# American Institute of Mathematical Sciences

October  2018, 14(4): 1349-1365. doi: 10.3934/jimo.2018010

## Modeling and computation of energy efficiency management with emission permits trading

 1 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China 2 Department of Mathematics and Statistics, Curtin University, Perth, WA6845, Australia

* Corresponding author: Shuhua Zhang

Received  September 2016 Revised  October 2017 Published  October 2018 Early access  January 2018

Fund Project: This project was supported in part by the National Basic Research Program (2012CB955804), the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322), and the Major Program of Tianjin University of Finance and Economics (ZD1302).

In this paper, we present an optimal feedback control model to deal with the problem of energy efficiency management. Especially, an emission permits trading scheme is considered in our model, in which the decision maker can trade the emission permits flexibly. We make use of the optimal control theory to derive a Hamilton-Jacobi-Bellman (HJB) equation satisfied by the value function, and then propose an upwind finite difference method to solve it. The stability of this method is demonstrated and the accuracy, as well as the usefulness, is shown by the numerical examples. The optimal management strategies, which maximize the discounted stream of the net revenue, together with the value functions, are obtained. The effects of the emission permits price and other parameters in the established model on the results have been also examined. We find that the influences of emission permits price on net revenue for the economic agents with different initial quotas are quite different. All the results demonstrate that the emission permits trading scheme plays an important role in the energy efficiency management.

Citation: Shuhua Zhang, Xinyu Wang, Song Wang. Modeling and computation of energy efficiency management with emission permits trading. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1349-1365. doi: 10.3934/jimo.2018010
##### References:
 [1] International Energy Agency, World Energy Outlook 2006, http://www.iea.org/, 2007. [2] A. Bernard, A. Haurie, M. Vielle and L. Viguier, A two-level dynamic game of carbon emission trading between Russia, China, and Annex B countries, Journal of Economic Dynamics and Control, 32 (2008), 1830-1856.  doi: 10.1016/j.jedc.2007.07.001. [3] S. Chang, X. Wang and Z. Wang, Modeling and computation of transboundary industrial pollution with emission permits trading by stochastic differential game, PLoS ONE, 10 (2015), e0138641. [4] S. Chang and X. Wang, Modelling and computation in the valuation of carbon derivatives with stochastic convenience yields, PLoS ONE, 10 (2015), e0125679. doi: 10.1371/journal.pone.0125679. [5] K. Chang, S. Wang and K. Peng, Mean reversion of stochastic convenience yields for CO$_2$ emissions allowances: Empirical evidence from the EU ETS, The Spanish Review of Financial Economics, 11 (2013), 39-45. [6] C. Cobb and P. Douglas, A Theory of Production, American Economic Review, 8 (1928), 139-165. [7] G. Daskalakis, D. Psychoyios and R. Markellos, Modeling CO2 emission allowance prices and derivatives: Evidence from the European trading scheme, Journal of Banking&Finance, 33 (2009), 1230-1241. [8] E. Dockner, S. Jorgensen, N. Long and G. Sorger, Differential Games in Economics and Management Science Cambridge University Press, 2000. [9] L. Greening, D. Greene and C. Difiglio, Energy efficiency and consumption -the rebound effect -a survey, Energy Policy, 28 (2000), 389-401.  doi: 10.1016/S0301-4215(00)00021-5. [10] S. Hitzemann and M. Uhrig-Homburg, Empirical performance of reduced form models for emission permit prices Working paper Available at SSRN, (2013), 38pp. doi: 10.2139/ssrn.2297121. [11] R. Howarth, B. Haddad and B. Paton, The economics of energy efficiency: Insights from voluntary participation programs, Energy Policy, 28 (2000), 477-486.  doi: 10.1016/S0301-4215(00)00026-4. [12] J. Hu and S. Wang, Total-factor energy efficiency of regions in China, Energy Policy, 34 (2006), 3206-3217.  doi: 10.1016/j.enpol.2005.06.015. [13] A. Jaffe and R. Stavins, The energy-efficiency gap: What does it mean?, Energy Policy, 22 (1994), 804-810.  doi: 10.1016/0301-4215(94)90138-4. [14] W. Jin and Z. Zhang, On the mechanism of international technology diffusion for energy technological progress, Working Paper 2015. Available at SSRN: http://ssrn.com/abstract=2584473 [15] S. Li, A differential game of transboundary industrial pollution with emission permits trading, Journal of Optimization Theory and Applications, 163 (2014), 642-659.  doi: 10.1007/s10957-013-0384-7. [16] S. Osher and F. Solomon, Upwind difference schemes for hyperbolic system of conservation laws, Mathematics of Computation, 38 (1982), 339-374.  doi: 10.1090/S0025-5718-1982-0645656-0. [17] M. Pandian, A partial upwind difference scheme for nonlinear parabolic equations, Journal of Computational and Applied Mathematics, 26 (1989), 219-233.  doi: 10.1016/0377-0427(89)90295-1. [18] M. Patterson, What is energy efficiency? concepts, indicators and methodological issues, Energy efficiency, 24 (1996), 377-390. [19] U. Risch, An upwind finite element method for singularly perturbed elliptic problems and local estimates in the $L^{∞}$-norm, Matehmatical Modelling and Numerical Analysis, 24 (1990), 235-264.  doi: 10.1051/m2an/1990240202351. [20] S. Schurr, Energy use, technological change, and productive efficiency: An economic-historical interpretation, Annual Review of Energy, 9 (1984), 409-425.  doi: 10.1146/annurev.eg.09.110184.002205. [21] J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO2 spot prices, Journal of Environmental Economics and Management, 56 (2008), 180-194. [22] J. Strikwerda, Finite Difference Schemes and Partial Differential Equations Society for Industrial and Applied Mathematics, 2004. [23] S. Wang, F. Gao and K. Teo, An upwind finite-difference method for the approximate of viscosity solutions to Hamilton-Jacobi-Bellman equations, IMA Journal of Mathematics Control and Information, 17 (2000), 167-178.  doi: 10.1093/imamci/17.2.167. [24] E. Worrell, L. Bernstein, J. Roy, L. Price and J. Harnisch, Industrial energy efficiency and climate change mitigation, Energy Efficiency, 2 (2009), 109-123.  doi: 10.2172/957331. [25] S. Zhang, X. Wang and A. Shananin, Modeling and computation of mean field equilibria in producers' game with emission permits trading, Communications in Nonlinear Science and Numerical Simulation, 37 (2016), 238-248.  doi: 10.1016/j.cnsns.2016.01.020.

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##### References:
 [1] International Energy Agency, World Energy Outlook 2006, http://www.iea.org/, 2007. [2] A. Bernard, A. Haurie, M. Vielle and L. Viguier, A two-level dynamic game of carbon emission trading between Russia, China, and Annex B countries, Journal of Economic Dynamics and Control, 32 (2008), 1830-1856.  doi: 10.1016/j.jedc.2007.07.001. [3] S. Chang, X. Wang and Z. Wang, Modeling and computation of transboundary industrial pollution with emission permits trading by stochastic differential game, PLoS ONE, 10 (2015), e0138641. [4] S. Chang and X. Wang, Modelling and computation in the valuation of carbon derivatives with stochastic convenience yields, PLoS ONE, 10 (2015), e0125679. doi: 10.1371/journal.pone.0125679. [5] K. Chang, S. Wang and K. Peng, Mean reversion of stochastic convenience yields for CO$_2$ emissions allowances: Empirical evidence from the EU ETS, The Spanish Review of Financial Economics, 11 (2013), 39-45. [6] C. Cobb and P. Douglas, A Theory of Production, American Economic Review, 8 (1928), 139-165. [7] G. Daskalakis, D. Psychoyios and R. Markellos, Modeling CO2 emission allowance prices and derivatives: Evidence from the European trading scheme, Journal of Banking&Finance, 33 (2009), 1230-1241. [8] E. Dockner, S. Jorgensen, N. Long and G. Sorger, Differential Games in Economics and Management Science Cambridge University Press, 2000. [9] L. Greening, D. Greene and C. Difiglio, Energy efficiency and consumption -the rebound effect -a survey, Energy Policy, 28 (2000), 389-401.  doi: 10.1016/S0301-4215(00)00021-5. [10] S. Hitzemann and M. Uhrig-Homburg, Empirical performance of reduced form models for emission permit prices Working paper Available at SSRN, (2013), 38pp. doi: 10.2139/ssrn.2297121. [11] R. Howarth, B. Haddad and B. Paton, The economics of energy efficiency: Insights from voluntary participation programs, Energy Policy, 28 (2000), 477-486.  doi: 10.1016/S0301-4215(00)00026-4. [12] J. Hu and S. Wang, Total-factor energy efficiency of regions in China, Energy Policy, 34 (2006), 3206-3217.  doi: 10.1016/j.enpol.2005.06.015. [13] A. Jaffe and R. Stavins, The energy-efficiency gap: What does it mean?, Energy Policy, 22 (1994), 804-810.  doi: 10.1016/0301-4215(94)90138-4. [14] W. Jin and Z. Zhang, On the mechanism of international technology diffusion for energy technological progress, Working Paper 2015. Available at SSRN: http://ssrn.com/abstract=2584473 [15] S. Li, A differential game of transboundary industrial pollution with emission permits trading, Journal of Optimization Theory and Applications, 163 (2014), 642-659.  doi: 10.1007/s10957-013-0384-7. [16] S. Osher and F. Solomon, Upwind difference schemes for hyperbolic system of conservation laws, Mathematics of Computation, 38 (1982), 339-374.  doi: 10.1090/S0025-5718-1982-0645656-0. [17] M. Pandian, A partial upwind difference scheme for nonlinear parabolic equations, Journal of Computational and Applied Mathematics, 26 (1989), 219-233.  doi: 10.1016/0377-0427(89)90295-1. [18] M. Patterson, What is energy efficiency? concepts, indicators and methodological issues, Energy efficiency, 24 (1996), 377-390. [19] U. Risch, An upwind finite element method for singularly perturbed elliptic problems and local estimates in the $L^{∞}$-norm, Matehmatical Modelling and Numerical Analysis, 24 (1990), 235-264.  doi: 10.1051/m2an/1990240202351. [20] S. Schurr, Energy use, technological change, and productive efficiency: An economic-historical interpretation, Annual Review of Energy, 9 (1984), 409-425.  doi: 10.1146/annurev.eg.09.110184.002205. [21] J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO2 spot prices, Journal of Environmental Economics and Management, 56 (2008), 180-194. [22] J. Strikwerda, Finite Difference Schemes and Partial Differential Equations Society for Industrial and Applied Mathematics, 2004. [23] S. Wang, F. Gao and K. Teo, An upwind finite-difference method for the approximate of viscosity solutions to Hamilton-Jacobi-Bellman equations, IMA Journal of Mathematics Control and Information, 17 (2000), 167-178.  doi: 10.1093/imamci/17.2.167. [24] E. Worrell, L. Bernstein, J. Roy, L. Price and J. Harnisch, Industrial energy efficiency and climate change mitigation, Energy Efficiency, 2 (2009), 109-123.  doi: 10.2172/957331. [25] S. Zhang, X. Wang and A. Shananin, Modeling and computation of mean field equilibria in producers' game with emission permits trading, Communications in Nonlinear Science and Numerical Simulation, 37 (2016), 238-248.  doi: 10.1016/j.cnsns.2016.01.020.
Computed errors in the $L^{\infty}$-norm at $t = 0$
The results at $t = 0$
phase portrait of $A$ v.s. $K$
The effects of $S$ on the results
The effect of $S$ on the value function when $E_0 = 0.2$
The effects of $E_0$ on the results
The effects of $\alpha$ on the results
The effects of $\beta$ on the results
The effects of $\delta$ on the results
The effects of $A_0$ and $K_0$ on the results
Some results values at $t = 0$
 Energy efficiency $A$ Capital stock $K$ Value function $V$ Indigenous innovation $\lambda$ Absorbed knowledge $\sigma$ Emission $E$ 0.3 0.3 -0.5658 0.1020 0.2380 0.5320 0.5 -0.3742 0.1007 0.2350 0.5190 0.7 -0.2476 0.0999 0.2332 0.5112 0.5 0.3 -0.5052 0.1116 0.1116 0.5464 0.5 -0.3201 0.1100 0.1100 0.5320 0.7 -0.1940 0.1090 0.1090 0.5233 0.7 0.3 -0.4728 0.1152 0.0494 0.5567 0.5 -0.2825 0.1134 0.0486 0.5413 0.7 -0.1567 0.1124 0.0482 0.5320
 Energy efficiency $A$ Capital stock $K$ Value function $V$ Indigenous innovation $\lambda$ Absorbed knowledge $\sigma$ Emission $E$ 0.3 0.3 -0.5658 0.1020 0.2380 0.5320 0.5 -0.3742 0.1007 0.2350 0.5190 0.7 -0.2476 0.0999 0.2332 0.5112 0.5 0.3 -0.5052 0.1116 0.1116 0.5464 0.5 -0.3201 0.1100 0.1100 0.5320 0.7 -0.1940 0.1090 0.1090 0.5233 0.7 0.3 -0.4728 0.1152 0.0494 0.5567 0.5 -0.2825 0.1134 0.0486 0.5413 0.7 -0.1567 0.1124 0.0482 0.5320
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