Modeling and computation of energy efficiency management with emission permits trading

Shuhua Zhang; Xinyu Wang; Song Wang

doi:10.3934/jimo.2018010

Article Contents

2018, Volume 14, Issue 4: 1349-1365. Doi: 10.3934/jimo.2018010

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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
* Corresponding author: Shuhua Zhang

Received: August 31, 2016

Revised: September 30, 2017

Early access: January 2018

Published: October 2018

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)

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Abstract

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:

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

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

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Figure 2. The results at $t = 0$

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

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

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

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

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

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

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

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

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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

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