1

Optimal Economic Decision Rules in the Biomass Supply

Chain with CO2 Considerations 

Peter LohmanderProfessor Dr., SUAS, Umea, SE-90183, Sweden

Peter@Lohmander.com

ALIO-INFORMS International 2010Buenos Aires, Monday, June 07, 08:30 - 10:00

2

Optimal Economic Decision Rules in the Biomass Supply Chain with CO2 Considerations 

Peter Lohmander

Abstract: Decisions in the biomass supply chain influence the size of the

renewable energy feedstock. Indirectly, the use of fossil fuels, CO2 uptake from the atmosphere, and emissions of CO2 to the atmosphere are affected. CCS, carbon capture and storage, is one method to limit total CO2 emissions. The total decision problem of this system is defined and general economic decision rules are derived. In typical situations, a unique global cost minimum can be obtained.

3

The role of the forest?

• The best way to reduce the CO2 in the atmosphere may be to increase harvesting of the presently existing forests (!), to produce energy with CCS and to increase forest production in the new forest generations.

• We capture and store more CO2!

4Permanent storage of CO2

Coal mine

Oil field

Natural gas

CCS, Carbon Capture and Storage, has alreadybecome the main future emissionreduction method of the fossile fuel energy industry

Energy plant with CO2 capture and separation

5

CO2

Permanent storage of CO2

How to reduce the CO2 level in the atmosphere,

not only to decrease the emission of CO2

Energy plant with CO2 capture and separation

6

The role of the forest in the CO2 and energy system

• The following six pictures show that it is necessary to intensify the use of the forest for energy production in combination with CCS in order to reduce the CO2 in atmosphere!

• All figures and graphs have been simplified as much as possible, keeping the big picture correct, in order to make the main point obvious.

• In all cases, we keep the total energy production constant.

7

CO2

Permanent storage of CO2

Coal, oil, gas

The present situation.

4

1

5

0

1

CO2 increase in the atmosphere:

5-1 = 4

8

CO2

Permanent storage of CO2

Coal, oil, gas

If we do not use the forest for energy production but use it as a carbon sink. Before the forest has reached equilibrium, this happens:

5

5

0

1

CO2 increase in the atmosphere:

5-1 = 4

9

CO2

Permanent storage of CO2

Coal, oil, gas

If we do not use the forest for energy production but use it as a carbon sink. When the forest has reached equilibrium, this happens:

5

1

5

0

1

CO2 increase in the atmosphere:

5+1-1 = 5

10

CO2

Permanent storage of CO2

Coal, oil, gas

If we use CCS with 80% efficiency and let the forest grow until it reaches equilibrium.

5

11

4

1

CO2 increase in the atmosphere:

1+1-1 = 1

11

CO2

Permanent storage of CO2

Coal, oil, gas

If we use CCS with 80% efficiency and use the forest with ”traditional” low intensity harvesting and silviculture.

4

1

1

4

1

CO2 increase in the atmosphere:

1-1 = 0

12

CO2

Permanent storage of CO2

Coal, oil, gas

If we use CCS with 80% efficiency and use the forest with increased harvesting and high intensity silviculture.

3

2

1

4

2

CO2 ”increase” in the atmosphere:

1-2 = -1

13

General conclusions:

• The best way to reduce the CO2 in the atmosphere may be to increase harvesting of the presently existing forests (!), to produce energy with CCS and to increase forest production in the new forest generations.

• We capture and store more CO2!

14

15

,min ( ) ( ) ( ) ( )

. .

u f a wu wC C u C f C a v C w

s t

u f K f K u

w a K a K w

v u

16

,min ( ) ( ) ( ) ( )u f a wu wC C u C K u C K u w C w

17

18

( ) ( ) ( ) 0

( ) ( ) 0

u f a

a w

CC u C K u C K u w

uC

C K u w C ww

u f a

a w

C C C

C C

19

Observation 1

• In optimum, the marginal cost for forest biomass utilization equals the marginal cost of fossil fuel utilization plus the marginal cost of global warming.

20

Observation 2

• In optimum, the marginal cost of global warming equals the marginal cost of CCS.

21

2

2

2

2

2

2

( ) ( ) ( )

( )

( )

( ) ( )

u f a

a

a

a w

CC u C K u C K u w

u

CC K u w

u w

CC K u w

w u

CC K u w C w

w

22

Second order minimum conditions:

2

20

C

u

2 2

2

2 2

2

0

C C

u u w

C C

w u w

23

0u f aC C C

0u f a a

a a w

C C C C

C C C

24

0u f a u f aC C C C C C

2u f a a

u f a a w a

a a w

C C C CC C C C C C

C C C

25

2

2 2

0

u f a a w a

u a u w f a f w a a w a

u a u w f a f w a w

C C C C C C

C C C C C C C C C C C C

C C C C C C C C C C

26

Observations of the first and second order conditions:

( , )

0

0

( , ) 0

x

y

x y

f x y

f

f

df x y f dx f dy

27

2 2 2( , ) ( ) ( )xx xy yx yyd f x y f dx f dxdy f dydx f dy

xy yxf f

2 2 2

2 2 2

( , ) ( ) ( )

( , ) 2

xx xy yx yyd f x y f dx f dxdy f dydx f dy

d f x y au huv bv

28

2 2 22d f au huv bv

2 2 22h

d f a u uv bva

2 22 2 2 2 2

22h h h

d f a u uv v bv va a a

29

2 22 2h ab hd f a u v v

a a

xxf a a

2xx xy

yx yy

f f a hab h

f f h b

30

2 20 0 0a ab h d f

2 20 0 0 0h

a ab h u v d fa

2 20 0 0 0a ab h v d f

2 20 0 0 0 0a ab h u v d f

31

So, if 0xxf and 0xx xy

yx yy

f f

f f

and 0 0dx dy

or 0 0dx dy

or 0 0dx dy

then 2 0d f

32

Then, the solution to

represents a (locally) unique minimum.

0

0x

y

f

f

33

34

A numerically specified example:

1( ) 5

20uC u u

1( ) 10

300fC f f

1( ) 0

20aC K u w K u w

1( ) 14

100wC w w

35

36

Comparative statics analysis:

( ) ( ) ( ) 0

( ) ( ) 0

u f a

w a

CC u C K u C K u w

uC

C w C K u ww

37

u f a a f a

a w a a

C C C du C dw C C dK

C du C C dw C dK

38

1 1 1 1 1 1

20 300 20 20 300 20

1 1 1 1

20 100 20 20

du dw dK

du dw dK

39

31 1 16

300 20 300

1 6 1

20 100 20

du dw dK

du dw dK

40

16 1300 20

1 620 100 0.0007

0.1891891890.003731 1

300 20

1 620 100

du

dK

41

31 16300 300

1 120 20 0.0025

0.6756756750.003731 1

300 20

1 620 100

dw

dK

42

Explicit solution of the example for alternative values of K

( ) ( ) ( ) 0

( ) ( ) 0

u f a

w a

CC u C K u C K u w

uC

C w C K u ww

43

1 1 15 10 ( ) 0 ( ) 020 300 20

1 114 0 ( ) 0

100 20

Cu K u K u w

u

Cw K u w

w

44

31 15 16 15000

300 300 300 3005 6 5 1400

0100 100 100 100

Cu w K

uC

u w Kw

45

0 31 15 16 1500 0

0 5 6 5 1400 0

Cu w K

u

Cu w K

w

46

0 31 15 16 1500 0

0 5 6 5 1400 0

Cu w K

u

Cu w K

w

47

0 31 15 1500 16

0 5 6 1400 5

Cu w K

u

Cu w K

w

48

49

1500 16 15

6 1500 16 15 1400 51400 5 6

31 15 31 6 5 15

5 6

30000 21

111

K

K KKu

Ku

50

31 1500 16

5 1400 5 31( 1400 5 ) 5(1500 16 )31 15 31 6 5 15

5 6

50900 75

111

K

K K Kw

Kw

51

Dynamic approach analysis

du Cu

dt udw C

wdt w

1

52

eq

eq

x u u

y w w

53

u f a a

a w a

x C C C x C y

y C x C C y

54

xx xy

yx yy

x m x m y

y m x m y

55

( ) ; ( )kt ktx t Ae y t Be

kt kt ktxx xy

kt kt ktyx yy

kAe m Ae m Be

kBe m Ae m Be

56

xx xy

yx yy

kA m A m B

kB m A m B

57

0

0xx xy

yx yy

k m m A

m k m B

58

• k is selected in way such that the two equations become identical. This way, the equations only determine the ratio B/A, not the values of A and B. This is necessary since we must have some freedom to determine A and B such that they fit the initial conditions.

• With two roots (that usually are different), we (usually) get two different ratios B/A. This makes it possible to fit the parameters to the (two dimensional) initial conditions (x(0),y(0)).

59

One way to determine the value(s) of k is to use this equation:

0xx xy

xx yy xy yxyx yy

k m mk m k m m m

m k m

60

xy yxm m

2 0xx yy xyk m k m m

22 0xx yy xx yy xyk m m k m m m

61

Another way to get to the same equation, is to make sure that the two equations give the

same value to the ratio B/A.

0

0

xxxx xy

xy

xy yx

xyxy yy

yy

k mBk m A m B

A m

m m

mBm A k m B

A k m

62

2

22

0

0

xyxx

xy yy

xx yy xy

xx yy xx yy xy

mk mB

A m k m

k m k m m

k m m k m m m

63

Lets us solve the equation!

2

2

2 2xx yy xx yy

xx yy xy

m m m mk m m m

2

2

2 2xx yy xx yy

xy

m m m mk m

64

No cyclical solutions!

• Observe that the expression within the square root sign is positive.

• As a consequence, only real roots, k, exist.• For this reason, cyclical solutions to the

differential equation system can be ruled out.

65

xx u f a

xy yx a

yy w a

m C C C

m m C

m C C

66

2

2

2 2

u f a w au f a w a

a

C C C C CC C C C Ck C

2

22

2 2

u f w a u f w

a

C C C C C C Ck C

67

• We may observe that

• As a consequence, both roots to to the equation are strictly negative.

• Therefore, divegence from the equilibrium solution is ruled out.

u f w u f wABS C C C ABS C C C

2

22

2 2

u f w a u f w

a

C C C C C C Ck C

68

• With only strictly negative roots, we have a guaranteed convergence to the equilibrium.

• However, this does not have to be monotone.

• With two different roots (k1 and k2) and with parameters A1 and A2 with different signs (and/or parameters B1 and B2 with different signs), the sign(s) of the deviation(s) from the equilibrium value(s) may change over time.

69

Derivation of the roots in the example:

2

21 1 1 1 1 1 1

120 300 100 10 20 300 1002 2 20

k

70

249 13 1

600 600 400k

1

1

2

2

0.081667 0.054493

0.13616

0.081667 0.054493

0.027174

k

k

k

k

71

1 2

1 2

1 2

1 2

( )

( )

k t k t

k t k t

x t Ae A e

y t B e B e

72

We may determine the path completely using the initial

conditions

0 0(0), (0) ,x y x y

73

We also use the earlier derived results:

xyxx

xy yy

mk mB

A m k m

74

Using the derived roots, we get:

11 1

xx

xy

k mB A

m

2 22

xy

xx

mA B

k m

75

1 2

1 2

1 22

11 2

( )

( )

xyk t k t

xx

xx k t k t

xy

mx t Ae B e

k m

k my t Ae B e

m

76

Let us use the initial conditions and determine the parameters!

0 1 22

10 1 2

xy

xx

xx

xy

mx A B

k m

k my A B

m

77

2 01

021

1

1

xy

xx

xx

xy

m

k m xA

yBk m

m

78

02

0 020

11

22

1

1

11

1

xy

xx xy

xx

xxxy

xxxx

xx

xy

mx

k m mx y

k myA

k mm

k mk m

k m

m

79

0

1 10 0 0

21

22

1

1

11

1

xx xx

xy xy

xxxy

xxxx

xx

xy

x

k m k my y x

m mB

k mm

k mk m

k m

m

80

Using the figures from the example, we get:

0 01

0.6565

1.431

x yA

0 02

0.6565

1.431

y xB

81

The solutions to the numerically specified example

X(t) = -106.63·EXP(- 0.13612·t) + 6.61·EXP(- 0.02718·t)

82

Y(t) = - 69.95·EXP(- 0.13612·t) - 10.07·EXP(- 0.02718·t)

83

84

The cost function from different perspectives:

(based on the numerically specified example)

85

86

87

Numerical solution of the example problem using direct

minimization:

88

K = 600

• model:• min = C;• k = 600;• C = cu + cf + ca + cw;• cu = 5*u+1/40*u^2;• cf = 10*f + 1/600*f^2;• ca = 1/40*(k-u-w)^2;• cw = 14*w+1/200*w^2;• f = k-u;• a = k-w;• @free(anet);• anet = a - u;• @free(eqw);• 31*equ+15*eqw=1500 + 16*k;• 5*equ+6*eqw = -1400+5*k;• end

89

• Variable Value Reduced Cost• C 8975.806 0.000000• K 600.0000 0.000000• CU 4995.578 0.000000• CF 2516.909 0.000000• CA 1463.319 0.000000• CW 0.000000 0.000000• U 358.0645 0.000000• F 241.9355 0.000000• W 0.000000 1.903226• A 600.0000 0.000000• ANET 241.9355 0.000000• EQW -53.15315 0.000000• EQU 383.7838 0.000000

90

K = 800

• model:• min = C;• k = 800;• C = cu + cf + ca + cw;• cu = 5*u+1/40*u^2;• cf = 10*f + 1/600*f^2;• ca = 1/40*(k-u-w)^2;• cw = 14*w+1/200*w^2;• f = k-u;• a = k-w;• @free(anet);• anet = a - u;• @free(eqw);• 31*equ+15*eqw=1500 + 16*k;• 5*equ+6*eqw = -1400+5*k;• end

91

• Variable Value Reduced Cost• C 13952.25 0.000000• K 800.0000 0.000000• CU 6552.228 0.000000• CF 4022.401 0.000000• CA 2196.271 0.000000• CW 1181.353 0.000000• U 421.6216 0.000000• F 378.3784 0.000000• W 81.98198 0.000000• A 718.0180 0.000000• ANET 296.3964 0.000000• EQW 81.98198 0.000000• EQU 421.6216 0.000000

92

K = 1000• model:• min = C;• k = 1000;• C = cu + cf + ca + cw;• cu = 5*u+1/40*u^2;• cf = 10*f + 1/600*f^2;• ca = 1/40*(k-u-w)^2;• cw = 14*w+1/200*w^2;• f = k-u;• a = k-w;• @free(anet);• anet = a - u;• @free(eqw);• 31*equ+15*eqw=1500 + 16*k;• 5*equ+6*eqw = -1400+5*k;• end

93

• Variable Value Reduced Cost• C 19357.66 0.000000• K 1000.000 0.000000• CU 7574.872 0.000000• CF 5892.379 0.000000• CA 2615.068 0.000000• CW 3275.339 0.000000• U 459.4595 0.000000• F 540.5405 0.000000• W 217.1171 0.000000• A 782.8829 0.000000• ANET 323.4234 0.000000• EQW 217.1171 0.000000• EQU 459.4595 0.000000

94

Numerical approximation of the dynamics:

• ! dynsim;• ! Peter Lohmander Valencia 20100222;• model:• sets:• time/1..100/:x,y,dx,dy;• endsets• cxx = 31/300;• cxy = 15/300;• cyx = 15/300;• cyy = 18/300;• x(1) = -100;• y(1) = -80;

95

• @FOR( time(t): dx(t)= -( cxx*x(t) + cxy*(y(t)) ));• @FOR( time(t): dy(t)= -( cyx*x(t) + cyy*(y(t)) ));

• @FOR( time(t)| t#GT#1: x(t)= x(t-1) + dx(t-1) );• @FOR( time(t)| t#GT#1: y(t)= y(t-1) + dy(t-1) );

• @for(time(t): @free(x(t))); • @for(time(t): @free(y(t)));• @for(time(t): @free(dx(t))); • @for(time(t): @free(dy(t)));

• end

96

Conclusions

Global warming, forest policy, energy policy and CCS should be studied as one system. This way, the economically most efficient solution can be obtained.

General and optimal decision rules have been derived.

In typical situations, a unique global cost minimum can be obtained.

We should, in the optimally coordinated way:

- Increase harvesting of the presently existing forests.

- Use more biomass from the forests to produce energy.

- Increase forest production in the new forest plantations.

- Increase the use of CCS.

97

References

• Lohmander, P., Adaptive Optimization of Forest Management in a Stochastic World, in Weintraub A. et al (Editors), Handbook of Operations Research in Natural Resources, Springer, Springer Science, International Series in Operations Research and Management Science, New York, USA, pp 525-544, 2007 http://www.amazon.ca/gp/reader/0387718141/ref=sib_dp_pt/701-0734992-1741115#reader-link

• Lohmander, P,. Energy Forum, Stockholm, 6-7 February 2008, Conference program with links to report and software by Peter Lohmander:http://www.energyforum.com/events/conferences/2008/c802/program.phphttp://www.lohmander.com/EF2008/EF2008Lohmander.htm

• Lohmander, P., Ekonomiskt rationell utveckling för skogs- och energisektorn i Sverige, Nordisk Papper och Massa, Nr 3, 2008

• Lohmander, P., Guidelines for Economically Rational and Coordinated Dynamic Development of the Forest and Bio Energy Sectors with CO2 constraints, Proceedings from the 16th European Biomass Conference and Exhibition, Valencia, Spain, 02-06 June, 2008 (In the version in the link, below, an earlier misprint has been corrected. ) http://www.Lohmander.com/Valencia2008.pdf

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• Lohmander, P., Economically Optimal Joint Strategy for Sustainable Bioenergy and Forest Sectors with CO2 Constraints, European Biomass Forum, Exploring Future Markets, Financing and Technology for Power Generation, CD, Marcus Evans Ltd, Amsterdam, 16th-17th June, 2008 http://www.Lohmander.com/Amsterdam2008.ppt

• Lohmander, P., Ekonomiskt rationell utveckling för skogs- och energisektorn, Nordisk Energi, Nr. 4, 2008

• Lohmander, P., Optimal resource control model & General continuous time optimal control model of a forest resource, comparative dynamics and CO2 consideration effects, SLU Seminar in Forest Economics, Umea, Sweden, 2008-09-18 http://www.lohmander.com/CM/CMLohmander.ppt

• Lohmander, P., Tools for optimal coordination of CCS, power industry capacity expansion and bio energy raw material production and harvesting, 2nd Annual EMISSIONS REDUCTION FORUM: - Establishing Effective CO2, NOx, SOx Mitigation Strategies for the Power Industry, CD, Marcus Evans Ltd, Madrid, Spain, 29th & 30th September 2008 http://www.lohmander.com/Madrid08/Madrid_2008_Lohmander.ppt

• Lohmander, P., Optimal CCS, Carbon Capture and Storage, Under Risk, International Seminars in Life Sciences, Universidad Politécnica de Valencia, Thursday 2008-10-16 http://www.lohmander.com/OptCCS/OptCCS.ppt

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• Lohmander, P., Economic forest production with consideration of the forest and energy industries, E.ON International Bioenergy Conference, Malmo, Sweden, 2008-10-30 http://www.lohmander.com/eon081030/eon081030.ppt

• Lohmander, P., Optimal dynamic control of the forest resource with changing energy demand functions and valuation of CO2 storage, UE2008.fr, The European Forest-based Sector: Bio-Responses to Address New Climate and Energy Challenges? Nancy, France, November 6-8, 2008 http://www.lohmander.com/Nancy08/Nancy08.ppt (See also later versions 2009)

• Lohmander, P., Optimal dynamic control of the forest resource with changing energy demand functions and valuation of CO2 storage, The European Forest-based Sector: Bio-Responses to Address New Climate and Energy Challenges, Nancy, France, November 6-8, 2008, Proceedings: (forthcoming) in French Forest Review (2009) Abstract: Page 65 of: http://www.gip-ecofor.org/docs/34/rsums_confnancy2008__20081105.pdfPresentation as pdf: http://www.gip-ecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdfConference: http://www.gip-ecofor.org/docs/34/nancy2008englishprogramme20081106.pdf

• ECOFOR, (in French) Summary of results by Peter Lohmander (on page 8) in “Evaluation du developpement de la bioenergie”, in Bulletin d’information sur les forets europeennes, l’energie et climat, Volume 157, Numero 1, Lundi 10 novembre 2008 http://www.gip-ecofor.org/docs/34/nancy2008synthseiisd.pdf

• IISD, Summary of results by Peter Lohmander (on page 6) in “Evaluation of Bioenergy Development”, in European Forests, Energy and Climate Bulletin, Published by the International Institute for Sustainable Development (IISD) http://www.iisd.org/ , Vol. 157, No. 1, Monday, 10 November, 2008 http://www.iisd.ca/download/pdf/sd/ymbvol157num1e.pdf

100

• Lohmander, P., Integrated Regional Study Stage 1., Presentation at the E.ON - Holmen - Sveaskog - SLU Research Meeting, Norrköping, Sweden, 2008-12-10 – 2008-12-11, http://www.lohmander.com/NorrDec08/NorrDec08.ppt , http://www.lohmander.com/NorrDec08/NorrDec08.pdf , http://www.lohmander.com/NorrDec08/NorrDec08RawData.xls

• Lohmander, P., Öka avverkningen och hjälp Sverige ur krisen, VI SKOGSÄGARE, Debatt, Nr. 1, 2009 http://www.lohmander.com/PLdebattVIS2009nr1.pdf

• Lohmander, P., Economic Forest Production with Consideration of the Forest and Energy Industries (SLU 2009-01-29), http://www.lohmander.com/SLU09/SLU09.pdf http://www.lohmander.com/SLU09/SLU09.ppt

• Lohmander, P., Rational and sustainable international policy for the forest sector with consideration of energy, global warming, risk, and regional development, SLU, Umea, 2009-02-18, http://www.lohmander.com/IntPres090218.ppt

• Lohmander, P., Strategic options for the forest sector in Russia with focus on economic optimization, energy and sustainability (Full paper in English with short translation to Russian), ICFFI News, Vol. 1, Number 10, March 2009http://www.Lohmander.com/RuMa09/RuMa09.htm

101

• International seminar, ECONOMICS OF FORESTRY AND FOREST SECTOR: ACTUAL PROBLEMS AND TRENDS, St Petersburg, Russia, March 2009, http://www.lohmander.com/RuMa09/ProgramRuMa09.pdf

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• Lohmander, P., Stor potential för svensk skogsenergi, Nordisk Energi, Nr. 2, 2009 http://www.Lohmander.com/Information/ne1.jpghttp://www.Lohmander.com/Information/ne2.jpghttp://www.Lohmander.com/Information/ne3.jpghttp://www.Lohmander.com/PL_SvSE_090205.pdfhttp://www.Lohmander.com/PL_SvSE_090205.doc

• Lohmander, P., Strategiska möjligheter för skogssektorn i Ryssland Nordisk Papper och Massa, Nr 2, 2009 http://www.Lohmander.com/PL_NPM_2_2009.pdf http://www.Lohmander.com/PL_RuSwe_09.pdf http://www.Lohmander.com/PL_RuSwe_09.doc

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• Lohmander, P., Economic forest production with consideration of the forest- and energy industries, Project meeting presentation, Stockholm, Sweden, 2009-05-11, http://www.lohmander.com/EON_090511.ppt

• Lohmander, P., Derivation of the Economically Optimal Joint Strategy for Development of the Bioenergy and Forest Products Industries, European Biomass and Bioenergy Forum, MarcusEvans, London, UK, 8-9 June, 2009, http://www.lohmander.com/London09/London_Lohmander_09.ppt & ttp://www.lohmander.com/London09.pdf

• Lohmander, P., Rational and sustainable international policy for the forest sector - with consideration of energy, global warming, risk, and regional development, Preliminary plan, 2009-08-05, http://www.lohmander.com/ip090805.pdf