KSCE Journal of Civil Engineering (2013) 17(5):1139-1148 DOI 10.1007/s12205-013-0037-2
Water Engineering
www.springer.com/12205
Application of Excel Solver for Parameter Estimation of the Nonlinear Muskingum Models Reza Barati* Received February 2, 2011/Revised January 13, 2012/Accepted October 21, 2012
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Abstract The Muskingum model continues to be a popular procedure for river flood routing. An important aspect in nonlinear Muskingum models is the calibration of the model parameters. The current study presents the application of commonly available spreadsheet software, Microsoft Excel 2010, for the purpose of estimating the parameters of nonlinear Muskingum routing models. Main advantage of this approach is that it can calibrate the parameters using two different ways without knowing the exact details of optimization techniques. These procedures consist of (1) Generalized Reduced Gradient (GRG) solver and (2) evolutionary solver. The first one needs the initial values assumption for the parameter estimation while the latter requires the determination of the algorithm parameters. The results of the simulation of an example that is a benchmark problem for parameter estimation of the nonlinear Muskingum models indicate that Excel solver is a promising way to reduce problems of the parameter estimation of the nonlinear Muskingum routing models. Furthermore, the results indicate that the efficiency of Excel solver for the parameter estimation of the models can be increased, if both GRG and evolutionary solvers are used together. Keywords: flood routing, hydrologic model, spreadsheets, parameter, estimation ··································································································································································································································
1. Introduction River flood routing is of paramount importance to water resources engineers (Choudhury et al., 2002). From one point of view two general approaches, “hydrologic routing” and “hydraulic routing”, are employed to route the flood wave in natural channels (Choudhury, 2007). The former is based on the storage-continuity equation, and the momentum and continuity equations, the SaintVenant equations, governing the phenomenon are solved numerically in the latter. In other words, the momentum equation is not used in the hydrologic procedures. This simplification causes that the procedures need to calibrate the hydrologic parameters based on recorded data. Therefore, a flooding is modeled in a hydrologic routing procedure by the hydrologic practitioners. In most of cases the field data scarcity prevents the use of the Saint-Venant equations to route floods in rivers (Akbari et al., 2012; Akbari and Barati, 2012). The Muskingum model is one of the more sophisticated of the hydrologic flood routing procedures. The model uses the following hydrologic budget equation that is the basis for a hydrologic procedure of routing: dS ------ ≈ ∆ -----S- = I – Q dt ∆t
downstream, respectively (L3T−1); S is the channel storage (L3); t is the time (T); and ∆S/∆t is the change in storage during a time interval ∆t. The value of ∆S/∆t is positive when the storage is increasing and negative when the storage is decreasing (Chow, 1959). The channel storage that depends on the inflow and outflow discharges and on the geometric and hydraulic characteristics of the channel is commonly modeled by using either a linear form or any of the nonlinear forms of Muskingum models. In the linear Muskingum model, the following storage equation is used: S = K[ XI + ( 1 – X )Q ]
(2)
Where K is a proportionality coefficient, which has a value close to the flow travel time through the river reach (T); and X is a dimensionless weighting factor commonly varying between 0 and 0.5. The value of ∆S (i.e., the change in storage over time interval ∆t) between time t and t+1 can be written as:
(1)
Where I and Q denote the flow rates of upstream and
∆S = St + 1 – St = K{ [ XIt + 1 + ( 1 – X )Qt + 1] – [ XI t + ( 1 – X )Qt ]}
(3)
Eq. (1), the change in storage, can be rewritten as: ( I t + It + 1 ) ( Qt + Qt + 1 ) ∆t –------------------------ ∆t St + 1 – St = -------------------2 2
(4)
*P.G. Researcher, Young Researchers Club and Elites, Mashhad Branch, Islamic Azad University, Mashhad 9187147578, Iran (Corresponding Author, Email:
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Combining Eqs. (3) and (4) and simplifying gives: (5)
Qt + 1 = C 1It + 1 + C2 It + C3 Qt
the reach, and SU and SD are the storages related to the depths at the upstream and downstream end sections, respectively. By eliminating y from Eq. (10), SU and SD can be expressed as:
Where,
M -----
∆t – 2KX , C = --------------------------------∆t + 2KX , C = 1 – ( C + C ) (6) C1 = --------------------------------2 3 1 2 2K ( 1 – X ) + ∆t 2K ( 1 – X ) + ∆t
Traditionally, K and X are estimated by plotting accumulated storage versus [XIt + (1 − X)Qt] of a given reach for different values of X. After X is assumed, the values of [XIt + (1 − X)Qt] are computed using recorded data and plotted against St. The value of X that minimizes the width of the plotted loop can be chosen as the correct value of X, and the line slope for the correct value of X can be chosen as K. In other words, these coefficients can be estimated physically. It is notable that the curve of the storage versus the weighted-discharge is a loop because the storage for a given discharge on the rising (or falling) limb of the flood wave will be greater than (or less than) the storage corresponding to the condition of steady flow (Chow, 1959). This trial and error graphical procedure is somewhat subjective and time consuming. Thus, several alternate numerical methods have been developed. These methods which have no physical base consist of linear and nonlinear regression schemes based on the least-squares method, orthogonal least-squares methods, piecewise linear iterative regression methods, iterative optimization techniques, methods of moments and cumulates, and etc., (AlHumoud and Esen, 2006). In natural channel reaches, it is common to observe a nonlinear storage-discharge relationship that causes the use of the linear Muskingum model could result in significant error in the prediction of flood behavior during its propagation along a river (Gill, 1978; Tung, 1985). Three forms of nonlinear Muskingum models have been suggested in the literature for taking into account the nonlinearity (Chow, 1959; Gill, 1978; Papamichail and Georgiou, 1994; Mohan, 1997; Luo and Xie, 2010) and they are: p
p
S = K [XI + ( 1 – X )Q ] p1
(7)
p2
S = K [XI + ( 1 – X )Q ] S = K [XI + ( 1 – X )Q ]
(8)
m
(9)
These models have additional fitting parameters as exponents (p1 and p2; p; and m) for considering the effects of nonlinearity between weighted-flow and storage volume, respectively. The basic concepts of the development of the storage equations can be investigated. For example, for the Eq. (7), it can be assumed that the upstream and downstream end sections of the reach have the same mean discharge and storage relationships with respect to the depth of flow y as (Chow, 1959): N
N
M
I = Ay , Q = Ay , SU = By , SD = By
M
(10)
Where A and N are the depth-discharge characteristics of the sections, B and M are the mean depth-storage characteristics of
M -----
I N Q N SU = B ⎛ ---⎞ , SD = B ⎛ ----⎞ ⎝ A⎠ ⎝ A⎠
(11)
Then, the storage at any given time can be expressed as: p
p
S = XSU + ( 1 – X )SD = K [XI + ( 1 – X )Q ]
(12)
Where, K = B/AM/N, p = M/N. It is notable that, for example, discharge varies with the five-thirds power of the depth on the basis of the Manning formula, and storage varies with the first power in prismatic rectangular channels (i.e. N = 5/3 and M = 1, therefore, p = 0.6). However, in natural channels, M may be considerably greater than unity and hence p is larger than 0.6. The other storage equations [i.e., Eqs. (8) and Eq. (9)] have similar basic concepts.
2. Literature Review Several methods for estimating parameters of the nonlinear Muskingum models have been attempted by various researchers (Gill, 1978; Tung, 1985; Yoon and Padmanabhan, 1993; Mohan, 1997; Kim et al., 2001; Das, 2004; Geem, 2006; Das, 2007; Chu, 2009; Chu and Chang, 2009; and Luo and Xie, 2010). Gill (1978) used a Least-Squares Method (LSM) to find the values of the parameters in the nonlinear Muskingum model. However, LSM technique arbitrarily selected points to solve the simultaneous nonlinear equations (Tung, 1985). Tung (1985) offered parameter estimation using the Hook-Jeeves (HJ) pattern search in conjunction with the Linear Regression (LR), the Conjugate Gradient (CG), and Davidon-Fletcher-Powell (DFP) algorithms. The performances of these methods were compared with Gill’s procedure and (HJ+CG) and (HJ+DFP) were found to produce better results. Yoon and Padmanabhan (1993) discussed six methods for estimating the parameters of linear and nonlinear Muskingum models. The Nonlinear Least Squares Regression (NONLR) method that was proposed for the nonlinear routing model is an iterative procedure. Mohan (1997) suggested a calibration technique for the nonlinear Muskingum models by using a Genetic Algorithm (GA). The results indicated that GA is better than the above methods and does not require the process of assuming initial values close to the optimal solution. Kim et al. (2001) implemented the Harmony Search (HS) algorithm to the same parameter calibration problem. HS algorithm found the best values of the parameters compared to the previously mentioned methods. The technique also did not require initial guess for the hydrologic parameters. However, this method needs careful attention for the harmony memory considering rate and pitch adjusting rate. Das (2004) estimated parameters for linear and nonlinear Muskingum models using the Lagrange Multiplier (LM) method. This algorithm transforms the constrained parameter optimization problem into an unconstrained problem.
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Application of Excel Solver for Parameter Estimation of the Nonlinear Muskingum Models
However, the results obtained by the Lagrange multiplier were not as good as the results obtained by other techniques. Geem (2006) suggested the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique which searches the solution area based on mathematical gradients. This algorithm requires assuming initial value and requires complicated calculus. Das (2007) offered a chanceconstrained optimization-based model for Muskingum model parameter estimation. This model is very complex and requires massive computation for parameter estimation of Muskingum model (Luo and Xie, 2010). Chu (2009) estimated parameters for Muskingum model using a Neuro-Fuzzy approach that has no physical base. The method can be an alternative in application of the Muskingum model. Chu and Chang (2009) used PSO for calibration of nonlinear Muskingum model. They compared PSO results with other traditional methods. The results obtained by PSO were not as good as the results obtained by other algorithms such as HS. Luo and Xie (2010) recommended an Immune Clonal Selection Algorithm (ICSA) to find the values of the hydrologic parameters in the nonlinear Muskingum model. ICSA is a new intelligent algorithm which can effectively overcome the prematurity and slow convergence speed of traditional evolution algorithm (Luo and Xie, 2010). However, this algorithm requires careful attention for the algorithm parameters such as the clonal scale, mutation probability, and crossover probability (Barati, 2011). On the other hand, in the past several years, the Microsoft Excel has been successfully used in hydraulic and hydrologic modeling. For example, the Excel solver was used to solve reservoir optimization problems for water supply and energy generation by Fontane (2001). Huddleston et al. (2004) applied the Excel to the water distribution network analysis. The results showed that the application of this technology is an efficient way to solve a relatively complex engineering system while minimizing the computational burden. On the other hand, many problems in water engineering were used to solve by a trial-and-error method, however by using the Excel solver, these problems can be solved very easily, with high accurate results, and shortest operating time. For example, the Excel solver can compute normal and critical depths for different cross sections of the channel flow (Wong and Zhou, 2004), water depths before and after a hydraulic jump, Horton’s infiltration parameters, constant rainfall loss, and aquifer constant for confined aquifers. Other applications of Excel software are Lee (2003), Lee and Noh (2003), Yidana and Ophori (2008), Grabow and McCornick (2007), and Bhattacharjya (2011). In conclusion, in former researches, the Excel solver has been successfully applied including water engineering problems (Fontane, 2001; Huddleston et al., 2004; Wong and Zhou, 2004). In this study, in order to eliminate the limitation of the aforementioned parameter estimation procedures, parameter estimation for the nonlinear Muskingum models using Excel solver is developed. The performance of this approach is compared with other reported techniques (Gill, 1978; Tung, 1985; Mohan, 1997; and Luo and Xie, 2010) through several Vol. 17, No. 5 / July 2013
performance evaluation criteria, and several aspects of the procedures.
3. Simulation Procedure of the Nonlinear Muskingum Models Based on nonlinear storage equation [Eqs. (7), (8) or (9)], procedure of the nonlinear Muskingum model is performed. Although, each one represents a nonlinear Muskingum model, because the Eqs. (7) and (8) are not as popular as Eq. (9), this paper is focused on the latter model for comparison. By reorganizing Eq. (5), the rate of the outflow Qt can be expressed as: S 1 Qt = ⎛ -----------⎞ ⎛ ----t⎞ ⎝ 1 – X⎠ ⎝ K⎠
1⁄m
X – ⎛ -----------⎞ It ⎝ 1 – X⎠
(13)
With combining Eq. (13) and Eq. (1), the change in storage during a time interval can be declared as: 1⁄m ∆S 1 ⎞ ⎛ St⎞ 1 -------t = – ⎛ ---------- ---+ ⎛ -----------⎞ It ⎝ 1 – X⎠ ⎝ K⎠ ⎝ 1 – X⎠ ∆t
(14)
The next accumulated storage can be expressed as: S t + 1 = S t + ∆S t
(15)
The routing procedure involves the following steps (Tung, 1985; Kim et al., 2001): Step 1: Determine values of the hydrologic parameters (K, X, and m) by implementing optimization algorithm. Step 2: Calculate the initial storage volume by Eq. (9), where the initial outflow is the same as the initial inflow. Step 3: Calculate the change in storage during a time interval by Eq. (14). Step 4: Estimate the next accumulated storage by Eq. (15). Step 5: Calculate the magnitude of the outflow at the next time by Eq. (13). The routing procedure executes unit all of the discharge hydrograph ordinates are simulate. Similar procedures may be developed for the nonlinear models of Eqs. (7) and (8).
4. Excel Solver In general, a user can find an optimal value for an objective function in target cell on a worksheet by using Excel solver. Excel solver acts on a group of cells that are directly or indirectly related to the objective function in the target cell, and it adapts the values in the user identified adjustable cells to optimize the value of the target cell. It is notable that the constraints that refer to adjustable cells effect on the target cell formula (i.e., objective function). Parameter estimation of the nonlinear Muskingum models is a highly nonlinear optimization problem (Luo and Xie, 2010). In this study, Excel solver is suggested for this scope. For parameter estimation of the nonlinear Muskingum models, Excel software can be used in calibrating the design parameters with two powerful search methods. These procedures consist of (1)
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Generalized Reduced Gradient (GRG) solver (Lasdon et al., 1978; Murtagh and Saunders, 1978; Lasdon and Smith, 1992) and (2) evolutionary solver (Premium Solver Platform, 2010). GRG solver is nonlinear optimization code developed by Leon Lasdon from University of Texas at Austin and Allan Waren from Cleveland State University. GRG and its specific implementation have been proven in use over many years as one of the most robust and reliable approaches to solve difficult nonlinear programming problems (Lasdon and Smith, 1992). Several techniques which consist of the steepest descent, conjugate gradient, quasi-Newton, and the Newton methods can be used to determine the search direction in the optimization procedure. Among these methods, the Newton method is the most intensive technique and the steepest descent method is the simplest technique while two others are mediatory techniques. In GRG solver two techniques are available to determine the search direction. The default choice is the quasi-Newton method which is a gradient-based technique and involves complicated calculus and the alternative choice is the conjugate gradient method (Premium Solver Platform, 2010). The former method maintains an approximation to the Hessian matrix which requires more storage while the later method does not require storage for the Hessian matrix and still performs well in most cases. GRG solver is capable of switching automatically between the quasiNewton and conjugate gradient methods depending on the available storage. On the other hand, Evolutionary solver is a hybrid of genetic and evolutionary algorithms and classical optimization methods, including gradient-free direct search methods, classical gradient-based quasi-Newton methods, and even the simplex method (Premium Solver Platform, 2010). The comparisons of different aspects of two procedure show that Evolutionary solver relies in part on random sampling. This makes it a stochastic method, which may yield different solutions on different runs. In contrast, GRG Solver is a deterministic method. On the other hand, GRG solver needs the initial values assumption for the parameter estimation while evolutionary solver requires the determination of the algorithm parameters such as mutation rate, population size, and random seed. GRG solver stops if the absolute value of the relative change in the objective function is less than the value of tolerance for the last five iterations while the stopping condition for evolutionary solver is satisfied if 99% of the population members all have fitness values that are within the convergence tolerance of each other (Premium Solver Platform, 2010). For more details on the optimization algorithms of Excel software, the reader is referred to Premium Solver Platform (2010).
2009; Toprak et al., 2009; Toprak, 2009; Luo and Xie, 2010). The following evaluation criteria are adopted for verifying the efficacy of the Excel solver and the other parameter estimation procedures. 5.1 Accuracy of Procedure Consideration Accuracy of the procedure can be measured by the sum of the square of the deviation between the routed and observed outflows (SSQ) [L6T−2] (Mohan, 1997). SSQ is given as: N
SSQ =
ˆ
∑ { Ot – O t }
2
(16)
t=1
ˆ = routed Where Ot = observed outflow at time t (L3T−1); O t 3 −1 outflow at time t (L T ); and N = total number of discharge hydrograph ordinates to be simulated. 5.2 Magnitude of Peak Consideration The absolute value of the deviations of peak of routed and observed outflows (DPO) [L3T−1] is considered as accuracy of the amount of peak outflow (Yoon and Padmanabhan, 1993). DPO is given as: DPO = Peakrouted – Peakobserved
(17)
5.3 Mean Absolutely Relative Error Consideration The mean absolutely relative error between the routed and observed outows (MARE) [−] is considered as error mode (Toprak, 2009). MARE is given as: N ˆ 1 Ot – O (18) MARE = ---- ∑ ----------------t N t = 1 Ot 5.4 Closeness of Shape and Size of the Hydrograph Consideration The closeness of shape and size of the hydrograph with the observed data can be measured by the Nash-Sutcliffe criterion of the variance explained (ηq) [%] (McCuen et al., 2006). The ASCE Task Committee (1993) suggested the use of the NashSutcliffe criterion for testing the goodness of fit of the flood hydrograph simulation model. The variance explained in percentage is given as: N
ˆ
∑ ( Ot – Ot )
2
t=1 × 100 ηq = 1 – --------------------------------N 2 ( O – O ) Obs t ∑
(19)
t=1
Where, O Obs = mean of the observed outflow.
5. Performance Evaluation Criteria 6. Simulation Methodology In order to comparison between results of different methods, several performance evaluation criteria were proposed (Yoon and Padmanabhan, 1993; Mohan, 1997; McCuen et al., 2006; Toprak and Savci, 2007; Toprak and Cigizoglu, 2008; Chu,
The discrete time-state variable modeling of the nonlinear Muskingum model is modeled in spreadsheet format. The optimization model may be written as:
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Application of Excel Solver for Parameter Estimation of the Nonlinear Muskingum Models
Minimize, SSQ =
N
⎧
t=1
⎩
∑ ⎨ Ot –
1 ⎞ ⎛ St⎞ ⎛ ---------- ---⎝ 1 – X⎠ ⎝ K⎠
1⁄m
⎫ X – ⎛ -----------⎞ It ⎬ ⎝ 1 – X⎠ ⎭
2
(20)
Subject to, 0 ≤ X ≤ 0.5
(21)
This constrain is considered because the storage is a function of the outflow (i.e., X = 0) when the stages in a reach are determined by the control at its downstream end, for example, at the spillway of a level-pool reservoir. On the other hand, in uniform channels, equal weight is given to inflow and outflow (i.e., X = 0.5) (Chow, 1959). In general, when the storage due to backwater effect at the upstream end of the reservoir is significant, X will be changed between 0 and 0.5. It is notable that other hydrologic parameters (i.e., K and m) have not a general constrain. A simple spreadsheet methodology that is shown in Fig. 1 is used for parameter estimation of the nonlinear Muskingum model. In this procedure, the numerical simulation of the flood routing process is carried out by using the embedded optimization approaches and spreadsheet solver. It can be seen that the data of time, inflow, and observed outflow are input in A2 to A23, B2 to B23, and C2 to C23 cells, respectively. Also, the upper and lower limits for X, K and m and their initial values are input in J2 to L2, J3 to L3, and J4 to L4 cells, respectively. The J4 to L4 cells that refer to the hydrologic parameters are adjustable cells. It is notable that the initial input values of these parameters are changed by Excel Solver to determine their optimal values which are shown in Fig. 1. The lower and upper limits of the hydrologic parameters are considered in the wide ranges. This capacity shows that Excel Solver is a powerful tool for parameter estimation. On the other hand, the values of ∆S/∆t, ∆S, S, and Q are stored in D3 to D23, E3 to E23, F2 to F23, and G2 to G23 cells, respectively. Also, the values of the performance evaluation
criteria which consist of ηq (%), MARE, and DPO which are calculated using Eq. (19), Eq. (18), and Eq. (17) are stored in O2, P2, Q2, and R2 cells, respectively. Furthermore, the value of the objective function (SSQ) is stored in merged cells of N3 and N4 (i.e. the target cell). It is notable that the archive functions of Excel software are used in different steps of this procedure. For example, the value of SSQ is calculated by using the archive function‘‘SUMXMY2’’ of the software. In order to run the program, the engineer specifies the objective function, the parameters to be optimized, the constraints related to the lower and upper limits of the parameters, and the solving method (i.e., GRG solver or evolutionary solver) in the solver parameters window of Excel software.
7. Results and Discussion For calibration, testing and comparison of the results of Excel solver, the data set from Wilson (1974) was used. Data reported by Wilson are known to present a nonlinear relationship between storage volume and weighted-flow (see Fig. 2). This data set has also been extensively studied by others (Gill, 1978; Tung, 1985;
Fig. 2. The Illustration of the Nonlinearity between Storage Volume and Weighted-flow of the Application Example
Fig. 1. The Representation of the Procedure of the Study in Spreadsheet Format Vol. 17, No. 5 / July 2013
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Table 1. Comparison of the Routing Results of the Various Methods 3
Time (hr) 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126
Actual data (m /s) Inflow Outflow 22 22 23 21 35 21 71 26 103 34 111 44 109 55 100 66 86 75 71 82 59 85 47 84 39 80 32 73 28 64 24 54 22 44 21 36 20 30 19 25 19 22 18 19
LSM 22.000 22.000 22.814 29.682 39.304 47.977 58.352 67.526 75.130 80.733 83.492 82.968 80.129 74.454 66.971 57.758 47.627 36.968 27.661 21.569 19.005 19.000
Calculated outflow (m3/s) GA ICSA 22.000 22.000 22.000 22.000 22.393 22.419 26.369 26.585 34.206 34.442 44.180 44.200 56.999 56.912 68.237 68.122 77.189 77.124 83.295 83.346 85.714 85.898 84.228 84.507 80.180 80.527 73.307 73.644 65.073 65.338 55.824 55.941 46.709 46.634 38.034 37.752 30.918 30.495 25.721 25.269 22.164 21.779 20.260 20.020
HJ+DFP 22.000 22.000 22.427 26.689 34.847 44.665 56.939 67.683 76.280 82.218 84.697 83.476 79.785 73.339 65.478 56.503 47.482 38.727 31.379 25.902 22.136 20.181
Fig. 3. Inflow Hydrograph, Observed and Computed Outflows Hydrographs with Different Methods
Yoon and Padmanabhan, 1993; Mohan, 1997; Kim et al., 2001; Al-Humoud and Esen, 2006; Geem, 2006; Chu, 2009; Chu and Chang, 2009; and Luo and Xie, 2010). The routed outflows obtained by Eq. (9) using different methods consist of LSM (Gill, 1978), HJ+DFP (Tung, 1985), GA (Mohan, 1997), ICSA (Luo and Xie, 2010), GRG solver, and evolutionary solver are listed in Table 1. The computed outflow hydrographs for different methods along with the observed inflow and outflow hydrographs are shown in Fig. 3. It can be seen that the computed outflow hydrograph of both GRG solver and evolutionary solver, closely follow the observed outflow hydrograph. The differences between the observed and computed outflows of different methods are compared in Fig. 4. The results of the
GRG 22.000 22.000 22.422 26.611 34.456 44.167 56.855 68.059 77.071 83.318 85.901 84.536 80.581 73.711 65.407 55.997 46.667 37.754 30.469 25.228 21.739 19.994
Evolutionary 22.000 22.000 22.422 26.609 34.450 44.156 56.842 68.044 77.057 83.306 85.892 84.532 80.581 73.716 65.416 56.010 46.682 37.769 30.481 25.237 21.744 19.997
Fig. 4. Difference between the Computed and Observed of Outflows Hydrograph Ordinates Table 2. Results of the Hydrologic Parameters Values Method LSM HJ+DFP GA ICSA GRG Evolutionary
K 0.0100 0.0764 0.1033 0.0884 0.0862 0.0862
X 0.2500 0.2677 0.2813 0.2862 0.2869 0.2869
m 2.3470 1.8978 1.8282 1.8624 1.8681 1.8683
hydrologic parameters values and performance evaluation criteria are presented in Table 2 and Table 3, respectively. It is notable that SSQ is the objective function since it is premier
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Application of Excel Solver for Parameter Estimation of the Nonlinear Muskingum Models
Table 3. Results of the Performance Evaluation Criteria SSQ (m3/s)2 145.71 45.62 38.24 36.80 36.77 36.77
Method LSM HJ+DFP GA ICSA GRG Evolutionary
DPO (m3/s) 1.51 0.30 0.71 0.90 0.90 0.89
MARE (−) 0.057 0.030 0.026 0.025 0.025 0.025
Table 4. Results of the Sensitivity Analysis of the Initial Guess in Terms of the SSQ for GRG Solver
ηq (−) 98.81 99.63 99.69 99.70 99.70 99.70
measure in Table 3. The results indicate that the Excel solver for two ways give the lowest value of SSQ between these methods. Although the Excel solver has not the lowest value for DPO, it shows good results for all of the criteria. It is notable that the value of SSQ for the optimal solution is SSQ = 36.77 (more accurate: SSQ = 36.767888 with considering six digits after decimal point accuracy for the parameter vector K = 0.086249, X = 0.286916, and m = 1.868087; GRG solver). The Excel solver is also applied for the models of Eqs. (7) and (8). The results indicate that the Excel solver has better results [SSQ = 245.59 for the parameter vector (K = 0.4608, X = 0.2287, and p = 1.5011)] than [SSQ = 1038.2 for the parameter vector (0.0500, 0.2200, 2.3500)] of NONLR (Yoon and Padmanabhan, 1993) and [SSQ = 264.8 for the parameter vector (0.1773, 0.2305, and 1.7105)] of GA (Mohan, 1997) for the nonlinear model of Eq. (7). Also, it funds SSQ = 184.36 for the parameter vector (K = 1.6840, X = 0.0003, p1 = 2.9990, and p2 = 1.5595) and the nonlinear model of Eq. (8). Based on these results, it can be concluded that the Excel solver has the lowest value of SSQ regardless of the nonlinear storage equation. An alternative objective function for the hydrologic parameters can be considered to minimize the absolute value of the deviations between the routed and observed outflows (SAD) [L3T−1] (Mohan, 1997): N
SAD =
ˆ
∑ Ot – Ot
(22)
t=1
In this condition, the results for the model of Eq. (9) are SAD = 22.88, and SSQ = 42.21 for the parameter vector (K = 0.0783, X = 0.2795, and m = 1.8935). Based on these results, it can be concluded that Excel solver has good results regardless of the objective function equation. The comparison between different methods indicate that the method such as HJ+DFP and GRG solver require needs initial guess for the parameter estimation while GA, ICSA, and evolutionary solver need careful attention for determination of the algorithm parameters. The results of these methods are sensitive to the algorithm parameters or initial guess. For the model of Eq. (9) and the objective function of Eq. (20), a sensitivity analysis of the initial guess of the hydrologic parameters for GRG solver was performed. For all combinations of K = (0.01, 0.05, 0.1, 0.15, 0.2), X = (0.25, 0.275, 0.3) and m = Vol. 17, No. 5 / July 2013
m K X=0.25 X=0.275 X=0.3 0.01 0.05 0.1 0.15 1.5 Inf. Inf. Opt. Opt. 1.75 Inf. Opt. Opt. Opt. 2 Inf. Div. Div. Div. 2.25 Opt. Opt. Opt. Opt. 2.5 Opt. Opt. Opt. Opt. 1.5 Inf. Inf. Opt. Opt. 1.75 Inf. Opt. Opt. Opt. 2 Inf. Div. Div. Div. 2.25 Opt. Opt. Opt. Opt. 2.5 Opt. Opt. Opt. Opt. 1.5 Inf. Inf. Opt. Opt. 1.75 Inf. Opt. Opt. Opt. 2 Inf. Div. Div. Div. 2.25 Opt. Opt. Opt. Opt. 2.5 Opt. Opt. Opt. Opt. Note: Inf. = infeasible; Div. = divergent; and Opt. = optimal.
0.2 Opt. Opt. Div. Opt. Opt. Opt. Opt. Div. Opt. Opt. Opt. Opt. Div. Opt. Opt.
(1.5, 1.75, 2.0, 2.25, 2.5) as initial values assumption, the parameters have been optimized. The results indicate that the optimum (SSQ = 36.77), divergent, and infeasible solutions are achieved in 68, 16, and 16 percent of the cases, respectively (Table 4). Therefore, GRG solver found optimal solution in most of cases. Investigation of the results of the sensitivity analysis also shows that (1) by increasing K and m parameters the attenuation and lag are increased although the severity of these changes in K parameter is greater than m parameter; (2) by increasing X parameter the attenuation in peak discharge is decreased, but this parameter is no effect in the lag of floods; and (3) the initial value of X parameter than K and m parameters is less effective in finding the optimal solution. After a study of the sensitivity of evolutionary solver, the algorithm parameters are determined. The values of these parameters are considered as the mutation rate = 0.05; the population size = 50; the random seed = 0; and convergence = 0.00001. These set parameters have shortest operating time for convergence. It is notable that Evolutionary solver is a randomly search for the optimal solution. Therefore, some uncertainties are existed when using evolutionary solver for the parameter estimation. For escape from these conditions, it is recommended that the parameter calibration should be performed for several times. However, the effects of these uncertainties are not significant in the evolutionary solver because of the hybrid structure of the algorithm. The improvement procedure of the best solution in GRG solver (for the parameter vector [K = 0.01, X = 0.25, and m = 2.5] as initial guess) and evolutionary solver as number of iterations are presented in Figs. 5 and 6, respectively. It can be seen that GRG solver found the optimal solution in 39 iterations while evolutionary solver found the optimal solution in less than 5,700
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Fig. 5. Progression of Solution with Number of Iterations for GRG Solver
Fig. 6. Progression of Solution with Number of Iterations for Evolutionary Solver
iterations. For the comparison of the results, the several aspects of the different procedures (i.e., LSM, HJ+DFP, GA, ICSA, GRG solver, and evolutionary solver) can be investigated. With a glance of the results LSM and HJ+DFP can be eliminated from the challenge because the values of SSQ of two methods are far away from others. On the other hand, the stochastic algorithms that may yield different solutions in different runs need to determine the algorithm parameters such as the crossover rate and mutation rate in GA; and the clonal scale, mutation probability, and crossover probability in ICSA for each case study and also some uncertainties are existed in the random algorithms by probability distribution. Also, the number of iterations for convergence of these random algorithms is more than the proposed methodology. These disadvantages confine the application of these procedures for the parameter estimation of the nonlinear Muskingum models although some of these procedures have a good performance in terms of SSQ, MARE, DPO, and ηq. Based on the above results and discussions, it can be concluded that the both GRG and evolutionary solvers have best results in terms of SSQ. Also, the sensitivity analyses show that GRG solver finds the optimal solution in most of initial gauss of the hydrologic parameters, and the effects of the uncertainties of the algorithm parameters of evolutionary solver is less than the other stochastic algorithms. Furthermore, Excel solver has the
best accuracy regardless of the objective function equation and the nonlinear storage equation, and the time of operation of the procedure is less than other algorithms. However, the best advantage of GRG solver and evolutionary solver than the other procedure is the simplicity of the parameter estimation through spreadsheets, and powerful optimization algorithms. The advantage causes that the hydrologic engineers prefer the nonlinear Muskingum models than the linear Muskingum model for using in the simulation of flood events which have a nonlinear relationship between weighted-flow and storage volume in nonprismatic channels. Therefore, it can be said that the performance of Excel solver for parameter estimation of nonlinear Muskingum models is as good as or better than the other methods. The skewness of the discharge hydrograph has important effects on the parameter estimation of the nonlinear Muskingum models. Its variations are due to the variations of the characteristics of catchments such as the catchment area, catchment shape, river morphology, lithology, and vegetation. For example, the duration of hydrographs in vast catchments is longer than small catchments (i.e., higher values of the skewness). On the other hand, the characteristics of flood and rain such as the rainfall intensity and rainfall duration are important in hydrograph shapes. For example, the hydrograph shape in flash flood with high intensity is taper than rainfalls with lower intensity (i.e., lower values of the skewness). The variations of environmental conditions can change the skewness of the discharge hydrograph. Therefore, the values of the hydrologic parameters for a given river reach must be updated after any significant variations in environmental conditions. In real design projects, for a given river, available data of flood events which have the both upstream and downstream hydrographs can be used in the calibration procedure of the hydrologic parameters if the characteristics of the river such as the geometric and hydraulic aspects that have important effects on the channel storage were not changed in the period of the occurring of the flood events by changing the environmental conditions or human beings activities. However, the simulation results in the design step are more accordance with actual conditions if the numbers of flood events that use in the calibration procedure are increased. It is notable that the scope of the present study is that the efficiency of Excel solver for parameter estimation of the nonlinear Muskingum models is investigated through the benchmark problem.
8. Conclusions Although the nonlinear Muskingum models have special advantages than the linear Muskingum model, the hydrologic practitioners avoid from the nonlinear Muskingum models, because of the difficulties of the estimation of the additional parameters of the nonlinear models. Therefore, various researchers attempted to reduce these difficulties through alternative optimization algorithms. These procedures can be classified as two groups consist of 1) mathematical techniques such as LSM (Gill, 1978), HJ+DFP (Tung, 1985), NONLR (Yoon and Padmanabhan,
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Application of Excel Solver for Parameter Estimation of the Nonlinear Muskingum Models
1993), LM (Das, 2004), and BFGS (Geem, 2006), and 2) phenomenon mimicking algorithms such as GA (Mohan, 1997), HS (Kim et al., 2001), PSO (Chu and Chang, 2009), and ICSA (Luo and Xie, 2010). However, the methods of the former group have drawbacks of the complex derivative requirement and/or initial vector assumption, and the trouble of the methods of the latter group is that these methods need the determination of the algorithm parameters, and also some uncertainties are existed in these algorithms by probability distribution. In this paper, in order to reduce the drawbacks of the aforementioned techniques, the application of solver add-in of Microsoft Excel 2010 software for the purpose of estimating the parameters of nonlinear Muskingum models is presented. This software can calibrate the parameters using 1) Generalized Reduced Gradient (GRG) solver and 2) evolutionary solver. GRG solver needs initial guess for the parameter estimation although the results of the sensitivity analysis of the initial guess of the hydrologic parameters show that GRG solver found optimal solution in most of cases. Also, it found a solution in seconds. On the other hand, efficiency of evolutionary solver dependent on the algorithm parameters such as mutation rate, population size, and random seed. The performance of these approaches was compared with the different methods using the different performance evaluation criteria for an example with high nonlinearity between storage volume and weighted-flow. The results indicate that the Excel solver is efficient approach for estimating the parameters of nonlinear Muskingum routing models. Although the both GRG solver and evolutionary solver than other methods are good results separately, an alternative way is suggested for improvement of the efficiency of the Excel solver. In this way, first the evolutionary solver run and after a few second in meddle of the operation, the model can be stopped. Then, the values obtained for the hydrologic parameters from evolutionary solver are considered as the initial guess for GRG solver. In other words, the procedure uses preliminary values of the algorithm parameters for evolutionary solver, and results of the algorithm are used as initial guess of the hydrologic parameters for the GRG solver. In this condition, the sensitivity analyses of either algorithm parameters or initial values assumption are not necessary, and also the effects of the uncertainties are very little in the procedure because of using the deterministic algorithm at final stage. Therefore, finding the optimal solution can be done in less time. In conclusion, with following reasons, Excel solver is a promising procedure to use in the estimation of nonlinear Muskingum routing parameters: (1) it can shows calculation procedure; (2) it can uses by engineers without knowing the exact details of optimization mathematics; (3) it can calibrates the parameters using different ways; (4) it is widely available for engineers; and (5) it is efficient, simple and convenient for parameter estimation. In this way, the nonlinear Muskingum model is a very practical approach especially for unmanaged catchments which have not the required data (such as: the geometries of cross sections and bed slopes of river, the Vol. 17, No. 5 / July 2013
roughness coefficients in each reach of river and etc., which can be determined using expensive projects) for the simulation of flood events through the dynamic wave model that uses the full Saint-Venant equations. In addition, it is expected that the Excel solver can be successfully applied to other problems in civil engineering.
Acknowledgements The writer thanks the editors, anonymous reviewers, and helpers. The author would like to thank the Young Researchers Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran, for financially supporting this research.
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