Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Optimal Economic Decision Rules in the Biomass Supply Chain with CO2 Considerations Peter Lohmander Professor Dr., SUAS, Umea, SE-90183, Sweden [email protected] ALIO-INFORMS International 2010 Buenos Aires, Monday, June 07, 08:30 - 10:00 1 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. 2 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! 3 Energy plant CCS, with CO2 capture and Carbon separation Capture and Storage, has already become Oil field the main Coal future mine Natural emission gas reduction method of the fossile fuel energy Permanent storage of CO2 industry 4 How to reduce the CO2 level in the atmosphere, Energy plant with CO2 capture and separation CO2 not only to decrease the emission of CO2 Permanent storage of CO2 5 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. 6 The present situation. CO2 5 1 1 CO2 increase in the atmosphere: 5-1 = 4 4 0 Coal, oil, gas Permanent storage of CO2 7 CO2 CO2 increase If we do not in the 5 use the 1 atmosphere: forest for 5-1 = 4 energy production but use it as 5 a carbon sink. Before the forest Coal, 0 has reached oil, gas equilibrium, this happens: Permanent storage of CO2 8 CO2 CO2 increase If we do not in the 5 use the 1 atmosphere: forest for 1 5+1-1 = 5 energy production but use it as 5 a carbon sink. When the forest Coal, 0 has reached oil, gas equilibrium, this happens: Permanent storage of CO2 9 If we use CCS with 80% efficiency and let the forest grow until it reaches equilibrium. CO2 1 1 1 CO2 increase in the atmosphere: 1+1-1 = 1 5 4 Coal, oil, gas Permanent storage of CO2 10 If we use CCS with 80% efficiency and use the forest with ”traditional” low intensity harvesting and silviculture. CO2 1 1 1 CO2 increase in the atmosphere: 1-1 = 0 4 4 Coal, oil, gas Permanent storage of CO2 11 If we use CCS with 80% efficiency and use the forest with increased harvesting and high intensity silviculture. CO2 1 2 2 CO2 ”increase” in the atmosphere: 1-2 = -1 3 4 Coal, oil, gas Permanent storage of CO2 12 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! 13 14 min C Cu (u ) C f ( f ) Ca ( a v) Cw ( w) u ,w s.t. u f K f K u wa K a K w vu 15 min C Cu (u) C f ( K u) Ca ( K u w) Cw (w) u ,w 16 17 C Cu (u ) C f ( K u ) Ca ( K u w) 0 u C Ca ( K u w) Cw ( w) 0 w Cu C f Ca Ca Cw 18 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. 19 Observation 2 • In optimum, the marginal cost of global warming equals the marginal cost of CCS. 20 C C ( u ) C ( K u ) C ( K u w ) u f a 2 u 2 C Ca ( K u w) uw 2 C Ca ( K u w) wu 2 C C ( K u w ) C a w ( w) 2 w 2 21 Second order minimum conditions: C 0 2 u 2 C 2 u 2 C wu 2 C uw 0 2 C 2 w 2 22 Cu C f Ca 0 Cu C f Ca Ca Ca Ca Cw 0 23 Cu C f Ca Cu C f Ca 0 C C C C C C C C C C C C C u f a a u a a f a a w 2 a w 24 C C C C C C u f a a w a 2 C C C CuCa CuCw C f Ca C f Cw Ca 2 a w 2 a CuCa CuCw C f Ca C f Cw CaCw 0 25 Observations of the first and second order conditions: f ( x, y ) fx 0 f 0 y df ( x, y ) f x dx f y dy 0 26 d f ( x, y) f xx (dx) f xy dxdy f yx dydx f yy (dy) 2 2 2 f xy f yx d 2 f ( x, y ) f xx (dx) 2 f xy dxdy f yx dydx f yy (dy ) 2 d f ( x, y ) 2 au 2 2huv bv 2 27 d f au 2huv bv 2 2 2 h 2 2 d f a u 2 uv bv a 2 2 h h 2 h 2 2 d f a u 2 uv 2 v bv v a a a 2 2 2 28 2 h d f au v a 2 ab h 2 v a 2 f xx a a f xx f xy f yx f yy a h h b ab h 2 29 a 0 ab h 2 0 d f 0 2 h 2 a 0 ab h 0 u v 0 d f 0 a 2 a 0 ab h a 0 ab h 2 2 0 v 0 d f 0 2 0 u 0 v 0 d f 0 2 30 So, if and then f xx 0 and f xx f xy f yx f yy 0 dx 0 dy 0 or dx 0 dy 0 or dx 0 dy 0 d2 f 0 31 Then, the solution to fx 0 f 0 y represents a (locally) unique minimum. 32 33 A numerically specified example: 1 Cu (u ) 5 u 20 1 C f ( f ) 10 f 300 1 Ca ( K u w) 0 K u w 20 1 Cw ( w) 14 w 100 34 35 Comparative statics analysis: C Cu (u ) C f ( K u ) Ca ( K u w) 0 u C Cw ( w) Ca ( K u w) 0 w 36 C C C du C dw C C dK f a a f a u Ca du Cw Ca dw Ca dK 37 1 1 1 1 1 1 20 300 20 du 20 dw 300 20 dK 1 1 1 1 du dw dK 20 100 20 20 38 31 1 16 du dw dK 300 20 300 1 du 6 dw 1 dK 20 100 20 39 16 300 1 du 20 dK 31 300 1 20 1 20 6 100 0.0007 0.189189189 0.0037 1 20 6 100 40 31 300 1 dw 20 dK 31 300 1 20 16 300 1 0.0025 20 0.675675675 0.0037 1 20 6 100 41 Explicit solution of the example for alternative values of K C Cu (u ) C f ( K u ) Ca ( K u w) 0 u C Cw ( w) Ca ( K u w) 0 w 42 C 1 1 1 5 u 10 ( K u ) 0 ( K u w) 0 u 20 300 20 C 1 1 14 w 0 ( K u w) 0 w 100 20 43 C 31 15 16 1500 u w K 0 u 300 300 300 300 C 5 6 5 1400 u w K 0 w 100 100 100 100 44 C 0 31u 15w 16 K 1500 0 u C 0 5u 6 w 5 K 1400 0 w 45 C 0 31u 15w 16 K 1500 0 u C 0 5u 6 w 5 K 1400 0 w 46 C 0 31u 15w 1500 16 K u C 0 5u 6w 1400 5 K w 47 48 1500 16 K 15 1400 5K u 31 15 6 5 6 1500 16 K 15 1400 5 K 31 6 5 15 6 30000 21K u 111 49 31 1500 16 K w 5 1400 5 K 31(1400 5 K ) 5(1500 16 K ) 31 15 31 6 5 15 5 6 50900 75K w 111 50 Dynamic approach analysis C du u u dt C dw w w dt 1 51 x u ueq y w w eq 52 x C C C x C y u f a a y Ca x Cw Ca y 53 x m x m y xx xy y m yx x m yy y 54 x(t ) Ae ; kt y(t ) Be kt kAe mxx Ae mxy Be kt kt kt kBe m Ae m Be yx yy kt kt kt 55 kA mxx A mxy B kB m A m B yx yy 56 k mxx m yx mxy A 0 k myy B 0 57 • 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)). 58 One way to determine the value(s) of k is to use this equation: k mxx mxy myx k myy k mxx k myy mxy myx 0 59 mxy myx k mxx k myy mxy 2 0 k mxx myy k mxx myy mxy 0 2 2 60 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. B k mxx k mxx A mxy B 0 A m xy mxy myx B mxy mxy A k myy B 0 A k myy 61 k mxx B A mxy k m k m k m m 0 k m m k m m m mxy yy 2 xx yy xy 2 xx yy xx yy xy 2 0 62 Lets us solve the equation! m k xx myy mxx myy 2 mxx myy mxy 2 2 2 m k m m m 2 xx yy xx yy mxy 2 2 2 63 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. 64 mxx Cu C f Ca mxy m yx Ca m yy Cw Ca 65 C C C C C C C C C C k C 2 2 2 u f a w u a f a w a a Cu C f Cw 2Ca Cu C f Cw k 2 2 C 2 2 a 66 2 C C C 2C C C C C k 2 2 2 u f w a u f w 2 a • We may observe that ABS Cu C f Cw ABS Cu C f Cw • As a consequence, both roots to to the equation are strictly negative. • Therefore, divegence from the equilibrium solution is ruled out. 67 • 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. 68 Derivation of the roots in the example: 1 1 1 1 20 300 100 10 k 2 2 1 1 1 20 300 100 1 2 2 20 69 49 13 1 k 600 600 400 2 k1 0.081667 0.054493 k1 0.13616 k2 0.081667 0.054493 k2 0.027174 70 x(t ) A1e A2e k1t k2 t y ( t ) B e B e 1 2 k1t k2 t 71 We may determine the path completely using the initial conditions x(0), y(0) x0 , y0 72 We also use the earlier derived results: B A mxy k mxx mxy k myy 73 Using the derived roots, we get: k1 mxx B1 A1 mxy A2 mxy k2 mxx B2 74 m xy k1t k2t x ( t ) A e B e 1 2 k m 2 xx k m 1 xx k1t k2 t y (t ) A1e B2e m xy 75 Let us use the initial conditions and determine the parameters! mxy x A B 1 2 0 k m 2 xx k m 1 xx y A B 0 1 2 m xy 76 1 k m 1 xx mxy k2 mxx A1 x0 B2 y0 1 mxy 77 x0 A1 mxy k2 mxx y0 1 k1 mxx mxy 1 mxy k2 mxx mxy x0 y0 k2 mxx k1 mxx 1 k2 mxx 1 78 1 B2 k1 mxx mxy 1 k1 mxx mxy x0 y0 mxy k2 mxx k1 mxx y0 x0 m xy k1 mxx 1 k2 mxx 1 79 Using the figures from the example, we get: x0 0.6565 y0 A1 1.431 y0 0.6565 x0 B2 1.431 80 The solutions to the numerically specified example X(t) = -106.63·EXP(- 0.13612·t) + 6.61·EXP(- 0.02718·t) 81 Y(t) = - 69.95·EXP(- 0.13612·t) - 10.07·EXP(- 0.02718·t) 82 83 The cost function from different perspectives: (based on the numerically specified example) 84 85 86 Numerical solution of the example problem using direct minimization: 87 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 88 • • • • • • • • • • • • • • Variable C K CU CF CA CW U F W A ANET EQW EQU Value 8975.806 600.0000 4995.578 2516.909 1463.319 0.000000 358.0645 241.9355 0.000000 600.0000 241.9355 -53.15315 383.7838 Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.903226 0.000000 0.000000 0.000000 0.000000 89 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 90 • • • • • • • • • • • • • • Variable C K CU CF CA CW U F W A ANET EQW EQU Value 13952.25 800.0000 6552.228 4022.401 2196.271 1181.353 421.6216 378.3784 81.98198 718.0180 296.3964 81.98198 421.6216 Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 91 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 92 • • • • • • • • • • • • • • Variable C K CU CF CA CW U F W A ANET EQW EQU Value 19357.66 1000.000 7574.872 5892.379 2615.068 3275.339 459.4595 540.5405 217.1171 782.8829 323.4234 217.1171 459.4595 Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 93 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; 94 • @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 95 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. 96 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/701-07349921741115#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.php http://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 97 • 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 98 • 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.pdf Presentation as pdf: http://www.gipecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdf Conference: 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 99 • 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 2009 http://www.Lohmander.com/RuMa09/RuMa09.htm 100 • International seminar, ECONOMICS OF FORESTRY AND FOREST SECTOR: ACTUAL PROBLEMS AND TRENDS, St Petersburg, Russia, March 2009, http://www.lohmander.com/RuMa09/ProgramRuMa09.pdf • Lohmander, P., Satsa på biobränsle, Skogsvärden, Nr 1, 2009 http://www.Lohmander.com/PL_SV_1_09.jpg • Lohmander, P., Stor potential för svensk skogsenergi, Nordisk Energi, Nr. 2, 2009 http://www.Lohmander.com/Information/ne1.jpg http://www.Lohmander.com/Information/ne2.jpg http://www.Lohmander.com/Information/ne3.jpg http://www.Lohmander.com/PL_SvSE_090205.pdf http://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 101 • 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.p pt & 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 102