Ecoomic Aalysis ad Optimizatio Assess ecoomic feasibility of eergy systems Idetify aticipated cost of eergy (COE) ad other measures of ecoomic performace usig cosistet methodologies Compare alteratives over differet time frames Evaluate ecoomies of scale ad potetial for optimizatio EBS 26 Eergy Systems Ecoomic Aalysis Ecoomic Aalysis ad Optimizatio Review of ecoomic aalysis Cash flow equivalece Time value of moey Depreciatio Taxes Escalatio ad iflatio Curret ad costat dollar aalysis Evaluatio of alteratives Preset worth Levelized cost Rate of retur Beefit/Cost Reveue Requiremets method Sesitivity aalysis Optimizatio EBS 26 Eergy Systems Ecoomic Aalysis 2
Ecoomic Aalysis ad Optimizatio Review of ecoomic aalysis Cash flow equivalece Time value of moey Depreciatio Taxes Escalatio ad iflatio Curret ad costat dollar aalysis Evaluatio of alteratives Preset worth Levelized cost Rate of retur Beefit/Cost Reveue Requiremets method Sesitivity aalysis Optimizatio EBS 26 Eergy Systems Ecoomic Aalysis 3 Cash Flow Equivalece Ivestmet of moey is iteded to geerate additioal moey ("retur or iterest ) Moey placed i a bak accout ("loaed" to the bak) is aticipated to ear iterest If the iterest eared is reivested (e.g., left i the bak accout), the additioal iterest is to be eared o the iterest already eared. This is kow as compoudig. Uder compoudig, cash flows of various types ca be foud to be equivalet A future sum of moey resultig from the ivestmet of a preset sum of moey is to be greater tha the preset sum (exclusive of iflatio) due to the effects of iterest EBS 26 Eergy Systems Ecoomic Aalysis 4 2
Future amout A future sum, F ($) ca be foud from the preset sum, P ($), kow as the preset worth or preset value, give a iterest rate, i (period - ), ad a umber of compoudig periods,, as F = P(+i) time 2 The factor (+i) is called the compoud amout factor, F/P 0 P F EBS 26 Eergy Systems Ecoomic Aalysis 5 Cash Flow Diagram 0 P time 2 By covetio, a arrow poitig dowwards represets a ivestmet, a arrow poitig upwards represets a retur. Note that for future amouts, a ed of period covetio is utilized for a preset worth occurrig at time "zero" F EBS 26 Eergy Systems Ecoomic Aalysis 6 3
Preset worth 0 P time 2 F The preset worth, P ($), of a future sum of moey is P F i ( ) The factor (+i) - worth factor, P/F is called the preset EBS 26 Eergy Systems Ecoomic Aalysis 7 Uiform (Level) Cash Flow 0 P A time 2 F A equivalece ca be foud betwee a preset worth or a future worth ad a uiform series of cash flows, desigated A (for auity, although the periods ca be other tha years) The uiform series of cash flows begis at the ed of period i the followig aalysis EBS 26 Eergy Systems Ecoomic Aalysis 8 4
Uiform (Level) Cash Flow 0 P A time 2 F A i( i) P ( i) the term i brackets is kow as the capital recovery factor, A/P, or sometimes, CRF A i F ( i) the term i brackets is kow as the sikig fud factor, A/F Note that A/P = A/F + i Note that if =, the A/P = i EBS 26 Eergy Systems Ecoomic Aalysis 9 Iterest rate The omial iterest rate, r, is the aual iterest rate ot icludig compoudig, expressed as r = i m m is the umber of compoudig periods per year, i is the iterest rate per iterest period (i = r/m), as before The effective iterest rate is the iterest rate per year icludig compoudig, expressed as i e = (+i) m - Uder cotiuous compoudig m = ad i e = e r - EBS 26 Eergy Systems Ecoomic Aalysis 0 5
Depreciatio Depreciatio accouts for the loss i value of assets over time, due to wear-out, obsolescece, or ecoomic maagemet. Accoutig for depreciatio is ecessary to esure that replacemet assets ca be purchased at the ed of the useful life of the existig asset. Depreciatio is also a importat tax deductio. Several depreciatio methods may be used. For tax purposes the tax code ormally specifies the method to be used. The three most commo methods are the straight lie, sum of year's digits, ad decliig balace. The latter two methods are said to be "accelerated" (depreciate faster iitially) relative to straight lie. Acceleratig the depreciatio allowace is beeficial for tax purposes if the busiess is profitable. Modified accelerated cost recovery systems (MACRS) are defied for federal tax purposes. EBS 26 Eergy Systems Ecoomic Aalysis Depreciatio: Straight Lie Depreciatio allowace to be take i each year, D, by the straight lie method ca be foud as B S D l B S l = Book value at the begiig of the year = Salvage value of asset = remaiig useful life at the begiig of the year EBS 26 Eergy Systems Ecoomic Aalysis 2 6
Depreciatio: Straight Lie The straight lie method will depreciate the asset to its salvage value. The book value is the value of the asset, equal to the first cost less accumulated depreciatio. Commoly, the value D is the same i each year ad computed as (P - S)/N P N = first cost (or purchase cost) of asset = total useful or ecoomic life of asset EBS 26 Eergy Systems Ecoomic Aalysis 3 Depreciatio: Sum of year s digits Sum-of-the-year s digits depreciatio allowace take i each year is D l SOYD ( P S) N SOYD = sum of year's digits = + 2 + 3 +... + N = (N ) 2 Note that this method will also depreciate to the salvage value EBS 26 Eergy Systems Ecoomic Aalysis 4 7
Depreciatio: Decliig Balace Decliig balace depreciatio allowace take i each year is D B N = declie rate Double (or 200%) decliig balace is commoly used, i which case, = 2 Oly by coicidece will decliig balace depreciate to the salvage value EBS 26 Eergy Systems Ecoomic Aalysis 5 Depreciatio: Decliig Balace The book value at the ed of years, (begiig of + years), is P( - /N), so that D P N N At some poit, the depreciatio allowace by decliig balace may fall below that by straight lie or SOYD based o the remaiig book value ad ecoomic life The depreciatio method ca be switched at that time to use straight lie or SOYD o remaiig book value while maitaiig the highest depreciatio allowace i each year EBS 26 Eergy Systems Ecoomic Aalysis 6 8
ACRS EBS 26 Eergy Systems Ecoomic Aalysis 7 Taxes Taxes are fees assessed o earigs ad other trasactios, geerally determied as T = (t)(i t ) T = taxes ($) t = tax rate (decimal) I t = taxable icome ($) EBS 26 Eergy Systems Ecoomic Aalysis 8 9
Taxable icome Taxable icome is gross icome adjusted by deductios I t = I g -E -D t - D i I g = gross icome ($) E = expeses ($) D t = depreciatio for tax purposes ($) D i = iterest o debt ($) EBS 26 Eergy Systems Ecoomic Aalysis 9 Depreciatio for tax purposes Tax depreciatio procedures will ormally specify the schedule of depreciatio, icludig how to hadle salvage, although they are usually based o oe or more of the previous methods e.g., double decliig balace switchig to straight lie possibly with restrictios o the amout of depreciatio allowed i the first year (e.g. halfyear covetio i first ad last years) EBS 26 Eergy Systems Ecoomic Aalysis 20 0
Combied Tax Rate For federal purposes, state tax is a deductio, but federal tax is ot ormally cosidered a deductio for state tax A combied tax rate, t, ca be computed: t = t F ( - t S ) + t S t F = federal tax rate (decimal) t S = state tax rate (decimal) EBS 26 Eergy Systems Ecoomic Aalysis 2 Escalatio ad Iflatio Iflatio results i a icrease i the cost of a good or service over time ad a decrease i purchasig power, ad is geerally modeled i a maer similar to iterest rate Defiig: f = iflatio rate (period - ) i = apparet (or quoted) iterest rate, as before i = real iterest rate (accoutig for the effect of iflatio) EBS 26 Eergy Systems Ecoomic Aalysis 22
Iflatio 0 time 2 F F o Uder iflatio, a future amout of moey, F, will ot have the same purchasig power as if there had bee o iflatio. Expressed i the same value uits as whe the ivestmet was made, that is, i time zero dollars, the year zero equivalet future value is F o F f EBS 26 Eergy Systems Ecoomic Aalysis 23 Iflatio 0 P time 2 F F o The preset worth of F o is P F o F i' i' ( f ) ( i) F P is related to F i the same way as before EBS 26 Eergy Systems Ecoomic Aalysis 24 2
Iflatio The apparet iterest rate is i ( i')( f ) i' f i' f The real iterest rate is i' i f i f f EBS 26 Eergy Systems Ecoomic Aalysis 25 Curret ad Costat Dollars A curret dollar aalysis icludes the effect of iflatio A costat dollar aalysis attempts to adjust for the effect of iflatio so that ecoomic values may be compared o a equivalet basis typically expressed for some base year for example, comparisos of the cost of alterative fuel resources i the future to the preset cost of existig resources, exclusive of geeral iflatio i the ecoomy EBS 26 Eergy Systems Ecoomic Aalysis 26 3
Curret ad Costat Dollars Example: cosider that the cost of fuel, C, over time will escalate at a apparet escalatio rate, e, such that C = C o (+e) C o = preset cost of fuel C = escalated cost of fuel i the future The preset worth of C is P C ( i) C o ( e) ( i) k = (+e)/(+i) C o k EBS 26 Eergy Systems Ecoomic Aalysis 27 Curret ad Costat Dollars The total preset worth of the fuel cost over the etire aalysis time is Factorig k Combiig N P C o k P N kc o P kc o k N ( k ) ( k) EBS 26 Eergy Systems Ecoomic Aalysis 28 4
Curret Dollar LAC The curret dollar levelized aual cost, LAC, is LAC ( P)( A/ P) kc o N ( k ) ( k) N i( i) N ( i) EBS 26 Eergy Systems Ecoomic Aalysis 29 Costat Dollar LAC The costat dollar levelized aual cost adjusts for iflatio by usig the real escalatio ad iterest rates ( e') ( e) ( i) k' k e' i' ( i') ( f ) ( f ) LAC' k' C o N ( k' ) ( k') N i'( i') N ( i') EBS 26 Eergy Systems Ecoomic Aalysis 30 5
Evaluatio of Alteratives Alterative eergy techologies or strategies may be compared ecoomically usig ay of a umber of differet ratioal methods, all of which are equivalet ad yield the same decisio if employed properly. Other techiques, such as simple payback aalysis, have differet objectives ad do ot ecessarily yield the same decisio. Payback is useful whe the rapid recovery of capital ivestmet is required, however, it will ot ecessarily select the alterative that maximizes the amout of moey made over the ecoomic life (more precisely, that maximizes et preset worth, NPW). Net Preset Worth, Levelized Cost, ad Beefit/Cost aalysis assume a iterest or discout rate. Rate of retur aalysis computes the iterest rate eared o ivestmet equal to that rate which makes NPW=0 EBS 26 Eergy Systems Ecoomic Aalysis 3 Evaluatio of Alteratives Method Objective Preset Worth Compute et preset worth (preset worth of beefits less preset worth of costs) of alteratives. Maximize Net Preset Worth (NPW) Levelized Cost Rate of Retur (ROR) Beefit-Cost Ratio (B/C) Compute et level (typically aual) cash flow (level beefits less level costs) of alteratives. Maximize Net Level Beefits Miimize Net Level Costs Fid i=ror such that NPW=0 For feasible alteratives with ROR Miimum attractive rate of retur (MARR), compute icremetal ROR ( ROR) betwee alteratives: If ROR MARR, choose higher cost alterative. If ROR<MARR, choose lower cost alterative. Fid feasible alteratives with ratio of preset worth of beefits (PWB) to preset woth of costs (PWC) greater tha uity, i.e., PWB/PWC. Compute icremetal B/C ( B/C) betwee feasible alteratives. If B/C, choose higher cost alterative. If B/C<, choose lower cost alterative. EBS 26 Eergy Systems Ecoomic Aalysis 32 6
Evaluatio of Alteratives Example: Choice of two equal-outcome alteratives i a productio activity Category: Alterative Alterative 2 Capital Cost ($) 8,000 23,500 Operatig Cost ($/y) 00,000 Eergy Cost ($/y) 7,935 2,267 Gross Reveue ($/y) 9,85 9,85 Net Reveue ($/y),780 6,548 Life (y) 0 0 Iterest (Discout) Rate (%/y) 2 2 EBS 26 Eergy Systems Ecoomic Aalysis 33 Evaluatio of Alteratives: Example Category: Alterative Alterative 2 Capital Cost ($) 8,000 23,500 Operatig Cost ($/y) 00,000 Eergy Cost ($/y) 7,935 2,267 Gross Reveue ($/y) 9,85 9,85 Net Reveue ($/y),780 6,548 Life (y) 0 0 Iterest (Discout) Rate (%/y) 2 2. Preset Worth Aalysis Objective: Max NPW Alterative : NPW (Alt) = -8,000 +,780(P/A, 2%, 0) = -8,000 +,780(5.65) = +2,057 Alterative 2: NPW (Alt2) = -23,500 + 6,548(5.65) = +3,496 Decisio: Max NPW--choose Alterative 2 EBS 26 Eergy Systems Ecoomic Aalysis 34 7
Evaluatio of Alteratives: Example 2. Levelized Cost Objective: Max Net Level Beefits Alterative : Level Aual Beefits (LAB) =,780/y Level Aual Costs (LAC) = -8,000(A/P, 2%, 0) = -8,000(0.77) = -,46/y LAB-LAC = +364/y Alterative 2: Level Aual Beefits (LAB) = 6,548/y Level Aual Costs (LAC) = -23,500(0.77) = -4,60/y LAB-LAC = +2,388/y Decisio: Max Net Level Beefits--choose Alterative 2 EBS 26 Eergy Systems Ecoomic Aalysis 35 Evaluatio of Alteratives: Example 3. Rate of Retur Aalysis Miimum Attractive Rate of Retur (MARR) = 2%/y Check for feasibility: Solve ROR by fidig iterest rate that forces NPW=0 Alterative : NPW = 0 = -8,000 +,780(P/A, i, 0) (P/A, i, 0) = 4.494 i = 8%/y > MARR : OK Alterative 2: NPW = 0 = -23,500 + 6,548(P/A, i, 0) (P/A, i, 0) = 3.589 i 25%/y > MARR : OK Use icremetal ( ROR) aalysis to choose best alterative EBS 26 Eergy Systems Ecoomic Aalysis 36 8
Evaluatio of Alteratives: Example 3. Rate of Retur Aalysis (cotiued) Use icremetal ( ROR) aalysis to choose best alterative Category Differece Alt 2 Alt Capital Cost ($) = -23,500 (-8,000) = -5,500 Net Reveue ($/y) = 6,548,780 = 4,768 Solve ROR by fidig iterest rate that forces NPW = 0 NPW = 0 = -5,500 + 4,768(P/A, i, 0) (P/A, i, 0) = 3.25 i 25%/y > MARR Choose higher cost alterative, choose Alterative 2 EBS 26 Eergy Systems Ecoomic Aalysis 37 Evaluatio of Alteratives: Example 4. Beefit-Cost Ratio Aalysis PWB = preset worth of beefits PWC = preset worth of costs Check for feasibility: Alterative : PWB =,780(P/A, 2%, 0) =,780(5.65) = 0,057 PWC = 8,000 PWB/PWC =.257 > : OK Alterative 2: PWB = 6,548(P/A, 2%, 0) = 6,548(5.65) = 36,996 PWC = 23,500 PWB/PWC =.574 > : OK Use icremetal ( B/C) aalysis to choose best alterative EBS 26 Eergy Systems Ecoomic Aalysis 38 9
Evaluatio of Alteratives: Example 4. Beefit-Cost Ratio Aalysis (cotiued) Use icremetal ( B/C) aalysis to choose best alterative Category Differece Alt 2 Alt Capital Cost ($) = -23,500 (-8,000) = -5,500 Net Reveue ($/y) = 6,548,780 = 4,768 PWB = 4,768(P/A, 2%, 0) = 4,768(5.65) = 26,939 PWC = 5,500 PWB/ PWC =.738 > Choose higher cost alterative, choose Alterative 2 EBS 26 Eergy Systems Ecoomic Aalysis 39 Ecoomic Aalysis ad Optimizatio Review of ecoomic aalysis Cash flow equivalece Time value of moey Depreciatio Taxes Escalatio ad iflatio Curret ad costat dollar aalysis Evaluatio of alteratives Preset worth Levelized cost Rate of retur Beefit/Cost Reveue Requiremets method Sesitivity aalysis Optimizatio EBS 26 Eergy Systems Ecoomic Aalysis 40 20
Reveue Requiremets Method Approach is similar to the levelized cost method A reveue requiremets aalysis attempts to determie the ecessary eergy price to yield the stipulated rate of retur The approach has commoly bee used by regulated utilities to establish prices, but has geeral applicatio EBS 26 Eergy Systems Ecoomic Aalysis 4 Reveue Requiremets Method The reveue requiremets fall geerally ito four categories: Capital repaymet Retur o ivestmet Expeses Taxes Other categories may iclude tax credits ad other subsidies Capital repaymet recovers the capital cost of the project over the ecoomic life. The retur represets the iterest eared o the ivestmet over the ecoomic life. EBS 26 Eergy Systems Ecoomic Aalysis 42 2
Reveue Requiremets Method: Taxes Because the reveue requiremets method specifies the rate of retur to be eared, taxes, which are part of the reveue requiremet, are computed i a special way, rather tha directly as if the reveues were kow a priori. The tax paymet i each year of the aalysis is The taxable icome, I t, is T = (t)(i t ) The reveue requiremets, I g, are I t = I g -E -D t - D i I g = C r + I r + T + E C r = capital (pricipal) repaymet ($) I r = retur o ivestmet ($) EBS 26 Eergy Systems Ecoomic Aalysis 43 Reveue Requiremets Method: Taxes The taxes are T = t(c r + I r + T + E - D t -E - D i ) = t(c r + I r + T - D t -D i ) Solve for T T t t ( C r D t I r D i ) The term I r - D i, the differece betwee the total retur o the urecovered ivestmet ad the iterest o debt, is simply the retur o equity. Defiig the debt ratio, r D (decimal), as the fractio of the cost of the project covered by debt (as opposed to equity), ad C as the urecovered ivestmet to date, the total retur at rate of retur, i, is I r = ic EBS 26 Eergy Systems Ecoomic Aalysis 44 22
Reveue Requiremets Method: Taxes the iterest or retur o debt is D i =i D r D C i D is the iterest rate o debt The D i I r i D r D i Taxes are T t t i Dr Cr Dt I r i D EBS 26 Eergy Systems Ecoomic Aalysis 45 Reveue Requiremets Method: Example Biomass Power Plat Power plat is assumed to bur biomass i a fluidized bed furace, with a et electrical capacity of 25 MW e ad a availability of 85% equal to the capacity factor The cost of the plat ($70M) is covered by both debt ad equity fuds 75% borrowed from the bak, 25% equity supplied from the owers The rate of retur o equity is assumed to be 5% The debt iterest rate is 5% As part of the fiacig pla, the ower is required to place the equivalet of oe year of debt repaymet i a savigs accout i case techical problems cause a suspesio of plat operatios ad reveue is ot geerated. This "debt reserve" accout ears simple iterest, which is available to the plat as reveue. The plat also receives a paymet for "capacity," whe it ca guaratee a certai miimum level of performace. This paymet ad the iterest o the debt reserve, reduce the amout of reveue required to cover the cost of geeratig the electrical eergy. A federal accelerated depreciatio schedule (MACRS) based o double decliig balace switchig to straight lie is used for tax purposes. Negative taxes are permitted uder the assumptio that the ower-compay is large, ad egative taxes merely reduce the tax burde of the compay as a whole. If this is ot the case, the egative taxes are ot permitted. Carry-forward of taxes is ot cosidered for this example. A productio tax credit for ope loop biomass is icluded Fuel costs ad expeses are assumed to escalate at 2.% aually, with iflatio also ruig at 2.%. EBS 26 Eergy Systems Ecoomic Aalysis 46 23
Reveue Requiremets Method: Example Biomass Power Plat Model calculatio i Excel Web ecoomic calculators for biomass see http://biomass.ucdavis.edu EBS 26 Eergy Systems Ecoomic Aalysis 47 Ecoomic Aalysis ad Optimizatio Review of ecoomic aalysis Cash flow equivalece Time value of moey Depreciatio Taxes Escalatio ad iflatio Curret ad costat dollar aalysis Evaluatio of alteratives Preset worth Levelized cost Rate of retur Beefit/Cost Reveue Requiremets method Sesitivity aalysis Optimizatio EBS 26 Eergy Systems Ecoomic Aalysis 48 24
Sesitivity Aalysis 0.6 0.4 COE ($/kwh, Costat) 0.2 0.0 0.08 0.06 0.04 0.02 0.00-200 -00 0 00 200 300 400 Relative Chage (%) Capital Cost Fuel Cost Debt Ratio Debt Iterest Rate Cost of Equity Net Efficiecy Capacity Factor EBS 26 Eergy Systems Ecoomic Aalysis 49 Sesitivity Aalysis Relative Cost 60 40 20 Relative Chage i COE (%) 00 80 60 40 20 0-20 -40-60 -200-00 0 00 200 300 400 Relative Chage (%) Capital Cost Fuel Cost Debt Ratio Debt Iterest Rate Cost of Equity Net Efficiecy Capacity Factor EBS 26 Eergy Systems Ecoomic Aalysis 50 25
Ecoomic Aalysis ad Optimizatio Review of ecoomic aalysis Cash flow equivalece Time value of moey Depreciatio Taxes Escalatio ad iflatio Curret ad costat dollar aalysis Evaluatio of alteratives Preset worth Levelized cost Rate of retur Beefit/Cost Reveue Requiremets method Sesitivity aalysis Optimizatio EBS 26 Eergy Systems Ecoomic Aalysis 5 Optimizatio The opportuity ofte exists to optimize the eergy system i some way, for example, the size or capacity of the system A optimizatio requires a objective fuctio expressig explicitly how the optimum value is to be determied Tradeoffs amog capital, operatig, ad fuel costs iflueced by size or scale frequetly lead to a optimum providig either miimum cost or maximum profit EBS 26 Eergy Systems Ecoomic Aalysis 52 26
Ecoomy of Scale A ecoomy of scale is said to occur whe the cost per uit output of productio varies depedig o the size or scale of the facility. A commo expressio is: C p C o M p M o C p = capital cost of the plat ($) C o = capital cost of some base or referece size plat ($) M p = size of the plat (e.g., kw) M o = referece size (e.g., kw) s = scale factor (dimesioless) s EBS 26 Eergy Systems Ecoomic Aalysis 53 Ecoomy of Scale For capital costs expressed per uit of capacity: K p K o M p M o s K is the cost per uit of istalled capacity (e.g., $ kw - ), K = C/M EBS 26 Eergy Systems Ecoomic Aalysis 54 27
Objective Fuctio Aalytic fuctio defiig how the optimum coditio is to be determied Example: Miimizatio of electricity productio cost mi[cost of Electricity = Capital Cost + Operatig Cost + Fuel Cost] Note this does ot address delivery ito the fial demad sector. Full optimizatio should cosider this as well. EBS 26 Eergy Systems Ecoomic Aalysis 55 Optimizatio: Example Biomass power plat subject to ecoomy of scale i both capital ad o-fuel operatig costs Optio : s = costat Optio 2: s = variable to accout for greater risk at large scale EBS 26 Eergy Systems Ecoomic Aalysis 56 28
Capital Cost C p K p M p K o M M s s p o Aualized capital cost is C p (A/P) where (A/P) is the capital recovery factor Aual eergy productio is M p h where h is the umber of hours per year the plat operates at its average capacity M p Capital cost cotributio to the levelized cost of eergy, C ($ kwh - ): C (A / P)C p M p h (A / P)K o h M p M o EBS 26 Eergy Systems Ecoomic Aalysis 57 s No-fuel O&M Costs Subject to ecoomy of scale, operatig ad maiteace costs scale as: R M s p R o M o the prime deotes that the costs are total aual costs (i.e., $ y - ) The value of s for O&M eed ot be the same as that for the capital cost EBS 26 Eergy Systems Ecoomic Aalysis 58 29
No-fuel O&M Costs Per uit eergy output: R R M p h R is the cost per uit eergy ($ kwh - ) R R o M p M o s EBS 26 Eergy Systems Ecoomic Aalysis 59 Fuel Cost Fuel cost derived as described previously for a rectagular road grid: D avg q /2 T Q/2 2T / 2 q = average fuel desity i the area (t km -2 ) Q = aual plat fuel demad (t) T = average truck payload (t) EBS 26 Eergy Systems Ecoomic Aalysis 60 30
Fuel Cost Total fuel cost (F', $ t - ) is composed of two compoets: acquisitio cost (F' p, $ t - ) icludig the cost of productio, harvestig, storage, ad processig, ad trasportatio cost (F' t, $ t - ) Defie F" t as the uit cost of trasportatio ($ t - km - ) EBS 26 Eergy Systems Ecoomic Aalysis 6 Fuel Cost Total fuel cost cotributio to COE: F F p F t F p F t Davg T Fuel cost cotributio to the price of output electrical eergy, F, $ kwh - : F 3.6F Q f Q f = heatig value (or heat of combustio of the fuel), MJ t - = plat thermal efficiecy (dimesioless) EBS 26 Eergy Systems Ecoomic Aalysis 62 3
Fuel Cost Total fuel demad of the plat, Q, is a fuctio of the plat size: Q 3.6M h p Q f If Q>>T: D avg T Q q / 2 E = Q f /3.6 (kwh t - ) is the reciprocal of the fuel rate M h p qe / 2 EBS 26 Eergy Systems Ecoomic Aalysis 63 Fuel Cost Fuel cost cotributio to COE: F F p E M h p q / 2 F t E 3/2 EBS 26 Eergy Systems Ecoomic Aalysis 64 32
Objective Fuctio Miimize COE: mi P e C R F am p s bm p /2 c a M o s (A / P)K o h R o b F t h /2 q /2 E 3/2 c F p E EBS 26 Eergy Systems Ecoomic Aalysis 65 Optimum Capacity The miimum, or optimal, value of P e is foud by takig the derivative of P e with respect to M p ad settig equal to zero (dp e /dm p = 0) With F' p ad F" t costat: M opt 2a( s) b.5 s where M opt is the optimal size (kw) which miimizes P e. EBS 26 Eergy Systems Ecoomic Aalysis 66 33
Example: fixed s M o = 25,000 kw K o = $2,000 kw - R o = $0.02 kwh - h = 6,570 h y - 75% capacity factor F" t = $0. t - km - q = 000 t km -2 E =,389 kwh t - Q f = 20,000 MJ t - = 0.25 i= 0.06 l = 20 years s = 0.979* Capital/O&M M opt =,253 MW ($/kwh) 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.0 Capital + O&M Cost Feedstock Cost Total Cost s = costat Miimum COE Note limited curvature i regio of miimum 0.00 0 00,000 0,000 M (MW) *derived from variable s model, see followig EBS 26 Eergy Systems Ecoomic Aalysis 67 Ifluece of Scale Factor o Optimal Capacity ad Sesitivity of Solutio (s = costat) Mopt (MW) dmopt/ds (GW) 2500 2000 500 000 500 0 5 0-5 -0-5 -20-25 0 0.2 0.4 0.6 0.8 s EBS 26 Eergy Systems Ecoomic Aalysis 68 34
Relative Mopt Sesitivity Fuctios 5 4 3 2 z= h s Qf H Y/X b F =Ct 0 0 2 3 4 5 Relative z Ko Ro q Mo z q K o R o H Q f =H M opt z 3 2s a / (.5 s) 4 q (s )/(.5 s) ; a 4 2 a s sm a 2 q /2 o a 5.5 s a 2 / (s.5) a 5 K o a 6 (s0.5)/ (.5s) ; a 5 2 f h sm o s s, a 6 2R o sm o a 7.5 s a 2 / (s.5) a 7 R o a 8 (s0.5)/(.5s) ; s a 7 2 sm o, a8 2 f h K o sm o s 2 a /(.5s) 9 H (s)/(.5 s) a ; a 9 2 3 s a 2 H 3/2 2 /(.5 a s) 0 (s)/(.5 s) a ; a 0 2 3 s a 2 3/2 sm o s s M o s CF =C t s.5 a /(.5 s) (2.5s)/(s C.5) t ; a 2 a t s a 2 C t s.5 a /(.5 s) 2 (2.5s)/(s.5) ; a 2 2 a a 2 sm o f a 3.5 s a 2 / (s.5) a 3 f a 6 (s 0.5)/ (.5s) ; a 3 2 K o h sm o s h. 5 s a h 3/2 4 a 5 h /2 2 a 4 a 2 h fk sm 2 /2 o o s, a 5 a 2 h R sm / 2 o o M o s.5 s a 6 /(.5 s) / (2s3) M o ; a6 2 a s a 2 Y/X=b b.5 s M b opt 2b b s.5 s M opt.5 s l M.5 s opt s l M o z idicates referece value to be iserted. M o s (s0.5)/(.5 s) 3 2 a 4 h 5/2 2 a 5 h 3/2 s s ; EBS 26 Eergy Systems Ecoomic Aalysis 69 Variable s Use of fixed scalig factor, s, for all sizes igores potetial risks as the size becomes very large Large coal ad uclear plats (> 400 MW) tred toward s > 0.9 Model with s = variable yields differet optimal capacity depedig o fuctio used, s = f(m) Commet EBS 26 Eergy Systems Ecoomic Aalysis 70 35
Example: variable s.0 s ta ( M ) 0.9 0.8 (0.8,-2, 2, 0) 0.7 0.6 s (--) 0.5 0.4 0.3 0.2 0. 0.0 0 00,000 0,000 M (MW) EBS 26 Eergy Systems Ecoomic Aalysis 7 Fixed ad Variable s K ($/kw) s = variable s = costat 8,000 7,000 6,000 5,000 4,000 3,000 2,000,000 0 0 00,000 0,000 M (MW) EBS 26 Eergy Systems Ecoomic Aalysis 72 36
Optimizatio uder variable s dp dm 0 M M ( s ) o M o s2 Term l M M o M M o s M ( M ) 2 Term 2 / 2 2 a b M Solved via root-fidig techique Term 3 Excel Model EBS 26 Eergy Systems Ecoomic Aalysis 73 Cost Fuctios (costat/variable s) 0.20 Mopt=252 MW costat s Mopt=305 MW variable s 0.5 ($/kwh) 0.0 Total Cost Total Cost 0.05 Capital + O&M Cost Capital + O&M Cost Fuel Cost Fuel Cost 0.00 0 00 000 0000 0 00 000 0000 M (MW) M (MW) EBS 26 Eergy Systems Ecoomic Aalysis 74 37