Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7919324, 19 pages

https://doi.org/10.1155/2017/7919324

## Automatic Calibration of an Unsteady River Flow Model by Using Dynamically Dimensioned Search Algorithm

^{1}Department of Civil Engineering, National Taiwan University, Taipei City 10617, Taiwan^{2}Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi City 60004, Taiwan^{3}MWH Americas Incorporated, Taiwan Branch, Taipei City 10549, Taiwan

Correspondence should be addressed to Nan-Jing Wu; wt.ude.uycn.liam@uwjn

Received 7 July 2016; Revised 19 October 2016; Accepted 18 December 2016; Published 1 February 2017

Academic Editor: Paolo Lonetti

Copyright © 2017 Fu-Ru Lin 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.

#### Abstract

Dynamically dimensioned search (DDS) algorithm is a new-type heuristic algorithm which was originally developed by Tolson and Shoemaker in 2007. In this study, the DDS algorithm is applied to automate the calibration process of an unsteady river flow model in the Tamsui River basin, which was developed by Wu et al. (2007). Data observed during 2012 and 2013 are collected in this study. They are divided into three groups, one for the test case, one for calibration, and one for the validation. To prove that the DDS algorithm is capable of solving this research problem and the convergence property, a test simulation is first performed. In the studied area, the whole river systems are divided into 20 reaches, and each reach has two parameters ( and ) to be determined. These two parameters represent resistance coefficients for low- and high-water conditions. Comparing with another algorithm, it is shown that the DDS algorithm has not only improved on the efficiency but also increased the stability of calibrated results.

#### 1. Introduction

The river resistance coefficient, also known as the value in the Gauckler–Manning formula, is an important hydraulic parameter used in flow calculations for planning and design in hydraulic engineering. Proper hydraulic parameters, coupled with appropriate flood simulation models, can simulate water level variations in the calculation for basins during typhoon floods and thereby help mitigate the potential loss of life and property. Nowadays, one-dimensional (1D) and two-dimensional (2D) hydraulic models seem to be crucial tools for the development of flood propagation and for supporting flood risk assessment [1–3]. Whether or not river resistance coefficients are correctly estimated, they affect the result of water level calculation, planning, design, and operation of channels, and accuracy of various hydraulic calculations including water transport efficiency. Since different watersheds have different channel characteristics, each reach of a river has its own resistance coefficients, which is generally computed by an experienced person.

Strickler [4] used an empirical formula with of the riverbed to estimate the value of . Cowan [5] proposed a formula to estimate the river resistance coefficient based on the material of the river bottom, windiness of the channel, distribution of plantation, and man-made structures. In addition, textbooks on canal water mechanics offer their own methods for estimating river resistance under various canal conditions, for example, Chow [6] and Henderson [7]. Yen [8] indicated that, because of vegetation and other obstacles, the local roughness of a typical composite channel cross section will be higher or lower for the floodplains due to different bed texture. Wilson [9] highlighted that Manning’s coefficient decreases with increasing flow depth, reaching an asymptotic constant at relatively high flow depths. Vegetation was also pointed out by other references [10–12] that will affect the water level. However, these types of estimates are subjective in nature and thus are difficult to pass on.

The trial and error method has also been used to adjust the river resistance coefficients so as to match the water levels with the observed values as closely as possible. However, calibration using this process is usually very time-consuming. Additionally, there is no guarantee that the resistance coefficients calibrated in this manner will reach the optimal values and are thus unable to pass the validation test. Therefore, research on automatic calibration has been conducted recently. Unlike traditional, labor-intensive trial and error methods, automated calibration is a systematic optimization process. The parameter to be calibrated is fed into the model, and then the numerically simulated value is compared to empirical measurements to determine the right direction for parameter adjustment. The iterative process is executed until the results converge.

Becker and Yeh [13] used the influence coefficient algorithm in an attempt to calibrate the river resistance coefficients and hydraulic radius index for a one reach river. The same researchers expanded their method to calibrate river resistance coefficients and hydraulic radius indexes for multiple reach rivers in 1973 [14].

Lal [15] investigated the use of singular value decomposition (SVD) to calibrate the river resistance coefficients for the upper reaches of the Niagara River. It was suggested that an underdetermined parameter validation problem (the number of parameters to be calibrated exceeds the number of water level or flow rate measurement stations) could be simplified by grouping parameters, where all parameters in a given group had the same value. However, the resistance coefficients calculated using different numbers of groups were not alike, thus suggesting that the method was not stable.

Currently, automatic calibration methods can be divided into two types: gradient search methods and heuristic algorithms. Although gradient search methods are based on a theoretical foundation rigorously, they are not suitable for solving multidimensional optimization problems, for two reasons: the solution space is limited to the neighborhood of the initial solution; if providing a good initial solution is not possible, the calculation mechanism is trapped by the local optimal solution in most cases; thus, the search mechanism lacks diversity; the gradient search method can only handle problems where the gradient exists. However, there are many real problems in nature that contain discontinuities. This further limits the possibility of finding a solution with the gradient method. For instance, Yang [16] investigated the use of the conjugate gradients method to calibrate the resistance coefficients of the Shimen Dazhen Channel and showed that the method was only suitable for simple river reaches.

Thus, Heuristic algorithm methods were developed to overcome these deficiencies. With the help of artificial intelligence (AI) technologies, these methods are mainly adopted effectively and efficiently to find an optimal solution within a feasible solution space. Researchers have developed various algorithms and calculation mechanisms by imitating various examples of natural intelligence.

Nowadays, heuristic algorithms are widely employed to solve optimization problems in many fields, such as electronics, IT, engineering, and economics [17]. The most well-known heuristic algorithms currently in use include tabu search, genetic algorithms, simulated annealing algorithms, neural networks, and fuzzy methods [18–21]. The most beneficial feature of heuristic algorithms is that they can find good solutions quickly. They also have built-in mechanisms to avoid falling into local optimum solution traps, such as the tabu list mechanism in the tabu search and the mutation operand in the genetic algorithm. Simulated annealing algorithms, in particular, escape local optimal solutions by adjusting the probability of the solution moving to neighboring regions based on temperature changes. These search methods are nothing more than intelligent trial and error methods. They combine several natural laws, the ability to learn, probability characteristics, fuzzy concepts, memory functions, and so on to construct calculation methods that are most capable of solving optimization problems.

Chan [22] and Huang [23] both used unsteady-flow simulation models for basin-wide river systems coupled with a real-value-coding of genetic algorithm or an additional simulated annealing algorithm (SARvcNGA, Simulated Annealing Real-value-coding Niche Genetic Algorithm). The root-mean-square error (RMSE) between the calculated water level and the measured water level was used as the objective function to ascertain the river resistance coefficients that could best describe the actual flow conditions. Though SARvcNGA has been proven to be more efficient and accurate than genetic algorithm with a real-value-coding, it still requires lots of time to complete the computations. Clearly, there is room for improvement in terms of efficiency, especially in support of flood mitigation efforts, where timely information is paramount.

The dynamically dimensioned search algorithm (DDS), which is a new type of heuristic algorithm, was developed by Tolson and Shoemaker [24]. It was designed for calibration problems with many parameters. Many studies related to this method have been conducted, such as [25–27]. Tu [28] used DDS algorithm in automatic calibration of an unsteady river flow model.

Present study focuses on optimizing the river resistance coefficients by DDS algorithm. It is the extension of the previous research efforts [28]. Because one of the branches of study area was dredged for flood control in 2010, the coefficients in the reaches need to be renewed; thus the flood forecasting model can keep its accuracy.

These resistant coefficients will be used in combination with an unsteady-flow simulation model for basin-wide river systems [29] to simulate water levels in flood research. The results will also be compared with those by using the algorithm proposed in Huang [23]. It is illustrated that the DDS algorithm will provide an accurate result and is a diverse and robust method for automatic calibration of the unsteady river flow model.

#### 2. Dynamically Dimensioned Search Algorithm

This section is mainly drawn from “dynamically dimensioned search algorithm for computationally efficient watershed model calibration” [24], for completeness.

##### 2.1. Overview

The dynamically dimensioned search algorithm (DDS) is mainly designed to solve multiparameter calibration problems. When applied to optimization problems requiring large amounts of computational time, this search method does not demand complicated parameter adjustment. That is, there are no additional control parameters in the algorithm that require separate adjustment, such as the initial temperature, temperature reduction factor in the simulated annealing algorithm, mutation probability, or number of ethnic groups in the genetic algorithm. Thus, the number of uncertainty factors decreases, and the greatest possible portion of the calculation time is used for searching the optimal solution. Other features of this algorithm include the option to limit the number of searches to be performed to suit the user’s time limit. The range of feasible solutions to be searched can also be adjusted based on the maximum number of searches to be performed. These design features do not affect the diversity or robustness of the search.

##### 2.2. Calculation Mechanism

One special characteristic of the DDS algorithm is that, regardless of the number of searches set by the user, the search for candidate solutions proceeds from global to local. This is made possible by an adjustment mechanism that decreases the number of algorithm parameters to be determined based on the changes in the probability. Moreover, a new candidate solution is created only by modifying the best current solution. Both solutions are substituted into the objective function for comparison. The candidate solution only replaces the best current solution if it decreases the value of the objective function. Otherwise, it is rejected, and another candidate solution is created from the best current solution. This process is repeated until convergence conditions are achieved or the maximum number of searches has been performed.

The DDS algorithm includes three main calculation steps: setting up the initial solution; selecting a candidate solution; and updating the best current solution. Following are detailed descriptions of each of these steps.

*(1) Setting up the Initial Solution.* Consider that the model being calibrated contains parameters (called decision variables). The only three algorithm parameters to set in the DDS algorithm which must be initialized before are as follows: the neighborhood perturbation size parameter ; the maximum number of searches ; and the upper and lower limits of the parameters to be determined (). Next, a solution set can be chosen at random from the feasible solution space to serve as the initial solution for the calibration process. And the initial solution is substituted into the objective function as , see (1), to complete the setup.

*(2) Selecting a Candidate Solution.* During the search for the optimal solution, the solution currently being searched and modified is called the “current best solution.” The current best solution is modified according to calculation mechanism to search for the next candidate solution. During this process, all of the solutions that could possibly become the next candidate solution are known as “neighboring solutions.” The set of all neighboring solutions is called the “neighborhood” . The solution that is actually chosen from the neighborhood is called the “candidate solution.” The heuristic algorithm methods are distinguished from each other by comparing the manner through which they simultaneously satisfy both diversity and robustness while selecting the candidate solution. The following are the rules used by the DDS algorithms for selecting a candidate solution.

Totally there are parameters to be determined. Among them, parameters are selected in neighborhood, , based on the probability as shown in (2), where is the number of the current iterations. As the number of iterations increases, the number of chosen parameters decreases. This is equivalent to the gradual progression from a global to a local search. This continues until the convergence conditions are satisfied or the maximum number of searches has been performed.where is the maximum numbers of searches.

For example, when the number of parameters is 10 (i.e., ) and the maximum number of searches is 1000 (i.e., ), the probability of the first search () can be calculated as

Therefore, during the first search, the probability that each parameter to be determined will be chosen for updating is 100%, and so forth.

*(3) Updating the Best Current Solution.* The parameters chosen in the previous step are updated according towhere and is standard normal random variable.

If the value of any updated parameter is less than the lower limit or greater than the upper limit, it is adjusted according to

Further, the objective function is evaluated and the best current solution is updated if necessary, as shown in

#### 3. Problem Description and Research Zone

##### 3.1. Overview of Tamsui River Basin

This study attempts to ascertain the river resistance coefficients for the unsteady-flow model of the Tamsui River basin. The Tamsui River basin is located at the northern extremity of Taiwan. It has an area of 2,726 km^{2} and a main stream length of 158.7 km, making it the third largest river in Taiwan. The Tamsui River has three principal tributaries: the Dahan Stream in the south and the Sindian Stream in the middle, both converging at Jiangzihcuei near Taipei City; and in the north, the other tributary Keelung River converges with the Tamsui River at Guandu.

The Dahan Stream is 135 km long, and its watershed has an area of 1,163 km^{2} and an average gradient of 1/37. The upper reaches are the Shimen Reservoir catchment, which has an area of 759 km^{2}. There are mountain valleys in the upper reaches of the Dahan Stream and terraces and alluvial plains in the middle and lower reaches. Further, there is well-developed transportation infrastructure within the Dahan Stream watershed boundaries. It is located in the core region of the greater Taipei metropolis.

The Sindian Stream is 82 km long, and its watershed has an area of 910 km^{2}, 89% of which is mountain slopes. The river’s abundant water flow is an important source of water for Taipei.

The Keelung River originates at Jingtong Mountain in Pingci Village of Taipei County. Its main stream is 89.4 km long and its watershed has an area of 490.77 km^{2}. The upper reaches of the Keelung River are between the Jieshou Bridge in Houdong and the Dahua Bridge in Cidu. Here, the average gradient is approximately 1/250. The middle reaches, from the Dahua Bridge to the Nanhu Bridge, have an average gradient of around 1/4900. The lower reaches are from the Nanhu Bridge to the mouth of the river, with an average gradient of around 1/6700. The riverbed within the basin is fairly flat and windy (data source: the 10th River Management Office, Water Resource Agency, Ministry of Economic Affairs, Taiwan).

##### 3.2. Study Area

This research focuses on the Tamsui River basin because of its rather comprehensive profile data and the availability of several water level measurement stations for data collection. The research zone includes the Keelung River downstream of the Dahua Bridge, the Dahan Stream downstream of the Sinhai Bridge, the Sindian Stream downstream of the Zhongzheng Bridge, and the Tamsui River from the mouth of the river and up, as shown in Figure 1.