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Modeling and computation of energy efficiency management with emission permits trading

  • * Corresponding author: Shuhua Zhang

    * Corresponding author: Shuhua Zhang 
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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  • 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.

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


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

    Figure 2.  The results at $t = 0$

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

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

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

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

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

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

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

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

    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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