# “In-Process Type” Dynamic Muskingum Model Parameter Estimation Method

^{1}

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## Abstract

**:**

## 1. Introduction

- (1)
- At present, most methods are developed with the sole focus on improving the simulation accuracy of one flood [16,17,18], such as in [16], where the average relative errors obtained from four algorithms were 1.16%, 1.11%, 1.13%, and 1.10%, respectively, and the improvement was inherently limited. Also, although the obtained flood parameters could forecast the selected flood, they did not validate whether the parameters can be applied to the other floods, and the practicality of the model is generally ignored.
- (2)

- (1)
- The Muskingum model parameters should not be unique to a given river course, and should change dynamically with the characteristics of different floods (e.g., flood flow, velocity, flood volume).
- (2)
- The Muskingum model parameters should not be determined by one flood or the mean of several floods, it should be obtained from the real-time dynamic parameter estimation.

## 2. Materials and Methods

#### 2.1. Muskingum Model

#### 2.1.1. Introduction to Muskingum Model Theories

#### 2.1.2. Establishing the Muskingum Parameter Optimization Model

^{6}.

#### 2.2. Single-Flood Muskingum Model Parameter Estimation Based on Pigeon-Inspired Optimization Algorithm

#### 2.2.1. Pigeon-Inspired Optimization (PIO) Algorithm

- Step 1
- Generate the initial population, in which the position of each individual i is ${X}_{i}=\left\{{x}_{i1},{x}_{i2},\dots ,{x}_{iD}\right\}$, which means the values of ${c}_{0},{c}_{1},{c}_{2}$, the initial velocity ${V}_{i}=\left\{{v}_{i1},{v}_{i2},\dots ,{v}_{iD}\right\},\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}{V}_{i}\in \left[0,1\right]$, and the position ${X}_{i}$ satisfy the upper and lower bound (i.e., range = [0, 1]) of the problem to be solved. That is to say, the value range of ${c}_{0},{c}_{1},{c}_{2}$ is from 0 to 1. The maximum iteration of the Map and Compass operator is $M{C}_{\mathrm{max}}$, the maximum iteration of the Landmark operator is ${L}_{\mathrm{max}}$, the population size is N, the dimension of the problem is D, and the compass factor is R.
- Step 2
- Set the best position of the individuals ${X}_{p}={X}_{i}$, in which ${X}_{p}$ means the optimal value of ${c}_{0},{c}_{1},{c}_{2}$, and iterations $k=1$, then calculate the fitness value of all individuals of the population (i.e., the solution of Equation (6)) and obtain the optimum value ${X}_{best}$, which is the optimal value of the optimization function.
- Step 3
- Start the iteration procedure of the Map and Compass operator. Update the position ${X}_{i}$ and velocity ${V}_{i}$ of each individual in each iteration procedure, and note that each ${X}_{i}$ should satisfy the range. Run this iteration until the iterations $k=M{C}_{\mathrm{max}}$, then update ${X}_{p}$ and ${X}_{best}$.
- Step 4
- Take the population obtained in Step 3 as the input of the iteration procedure of the Landmark operator. In each iteration, the individual population is first ranked by the fitness value; the first half is selected as the reference population, then the center position of the better population ${X}_{center}$ is calculated as the landmark. Next, update the position of each individual until the iterations reach the maximum $M{C}_{\mathrm{max}}$, then stop the iteration procedure and update the best position ${X}_{p}$ and best fitness ${X}_{best}$.
- Step 5
- ${X}_{best}$ is the required global optimum.

#### 2.2.2. Single-Flood Muskingum Model Parameter Estimation

#### 2.3. “In-Process Type” Dynamic Parameter Estimation of the Muskingum Model Based on BP-Neural Network

#### 2.3.1. Basic Principles of BP-Neural Network

- Step 1
- Network initialization. The number of node in the input layer is n, in the hidden layer is p, and in the output layer the node number is m. The connection weights between the input layer and the hidden layer, the hidden layer and the output layer is ${W}_{i}^{j}$ and ${V}_{j}^{k}$, respectively. The threshold of hidden layer is ${\theta}_{j}(i=1,2,\dots ,p)$ and the threshold of output layer is ${\mu}_{k}(i=1,2,\dots ,m)$
- Step 2
- Calculate the output ${h}_{j}$ of hidden layer. The calculation formula is:$${h}_{j}=f\left({\displaystyle \sum _{i=1}^{n}{W}_{i}^{j}{X}_{i}-{\theta}_{j}}\right),j=1,2,\cdots ,p$$
- Step 3
- Calculate the output ${Y}_{k}$ of output layer. It is calculated by$${Y}_{k}={\displaystyle \sum _{j=1}^{p}{h}_{j}{V}_{j}^{k}}-{\mu}_{k},k=1,2,\cdots ,m$$
- Step 4
- Update the weights through the equations as follows:$${W}_{i}^{j}\left(t+1\right)={W}_{i}^{j}\left(t\right)+\eta \left[\left(1-\beta \right)D\left(t\right)+\beta D\left(t-1\right)\right],i=1,2,\cdots ,n$$$${V}_{j}^{k}\left(t+1\right)={V}_{j}^{k}\left(t\right)+\eta \left[\left(1-\beta \right){D}^{\prime}\left(t\right)+\beta {D}^{\prime}\left(t-1\right)\right]$$
- Step 5
- Threshold update. Update the value of ${\theta}_{j}$ and ${\mu}_{k}$ according to the error between the network output ${Y}_{k}$ and the desired output ${y}_{k}$. The formulas to calculate ${\theta}_{j}$ and ${\mu}_{k}$ are:$${\theta}_{j}\left(t+1\right)={\theta}_{j}\left(t\right)+\eta {h}_{j}\left(1-{h}_{j}\right){\displaystyle \sum _{k=1}^{m}{V}_{j}^{k}}\left({y}_{k}-{Y}_{k}\right)$$$${\mu}_{k}\left(t+1\right)={\mu}_{k}\left(t\right)+\left({y}_{k}+{Y}_{k}\right)$$
- Step 6
- Determine whether the iteration ends. If it doesn’t end, go back to step 2.

#### 2.3.2. Basic Theory of “In-Process Type” Dynamic Parameter Estimation

- Step 1
- Select multiple floods and use the PIO algorithm to estimate their respective parameters.
- Step 2
- Extract the characteristic attributes of each flood according to the specific river conditions.
- Step 3
- Take the extracted characteristic attributes of each flood as the input, and parameters $K$ and $x$ of the Muskingum model as outputs to train the Neural Network model; use the PIO algorithm to estimate the parameters.
- Step 4
- Extract the characteristic attributes of the flood to be forecasted and plug them into the Neural Network to calculate parameters $K$ and $x$ of the Muskingum model, then use the Muskingum model to perform the flood routing computation.

#### 2.3.3. Study River Channel

^{2}, making it the fifth longest river in the world and the second longest river in China. The annual precipitation in most parts of the Yellow River Basin is about 200 to 650 mm, and the annual precipitation in the lower reach is more than 650 mm. The Sunkou and Gaocun Hydrometric Stations, which are the national key hydrometric stations in the lower reaches of the Yellow River, are located in Shandong Province. There are about 197 km between the two hydrometric stations, in which there are no tributary imports or exports. We selected this river channel to validate the proposed dynamic parameter estimation method of the Muskingum model (Figure 5).

#### 2.3.4. Extracting Flood Characteristic Attributes

- (1)
- The initial water level (IWL), which corresponds to the rising point (the time at ${t}_{1}$) of the flood, reflects the rising conditions of the flood and the previous water volume of the river. The larger the previous water volume, the more prone the area of interest is to flooding. The water level also reflects flow velocity information and thus exerts some influence on flood routing.
- (2)
- The peak discharge (PD), which reflects the magnitude of the flood and has significant influence on flood routing.
- (3)
- The flood peak emergence time (FPET), i.e., the time at which the flood appears. FPET is usually an instantaneous value, so it cannot exactly reflect the characteristics of each flood along the entire route. Here, we consider the point at which the flood peak appears as the FPET; that is to say, $the\text{\hspace{0.05em}\hspace{0.05em}}FPET=1,2,\dots ,T$, in which T is the duration time of the flood.
- (4)
- The total flood volume on the first two days after a flood appears (FVFT) reflects the rising conditions of the flood.
- (5)
- The flood volume on the day before the flood peak appears (FVBFP), which reflects the width of the flood peak.
- (6)
- The average discharge before the flood peak (ADBFP), i.e., the mean discharge prior to the flood peak, which reflects the ratio between the total flood volume and time.

#### 2.3.5. The Flow Chart of the IP-DPE Method

## 3. Results

- (1)
- The mean absolute error (MAE) of the flood (m
^{3}/s):$$MAE={\displaystyle \sum _{i=1}^{T}\left|{Y}_{o}{}_{i}-{Y}_{ci}\right|}/T$$ - (2)
- The average relative error (ARE) of the flood (%):$$ARE={\displaystyle \sum _{i=1}^{T}\left|({Y}_{oi}-{Y}_{ci})/{Y}_{oi}\times 100\right|}/T$$
- (3)
- The peak flow relative error (PFRE) of the flood (%):$$PFRE=({P}_{o}-{P}_{c})/{P}_{o}\times 100$$

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Simulation results of four algorithms. Abbreviations: PIO, Pigeon-Inspired Optimization algorithm; L-S, Least-Square method; AGA, Accelerated Genetic Algorithm; ACO, Ant Colony Optimization algorithm.

**Figure 7.**Correlation between each flood characteristic attribute and Muskingum model parameters. (

**a**) Correlation between IWL and parameter K; (

**b**) Correlation between IWL and parameter x; (

**c**) Correlation between PD and parameter K; (

**d**) Correlation between PD and parameter x; (

**e**) Correlation between FPET and parameter K; (

**f**) Correlation between FPET and parameter x; (

**g**) Correlation between FVFT and parameter K; (

**h**) Correlation between FVFT and parameter x; (

**i**) Correlation between FVBFP and parameter K; (

**j**) Correlation between FVBFP and parameter x; (

**k**) Correlation between ADBFP and parameter K; (

**l**) Correlation between ADBFP and parameter x.

**Figure 10.**Performance comparison between the three methods. (

**a**) Mean absolute error comparison between three methods; (

**b**) Average relative error comparison between three methods; (

**c**) Peak flow relative error comparison between three methods.

**Figure 11.**Prediction results of three methods. (

**a**) Prediction results of No. 071008 flood; (

**b**) Prediction results of No. 080620 flood; (

**c**) Prediction results of No. 090618 flood; (

**d**) Prediction results of No. 090727 flood; (

**e**) Prediction results of No. 090920 flood; (

**f**) Prediction results of No. 091108 flood; (

**g**) Prediction results of No. 100619 flood; (

**h**) Prediction results of No. 100726 flood; (

**i**) Prediction results of No. 100812 flood; (

**j**) Prediction results of No. 100904 flood.

PIO | L-S | AGA | ACO | |
---|---|---|---|---|

K | 12.536 | 11.7916 | 12.4508 | 12.7915 |

x | −0.4189 | −0.352 | −0.356 | −0.2592 |

Average absolute error | 4.76 | 5.63 | 5.11 | 5.23 |

Average relative error | 1.07% | 1.29% | 1.15% | 1.18% |

**Table 2.**Parameter estimation results obtained from mean value (MV) method, flow classification (FC) method, and IP-DPE method.

Flood Code | 071008 | 080620 | 090618 | 090727 | 090920 | 091108 | 100619 | 100726 | 100812 | 100904 | |
---|---|---|---|---|---|---|---|---|---|---|---|

MV method | $K$ | 22.318 | 22.318 | 22.318 | 22.318 | 22.318 | 22.318 | 22.318 | 22.318 | 22.318 | 22.318 |

$x$ | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | |

FC method | $K$ | 21.851 | 10.066 | 10.866 | 31.360 | 21.409 | 23.118 | 9.005 | 14.049 | 13.690 | 20.557 |

$x$ | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | 0.1998 | |

IP-DPE method | $K$ | 22.996 | 12.415 | 14.601 | 27.809 | 24.961 | 12.004 | 11.943 | 15.720 | 17.220 | 17.6342 |

$x$ | 0.2002 | 0.1278 | 0.1170 | 0.2392 | 0.0039 | 0.3337 | 0.2203 | 0.1533 | 0.1608 | 0.0084 |

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**MDPI and ACS Style**

Zhang, G.; Xie, T.; Zhang, L.; Hua, X.; Wu, C.; Chen, X.; Li, F.; Zhao, B.
“In-Process Type” Dynamic Muskingum Model Parameter Estimation Method. *Water* **2017**, *9*, 849.
https://doi.org/10.3390/w9110849

**AMA Style**

Zhang G, Xie T, Zhang L, Hua X, Wu C, Chen X, Li F, Zhao B.
“In-Process Type” Dynamic Muskingum Model Parameter Estimation Method. *Water*. 2017; 9(11):849.
https://doi.org/10.3390/w9110849

**Chicago/Turabian Style**

Zhang, Gang, Tuo Xie, Lei Zhang, Xia Hua, Chen Wu, Xi Chen, Fangfeng Li, and Bin Zhao.
2017. "“In-Process Type” Dynamic Muskingum Model Parameter Estimation Method" *Water* 9, no. 11: 849.
https://doi.org/10.3390/w9110849