Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 634852, 7 pages http://dx.doi.org/10.1155/2014/634852 Research Article A Novel Parameter Estimation Method for Muskingum Model Using New Newton-Type Trust Region Algorithm Zhou Sheng,1 Aijia Ouyang,2,3 Li-Bin Liu,4 and Gonglin Yuan1 1 College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China 2Department of Electrical and Information Engineering, Hunan Institute of Traffic Engineering, Hengyang, Hunan 421001, China 3School of Information Science and Engineering, Hunan City University, Yiyang, Hunan 413000, China 4Department of Mathematics and Computer Science, Chizhou College, Chizhou, Anhui 247000, China Correspondence should be addressed to Li-Bin Liu; [email protected] Received 28 August 2014; Accepted 4 December 2014; Published 21 December 2014 Academic Editor: Valder Steffen Jr. Copyright © 2014 Zhou Sheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Parameters estimation of Muskingum model is very significative in both exploitation and utilization of water resources and hydrological forecasting. The optimal results of parameters directly affect the accuracy of flood forecasting. This paper considers the parameters estimation problem of Muskingum model from the following two aspects. Firstly, based on the general trapezoid formulas, a class of new discretization methods including a parameter to approximate Muskingum model is presented. The accuracy of these methods is second-order, when =1/3̸ .Particularly,ifwechoose=1/3, the accuracy of the presented method can be improved to third-order. Secondly, according to the Newton-type trust region algorithm, a new Newton-type trust region algorithm is given to obtain the parameters of Muskingum model. This method can avoid high dependence on the initial parameters. The average absolute errors (AAE) and the average relative errors (ARE) of the proposed algorithm of parameters estimation for Muskingum model are 8.208122 and 2.462438%, respectively, where =1/3. It is shown from these results that the presented algorithm has higher forecasting accuracy and wider practicability than other methods. 1. Introduction where () represents the channel storage at time ; () and () represent the rates of inflow and outflow at time , Flood routing in open channels is a very important tool in respectively. The linear Musikingum model is the design of flood protection measures to estimate how the proposedmeasureswillaffectthebehavioroffloodwaves () =[ () + (1−) ()] , (2) in rivers so that enough protection and economic solutions can be found [1]. In general, flood routing procedures can where is a storage time constant for the river reach and be classified as either hydrologic or hydraulic. Hydrologic is a weighting factor commonly varying between 0.0 and 0.3 routing method is on the basis of the storage-continuity fortheriverchannel. equation, whereas hydraulic routing method is on the basis of It is worth mentioning that the parameters and in (2) both continuity and momentum equations. One of the hydro- are graphically estimated by a trial-and-error procedure [2]. If logic routing approaches, the Muskingum method, was first is obtained, the values of [()+(1−)()] are calculated developed by McCarthy for flood control studies in the by using observed data in both upstream and downstream Muskingum river basin in Ohio. and plotted against . The particular value which generates The following continuity and storage equations are used the loop is accepted as the best estimation of .Theslopeof to describe the Muskingum model: the straight line fits through the loop derives . Although the trial-and-error procedure has been used for many years, it is d () =() −() , (1) time consuming and subjective interpretation. Therefore, in d order to avoid subjective interpretations of observed data in 2 Mathematical Problems in Engineering estimating and , the following finite difference scheme [3] are that the global search ability is strong and the program is used to the resulting ordinary differential equation (1): is relatively simple to design. However, these methods have unavoidable disadvantages of slow convergence precision, ()−(−1) low precision solution in limited generations, and premature Δ convergence. (3) ()+( ) ()+( ) In a word, almost all of the above methods used to =[ −1 ]−[ −1 ], estimate parameters and are based on (3).Weknowthat 2 2 2 the truncation error of (3) is only ((Δ) ).Thus,itisvery necessary to design a higher accuracy method to discrete where differential equation (1). Based on the generalized trapezoidal () = [ ( )+(1−) ()] , formula [16], this paper will first develop a class of new differ- ence scheme which contains a parameter to approximate the (−1) = [−1 ( )+(1−) (−1)], =2,...,, differential equation (1).Then,similartotheaboveoptimiza- (4) tion problem (7), we can obtain a class of new unconstrained nonlinear optimization problems which contain parameter and () represents observed outflow discharges at time , . It is noted that the accuracy of the presented difference () represents the inflow discharge at time interval , Δ is schemes to approximate the differential equation (1) can be thetimestep,and is the total time number. Substituting (4) improved to third order, when =1/3. In other words, we into (3),wehave can get a higher accuracy parameter estimation model, if = 1/3. In addition, based on the Newton-type trust region algo- (+1) =0 (+1) +1 () +2 () , (5) rithm [17], this paper will design a new Newton-type trust region algorithm (NN-TTRA) for estimating the Muskingum where model parameters. Briefly speaking, this algorithm can avoid − + (Δ/2) high dependence on the initial parameters and find the global = , 0 (1−) + (Δ/2) optimization solution quickly. + (Δ/2) = , 1 (1−) + (Δ/2) (6) 2. A New Parameter Estimation of Muskingum Model (1−) − (Δ/2) = . 2 (1−) + (Δ/2) 2.1. Model Description. At first, (1) can be rewritten as follows: By minimizing the sum of the square of the deviations d () between observed and calculated outflows, we obtain the =() +() , (8) d following objective function: −1 () = () − () −() 2 where . min =∑ [(+1)−0(+1)−1()−2()] . (7) Then, applying the generalized trapezoidal formula [16] =1 GTF ()ofChawlaetal.to(8),wecanobtain In the past two decades, in order to obtain the parameters ( ) and , some researchers adopted many optimization ̃ d +1 ()=(+1)−Δ , (9) methods to solve the above optimization problem (7).Until d now, these methods can be classified into traditional opti- Δ () mization methods and intelligent algorithms. The existing d (+1)=()+ [(1−) traditional methods include nonlinear programming method 2 d (NPM) [4], the least-square method (L-SM) [5, 6], method (10) (̃ ) ( ) of trial-and-error (TAE) [2], the minimum area method +d + d +1 ]. (MAM) [1, 7], the Broydene-Fletchere-Goldfarbe-Shanno d d (BFGS) technique [8], test-method and least residual square method [9], and Nelder-Mead simplex method [10]. These Combining (8), (9),and(10),weget traditional methods have their own advantages, but there also exist complex calculations and poor generality disadvantages, ̃ and some of these methods are related to the selection ()=(1−Δ) (+1)−Δ(+1), (11) of initial point, which is easy to fall into local optimum. Δ The existing intelligent algorithms include genetic algorithm (+1)=()+ [ +1( )+()+(1−) () [11], harmony search [12], Gray-encoded accelerating genetic 2 algorithm [9], particle swarm optimization [13], immune +(̃ )+( )] . clonal selection algorithm [14], and differential evolution +1 algorithm [15]. The advantages of those intelligent algorithms (12) Mathematical Problems in Engineering 3 Substituting (11) into (12),wefinallyobtain 3. Algorithm Construction Δ 3.1. Newton-Type Trust Region Algorithm. As far as we know, (1 − ) +1 ( ) 2 the basic trust region algorithm used to solve the following Δ Δ unconstrained optimization problem =(1− ) ( )+ [ ( )−()] 2 2 (13) min () Δ ∈ (20) + (1−Δ) [ ( )−( )] . 2 +1 +1 =0 was first presented clearly in [19]. Recently, many researchers Obviously, when , (13) becomes (3). have proposed some improved trust region algorithm to solve At last, substituting (4) into (13),wehave (20); see for example Esmaeili and Kimiaei [20]andAmini (+1)=0(+1)+1()+2(), (14) and Ahookhosh [21]. Here, we assume that is the th iterative point, = where (), =∇(),and is the th iteration of Hesse matrix − (1−(Δ/2) ) + (Δ/2)(1−Δ) ∇2( ) = , ; the trust region subproblem, which is at the th 0 (1−)(1−(Δ/2) ) + (Δ/2)(1−Δ) iterative step of problem (20), can be formulated as follows: (1−(Δ/2) ) + (Δ/2) 1 1 = , (15) () = + , (1−)(1−(Δ/2) ) + (Δ/2)(1−Δ) min 2 (21) (1−)(1−(Δ/2) ) − (Δ/2) = . s.t. ‖‖ ≤Δ, 2 (1−)(1−(Δ/2) ) + (Δ/2)(1−Δ) By minimizing the residual sum of squares between where Δ is a trust region radius and ∈ is an iterative step observed and calculated outflows, the objective function can and ‖⋅‖denotes the Euclidian norm of vectors or its induced be given as follows: matrix norm. Denote −1 2 min =∑ [(+1)−0(+1)−1()−2()] . (16) =1 Δ =()−( +), (22) Δ = (0) − ( ), 2.2. Truncation Errors Analysis. Here, we analyze the local (23) truncation error of (13) at time direction [16]. Firstly, from Δ (9),wehave = ; Δ (24) ̃ 2 d() d(+1) d (+1) = −Δ . (17) 2 d d d the general trust region algorithm [17]forsolvingtheuncon- Then, by using Taylor expansion, (10) can be written as strained optimization problem (20) is given as follows. follows: Algorithm 1. (Δ)3 3() (Δ)4 4() − ( )− ( )+((Δ)5) 4 4 12 24 Step 0.Givenastartingpoint0 ∈, Δ 0 >0is the initial trust region radium, >0, 0 = Hesse(0).Set:=0. ̃ (Δ) d() d() 5 = [ − ]+((Δ) ). ‖ ‖≤ 2 d d Step 1.Calculate ,if , and stop iteration; otherwise, (18) go to Step 2. ̃ Substituting for d()/d from (17) into (18), the truncation Step 2. Utilize the smoothing Newton method to obtain the errors of generalized trapezoidal formula can be obtained as solution of the subproblem (21).