Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt 299 A NONLINEAR OPTIMIZATION MODEL FOR ESTIMATING MANNING S ROUGHNESS COEFFICIENT Maysoon Kh. Askar and K. K. Al-Jumaily * Dr., University of Salahaddin, Erbil, Iraq ** Prof. Dr., University of Technology, Baghdad, Iraq ABSTRACT The present research formulated mathematical model for estimating Manning's roughness coefficient in open canal networks. The inverse problem was formulated as a nonlinear optimization model with n value as an unknown variable by embedding the finite difference approximation of the Saint-Venant equations for unsteady flow in an open canal as equality constraints. The Sequential Quadratic Program was used to solve the optimization model and to minimize least square objective function. The Hilla-Kifl irrigation project was selected as a field case study for this work. The project was situated in the southern sector of Iraq. Six sections on Hilla Main Canal and Three sections on Distributary canal were chosen. The Root Mean Square Error of n values is found between 0.00075 and 0.0030 for different time increment. Optimization model showed a deviation between calculated and designed n values of about 6.4%. INTRODUCTION The objective of the present research is to formulate a new approach to estimate Manning's roughness in a canal network with different types of hydraulic structures such as: gates, drops, tail and side escapes, sudden change in cross sections, and combination of such structures which are performed by embedding the implicit finite difference method of the Saint-Venant equations for flow into a nonlinear optimization model. The determination of an optimum value of the flow resistance coefficient and other parameters such as weighting factor, and distance increment (x) are the primary task in the calibration of one-dimensional unsteady flow model. An obvious and widely used method to determine the parameters is by a trial and error technique in which the continuity and momentum equations (Saint-Venant equations) can be repeatedly solved by a finite difference technique with certain boundary conditions for different assumed function of the parameters.
300 Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt The best fit between observed and calculated values of discharge or depth is the optimum value. The literature relevant to estimate or calibrate n values for unsteady open-channel flow is sparse. In the present research, the sequential quadratic programming algorithm description by Nash et al. (996) was used to solve the nonlinear optimization model to estimate value of Manning's roughness coefficient. GOVERNING EQUATIONS An optimization model was formulated by using Sequential quadratic programming with inverse problem to estimate the Manning's roughness coefficient as unknown variable. The Saint-Venant equations for trapezoidal canal with external and internal boundary conditions as discussed by Askar (2005) by a single weighted roughness coefficient at each cross section were used as governing equations. The governing equations can be written as follows: y v y B + A + vb = t x x q () g y x + v v x + v t = g ( S S ) o f q. v A (2) where B is the free surface flow width; x = distance along the longitudinal axis of the canal; A = active cross-sectional area; t = time; y = depth of flow; q = lateral inflow (positive) or outflow (negative); g = acceleration due to gravity; S f = the energy line slope computed from Manning equation; S o = the canal bed slope and v = velocity. MODEL FORMULATION The inverse problem is formulated as a nonlinear optimization model with Manning's roughness coefficient as the unknown variable. The formulation of the inverse problem for canal network can be summarized as follows: - Objective Function The objective function is to minimize the sum of squares of the difference between the simulated and observed values of the flow depth.
Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt 30 Minimize f ( y) = N m j= i= y (3) j+ j+ si+ yoi+ j+ yoi+ where y s = simulated depth values at several observation stations; y o = observed depth at the corresponding points; i = grid point in space; m = total number of sections for every channel; j = time level; and N = total number of times. The finite difference method was used to solve the inverse problem by using n values as the unknown. However, the simulated depth used in the objective function can be calculated from the equation: 2 y k + s = y k s y + dn n (4) where n = Manning's roughness coefficient. 2- Nonlinear Constraints In this research, the Saint-Venant continuity and momentum equations for a trapezoidal canal were used as nonlinear constraint. The Saint-Venant equations are written as shown in equations (), and (2), respectively. The downstream boundary condition is used also as nonlinear constrains. It can be written as follows: Where J + J + J + G N = v N R n S f = 0 (5) n S f = friction slope of canal which is given by Manning's equation: 4 2 2 3 S f = v n / R (6) Internal boundary conditions such as gates, drops, escapes, looped, divergent, etc used the empirical equation of each control structure, flow and conservation equations as nonlinear constraint also, Askar (2005). The specified upstream boundary condition is included as a linear constraint. This can be given as follows: + ( ) f ( t ) Q j = (7)
302 Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt where f ( t) = specified inflow hydrograph. The nonlinear optimization problem described in this section was solved using the Sequential Quadratic Programming (SQP) algorithm. The basic idea for (SQP) framework for equality constraints is analogous to Newton's method for unconstraint minimization: At each step, a local model of optimization problem is constructed and solved, yielding a step towards the solution of the original problem. In the nonlinear optimization programming both the objective function and constraints must be modeled as: Minimize f ( x) (8) Subject to ( x ) = 0 g (9) FIELD INVESTIGATIONS The following constituents of canals network, Fig. (), were chosen to be used in the measurements: Main canal: a part of the Hilla Main Canal, HC was chosen from km 2 + 000 up to km 3+500 with base width of 2.5m, side slope of.5h: V, design longitudinal slope of 5 cm/km, design Manning's n of 0.05, design discharge of 4.2 m 3 /s, and design depth of 3.5 m. The Hilla Main Canal takes off from the Kifl Main Canal about 4.9 km downstream of the main intake structure of the project. Distributary canal: a part of distributary canal, R from km 0+035 to 0+500 was chosen with base width of.5 m, side slope of.5h: V, design longitudinal slope of 0 cm/km, design discharge of 2.5 m 3 /s, design depth of 2m, and design Manning's n of 0.05. The following steps were followed in gathering the field data:. Six sections on the main canal were selected as shown in Fig. (). The first and sixth sections represent the upstream and downstream boundary conditions, respectively. 2. Three sections were chosen at distributary canal, the first one at km 0+035, the second at km 0+250, and the third at km 0+500. 3. The rating curves were plotted for sections No., 3, 5, 6, 7, and 9, using a set of measurements of discharge and depth of the flow. These measurements were made over a period from 2/2/2002 up to 5//2003 so as to obtain an
Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt 303 accurate relationship between the discharge and depth of flow. The depth of water was measured with a graduated rod. 4. The water level at each section was measured instantaneously (every 5 minutes over a period up to six hrs) at fixed time from 6//2002 up to 22//2003 for calibration purposes; this was achieved by changing the height of the head gate (h g ). 5. The discharge was determined by using the rating curve for each cross section. 6. For verification purposes, the discharges and the depths at section, 3, 5, 6, and 9 were measured every hr up to 6 hr and continued for several days from 23//2003 to 30//2003. km 0 + 500 Q U/s t Hilla main canal Canal no 2 Canal no 3 km 0 + 750 km 0+ 035 km 0 + 250 km 0 +500 km + 000 Canal no 3 km + 250 3 4 5 7 8 9 Q D/s t 6 km +500 Q t D/s Fig. () Schematic layout of Hilla-Kifl network EVALUATION OF THE MODEL Performance of the proposed optimization model for the calibration of Manning's roughness coefficient was evaluated using hypothetical open canal network.
304 Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt This example problem, as shown in Fig. (2), includes flow in simple hypothetical dendritic canals with multiple n values corresponding to different reaches. The observation data for these cases were calculated by using model development (Canal Network Optimization Program), Askar (2005) for assumed true values of n. These calculated observation data for discharge and flow depth were then used in the optimization model to estimate the actual Manning's roughness coefficient. Identical initial and boundary conditions were applied while obtaining the calculated observation data and while solving the optimization model. This approach was used to evaluate the performance of the methodology. The advantage of using observation data is that to know the actual deviation between the true values and calculated values of n. The characteristics of simple hypothetical network are shown in Table (). The calculated (simulated) flow measurement data were used in the objective function of the optimization model to correct n values for all three canals. The initial estimates of n for all three canals were specified to be 0.02. Optimization program was carried out to estimate n values applied to Al-Kifl canal system. The arrangement shown in Fig. (2) was used for this purpose. The initial assumed n value was taken in the optimization model equal to 0.0. The root mean square error of n values were taken as indicator to comparison between estimated and observed n value Table (2) shows the Root Mean Square Error (RMSE) of Manning's roughness coefficient with different time increment for the arrangement of Fig. (2). It can be observed from Table (2) that errors in estimated values of n increase as t was increased. Identical results were obtained when the initial estimates for n was taken equal to 0.05. The result of the optimization program showed that the values of n = 0.0432, 0.04, 0.0379 for canals no., 2, 3 which were calculated respectively, concerning Fig. (). The Root Mean Square Error (RMSE) of n values in sections 3 and 5 = 0.054, and 0.057 respectively. Canal No.2 Q t Fig. (2) Schematic layout of simple network
Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt 305 Identical initial and boundary conditions were applied while obtaining the calculated observation data and while solving the optimization model. This approach was used to evaluate the performance of the methodology. The advantage of using observation data was that to know the actual deviation between the true values and calculated values of n. Table () Characteristics of simple network Characteristic Canal Canal 2 Canal 3 Length of canal (m) 000 500 500 Bed width (m) 2.8.6 Side slope (Z).5.5.5 Longitudinal slope (cm/km) 0 0 0 Manning's Roughness Coefficient (n) 0.06 0.022 0.08 Table (2) Root Mean Square Error (RMSE) of n values in canals with different time increment X t RMSE of Manning's Roughness Coefficient (m) (min) Canal Canal 2 Canal 3 00 5 0.030 0.035 0.035 00 30 0.037 0.043 0.047 00 60 0.038 0.04 0.049 00 20 0.048 0.044 0.052 SUMMARY AND CONCLUSIONS In this research, the Sequential Quadratic Programming was used for estimating the Manning's roughness coefficient from unsteady flow measurement data in open canal network. Four-point implicit finite difference method was used to solve the governing equations. Performance of the proposed model was evaluated for small dendritic network and for Al-Kifl canal. Results of these evaluations showed that potential applicability of the proposed model for estimating Manning's roughness coefficient in open canal network with different types of hydraulic structures as internal boundary conditions. It was found that the model performs satisfactorily when discharge measurements were available. Further testing of the methodology using field data is necessary to establish its practical utility for natural channels.
306 Twelfth International Water Technology Conference, IWTC2 2008, Alexandria, Egypt REFERENCES. Askar, M. Kh., 2005, "A Hydrodynamic Model for Simulation of Flow in Open Canal Network", Ph.D. Thesis, University of Technology, Baghdad. 2. Backer, L., and Yeh, W.W.G., 972, "The Identification of Parameters in Unsteady Open Channel Flows", Water Resource Research, Vol. 8, No. 4, pp. 956-965. 3. Backer, L., and Yeh, W.W.G., 973, "The Identification of Multiple Reach Channel Parameters", Water Resource Research, Vol. 9, No. 2, pp. 326-335. 4. Fread, D.L., and Smith, G.F., 978, "Calibration Technique for -D Unsteady Flow Models", Journal of the Hydraulics Division, ASCE, Vol. 04, No. 7, pp. 027-044. 5. Nash, S.G., and Sofer, A., 2000, "Linear and Nonlinear Programming", McGraw- Hill Company, Inc., New York. 6. Powell, M.J.D., 974, "Introduction to Constrained Optimization", Numerical Methods for Constrained Optimization, P.E. Gill and W. Murray, eds., Academic Press, London, pp. -28. 7. Ramesh, R., Datta, B., Bhallamudi, M., and Narayana, A., 2000, "Optimal Estimation of Roughness in Open Channel Flows", Journal of Hydraulic Engineering, Vol. 26, No. 4, pp. 299-303. 8. Venutelli, M., 2002, "Stability and Accuracy of Weighted Four-Point Implicit Finite Difference Schemes for Open Channel Flow", Journal of Hydraulic Engineering, ASCE, Vol. 28, No. 3, pp. 28-288.