Transcript Document

Tools for optimal coordination of
CCS, power industry capacity
expansion and bio energy raw
material production and harvesting
Peter Lohmander
Prof. Dr. SLU Sweden
[email protected]
2nd Annual EMISSIONS REDUCTION FORUM:
Establishing Effective CO2, NOx, SOx Mitigation
Strategies for the Power Industry, Marcus Evans,
29th & 30th September 2008, Madrid, Spain
Abstract (Peter Lohmander, Madrid 2008)
• If we plan the future carbon capture and storage
strategy, the power industry capacity expansion and bio
energy raw material production and harvesting
separately, we will never reach the best possible
solution.
• Alternative approaches to the optimal coordination
problem are presented.
• A new continuous time optimal control model is
described and analyzed.
• The optimal time path of resource extraction is explicitly
determined as a function of time dependent CO2 storage
revenues and other relevant parameters, including the
time dependent parameters of the energy demand
function.
e-mail:
[email protected]
References:
http://www.lohmander.com/Information/Ref.htm
Software:
http://www.lohmander.com/Program/Program.htm
Conferences:
http://www.lohmander.com/Kurser/Kurser.htm
• If we plan the future carbon capture and
storage strategy, the power industry
capacity expansion and bio energy raw
material production and harvesting
separately, we will never reach the best
possible solution.
• We may, with alternative definitions and
methods, investigate the system that includes
the three ”sectors” Forestry (F), the Forest
Products Industry (FPI) and the Forest Raw
Material Based Energy Industry (FRMBEI).
• The ambition is to find the dynamic strategy for
the management of this system that leads to the
best possible total economic surplus (present
value), when we simultaneously consider the
CO2 issue.
The ”Small Unit Raw Material
Perspective” and Optimization
• You may instantly calculate the
economically optimal decisions, from a
”small unit raw material perspective”, using
software available from the Internet:
• http://www.lohmander.com/program/Faust_Slut/InFaust3.html
• http://www.lohmander.com/program/Stump02/InStump022.html
Present Value (SEK/Hectare)
Number of Years from the Present
Figure 1.
The Present Value
EXP(- 0.03·t)·(20000 + 1000·t + 2000)
Web Software for Economic Optimization
from a Raw Material Perspective
= Stock level
= Growth
= Net Price
= Net Price Growth
= Land Value
= Interest Rate(%)
Optimize!
Optimal Results
Optimal Harvest Year
Optimal Present Value
250
200
150
A large part of the forest is much older
than the economically optimal harvest
age
100
50
0
02
310
11
-2
0
21
-3
0
31
-4
0
41
-5
0
51
-6
0
61
-7
0
71
-8
0
81
-9
91 0
-1
10 00
11
12 20
11
14 40
116
0
16
1-
tusentals hektar
Åldersklassfördelning i Gävleborgs län
(perioden 2001-2005)
Åldersklass (år)
Age distribution in the county of Gävleborg (2001-2005).
Thousands of hectares in different age classes (years).
• If we invest in CO2 capture and storage
technology (CCS) in the energy industry, it is
even more important to rapidly harvest the
old forest stands!
• Then, if we increase the area that is replanted
with more rapidly growing trees, we will absorb
more CO2 from the atmosphere, separate it and
store it permanently.
http://www.lohmander.com/co2ill2/co2ill2.htm
• A growing forest captures and stores
CO2 from the atmosphere.
• However, trees do not grow for
ever.
• Sooner or later, the growth level is
reduced, because the age of the trees
becomes too high and the competition
between neighbour trees too strong for
continued growth.
• Furthermore, at some age, trees die,
and once again release the stored CO2
to the atmosphere.
• The total amount of CO2 that
may be captured from the
atmosphere and permanently
stored is much higher if we, in
a repeated sequence, harvest
the forest, use some part of the
biomass for instant energy
production, capture and store
the CO2 and replant the area
again.
• With repeated harvesting an CCS, we
may, in the long run capture and store
any amount of CO2 from the
atmosphere!
• We do not only reduce the new emissions
of CO2 but we really make sure that the
CO2 level of the atmosphere is reduced!
• If you just leave the forest for ever,
without harvesting, you will never be
able to store more than what you find in
an old forest stand.
General continuous time optimal
control model of a forest resource,
comparative dynamics and CO2
storage consideration effects
Economic Valuation
of the Production of
Energy and Other
Industrial Products
Economic valuation of CO2
storage in the natural
resource
t2




 rt
2
max  J   e   f1  f 2t  x   k1  k2t  u  k3u  dt 
t1




The Total
Economic
Result
(Present Value)
The Stock Level
The ”Control” Level

x  f ( x, u, t ) ; x(t1 )  x1 , x(t2 )  x2
Initial stock level
The change of
the stock level
during a marginal
time interval
Terminal stock level
Lohmander, P., Optimal resource control
model & General continuous time
optimal control model of a forest
resource, comparative dynamics and
CO2 consideration effects, Seminar at
SLU, Umea, Sweden, 2008-09-18
http://www.lohmander.com/CM/CMLohmander.ppt
Software:
http://www.lohmander.com/CM/CM.htm
t2




 rt
2
max  J   e   f1  f 2t  x   k1  k2t  u  k3u  dt 
t1





x  g0  g1 x  g 2t  u
x(t1 )  x1; x(t2 )  x2
Optimal strategy and
dynamic effects of
the rate of interest
Optimal Objective Function Values
1800
Objective Value (Relevant Currency)
1600
1400
1200
1000
800
600
400
200
0
J_3%
J_5%
Alternative
J_7%
Optimal Stock Path
3500
Optimal Stock (Mm3sk)
3000
2500
2000
x_3%
x_5%
1500
x_7%
1000
500
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Control Path
160
140
Optimal Control (Mm3sk)
120
100
u_3%
80
u_5%
u_7%
60
40
20
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Shadow Price Path
250
Shadow Price (Relevant Currency)
200
150
SP_3%
SP_5%
SP_7%
100
50
0
0
5
10
15
20
Tim e (Years)
25
30
35
Comparisions with alternatives that
are not optimal:
30
N1   e
.05t
1000  3*106106 dt 
6
1.123229489·10
0
30
N2   e
0
.05t
1000  3*86 86 dt  9.914723644·10
5
Optimal strategy and
dynamic effects of
the slope of the demand function
Optimal Stock Path
3500
Optimal Stock (Mm3sk)
3000
2500
2000
x_-4
x_-3
1500
x_-2
1000
500
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Control Path
200
180
Optimal Control (Mm3sk)
160
140
120
u_-4
100
u_-3
u_-2
80
60
40
20
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Shadow Price Path
300
Shadow Price (Relevant Currency)
250
200
SP_-4
150
SP_-3
SP_-2
100
50
0
0
5
10
15
20
Tim e (Years)
25
30
35
Comparisions with two alternatives:
30
N3   e.05t 1000  2*106 106 dt  1.297807679·106
0
30
N4   e.05t 1000  2*(181  5* t )  (181  5* t ) dt  1.397224592·106
0
Optimal strategy and
dynamic effects of
the terminal condition
Optimal Objective Function Values
1240
Objective Value (Relevant Currency)
1220
1200
1180
1160
1140
1120
1100
J_2500
J_2800
Alternative
J_3100
Optimal Stock Path
3500
Optimal Stock (Mm3sk)
3000
2500
2000
x_3100
x_2500
1500
x_2800
1000
500
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Control Path
160
140
Optimal Control (Mm3sk)
120
100
u_3100
80
u_2500
u_2800
60
40
20
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Shadow Price Path
250
Shadow Price (Relevant Currency)
200
150
SP_3100
SP_2500
SP_2800
100
50
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal strategy and
dynamic effects of
continuous stock level valuation,
for instance because we
want to reward the storage
of CO2
Optimal Objective Function Values
1800
Objective Value (Relevant Currency)
1600
1400
1200
1000
800
600
400
200
0
J_f1=0
J_f1=5
Alternative
J_f1=10
Optimal Stock Path
3500
Optimal Stock (Mm3sk)
3000
2500
2000
x_f1=5
x_f1=0
1500
x_f1=10
1000
500
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Control Path
160
140
Optimal Control (Mm3sk)
120
100
u_f1=5
80
u_f1=0
u_f1=10
60
40
20
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Shadow Price Path
300
Shadow Price (Relevant Currency)
250
200
SP_f1=5
150
SP_f1=0
SP_f1=10
100
50
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal strategy and
dynamic effects of
continuous stock level valuation
in combination with variations of
the the slope of the demand
function
Optimal Objective Function Values
Objective Value (Relevant Currency)
2500
2000
1500
1000
500
0
J_f_-3
J_f_-2.3
Alternative
J_f_-1.6
Optimal Stock Path
3500
Optimal Stock (Mm3sk)
3000
2500
2000
x_f_-2.3
x_f_-3
1500
x_f_-1.6
1000
500
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Control Path
200
180
Optimal Control (Mm3sk)
160
140
120
u_f_-2.3
100
u_f_-3
u_f_-1.6
80
60
40
20
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Shadow Price Path
500
Shadow Price (Relevant Currency)
450
400
350
300
SP_f_-2.3
250
SP_f_-3
SP_f_-1.6
200
150
100
50
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal strategy and
dynamic effects of
the growth function
Optimal Objective Function Values
1250
Objective Value (Relevant Currency)
1245
1240
1235
1230
1225
1220
1215
1210
1205
1200
J_-.01
J_.001
Alternative
J_.01
Optimal Stock Path
3500
Optimal Stock (Mm3sk)
3000
2500
2000
x_.001
x_.01
1500
x_-.01
1000
500
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Control Path
180
160
Optimal Control (Mm3sk)
140
120
u_.001
100
u_.01
80
u_-.01
60
40
20
0
0
5
10
15
20
Tim e (Years)
25
30
35
Optimal Shadow Price Path
160
Shadow Price (Relevant Currency)
140
120
100
SP_.001
80
SP_.01
SP_-.01
60
40
20
0
0
5
10
15
20
Tim e (Years)
25
30
35
Total perspective I
Viirkesförrådets utveckling senda 1920-talet. Alla ägoslag 1
Trend for total standing volume since 1920, all land-use 1
Stock
3500
V0
Milj m3sk
3000
Döda träd Dead or
windthrown trees
2500
Lövträd Broad-leaved
2000
Gran Norway spruce
1500
1000
Tall Scots pine
500
Time
19
96
19
86
19
76
19
66
19
56
19
46
19
36
19
26
0
år year
0
1 Exkl. fjäll, fridlyst mark, militära impediment, bebyggd mark samt söt- och saltvatten.
Excl. high mountains, restricted military areas, urban land and water surfaces.
Milj. M3sk Millions cubic metre standing volume (stem volume over bark from stump to
tip)
h0 < g
h1 > g
h2 = g
t1
t2

max    z0 h0 e dt   z1h1e dt   z2 h2e dt
h1
0
 rt
t1
 rt
t2
 rt
t1
t2

max    z0 h0 e dt   z1h1e dt   z2 h2e dt
h1
 rt
0
V0  3000
V1  V0  ( g  h0 )t1
V  ?
h1  ?
(V1  V )
t2  t1 
(h1  g )
t1
 rt
t2
 rt
Derivations and parameters (I)
Rate of interest
r
0,06
Growth (now)
Growth (future)
g
g
t1
v0
106
106
5
3000
h0
z0
z1
86
1
1
z2
1
Stock (now)
Harvest (before t1)
Web software for Total Perspective I
http://www.lohmander.com/EF2008/EF2008.htm
Report (in Swedish) with appendix
(in English) describing Total
Perspective I and II
http://www.lohmander.com/EF2008/EF2008Lohmander.pdf
vfuture
h1
t2
totval
3000
106
inf
1680
3000
108
55
1703
3000
110
30
1718
3000
112
21,6
1727
3000
114
17,5
1732
3000
116
15
1735
Observations
• Even if we do not accept to decrease
the stock level below the very high
level of today, we should strongly
increase harvesting during a
considerable time interval.
• In this first derivation, the improved growth
rate in new plantations has not been
considered.
vfuture
h1
t2
totval
2500
106
inf
1680
2500
116
65
1800
2500
126
35
1886
2500
136
25
1939
2500
146
20
1973
2500
156
17
1997
Observations
• If we are prepared to adjust the stock level
to the stock level of the year 1985,
(approximately 2 500 Mm3sk), we should
increase harvesting very much during a
long time period.
• Then, the total economic value strongly
improves.
• In this derivation, the improved growth rate in
new plantations has not been considered.
Total perspective II
If harvested areas are
replanted with
more rapidly growing
seedlings,
the stock path becomes
strictly convex
(during time periods with
constant harvesting)
t1
V1  V0   ((1  s0t ) g0  s0tg1 )dt  h0t1
0
t1
t1
0
0
V1  V0   g 0 dt   s0 ( g1  g 0 )t dt  h0t1
s0 ( g1  g 0 )t
V1  V0  g 0t1 
 h0t1
2
2
1
Derivations and parameters (II)
Rate of interest
r
0,06
Growth (now)
Growth (new seedlings)
g0
g1
t1
v0
106
126
5
3000
h0
z0
z1
86
1
1
z2
ATKvot
1
80
Stock (now)
Harvest (before t1)
Web software for Total Perspective II
http://www.lohmander.com/EF2008/EFchange2008.htm
h1
t2
totval
vfuture
116
65
1806
3055
126
35
1906
2666
136
25
1966
2586
146
20
2004
2556
156
17
2029
2541
Conclusions
• If we plan the future carbon capture and storage strategy, the
power industry capacity expansion and bio energy raw material
production and harvesting separately, we will never reach the
best possible solution.
• The presented tools may be used to determine what should be
done.
• Peter Lohmander is organizing the conference stream “Optimal
Forest Management with Increasing Bioenergy Demand” within
The 23rd European Conference on Operational Research
(EURO XXIII), July 5-8, 2009, Bonn, Germany.
http://www.lohmander.com/Bonn2009/Bonn2009.pdf
• Let us continue our discussions and meet there!
My warmest ”Thanks” to E.ON Sweden
for economic support to the project
”Economic forest production with
consideration of the forest- and
energy- industries”!
Peter Lohmander
Professor of Forest Management and Economic Optimization, Swedish University of
Agricultural Sciences
http://www.Lohmander.com
[email protected]