September  2015, 35(9): 4503-4525. doi: 10.3934/dcds.2015.35.4503

Expediting the transition from non-renewable to renewable energy via optimal control

1. 

Institute of Computational and Applied Mathematics, University of Muenster, Einsteinstr. 62, D-48149 Muenster

2. 

New School for Social Research, Department of Economics, 6 East 16th Street, D-1123, New York, NY 10003, United States

Received  April 2014 Revised  October 2014 Published  April 2015

Much recent climate research suggests that the transition from non-renewable to renewable energy should be expedited. To address this issue we use an optimal control model, based on an integrated assessment model of climate change that includes two types energy production. After setting up the model, we derive necessary optimality conditions in the form of a Pontryagin type Maximum Principle. We use a numerical discretization method for optimal control problems to explore various policy scenarios. The algorithm allows to compute both state and co-state variables by providing a consistent numerical approximation for the adjoint variables of the various scenarios. Our numerical method applies to control and state-constrained control problems as well as to delayed control problems. In the policy scenarios, we explore ways how the transition from non-renewable to renewable energy can be expedited.
Citation: Helmut Maurer, Willi Semmler. Expediting the transition from non-renewable to renewable energy via optimal control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4503-4525. doi: 10.3934/dcds.2015.35.4503
References:
[1]

K. J. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy, The John Hopkins Press, Baltimore, 1970.

[2]

T. Brechet, C. Carmacho and V. M. Veliov, Model predictive control, the economy, and the issue of global warming, Annals of Operations Research, 220 (2014), 25-48. doi: 10.1007/s10479-011-0881-8.

[3]

M. M. Byrne, Is growth a dirty word? pollution, abatement and endogenous growth, Journal of Development Economics, 54 (1997), 261-284. doi: 10.1016/S0304-3878(97)00043-6.

[4]

C. W. Clark, Mathematical Bioeconomis: The Optimal Management of Renewable Resources, John Wiley $&$ Sons, New York, 1976.

[5]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.

[6]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control and mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365. doi: 10.1002/oca.843.

[7]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Special Issue on Computational Methods for Optimization and Control, J. of Industrial and Management Optimization, 10 (2014), 413-441.

[8]

A. Greiner, L. Grüne and W. Semmler, Growth and climate change: Threshold and multiple equilibria, in J. Crespo Cuaresma, T. Palokangas and A. Tarasyev (eds.) Dynamic Systems, Economic Growth, and the Environment,Berlin, Springer, 12 (2010), 63-78. doi: 10.1007/978-3-642-02132-9_4.

[9]

A. Greiner, L. Grüne and W. Semmler, Economic growth and the transition from non-renewable to renewable energy, Environment and Development Economics, 19 (2014), 417-439. doi: 10.2139/ssrn.2098707.

[10]

A. Greiner, W. Semmler and T. Mette, An economic model of oil exploration and extraction, Computational Economics, 40 (2012), 387-399. doi: 10.1007/s10614-011-9272-0.

[11]

R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the Maximum Principles for optimal control problems with state constraints, SIAM Review, 37 (1995), 181-218. doi: 10.1137/1037043.

[12]

C. Heinzel and R. Winkler, Distorted time preferences and time-to-build in the transition to a low-carbon energy industry, Environmental and Resource Economics, 49 (2011), 217-241. doi: 10.1007/s10640-010-9431-0.

[13]

M. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.

[14]

M. Hoel and S. Kverndokk, Depletion of fossil fuels and the impacts of global warming, Resource and Energy Economics, 18 (1996), 115-136. doi: 10.1016/0928-7655(96)00005-X.

[15]

H. Hotelling, The economics of exhaustible resources, The Journal of Political Economy, 39 (1931), 137-175. doi: 10.1086/254195.

[16]

J. A. Krautkraemer, Optimal growth, resource amenities and the preservation of natural environments, Review of Economic Studies, 52 (1985), 153-170. doi: 10.2307/2297476.

[17]

J. A. Krautkraemer, Nonrenewable resource scarcity, Journal of Economic Literature, 36 (1998), 2065-2107.

[18]

K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints, Computational Optimization and Applications, 5 (1996), 253-283. doi: 10.1007/BF00248267.

[19]

H. Maurer, On the Minimum Principle for Optimal Control Problems with State Constraints, Schriftenreihe des Rechenzentrums, Report no. 41, 1979, Universität Münster, Germany.

[20]

H. Maurer and S. Pickenhain, Second-order sufficient conditions for control problems with mixed control-state constraints, J. of Optimization Theory and Applications, 86 (1995), 649-667. doi: 10.1007/BF02192163.

[21]

S. Mittnik, M. Kato, D. Samaan and W. Semmler, Climate policies and structural change - employment and output effects of sustainable growth, in The Macroeconomics of Climate Change, L. Bernard and W. Semmler (eds.), Oxford University Press, New York, forthcoming 2014.

[22]

W. D. Nordhaus, A Question of Balance. Weighing the Options on Global Warming Policies, Yale University Press, New Haven, 2008.

[23]

F. van der Ploeg and C. Withagen, Too much coal, too few oil, Journal of Public Economics, 96 (2012), 62-77.

[24]

F. van der Ploeg and C. Withagen, Growth, renewables and the optimal carbon tax, International Economic Review, 55 (2014), 283-311. doi: 10.1111/iere.12049.

[25]

L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes [in Russian], Fitzmatgiz, Moscow; English translation: Pergamon Press, New York, 1964.

[26]

S. Smulders and R. Gradus, Pollution abatement and long-term growth, European Journal of Political Economy, 12 (1996), 505-532. doi: 10.1016/S0176-2680(96)00013-4.

[27]

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; cf. Ipopt home page (C. Laird and A. Wächter): https://projects.coin-or.org/Ipopt. doi: 10.1007/s10107-004-0559-y.

[28]

F. Wirl and Y. Yegorov, Renewable Energy - Models, Implications and Prospects, in The Macroeconomics of Climate Change, L. Bernard and W. Semmler, eds., Oxford University Press, New York, forthcoming 2014.

show all references

References:
[1]

K. J. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy, The John Hopkins Press, Baltimore, 1970.

[2]

T. Brechet, C. Carmacho and V. M. Veliov, Model predictive control, the economy, and the issue of global warming, Annals of Operations Research, 220 (2014), 25-48. doi: 10.1007/s10479-011-0881-8.

[3]

M. M. Byrne, Is growth a dirty word? pollution, abatement and endogenous growth, Journal of Development Economics, 54 (1997), 261-284. doi: 10.1016/S0304-3878(97)00043-6.

[4]

C. W. Clark, Mathematical Bioeconomis: The Optimal Management of Renewable Resources, John Wiley $&$ Sons, New York, 1976.

[5]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.

[6]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control and mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365. doi: 10.1002/oca.843.

[7]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Special Issue on Computational Methods for Optimization and Control, J. of Industrial and Management Optimization, 10 (2014), 413-441.

[8]

A. Greiner, L. Grüne and W. Semmler, Growth and climate change: Threshold and multiple equilibria, in J. Crespo Cuaresma, T. Palokangas and A. Tarasyev (eds.) Dynamic Systems, Economic Growth, and the Environment,Berlin, Springer, 12 (2010), 63-78. doi: 10.1007/978-3-642-02132-9_4.

[9]

A. Greiner, L. Grüne and W. Semmler, Economic growth and the transition from non-renewable to renewable energy, Environment and Development Economics, 19 (2014), 417-439. doi: 10.2139/ssrn.2098707.

[10]

A. Greiner, W. Semmler and T. Mette, An economic model of oil exploration and extraction, Computational Economics, 40 (2012), 387-399. doi: 10.1007/s10614-011-9272-0.

[11]

R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the Maximum Principles for optimal control problems with state constraints, SIAM Review, 37 (1995), 181-218. doi: 10.1137/1037043.

[12]

C. Heinzel and R. Winkler, Distorted time preferences and time-to-build in the transition to a low-carbon energy industry, Environmental and Resource Economics, 49 (2011), 217-241. doi: 10.1007/s10640-010-9431-0.

[13]

M. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.

[14]

M. Hoel and S. Kverndokk, Depletion of fossil fuels and the impacts of global warming, Resource and Energy Economics, 18 (1996), 115-136. doi: 10.1016/0928-7655(96)00005-X.

[15]

H. Hotelling, The economics of exhaustible resources, The Journal of Political Economy, 39 (1931), 137-175. doi: 10.1086/254195.

[16]

J. A. Krautkraemer, Optimal growth, resource amenities and the preservation of natural environments, Review of Economic Studies, 52 (1985), 153-170. doi: 10.2307/2297476.

[17]

J. A. Krautkraemer, Nonrenewable resource scarcity, Journal of Economic Literature, 36 (1998), 2065-2107.

[18]

K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints, Computational Optimization and Applications, 5 (1996), 253-283. doi: 10.1007/BF00248267.

[19]

H. Maurer, On the Minimum Principle for Optimal Control Problems with State Constraints, Schriftenreihe des Rechenzentrums, Report no. 41, 1979, Universität Münster, Germany.

[20]

H. Maurer and S. Pickenhain, Second-order sufficient conditions for control problems with mixed control-state constraints, J. of Optimization Theory and Applications, 86 (1995), 649-667. doi: 10.1007/BF02192163.

[21]

S. Mittnik, M. Kato, D. Samaan and W. Semmler, Climate policies and structural change - employment and output effects of sustainable growth, in The Macroeconomics of Climate Change, L. Bernard and W. Semmler (eds.), Oxford University Press, New York, forthcoming 2014.

[22]

W. D. Nordhaus, A Question of Balance. Weighing the Options on Global Warming Policies, Yale University Press, New Haven, 2008.

[23]

F. van der Ploeg and C. Withagen, Too much coal, too few oil, Journal of Public Economics, 96 (2012), 62-77.

[24]

F. van der Ploeg and C. Withagen, Growth, renewables and the optimal carbon tax, International Economic Review, 55 (2014), 283-311. doi: 10.1111/iere.12049.

[25]

L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes [in Russian], Fitzmatgiz, Moscow; English translation: Pergamon Press, New York, 1964.

[26]

S. Smulders and R. Gradus, Pollution abatement and long-term growth, European Journal of Political Economy, 12 (1996), 505-532. doi: 10.1016/S0176-2680(96)00013-4.

[27]

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; cf. Ipopt home page (C. Laird and A. Wächter): https://projects.coin-or.org/Ipopt. doi: 10.1007/s10107-004-0559-y.

[28]

F. Wirl and Y. Yegorov, Renewable Energy - Models, Implications and Prospects, in The Macroeconomics of Climate Change, L. Bernard and W. Semmler, eds., Oxford University Press, New York, forthcoming 2014.

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