(()()(min kkkxEkxf =+ ρµ
Study methodsNLP-formulation of Muskingum flood routing
The NLP -problem for flood routing is formulated here as follows
under the constraints
for the input and output discharges measured atthe discrete times j=1,2,…,m.
13
VVxf ˆ)(min −= tV QQi
n
ii
∆+=+
=∑ )(5.0
11
mjQCICICQ jjjj ,...,2,1,32111 =++= ++
Study methodsMultiple linear regression (MLR) method
In the multiple linear regression (MLR) model, theMuskingum equation is read like a linear regressionequation for the dependent output variable Qi+1 as afunction of the three independent variables Ii+1, Ii(measured input hydrograph) and Qi. (measuredoutput hydrograph). With this the MLR-model can bestated as:
14mjQCICICQ jjjj ,...,2,1,32111 =+++= ++ ε
Study area and flood events used
The study area is located along the reach of theMehranrood River in the Azarbayejan-e-Sharghi province innorthwestern Iran between the two hydrometer stationsHervi (upstream) and Lighvan (downstream). The streamdistance between these two stations is 12280 meters
Three flood events that occurred on April 6, 2003, June 9,2005 and May 4, 2007, respectively, were selected for theflood routing experiments. Input hydrographs for thesimulations are the observed dis-charges at Hervi gagestation and the output hydrographs those at station Lighvan
15
Results and discussionGeneral set-up of the flood routing computations
The NLP- and the MLR- flood routing method have been applied to the three flood hydrographs
the optimal calibration of the three routingcoefficients Ci=1,2,3 (the decision variables in NLP,or the regression coefficients in MLR) have beendone with the observed input and the routed outputhydrographs of the April 6, 2003 flood event
After calibration, these routing coefficients havebeen used in the subsequent verification of theother two flood events. 16
Results and discussionOptimal calibration of routing coefficients using the April 6, 2003 flood event
Parameters used in the NLP-penalty function method:
Optimal NLP-, MLR-, and classical Muskingum(Oladghaffariet al. (2009), who used a classical graphical procedure)routing coefficients for the April 6, 2003 flood event
17
X(1) µ(1) β δ ρ Parameter 0 1 0.1 0.0001 2 Value
C3 C2 C1 Method
0.6239 0.2886 0.0877 NLP
0.6612 0.2347 0.1043 MLR
0.6636 0.0758 0.2606 Muskingum
Results and discussionOptimal calibration of routing coefficients using the April 6, 2003 flood event
18
Results and discussionVerification of the flood routing methods with the June 9, 2005 flood event
19
Results and discussionVerification of the flood routing methods with the May 4, 2007 flood event
20
Results and discussion
Observed and calculated peak discharges and errorsfor NLP, MLR and Muskingum flood routing
21
Muskingum-error (%) Muskingum MLR-error (%) MLR NLP-error (%) NLP Observed Flood event
0.47 4.19 0.47 4.19 0.47 4.19 4.21 April 6, 2003
8.01 2.87 7.37 2.89 8.33 2.86 3.12 June 9, 2005
2.65 3.31 2.35 3.32 2.35 3.32 3.40 May 4, 2007
Conclusions
Based on the results it can be concluded that the NLPand MLR methods proposed here for the automaticcalibration of the routing coefficients in the widely usedMuskingum flood routing method, are powerful andreliable procedures for flood routing in rivers.
These two methods may be more conveniently usedthan Muskingum, where suitable routing coefficients(usually the storage parameter K and the weightingfactor X) are often obtained only after some lengthytrial and error process
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