An analysis of MLR and NLP for use in river flood routing and comparison

with the Muskingum method

Mohammad Zare

Manfred 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

Conclusions11

Introduction

occurrence of floods has resulted in tremendous economic damages and life losses.

the correct prediction of the rise and fall of a flood (flood routing) is important.

The fundamental differential equation to describe one-dimensional unsteady river flow is the Saint-Venant equation.

22

Introduction

Because of the nonlinearity of the convective acceleration term in the Saint-Venant equation, in its most complete form it can only be solved numerically.

Nowadays, dynamic wave, diffusion wave and kinematic wave method are widely used in practice.

33

Introduction

The numerical implementation of the kinematic wave approximation is usually the Muskingum or the Muskingum-Cunge method.

Although Muskingum method is not a very accurate Method, this routing method is alive and well and by no means exhausted.

44

Introduction

the determination of the routing coefficients in the Muskingum model is solved by trial and error method and graphical method.

In this study, two new parameter estimation techniques, namely, nonlinear programming (NLP) and multiple linear regression (MLR), will be applied to the routing of three flood events in a river section.

55

Literature Review

66

Year Researcher Description

1951 Hayami Firstly, presented Diffusive wave theory

1955 to

1989

Various researcher

There have been a lot of investigations since then to what extent the various simplifications in the Saint-Venant Eqs. are valid for routing in a particular channel

1978 Gillused linear least squares to determine the two unknown parameters in the prism/wedge storage term which is the basis of the Muskingum method

Literature Review

77

Year Researcher Description

1997 Mohanapplied a genetic algorithm to estimate the parameters in a nonlinear Muskingum model

2004 Das

estimated parameters for Muskingum models using a Lagrange multiplier formulation to transform the constrained parameter optimization problem into an unconstrained one

2009Oladghafari & Fakheri

determined the routing parameters of the Muskingum model for three flood events (which are also at the focus of the present study) in a reach of the Mehranrood river in northwestern Iran by 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

88

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 the travel-time of a flood wave through the stream reach), and X a weighting factor, both of which a usually determined 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 and C3 are the routing coefficients, which include the two constants K and X , as well as the time step Δt

99

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 functional variation, while respecting certain constraints.

The NLP-problem is generally formulated as

Eq.4

1010

E

hhhn

n

x

xxxts

xf

0)(,...,0)(,0)(:.

)(min

21

Study methodsGeneral formulation of a constrained nonlinear programming problem (NLP)

There are lot of methods for solving the NLP problems.

One of these methods that is used by WinQSB-software is penalty function method

Penalty function method is transformed constrained problem into an unconstrained one.

1111

Study methodsGeneral formulation of a constrained nonlinear programming problem (NLP)

In penalty function method the constrained problem changes to

Eq.6

for each stage k, where x(k) is the solution at that stage; μ(k) is the penalty parameter; and E(x(k)) is the sum 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, otherwise μ(k+1) = β*μ(k), with β a constant

1212

max,....,2,1))(()()(min kkkxEkxf

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 at the discrete times j=1,2,…,m.

1313

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, the Muskingum equation is read like a linear regression equation for the dependent output variable Q i+1 as a function of the three independent variables I i+1, Ii (measured input hydrograph) and Qi. (measured output hydrograph). With this the MLR-model can be stated as:

1414mjQCICICQ jjjj ,...,2,1,32111

Study area and flood events used

The study area is located along the reach of the Mehranrood River in the Azarbayejan-e-Sharghi province in northwestern Iran between the two hydrometer stations Hervi (upstream) and Lighvan (downstream). The stream distance 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 the flood routing experiments. Input hydrographs for the simulations are the observed dis-charges at Hervi gage station and the output hydrographs those at station Lighvan

1515

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 routing coefficients Ci=1,2,3 (the decision variables in NLP, or the regression coefficients in MLR) have been done with the observed input and the routed output hydrographs of the April 6, 2003 flood event

After calibration, these routing coefficients have been used in the subsequent verification of the other two flood events. 1616

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(Oladghaffari et al. (2009), who used a classical graphical procedure) routing coefficients for the April 6, 2003 flood event

1717

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

1818

Results and discussionVerification of the flood routing methods with the June 9, 2005 flood event

1919

Results and discussionVerification of the flood routing methods with the May 4, 2007 flood event

2020

Results and discussion

Observed and calculated peak discharges and errors for NLP, MLR and Muskingum flood routing

2121

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 NLP and MLR methods proposed here for the automatic calibration of the routing coefficients in the widely used Muskingum flood routing method, are powerful and reliable procedures for flood routing in rivers.

These two methods may be more conveniently used than Muskingum, where suitable routing coefficients (usually the storage parameter K and the weighting factor X) are often obtained only after some lengthy trial and error process

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