JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (1): 124-131, 138.doi: 10.6040/j.issn.1671-9352.0.2022.542

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Delay optimal control of 1,3-propanediol batch fermentation

Xiao WANG1(),Chongyang LIU2,*(),Dianzhong HU3,Gang LIU3   

  1. 1. School of Information and Electronical Engineering, Shandong Technology and Business University, Yantai 264005, Shandong, China
    2. School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, Shandong, China
    3. Yantai Haiyi Software Co., Ltd., Yantai 264000, Shandong, China
  • Received:2022-10-10 Online:2024-01-20 Published:2024-01-19
  • Contact: Chongyang LIU E-mail:13854812465@163.com;chongyangliu@aliyun.com

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Abstract:

In the batch process of glycerol bioconversion to 1,3-propanediol, the initial biomass and glycerol concentration will affect the productivity of 1,3-propanediol. This paper proposes a constrained delay optimal control model to maximize the productivity of 1,3-propanediol. For this problem, the time-scaling transformation is applied to convert it to an optimal control problem with fixed terminal time. Then, the penalty method is used to deal with the constraints in the optimal control problem. Finally, a hybrid algorithm of simulated annealing and genetic algorithm is developed to solve the resulting problem. Numerical results indicate that the productivity of 1,3-propanediol increases by 21.12% compared with the previous result.

Key words: nonlinear time-delay system, optimal control, batch fermentation, simulated annealing algorithm, genetic algorithm

CLC Number: 

  • O232

Table 1

Parameter values and critical concentrations in System (1)"

i mi Yi x*i xi* ki μm
1 — — 0.01 6 69.927 3 0.994
2 9.34 0.008 51 0 2 039 — —
3 7.3 76 0 939.5 — —
4 -0.368 35.54 0 1 026 — —
5 -0.45 14.78 0 360.9 — —

Table 2

Information of test functions"

函数 函数表达式 搜索空间
Schaffer $f_1\left(x_1, x_2\right)=\left(x_1^2+x_2^2\right)^{0.25}\left[\sin ^2\left(50\left(x_1^2+x_2^2\right)^{0.1}\right)+1\right]$ [-100, 100]
Alpine $f_2(x)=\sum\limits_{i=1}^n\left|x_i \sin \left(x_i\right)+0.1 x_i\right|$ [-10, 10]

Table 3

Comparison of four optimization algorithms"

算法 准则 SA[14] GA[14] GAEA[14] 本文算法
平均值 0.222 0 0.068 1 8.49×10-8 0
f1 最差解 0.515 3 0.240 0 1.57×10-7 0
最优解 0.042 1 0.015 2 1.21×10-8 0
最优解次数 1 2 11 60
平均值 1.9×10-4 5.67×10-16 2.34×10-16 1.654×10-16
f2 最差解 7.74×10-4 9.43×10-5 3.89×10-15 1.221×10-15
最优解 2.21×10-7 2.30×10-7 0 0
最优解次数 4 5 12 38

Fig.1

Change of 1,3-propanediol productivity(Annotation: the productivity of 1,3-PD does not exist at time t=0, and the first time point starts from t=2.65×10-6)"

Fig.2

Time varying curve of each substance concentration"

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