# 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 & Continuous Dynamical Systems - S, 2022, 15 (3) : 587-601. doi: 10.3934/dcdss.2021153
##### References:

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##### References:
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
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