Modeling and computation of energy efficiency management with emission permits trading

Shuhua Zhang; Xinyu Wang; Song Wang

doi:10.3934/jimo.2018010

Previous Article
Disaster relief routing in limited capacity road networks with heterogeneous flows
JIMO Home
This Issue
Next Article
Optimal investment and dividend payment strategies with debt management and reinsurance

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

Modeling and computation of energy efficiency management with emission permits trading

Shuhua Zhang ^1,, , Xinyu Wang ^1, and Song Wang ^2,

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.

Keywords: Energy efficiency management, emission permits trading, upwind finite difference method, optimal management strategies.

Mathematics Subject Classification: Primary: 49J20, 65M08, 91B06.

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.

show all references

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.

Figure 1. Computed errors in the $L^{\infty}$-norm at $t = 0$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. The results at $t = 0$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. phase portrait of $A$ v.s. $K$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. The effects of $S$ on the results

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. The effect of $S$ on the value function when $E_0 = 0.2$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. The effects of $E_0$ on the results

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. The effects of $\alpha$ on the results

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. The effects of $\beta$ on the results

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9. The effects of $\delta$ on the results

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10. The effects of $A_0$ and $K_0$ on the results

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. 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

Table Options

Download as excel

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

[1]	Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial and Management Optimization, 2022, 18 (2) : 825-841. doi: 10.3934/jimo.2020180
[2]	Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009
[3]	Harald Held, Gabriela Martinez, Philipp Emanuel Stelzig. Stochastic programming approach for energy management in electric microgrids. Numerical Algebra, Control and Optimization, 2014, 4 (3) : 241-267. doi: 10.3934/naco.2014.4.241
[4]	Jemal Mohammed-Awel, Ruijun Zhao, Eric Numfor, Suzanne Lenhart. Management strategies in a malaria model combining human and transmission-blocking vaccines. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 977-1000. doi: 10.3934/dcdsb.2017049
[5]	Andrew J. Whittle, Suzanne Lenhart, Louis J. Gross. Optimal control for management of an invasive plant species. Mathematical Biosciences & Engineering, 2007, 4 (1) : 101-112. doi: 10.3934/mbe.2007.4.101
[6]	Guirong Pan, Bing Xue, Hongchun Sun. An optimization model and method for supply chain equilibrium management problem. Mathematical Foundations of Computing, 2022, 5 (2) : 145-156. doi: 10.3934/mfc.2022001
[7]	Yumo Zhang. Mean-variance asset-liability management under CIR interest rate and the family of 4/2 stochastic volatility models with derivative trading. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022121
[8]	Zhengyan Wang, Guanghua Xu, Peibiao Zhao, Zudi Lu. The optimal cash holding models for stochastic cash management of continuous time. Journal of Industrial and Management Optimization, 2018, 14 (1) : 1-17. doi: 10.3934/jimo.2017034
[9]	Ximin Huang, Na Song, Wai-Ki Ching, Tak-Kuen Siu, Ka-Fai Cedric Yiu. A real option approach to optimal inventory management of retail products. Journal of Industrial and Management Optimization, 2012, 8 (2) : 379-389. doi: 10.3934/jimo.2012.8.379
[10]	C.E.M. Pearce, J. Piantadosi, P.G. Howlett. On an optimal control policy for stormwater management in two connected dams. Journal of Industrial and Management Optimization, 2007, 3 (2) : 313-320. doi: 10.3934/jimo.2007.3.313
[11]	Mourad Azi, Mohand Ouamer Bibi. Optimal control of a dynamical system with intermediate phase constraints and applications in cash management. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 279-291. doi: 10.3934/naco.2021005
[12]	Yu Yuan, Hui Mi. Robust optimal asset-liability management with penalization on ambiguity. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3461-3485. doi: 10.3934/jimo.2021121
[13]	Xiaoyong Mao, Baoguo Yu, Yingjing Shi, Rui Li. Real-time online trajectory planning and guidance for terminal area energy management of unmanned aerial vehicle. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022026
[14]	Luis F. Gordillo. Optimal sterile insect release for area-wide integrated pest management in a density regulated pest population. Mathematical Biosciences & Engineering, 2014, 11 (3) : 511-521. doi: 10.3934/mbe.2014.11.511
[15]	Jun Li, Hairong Feng, Kun-Jen Chung. Using the algebraic approach to determine the replenishment optimal policy with defective products, backlog and delay of payments in the supply chain management. Journal of Industrial and Management Optimization, 2012, 8 (1) : 263-269. doi: 10.3934/jimo.2012.8.263
[16]	Ruopeng Wang, Jinting Wang, Chang Sun. Optimal pricing and inventory management for a loss averse firm when facing strategic customers. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1521-1544. doi: 10.3934/jimo.2018019
[17]	Zheng Dou, Shaoyong Lai. Optimal contracts and asset prices in a continuous-time delegated portfolio management problem. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022083
[18]	Xue Qiao, Zheng Wang, Haoxun Chen. Joint optimal pricing and inventory management policy and its sensitivity analysis for perishable products: Lost sale case. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2533-2552. doi: 10.3934/jimo.2021079
[19]	Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012
[20]	Lisa C Flatley, Robert S MacKay, Michael Waterson. Optimal strategies for operating energy storage in an arbitrage or smoothing market. Journal of Dynamics and Games, 2016, 3 (4) : 371-398. doi: 10.3934/jdg.2016020

2021 Impact Factor: 1.411

Tools

Metrics

Tools

Article outline

Figures and Tables

2021 Impact Factor: 1.411

Tools

Figures and Tables

2021 Impact Factor: 1.411

American Institute of Mathematical Sciences

Modeling and computation of energy efficiency management with emission permits trading

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Modeling and computation of energy efficiency management with emission permits trading

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors