Mathematical modeling and optimal control of carbon dioxide emissions from energy sector

Abstract

Energy demand is rising day by day and will continue to increase to meet the demand of the growing population. A major portion of global energy production comes from fossil fuel burning, resulting in the increase in the atmospheric burden of global warming gas carbon dioxide (\(CO _{2}\)). Cutting down \(CO _{2}\) emission from the energy sector is crucial to meet the climate change mitigation target. This paper is focused on fulfilling two objectives: The first objective is to present a mathematical model that captures the dynamical relationship between the human population, energy use, and atmospheric carbon dioxide, and the second aim is to derive a mathematical framework to effectively utilize the available mitigation options to curtail \(CO _{2}\) emission from energy use by proposing an optimal control problem. The mitigation options that reduce the \(CO _{2}\) emission rate from energy production, as well as the options that reduce the energy consumption rate, are considered in the modeling process. The proposed mathematical model is analyzed qualitatively to comprehend the system’s long-term behavior. The model parameters are fitted to real data of global energy use, population, and \(CO _{2}\) concentration. It is shown that the equilibrium level of \(CO _{2}\) reduces with the increase in the efficiencies of mitigation options to reduce the \(CO _{2}\) emission rate per unit energy use and energy consumption rate. The optimality system is derived analytically by taking the efficiencies of the mitigation options to reduce the \(CO _{2}\) emission rate and energy consumption rate as control variables. Numerical simulations are conducted to validate the theoretical findings and identify the optimal profiles of control variables under different settings of \(CO _{2}\) emission rate, energy consumption rate, and maximum efficiencies of available mitigation options to cut down \(CO _{2}\) emission rate and energy consumption rate. It is found that the development and implementation of more efficient mitigation options and switching to low carbon energy sources bring reduction in the mitigation cost.

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References

  1. AlFarra, H. J., & Abu-Hijleh, B. (2012). The potential role of nuclear energy in mitigating \(\text{ CO}_{2}\) emissions in the United Arab Emirates. Energy, 42, 272–285.

    Google Scholar 

  2. Allen, R. C. (2009). The British Industrial Revolution in Global Perspective. Cambridge: Cambridge University Press.

    Google Scholar 

  3. Anser, M. K. (2019). Impact of energy consumption and human activities on carbon emissions in Pakistan: application of STIRPAT model. Environmental Science and Pollution Research, 26(1), 13453–13463.

    CAS  Article  Google Scholar 

  4. Bortz, D. M., & Nelson, P. W. (2004). Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics. Bulletin of Mathematical Biology, 66(5), 1009–1026.

    CAS  Article  Google Scholar 

  5. BP (2019). BP Statistical Review of World Energy, 2019, 68th Edition. London: BP. https://www.bp.com/content/dam/bp/business-sites/en/global/corporate/pdfs/energy-economics/statistical-review/bp-stats-review-2019-full-report.pdf. Accessed 14 March 2020.

  6. Caetano, M. A. L., Gherardi, D. F. M., & Yoneyama, T. (2008). Optimal resource management control for \(\text{ CO}_2\) emission and reduction of the greenhouse effect. Ecological Modelling, 213(1), 119–126.

    CAS  Article  Google Scholar 

  7. Caetano, M. A. L., Gherardi, D. F. M., & Yoneyama, T. (2011). An optimized policy for the reduction of \(\text{ CO}_2\) emission in the Brazilian Legal Amazon. Ecological Modelling, 222(15), 2835–2840.

    Article  Google Scholar 

  8. Casper, J. K. (2010). Changing ecosystems: effects of global warming. New York: Facts on File Inc.

    Google Scholar 

  9. Casper, J. K. (2010). Greenhouse Gases: Worldwide Impacts. New York: Facts on File Inc.

    Google Scholar 

  10. Cheng, B., Dai, H., Wang, P., Xie, Y., Chen, L., Zhao, D., et al. (2016). Impacts of lowcarbon power policy on carbon mitigation in Guangdong Province, China. Energy Policy, 88, 515e527.

    Article  Google Scholar 

  11. Chang, Z., Wu, H., Pan, K., Zhu, H., & Chen, J. (2017). Clean production pathways for regional power-generation system under emission constraints: A case study of Shanghai, China. Journal of Cleaner Production, 143, 989e1000.

    Google Scholar 

  12. Devi, S., & Gupta, N. (2018). Dynamics of carbon dioxide gas (\(\text{ CO}_2\) ): Effects of varying capability of plants to absorb \(CO_2\). Natural Resource Modeling, 32(1), e12174.

    Article  Google Scholar 

  13. Devi, S., & Gupta, N. (2020). Comparative study of the effects of different growths of vegetation biomass on \(\text{ CO}_2\) in crisp and fuzzy environments. Natural Resource Modeling, 33(2), e12263.

    Article  Google Scholar 

  14. DeLong, J. P., & Burger, O. (2015). Socio-economic instability and the scaling of energy use with population size. PLoS ONE, 10(6), e0130547.

    Article  CAS  Google Scholar 

  15. El-Fadel, M., Chedid, R., Zeinati, M., & Hmaidan, W. (2003). Mitigating energy-related GHG emissions through renewable energy. Renewable Energy, 28(8), 1257–1276.

    CAS  Article  Google Scholar 

  16. Feng, Y. Y., & Zhang, L. X. (2012). Scenario analysis of urban energy saving and carbon abatement policies: A case study of Beijing city, China. Procedia Environmental Sciences, 13, 632–644.

    CAS  Article  Google Scholar 

  17. Feng, Y. Y., Chen, S. Q., & Zhang, L. X. (2013). dynamics modeling for urban energy consumption and \(\text{ CO } _{2}\) emissions: A case study of Beijing, China. Ecological Modelling, 252, 44–52.

    Article  Google Scholar 

  18. Fleming, W. H., & Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. New York: Springer-Verlag.

    Google Scholar 

  19. Gibbins, J., & Chalmers, H. (2008). Carbon capture and storage. Energy Policy, 36(12), 4317–4322.

    Article  Google Scholar 

  20. Harris, T., Devkota, J. P., Khanna, V., Eranki, P. L., & Landis, A. E. (2018). Logistic growth curve modelling of US energy production and consumption. Renewable and Sustainable Energy Reviews, 96, 46–57.

    Article  Google Scholar 

  21. Hashim, H., Douglas, P., Elkamel, A., & Croiset, E. (2005). Optimization model for energy planning with \(\text{ CO}_{2}\) emission considerations. Industrial & Engineering Chemistry Research, 44(4), 879–890.

    CAS  Article  Google Scholar 

  22. Huang, L., Kelly, S., Lv, K., & Giurco, D. (2019). A systematic review of empirical methods for modelling sectoral carbon emissions in China. Journal of Cleaner Production, 215, 138e21401.

    Google Scholar 

  23. IEA. (2019). Global energy & \(\text{ CO}_{2}\) status report 2019. https://www.iea.org/reports/global-energy-co2-status-report-2019/emissions. Accessed 14 March 2020.

  24. EIA. (2013). International Energy Outlook 2013. https://www.eia.gov/outlooks/ieo/pdf/0484(2013).pdf. Accessed 14 March 2020.

  25. Metz, B., Davidson, O., De Coninck, H., Loos, M., & Meyer, L. (2005). IPCC special report on carbon dioxide capture and storage. Cambridge: Cambridge University Press.

    Google Scholar 

  26. IPCC. (2001). The carbon cycle and atmospheric carbon dioxide. In J. T. Houghton, Y. Ding, D. J. Griggs, M. Noguer, P. J. van der Linden, X. Dai, K. Maskell, & C. A. Johnson (Eds.), Climate Change 2001: The Scientific Basis. Cambridge: Cambridge University Press.

  27. Jorgenson, A. K., & Clark, B. (2013). The relationship between national-level carbon dioxide emissions and population size: an assessment of regional and temporal variation, 1960–2005. PLoS ONE, 8(2), e57107.

    CAS  Article  Google Scholar 

  28. Khazalah, M., & Gopalan, B. (2019). Climate change - causes, impacts, mitigation: a review. GCEC, 2017(9), 715–721.

    Google Scholar 

  29. Kurane, I. (2010). The effect of global warming on infectious diseases. Osong Public Health and Research Perspectives, 1(1), 4–9.

    Article  Google Scholar 

  30. Lin, B., & Agyeman, S. D. (2019). Assessing Ghana’s carbon dioxide emissions through energy consumption structure towards a sustainable development path. Journal of Cleaner Production, 238, 117941.

    Article  Google Scholar 

  31. Lin, B., & Zhu, J. (2019). The role of renewable energy technological innovation on climate change: Empirical evidence from China. Science of the Total Environment, 659, 1505–1512.

    CAS  Article  Google Scholar 

  32. Lonngren, K. E., & Bai, E. W. (2008). On the global warming problem due to carbon dioxide. Energy Policy, 36(4), 1567–1568.

    Article  Google Scholar 

  33. Lu, C., Zhang, X., & He, J. (2010). A CGE analysis to study the impacts of energy investment on economic growth and carbon dioxide emission: A case of Shaanxi Province in western China. Energy, 35(11), 4319–4327.

    Article  Google Scholar 

  34. Lukes, D. L. (1982). Differential equations: classical to controlled. Edinburgh: Academic Press.

    Google Scholar 

  35. McMichael, A. J., Woodruff, R. E., & Hales, S. (2006). Climate change and human health: present and future risks. Lancet, 367, 859–869.

    Article  Google Scholar 

  36. Misra, A. K. (2014). Climate change and challenges of water and food security. International Journal of Sustainable Built Environment, 3(1), 153–165.

    Article  Google Scholar 

  37. Misra, A. K., & Verma, M. (2013). A mathematical model to study the dynamics of carbon dioxide gas in the atmosphere. Applied Mathematics Computation, 219(16), 8595–8609.

    Article  Google Scholar 

  38. Misra, A. K., & Verma, M. (2015). Impact of environmental education on mitigation of carbon dioxide emissions: a modelling study. International Journal of Global Warming, 7(4), 466–486.

    Article  Google Scholar 

  39. Misra, A. K., Verma, M., & Venturino, E. (2015). Modeling the control of atmospheric carbon dioxide through reforestation: effect of time delay. Modeling Earth Systems and Environment, 1(3), 24.

    Article  Google Scholar 

  40. Mohiuddin, O., Asumadu-Sarkodie, S., & Obaidullah, M. (2016). The relationship between carbon dioxide emissions, energy consumption, and GDP: A recent evidence from Pakistan. Cogent Engineering, Cogent Engineering, 3(1), 1210491.

    Article  Google Scholar 

  41. Newell, N. D., & Marcus, L. (1987). Carbon dioxide and people. Palaios, 2(1), 101–103.

    CAS  Article  Google Scholar 

  42. Nikol’skii, M. S. (2010). A controlled model of carbon circulation between the atmosphere and the ocean. Computational Mathematics and Modeling, 21, 414–424.

    Article  Google Scholar 

  43. NOAA. (2019). The NOAA Annual Greenhouse Gas Index (AGGI). http://www.esrl.noaa.gov/gmd/aggi/aggi.html. Accessed 14 March 2020.

  44. NOAA. (2020). Trends in Atmospheric Carbon Dioxide, Mauna Loa \(\text{ CO}_{2}\) annual mean data. https://www.esrl.noaa.gov/gmd/ccgg/trends/data.html. Accessed 14 March 2020.

  45. Onozaki, K. (2009). Population is a critical factor for global carbon dioxide increase. Journal of Health Science, 55, 125–127.

    CAS  Article  Google Scholar 

  46. Our World in Data. (2020). Global primary energy consumption. https://ourworldindata.org/grapher/global-primary-energy. Accessed 14 March 2020.

  47. Pao, H. T., & Tsai, C. M. (2011). Modeling and forecasting the \(\text{ CO}_{2}\) emissions, energy consumption, and economic growth in Brazil. Energy, 36(5), 2450–2458.

    Article  Google Scholar 

  48. Pekala, L. M., Tan, R. R., Foo, D. C. Y., & Jezowski, J. M. (2010). Optimal energy planning models with carbon footprint constraints. Applied Energy, 87(6), 1903–1910.

    CAS  Article  Google Scholar 

  49. Peng, L., Zhang, Y., Li, F., Wang, Q., Chen, X., & Yu, A. (2019). Policy implication of nuclear energy’s potential for energy optimization and \(\text{ CO}_{2}\) mitigation: A case study of Fujian. China. Nuclear Engineering and Technology, 51(4), 1154–1162.

    Article  Google Scholar 

  50. Phdungsilp, A. (2010). Integrated energy and carbon modeling with a decision support system: Policy scenarios for low-carbon city development in Bangkok. Energy Policy, 38(9), 4808–4817.

    Article  Google Scholar 

  51. Pires, J. C. M., Martins, F. G., Alvim-Ferraz, M. C. M., & Simes, M. (2011). Recent developments on carbon capture and storage: An overview. Chemical Engineering Research and Design, 89(9), 1446–1460.

    CAS  Article  Google Scholar 

  52. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., & Mishchenko, E. F. (1962). The Mathematical Theory of Optimal Processes. London: Wiley.

    Google Scholar 

  53. Poudyal, R., Loskot, P., Nepal, R., Parajuli, R., & Khadka, & S.K., (2019). Mitigating the current energy crisis in Nepal with renewable energy sources. Renewable and Sustainable Energy Reviews, 116, 109388.

  54. Roser, M., Ritchie, H., & Ortiz-Ospina, E. (2019). World Population Growth. https://ourworldindata.org/world-population-growth. Accessed 10 November 2020

  55. Robalino-Lopez, A., Mena-Nieto, A., & García-Ramos, J. E. (2014). System dynamics modeling for renewable energy and CO2 emissions: A case study of Ecuador. Energy for Sustainable Development, 20, 11–20.

    CAS  Article  Google Scholar 

  56. Shukla, J. B., Chauhan, M. S., Sundar, S., & Naresh, R. (2015). Removal of carbon dioxide from the atmosphere to reduce global warming: A modeling study. International Journal of Global Warming, 7(2), 270–292.

    Article  Google Scholar 

  57. Singh, J., & Dhar, D. W. (2019). Overview of Carbon Capture Technology: Microalgal Biorefinery Concept and State-of-the-Art. Frontiers in Marine Science, 6, 29.

    Article  Google Scholar 

  58. Stern, D. I., & Kander, A. (2012). The Role of Energy in the Industrial Revolution and Modern Economic Growth. The Energy Journal, 33(3), 125–152.

    Article  Google Scholar 

  59. The World Bank. (2019a). Population, Total. https://data.worldbank.org/indicator/sp.pop.totl. Accessed 14 March 2020.

  60. The World Bank. (2019b). \(\text{ CO}_{2}\) emissions (metric tons per capita). https://data.worldbank.org/indicator/EN.ATM.CO2E.PC. Accessed 14 March 2020.

  61. UNEP. (2012). UNEP Global Environmental Alert Service, One planet, How many people? A review of Earth’s carrying capacity, A discussion paper for the year of RIO+20. https://na.unep.net/geas/archive/pdfs/geas_jun_12_carrying_capacity.pdf. Accessed 14 March 2020.

  62. USEPA. (2019). U.S. Environmental Protection Agency, Inventory of U.S. Greenhouse Gas Emissions and Sinks: 1990-2017, Executive Summary. https://www.epa.gov/ghgemissions/inventory-us-greenhouse-gas-emissions-and-sinks-1990-2017. Accessed 14 March 2020.

  63. Verma, M., & Misra, A. K. (2018). Optimal control of anthropogenic carbon dioxide emissions through technological options: a modeling study. Comp. Appl. Math., 37, 605–626.

    Article  Google Scholar 

  64. Wee, J. H. (2013). A review on carbon dioxide capture and storage technology using coal fly ash. Applied Energy, 106, 143–151.

    CAS  Article  Google Scholar 

  65. WHO. (2020). Health and Environment Linkages Initiative (HELI), Priority environment and health risk. https://www.who.int/heli/risks/climate/climatechange/en/. Accessed 14 March 2020.

  66. Zabel, G. (2009). Peak people: The interrelationship between population growth and energy resources. https://www.resilience.org/stories/2009-04-20/peak-people-interrelationship-between-population-growth-and-energy-resources/. Accessed 14 March 2020.

  67. Zhang, D., Ma, L., Liu, P., Zhang, L., & Li, Z. (2012). A multi-period superstructure optimisation model for the optimal planning of China’s power sector considering carbon dioxide mitigation: discussion on China’s carbon mitigation policy based on the model. Energy Policy, 41, 173e183.

    Google Scholar 

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Acknowledgments

Authors are thankful to the handling editor and the anonymous reviewers for their useful suggestions. The first author (Maitri Verma) thankfully acknowledges University Grants Commission, New Delhi, India for financial support in form of UGC-BSR Research Start-Up Grant (No.F.30-442/2018(BSR)). The second author (Alok Kumar Verma) thankfully acknowledges Council of Scientific & Industrial Research (CSIR), New Delhi, India for financial support in form of junior research fellowship (09/961(0014)/2019-EMR-1).

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Appendices

Appendix

A Proof of Theorem 1

Let \(J_{S_i}\) denote the Jacobian matrix of system (4) evaluated at \(S_i\). The eigenvalues of \(J_{S_1}\) are \(-\alpha\), r, and 0. As one eigenvalue is \(r>0\), therefore \(S_1\) is unstable.

One of the eigenvalues of \(J_{S_2}\) is \(\frac{\gamma (1-\mu _1) N_2}{K+N_2}\). The other two eigenvalues are root of equation \(x^2+\left( \alpha +\frac{r N_2}{L}\right) x+ \left( \frac{\alpha r}{L}+\theta \lambda _1\right) N_2=0\), which are either negative or with negative real part. Since one eigenvalue of \(J_{S_2}\) is positive, therefore \(S_2\) is also unstable.

The local stability of \(S^*\) is examined using Lyapunov’s direct method. Consider the following function:

$$\begin{aligned} W=\frac{1}{2}c^2+\frac{p_1}{2} \frac{n^2}{N^*}+\frac{p_2}{2} \frac{e^2}{E^*}, \end{aligned}$$
(23)

where \(p_1\) and \(p_2\) are positive constants. Here c, n and e are small perturbations in C, N and E about the steady state \(S^*\).

The function W is positive definite function. The time derivative of ‘W’ along the linearized system of (4) corresponding to \(S^*\) is given by

$$\begin{aligned} \dot{W}= & {} -\alpha c^2-p_1\left( \frac{r}{L}-\beta _2E^*\right) n^2-p_2\gamma _0 e^2+(\lambda _1-p_1 \theta )c n\\&+\lambda _2(1-\mu _2) c \ e+p_1(\beta _1 +\beta _2N^*)n \ e+\frac{p_2(1-\mu _1)\gamma K}{(K+N^*)^2}n \ e. \end{aligned}$$

After taking \(p_1=\frac{\lambda _1}{\theta }\), \(\dot{W}\) is negative definite if the following inequalities hold:

$$\begin{aligned}&p_2 >\frac{3}{4}\frac{\lambda _2^2(1-\mu _2)^2}{\alpha \gamma _0} \end{aligned}$$
(24)
$$\begin{aligned}&p_2 > \frac{3}{2}\frac{\lambda _1(\beta _1+\beta _2 N^*)^2}{\theta \left( \frac{r}{L}-\beta _2 E^*\right) \gamma _0} \end{aligned}$$
(25)
$$\begin{aligned}&p_2 < \frac{2}{3}\frac{\lambda _1 \gamma _0(K+N^*)^4\left( \frac{r}{L}-\beta _2 E^*\right) }{\theta (1-\mu _1)^2K^2\gamma ^2} \end{aligned}$$
(26)

From the above inequalities, it is found that \(\dot{W}\) is negative definite, and hence, W is a Lyapunov function, under the condition (12).

B Proof of Theorem 2

Consider the following scalar valued positive definite function:

$$\begin{aligned} V=\frac{1}{2}(C-C^*)^2+m_1\left( N-N^*-N^* ln \frac{N}{N^*}\right) +m_2\left( E-E^*-E^*ln \frac{E}{E^*}\right) , \end{aligned}$$
(27)

where \(m_1\) and \(m_2\) are positive constants.

The time derivative of ‘V’ is given as

$$\begin{aligned} \dot{V}= & {} -\alpha (C-C^*)^2-m_1\left( \frac{r}{L}-\beta _2E^*\right) (N-N^*)^2-m_2\gamma _0(E-E^*)^2\\&+(\lambda _1-m_1 \theta )(C-C^*)(N-N^*)+\lambda _2(1-\mu _2)(C-C^*)(E-E^*)\\&+m_1(\beta _1 +\beta _2N)(N-N^*)(E-E^*)+\frac{m_2(1-\mu _1)\gamma K}{(K+N)(K+N^*)}(N-N^*)(E-E^*). \end{aligned}$$

Choosing \(m_1=\frac{\lambda _1}{\theta }\), \(\dot{V}\) is negative definite if the following inequalities hold:

$$\begin{aligned}&m_2 >\frac{3}{4}\frac{\lambda _2^2(1-\mu _2)^2}{\alpha \gamma _0} \end{aligned}$$
(28)
$$\begin{aligned}&m_2 > \frac{3}{2}\frac{\lambda _1(\beta _1+\beta _2 N_m)^2}{\theta \left( \frac{r}{L}-\beta _2 E^*\right) \gamma _0} \end{aligned}$$
(29)
$$\begin{aligned}&m_2 < \frac{2}{3}\frac{\lambda _1 \gamma _0(K+N^*)^2\left( \frac{r}{L}-\beta _2 E^*\right) }{\theta (1-\mu _1)^2\gamma ^2} \end{aligned}$$
(30)

The above inequalities are reduced to the condition (13). Thus, V is Lyapunov function on \(\varOmega\) provided the condition (13) holds.

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Verma, M., Verma, A.K. & Misra, A.K. Mathematical modeling and optimal control of carbon dioxide emissions from energy sector. Environ Dev Sustain (2021). https://doi.org/10.1007/s10668-021-01245-y

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Keywords

  • Mathematical model
  • Energy use
  • Atmospheric carbon dioxide
  • Climate change mitigation
  • Optimal control