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June  2021, 26(6): 2957-2976. doi: 10.3934/dcdsb.2020215

## 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

Received  October 2019 Revised  March 2020 Published  July 2020

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.

Citation: Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215
##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723.  doi: 10.1002/bit.260100602.  Google Scholar [2] S. Anita, V. Capasso and V. Arnautu, An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8098-5.  Google Scholar [3] O. Bara, S. M. Djouadi, J. D. Day and S. Lenhart, Immune therapeutic strategies using optimal controls with L1 and L2 type objectives, Math. Biosci., 290 (2017), 9-21.  doi: 10.1016/j.mbs.2017.05.010.  Google Scholar [4] T. Bayen, O. Cots and P. Gajardo, Analysis of an optimal control problem related to anaerobic digestion process, J. Optimiz. Theory App., 178 (2018), 627-659.  doi: 10.1007/s10957-018-1292-7.  Google Scholar [5] T. Bayen and P. Gajardo, On the steady state optimization of the biogas production in a two-stage anaerobic digestion model, J. Math. Biol., 78 (2019), 1067-1087.  doi: 10.1007/s00285-018-1301-3.  Google Scholar [6] T. Bayen, J. Harmand and M. Sebbah, Time-optimal control of concentration changes in the chemostat with one single species, Appl. Math. Model., 50 (2017), 257-278.  doi: 10.1016/j.apm.2017.05.037.  Google Scholar [7] S. Bozkurt, L. Moreno and I. Neretnieks, Long-term processes in waste deposits, Sci. Total Environ., 250 (2000), 101-121.  doi: 10.1016/S0048-9697(00)00370-3.  Google Scholar [8] S.-J. Feng, B.-Y. Cao and H.-J. Xie, Modeling of leachate recirculation using spraying-vertical well systems in bioreactor landfills, Int. J. Geomech., 17 (2017), 04017012. doi: 10.1061/(ASCE)GM.1943-5622.0000887.  Google Scholar [9] H. V. M. Hamelers, A Mathematical Model for Composting Kinetics, Ph.D thesis, Wageningen University, 2001. Google Scholar [10] J. Harmsen, Identification of organic compounds in leachate from a waste tip, Water Res., 17 (1983), 699-705.  doi: 10.1016/0043-1354(83)90239-7.  Google Scholar [11] R. T. Haug, The Practical Handbook of Compost Engineering, Lewis Publishers, Boca Raton, FL, 1993. doi: 10.1201/9780203736234.  Google Scholar [12] M. M. Haydar and M. V Khire, Leachate recirculation using permeable blankets in engineered landfills, J. Geotech. GeoEnviron., 133 (2007), 360-371.  doi: 10.1061/(ASCE)1090-0241(2007)133:4(360).  Google Scholar [13] A. Husain, Mathematical models of the kinetics of anaerobic digestion - a selected review, Biomass Bioenergy, 14 (1998), 561-571.  doi: 10.1016/S0961-9534(97)10047-2.  Google Scholar [14] U. Ledzewicz, T. Brown and H. Schättler, Comparison of optimal controls for a model in cancer chemotherapy with $L_1$- and $L_2$-type objectives, Optim. Method. Softw., 19 (2004), 339-350.  doi: 10.1080/10556780410001683104.  Google Scholar [15] P. J. Maris, D. W. Harrington and F. E. Mosey, Treatment of landfill leachate; management options, Water Qual. Res. J. Can., 20 (1985), 25-42.  doi: 10.2166/wqrj.1985.026.  Google Scholar [16] G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri, Mathematical modeling of oxygen control in biocell composting plants, Math. Comput. Simulat., 177 (2020), 105-119.  doi: 10.1016/j.matcom.2020.04.011.  Google Scholar [17] G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri, On the role of inhibition processes in modeling control strategies for composting plants, in Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, Volume 21 (editors M. Aguiar, C. Braumann, B. Kooi, A. Pugliese, N. Stollenwerk and E. Venturino), Springer, Cham, (2020), 125–145. Google Scholar [18] J. Monod, The growth of bacterial cultures, Annu. Rev. Microbiol., 3 (1949), 371-394.  doi: 10.1146/annurev.mi.03.100149.002103.  Google Scholar [19] J. Nygren, Output Feedback Control: Some Methods and Applications, Doctoral dissertation, Uppsala Universitet, 2014. Google Scholar [20] J. Pacey, D. Augenstein, R. Morck, D. Reinhart and R. Yazdani, The bioreactor landfill - An innovation in solid waste management, MSW Management (1999). Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.2984&rep=rep1&type=pdf Google Scholar [21] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar [22] C. Polprasert, Organic Waste Recycling, John Wiley and Sons Ltd., 1989. Google Scholar [23] L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.   Google Scholar [24] X. Qian, R. M. Koerner and D. H. Gray, Geotechnical Aspects of Landfill Design and Construction, Pearson College Div., 2001. Google Scholar [25] A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico, Dynamical modeling and optimal control of landfills, Math. Models Methods Appl. Sci., 26 (2016), 901-929.  doi: 10.1142/S0218202516500214.  Google Scholar [26] S. Revollar, P. Vega, R. Vilanova and M. Francisco, Optimal control of wastewater treatment plants using economic-oriented model predictive dynamic strategies, App. Sci., 7 (2017), 813. doi: 10.3390/app7080813.  Google Scholar [27] H. Schaettler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Science & Business Media, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar [28] D. T. Sponza and O. N. Agdag, Impact of leachate recirculation and recirculation volume on stabilization of municipal solid wastes in simulated anaerobic bioreactors, Process Biochem., 39 (2004), 2157-2165.  doi: 10.1016/j.procbio.2003.11.012.  Google Scholar [29] R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace, Optimal control of innate immune response, Optim. Control Appl. Meth., 23 (2002), 91-104.  doi: 10.1002/oca.704.  Google Scholar [30] V. A. Vavilin, S. V. Rytov, L. Y. Lokshina, S. G. Pavlostathis and M. A. Barlaz, Distributed model of solid waste anaerobic digestion: effects of leachate recirculation and pH adjustment, Biotechnol. Bioeng., 81 (2003), 66-73.  doi: 10.1002/bit.10450.  Google Scholar

show all references

##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723.  doi: 10.1002/bit.260100602.  Google Scholar [2] S. Anita, V. Capasso and V. Arnautu, An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8098-5.  Google Scholar [3] O. Bara, S. M. Djouadi, J. D. Day and S. Lenhart, Immune therapeutic strategies using optimal controls with L1 and L2 type objectives, Math. Biosci., 290 (2017), 9-21.  doi: 10.1016/j.mbs.2017.05.010.  Google Scholar [4] T. Bayen, O. Cots and P. Gajardo, Analysis of an optimal control problem related to anaerobic digestion process, J. Optimiz. Theory App., 178 (2018), 627-659.  doi: 10.1007/s10957-018-1292-7.  Google Scholar [5] T. Bayen and P. Gajardo, On the steady state optimization of the biogas production in a two-stage anaerobic digestion model, J. Math. Biol., 78 (2019), 1067-1087.  doi: 10.1007/s00285-018-1301-3.  Google Scholar [6] T. Bayen, J. Harmand and M. Sebbah, Time-optimal control of concentration changes in the chemostat with one single species, Appl. Math. Model., 50 (2017), 257-278.  doi: 10.1016/j.apm.2017.05.037.  Google Scholar [7] S. Bozkurt, L. Moreno and I. Neretnieks, Long-term processes in waste deposits, Sci. Total Environ., 250 (2000), 101-121.  doi: 10.1016/S0048-9697(00)00370-3.  Google Scholar [8] S.-J. Feng, B.-Y. Cao and H.-J. Xie, Modeling of leachate recirculation using spraying-vertical well systems in bioreactor landfills, Int. J. Geomech., 17 (2017), 04017012. doi: 10.1061/(ASCE)GM.1943-5622.0000887.  Google Scholar [9] H. V. M. Hamelers, A Mathematical Model for Composting Kinetics, Ph.D thesis, Wageningen University, 2001. Google Scholar [10] J. Harmsen, Identification of organic compounds in leachate from a waste tip, Water Res., 17 (1983), 699-705.  doi: 10.1016/0043-1354(83)90239-7.  Google Scholar [11] R. T. Haug, The Practical Handbook of Compost Engineering, Lewis Publishers, Boca Raton, FL, 1993. doi: 10.1201/9780203736234.  Google Scholar [12] M. M. Haydar and M. V Khire, Leachate recirculation using permeable blankets in engineered landfills, J. Geotech. GeoEnviron., 133 (2007), 360-371.  doi: 10.1061/(ASCE)1090-0241(2007)133:4(360).  Google Scholar [13] A. Husain, Mathematical models of the kinetics of anaerobic digestion - a selected review, Biomass Bioenergy, 14 (1998), 561-571.  doi: 10.1016/S0961-9534(97)10047-2.  Google Scholar [14] U. Ledzewicz, T. Brown and H. Schättler, Comparison of optimal controls for a model in cancer chemotherapy with $L_1$- and $L_2$-type objectives, Optim. Method. Softw., 19 (2004), 339-350.  doi: 10.1080/10556780410001683104.  Google Scholar [15] P. J. Maris, D. W. Harrington and F. E. Mosey, Treatment of landfill leachate; management options, Water Qual. Res. J. Can., 20 (1985), 25-42.  doi: 10.2166/wqrj.1985.026.  Google Scholar [16] G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri, Mathematical modeling of oxygen control in biocell composting plants, Math. Comput. Simulat., 177 (2020), 105-119.  doi: 10.1016/j.matcom.2020.04.011.  Google Scholar [17] G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri, On the role of inhibition processes in modeling control strategies for composting plants, in Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, Volume 21 (editors M. Aguiar, C. Braumann, B. Kooi, A. Pugliese, N. Stollenwerk and E. Venturino), Springer, Cham, (2020), 125–145. Google Scholar [18] J. Monod, The growth of bacterial cultures, Annu. Rev. Microbiol., 3 (1949), 371-394.  doi: 10.1146/annurev.mi.03.100149.002103.  Google Scholar [19] J. Nygren, Output Feedback Control: Some Methods and Applications, Doctoral dissertation, Uppsala Universitet, 2014. Google Scholar [20] J. Pacey, D. Augenstein, R. Morck, D. Reinhart and R. Yazdani, The bioreactor landfill - An innovation in solid waste management, MSW Management (1999). Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.2984&rep=rep1&type=pdf Google Scholar [21] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar [22] C. Polprasert, Organic Waste Recycling, John Wiley and Sons Ltd., 1989. Google Scholar [23] L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.   Google Scholar [24] X. Qian, R. M. Koerner and D. H. Gray, Geotechnical Aspects of Landfill Design and Construction, Pearson College Div., 2001. Google Scholar [25] A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico, Dynamical modeling and optimal control of landfills, Math. Models Methods Appl. Sci., 26 (2016), 901-929.  doi: 10.1142/S0218202516500214.  Google Scholar [26] S. Revollar, P. Vega, R. Vilanova and M. Francisco, Optimal control of wastewater treatment plants using economic-oriented model predictive dynamic strategies, App. Sci., 7 (2017), 813. doi: 10.3390/app7080813.  Google Scholar [27] H. Schaettler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Science & Business Media, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar [28] D. T. Sponza and O. N. Agdag, Impact of leachate recirculation and recirculation volume on stabilization of municipal solid wastes in simulated anaerobic bioreactors, Process Biochem., 39 (2004), 2157-2165.  doi: 10.1016/j.procbio.2003.11.012.  Google Scholar [29] R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace, Optimal control of innate immune response, Optim. Control Appl. Meth., 23 (2002), 91-104.  doi: 10.1002/oca.704.  Google Scholar [30] V. A. Vavilin, S. V. Rytov, L. Y. Lokshina, S. G. Pavlostathis and M. A. Barlaz, Distributed model of solid waste anaerobic digestion: effects of leachate recirculation and pH adjustment, Biotechnol. Bioeng., 81 (2003), 66-73.  doi: 10.1002/bit.10450.  Google Scholar
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
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 $b^{*}\simeq0.705$
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 $c = 0.417$, $c_h = 0.245$, $\theta = 0.3$
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)
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
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
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
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 $\alpha = 0.01$
Optimal controls for objective functional (21), $\alpha = 0.001, 0.01, 0.1, 1$ and given initial configuration $(s_1^0, s_2^0) = (0.1, 0.5)$
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)
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)
State variables and optimal control for objective functional (24), for different values of $s_2^0$ ($\alpha = 0.01$, $s_1^0 = 0.1$)
State variables and optimal control for objective functional (24), for different values of $s_1^0$ ($\alpha = 0.01$, $s_2^0 = 0.5$)
Optimal controls for objective functional (24) and varying $\alpha = 0.001, 0.01, 0.1, 1$, when $(s_1^0, s_2^0) = (0.1, 0.5)$
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
 $s_2^{0}$ $t_s$ $s_1(t_f)$ $s_2(t_f)$ 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
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
 $s_1^{0}$ $t_s$ $s_1(t_f)$ $s_2(t_f)$ 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
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 $(s_1^0, s_2^0) = (0.1, 0.5)$
 $\alpha$ $t_s$ $I= {\int_0^{t_f}}u(t)dt$ $s_1(t_f)$ $s_2(t_f)$ $0.001$ $2.91$ $7.0900$ $0.3943$ $0.0989$ $0.01$ $5.25$ $4.7500$ $0.4063$ $0.1378$ $0.1$ $9.26$ $0.7479$ $0.7037$ $0.1222$ $1$ $-$ $0$ $0.8420$ $0.0001$
 $\alpha$ $t_s$ $I= {\int_0^{t_f}}u(t)dt$ $s_1(t_f)$ $s_2(t_f)$ $0.001$ $2.91$ $7.0900$ $0.3943$ $0.0989$ $0.01$ $5.25$ $4.7500$ $0.4063$ $0.1378$ $0.1$ $9.26$ $0.7479$ $0.7037$ $0.1222$ $1$ $-$ $0$ $0.8420$ $0.0001$
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
 $s_2^{0}$ $\tilde{t}$ $s_1(t_f)$ $s_2(t_f)$ 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
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
 $s_1^{0}$ $\tilde{t}$ $s_1(t_f)$ $s_2(t_f)$ 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
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 $(s_1^0, s_2^0) = (0.1, 0.5)$
 $\alpha$ $\tilde{t}$ $I= {\int_0^{t_f}}u(t)dt$ $s_1(t_f)$ $s_2(t_f)$ $0.001$ $3.00$ $8.1174$ $0.3942$ $0.0961$ $0.01$ $5.88$ $5.8249$ $0.4018$ $0.1157$ $0.1$ $-$ $2.6023$ $0.5600$ $0.1186$ $1$ $-$ $0.5777$ $0.7659$ $0.0383$
 $\alpha$ $\tilde{t}$ $I= {\int_0^{t_f}}u(t)dt$ $s_1(t_f)$ $s_2(t_f)$ $0.001$ $3.00$ $8.1174$ $0.3942$ $0.0961$ $0.01$ $5.88$ $5.8249$ $0.4018$ $0.1157$ $0.1$ $-$ $2.6023$ $0.5600$ $0.1186$ $1$ $-$ $0.5777$ $0.7659$ $0.0383$
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