# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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, (1970).   Google Scholar [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.  doi: 10.1007/s10479-011-0881-8.  Google Scholar [3] M. M. Byrne, Is growth a dirty word? pollution, abatement and endogenous growth,, Journal of Development Economics, 54 (1997), 261.  doi: 10.1016/S0304-3878(97)00043-6.  Google Scholar [4] C. W. Clark, Mathematical Bioeconomis: The Optimal Management of Renewable Resources,, John Wiley $&$ Sons, (1976).   Google Scholar [5] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993).   Google Scholar [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.  doi: 10.1002/oca.843.  Google Scholar [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, 10 (2014), 413.   Google Scholar [8] A. Greiner, L. Grüne and W. Semmler, Growth and climate change: Threshold and multiple equilibria,, in J. Crespo Cuaresma, 12 (2010), 63.  doi: 10.1007/978-3-642-02132-9_4.  Google Scholar [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.  doi: 10.2139/ssrn.2098707.  Google Scholar [10] A. Greiner, W. Semmler and T. Mette, An economic model of oil exploration and extraction,, Computational Economics, 40 (2012), 387.  doi: 10.1007/s10614-011-9272-0.  Google Scholar [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.  doi: 10.1137/1037043.  Google Scholar [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.  doi: 10.1007/s10640-010-9431-0.  Google Scholar [13] M. Hestenes, Calculus of Variations and Optimal Control Theory,, John Wiley, (1966).   Google Scholar [14] M. Hoel and S. Kverndokk, Depletion of fossil fuels and the impacts of global warming,, Resource and Energy Economics, 18 (1996), 115.  doi: 10.1016/0928-7655(96)00005-X.  Google Scholar [15] H. Hotelling, The economics of exhaustible resources,, The Journal of Political Economy, 39 (1931), 137.  doi: 10.1086/254195.  Google Scholar [16] J. A. Krautkraemer, Optimal growth, resource amenities and the preservation of natural environments,, Review of Economic Studies, 52 (1985), 153.  doi: 10.2307/2297476.  Google Scholar [17] J. A. Krautkraemer, Nonrenewable resource scarcity,, Journal of Economic Literature, 36 (1998), 2065.   Google Scholar [18] K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253.  doi: 10.1007/BF00248267.  Google Scholar [19] H. Maurer, On the Minimum Principle for Optimal Control Problems with State Constraints,, Schriftenreihe des Rechenzentrums, (1979).   Google Scholar [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.  doi: 10.1007/BF02192163.  Google Scholar [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, (2014).   Google Scholar [22] W. D. Nordhaus, A Question of Balance. Weighing the Options on Global Warming Policies,, Yale University Press, (2008).   Google Scholar [23] F. van der Ploeg and C. Withagen, Too much coal, too few oil,, Journal of Public Economics, 96 (2012), 62.   Google Scholar [24] F. van der Ploeg and C. Withagen, Growth, renewables and the optimal carbon tax,, International Economic Review, 55 (2014), 283.  doi: 10.1111/iere.12049.  Google Scholar [25] L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes [in Russian],, Fitzmatgiz, (1964).   Google Scholar [26] S. Smulders and R. Gradus, Pollution abatement and long-term growth,, European Journal of Political Economy, 12 (1996), 505.  doi: 10.1016/S0176-2680(96)00013-4.  Google Scholar [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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar [28] F. Wirl and Y. Yegorov, Renewable Energy - Models, Implications and Prospects,, in The Macroeconomics of Climate Change, (2014).   Google Scholar

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##### References:
 [1] K. J. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy,, The John Hopkins Press, (1970).   Google Scholar [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.  doi: 10.1007/s10479-011-0881-8.  Google Scholar [3] M. M. Byrne, Is growth a dirty word? pollution, abatement and endogenous growth,, Journal of Development Economics, 54 (1997), 261.  doi: 10.1016/S0304-3878(97)00043-6.  Google Scholar [4] C. W. Clark, Mathematical Bioeconomis: The Optimal Management of Renewable Resources,, John Wiley $&$ Sons, (1976).   Google Scholar [5] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993).   Google Scholar [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.  doi: 10.1002/oca.843.  Google Scholar [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, 10 (2014), 413.   Google Scholar [8] A. Greiner, L. Grüne and W. Semmler, Growth and climate change: Threshold and multiple equilibria,, in J. Crespo Cuaresma, 12 (2010), 63.  doi: 10.1007/978-3-642-02132-9_4.  Google Scholar [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.  doi: 10.2139/ssrn.2098707.  Google Scholar [10] A. Greiner, W. Semmler and T. Mette, An economic model of oil exploration and extraction,, Computational Economics, 40 (2012), 387.  doi: 10.1007/s10614-011-9272-0.  Google Scholar [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.  doi: 10.1137/1037043.  Google Scholar [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.  doi: 10.1007/s10640-010-9431-0.  Google Scholar [13] M. Hestenes, Calculus of Variations and Optimal Control Theory,, John Wiley, (1966).   Google Scholar [14] M. Hoel and S. Kverndokk, Depletion of fossil fuels and the impacts of global warming,, Resource and Energy Economics, 18 (1996), 115.  doi: 10.1016/0928-7655(96)00005-X.  Google Scholar [15] H. Hotelling, The economics of exhaustible resources,, The Journal of Political Economy, 39 (1931), 137.  doi: 10.1086/254195.  Google Scholar [16] J. A. Krautkraemer, Optimal growth, resource amenities and the preservation of natural environments,, Review of Economic Studies, 52 (1985), 153.  doi: 10.2307/2297476.  Google Scholar [17] J. A. Krautkraemer, Nonrenewable resource scarcity,, Journal of Economic Literature, 36 (1998), 2065.   Google Scholar [18] K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253.  doi: 10.1007/BF00248267.  Google Scholar [19] H. Maurer, On the Minimum Principle for Optimal Control Problems with State Constraints,, Schriftenreihe des Rechenzentrums, (1979).   Google Scholar [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.  doi: 10.1007/BF02192163.  Google Scholar [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, (2014).   Google Scholar [22] W. D. Nordhaus, A Question of Balance. Weighing the Options on Global Warming Policies,, Yale University Press, (2008).   Google Scholar [23] F. van der Ploeg and C. Withagen, Too much coal, too few oil,, Journal of Public Economics, 96 (2012), 62.   Google Scholar [24] F. van der Ploeg and C. Withagen, Growth, renewables and the optimal carbon tax,, International Economic Review, 55 (2014), 283.  doi: 10.1111/iere.12049.  Google Scholar [25] L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes [in Russian],, Fitzmatgiz, (1964).   Google Scholar [26] S. Smulders and R. Gradus, Pollution abatement and long-term growth,, European Journal of Political Economy, 12 (1996), 505.  doi: 10.1016/S0176-2680(96)00013-4.  Google Scholar [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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar [28] F. Wirl and Y. Yegorov, Renewable Energy - Models, Implications and Prospects,, in The Macroeconomics of Climate Change, (2014).   Google Scholar
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