Optimal control of leachate recirculation for anaerobic processes in landfills

Marzia Bisi; Maria Groppi; Giorgio Martalò; Romina Travaglini

doi:10.3934/dcdsb.2020215

Article Contents

2021, Volume 26, Issue 6: 2957-2976. Doi: 10.3934/dcdsb.2020215

This issue Previous Article Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials Next Article Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations

Optimal control of leachate recirculation for anaerobic processes in landfills

Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze 53/A, 43124, Parma, Italy

^* Corresponding author: Giorgio Martalò
^* Corresponding author: Giorgio Martalò

Received: September 30, 2019

Revised: February 29, 2020

Early access: July 2020

Published: June 2021

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Abstract

A mathematical model for the degradation of the organic fraction of solid waste in landfills, by means of an anaerobic bacterial population, is proposed. Additional phenomena, like hydrolysis of insoluble substrate and biomass decay, are taken into account. The evolution of the system is monitored by controlling the effects of leachate recirculation on the hydrolytic process. We investigate the optimal strategies to minimize substrate concentration and recirculation operation costs. Analytical and numerical results are presented and discussed for linear and quadratic cost functionals.

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Mathematics Subject Classification: Primary: 37N35, 37N40, 49J15.

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Figure 1. Phase portrait in absence of solubilization of the insoluble substrate ($ u\left(t\right) = 0 $ for any $ t>0 $). Bacterial growth is described by a Monod response function (10); parameters $ c = 0.417 $ and $ b = 0.19 $ are purely illustrative

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Figure 2. Second component of equilibria versus the bifurcation parameter $ b $ in the case of Monod response function (10). Continuous and dashed lines denote stability and instability of equilibria, respectively. The bifurcation value is $ b^{*}\simeq0.705 $

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Figure 3. Phase portrait for system (7), when the bifurcation parameter $ b $ is less (panel (a)) or greater (panel (b)) than the bifurcation value $ b^*\simeq 0.705 $. Bacterial growth is modeled by Monod growth function (10); parameters $ c = 0.417 $, $ c_h = 0.245 $, $ \theta = 0.3 $

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Figure 4. State variables profiles and optimal control for objective functional (21), Monod response function (10) and configuration (22). Parameters are given in (20). The optimal control is of bang-bang type with a unique switch from 0 to 1 for $ t = t_s\simeq 5.25 $ (dashed line)

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Figure 5. Optimal control $ u $ and scaled ($ \times 20 $) switching function $ \phi_1 $ for objective functional (21), when parameters and initial configuration are given in (20) and (22), respectively

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Figure 6. State variables and optimal control for objective functional (21); initial soluble concentrations $ s_2^0 $ varies from $ 0.1 $ to $ 0.7 $, while parameter $ \alpha = 0.01 $ and initial insoluble substrate $ s_1^0 = 0.1 $ remain fixed

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Figure 7. State variables and optimal control for objective functional (21); initial insoluble concentration $ s_1^0 $ varies from $ 0.1 $ to $ 0.4 $, while parameter $ \alpha = 0.01 $ and initial soluble substrate $ s_2^0 = 0.5 $ remain fixed

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Figure 8. State variables and optimal control for objective functional (21), when the initial configuration is given by $ (s_1^0, s_2^0) = (0.8, 0.1) $ and $ \alpha = 0.01 $

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Figure 9. Optimal controls for objective functional (21), $ \alpha = 0.001, 0.01, 0.1, 1 $ and given initial configuration $ (s_1^0, s_2^0) = (0.1, 0.5) $

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Figure 10. State variables profiles and optimal control objective functional (24) with Monod response function (10). Initial configuration and parameter $ \alpha $ are given in (25); other parameters are given in (20)

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Figure 11. Optimal control $ u $ and scaled ($ \times 20 $) switching function $ \phi_2 $ for objective functional (24), when parameter $ \alpha $ and initial configuration are given in (25); other parameters are given in (20)

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Figure 12. State variables and optimal control for objective functional (24), for different values of $ s_2^0 $ ($ \alpha = 0.01 $, $ s_1^0 = 0.1 $)

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Figure 13. State variables and optimal control for objective functional (24), for different values of $ s_1^0 $ ($ \alpha = 0.01 $, $ s_2^0 = 0.5 $)

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Figure 14. Optimal controls for objective functional (24) and varying $ \alpha = 0.001, 0.01, 0.1, 1 $, when $ (s_1^0, s_2^0) = (0.1, 0.5) $

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Table 1. Switching times and final substrate concentrations for objective functional (21), when soluble substrate $ s_2^0 $ varies ($ \alpha = \; 0.01 $, $ s_1^0 = 0.1) $

$ s_2^{0} $	$ t_s $	$ s_1(t_f) $	$ s_2(t_f) $
0.1	5.03	0.4031	0.1454
0.3	5.10	0.4043	0.1428
0.5	5.25	0.4063	0.1378
0.7	5.58	0.4084	0.1262

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Table 2. Switching times and final substrate concentrations for objective functional (21), when insoluble substrate $ s_1^0 $ varies ($ \alpha = 0.01 $, $ s_2^0 = 0.5 $)

$ s_1^{0} $	$ t_s $	$ s_1(t_f) $	$ s_2(t_f) $
0.1	5.25	0.4063	0.1378
0.2	5.16	0.4046	0.1406
0.3	5.10	0.4021	0.1422
0.4	5.09	0.3933	0.1414

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Table 3. Switching times, global effort required to control the system and final substrate concentrations for objective functional, when $ \alpha = 0.001, \, 0.01, \, 0.1, \, 1 $ and $ (s_1^0, s_2^0) = (0.1, 0.5) $

$ \alpha $	$ t_s $	$ I= {\int_0^{t_f}}u(t)dt $	$ s_1(t_f) $	$ s_2(t_f) $
$ 0.001 $	$ 2.91 $	$ 7.0900 $	$ 0.3943 $	$ 0.0989 $
$ 0.01 $	$ 5.25 $	$ 4.7500 $	$ 0.4063 $	$ 0.1378 $
$ 0.1 $	$ 9.26 $	$ 0.7479 $	$ 0.7037 $	$ 0.1222 $
$ 1 $	$ - $	$ 0 $	$ 0.8420 $	$ 0.0001 $

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Table 4. First time $ \tilde{t} $ at which the control assumes constantly its maximal value and final substrate concentrations for objective functional (24) and different values of $ s_2^0 $ ($ \alpha = 0.01 $, $ s_1^0 = 0.1) $

$ s_2^{0} $	$ \tilde{t} $	$ s_1(t_f) $	$ s_2(t_f) $
0.1	5.72	0.4008	0.1210
0.3	5.78	0.4012	0.1192
0.5	5.88	0.4018	0.1157
0.7	6.16	0.4010	0.1075

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Table 5. First time $ \tilde{t} $ at which the control assumes constantly its maximal value and final substrate concentrations for objective functional (24) and different values of $ s_1^0 $ ($ \alpha = 0.01 $, $ s_2^0 = 0.5) $

$ s_1^{0} $	$ \tilde{t} $	$ s_1(t_f) $	$ s_2(t_f) $
0.1	5.88	0.4018	0.1157
0.2	5.84	0.4009	0.1171
0.3	5.86	0.3990	0.1169
0.4	6.04	0.3902	0.1135

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Table 6. First time $ \tilde{t} $ at which the control assumes constantly its maximal value, global effort $ I $ required to control the process, final substrate concentrations for objective functional (24), when $ \alpha = 0.001, 0.01, 0.1, 1 $ and $ (s_1^0, s_2^0) = (0.1, 0.5) $

$ \alpha $	$ \tilde{t} $	$ I= {\int_0^{t_f}}u(t)dt $	$ s_1(t_f) $	$ s_2(t_f) $
$ 0.001 $	$ 3.00 $	$ 8.1174 $	$ 0.3942 $	$ 0.0961 $
$ 0.01 $	$ 5.88 $	$ 5.8249 $	$ 0.4018 $	$ 0.1157 $
$ 0.1 $	$ - $	$ 2.6023 $	$ 0.5600 $	$ 0.1186 $
$ 1 $	$ - $	$ 0.5777 $	$ 0.7659 $	$ 0.0383 $

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