# American Institute of Mathematical Sciences

March  2022, 15(3): 587-601. doi: 10.3934/dcdss.2021153

## An optimal control problem applied to a wastewater treatment plant

 1 ALGORITMI Center, University of Minho, Department of Production and Systems, University of Minho, Campus Gualtar, 4710-057 Braga, Portugal 2 Instituto Politécnico de Viana do Castelo, Portugal, CIDMA, Department of Mathematics, University of Aveiro, Portugal

* Corresponding author: M. Teresa T. Monteiro

Received  January 2020 Revised  June 2021 Published  March 2022 Early access  November 2021

This paper aims to present a mathematical model that describes the operation of an activated sludge system during one day. Such system is used in the majority of wastewater treatment plants and depends strongly on the dissolved oxygen, since it is a biological treatment. To guarantee the appropriate amount of dissolved oxygen, expensive aeration strategies are demanded, leading to high costs in terms of energy consumption. It was considered a typical domestic effluent as the wastewater to test the mathematical model and it was used the ASM1 to describe the activated sludge behaviour. An optimal control problem was formulated whose cost functional considers the trade-off between the minimization of the control variable herein considered (the dissolved oxygen) and the quality index that is the amount of pollution. The optimal control problem is treated as a nonlinear optimization problem after discretization by direct methods. The problem was then coded in the AMPL programming language in order to carry out numerical simulations using the NLP solver IPOPT from NEOS Server.

Citation: M. Teresa T. Monteiro, Isabel Espírito Santo, Helena Sofia Rodrigues. An optimal control problem applied to a wastewater treatment plant. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 587-601. doi: 10.3934/dcdss.2021153
##### References:
 [1] L. Amand and B. Carlsson, Optimal aeration control in a nitrifying activated sludge process, Water Research, 46 (2012), 2101-2110. [2] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York-Chichester-Brisbane, 1979. [3] C. Binnie, M. Kimber and G. Smethurst, Basic Water Treatment, Royal Society of Chemistry, Cambridge, UK, 2002. [4] S. Cristea, C. de Prada, D. Sarabia and G. Gutierrez, Aeration control of a wastewater treatment plant using hybrid NMPC, Computers & Chemical Engineering, 35 (2011), 638-650. [5] J. Czyzyk, M. P. Mesnier and J. J. Moré, The EOS server, IEEE Computational Science and Engineering, 5 (1998), 68-75.  doi: 10.1109/99.714603. [6] I. Espírito Santo, Desenho Óptimo de Estações de Águas Residuais Através da ModelaÇão dee FunÇões Custo, Ph.D thesis, Universidade do Minho, Braga, 2007. [7] M. Fikar, B. Chachuat and M. A. Latifi, Optimal operation of alternating activated sludge processes, Control Engineering Practice, 13 (2005), 853-861.  doi: 10.1016/j.conengprac.2004.10.003. [8] R. Fourer, D. M. Gay and B. Kernighan, AMPL: A mathematical programming language, Algorithms and Model Formulations in Mathematical Programming, 51 (1989), 150-151.  doi: 10.1007/978-3-642-83724-1_12. [9] G. Fu, D. Butler and S. T. Khu, Multiple objective optimal control of integrated urban wastewater systems, Environmental Modelling & Software, 23 (2008), 225-234. [10] M. Henze, C. P. L. Grady Jr, W. Gujer, G. V. R. Marais and T. Matsuo, Activated Sludge Model No.1, Technical Report 1, IAWPRC Task Group on Mathematical Modelling for design and operation of biological wastewater treatment, 1986. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. [12] B. Holenda, E. Domokos, Á. Rédey and J. Fazakas, Aeration optimization of a wastewater treatment plant using genetic algorithm, Optimal Control Applications and Methods, 28 (2007), 191-208.  doi: 10.1002/oca.796. [13] R. Hreizab, N. Roche, B. Benyahiac and M. A. Latifia, Multi-objective optimal control of small-size wastewater treatment plants, Chemical Engineering Research and Design, 102 (2015), 345-353. [14] L. S. Pontryagin, The Mathematical Theory of Optimal Processes, Mathematics & Statistics for Engineers, 2017. doi: 10.1201/9780203749319. [15] J. F. Qiao, Y. C. Bo, W. Chai and H. G. Han, Adaptive optimal control for a wastewater treatment plant based on a data-driven method, Water Sci Techno, 67 (2013), 2314-2320.  doi: 10.2166/wst.2013.087. [16] H. S. Rodrigues, Optimal Control and Numerical Optimization Applied to Epidemiological Models, PhD thesis, University of Aveiro, 2012. [17] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Optimization of dengue epidemics: A test case with different discretization schemes, AIP Conference Proceedings, 1168 (2009), 1385-1388.  doi: 10.1063/1.3241345. [18] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Optimal Control and numerical software: An overview, Systems Theory: Perspectives, Applications and Developments, (2014), 93–110. [19] C. Sweetapple, G. Fu and D. Butler, Multi-objective optimisation of wastewater treatment plant control to reduce greenhouse gas emissions, Water Research, 55 (2014), 52-62.  doi: 10.1016/j.watres.2014.02.018. [20] A. Wätcher and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.

show all references

##### References:
 [1] L. Amand and B. Carlsson, Optimal aeration control in a nitrifying activated sludge process, Water Research, 46 (2012), 2101-2110. [2] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York-Chichester-Brisbane, 1979. [3] C. Binnie, M. Kimber and G. Smethurst, Basic Water Treatment, Royal Society of Chemistry, Cambridge, UK, 2002. [4] S. Cristea, C. de Prada, D. Sarabia and G. Gutierrez, Aeration control of a wastewater treatment plant using hybrid NMPC, Computers & Chemical Engineering, 35 (2011), 638-650. [5] J. Czyzyk, M. P. Mesnier and J. J. Moré, The EOS server, IEEE Computational Science and Engineering, 5 (1998), 68-75.  doi: 10.1109/99.714603. [6] I. Espírito Santo, Desenho Óptimo de Estações de Águas Residuais Através da ModelaÇão dee FunÇões Custo, Ph.D thesis, Universidade do Minho, Braga, 2007. [7] M. Fikar, B. Chachuat and M. A. Latifi, Optimal operation of alternating activated sludge processes, Control Engineering Practice, 13 (2005), 853-861.  doi: 10.1016/j.conengprac.2004.10.003. [8] R. Fourer, D. M. Gay and B. Kernighan, AMPL: A mathematical programming language, Algorithms and Model Formulations in Mathematical Programming, 51 (1989), 150-151.  doi: 10.1007/978-3-642-83724-1_12. [9] G. Fu, D. Butler and S. T. Khu, Multiple objective optimal control of integrated urban wastewater systems, Environmental Modelling & Software, 23 (2008), 225-234. [10] M. Henze, C. P. L. Grady Jr, W. Gujer, G. V. R. Marais and T. Matsuo, Activated Sludge Model No.1, Technical Report 1, IAWPRC Task Group on Mathematical Modelling for design and operation of biological wastewater treatment, 1986. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. [12] B. Holenda, E. Domokos, Á. Rédey and J. Fazakas, Aeration optimization of a wastewater treatment plant using genetic algorithm, Optimal Control Applications and Methods, 28 (2007), 191-208.  doi: 10.1002/oca.796. [13] R. Hreizab, N. Roche, B. Benyahiac and M. A. Latifia, Multi-objective optimal control of small-size wastewater treatment plants, Chemical Engineering Research and Design, 102 (2015), 345-353. [14] L. S. Pontryagin, The Mathematical Theory of Optimal Processes, Mathematics & Statistics for Engineers, 2017. doi: 10.1201/9780203749319. [15] J. F. Qiao, Y. C. Bo, W. Chai and H. G. Han, Adaptive optimal control for a wastewater treatment plant based on a data-driven method, Water Sci Techno, 67 (2013), 2314-2320.  doi: 10.2166/wst.2013.087. [16] H. S. Rodrigues, Optimal Control and Numerical Optimization Applied to Epidemiological Models, PhD thesis, University of Aveiro, 2012. [17] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Optimization of dengue epidemics: A test case with different discretization schemes, AIP Conference Proceedings, 1168 (2009), 1385-1388.  doi: 10.1063/1.3241345. [18] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Optimal Control and numerical software: An overview, Systems Theory: Perspectives, Applications and Developments, (2014), 93–110. [19] C. Sweetapple, G. Fu and D. Butler, Multi-objective optimisation of wastewater treatment plant control to reduce greenhouse gas emissions, Water Research, 55 (2014), 52-62.  doi: 10.1016/j.watres.2014.02.018. [20] A. Wätcher and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.
Schematic view of the activated sludge system
Hourly results during one day with $(\omega_1,\omega_2) = (1,0)$
Hourly results during one day with $(\omega_1,\omega_2) = (0.5,0.5)$
Hourly results during one day with $(\omega_1,\omega_2) = (0,1)$
Hourly results during one day with $S_{inf} = 125$
Hourly results for the law limits during one day with $S_{inf} = 125$
Variables of the problem and initial values
 Variable Init.val. Variable Init.val. Variable Init.val. $Q$ 4000 $Q_w$ 100 $Q_r$ 2000 $Q_{ef}$ 1900 $X_I$ 727.3 $X_{I_r}$ 950.571 $X_{I_{ef}}$ $10^{-5}$ $S_{S_{in}}$ 50 $S_S$ 10 $S_{O_{in}}$ 1 $S_{NO_{in}}$ $10^{-6}$ $S_{NO}$ $10^{-6}$ $X_{BH_{in}}$ 0.2 $X_{BH}$ 350 $X_{BH_{r}}$ 711.2 $X_{BH_{ef}}$ $10^{-5}$ $X_{S_{in}}$ 3000 $X_S$ 350 $X_{S_r}$ 806.714 $X_{S_{ef}}$ $10^{-5}$ $X_{BA_{in}}$ $10^{-5}$ $X_{BA}$ $10^{-6}$ $X_{BA_r}$ 1.9$\times 10^{-6}$ $X_{BA_{ef}}$ $10^{-5}$ $S_{NH_{in}}$ 10 $S_{NH}$ 7.5 $X_{P_{in}}$ 1500 $X_{P}$ 90 $X_{P_{r}}$ 174.52 $X_{P_{ef}}$ $10^{-5}$ $S_{ND_{in}}$ 0.5 $S_{ND}$ 0.5 $X_{ND_{in}}$ 200 $X_{ND}$ 20 $X_{ND_{r}}$ 20 $X_{ND_{ef}}$ 0.5 $G_S$ 10000 $SSI$ 1500 $SSI_{ef}$ 0.1 $SSI_r$ 3500 $HRT$ 3.5 $r$ 1 $S_O$ 2
 Variable Init.val. Variable Init.val. Variable Init.val. $Q$ 4000 $Q_w$ 100 $Q_r$ 2000 $Q_{ef}$ 1900 $X_I$ 727.3 $X_{I_r}$ 950.571 $X_{I_{ef}}$ $10^{-5}$ $S_{S_{in}}$ 50 $S_S$ 10 $S_{O_{in}}$ 1 $S_{NO_{in}}$ $10^{-6}$ $S_{NO}$ $10^{-6}$ $X_{BH_{in}}$ 0.2 $X_{BH}$ 350 $X_{BH_{r}}$ 711.2 $X_{BH_{ef}}$ $10^{-5}$ $X_{S_{in}}$ 3000 $X_S$ 350 $X_{S_r}$ 806.714 $X_{S_{ef}}$ $10^{-5}$ $X_{BA_{in}}$ $10^{-5}$ $X_{BA}$ $10^{-6}$ $X_{BA_r}$ 1.9$\times 10^{-6}$ $X_{BA_{ef}}$ $10^{-5}$ $S_{NH_{in}}$ 10 $S_{NH}$ 7.5 $X_{P_{in}}$ 1500 $X_{P}$ 90 $X_{P_{r}}$ 174.52 $X_{P_{ef}}$ $10^{-5}$ $S_{ND_{in}}$ 0.5 $S_{ND}$ 0.5 $X_{ND_{in}}$ 200 $X_{ND}$ 20 $X_{ND_{r}}$ 20 $X_{ND_{ef}}$ 0.5 $G_S$ 10000 $SSI$ 1500 $SSI_{ef}$ 0.1 $SSI_r$ 3500 $HRT$ 3.5 $r$ 1 $S_O$ 2
Parameters
 Kinetic Operational Stoichiometric $\mu_H$ 6 $T$ 20 $Y_A$ 0.24 $\mu_A$ 0.8 $P_{O_2}$ 0.21 $Y_H$ 0.666 $k_h$ 3 $SRT$ 20 $f_P$ 0.08 $k_a$ 0.08 $\theta$ 1.024 $i_{X_B}$ 0.086 $b_h$ 0.62 $\alpha$ 0.8 $i_{X_P}$ 0.06 $b_a$ 0.04 $\eta$ 0.07 $\eta_g$ 0.8 $\beta$ 0.95 $\eta_h$ 0.4 $K_S$ 20 $K_X$ 0.03 $K_{OH}$ 0.2 $K_{NO}$ 0.5 $K_{NH}$ 1 $K_{OA}$ 0.4
 Kinetic Operational Stoichiometric $\mu_H$ 6 $T$ 20 $Y_A$ 0.24 $\mu_A$ 0.8 $P_{O_2}$ 0.21 $Y_H$ 0.666 $k_h$ 3 $SRT$ 20 $f_P$ 0.08 $k_a$ 0.08 $\theta$ 1.024 $i_{X_B}$ 0.086 $b_h$ 0.62 $\alpha$ 0.8 $i_{X_P}$ 0.06 $b_a$ 0.04 $\eta$ 0.07 $\eta_g$ 0.8 $\beta$ 0.95 $\eta_h$ 0.4 $K_S$ 20 $K_X$ 0.03 $K_{OH}$ 0.2 $K_{NO}$ 0.5 $K_{NH}$ 1 $K_{OA}$ 0.4
Characteristics of the wastewater entering the system
 $Q_{inf}$ 530 $S_{I_{inf}}$ 5.45, 12.5 and 25 $S_S$ 44.55,112.5 and 225 $S_{inf}$ 50,125 and 250 $X_{BH_{inf}}$ 0 $X_{BA_{inf}}$ 0 $X_{P_{inf}}$ 0 $S_{O_{inf}}$ 0 $S_{NO_{inf}}$ 0 $S_{alk_{inf}}$ 7 $X_{I_{inf}}$ 90 $X_{S_{inf}}$ 168.75 $S_{NH_{inf}}$ 11.7 $X_{S_{inf}}$ 168.75 $S_{NH_{inf}}$ 11.7 $S_{ND_{inf}}$ 0.63 $X_{ND_{inf}}$ 1.251 $X_{II}$ 18.3
 $Q_{inf}$ 530 $S_{I_{inf}}$ 5.45, 12.5 and 25 $S_S$ 44.55,112.5 and 225 $S_{inf}$ 50,125 and 250 $X_{BH_{inf}}$ 0 $X_{BA_{inf}}$ 0 $X_{P_{inf}}$ 0 $S_{O_{inf}}$ 0 $S_{NO_{inf}}$ 0 $S_{alk_{inf}}$ 7 $X_{I_{inf}}$ 90 $X_{S_{inf}}$ 168.75 $S_{NH_{inf}}$ 11.7 $X_{S_{inf}}$ 168.75 $S_{NH_{inf}}$ 11.7 $S_{ND_{inf}}$ 0.63 $X_{ND_{inf}}$ 1.251 $X_{II}$ 18.3
Results for three different amounts of pollution in the wastewater
 Scenario A Scenario B Scenario C $(w_1,w_2)=(1,0)$ ($w_1,w_2)=(0.5,0.5)$ $(w_1,w_2)=(0,1)$ $S_{inf}$ 50 125 250 50 125 250 50 125 250 $S_O$ 60.6 66.4 80.2 71.5 73.8 77.9 72.0 73.4 80.2 $QI$ 2390.6 2849.9 813.1 692.3 795.4 857.2 750.2 750.4 813.1
 Scenario A Scenario B Scenario C $(w_1,w_2)=(1,0)$ ($w_1,w_2)=(0.5,0.5)$ $(w_1,w_2)=(0,1)$ $S_{inf}$ 50 125 250 50 125 250 50 125 250 $S_O$ 60.6 66.4 80.2 71.5 73.8 77.9 72.0 73.4 80.2 $QI$ 2390.6 2849.9 813.1 692.3 795.4 857.2 750.2 750.4 813.1
 [1] Andrew J. Whittle, Suzanne Lenhart, Louis J. Gross. Optimal control for management of an invasive plant species. Mathematical Biosciences & Engineering, 2007, 4 (1) : 101-112. doi: 10.3934/mbe.2007.4.101 [2] Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311 [3] Loïc Louison, Abdennebi Omrane, Harry Ozier-Lafontaine, Delphine Picart. Modeling plant nutrient uptake: Mathematical analysis and optimal control. Evolution Equations and Control Theory, 2015, 4 (2) : 193-203. doi: 10.3934/eect.2015.4.193 [4] Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 [5] Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639 [6] Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067 [7] Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971 [8] Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009 [9] Cristiana J. Silva, Delfim F. M. Torres. Optimal control strategies for tuberculosis treatment: A case study in Angola. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 601-617. doi: 10.3934/naco.2012.2.601 [10] Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621 [11] Djamila Moulay, M. A. Aziz-Alaoui, Hee-Dae Kwon. Optimal control of chikungunya disease: Larvae reduction, treatment and prevention. Mathematical Biosciences & Engineering, 2012, 9 (2) : 369-392. doi: 10.3934/mbe.2012.9.369 [12] Holly Gaff, Elsa Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences & Engineering, 2009, 6 (3) : 469-492. doi: 10.3934/mbe.2009.6.469 [13] Kbenesh Blayneh, Yanzhao Cao, Hee-Dae Kwon. Optimal control of vector-borne diseases: Treatment and prevention. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 587-611. doi: 10.3934/dcdsb.2009.11.587 [14] Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial and Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73 [15] Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial and Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275 [16] Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097 [17] Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 [18] Nasser H. Sweilam, Taghreed A. Assiri, Muner M. Abou Hasan. Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1247-1268. doi: 10.3934/dcdss.2021085 [19] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial and Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [20] Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456

2021 Impact Factor: 1.865