Hybrid modeling and distributed optimization control method for the iron removal process

Ning Chen; Yan Xia Zhao; Jia Yang Dai; Yu Qian Guo; Wei Hua Gui; Jun Jie Peng

doi:10.3934/jimo.2022003

Article Contents

2023, Volume 19, Issue 2: 1486-1512. Doi: 10.3934/jimo.2022003

This issue Previous Article Optimal financing strategy in a closed-loop supply chain for construction machinery remanufacturing with emissions abatement Next Article Optimality and duality for

$ E $

-differentiable multiobjective programming problems involving

$ E $

-type Ⅰ functions

Hybrid modeling and distributed optimization control method for the iron removal process

1.
School of Automation, Central South University, Changsha Hunan 410083, China
2.
School of Electrical Engineering, Guangxi University, Nanning Guangxi 53004, China

^* Corresponding author: Ning Chen
^* Corresponding author: Ning Chen

Received: June 30, 2021

Revised: October 31, 2021

Early access: January 2022

Published: February 2023

The research is supported in part by the Program of National Natural Science Foundation of China (61673399), and in part by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61621062)

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Abstract

Iron removal by goethite is a key procedure in zinc hydrometallurgy. Due to its complex chemical reaction mechanism, multiple reactors cascade, and high uncertainty, it is difficult to optimize and control in industrial process. In this paper, a distributed optimization control method which is based on a novel hybrid model of the iron removal process is proposed. By combining the mechanism model and a data-driven oxygen mass transfer coefficient model, a hybrid model is first established. Then, to overcome the influence of the former reactor on the latter reactor, the ratio of the status in each subsystem to the set point is taken as a new status, and a distributed optimization control problem is constructed. Considering the high dimensionality of this problem, it is necessary to reconstruct it by Virtual Motion Camouflage (VMC), so that the optimal control problem is transformed into a nonlinear constrained optimal trajectory planning problem. And a Legendre pseudo-spectral method is used to solve the problem accurately to obtain the optimal trajectory of the ion concentration. Finally, simulation results show that the proposed method can effectively reflect the industrial process, and track the fluctuation of inlet ion concentrations with a nice real-time performance.

Keywords:

Mathematics Subject Classification: Primary: 49M37; Secondary: 93C15.

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Figure 1. Process diagram of the iron removal process

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Figure 2. The dissolution and reaction process of oxygen

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Figure 3. The hybrid modeling block diagram

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Figure 4. Block diagram of VMC based optimized control of double-layer structure

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Figure 5. The prey-predator relationship in MC

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Figure 6. Variation of ferrous ion concentration in #1 reactor

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Figure 7. Variation of ferric ion concentration in #1 reactor

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Figure 8. Variation of ferrous ion concentration in #2 reactor

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Figure 9. Variation of ferric ion concentration in #2 reactor

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Figure 10. Variation of ferrous ion concentration in #3 reactor

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Figure 11. Variation of ferric ion concentration in #3 reactor

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Figure 12. Variation of ferrous ion concentration in #4 reactor

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Figure 13. Variation of ferric ion concentration in #4 reactor

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Figure 14. Variation of ferrous ion concentration in #5 reactor

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Figure 15. Variation of ferric ion concentration in #5 reactor

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Figure 16. The concentration of ferrous ions at the outlet of #5 reactor within 80 hours

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Figure 17. The ferrous ion concentration curve at the outlet of the #1 reactor

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Figure 18. The ferrous ion concentration curve at the outlet of the #2 reactor

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Table 1. Optimal control of the iron removal process based on VMC

Initialization	Step0: Determine the expected ion concentration decline curve from process history data. Define the initial time $ {t_0} $, the end time $ {t_f} $ and the number of LGL points $ N $. Determine the LGL points based on Quasi Newton method [27]. Calculate the differential matrix $ {D_{kj}} $.
Iteration	Step1: Initialize path control parameter $ v $.
	Step2: Disperse $ \mathit{\boldsymbol{{c_{pi}} }}$ at the LGL points to obtain $ \mathit{\boldsymbol{c_{pj}^i({\tau _k}) }}$.
	Step3: Disperse the path control parameter $ v $ at the LGL points
	to obtain $ v({\tau _k})(k = 1,2,...,N - 1) $.
	Step4: Disperse the constraints and performance functions in Eq.
	(52) at the LGL points.
	Step5: Calculate the optimal path control parameter $ v({\tau _k}) $ by the
	fmincon solution tool in MATLAB(the fmincon function uses the
	sequential quadratic programming method, and solve the quadratic
	programming sub-problem in each iteration).
	Step6: Calculate $ \mathit{\boldsymbol{c_j^i({\tau _k}) }}$ in Eq.(48).
	Step7: If the convergence criterion is satisfied or the maximum num-
	ber of iterations has reached, the optimization is terminated. Other-
	wise, recalculate path control parameter $ v $ and go back to Step1.

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Table 2. Comparison of accuracy and efficiency of D-LWKPCR when $ \bar N $ changes

$ \bar N $	RMSE(g/L)	MAE(g/L)	Time(s)
5	0.8303	0.6880	0.4840
6	0.6933	0.4977	0.5300
8	0.6378	0.4142	0.5770
10	0.6168	0.3732	0.6860
12	0.6269	0.3887	0.6870
15	0.6445	0.4352	0.7330
20	0.6694	0.4719	0.8110

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Table 3. Parameter identification results of the hybrid model based on D-LWKPCR

Parameter	$ k_1^1 $	$ k_2^1 $	$ k_3^1 $	$ \alpha $	$ \beta $	$ \gamma $	$ \delta $	$ \sigma $
Identification results	1.5963	0.0013	20.0868	1.3592	1.2807	0.3764	4.4953	1.9341

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Table 4. Parameter identification results of the hybrid model based on JITL-SWPPLS

Parameter	$ k_1^1 $	$ k_2^1 $	$ k_3^1 $	$ \alpha $	$ \beta $	$ \gamma $	$ \bar \sigma $
Identification results	1.7113	0.3521	22.9975	1.5091	1.8809	0.3764	0.6421

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Table 5. The set-points at the outlet of #1-#5 reactors

Ion concentration	#1 reactor	#2 reactor	#3 reactor	#4 reactor	#5 reactor
$ F{e^{2 + }} $	9.61	5.89	3.66	1.57	0.68
$ F{e^{3 + }} $	1.08	0.92	0.8	0.67	0.5

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Table 6. The ranges of oxygen addition and pH value of #1-#5 reactors

Reactor	#1 reactor	#2 reactor	#3 reactor	#4 reactor	#5 reactor
Oxygen addition($ {m^3}/h $)	[10, 50]	[50,100]	[60,115]	[70,135]	[40, 90]
pH	[3.0, 3.5]	[3.0, 3.5]	[3.0, 3.5]	[3.0, 3.5]	[3.0, 3.5]

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Table 7. Performance of the proposed method in #1 reactor

Performance	Control error of $ F{e^{2 + }} $(%)	Control error of $ F{e^{3 + }} $(%)	Oxygen addition ($ {m^3}/h $)	Time(s)
$ N = 4 $	4.37	21.96	31.3	1.85
$ N = 10 $	1.75	8.27	38.68	1.92
$ N = 20 $	0.87	4.05	30.11	2.08
$ N = 30 $	0.58	2.69	31.03	2.47
$ N = 40 $	0.19	1.41	29.67	2.63
$ N = 50 $	0.48	3.33	35.7	2.72

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Table 8. The performance of the proposed method in the overall system

Performance	Control error of $ F{e^{2 + }} $(%)	Control error of $ F{e^{3 + }} $(%)	Oxygen addition ($ {m^3}/h $)	Time(s)
$ N = 4 $	20.12	7.6	340.04	9.7
$ N = 10 $	8.72	2.96	338.62	10.29
$ N = 20 $	4.27	1.48	330.45	11.03
$ N = 30 $	3.63	1.38	350.24	12.7
$ N = 40 $	1.31	0.34	322.16	13.4
$ N = 50 $	3.6	0.94	350.34	13.96

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Table 9. The $ F{e^{2 + }} $ ion concentration values at the outlet of each reactor when $ N $ = 40

Reactor	The first method	The second method	The set points of $ F{e^{2 + }} $ ion concentration
#1	9.60	9.50	9.61
#2	5.89	5.66	5.89
#3	3.66	3.52	3.66
#4	1.57	1.44	1.57
#5	0.68	0.63	0.68

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Table 10. The $ F{e^{3 + }} $ ion concentration values at the outlet of each reactor when $ N $ = 40

Reactor	The first method	The second method	The set points of $ F{e^{3 + }} $ion concentration
#1	1.08	1.00	1.08
#2	0.92	0.91	0.92
#3	0.8	0.79	0.8
#4	0.67	0.67	0.67
#5	0.5	0.49	0.5

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Table 11. System performance comparison of the two methods

Algorithm	The first method		The second method
Performance	Oxygen addition	Time(s)	Oxygen addition	Time(s)
	(${m^3}/h$)		(${m^3}/h$)
#1	38.07	3.39	29.67	2.63
#2	68.32	3.25	75.43	2.58
#3	90.64	3.78	95.43	2.44
#4	80.25	3.56	79.24	2.32
#5	46.25	3.12	45.42	2.10

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