﻿ 灰狼优化算法在马斯京根模型参数估计中的应用
 计算机系统应用  2018, Vol. 27 Issue (12): 198-203 PDF

Application of Grey Wolf Optimizer to Parameter Estimation to Muskingum Routing Model
WANG Meng-Na, WANG Qiu-Ping, WANG Xiao-Feng
Faculty of Sciences, Xi’an University of Technology, Xi’an 710054, China
Foundation item: National Natural Science Foundation of China (61772416)
Abstract: In order to improve the computation accuracy of the Muskingum flood routing model, a new method of parameter estimation for Muskingum model based on the improved grey wolf optimizer (IGWO) is proposed and applied to the flood calculation in the south canal between Chenggouwan and Linqing River. The experimental results show that IGWO can effectively estimate the parameters of the Muskingum model. Compared with the other parameter estimation methods of Muskingum routing model, IGWO has higher calculation accuracy and better optimization performance.
Key words: flood routing     Muskingum routing model     grey wolf optimizer     parameter estimation

1 马斯京根模型理论

 \left\{ {\begin{aligned} & {\displaystyle\frac{{{\rm{d}}W}}{{{\rm{d}}t}} = I - Q} \\ & {W = K\left[ {xI + \left( {1 - x} \right)Q} \right]} \end{aligned}} \right. (1)

 $\left\{ {\begin{array}{*{20}{l}} {\tilde Q\left( 1 \right) = Q\left( 1 \right)} \\ {\tilde Q\left( i \right) = {C_0}I\left( i \right) + {C_1}I\left( {i - 1} \right) + {C_2}\tilde Q\left( {i - 1} \right),i = 2,3,\cdots,n} \end{array}} \right.$ (2)

 ${C_0} + {C_1} + {C_2} = 1$ (3)

 $\begin{split}& \min F = {\displaystyle\sum\limits_{i = 2}^n {\left( {\left| {{C_0}I\left( i \right) + {C_1}I\left( {i - 1} \right) + {C_2}\tilde Q\left( {i - 1} \right) - Q\left( i \right)} \right|} \right)} ^q}\\& {\rm{s}}.{\rm{t}}.\;\;\;\;\;\;{g_1}:{C_0} \in \left[ {0,1} \right]\\& \;\;\;\;\;\;\;\;\;\;\;{g_2}:{C_1} \in \left[ {0,1} \right]\\& \;\;\;\;\;\;\;\;\;\;\;{g_3}:1 - {C_0} - {C_1} \in \left[ {0,1} \right]\end{split}$ (4)

 $\min f = F + p\left( {{g_3}\left( {{C_0},{C_1}} \right)} \right)$ (5)

2 灰狼优化算法 2.1 基本灰狼优化算法

GWO算法[10]是通过模拟自然界中灰狼群体的社会等级机制和捕食行为而提出的一种新型群体智能优化算法. 灰狼种群具有严格的等级制度, 如图1所示.

 图 1 灰狼等级示意图

 ${{D}} = \left| {{{C}} \cdot {{{X}}_p}\left( t \right) - {{X}}\left( t \right)} \right|$ (6)
 ${{X}}\left( {t + 1} \right) = {{{X}}_p}\left( t \right) - {{A}} \cdot {{D}}$ (7)

 ${{A}} = 2a \cdot {{{r}}_1} - a$ (8)
 ${{C}} = 2 \cdot {{{r}}_2}$ (9)

 $\left\{ {\begin{array}{*{20}{c}}{{{{D}}_\alpha } = \left| {{{{C}}_1} \cdot {{{X}}_\alpha } - {{X}}} \right|,\;\;\;{{{X}}_1} = {{{X}}_\alpha } - {{{A}}_1} \cdot {{{D}}_\alpha }}\\{{{{D}}_\beta } = \left| {{{{C}}_2} \cdot {{{X}}_\beta } - {{X}}} \right|,\;\;\;{{{X}}_2} = {{{X}}_\beta } - {{{A}}_2} \cdot {{{D}}_\beta }}\\{{{{D}}_\delta } = \left| {{{{C}}_3} \cdot {{{X}}_\delta } - {{X}}} \right|,\;\;\;\;{{{X}}_3} = {{{X}}_\delta } - {{{A}}_3} \cdot {{{D}}_\delta }}\end{array}} \right.$ (10)
 ${{X}}\left( {t + 1} \right) = \frac{{{{{X}}_1} + {{{X}}_2} + {{{X}}_3}}}{3}$ (11)
2.2 改进的灰狼优化算法

GWO算法自提出以来已在许多领域得到了应用, 但其仍存在依赖初始种群和易陷入局部最优的缺点. 为了解决这些问题, 对基本GWO算法作如下改进:

(1) 混沌种群初始化

 ${x_{n + 1}} = \cos \left( {k{\rm{cos}^{ - 1}}\left( {{x_n}} \right)} \right)$ (12)

(2) 随机分布调整收敛因子

 $a = \left( {{a_{\max }} - {a_{\min }}} \right)rand() + \sigma randn()$ (13)

2.3 仿真实验

 图 2 Sphere函数收敛曲线

 图 3 Ackley函数收敛曲线

 图 4 Schwefel 1.2函数在不同初始化策略下的收敛曲线

3 马斯京根模型的参数估计步骤

 图 5 Quartic函数在不同初始化策略下的收敛曲线

 图 6 Schwefel 2.22函数在不同收敛因子下的收敛曲线

 图 7 Ackley函数在不同收敛因子下的收敛曲线

Step1. 设置种群规模N=30, 维数dim=2, 最大迭代次数tmax=500, 搜索区间为[0, 1], 随机生成 $a$ AC等参数.

Step2. 利用Chebyshev混沌映射策略初始化灰狼种群;

Step3. 将式(5)设置为适应度函数, 计算种群中所有灰狼个体的适应度值, 选择前三个最好的狼, 记录其位置XαXβXδ;

Step4. 利用公式(10)和(11)更新种群中其它灰狼个体的位置;

Step5. 利用公式(13)计算 $a$ , 然后利用公式(8)和(9)更新A, C的值;

Step6. 判断算法是否满足结束条件, 若达到预定的最大迭代次数tmax, 则停止计算, 输出最优位置 ${{{X}}_\alpha }$ 和最优函数值; 否则, 重复执行Step3–Step5.

4 应用实例

 图 8 IGWO算法洪水演算效果

 图 9 GWO和IGWO算法估计参数的迭代收敛曲线

5 结论

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