Derivation of the principles of optimal expansion of bioenergy
based on forest resources in combination with fossil fuels, CCS
and combined heat and power production with consideration of
economics and global warming
Lecture by Professor Dr. Peter Lohmander, Swedish
University of Agricultural Sciences, SLU, Sweden,
http://www.Lohmander.com
[email protected]
Operations Research for the Future Forest Management,
University of Guilan, Iran, March 8-12, 2014
Saturday March 8, 10.00-12.00
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Derivation of the principles of optimal expansion of bioenergy based on forest
resources in combination with fossil fuels, CCS and combined heat and power
production with consideration of economics and global warming
• In the production of district heating, electricity and many other energy outputs, we may use fossil fuels and renewable inputs, in
different combinations. The optimal input mix is a function of technological options, prices of different inputs and costs of
alternative levels of environmental consequences. Even with presently existing technology in combined heat and power plants, it is
usually possible to reduce the amount of fossil fuels such as coal and strongly increase the level of forest based energy inputs. In
large parts of the world, such as Russian Federation, forest stocks are close to dynamic equilibria, in the sense that the net growth
(and net carbon uptake) is close to zero. If these forests will be partly harvested, the net growth can increase and a larger part of the
CO2 emitted from the CHP plants will be captured by the forests. Furthermore, with CCS, Carbon Capture and Storage, the rest of
the emitted CO2 can be permanently stored. The lecture incudes the definition of a general optimization model that can handle these
problems in a consistent way. The model is used to derive general principles of optimal decisions in this problem area via
comparative statics analysis with general functions.
• The lecture also includes case studies from different parts of the world. In several countries, it is presently profitable to replace coal
by forest inputs in CHP plants. In Norway and UK, CCS has been applied and a commercially attractive option during many years
and the physical potential is large. Carbon taxes on fossil fuels explain this development. With increasing carbon taxes in all parts of
the world, such developments could be expected everywhere. With increasing levels of forest inputs in combination with CCS, it is
possible to reduce the CO2 in the atmosphere and the global warming problem can be managed. Furthermore, international trade in
forest based energy can improve international relations, regional development and environmental conditions.
•
Professor Dr. Peter Lohmander, Swedish University of Agricultural Sciences, SLU, Sweden, http://www.Lohmander.com
[email protected]
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The Lohmander Energy, Forest, Fossil Fuels, CCS and Climate System Optimization Model
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The Lohmander Energy, Forest, Fossil Fuels, CCS and Climate System Optimization Model
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Fuel mix 1991 - 2006
Oil (BLACK)
Coal (DARK GREY)
Ruber (LIGHT GREY)
Forest chips
(ORANGE)
Waste wood
(PINK)
Waste (RED)
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The Lohmander Energy, Forest, Fossil Fuels, CCS and Climate System Optimization Model
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𝑴𝒂𝒙 𝒁
s.t.
= −𝒌𝝅 𝑪𝑭 𝑭 + 𝑪𝒉 𝒉 + 𝑪𝑺 𝑺
𝑭+𝒉 ≥𝑴
− 𝒌𝑮 𝟏 + 𝜶𝑭 𝑭 + 𝜶𝒉 𝒉 − 𝜶𝑺 𝑺
16
The model does not include absolute constraints
on the availability of fossil fuels.
Motive:
Such constraints are assumed not to be binding
in the optimal total solution.
Hydraulic fracturing also called fracturing
and fracking has recently shown that
there are very large quantitites of fossil
fuels available in the world.
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Analysis:
𝑀𝑎𝑥 𝑍 = −𝑘𝜋 𝐶𝐹 𝐹 + 𝐶ℎ ℎ + 𝐶𝑆 𝑆
− 𝑘𝐺 1 + 𝛼𝐹 𝐹 + 𝛼ℎ ℎ − 𝛼𝑆 𝑆
s.t.
𝐹+ℎ ≥𝑀
19
Parameter assumptions:
𝑘𝜋 > 0
𝑘𝐺 > 0
𝛼𝐹 > 0
𝛼ℎ > 0
0 < 𝛼𝑆 < 1
𝛼ℎ < 1 + 𝛼𝐹
20
This constraint is not binding:
𝑆 <𝐹+ℎ
21
Cost function assumptions:
𝐶𝐹′ > 0
𝐶𝐹′′ > 0
𝐶ℎ′ > 0
𝐶ℎ′′ > 0
𝐶𝑆′ > 0
𝐶𝑆′′ > 0
22
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24
25
26
27
28
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The classical dynamic natural resource model:
𝒅𝒙
𝒙
= 𝒔𝒙 𝟏 −
𝒅𝒕
𝑲
𝑲
𝒙(𝒕) =
𝟏 + 𝑪𝒆−𝒔𝒕
𝒍𝒊𝒎 𝒙(𝒕) = 𝑲
𝒕→∞
(𝒔>𝟎)
30
−𝒔𝒕
𝑲𝒔𝑪𝒆
𝒅𝒙
=
𝒅𝒕
𝟏 + 𝑪𝒆−𝒔𝒕
𝟐
𝒅𝒙
𝒍𝒊𝒎
=𝟎
𝒕→∞ 𝒅𝒕
(𝒔>𝟎)
Such forests do not change the
carbon levels of the atmosphere.
31
The Lagrange function, L, 𝐢𝐬:
= −𝑘𝜋 𝐶𝐹 𝐹 + 𝐶ℎ ℎ + 𝐶𝑆 𝑆
− 𝑘𝐺 1 + 𝛼𝐹 𝐹 + 𝛼ℎ ℎ − 𝛼𝑆 𝑆
+𝜆 𝐹 + ℎ − 𝑀
32
The Kuhn Tucker conditions are:
𝑭 ≥ 𝟎; 𝒉 ≥ 𝟎; 𝑺 ≥ 𝟎
𝒅𝑳
𝒅𝑳
𝒅𝑳
≤ 𝟎;
≤ 𝟎;
≤𝟎
𝒅𝑭
𝒅𝒉
𝒅𝑺
𝒅𝑳
𝒅𝑳
𝒅𝑳
𝑭
= 𝟎; 𝒉
= 𝟎; 𝑺
=𝟎
𝒅𝑭
𝒅𝒉
𝒅𝑺
𝝀≥𝟎
𝒅𝑳
≥𝟎
𝒅𝝀
𝒅𝑳
𝝀
=𝟎
𝒅𝝀
33
Here, we have four of the constraints:
𝒅𝑳
=𝑭+𝒉−𝑴
𝒅𝝀
𝒅𝑳
= −𝒌𝝅 𝑪′𝑭 𝑭 − 𝒌𝑮 𝟏 + 𝜶𝑭 + 𝝀
𝒅𝑭
𝒅𝑳
′
= −𝒌𝝅 𝑪𝒉 𝒉 − 𝒌𝑮 𝜶𝒉 + 𝝀
𝒅𝒉
𝒅𝑳
′
= −𝒌𝝅 𝑪𝑺 𝑺 + 𝒌𝑮 𝜶𝑺
𝒅𝑺
≥𝟎
≤𝟎
≤𝟎
≤𝟎
34
Assumption:
The optimal solution is an interior solution.
𝐹 > 0; ℎ > 0; 𝑆 > 0; 𝜆 > 0
As a consequence, we know that:
𝑑𝐿
𝑑𝐿
𝑑𝐿
𝑑𝐿
= 0;
= 0;
= 0;
=0
𝑑𝜆
𝑑𝐹
𝑑ℎ
𝑑𝑆
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This means that the following equation system
should be solved:
𝑑𝐿
=𝐹+ℎ−𝑀
𝑑𝜆
𝑑𝐿
′
= −𝑘𝜋 𝐶𝐹 𝐹 − 𝑘𝐺 1 + 𝛼𝐹 + 𝜆
𝑑𝐹
𝑑𝐿
′
= −𝑘𝜋 𝐶ℎ ℎ − 𝑘𝐺 𝛼ℎ + 𝜆
𝑑ℎ
𝑑𝐿
= −𝑘𝜋 𝐶𝑆′ 𝑆 + 𝑘𝐺 𝛼𝑆
𝑑𝑆
=0
=0
=0
=0
36
The system with four equations and four endogenous
variables is partly separable.
It may be split into one system with tree equations
and three endogenous variables (𝑭, 𝒉 𝒂𝒏𝒅 𝝀) and a
separate equation with only one endogenous
variable, 𝑺 .
Stars denote optimal values.
37
First, we investigate 𝑺∗ .
𝒅𝑳
= −𝒌𝝅 𝑪′𝑺 𝑺 + 𝒌𝑮 𝜶𝑺
𝒅𝑺
𝒅𝑳
=𝟎
𝒅𝑺
≤𝟎
𝒌
𝜶
𝑮
𝑺
′
⇒ 𝑪𝑺 𝑺 =
𝒌𝝅
38
We differentiate the first order optimum condition:
′
∗
−𝒌𝝅 𝑪′′
𝑺
𝒅𝑺
−
𝑪
𝑺
𝑺 𝑺 𝒅𝒌𝝅 + 𝜶𝑺 𝒅𝒌𝑮 + 𝒌𝑮 𝒅𝜶𝑺 = 𝟎
𝟏
′
𝒅𝑺 =
−𝑪
𝑺 𝑺 𝒅𝒌𝝅 + 𝜶𝑺 𝒅𝒌𝑮 + 𝒌𝑮 𝒅𝜶𝑺
′′
𝒌𝝅 𝑪𝑺 𝑺
∗
39
∗
The derivatives of 𝑺 with respect to the
parameters are the following:
∗
𝒅𝑺
=
𝒅𝒌𝝅
′
−𝑪𝑺 𝑺
𝒌𝝅 𝑪′′
𝑺 𝑺
<𝟎
𝒅𝑺∗
𝜶𝑺
=
>𝟎
′′
𝒅𝒌𝑮 𝒌𝝅 𝑪𝑺 𝑺
𝒅𝑺∗
𝒌𝑮
=
>𝟎
′′
𝒅𝜶𝑺 𝒌𝝅 𝑪𝑺 𝑺
40
∗
𝒅𝑺
=
𝒅𝒌𝝅
′
−𝑪𝑺 𝑺
𝒌𝝅 𝑪′′
𝑺 𝑺
<𝟎
𝒅𝑺∗
𝜶𝑺
=
>𝟎
′′
𝒅𝒌𝑮 𝒌𝝅 𝑪𝑺 𝑺
𝒅𝑺∗
𝒌𝑮
=
>𝟎
′′
𝒅𝜶𝑺 𝒌𝝅 𝑪𝑺 𝑺
𝑴𝒂𝒙 𝒁
s.t.
= −𝒌𝝅 𝑪𝑭 𝑭 + 𝑪𝒉 𝒉 + 𝑪𝑺 𝑺
𝑭+𝒉 ≥𝑴
− 𝒌𝑮 𝟏 + 𝜶𝑭 𝑭 + 𝜶𝒉 𝒉 − 𝜶𝑺 𝑺
41
Next, we investigate 𝑭∗ , 𝒉∗ 𝒂𝒏𝒅 𝝀∗ .
The first order optimum conditions are found from this
three dimensional equation system:
𝒅𝑳
= 𝑭+𝒉−𝑴
=𝟎
𝒅𝝀
𝒅𝑳
= −𝒌𝝅 𝑪′𝑭 𝑭 − 𝒌𝑮 𝟏 + 𝜶𝑭 + 𝝀 = 𝟎
𝒅𝑭
𝒅𝑳
′
= −𝒌𝝅 𝑪𝒉 𝒉 − 𝒌𝑮 𝜶𝒉 + 𝝀
=𝟎
𝒅𝒉
42
We differentiate the equations:
𝟏
−𝒌𝝅 𝑪′′
𝑭
𝟎
𝟏
𝟎
−𝒌𝝅 𝑪′′
𝒉
𝒅𝑴
𝟎 𝒅𝑭∗
′
∗
𝑪
𝟏 𝒅𝒉 = 𝑭 𝒅𝒌𝝅 + 𝟏 + 𝜶𝑭 𝒅𝒌𝑮 + 𝒌𝑮 𝒅𝜶𝑭
′
∗
𝑪
𝟏 𝒅𝝀
𝒉 𝒅𝒌𝝅 + 𝜶𝒉 𝒅𝒌𝑮 + 𝒌𝑮 𝒅𝜶𝒉
43
When we apply Cramer’s rule, we need
to know 𝑫 .
𝑫 =
𝟏
−𝒌𝝅 𝑪′′
𝑭
𝟎
𝟏
𝟎
′′
−𝒌𝝅 𝑪𝒉
′′
𝒌𝝅 𝑪𝒉
+ 𝑪′′
𝑭 >𝟎
𝑫 =
𝟎
𝟏
𝟏
44
The derivatives of 𝑭∗ , 𝒉∗ 𝒂𝒏𝒅 𝝀∗ with respect to M are
determined via Cramer’s rule:
𝟏
𝟎
𝟎
∗
𝒅𝑭
=
𝒅𝑴
∗
𝟏
𝟎
′′
−𝒌𝝅 𝑪𝒉
𝑫
𝟎
𝟏
𝟏
′′
𝑪𝒉
𝒅𝑭
= ′′
′′ > 𝟎
𝒅𝑴 𝑪𝒉 + 𝑪𝑭
45
𝟏
′′
−𝒌𝝅 𝑪𝑭
𝒅𝒉∗
𝟎
=
𝒅𝑴
𝑫
𝟏 𝟎
𝟎 𝟏
𝟎 𝟏
𝒅𝒉∗
𝑪′′
𝑭
= ′′
′′ > 𝟎
𝒅𝑴 𝑪𝒉 + 𝑪𝑭
46
∗
𝒅𝝀
=
𝒅𝑴
𝟏
−𝒌𝝅 𝑪′′
𝑭
𝟎
𝟏
𝟎
−𝒌𝝅 𝑪′′
𝒉
𝟏
𝟎
𝟎
𝑫
′′
𝒅𝝀∗ 𝒌𝝅 𝑪′′
𝑪
𝑭 𝒉
= ′′
′′ > 𝟎
𝒅𝑴 𝑪𝒉 + 𝑪𝑭
47
𝒅𝑭∗
𝑪′′
𝒉
= ′′
′′ > 𝟎
𝒅𝑴 𝑪𝒉 + 𝑪𝑭
𝒅𝒉∗
𝑪′′
𝑭
= ′′
′′ > 𝟎
𝒅𝑴 𝑪𝒉 + 𝑪𝑭
∗
𝒅𝝀
=
𝒅𝑴
𝑴𝒂𝒙 𝒁
s.t.
= −𝒌𝝅 𝑪𝑭 𝑭 + 𝑪𝒉 𝒉 + 𝑪𝑺 𝑺
𝑭+𝒉 ≥𝑴
′′ ′′
𝒌𝝅 𝑪𝑭 𝑪𝒉
′′
𝑪′′
+
𝑪
𝑭
𝒉
>𝟎
− 𝒌𝑮 𝟏 + 𝜶𝑭 𝑭 + 𝜶𝒉 𝒉 − 𝜶𝑺 𝑺
48
The derivatives of 𝑭∗ , 𝒉∗ 𝒂𝒏𝒅 𝝀∗ with respect to 𝒌𝑮 are:
∗
𝒅𝑭
=
𝒅𝒌𝑮
𝟎
𝟏
(𝟏 + 𝜶𝑭 )
𝟎
′′
𝜶𝒉
−𝒌𝝅 𝑪𝒉
𝟎
𝟏
𝟏
𝑫
∗
𝒅𝑭
𝜶𝒉 − (𝟏 + 𝜶𝑭 )
=
′′
′′ < 𝟎
𝒅𝒌𝑮
𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
49
𝒅𝒉∗
=
𝒅𝒌𝑮
𝟏
−𝒌𝝅 𝑪′′
𝑭
𝟎
𝟎
(𝟏 + 𝜶𝑭 )
𝜶𝒉
𝟎
𝟏
𝟏
𝑫
𝒅𝒉∗
𝟏 + 𝜶𝑭 − 𝜶𝒉
=
>𝟎
′′
′′
𝒅𝒌𝑮
𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
50
𝟏
′′
−𝒌𝝅 𝑪𝑭
𝟎
𝒅𝝀∗
=
𝒅𝒌𝑮
𝟏
𝟎
′′
−𝒌𝝅 𝑪𝒉
𝟎
𝟏 + 𝜶𝑭
𝜶𝒉
𝑫
∗
𝒅𝝀
=
𝒅𝒌𝑮
′′
𝑪𝒉
𝟏 + 𝜶𝑭 +
′′
′′
𝑪𝒉 + 𝑪𝑭
′′ 𝜶
𝑪𝑭 𝒉
>𝟎
51
𝒅𝑭∗ 𝜶𝒉 − (𝟏 + 𝜶𝑭 )
=
′′
′′ < 𝟎
𝒅𝒌𝑮
𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
𝒅𝒉∗
𝟏 + 𝜶𝑭 − 𝜶𝒉
=
>𝟎
′′
′′
𝒅𝒌𝑮
𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
′′ 𝜶
𝒅𝝀∗ 𝑪′′
𝟏
+
𝜶
+
𝑪
𝑭
𝑭 𝒉
𝒉
=
>𝟎
′′
′′
𝒅𝒌𝑮
𝑪𝒉 + 𝑪𝑭
𝑴𝒂𝒙 𝒁
s.t.
= −𝒌𝝅 𝑪𝑭 𝑭 + 𝑪𝒉 𝒉 + 𝑪𝑺 𝑺
𝑭+𝒉 ≥𝑴
− 𝒌𝑮 𝟏 + 𝜶𝑭 𝑭 + 𝜶𝒉 𝒉 − 𝜶𝑺 𝑺
52
The derivatives of 𝑭∗ , 𝒉∗ 𝒂𝒏𝒅 𝝀∗ with respect to 𝒌𝝅 :
𝒅𝑭∗
=
𝒅𝒌𝝅
𝟎
𝑪′𝑭
𝑪′𝒉
𝟏
𝟎
−𝒌𝝅 𝑪′′
𝒉
𝟎
𝟏
𝟏
𝑫
>
𝒅𝑭
−
=
′′
′′ = 𝟎
𝒅𝒌𝝅 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
<
∗
′
𝑪𝒉
𝑪′𝑭
53
Observation:
The sign of
𝒅𝑭∗
𝒅𝒌𝝅
is expected to change over time, since
fossil fuels in the long run become more scarce and more
costly to extract.
In most cases, it is assumed that
𝒅𝑭∗
𝒅𝒌𝝅
2014. At some future point in time,
points in time
𝒅𝑭∗
𝒅𝒌𝝅
> 𝟎 in the year
𝒅𝑭∗
𝒅𝒌𝝅
= 𝟎 and at later
<𝟎.
54
𝒅𝒉∗
=
𝒅𝒌𝝅
𝟏
′′
−𝒌𝝅 𝑪𝑭
𝟎
𝟎
′
𝑪𝑭
′
𝑪𝒉
𝟎
𝟏
𝟏
𝑫
>
𝒅𝒉∗
𝑪′𝑭 − 𝑪′𝒉
=
′′
′′ = 𝟎
𝒅𝒌𝝅 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
<
55
Observation:
The sign of
𝒅𝒉∗
𝒅𝒌𝝅
is expected to change over time, since
fossil fuels in the long run become more scarce and
more costly to extract.
In most cases, it is assumed that
𝒅𝒉∗
𝒅𝒌𝝅
2014. At some future point in time,
later points in time
𝒅𝒉∗
𝒅𝒌𝝅
< 𝟎 in the year
𝒅𝒉∗
𝒅𝒌𝝅
= 𝟎 and at
>𝟎.
56
𝟏
−𝒌𝝅 𝑪′′
𝑭
𝟎
∗
𝒅𝝀
=
𝒅𝒌𝝅
𝟏
𝟎
−𝒌𝝅 𝑪′′
𝒉
𝟎
𝑪′𝑭
𝑪′𝒉
𝑫
∗
𝒅𝝀
=
𝒅𝒌𝝅
′′ ′
𝑪𝒉 𝑪𝑭
′′
𝑪𝒉
+
+
′′ ′
𝑪𝑭 𝑪𝒉
′′
𝑪𝑭
>𝟎
57
>
𝒅𝑭∗
𝑪′𝒉 − 𝑪′𝑭
=
′′
′′ = 𝟎
𝒅𝒌𝝅 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
<
∗
′
′
𝑪𝑭 − 𝑪𝒉
′′
𝒌𝝅 (𝑪′′
+
𝑪
𝑭)
𝒉
∗
′′ ′
𝑪𝒉 𝑪𝑭
𝑪′′
𝒉
𝒅𝒉
=
𝒅𝒌𝝅
𝒅𝝀
=
𝒅𝒌𝝅
𝑴𝒂𝒙 𝒁
s.t.
= −𝒌𝝅 𝑪𝑭 𝑭 + 𝑪𝒉 𝒉 + 𝑪𝑺 𝑺
𝑭+𝒉 ≥𝑴
+
+
′′ ′
𝑪𝑭 𝑪𝒉
𝑪′′
𝑭
>
= 𝟎
<
>𝟎
− 𝒌𝑮 𝟏 + 𝜶𝑭 𝑭 + 𝜶𝒉 𝒉 − 𝜶𝑺 𝑺
58
The derivatives of 𝑭∗ , 𝒉∗ 𝒂𝒏𝒅 𝝀∗ with respect to 𝜶𝑭 are:
𝒅𝑭∗
=
𝒅𝜶𝑭
𝟎
𝒌𝑮
𝟎
𝟏
𝟎
′′
−𝒌𝝅 𝑪𝒉
𝟎
𝟏
𝟏
𝑫
𝒅𝑭∗
−𝒌𝑮
=
′′
′′ < 𝟎
𝒅𝜶𝑭 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
59
𝟏
′′
−𝒌𝝅 𝑪𝑭
𝒅𝒉∗
𝟎
=
𝒅𝜶𝑭
𝑫
𝟎
𝒌𝑮
𝟎
𝟎
𝟏
𝟏
𝒅𝒉∗
𝒌𝑮
=
′′
′′ > 𝟎
𝒅𝜶𝑭 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
60
∗
𝒅𝝀
=
𝒅𝜶𝑭
𝟏
−𝒌𝝅 𝑪′′
𝑭
𝟎
𝟏
𝟎
−𝒌𝝅 𝑪′′
𝒉
𝟎
𝒌𝑮
𝟎
𝑫
𝒅𝝀∗
𝑪′′
𝒉 𝒌𝑮
= ′′
′′ > 𝟎
𝒅𝜶𝑭 𝑪𝒉 + 𝑪𝑭
61
𝒅𝑭∗
−𝒌𝑮
=
′′
′′ < 𝟎
𝒅𝜶𝑭 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
𝒅𝒉∗
𝒌𝑮
=
′′
′′ > 𝟎
𝒅𝜶𝑭 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
𝒅𝝀∗
𝑪′′
𝒉 𝒌𝑮
= ′′
′′ > 𝟎
𝒅𝜶𝑭 𝑪𝒉 + 𝑪𝑭
𝑴𝒂𝒙 𝒁
s.t.
= −𝒌𝝅 𝑪𝑭 𝑭 + 𝑪𝒉 𝒉 + 𝑪𝑺 𝑺
𝑭+𝒉 ≥𝑴
− 𝒌𝑮 𝟏 + 𝜶𝑭 𝑭 + 𝜶𝒉 𝒉 − 𝜶𝑺 𝑺
62
The derivatives of 𝑭∗ , 𝒉∗ 𝒂𝒏𝒅 𝝀∗ with respect to 𝜶𝒉 are:
∗
𝒅𝑭
=
𝒅𝜶𝒉
𝟎
𝟎
𝒌𝑮
𝟏
𝟎
′′
−𝒌𝝅 𝑪𝒉
𝑫
𝟎
𝟏
𝟏
∗
𝒅𝑭
𝒌𝑮
=
′′
′′ > 𝟎
𝒅𝜶𝒉 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
63
∗
𝒅𝒉
=
𝒅𝜶𝒉
𝟏
′′
−𝒌𝝅 𝑪𝑭
𝟎
𝟎
𝟎
𝒌𝑮
𝟎
𝟏
𝟏
𝑫
∗
𝒅𝒉
−𝒌𝑮
=
′′
′′ < 𝟎
𝒅𝜶𝒉 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
64
𝒅𝝀∗
=
𝒅𝜶𝒉
𝟏
′′
−𝒌𝝅 𝑪𝑭
𝟎
𝟏
𝟎
′′
−𝒌𝝅 𝑪𝒉
𝟎
𝟎
𝒌𝑮
𝑫
∗
𝒅𝝀
=
𝒅𝜶𝒉
′′
𝑪𝑭 𝒌𝑮
′′
′′
𝑪𝒉 + 𝑪𝑭
>𝟎
65
𝒅𝑭∗
𝒌𝑮
=
′′
′′ > 𝟎
𝒅𝜶𝒉 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
𝒅𝒉∗
−𝒌𝑮
=
′′
′′ < 𝟎
𝒅𝜶𝒉 𝒌𝝅 (𝑪𝒉 + 𝑪𝑭 )
∗
𝒅𝝀
=
𝒅𝜶𝒉
𝑴𝒂𝒙 𝒁
s.t.
= −𝒌𝝅 𝑪𝑭 𝑭 + 𝑪𝒉 𝒉 + 𝑪𝑺 𝑺
𝑭+𝒉 ≥𝑴
′′
𝑪𝑭 𝒌𝑮
′′
′′
𝑪𝒉 + 𝑪𝑭
>𝟎
− 𝒌𝑮 𝟏 + 𝜶𝑭 𝑭 + 𝜶𝒉 𝒉 − 𝜶𝑺 𝑺
66
67
68
69
70
71
References
http://www.lohmander.com/Information/Ref.htm
Lohmander, P. (Chair) TRACK 1: Global Bioenergy, Economy
and Policy, BIT’s 2nd World Congress on Bioenergy,
Xi'an, China, April 25-28, 2012
http://www.Lohmander.com/Schedule_China_2012.pdf
http://www.Lohmander.com/PLWCBE12intro.ppt
Lohmander, P., Economic optimization of sustainable energy
systems based on forest resources with consideration of the
global warming problem: International perspectives,
BIT’s 2nd World Congress on Bioenergy,
Xi'an, China, April 25-28, 2012
http://www.Lohmander.com/WorldCongress12_PL.pdf
http://www.Lohmander.com/WorldCongress12_PL.doc
http://www.Lohmander.com/PLWCBE12.ppt
72
Lohmander, P., Lectures at Shandong Agricultural University,
April 29 - May 1, 2012, Jinan, China,
http://www.Lohmander.com/PLJinan12.ppt
Lohmander, P. (Chair) TRACK: Finance, Strategic Planning,
Industrialization and Commercialization, BIT’s 1st World Congress on Bioenergy,
Dalian World Expo Center,
Dalian, China, April 25-30, 2011
http://www.bitlifesciences.com/wcbe2011/fullprogram_track5.asp
http://www.lohmander.com/PRChina11/Track_WorldCongress11_PL.pdf
http://www.lohmander.com/ChinaPic11/Track5.ppt
Lohmander, P., Economic forest management with consideration
of the forest and energy industries, BIT’s 1st World Congress on Bioenergy, Dalian
World Expo Center, Dalian, China, April 25-30, 2011
http://www.lohmander.com/PRChina11/WorldCongress11_PL.pdf
http://www.lohmander.com/ChinaPic11/LohmanderTalk.ppt
73
CONCLUSIONS:
With increasing levels of forest inputs in combination with CCS, it is
possible to reduce the CO2 in the atmosphere and the global warming
problem can be managed.
Furthermore, international trade in forest based energy can improve
international relations, regional development and environmental
conditions.
All suggestions concerning future cooperation projects are welcome!
Thank you!
Peter Lohmander
74
APPENDIX
Observation:
The classical dynamic natural resource model:
𝑚 + 𝑚𝛾 + 𝑛 𝑥 = 1∀𝑥
𝑑𝑥
𝑥
= 𝑠𝑥 1 −
𝑑𝑡
𝐾
𝑚=1
𝑚𝛾 + 𝑛 = 0
𝑑𝑥
𝑠
= 𝑠𝑥 − 𝑥 2
𝑑𝑡
𝐾
𝑑𝑥
= 𝑠𝑥 1 + 𝛾𝑥 ,
𝑑𝑡
1
𝑑𝑥 = 𝑠𝑑𝑡
𝑥 1 + 𝛾𝑥
1
𝑚
𝑛
= +
𝑥 1 + 𝛾𝑥
𝑥 1 + 𝛾𝑥
𝑚
𝑛
1 + 𝛾𝑥 𝑚 + 𝑛𝑥
+
=
𝑥 1 + 𝛾𝑥
𝑥 1 + 𝛾𝑥
1 + 𝛾𝑥 𝑚 + 𝑛𝑥
1
=
𝑥 1 + 𝛾𝑥
𝑥 1 + 𝛾𝑥
Solution:
𝛾 = −𝐾
−1
𝑚, 𝑛 = (1, −𝛾)
1
𝑑𝑥 = 𝑠𝑑𝑡
𝑥 1 + 𝛾𝑥
𝑚
𝑛
+
𝑑𝑥 = 𝑠𝑑𝑡
𝑥 1 + 𝛾𝑥
1
𝛾
−
𝑑𝑥 = 𝑠𝑑𝑡
𝑥 1 + 𝛾𝑥
𝑑𝐿𝑁(1 + 𝛾𝑥)
𝛾
=
𝑑𝑥
1 + 𝛾𝑥
1 + 𝛾𝑥 𝑚 + 𝑛𝑥 = 1
𝑚 + 𝑚𝛾𝑥 + 𝑛𝑥 = 1
𝑚 + 𝑚𝛾 + 𝑛 𝑥 = 1
75
1
𝛾
𝑑𝑥 −
𝑑𝑥 = 𝑠𝑑𝑡
𝑥
1 + 𝛾𝑥
1
𝑑𝑥 −
𝑥
𝛾
𝑑𝑥 =
1 + 𝛾𝑥
𝑠𝑑𝑡
𝐿𝑁 𝑥 − 𝐿𝑁 1 + 𝛾𝑥 = 𝑠𝑡 + 𝐶1
𝑥
𝐿𝑁
= 𝑠𝑡 + 𝐶1
1 + 𝛾𝑥
𝑥
𝐿𝑁
𝑒 1+𝛾𝑥
𝑥(𝑡) =
lim 𝑥(𝑡) = 𝐾
𝑡→∞
(𝑠>0)
𝑑𝑥
𝐾𝑠𝐶𝑒 −𝑠𝑡
=
𝑑𝑡
1 + 𝐶𝑒 −𝑠𝑡
lim
= 𝑒 𝑠𝑡+𝐶1
𝐾
1 + 𝐶𝑒 −𝑠𝑡
𝑡→∞
(𝑠>0)
2
𝑑𝑥
=0
𝑑𝑡
𝑥
= 𝐶2 𝑒 𝑠𝑡
1 + 𝛾𝑥
𝑥 = 1 + 𝛾𝑥 𝐶2 𝑒 𝑠𝑡
𝑥 1 − 𝐶2 𝛾𝑒 𝑠𝑡 = 𝐶2 𝑒 𝑠𝑡
𝐶2 𝑒 𝑠𝑡
𝑥=
1 − 𝐶2 𝛾𝑒 𝑠𝑡
𝑥=
1
1 −𝑠𝑡
𝑒
−𝛾
𝐶2
𝛾 = −𝐾 −1
𝑥=
1
1 −𝑠𝑡 1
+𝐾
𝐶2 𝑒
76