﻿ 三峡库区碳排放预测模型
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 应用科技  2017, Vol. 44 Issue (6): 83-88  DOI: 10.11991/yykj.201610006 0

### 引用本文

LU Zuliang, LI Lin, CAO Longzhou. A Carbon emission prediction model in Three Gorges reservoir area[J]. Applied Science and Technology, 2017, 44(6), 83-88. DOI: 10.11991/yykj.201610006.

### 文章历史

1. 重庆三峡学院 非线性科学与系统结构重点实验室, 重庆 404100;
2. 天津财经大学 数学与经济研究中心, 天津 300222

A Carbon emission prediction model in Three Gorges reservoir area
LU Zuliang1,2, LI Lin1, CAO Longzhou1
1. Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing 404100, China;
2. Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China
Abstract: In order to reduce the ecological environment effects by the Three Gorges Project in the lower reaches of Three Gorges areas, a carbon emission prediction model was adopted. This paper mainly studies the prediction of carbon dioxide emissions in the Three Gorges Reservoir Area. By using Pontryagin's maximum principle and the Markov process, the model of carbon dioxide emissions in the Three Gorges Reservoir was established. The carbon emissions problem was transformed into a first-order differential equation. Then, the four-step variable step Runge Kutta method was used to calculate the total carbon dioxide emissions in the Three Gorges Reservoir Area during 2015~2021. According to these data, it can be derived that the amount of energy use and carbon emissions has a direct relationship, and a lot of carbon dioxide emissions is likely to lead to global greenhouse effect, however, the reduction of energy consumption will influence the growth of economy. So how to achieve a balance between the amount of energy and carbon dioxide emissions is very important, and we also can provide data for the environmental governance in the Three Gorges Reservoir area.
Key words: Three Gorges reservoir area    carbon emission    environmental governance    carbon emissions prediction model    Hamiltonian function    pontryagin maximum principle    Markov chain    Runge-Kutta method

1 数学模型

 $u({c_t}) = \frac{{{{({c_t} - \bar c)}^{1 - \sigma }}}}{{1 - \sigma }}$

1.1 能源和经济之间动态关系的模型

 ${\rm{max }}U = \int_0^\infty {u({c_t})} {N_t}\exp ( - \rho t){\rm{d}}t$ (1)

 ${\rm{max }}U = \int_{{\rm{ }}0}^{{\rm{ }}\infty } {N_0^\sigma } \exp \left( {(n\sigma - \rho )t} \right){\displaystyle{{{{({c_t} - \bar c)}^{1 - \sigma }}} \over {1 - \sigma }}}{\rm{d}}t$ (2)

 $\begin{array}{*{20}{c}}{{Y_t} = {A_0}\exp (vt)K_t^\alpha E_t^{1 - \alpha }L_t^\gamma ,}&{0 < \alpha < }\end{array}1$ (3)

 ${\theta _t} = {p_{d,t}}{s_t} + {p_{a,t}}(1 - {s_t})$

 $\tau = \frac{E}{Y}$ (4)

 $\begin{array}{*{20}{c}}{{Y_t} = {{\left( {{A_0}\exp (vt)} \right)}^{{\textstyle{1 \over \alpha }}}}{K_t}{\tau _t}^{{\textstyle{{1 - \alpha } \over \alpha }}}{L_t}^{{\textstyle{\gamma \over \alpha }}}},&{0 < \alpha < 1}\end{array}$

 $\begin{split}{K_t} \!=\! & {Y_t} \!-\! \delta {K_t} \!-\! {\theta _t}{E_t} \!-\! {C_t} = ({Y_t} \!-\! {\theta _t}{\tau _t}{Y_t}) \!-\! \delta {K_t} \!-\! {C_t} = \\& (1\! -\! {\theta _t}{\tau _t}){\left( {{A_0}\exp (vt)} \right)^{{\textstyle{1 \over \alpha }}}}{K_t}{\tau _t}^{{\textstyle{{1 \!-\! \alpha } \over \alpha }}}{L_t}^{{\textstyle{\gamma \over \alpha }}}\! -\! \delta {K_t} \!-\! {C_t}\end{split}$ (5)

 ${\rm{max }}U = \int_{{\rm{ }}0}^{{\rm{ }}\infty } {N_0^\sigma } \exp \left( {(n\sigma - \rho )t} \right){\displaystyle{{{{({c_t} - \bar c)}^{1 - \sigma }}} \over {1 - \sigma }}}{\rm{d}}t$

 ${\dot K_t} = (1 - {\theta _t}{\tau _t}){\left( {{A_0}\exp (vt)} \right)^{{\textstyle{1 \over \alpha }}}}{K_t}{\tau _t}^{{\textstyle{{1 - \alpha } \over \alpha }}}{L_t}^{{\textstyle{\gamma \over \alpha }}} - \delta {K_t} - {C_t}$

 $H = N_0^\sigma \exp \left( {(n\sigma - \rho )t} \right){\displaystyle{{{{({c_t} - \bar c)}^{1 - \sigma }}} \over {1 - \sigma }}} + \lambda {\dot K_t}$

 $\frac{{\partial {H_t}}}{{\partial {C_t}}} = N_0^\sigma \exp \left( {(n\sigma - \rho )t} \right){({C_t} - \bar C)^{ - \sigma }} - \lambda = 0$ (6)

 $\dot \lambda \!=\! - \frac{{\partial H}}{{\partial {K_t}}} \!=\! - \lambda [(1 \!-\! {\theta _t}{\tau _t}){({A_0}\exp (vt))^{{\textstyle{1 \over \alpha }}}}{\tau _t}^{{\textstyle{{1 - \alpha } \over \alpha }}}{L_t}^{{\textstyle{\gamma \over \alpha }}} \!-\! \delta ]$ (7)

 \begin{aligned}& (n\sigma - \rho )N_0^\sigma \exp \left( {(n\sigma - \rho )t} \right){({C_t} - \bar C)^{ - \sigma }} + \\& N_0^\sigma \exp \left( {(n\sigma - \rho )t} \right)( - \sigma ){({C_t} - \bar C)^{ - \sigma - 1}}{{\dot C}_t} - \dot \lambda = 0\end{aligned} (8)

 $\begin{split} {{\dot C}_t} = & \frac{1}{\sigma }[(n\sigma - \rho - \delta ) + \\& (1 - {\theta _t}{\tau _t}){({A_0}\exp (vt))^{\textstyle\frac{1}{\alpha }}}{\tau _t}^{\textstyle\frac{{1 - \alpha }}{\alpha }}{L_t}^{\textstyle\frac{\gamma }{\alpha }}] \times ({C_t} - \bar C)\end{split}$ (9)

 \begin{aligned}& {g_c} = \\& \frac{{\displaystyle\frac{1}{\sigma }[(n\sigma \! - \! \rho \!- \!\delta ) \!+\! (1 \!-\! {\theta _t}{\tau _t}){{({A_0}\exp (vt))}^{\textstyle\frac{1}{\alpha }}}{\tau _t}^{\textstyle\frac{{1 - \alpha }}{\alpha }}{L_t}^{\textstyle\frac{\gamma }{\alpha }}]({C_t} \!-\! \bar C)}}{{{C_t}}}\end{aligned}

 ${g_t} = \frac{{{C_t} - {C_{t - 1}}}}{{{C_{t - 1}}}}$
1.2 马尔可夫模型

 $\mathit{\boldsymbol{P}} = \left( {\begin{array}{*{20}{c}}{{P_{11}}}&{{P_{12}}}& \ldots &{{P_{1n}}}\\[6pt]{{P_{21}}}&{{P_{22}}}& \ldots &{{P_{2n}}}\\[6pt] \ldots & \ldots & \ldots & \ldots \\[6pt]{{P_{n1}}}&{{P_{n2}}}& \ldots &{{P_{nn}}}\end{array}} \right)$

 ${\mathit{\boldsymbol{s}}^{(k)}} = {\mathit{\boldsymbol{s}}^{(0)}} \cdot {\mathit{\boldsymbol{P}}^k}$
2 模型中参数的估计 2.1 经济系统中的参数

 ${Y_t} = {A_0}\exp (vt)K_t^\alpha E_t^{1 - \alpha }L_t^\gamma ,$

 $Y' = {a_0} + vt + \alpha K' + \gamma L' + \zeta$

2.2 其他参数

2.3 转移概率矩阵

 ${\mathit{\boldsymbol{P}}_1} = \left( {\begin{array}{*{20}{c}}{0.987}&{0.016}&{0.000}&{0.000}\\[6pt]{0.000}&{0.811}&{0.029}&{0.160}\\[6pt]{0.253}&{0.000}&{0.725}&{0.021}\\[6pt]{0.000}&{0.496}&{0.000}&{0.504}\end{array}} \right)$

 ${\mathit{\boldsymbol{P}}_2} = \left( {\begin{array}{*{20}{c}}{0.977}&{0.008}&{0.015}\\[6pt]{0.000}&{0.977}&{0.023}\\[6pt]{0.000}&{0.023}&{0.977}\end{array}} \right)$
3 三峡库区的碳排放预测

 ${{\tau _1} = 27.658 \,\,7{{\rm{e}}^{ - 0.021{\rm{ }}t}}}\times 10^{-6} \ {\rm t\cdot Yuan}$
 ${{\tau _2} = 165.924 6{{\rm{e}}^{ - 0.052{\rm{ }}t}}}\times 10^{-6} \ {\rm t \cdot Yuan}$
 ${{\tau _3} = 51.768 3{{\rm{e}}^{ - 0.041{\rm{ t}}}}}\times 10^{-6} \ {\rm t \cdot Yuan}$

 $\begin{split}& \tau \!=\! {g_1}^t{\tau _1} \!+\! {g_2}^t{\tau _2} \!+\! {g_3}^t{\tau _3} \!=\! \left( {\begin{array}{*{20}{c}}{{g_1}^t}&{{g_2}^t}&{{g_3}^t}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{\tau _1}}\\[6pt]{{\tau _2}}\\[6pt]{{\tau _3}}\end{array}} \right) = \\& \quad \quad \left( {\begin{array}{*{20}{c}}{{g_{\rm{1}}}^0}&{{g_{\rm{2}}}^0}&{{g_{\rm{3}}}^0}\end{array}} \right){\rm{ }}{\mathit{\boldsymbol{P}}_2}^t\left( {\begin{array}{*{20}{c}}{{\tau _1}}\\[6pt]{{\tau _2}}\\[6pt]{{\tau _3}}\end{array}} \right)\end{split}$ (10)

 $\begin{array}{*{20}{l}}{{P_2} = V\Lambda {V^{ - 1}} = \left( {\begin{array}{*{20}{c}}1 & {0.577\,\,4} & { - 0.210\,\,4}\\[6pt]0 & {0.577\,\,4} & { - 0.691\,\,3}\\[6pt]0 & {0.577\,\,4} & {0.691\,\,3}\end{array}} \right) \times }\\[24pt]\begin{array}{l}\quad \quad \left( {\begin{array}{*{20}{c}}{0.977} & {0} & {0}\\[6pt]0 & {1} & {0}\\[6pt]0 & {0} & {0.954}\end{array}} \right) \times \\[24pt]\quad \quad {\left( {\begin{array}{*{20}{c}}1 & {0.577\,\,4} & { - 0.210\,\,4}\\[6pt]0 & {0.577\,\,4} & { - 0.691\,\,3}\\[6pt]0 & {0.577\,\,4} & {0.691\,\,3}\end{array}} \right)^{ - 1}}\end{array}\end{array}$

 $\begin{array}{l}P_2^t = \left( {\begin{array}{*{20}{c}}1 & {0.577\,\,4} & { - 0.210\,\,4}\\[6pt]0 & {0.577\,\,4} & { - 0.691\,\,3}\\[6pt]0 & {0.577\,\,4} & {0.691\,\,3}\end{array}} \right) \times \\[24pt]\quad \quad {\left( {\begin{array}{*{20}{c}}{0.977} & 0 & 0\\[6pt]0 & 1 & 0\\[6pt]0 & 0 & {0.954}\end{array}} \right)^t} \times \\[24pt]\quad \quad {\left( {\begin{array}{*{20}{c}}1 & {0.577\,\,4} & { - 0.210\,\,4}\\[6pt]0 & {0.577\,\,4} & { - 0.691\,\,3}\\[6pt]0 & {0.577\,\,4} & {0.691\,\,3}\end{array}} \right)^{ - 1}}\end{array}$

 $\begin{split}{\tau _t} = & 0.092 \times {0.977^t} \times 27.6587{\kern 1pt} {\kern 1pt} {{\rm{e}}^{ - 0.021t}} + \\[7pt] & ( - 0.06 \times {0.977^t} + 0.5 + 0.061 \times {0.954^t}) \times \\[7pt] & 165.9246{\kern 1pt} {\kern 1pt} {{\rm{e}}^{ - 0.052t}} + ( - 0.032 \times {0.977^t} + 0.5 - \\[7pt] & 0.061 \times {0.954^t}) \times 51.7683{\kern 1pt} {\kern 1pt} {{\rm{e}}^{ - 0.041t}}\end{split}$ (11)

 \left\{ {\begin{aligned}& {{{\dot C}_t} = \frac{1}{\sigma }[(n\sigma - \rho - \delta ) + }\\[7pt]& \quad \quad {(1 - {\theta _t}{\tau _t}){{({A_0}\exp (vt))}^{\textstyle\frac{1}{\alpha }}}{\tau _t}^{\textstyle\frac{{1 - \alpha }}{\alpha }}{L_t}^{\textstyle\frac{\gamma }{\alpha }}]({C_t} - \bar C)}\\[7pt]& {C\left( 0 \right) = 2 \,\,\,\, 112.7}\end{aligned}} \right. (12)

 \left\{ {\begin{aligned}& {y' = f(x,y)}\\[6pt]& {y({x_0}) = {y_0}}\end{aligned}} \right.

 \left\{ \begin{aligned}& {y_{n + 1}} = {y_n} + {\displaystyle{h \over 6}}({K_1} + 2{K_2} + 2{K_3} + {K_4})\\[7pt]& {K_1} = f({x_n},{y_n})\\[7pt]& {K_2} = f({x_n} + {\displaystyle{h \over 2}},{y_n} + {\displaystyle{h \over 2}}{K_1})\\[7pt]& {K_3} = f({x_n} + {\displaystyle{h \over 2}},{y_n} + {\displaystyle{h \over 2}}{K_2})\\[7pt]& {K_4} = f({x_n} + h,{y_n} + h{K_3})\end{aligned} \right.

 $E = \tau \cdot Y$

 $\begin{split}e = E(\begin{array}{*{20}{c}}{{g_1}^{(t)}}&{{g_2}^{(t)}}&{{g_3}^{(t)}}&{{g_4}^{(t)}}\end{array})\left( {\begin{array}{*{20}{c}}{1.005\,\,2}\\[6pt]{0.753}\\[6pt]{0.617\,\,3}\\[6pt]0\end{array}} \right) = \\[20pt]E(\begin{array}{*{20}{c}}{{g_1}^{(0)}}&{{g_2}^{(0)}}&{{g_3}^{(0)}}&{{g_4}^{(0)}}\end{array})\mathit{\boldsymbol{P}}_1^t\left( {\begin{array}{*{20}{c}}{1.005\,\,2}\\[6pt]{0.753}\\[6pt]{0.617\,\,3}\\[6pt]0\end{array}} \right)\end{split}$ (13)

 图 1 能源使用量和碳排放量的趋势

4 结论

1)应用庞特里亚金最大值原理和马尔可夫过程研究三峡库区的碳排放问题，建立了污染问题的最优控制问题模型。

2)利用经典的龙格-库塔方法求解实际的三峡库区污染问题，将对三峡库区的环境保护研究发挥越来越重要的作用。

3)根据数值结果，得出能源使用量与碳的排放量有着直接关系，如果大量排放二氧化碳很可能导致全球的温室效应，然而刻意减少能源使用量就会影响三峡库区经济的增长。

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