Powell's method, Rosenbrock's algorithm, and the Nelder and Meade simplex method. Powells' method performed poorly; however, both the Rosenbrock and ...
Parameter optimization in modelling unsteady compound channel flows HABIBABIDAAND RONALDD. TOWNSEND
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Department of Civil Engineering, University of Ottawa, Ottawa, Ont., Canada KIN 6N.5 Received July 18, 1991 Revised manuscript accepted November 13, 1991 Optimization methods are used to estimate data for routing floods through open compound channels (main channels with flood plain zones). These data include the irregular channel section geometry and the varying boundary roughness. Differences between simulated and observed stages and discharges are minimized using three optimization algorithms: Powell's method, Rosenbrock's algorithm, and the Nelder and Meade simplex method. Powells' method performed poorly; however, both the Rosenbrock and simplex methods yielded good results. The estimated data using the Rosenbrock and simplex methods were used to route different flood events observed in a laboratory channel. Simulated peak stages and discharges were in good agreement with those estimated using actual routing data. Key words: compound channel, flood routing, lateral momentum transfer, optimization, unsteady flow. Des methodes d'optimisation sont utilisees pour estimer les donnies de propagation des crues par des canaux ouverts mixtes (canaux principaux avec pkrimtttres d'inondation). Ces donnees incluent la geometrie des sections de canal irregulittres et la rugosite limitrophe. Les icarts entre les debits et les niveaux simulis et ceux observes sont reduits en utilisant trois algorithmes d'optimisation : la methode de Powell, I'algorithme de Rosenbrock et la methode Simplex de Nelder et Meade. La methode de Powell a produit des resultats mediocres alors que les methodes de Rosenbrock et Simplex ont permis d'obtenir de bons resultats. Les donnies estimies a I'aide des mithodes de Rosenbrock et Simplex ont Cte utilisees pour calculer differentes propagations des crues observies dans un canal en laboratoire. On a constate une bonne concordance entre les debits et les niveaux de pointe simulis et ceux estimes a I'aide de donnkes reelles. Mots c l h : canal mixte, calcul de la propagation des crues, transfert lateral de la force vive, optimisation, Ccoulement non permanent. [Traduit par la redaction] Can. J. Civ. Eng. 19, 441-446
(1992)
Introduction Unsteady open channel flow modelling is used in such activities as flood routing and prediction, stream flow modelling and river regulation, and in the analysis of estuarine flow phenomena. In such studies, good estimates of stage and discharge require not only a reliable unsteady flow model but also good data for the flood routing exercise. The data consist mainly of the river cross-section geometry and appropriate boundary roughness coefficients. Obtaining these data usually implies extensive field surveys, which can make a numerical modelling exercise expensive. On the other hand, the accuracy gained by using the complete St. Venant equations would suffer if suspect data were used. A possible alternative procedure that could significantly reduce the necessary quantitity of data is one employing optimization techniques. In this approach, an objective function, representing the difference between the simulated and observed values of discharge or flow depth, is minimized to yield the model parameters required for the flood routing exercise. Optimization techniques were successfully used by Becker and Yeh (1972a, 1972b), Fread and Smith (1978), and Wormleaton and Karmegam (1984) to identify parameters for regular prismatic channels having simple cross sections. These researchers used the same optimization algorithm (the so-called "influence coefficient" algorithm) which, mathematically, is closely related to both quasi-linearization and the gradient method. NOTE: Written discussion of this paper is welcomed and will be received by the Editor until October 31, 1992 (address inside front cover). Prinled in Canada / lun~rimtau Canada
In this paper three optimization algorithms, the Nelder and Meade simplex method, Rosenbrock's method, and Powell's algorithm, are used to estimate the data required for routing flood events through an experimental compound channel. The composite cross section comprised a deep rectangular main channel flanked by two adjoining shallow flood plains. The unsteady flow model used in this exercise was developed and validated in a closely related study. The model equations (a modified form of the St. Venant equations) were developed specifically for routing floods through compound channels. The numerical model accounts not only for flood plain contributions to system conveyance but also for lateral momentum transer (LMT) between adjacent deep and shallow zones of compound flow fields (Prinos and Townsend 1984). Unsteady flow model The continuity equation is written as
where A is the cross-sectional area of flow (m2), Q is the discharge (m3/s), q1 and q3 are the left- and right-bank lateral discharges, t is time (s), and x is the longitudinal distance along the channel (m). The momentum equation is
CAN. J. CIV. ENG. VOL. 19, 1992
442
where g is the acceleration due to gravity (m/s2), H i s the water surface elevation (m), Mf is a momentum correction factor given by
the objective function should include the differences in depth and discharge between the observed and simulated downstream hydrographs. The relative errors at the jth time increment can be defined as
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and F is a coefficient, defined as
In relationships [3] and [4] subscripts i and j take the values 1, 2, and 3, which refer to the left flood plain, the main channel, and the right flood plain respectively; K is the subsection conveyance; 6 is the Boussinesq or momentum correction factor for a single subsection; and q and X are coefficients that account for the dynamics of flood plain flow and LMT respectively. The parameter A, which provides a means of accounting for LMT between the main channel and flood plain flows, is estimated using the following relationship (Nicollet and Uan 1979):
-
[5]
X
=
' L (1
-
2
-
- A,) cos
rs
+
(1
+
A,)
1
where r is the ratio of flood plain to main channel hydraulic radii; rs = 0.3; and Xo = 0.9 (nl/n2)-"6, in which n, and n2 are Manning's roughness coefficients for the flood plain and main channel zones respectively. Bhowmilk and Demissie (1982) show that flood plain flows are a function of compound channel geometry and the flood return period and can be as large as the main channel flow. They also demonstrate that when flood plain depths exceed 35% of the main channel depth, the composite channel can be treated as a single unit. While our model accounts for flood plain contributions to system conveyance, because velocities are much higher in the main channel than elsewhere, different friction slopes are assigned to the main channel and flood plain zones as follows: [6]
Sfi = qiSf;
i = 1, 2, and 3
in which Sf is the composite friction slope, qt
=
1.O, and
where yri is the flood plain to main channel depth ratio. Flood routing through compound channels can be accomplished by solving equations [I]-[4]. Equations [l] and [2] are nonlinear partial differential equations which can only be solved numerically. The weighted four-point implicit finite difference scheme (Amein and Fang 1970) was selected for this exercise. Optimization model Objective function As in many optimization problem, an objective function, Fo, which includes all parameters to be estimated, needs to be specified. In this unconstrained optimization problem,
where subscripts c and o refer to the calculated and observed values respectively. Two broad classes of objective function can be considered (Wormleaton and Karmegam 1984): [8]
Minimize Fo
=
Max(/ E',
[9]
Minimize Fo
=
C (E',
m
I + I E$ 1)
+ E$)~
j= 1
Equation [8] represents the "minimax" criterion, particularly suitable when the error at a critical time is of interest (e.g., peak discharge or depth). The least-square criterion, represented by equation [9], is adopted in this study because it is considered more important that the depth and the discharge be modelled correctly over the entire hydrograph. For both Rosenbrock's method and Powell's algorithm, the convergence criterion which has to be satisfied before terminating the optimization procedure was based on the difference in the objective function values for two successive iterations. In this instance, the difference is compared with a given tolerance value for this application) and the procedure terminates if the former is less than or equal to the latter. Using the Nelder and Meade simplex method, the objective function is evaluated at the vertices of the simplex and the variance of these objective function values is estimated. If it is less than or equal to a given critical value (10-lo for this application), the optimization procedure terminates and the desired minimum is reached. Model parameters The optimization model parameters, i.e., the parameters that need to be estimated using optimization methods, are the roughness coefficients of the different flow subsections and those parameters describing the composite section geometry. T o keep the number of model parameters small, only the main channel is assumed to be irregular; the floodplain zones are approximated by rectangular sections. In describing the main channel geometry, the following relationship is assumed:
where T is the channel top width, y is the flow depth, and c and e are parameters to be estimated through the optimization exercise. The value of e varies with cross-sectional shape; it is equal to zero for the rectangular and unity for the triangular, and takes any value between 0 and 1 for convex parabolic shapes. The main channel cross-sectional area and wetted
ABIDA AND TOWNSEND
443
perimeter, which are both required for the routing exercise, must be evaluated. The cross-sectional area is given as
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The wetted perimeter for any irregular section is obtained from
In the special case of a rectangular section, e = 0; and equation [12] yields P2 = 2y2. Therefore, since the wetted perimeter includes the channel floor width, this component was added to the above solution. Description of the optimization algorithms Powell's method Powell's multivariable nonlinear optimization method is a pattern search method, in that it follows m unidirectional searches along the coordinate axes and then searches for the minimum along a pattern direction S; defined as
[13]
Si
=
Xi -
where X, is the point obtained at the end of m univariate steps and Xi-, is the starting point. In the next cycle, one of the coordinate directions is dropped in favor of the pattern direction, and in the subsequent cycle, another coordinate direction is discarded in favor of the newly generated pattern direction. After all the initial coordinate directions are discarded, they are recovered and the procedure is repeated. This iterative procedure continues until the desired minimum point is reached. Rosenbrock's method This direct search method proceeds as follows: 1. A starting point and initial step sizes are chosen and the objective function is evaluated. 2. The first variable, X I , is stepped a distance S1parallel to the axis and the function is evaluated. If the objective function value decreased, the move is termed a "success" and S1is increased by multiplying it by a factor y (y r 1.0); otherwise the move is termed a "failure" and S1 is decreased by a factor a (0 Ia I 1) and the direction of movement is reversed. 3. Step 2 is repeated for all variables in consecutive sequences until a success and failure have been encountered in all directions. 4. The axes are then related using the Gram-Schmidt orthogonalization process (Beaumont 1965). 5. A search is made in each direction of the new coordinate axes. 6. The procedure terminates when the convergence criterion is satisfied. Nelder and Meade simplex algorithm The geometric figure formed by a set of m + 1 points in m-dimensional space is called a simplex. The basic idea in the simplex method is to compare the values of the objective function in the m + 1 vertices of a general simplex and move this simplex gradually towards the optimum point during the iterative process. The movement of the simplex
FIG. 1. Definition sketch of Treske's prismatic channel: (a) cross section; (b) steady uniform flow profile.
is achieved by using three operations, known as reflection, contraction, and expansion. For further information on these and other optimization methods, the reader is referred to Wilde and Beightler (1967). Simulations and results The laboratory data set used in this study is that of Treske (1980). Treske's experimental compound channel has two flood plains of different widths (Fig. 1). The working length of the channel was 210 m, the bed slope was 0.019%, and Manning's n for the channel bed was calculated to be 0.012. Depth and discharge, as functions of time, were measured at stations located at either end of the channel. Measured discharge hydrographs at the upstream station were used as the upstream boundary condition for the unsteady flow model. Hydrographs recorded at the downstream station supplied the differences between the observed and simulated stages and discharges required for the objective function evaluation. It is also important to note that since Manning's n was assumed to be the same for the three different subsections, the number of optimization variables was reduced to 5. Powell's multivariable nonlinear optimization method was first applied to solve the problem and estimate the parameters. In this case, convergence was very slow and the objective function never reduced to an acceptable value. The poor performance of Powell's algorithm is explained by its major drawback, namely the fact that the search directions tend to be linearly dependent on each other. This results in the collapse of the search directions, which limits the search space and therefore inhibits convergence to the correct minimum. Two sets of initial guesses for the optimization variables were selected at random for both the Rosenbrock and simplex methods. The difference between the two sets of initial estimates lies in their corresponding initial objective
CAN. J. CIV.
ENG. VOL. 19, 1992
TABLE1. Summary of results
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Optimal values* Variable and symbol
Trial value (1)
R,
Manning's n Width, T I (m) Width, T, (m) Coefficient, c Exponent, e
0.015 3.20 1.60 1.OO 0.00
0.016 3.28 1.67 0.99 0.00
Optimal values*
sI
Trial value (2)
R,
s 2
Actual value
0.011 3.17 1.69 1.18 0.00
0.010 2.30 1.20 0.90 0.00
0.009 3.23 2.15 1.01 0.00
0.014 2.80 1.96 1.44 0.00
0.012 3.00 1.50 1.25 0.00
*R for Rosenbrock method; S for simplex method. m
j Observed
t\l
"?
Simulated
O
1
Observed
Simulated
- Optimal
Data Actual Data
o
o
,
0
Time, T (min)
, , 15
,
30
,
45
, , .
