化工学报 ›› 2021, Vol. 72 ›› Issue (3): 1512-1520.DOI: 10.11949/0438-1157.20201804
徐斌(
)
收稿日期:2020-12-07
修回日期:2020-12-15
出版日期:2021-03-05
发布日期:2021-03-05
通讯作者:
徐斌
基金资助:
XU Bin()
Received:2020-12-07
Revised:2020-12-15
Online:2021-03-05
Published:2021-03-05
Contact:
XU Bin
摘要：
质子交换膜燃料电池（proton exchange membrane fuel cell， PEMFC）模型中通常包含一些未知的难以确定的参数。为了有效确定这些参数，提出一种基于概率选择模型的改进差分进化算法。该概率选择模型为进化群体中的每个个体分配关于个体优劣性能的选择概率，然后基于该选择概率选择优秀的个体参与变异和交叉操作。为了验证改进算法的有效性，首先将其用于求解标准测试函数，实验结果表明，基于概率选择模型的变异、交叉操作能显著提高差分进化算法的收敛速度和求解精度。然后将改进算法用于质子交换膜燃料电池模型最优参数识别问题，得到的仿真结果和实验测试数据之间具有较高的拟合精度，表明本文改进算法是一种有效的求解系统模型参数优化识别的方法。
中图分类号:
徐斌. 基于改进差分进化算法的质子交换膜燃料电池模型参数优化识别[J]. 化工学报, 2021, 72(3): 1512-1520.
XU Bin. Parameter optimal identification of proton exchange membrane fuel cell model based on an improved differential evolution algorithm[J]. CIESC Journal, 2021, 72(3): 1512-1520.
图1 PEMFC原理
Fig.1 Principle of PEMFC
| 函数 | 算法 | 最小值 | 最大值 | 中位值 | 平均值 | 标准差 |
|---|---|---|---|---|---|---|
| Fsph | DE | 3.024×10-5 | 1.372×10-4 | 6.460×10-5 | 6.897×10-5 | 3.124×10-5 |
| IDE | 9.946×10-15 | 3.052×10-13 | 3.161×10-14 | 5.171×10-14 | 5.979×10-14 | |
| Fros | DE | 4.392×101 | 6.439×102 | 9.895×101 | 1.707×102 | 1.564×102 |
| IDE | 2.521×101 | 1.033×102 | 2.597×101 | 3.816×101 | 2.545×101 | |
| Fack | DE | 1.322×10-3 | 2.871×10-3 | 2.188×10-3 | 2.135×10-3 | 4.463×10-4 |
| IDE | 3.023×10-8 | 1.217×10-7 | 6.460×10-8 | 6.870×10-8 | 2.363×10-8 | |
| Fgwr | DE | 8.075×10-5 | 1.080×10-1 | 7.965×10-4 | 1.221×10-2 | 2.738×10-2 |
| IDE | 9.992×10-15 | 9.865×10-3 | 2.450×10-13 | 1.972×10-3 | 3.629×10-3 | |
| Fras | DE | 1.373×102 | 1.611×102 | 1.521×102 | 1.526×102 | 7.239×100 |
| IDE | 4.724×101 | 1.144×102 | 8.666×101 | 8.472×101 | 1.816×101 | |
| Fsch | DE | 2.716×102 | 4.542×103 | 3.022×103 | 2.804×103 | 1.194×103 |
| IDE | 7.440×10-10 | 2.369×102 | 1.184×102 | 8.054×101 | 8.863×101 | |
| Fsal | DE | 5.187×10-1 | 8.132×10-1 | 7.052×10-1 | 7.142×10-1 | 6.810×10-2 |
| IDE | 1.999×10-1 | 2.999×10-1 | 2.825×10-1 | 2.570×10-1 | 4.716×10-2 | |
| Fwht | DE | 7.465×102 | 1.008×103 | 8.378×102 | 8.446×102 | 5.373×101 |
| IDE | 3.124×102 | 5.419×102 | 4.383×102 | 4.327×102 | 5.412×10-1 |
表1 两种算法求解基准函数的测试结果对比
Table 1 The comparison results of two algorithms over benchmark functions
| 函数 | 算法 | 最小值 | 最大值 | 中位值 | 平均值 | 标准差 |
|---|---|---|---|---|---|---|
| Fsph | DE | 3.024×10-5 | 1.372×10-4 | 6.460×10-5 | 6.897×10-5 | 3.124×10-5 |
| IDE | 9.946×10-15 | 3.052×10-13 | 3.161×10-14 | 5.171×10-14 | 5.979×10-14 | |
| Fros | DE | 4.392×101 | 6.439×102 | 9.895×101 | 1.707×102 | 1.564×102 |
| IDE | 2.521×101 | 1.033×102 | 2.597×101 | 3.816×101 | 2.545×101 | |
| Fack | DE | 1.322×10-3 | 2.871×10-3 | 2.188×10-3 | 2.135×10-3 | 4.463×10-4 |
| IDE | 3.023×10-8 | 1.217×10-7 | 6.460×10-8 | 6.870×10-8 | 2.363×10-8 | |
| Fgwr | DE | 8.075×10-5 | 1.080×10-1 | 7.965×10-4 | 1.221×10-2 | 2.738×10-2 |
| IDE | 9.992×10-15 | 9.865×10-3 | 2.450×10-13 | 1.972×10-3 | 3.629×10-3 | |
| Fras | DE | 1.373×102 | 1.611×102 | 1.521×102 | 1.526×102 | 7.239×100 |
| IDE | 4.724×101 | 1.144×102 | 8.666×101 | 8.472×101 | 1.816×101 | |
| Fsch | DE | 2.716×102 | 4.542×103 | 3.022×103 | 2.804×103 | 1.194×103 |
| IDE | 7.440×10-10 | 2.369×102 | 1.184×102 | 8.054×101 | 8.863×101 | |
| Fsal | DE | 5.187×10-1 | 8.132×10-1 | 7.052×10-1 | 7.142×10-1 | 6.810×10-2 |
| IDE | 1.999×10-1 | 2.999×10-1 | 2.825×10-1 | 2.570×10-1 | 4.716×10-2 | |
| Fwht | DE | 7.465×102 | 1.008×103 | 8.378×102 | 8.446×102 | 5.373×101 |
| IDE | 3.124×102 | 5.419×102 | 4.383×102 | 4.327×102 | 5.412×10-1 |
图2 两种算法迭代收敛曲线图
Fig.2 Convergence graphs of two algorithms
| 函数 | 算法 | N=50 | N=100 | N=150 | N=200 | N=250 | N=300 | N=350 | N=400 |
|---|---|---|---|---|---|---|---|---|---|
| Fsph | DE | 6.90×10-5 | 1.96×10-4 | 2.57×10-4 | 2.86×10-4 | 3.35×10-4 | 3.64×10-4 | 3.54×10-4 | 3.92×10-4 |
| IDE | 5.17×10-14 | 5.33×10-12 | 2.38×10-11 | 5.29×10-11 | 7.19×10-11 | 1.14×10-10 | 1.44×10-10 | 1.49×10-10 | |
| Fros | DE | 1.71×102 | 2.41×102 | 2.32×102 | 2.89×102 | 2.60×102 | 2.56×102 | 2.67×102 | 2.73×102 |
| IDE | 3.82×101 | 2.75×101 | 2.73×101 | 2.47×101 | 2.49×101 | 2.48×101 | 2.48×101 | 2.46×101 | |
| Fack | DE | 2.14×10-3 | 4.13×10-3 | 4.70×10-3 | 5.15×10-3 | 5.41×10-3 | 5.71×10-3 | 5.63×10-3 | 5.78×10-3 |
| IDE | 6.87×10-8 | 5.82×10-7 | 1.27×10-6 | 2.03×10-6 | 2.75×10-6 | 2.99×10-6 | 3.15×10-6 | 3.52×10-6 | |
| Fgwr | DE | 1.22×10-2 | 1.71×10-2 | 1.77×10-2 | 1.08×10-2 | 5.41×10-3 | 1.49×10-2 | 6.71×10-3 | 1.01×10-2 |
| IDE | 1.97×10-3 | 2.96×10-4 | 2.96×10-4 | 3.11×10-10 | 5.59×10-10 | 5.03×10-10 | 2.96×10-4 | 7.25×10-10 | |
| Fras | DE | 1.53×102 | 1.43×102 | 1.47×102 | 1.44×102 | 1.46×102 | 1.43×102 | 1.43×102 | 1.41×102 |
| IDE | 8.47×101 | 1.06×102 | 1.10×102 | 1.13×102 | 1.17×102 | 1.16×102 | 1.20×102 | 1.18×102 | |
| Fsch | DE | 2.80×103 | 3.75×103 | 4.03×103 | 4.13×103 | 4.20×103 | 4.10×103 | 4.16×103 | 4.25×103 |
| IDE | 8.05×101 | 2.11×10-5 | 2.60×10-4 | 2.40×10-4 | 2.17×10-4 | 2.41×10-4 | 2.63×10-4 | 1.72×10-3 | |
| Fsal | DE | 7.14×10-1 | 7.90×10-1 | 7.86×10-1 | 7.89×10-1 | 7.95×10-1 | 8.01×10-1 | 7.97×10-1 | 7.96×10-1 |
| IDE | 2.57×10-1 | 2.72×10-1 | 2.73×10-1 | 2.82×10-1 | 2.89×10-1 | 2.89×10-1 | 2.96×10-1 | 2.97×10-1 | |
| Fwht | DE | 8.45×102 | 1.07×103 | 1.21×103 | 1.45×103 | 1.37×103 | 1.39×103 | 1.52×103 | 1.51×103 |
| IDE | 4.33×102 | 5.36×102 | 5.54×102 | 5.81×102 | 5.84×102 | 5.87×102 | 5.80×102 | 5.86×102 |
表2 不同群体规模下测试结果
Table 2 Experimental results with different population size
| 函数 | 算法 | N=50 | N=100 | N=150 | N=200 | N=250 | N=300 | N=350 | N=400 |
|---|---|---|---|---|---|---|---|---|---|
| Fsph | DE | 6.90×10-5 | 1.96×10-4 | 2.57×10-4 | 2.86×10-4 | 3.35×10-4 | 3.64×10-4 | 3.54×10-4 | 3.92×10-4 |
| IDE | 5.17×10-14 | 5.33×10-12 | 2.38×10-11 | 5.29×10-11 | 7.19×10-11 | 1.14×10-10 | 1.44×10-10 | 1.49×10-10 | |
| Fros | DE | 1.71×102 | 2.41×102 | 2.32×102 | 2.89×102 | 2.60×102 | 2.56×102 | 2.67×102 | 2.73×102 |
| IDE | 3.82×101 | 2.75×101 | 2.73×101 | 2.47×101 | 2.49×101 | 2.48×101 | 2.48×101 | 2.46×101 | |
| Fack | DE | 2.14×10-3 | 4.13×10-3 | 4.70×10-3 | 5.15×10-3 | 5.41×10-3 | 5.71×10-3 | 5.63×10-3 | 5.78×10-3 |
| IDE | 6.87×10-8 | 5.82×10-7 | 1.27×10-6 | 2.03×10-6 | 2.75×10-6 | 2.99×10-6 | 3.15×10-6 | 3.52×10-6 | |
| Fgwr | DE | 1.22×10-2 | 1.71×10-2 | 1.77×10-2 | 1.08×10-2 | 5.41×10-3 | 1.49×10-2 | 6.71×10-3 | 1.01×10-2 |
| IDE | 1.97×10-3 | 2.96×10-4 | 2.96×10-4 | 3.11×10-10 | 5.59×10-10 | 5.03×10-10 | 2.96×10-4 | 7.25×10-10 | |
| Fras | DE | 1.53×102 | 1.43×102 | 1.47×102 | 1.44×102 | 1.46×102 | 1.43×102 | 1.43×102 | 1.41×102 |
| IDE | 8.47×101 | 1.06×102 | 1.10×102 | 1.13×102 | 1.17×102 | 1.16×102 | 1.20×102 | 1.18×102 | |
| Fsch | DE | 2.80×103 | 3.75×103 | 4.03×103 | 4.13×103 | 4.20×103 | 4.10×103 | 4.16×103 | 4.25×103 |
| IDE | 8.05×101 | 2.11×10-5 | 2.60×10-4 | 2.40×10-4 | 2.17×10-4 | 2.41×10-4 | 2.63×10-4 | 1.72×10-3 | |
| Fsal | DE | 7.14×10-1 | 7.90×10-1 | 7.86×10-1 | 7.89×10-1 | 7.95×10-1 | 8.01×10-1 | 7.97×10-1 | 7.96×10-1 |
| IDE | 2.57×10-1 | 2.72×10-1 | 2.73×10-1 | 2.82×10-1 | 2.89×10-1 | 2.89×10-1 | 2.96×10-1 | 2.97×10-1 | |
| Fwht | DE | 8.45×102 | 1.07×103 | 1.21×103 | 1.45×103 | 1.37×103 | 1.39×103 | 1.52×103 | 1.51×103 |
| IDE | 4.33×102 | 5.36×102 | 5.54×102 | 5.81×102 | 5.84×102 | 5.87×102 | 5.80×102 | 5.86×102 |
| 噪声 | 算法 | ξ1 | ξ2 | ξ3 | ξ4 | λ | RC | B | f(x) |
|---|---|---|---|---|---|---|---|---|---|
| 无噪声 | RCGA | -0.938096 | 0.00316553 | 8.805×10-5 | -1.75×10-4 | 20.18 | 0.000596 | 0.02575428 | 7.2056×10-4 |
| PSO | -0.853200 | 0.00249636 | 4.716×10-5 | -9.54×10-4 | 10.00 | 0.000100 | 0.01360000 | 8.0614×10-2 | |
| ABC | -0.996693 | 0.00333755 | 8.716×10-5 | -1.91×10-4 | 17.52 | 0.000212 | 0.01771729 | 4.7300×10-3 | |
| DE | -1.090382 | 0.00297845 | 3.837×10-5 | -1.81×10-4 | 22.17 | 0.000468 | 0.02667588 | 3.3816×10-4 | |
| IDE | -0.901873 | 0.00295771 | 7.894×10-5 | -1.87×10-4 | 22.65 | 0.000184 | 0.02832236 | 1.4398×10-4 | |
| 低噪声 | RCGA | -1.132093 | 0.00368763 | 8.541×10-5 | -1.79×10-4 | 23.98 | 0.000575 | 0.02942129 | 1.7529×10-2 |
| PSO | -0.948038 | 0.00159462 | 4.970×10-5 | -1.55×10-4 | 21.41 | 0.000127 | 0.02989030 | 1.6181×10-1 | |
| ABC | -0.907133 | 0.00285576 | 6.745×10-5 | -1.92×10-4 | 17.85 | 0.000608 | 0.01716988 | 2.5533×10-2 | |
| DE | -1.127019 | 0.00366308 | 8.368×10-5 | -1.88×10-4 | 21.43 | 0.000144 | 0.02802703 | 1.7573×10-2 | |
| IDE | -1.036553 | 0.00317264 | 6.538×10-5 | -1.83×10-4 | 23.35 | 0.000484 | 0.02817283 | 1.7343×10-2 | |
| 高噪声 | RCGA | -0.908332 | 0.00261682 | 4.761×10-5 | -2.12×10-4 | 24.00 | 0.000482 | 0.01833886 | 9.4149×10-2 |
| PSO | -1.186572 | 0.00356386 | 6.002×10-5 | -1.57×10-4 | 10.23 | 0.000672 | 0.01417309 | 2.3739×10-1 | |
| ABC | -1.122471 | 0.00318047 | 4.510×10-5 | -2.01×10-4 | 20.03 | 0.000730 | 0.01376492 | 9.6348×10-2 | |
| DE | -1.087087 | 0.00304861 | 4.168×10-5 | -2.15×10-4 | 23.05 | 0.000182 | 0.01858317 | 9.4147×10-2 | |
| IDE | -1.022205 | 0.00294057 | 4.790×10-5 | -2.14×10-4 | 23.69 | 0.000211 | 0.01950286 | 9.4140×10-2 |
表3 五种算法在三种情况下求解最优结果比较
Table 3 The best results of five algorithms under three different conditions
| 噪声 | 算法 | ξ1 | ξ2 | ξ3 | ξ4 | λ | RC | B | f(x) |
|---|---|---|---|---|---|---|---|---|---|
| 无噪声 | RCGA | -0.938096 | 0.00316553 | 8.805×10-5 | -1.75×10-4 | 20.18 | 0.000596 | 0.02575428 | 7.2056×10-4 |
| PSO | -0.853200 | 0.00249636 | 4.716×10-5 | -9.54×10-4 | 10.00 | 0.000100 | 0.01360000 | 8.0614×10-2 | |
| ABC | -0.996693 | 0.00333755 | 8.716×10-5 | -1.91×10-4 | 17.52 | 0.000212 | 0.01771729 | 4.7300×10-3 | |
| DE | -1.090382 | 0.00297845 | 3.837×10-5 | -1.81×10-4 | 22.17 | 0.000468 | 0.02667588 | 3.3816×10-4 | |
| IDE | -0.901873 | 0.00295771 | 7.894×10-5 | -1.87×10-4 | 22.65 | 0.000184 | 0.02832236 | 1.4398×10-4 | |
| 低噪声 | RCGA | -1.132093 | 0.00368763 | 8.541×10-5 | -1.79×10-4 | 23.98 | 0.000575 | 0.02942129 | 1.7529×10-2 |
| PSO | -0.948038 | 0.00159462 | 4.970×10-5 | -1.55×10-4 | 21.41 | 0.000127 | 0.02989030 | 1.6181×10-1 | |
| ABC | -0.907133 | 0.00285576 | 6.745×10-5 | -1.92×10-4 | 17.85 | 0.000608 | 0.01716988 | 2.5533×10-2 | |
| DE | -1.127019 | 0.00366308 | 8.368×10-5 | -1.88×10-4 | 21.43 | 0.000144 | 0.02802703 | 1.7573×10-2 | |
| IDE | -1.036553 | 0.00317264 | 6.538×10-5 | -1.83×10-4 | 23.35 | 0.000484 | 0.02817283 | 1.7343×10-2 | |
| 高噪声 | RCGA | -0.908332 | 0.00261682 | 4.761×10-5 | -2.12×10-4 | 24.00 | 0.000482 | 0.01833886 | 9.4149×10-2 |
| PSO | -1.186572 | 0.00356386 | 6.002×10-5 | -1.57×10-4 | 10.23 | 0.000672 | 0.01417309 | 2.3739×10-1 | |
| ABC | -1.122471 | 0.00318047 | 4.510×10-5 | -2.01×10-4 | 20.03 | 0.000730 | 0.01376492 | 9.6348×10-2 | |
| DE | -1.087087 | 0.00304861 | 4.168×10-5 | -2.15×10-4 | 23.05 | 0.000182 | 0.01858317 | 9.4147×10-2 | |
| IDE | -1.022205 | 0.00294057 | 4.790×10-5 | -2.14×10-4 | 23.69 | 0.000211 | 0.01950286 | 9.4140×10-2 |
图3 实验数据和模型输出曲线比较
Fig.3 Comparisons between the simulated data and the model curves
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