# American Institute of Mathematical Sciences

March  2022, 18(2): 1439-1451. doi: 10.3934/jimo.2021027

## Optimizing micro-algae production in a raceway pond with variable depth

 School of Electrical Engineering, Computing, and Mathematical Sciences, Curtin University, Kent Street, Bentley, Perth, Western Australia, 6102

* Corresponding author: v.rehbock@curtin.edu.au

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

Fund Project: We acknowledge the support of Todd Hurst through the Australian Government Research Training Program (RTP) Scholarship

We present a modified model of algae growth in a raceway pond with the additional feature of variable pond depth. This requires an additional state variable to model depth as well as additional control to allow for variable outflow. We apply numerical optimal control methods to this model and show that the lipid yield of the process can be increased by 67% compared to that obtained with a fixed pond depth.

Citation: Todd Hurst, Volker Rehbock. Optimizing micro-algae production in a raceway pond with variable depth. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1439-1451. doi: 10.3934/jimo.2021027
##### References:
 [1] Q. Béchet, A. Shilton and B. Guieysse, Maximizing productivity and reducing environmental impacts of full-scale algal production through optimization of open pond depth and hydraulic retention time, Environmental Science & Technology, 50 (2016), 4102-4110. [2] M. A. Borowitzka, Energy from microalgae: A short history, in Algae for Biofuels and Energy (ed. M. A. Borowitzka), 5, Springer, 2013, 1–15. doi: 10.1007/978-94-007-5479-9_1. [3] M. A. Borowitzka and N. R. Moheimani, Algae for Biofuels and Energy, 5, Springer, 2013. [4] M. A. Borowitzka and N. R. Moheimani, Open pond culture systems, in Algae for Biofuels and Energy (ed. M. A. Borowitzka), 5, Springer, 2013,133–152. doi: 10.1007/978-94-007-5479-9_8. [5] K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 14, Siam, 1996. [6] C. Büskens, Optimierungsmethoden und sensitivitatsanalyse fur optimale steuerprozesse mit steuer-und zustands-beschrankungen, Westfalische Wilhelms-Universitat Munster. [7] P. H. Chen and W. J. Oswald, Thermochemical treatment for algal fermentation, Environment International, 24 (1998), 889-897.  doi: 10.1016/S0160-4120(98)00080-4. [8] R. J. Craggs, T. J. Lundquist and J. R. Benemann, Wastewater Treatment and Algal Biofuel Production, 5, Springer, 2013,153–163. [9] M. D. R. De Pinho, I. Kornienko and H. Maurer, Optimal control of a SEIR model with mixed constraints and L1 cost, in CONTROLO'2014–Proceedings of the 11th Portuguese Conference on Automatic Control, Springer, 135–145. doi: 10.1007/978-3-319-10380-8_14. [10] R. Fourer, D. Gay and B. Kernighan, Ampl: A modeling language for mathematical programming, Duxbury Press. [11] T. Hurst and V. Rehbock, Optimal control for micro-algae on a raceway model, Biotechnology progress, 34 (2018), 107-119. [12] S. C. James and V. Boriah, Modeling algae growth in an open-channel raceway, Journal of Computational Biology, 17 (2010), 895-906.  doi: 10.1089/cmb.2009.0078. [13] L. S. Jennings, M. Fisher, K. L. Teo and C. Goh, MISER 3: Optimal Control Software, Version 2.0. Theory and User Manual, Dept. of Mathematics, University of Western Australia, Nedlands, 2002. [14] L. S. Jennings, K. L. Teo, V. Rehbock and W. X. Zheng, Optimal control of singular systems with a cost on changing control, Dynamics and Control, 6 (1996), 63-89.  doi: 10.1007/BF02169462. [15] B.-H. Kim, J.-E. Choi, K. Cho, Z. Kang, R. Ramanan, D.-G. Moon and H.-S. Kim, Influence of water depth on microalgal production, biomass harvest, and energy consumption in high rate algal pond using municipal wastewater, J. Microbiol. Biotechnol., 28 (2018), 630-637.  doi: 10.4014/jmb.1801.01014. [16] F. Mairet, O. Bernard, T. Lacour and A. Sciandra, Modelling microalgae growth in nitrogen limited photobiorector for estimating biomass, carbohydrate and neutral lipid productivities, IFAC Proceedings Volumes, 44 (2011), 10591-10596.  doi: 10.3182/20110828-6-IT-1002.03165. [17] H. Maurer, J.-H. R. Kim and G. Vossen, On a state-constrained control problem in optimal production and maintenance, Optimal Control and Dynamic Games, Springer, 2005,289–308. [18] A. Meurer, et al., Sympy: Symbolic computing in python, PeerJ Computer Science, 3 (2017), e103. doi: 10.7717/peerj-cs.103. [19] R. Muñoz-Tamayo, F. Mairet and O. Bernard, Optimizing microalgal production in raceway systems, Biotechnology Progress, 29 (2013), 543-552. [20] A. K. Pegallapati and N. Nirmalakhandan, Modeling algal growth in bubble columns under sparging with $\text{CO}_2$-enriched air, Bioresource Technology, 124 (2012), 137-145. [21] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The mathematical theory of optimal processes, Wiley Interscience, New York. [22] R. Pytlak and T. Zawadzki, On solving optimal control problems with higher index differential-algebraic equations, Optimization Methods and Software, 29 (2014), 1139-1162.  doi: 10.1080/10556788.2014.892597. [23] J. Quinn, L. De Winter and T. Bradley, Microalgae bulk growth model with application to industrial scale systems, Bioresource Technology, 102 (2011), 5083-5092.  doi: 10.1016/j.biortech.2011.01.019. [24] S. Sawant, H. Khadamkar, C. Mathpati, R. Pandit and A. Lali, Computational and experimental studies of high depth algal raceway pond photo-bioreactor, Renewable Energy, 118 (2018), 152-159.  doi: 10.1016/j.renene.2017.11.015. [25] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y. [26] P. J. L. B. Williams and L. M. Laurens, Microalgae as biodiesel & biomass feedstocks: Review & analysis of the biochemistry, energetics & economics, Energy & Environmental Science, 3 (2010), 554-590.  doi: 10.1039/b924978h. [27] G. C. Zittelli, L. Rodolfi, N. Bassi, N. Biondi and M. R. Tredici, Photobioreactors for microalgal biofuel production, in Algae for biofuels and energy (ed. M. A. Borowitzka), 5, Springer, 2013, 115-131. doi: 10.1007/978-94-007-5479-9_7.

show all references

##### References:
 [1] Q. Béchet, A. Shilton and B. Guieysse, Maximizing productivity and reducing environmental impacts of full-scale algal production through optimization of open pond depth and hydraulic retention time, Environmental Science & Technology, 50 (2016), 4102-4110. [2] M. A. Borowitzka, Energy from microalgae: A short history, in Algae for Biofuels and Energy (ed. M. A. Borowitzka), 5, Springer, 2013, 1–15. doi: 10.1007/978-94-007-5479-9_1. [3] M. A. Borowitzka and N. R. Moheimani, Algae for Biofuels and Energy, 5, Springer, 2013. [4] M. A. Borowitzka and N. R. Moheimani, Open pond culture systems, in Algae for Biofuels and Energy (ed. M. A. Borowitzka), 5, Springer, 2013,133–152. doi: 10.1007/978-94-007-5479-9_8. [5] K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 14, Siam, 1996. [6] C. Büskens, Optimierungsmethoden und sensitivitatsanalyse fur optimale steuerprozesse mit steuer-und zustands-beschrankungen, Westfalische Wilhelms-Universitat Munster. [7] P. H. Chen and W. J. Oswald, Thermochemical treatment for algal fermentation, Environment International, 24 (1998), 889-897.  doi: 10.1016/S0160-4120(98)00080-4. [8] R. J. Craggs, T. J. Lundquist and J. R. Benemann, Wastewater Treatment and Algal Biofuel Production, 5, Springer, 2013,153–163. [9] M. D. R. De Pinho, I. Kornienko and H. Maurer, Optimal control of a SEIR model with mixed constraints and L1 cost, in CONTROLO'2014–Proceedings of the 11th Portuguese Conference on Automatic Control, Springer, 135–145. doi: 10.1007/978-3-319-10380-8_14. [10] R. Fourer, D. Gay and B. Kernighan, Ampl: A modeling language for mathematical programming, Duxbury Press. [11] T. Hurst and V. Rehbock, Optimal control for micro-algae on a raceway model, Biotechnology progress, 34 (2018), 107-119. [12] S. C. James and V. Boriah, Modeling algae growth in an open-channel raceway, Journal of Computational Biology, 17 (2010), 895-906.  doi: 10.1089/cmb.2009.0078. [13] L. S. Jennings, M. Fisher, K. L. Teo and C. Goh, MISER 3: Optimal Control Software, Version 2.0. Theory and User Manual, Dept. of Mathematics, University of Western Australia, Nedlands, 2002. [14] L. S. Jennings, K. L. Teo, V. Rehbock and W. X. Zheng, Optimal control of singular systems with a cost on changing control, Dynamics and Control, 6 (1996), 63-89.  doi: 10.1007/BF02169462. [15] B.-H. Kim, J.-E. Choi, K. Cho, Z. Kang, R. Ramanan, D.-G. Moon and H.-S. Kim, Influence of water depth on microalgal production, biomass harvest, and energy consumption in high rate algal pond using municipal wastewater, J. Microbiol. Biotechnol., 28 (2018), 630-637.  doi: 10.4014/jmb.1801.01014. [16] F. Mairet, O. Bernard, T. Lacour and A. Sciandra, Modelling microalgae growth in nitrogen limited photobiorector for estimating biomass, carbohydrate and neutral lipid productivities, IFAC Proceedings Volumes, 44 (2011), 10591-10596.  doi: 10.3182/20110828-6-IT-1002.03165. [17] H. Maurer, J.-H. R. Kim and G. Vossen, On a state-constrained control problem in optimal production and maintenance, Optimal Control and Dynamic Games, Springer, 2005,289–308. [18] A. Meurer, et al., Sympy: Symbolic computing in python, PeerJ Computer Science, 3 (2017), e103. doi: 10.7717/peerj-cs.103. [19] R. Muñoz-Tamayo, F. Mairet and O. Bernard, Optimizing microalgal production in raceway systems, Biotechnology Progress, 29 (2013), 543-552. [20] A. K. Pegallapati and N. Nirmalakhandan, Modeling algal growth in bubble columns under sparging with $\text{CO}_2$-enriched air, Bioresource Technology, 124 (2012), 137-145. [21] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The mathematical theory of optimal processes, Wiley Interscience, New York. [22] R. Pytlak and T. Zawadzki, On solving optimal control problems with higher index differential-algebraic equations, Optimization Methods and Software, 29 (2014), 1139-1162.  doi: 10.1080/10556788.2014.892597. [23] J. Quinn, L. De Winter and T. Bradley, Microalgae bulk growth model with application to industrial scale systems, Bioresource Technology, 102 (2011), 5083-5092.  doi: 10.1016/j.biortech.2011.01.019. [24] S. Sawant, H. Khadamkar, C. Mathpati, R. Pandit and A. Lali, Computational and experimental studies of high depth algal raceway pond photo-bioreactor, Renewable Energy, 118 (2018), 152-159.  doi: 10.1016/j.renene.2017.11.015. [25] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y. [26] P. J. L. B. Williams and L. M. Laurens, Microalgae as biodiesel & biomass feedstocks: Review & analysis of the biochemistry, energetics & economics, Energy & Environmental Science, 3 (2010), 554-590.  doi: 10.1039/b924978h. [27] G. C. Zittelli, L. Rodolfi, N. Bassi, N. Biondi and M. R. Tredici, Photobioreactors for microalgal biofuel production, in Algae for biofuels and energy (ed. M. A. Borowitzka), 5, Springer, 2013, 115-131. doi: 10.1007/978-94-007-5479-9_7.
$f_{in}$
$f_{out}$
$L$
$\bar{I}$
$s$
$x_l$
$\eta_{L}$
State Variable, Control Variable, System Parameter and Function Definitions
 Variables Definition Units $s$ Nitrogen concentration gN $\text{m}^{-3}$ $q_n$ Nitrogen quota gN $\text{(gC)}^{-1}$ $x$ Carbon biomass concentration gC $\text{m}^{-3}$ $x_l$ Lipid carbon concentration gC $\text{m}^{-3}$ $x_f$ Functional carbon concentration gC $\text{m}^{-3}$ $L$ Pond depth $\text{m}$ $\bar{I}$ Average light intensity μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ $C_{hl}$ Chlorophyll concentration g$C_{hl}$ $\text{m}^{-3}$ $I_0$ Incident light intensity μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ $T$ Raceway temperature ℃ $\phi_{T}$ Temperature factor affecting growth kinetics $\lambda$ Optical depth $\mu$ Growth rate $\text{h}^{-1}$ $\rho$ Nitrogen uptake rate gN $\text{(gC h)}^{-1}$ $\theta_N$ Chl:N ratio g$C_{hl}$ $\text{(gN)}^{-1}$ $\xi$ Attenuation factor $\text{m}^{-1}$ $R$ Overall respiration rate $\text{h}^{-1}$ $f_{in}$ Feeding flow rate $\text{m}^{3}$ $\text{h}^{-1}$ $f_{out}$ Extraction flow rate $\text{m}^{3}$ $\text{h}^{-1}$ $\eta_{L}$ Efficiency of light absorption $t_f$ final time point h
 Variables Definition Units $s$ Nitrogen concentration gN $\text{m}^{-3}$ $q_n$ Nitrogen quota gN $\text{(gC)}^{-1}$ $x$ Carbon biomass concentration gC $\text{m}^{-3}$ $x_l$ Lipid carbon concentration gC $\text{m}^{-3}$ $x_f$ Functional carbon concentration gC $\text{m}^{-3}$ $L$ Pond depth $\text{m}$ $\bar{I}$ Average light intensity μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ $C_{hl}$ Chlorophyll concentration g$C_{hl}$ $\text{m}^{-3}$ $I_0$ Incident light intensity μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ $T$ Raceway temperature ℃ $\phi_{T}$ Temperature factor affecting growth kinetics $\lambda$ Optical depth $\mu$ Growth rate $\text{h}^{-1}$ $\rho$ Nitrogen uptake rate gN $\text{(gC h)}^{-1}$ $\theta_N$ Chl:N ratio g$C_{hl}$ $\text{(gN)}^{-1}$ $\xi$ Attenuation factor $\text{m}^{-1}$ $R$ Overall respiration rate $\text{h}^{-1}$ $f_{in}$ Feeding flow rate $\text{m}^{3}$ $\text{h}^{-1}$ $f_{out}$ Extraction flow rate $\text{m}^{3}$ $\text{h}^{-1}$ $\eta_{L}$ Efficiency of light absorption $t_f$ final time point h
Parameter Definition and Values
 Parameters Definition Units Value $\alpha$ Protein synthesis coefficient gC$\text{(gN)}^{-1}$ 3.0 $\beta$ Fatty acid synthesis coefficient gC$\text{(gN)}^{-1}$ 3.80 $\epsilon_I$ Dissociation light constant μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 50 $\varphi$ Biosynthesis cost coefficient gC$\text{(gN)}^{-1}$ 1.30 $\gamma$ Fatty acid mobilization coefficient gC$\text{(gN)}^{-1}$ 2.90 $v$ Reduction factor of nitrogen uptake during night 0.19 $\tilde{\mu}$ Theoretical maximum specific growth rate $\text{h}^{-1}$ $8.7916\times10^{-2}$ $\bar{\rho}$ Maximum uptake rate gC$\text{(gN h)}^{-1}$ $4.16\times10^{-3}$ $a$ Light attenuation due to chlorophyll $\text{m}^{2}$ $(\text{g } C_{hl})^{-1}$ 2.0 $b$ Light attenuation due to background turbidity $\text{m}^{-1}$ 0.087 $g_1$ Coefficient (7) gN $(\text{g} C_{hl})^{-1}$ 16.74 $g_2$ Coefficient (7) gN $(\text{g} C_{hl} \text{ }^\circ \text{C})^{-1}$ 0.39 $g_3$ Coefficient (7) gN (g$C_{hl}$μmol photons $\text{m}^{-2}$ $\text{s}^{-1})^{-1}$ $1.4\times10^{-3}$ $g_4$ Coefficient (7) $\text{ }^\circ \text{C}^{-1}$ 0.0015 $K_s$ Nitrogen saturation constant gN $\text{m}^{-3}$ 0.018 $K_{sI}$ Light saturation constant μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ $1.5\times 10^{2}$ $m$ Hill coefficient 3.0 $Q_l$ Saturation cell quota gN$\text{(gC)}^{-1}$ 0.20 $Q_0$ Minimum nitrogen cell quota gN$\text{(gC)}^{-1}$ 0.05 $r_0$ Maintenance respiration rate $\text{h}^{-1}$ $4.16\times10^{-4}$ $T_{\min}$ Lower temperature for micro-algae growth $\text{ }^\circ \text{C}$ -0.20 $T_{\max}$ Upper temperature for micro-algae growth $\text{ }^\circ \text{C}$ 33.30 $T_{\text{opt}}$ Temperature at which growth rate is maximal $\text{ }^\circ \text{C}$ 26.70 $s_{in}$ Influent nitrogen concentration gN $\text{m}^{-3}$ 50.0 $T_a$ Coefficient (2) $\text{ }^\circ \text{C}$ -5.75 $T_b$ Coefficient (2) $\text{ }^\circ \text{C}$ 20.75 $t_a$ time point h 3.733 $t_b$ time point h 4.9 $t_c$ time point h 19.1 $t_d$ time point h 20.267 $I_{0a}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -3.890408560 $I_{0b}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -52.13043162 $I_{0c}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 141.6278134 $I_{0d}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 841.1792340 $I_{0e}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 358.8207660 $I_{0f}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 3.890411835 $I_{0g}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -52.13042142 $I_{0h}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -141.6278049
 Parameters Definition Units Value $\alpha$ Protein synthesis coefficient gC$\text{(gN)}^{-1}$ 3.0 $\beta$ Fatty acid synthesis coefficient gC$\text{(gN)}^{-1}$ 3.80 $\epsilon_I$ Dissociation light constant μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 50 $\varphi$ Biosynthesis cost coefficient gC$\text{(gN)}^{-1}$ 1.30 $\gamma$ Fatty acid mobilization coefficient gC$\text{(gN)}^{-1}$ 2.90 $v$ Reduction factor of nitrogen uptake during night 0.19 $\tilde{\mu}$ Theoretical maximum specific growth rate $\text{h}^{-1}$ $8.7916\times10^{-2}$ $\bar{\rho}$ Maximum uptake rate gC$\text{(gN h)}^{-1}$ $4.16\times10^{-3}$ $a$ Light attenuation due to chlorophyll $\text{m}^{2}$ $(\text{g } C_{hl})^{-1}$ 2.0 $b$ Light attenuation due to background turbidity $\text{m}^{-1}$ 0.087 $g_1$ Coefficient (7) gN $(\text{g} C_{hl})^{-1}$ 16.74 $g_2$ Coefficient (7) gN $(\text{g} C_{hl} \text{ }^\circ \text{C})^{-1}$ 0.39 $g_3$ Coefficient (7) gN (g$C_{hl}$μmol photons $\text{m}^{-2}$ $\text{s}^{-1})^{-1}$ $1.4\times10^{-3}$ $g_4$ Coefficient (7) $\text{ }^\circ \text{C}^{-1}$ 0.0015 $K_s$ Nitrogen saturation constant gN $\text{m}^{-3}$ 0.018 $K_{sI}$ Light saturation constant μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ $1.5\times 10^{2}$ $m$ Hill coefficient 3.0 $Q_l$ Saturation cell quota gN$\text{(gC)}^{-1}$ 0.20 $Q_0$ Minimum nitrogen cell quota gN$\text{(gC)}^{-1}$ 0.05 $r_0$ Maintenance respiration rate $\text{h}^{-1}$ $4.16\times10^{-4}$ $T_{\min}$ Lower temperature for micro-algae growth $\text{ }^\circ \text{C}$ -0.20 $T_{\max}$ Upper temperature for micro-algae growth $\text{ }^\circ \text{C}$ 33.30 $T_{\text{opt}}$ Temperature at which growth rate is maximal $\text{ }^\circ \text{C}$ 26.70 $s_{in}$ Influent nitrogen concentration gN $\text{m}^{-3}$ 50.0 $T_a$ Coefficient (2) $\text{ }^\circ \text{C}$ -5.75 $T_b$ Coefficient (2) $\text{ }^\circ \text{C}$ 20.75 $t_a$ time point h 3.733 $t_b$ time point h 4.9 $t_c$ time point h 19.1 $t_d$ time point h 20.267 $I_{0a}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -3.890408560 $I_{0b}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -52.13043162 $I_{0c}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 141.6278134 $I_{0d}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 841.1792340 $I_{0e}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 358.8207660 $I_{0f}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ 3.890411835 $I_{0g}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -52.13042142 $I_{0h}$ Coefficient (3) μmol photons $\text{m}^{-2}$ $\text{s}^{-1}$ -141.6278049
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