November 2014 F-1



# Appendix F: General Algebraic Representation of the Biogenic Assessment Factor Equations

Table of Contents 1. Introduction ......................................

................................................... ................................................... ..................... F-1 2. Simple Algebraic Representation ...................

................................................... .................................................. F-1 2.1. Example 1: Simple Carbon Flow with Point of Assessm ent at the Boiler/Fermenter Mouth

.................................................. ................................................... ................................................... ......................... F-3 2.2. Example 2: Changing the Point of Assessment to Stac k Emissions ....................................... ...... F-6 2.3. Example 3: Changing the Point of Assessment to Fore st/Farm Gate ...................................... ... F-8 2.4. Example 4: Combining Forms from Examples 2 and 3 ..

................................................... .............. F-10 2.5. Example 5: Extending to a More Complex Biogenic CO 2 Trail—Toward Fully Generalizing the NBE Equation .........................................

................................................... ............................................... F-1 1 

## List of Figures Figure F-1: Conceptual Diagram Illustrating the Cal culation of Potential Gross Emissions at Different Points of Assessment. ...................

................................................... ............................................... F-3 Figure F-2: Stack Emissions as the Point of Assessm ent. .............................................. ...................................... F-7 Figure F-3: Forest/Farm as the Point of Assessment.

.................................................. ........................................ F-8 Figure F-4: Conceptual Diagram Illustrating a More Complex Carbon Trail. ............................. .............. F-11



## 1. Introduction The equations and conceptual diagrams in this appen dix are intended to illustrate the various carbon flows that contribute to the calculation of net atmospheric contribution of biogenic CO 2 emissions associated with the production, processin g, and use of biogenic material at a stationary
source (NBE) and biogenic assessment factor ( BAF ) values for stationary source biogenic feedstock consumption. This appendix builds on the primary NBE equation provided in the main report to develop a generic algebraic formulation describing how the net atmospheric contribution of biogenic CO 2 emissions for a stationary source could be calcula ted at different points of assessment, depending on the specific policy or programmatic co ntext, and to provide simple concrete examples of how the generic equation could be applied. 

## 2. Simple Algebraic Representation As discussed throughout this report, the Net Biogen ic Emissions (NBE) from stationary source biomass consumption equation can be presented as th e following equation:

 =    	
 + 

 +  + 
      = 
 (EQ. F.1)November 2014 F-2

Where the BAF

is given by Equation F.2:


 = 

 = 	
 + 

 +  + 
     (EQ. F.2) The equations above are designed to transform a mea surable or estimated quantity (the carbon content of the biogenic feedstock used at the point of assessment [potential gross emissions or PGE ]) into a quantity that cannot be directly measured (the net atmospheric biogenic CO 2 contributions associated with different stages of b iogenic feedstock production, processing, and use at a stationary source [ NBE ])

. The terms in the NBE equation each play a specific role in this transformation:

• PGE is the carbon content of the biogenic feedstock us ed by a specific entity or generally consumed). This is a quantity that could be measure d or estimated at different points of assessment (e.g., at the boiler mouth, stationary s ource gate, feedstock production site, or at the stack: wherever the point of assessment needs t o be. Thus, this term can have different values indicated by subscripts, representing differ ent points along the supply chain).

• L is a unitless adjustment factor greater than or equ al to one that represents biogenic feedstock carbon that leaves the supply chain (e.g. , via transit or decomposition, deviated for use as a product) between the feedstock product ion site and input into the conversion process at a stationary source. L scales PGE , as it was measured at the point of assessment, up to account for any losses during transportation or storage between the feedstock production site and the point of assessment. PGE times L is thus the carbon content of the biomass that was grown at the feedstock production site in order to deliver the quantity of feedstock measured at the point of assessment.

• P is a unitless adjustment factor between zero and on e, equal to the share of the carbon content of the feedstock at the point of assessment that is emitted to the atmosphere by a stationary source (versus that which is embedded in products). In effect, this term also reflects the share of carbon that remains in produc ts, that is either not emitted to the atmosphere or is sold and eventually emitted to the atmosphere by a downstream user.

• (GROW + AVOIDEMIT + SITETNC + LEAK) represents the landscape emissions effect. This landscape emissions effect is the sum of four unitl ess factors that relate the total biogenic carbon content of the feedstock grown at the feedst ock production site, i.e. (PGE)*(L) , to related landscape biogenic carbon pools. The detail s of these terms are discussed elsewhere in this framework. For the purposes of this appendi x, we can think of the terms (GROW + AVOIDEMIT + SITETNC + LEAK)*(PGE)*(L) as the estimated net contribution to the atmosphere associated with growing, harvesting, pro ducing, processing, and using the feedstock that was measured at the point of assessm ent. This amount is then multiplied by P to determine the share that is actually emitted an d is the responsibility of a particular entity. The framework itself is designed to be flexible eno ugh such that it can be applied to a variety of programs with different requirements. For example, PGE in the NBE equation above is the potential gross emissions at the point of assessment for the purposes of applying the biogenic assessment factor. Depending on how the framework is applied, this point of assessment could be interpretedNovember 2014 F-3

as biogenic emissions at the boiler mouth, the stac k, or total potential biogenic emissions at the farm gate once biomass has been harvested. Addition ally, losses can occur and products can be produced at different points along the supply chain

. To understand how the NBE equation adapts to different points of assessment and accounts for products and losses at different points along the s upply chain, we need to understand how the L and P terms are calculated and what exactly they represe nt. In order to build this understanding, this appendix begins with a simple example and then shows how the generic version of the equation is able to capture more complexity in subs equent examples.


### 2.1. Example 1: Simple Carbon Flow with Point of Assessm ent at the Boiler/Fermenter Mouth Consider the following conceptual example provided by Figure F-1. This flow diagram shows the evolution of PGE along the production supply chain. This representat ion uses atmospheric accounting methods, so terms that represent an emis sion (or potential emission) to the atmosphere (e.g., LOSS, PGE

) are positive, and terms that represent sequestrat ion (e.g., PROD ) are negative.

Figure F-1: Conceptual Diagram Illustrating the Cal culation of Potential Gross Emissions at Different Points of Assessment. The first thing to note in this diagram is that it contains different variables from the variables in Equations F.1 and F.2 such as LOSS and PROD . The main equations discussed above are designed t o be applied with limited data requirements. The exam ple here is designed as a thought experiment to “follow the tons of biogenic CO 2 ,” and we show how these quantities can be used to calculate the variables that go into the BAF equation. Another difference is that the PGE variables gain subscripts to represent the different points of assessment at which PGE could be measured (e.g., at the forest/farm, at the boiler mouth, out the stack). T his can be generalized to PGE iin order cover any number of measurement points, where i indexes over all the points of measurement. The re d circle indicates where the point of assessment is in this example, so here PGE1 would be equal to PGE from Equations F.1 and F.2 as applied at the boiler/ferm enter mouth.
The terms LOSS and PROD represent the actual tons of carbon lost in transp ortation or storage along the supply chain ( LOSS ) or stored in final products or by-products ( PROD ) (rather than emitted during conversion). The subscripts indicate at which stage in the supply chain each occurs,November 2014 F-4

e.g., LOSS 1 occurs before PGE 1 , and PROD 2 occurs before PGE 2 . These subscripts will become necessary when we move to more complex supply chain examples and generalize the equations. In the schematic of this example (Figure F.1), the top row represents actual gross emissions to the atmosphere. Here losses are assumed to generate act ual emissions, and PGE 2 represents actual emissions out the stack (note that at this point th e emissions are no longer “potential,” but for notational reasons we still refer to these stack em issions as PGE ). The middle row is for potential emissions at various assessment points. These value s represent tons of biomass moving through the supply chain that have not yet been emitted or sequestered or contained in products. The bottom line of Figure F.1 is tons that go into prod ucts or by-products and are either not emitted to the atmosphere or not emitted by this entity. Note that because we are using atmospheric accounting, emissions and potential gross emissions have a positive value, and carbon contained in products or by-products is assigned a negative valu e.

• PGE 0 : Represents PGE at the forest/farm gate. This is the total harvest ed biogenic CO 2 with the potential to be emitted at the forest/farm gate and includes all biomass that is harvested and transported from the forest or farm to the stat ionary source facility.

• LOSS 1 : Represents the biogenic CO 2 lost in transportation or storage between the forest/farm gate ( PGE 0 ) and the boiler/fermenter mouth ( PGE 1 ).

• PGE 1 : Represents PGE at the boiler/fermenter mouth.

• PROD 2 : Represents biogenic CO 2 stored in long-term product pools (including lumbe r, ethanol, or other purely marketable products produc ed with a portion of the harvested biomass) or other nonmarketable industrial by-produ cts (including ash).

• PGE

2 : Represents emissions at the stack. Thus, the equations relating these variables are:
PGE 1 = PGE

0 – LOSS

1 = 10 – 4 = 6 tCO 2 e. (EQ. F.3)
PGE 2 = PGE

1 + PROD 2 = 6 + (-2) = 4 tCO 2 e. (EQ. F.4)
Now let us define how the terms in the NBE equation (GROW + AVOIDEMIT + SITETNC + LEAK), (L), and (P) are calculated and apply our example values. Note t hat in the equations below we adopt the notation convention that PGE at the point of assessment ( PGE1 in this example, or simply PGE in Equations F.1 and F.2) is written as PGE j, where the subscript “

j ” represents the point of assessment (in this example, j =1, the boiler/fermenter mouth). The landscape emissions effects terms ( GROW, AVOIDEMIT, SITETNC, and LEAK ) are discussed extensively elsewhere in the framework, including d etailed discussions of how they can be calculated in practice. It is important to note tha t GROW, AVOIDEMIT, SITETNC , and LEAK

are all unitless. 1 For the purposes of this appendix, we can define o ur landscape emissions effects terms from the overall BAF

equation as follows:

1 However, the framework can be adapted to use units instead of unitless values as needed for a specifi c application.November 2014 F-5

 	
 + 

 +  + 
  =  
   

. (EQ. F.5)
This is merely another way to specify the landscape effect. The original term is a unitless ratio of the contribution of landscape effects on the overal l BAF value. However, we can also think of each of these elements as being relative to the total amoun t of biomass that is harvested at the forest or farm ( PGE 0 ). G, A, S, and Lk are variables that represent the actual tons of lan dscape net emissions resulting from producing PGE 0 tons of biomass. 2 In that sense, (G+A+S+Lk) represents actual net emissions on the landscape caused by a harvest of PGE 0 . The landscape-level emissions are normalized by

PGE 0 to arrive at the original, unitless term.
Now let’s consider the terms L and P from the NBE

equations. L accounts for transportation or storage losses, relating the carbon content of biom ass used by the facility at the boiler fermenter mouth ( PGE1 ) to the carbon content of that biomass when it was grown and harvested at the forest/farm ( PGE 0 ).

L is defined as the ratio of harvested potential bio genic emissions ( PGE 0 ) and PGE evaluated at assessment point j

.  =       (EQ. F.6)
Note that this fraction will result in a positive n umber greater than or equal to one. Also note that although LOSS

does not appear in this equation, we could substit ute in from equation 0 to express L in terms of just one of the PGE variables and the LOSS

term. The purpose of L here is twofold. First, it represents the transfor mation of PGE j to PGE 0 , implicitly capturing any losses and products 3 that occur between the point of assessment and the forest/farm. Furthermore, if considered in combination with the landscape effects term, it serves to bring the landscape effects in relation to PGE j . That is, while landscape-level terms implicitly c apture the emissions impact of total biogenic feedstock harves ts ( PGE 0 ), L provides the necessary adjustment for total potential biogenic emissions within the s tationary source boundary.

P accounts for biogenic feedstock carbon embodied in process products that pass out of the supply chain as product prior to combustion or after combu stion by exiting the stationary source through forms other than as stack emissions. P is a unitless term between zero and one that scale s PGE down so that the portion of biogenic feedstock carb on embodied in products or byproducts is not included in the final results when calculating the assessment factor for biogenic CO 2 emissions. As shown in the subsequent sections of this appendix, the equation used to calculate P can become quite complex depending on the point of assessment and the production supply chain. This example

2 Where GROW = G / PGE0 ; AVOIDEMIT = A / PGE0 ; SITETNC = S / PGE0 ; and LEAK = Lk / PGE0 . 3 In the example presented here in section 1.1 only losses occur between the point of assessment and th e forest/farm,
so L only needs to account for these losses. However, if any products or by-products are produced between t he forest/farm and the point of assessment (as in the example presented later in section 1.5), then in or der to scale PGE i up to PGE0 , L needs to account for these products, as well as any losses. The responsibility for the carbon embodied in these products will separately be accounted for in the P term as discussed below.November 2014 F-6

is constructed to produce the most simplified versi on of the P equation. Equation F.7 defines P as calculated in this simplified example:

 = + ∑ 	
 "

 "#   . (EQ. F.7)
Note that for many stationary sources, biomass will be used to produce a number of final products, so the summation in equation 0 simply sums over all i products that are produced, indexed from 1 to S , where S is the last point on the carbon trail and represen ts the stack. In this example, PROD 2 is the only product, so this summation would simply be equal to PROD 2 . Because PROD i is always negative and the absolute value of the sum of all PROD imust be less than PGE j , the sum of all PROD i divided by PGE

j will be a fraction between negative one and zero, a nd P will be a positive number between zero and one.

P is one minus the share of PGE j that is sequestered in products or by-products and is either not emitted to the atmosphere or is sold and eventually emitted to the atmosphere by a different entity. (Note that because PROD is negative, technically this expression is one pl us a negative number.) In effect, we are taking away the portion of carbon th e products or by-products are responsible for and attributing the remainder to the facility. Alth ough this is a relatively simple example, the mathematical representation of this term will chang e as we consider more complicated scenarios. We can now rewrite the equation for NBE by substituting into Equation F.1. from Equations F.5, F.6, and F.7, resulting in the following:

 = 

  
      

    + ∑ 	
 "

 "#   

. (EQ. F.8)

The PGE 0 in numerator of the L term cancels with the PGE 0 in the ratio of ( G+A+S+Lk )/ PGE 0 , and the PGE j in the denominator of the L term cancels with the first PGE j in the NBE

equation, so that the term (GROW + AVOIDEMIT + SITETNC + LEAK)* (PGE)*(L) = G+ A+S+Lk . As discussed above, this represents the actual net emissions on the landscap e caused by a harvest of PGE 0 . All of this canceling is important, because G, A, S, and Lk are not observable, and even PGE 0 may not be readily observable, but we can estimate the unitless values of L, GROW, AVOIDEMIT, SITETNC, and LEAK through other means (such as retrospective referenc e point or future anticipated baseline modeling methods) and calculate net emissions while only nee ding to observe PGE j . Now, applying numerical values from Figure F-1 and assuming that ( G+A+S+Lk ) = 3 tCO 2 e (for the purpose of these examples), net biogenic emissions at point of assessment “ j =1” are calculated as: = $  %

    $   +  &' 

$  = %  (

$  = '	* ' + . (EQ. F.9)


### 2.2. Example 2: Changing the Point of Assessment to Stac k Emissions Continuing through the following examples, necessar y modifications are made to the BAF equation for alternative points of assessment. The biogenic CO 2 trail depicted by Figure F-2 is used in the following examples to demonstrate that the same NBE can be calculated regardless of the point ofNovember 2014 F-7

assessment. This provides a consistency check becau se this exercise merely alters the point of assessment but not the biogenic carbon trail. The second example also relies on the simple hypoth etical carbon trail from Figure F-1, with the only difference being that the new point of assessm ent is stack emissions (represented by PGE 2 ), i.e., “ j =2

. ”

Figure F-2: Stack Emissions as the Point of Assessme nt. In the first example, all losses occurred before th e point of assessment, and all products were produced after the point of assessment. In this exa mple, both losses and products occur before the point of assessment. This change will require us to refine our interpretation of L and rewrite our definition of P

in more general form, as shown in Equation F.10. E quation F.11 is the new more generalized NBE equation that covers this example and incorporates this revised definition of P

. Note that example one is a special case of this new equation.

 = + ∑ 	
 "

 "#   & ∑ 	
 "  "# (EQ. F.10)

 = 

  
      

   , + ∑ 	
 "

 "#   & ∑ 	
 "  "# - (EQ. F.11)
The first point to notice is that L remains the ratio of PGE 0 to PGE j . Although the expression for L has not changed, the numerical value and the interpreta tion will be different, because PGE j is now stack emissions. In our original interpretation, L was a unitless adjustment factor greater than or e qual to one, that scales PGE

as it was measured at the point of assessment up t o account for any losses during transportation or storage between the forest /farm and the point of assessment. In a more general representation, when there can be both prod ucts and losses before the point of assessment, L scales PGE

as it was measured at the point of assessment up t o account for any losses during transportation or storage between the forest/farm a nd the point of assessment and to account for any products produced between the forest/farm and t he point of assessment. In the general form, L does more than just account for losses, it serves a s a general scaling factor to relate the carbon content of the biomass feedstock at the point of as sessment ( PGE j ) to the carbon content of biomass that was grown and harvested at the forest/farm and that was required to generate that feedstock ( PGE 0 ), accounting for all differences between those two points of measurement.November 2014 F-8

As in the previous example, (GROW + AVOIDEMIT + SITETNC + LEAK)* (PGE)*(L) = G+ A+S+Lk represents the primary portion of the equation, the net emissions on the landscape caused by a harvest of PGE 0 . The remaining job for the P term is to determine what share of these net lands cape emissions a particular facility is responsible for. The P definition from Equation F.7 must be altered to ensure that products that are processed before the point of assessment are properly accounted for, resulting in the new definition of P in Equation F.10. To do so, the denominator in the fraction used to calculate P

must be adjusted. Instead of representing the rati o of all products produced to PGE j, this new fraction deducts contributions from prod ucts that occur prior to the point of assessment from PGE j in the denominator ( ./0 1 − ∑ .345 6 1 678 ). Thinking back to Equation F.4, PGE j = PGE

j-1 + PROD

j, rearranging this expression and generalizing, the new denominator is simply the expression for PGE j-1 , so even though the point of assessment is now at the stack, after the products have been produced, our expression for P still needs to consider products as a share of PGE at a point before the products were produced. The following numerical example illustrates this po int. Because all that has changed in this example is the point of assessment, the value calculated fo r NBE should be the same as in the first example. Substituting numerical values from Figure F-1, and again assuming that ( G+A+S+Lk ) = 3 tCO 2 e, NBE is calculated in equation 0. = (  %

    (  9 + &'

(&&' : = % 9 − '

$ : = %  (

$  = '	;* ' + (EQ. F.12)
Note that although the point of assessment has chan ged, net biogenic emissions remain consistent with the previous example that evaluated PGE at the boiler/fermenter mouth.


### 2.3. Example 3: Changing the Point of Assessment to Fore st/Farm Gate The third example (Figure F-3) also relies on the s imple hypothetical carbon trail from Figures F-1 and F-2, with the only difference being that the ne w point of assessment is the forest/farm (represented by PGE 0 ), i.e., j =0.

Figure F-3: Forest/Farm as the Point of Assessment. In this example, because the point of assessment is at PGE 0 , the L term is equal to one. The L term is still theoretically scaling between the point of as sessment and the forest/farm level, but as they areNovember 2014 F-9

the same, the L term has a value of one. PGE 0 is directly measured as it is PGE j in this instance, so PGE j does not need to be scaled up, and L is simply equal to one. In this example, losses ( LOSS 1 ) do not enter into the L term, but because the mass balance has not changed, losses still are occurring, but they are o ccurring after the point of assessment, so the question is how are losses accounted for now? In al l examples, ( PGE j )*( L ) = PGE 0 . The equation then assumes that all of PGE 0 is emitted, unless the P term specifies that a share of PGE 0 is sequestered or contained in products. This means that once PGE j is transformed to PGE 0 , unless there are products, the calculation does not change regardless of wheth er the emissions come from a loss or from the stack. Consider for a moment an example without pro ducts (i.e., P =1); here <=0 = ./0 1  >?@AB C>D E   C>D E

C>D F  , and whether the point of assessment is before or after the losses occur (i.e., j = 0 or 1; or, in this example, at the forest/farm or at the boiler/fermenter mouth), either way, all the PGE

terms cancel so that NBE = G+A+S+Lk

. Returning to the example shown in figure 3, althoug h losses do not directly enter the L term in this example, they do enter the P term. In this instance, the equation must account for losses that occur after the point of assessment j , but before the first source of products, in the P term (as LOSS 1 does in this example). Similar to the previous example, a more generic form of the P term is applied. The calculation of P

is modified such that the denominator accounts for the losses that occur after the point of assessment, but before the first product i s produced. Similar to the previous example, in the denominator of the fraction in the P term, losses are deducted from PGE j in the denominator ( ./0 1 − ∑ G4HH 6 CI 6718  , and as demonstrated in Equation 0F.13, this is es sentially the expression for PGE

j+1 . In this case, the term PR stands for the process stage i where the first product is made (that is, the first stage of the industrial process in which PROD i> 0). In effect, as the P term ensures that the facility is not held responsi ble for the portion of biogenic carbon that is passed on embodied in products or by -products, the P term also ensures that the facility is not held responsible for the portion of the losses that the products or by-products are responsible for. The responsibility for the losses is shared between the facility and the products or by-products in proportion to their respective share s or biomass that remains after the losses occurred. Equation F.14 presents the NBE equation for this scenario.

 = + ∑ 	
 "

 "#   & ∑ 
 "  "#J (EQ. F.13)

 = 

  
      

   , + ∑ 	
 "

 "#   & ∑ 
 "  "#J - (EQ. F.14)
Substituting numerical values from Figure F-1, and again assuming that (G+A+S+Lk) = 3 tCO 2 e, yields the following NBE calculation, which generates the same value for NBE using the more general equation in this example: =   %

      9 + &'

&( : = % 9 − '

$ : = %  (

$  = '	* ' + (EQ. F.15)November 2014 F-10



### 2.4. Example 4: Combining Forms from Examples 2 and 3 In order to write a more general form of the NBE equation that can either account for products that
occur before the point of assessment or any losses generated be fore any products are produced, both forms of the equation as established in exampl es 2 and 3 must be combined. Doing so results in Equation 0F.17. This generalized form can be app lied to all the previous examples of biogenic CO 2 calculation at different points of assessment. Note that S in the equations below represents the last point on the carbon trail; i.e. the stack, so PGE S would be stack emissions.

 = + ∑ 	
 "  "#   & ∑ 	
 "  "# & ∑ 
 "

 "# (EQ. F.16)

 = 

  
      

   K + ∑ 	
 "  "#   & ∑ 	
 "  "# & ∑ 
 "

 "# L. (EQ. F.17)
Equation F.17 is general enough to cover all points of assessment for any carbon chain so long as all losses ( LOSS

i ) occur before the first product ( PROD i ) is generated. As explained in the previous examples, the denomina tor in the fraction in the P term uses the relationships in Equations F.3 and F.4 to transform PGE j into PGE at another point in the carbon trail. Before providing a more complex carbon trail scenario in example 5, it will be useful to introduce some new subscript notation and be a bit more explicit about the subscript notation used so far.

• 0 : First point on the carbon trail; represents the f orest/farm. PGE

begins at 0

. • S : Last point on the carbon trail; represents the st ack. PGE S is stack emissions.

• PR : Index number for the first product produced on th e carbon trail. PROD PR is the first product. The indexing for points along the carbon trail begi ns at 0, the forest/farm, and ends at S , the stack. The convention for indexing LOSS and PROD is that they take an index number equal to the nex t PGE , and a LOSS

and a PROD cannot both occur at the same stage. 4 Additionally, any loss that occurs after all products are produced is indistinguishabl e from stack emissions PGE s in these equations. To simplify the equation, it is assumed that LOSS is not specified after the last product is produce d in the carbon trail, and instead any such losses ar e rolled into the calculation of PGE s . With this notation in mind, Equations F.16 and F.17 are rewritten as:

 = + 	
 N*OP

 	Q (EQ. F.18)

4 If LOSS i and PROD i occurred at the same stage i , it would be ambiguous which came first and whethe r that PROD i should share responsibility for LOSS i or not. For purposes of specifying the full theore tical carbon trail, a PGE would need to be inserted between LOSS i and a PROD i .November 2014 F-11

 = 

  
      

   9 + 	
 N*OP

 	Q :
, (EQ. F.19)

Where we also define a new variable, .345 RSTUV = ∑ .345 6 @ 678 , or the sum of all products produced along the carbon supply chain. PGE PR-1 is potential gross emissions at the point in the carbon supply chain before the last product is prod uced. In examples 1 through 3 above, PGE PR-1 = PGE1 . With this new notation, the expression for P in Equation F.18 does not vary with different points of assessment. The examples so far all assume that losses occur be fore any products are produced in the carbon supply chain. If a product is produced at a point i n the carbon chain before a loss occurs, then that product should not be held responsible for the subs equent loss. Example 5 below shows how the NBE equation can be fully generalized to cover this si tuation. 

### 2.5. Example 5: Extending to a More Complex Biogenic CO 2 

### Trail—Toward Fully Generalizing the NBE Equation To illustrate how one could apply the NBE equation to a more complex carbon trail, consider the more complex hypothetical carbon trail in Figure F- 4. The primary differences between this conceptual diagram and the previous examples are th at this example includes four potential points of assessment for PGE j, and multiple points where products are produced a nd losses occur.

Figure F-4: Conceptual Diagram Illustrating a More Complex Carbon Trail. If one attempts to calculate the P term in the NBE

equation for this example using Equation F.180, the resulting value would be P = 1 + PROD Total

/ PGE 1 = 1+(−3/10) = 7/10. This would be incorrect though, because it assigns the facility full respon sibility for the LOSS 3 term and thus does not account for the portion of LOSS 3 that PROD 4 is accountable for. To properly account for how responsibility for LOSS 3 is shared, one should apply the following equation s for P and NBE :

 = + 	
 N*OP  ∑ K 
 " W ∑ 	
   #"  " XL  "#  	Q (EQ. F.20)

 = 

  
      

   Y + 	
 N*OP  ∑ K 
 " W ∑ 	
   #"  " XL  "#  	Q Z (EQ. F.21)November 2014 F-12

Assuming j =1, adding numerical values, and again assuming tha t (G+A+S+Lk) = 3 tCO 2 e, gives the following calculation of NBE: =   %

'   '   K + &%  %   Q' $ 
 L = % K − %% % 
 L = % 9 − (

 : = % 9 $

 : = . [ (EQ. F.22)
This example is more complicated than the earlier e xamples but can still be thought of intuitively. At the end of the carbon trail, PGE 4 and PROD 4 are responsible for 4 tCO 2 e and 2 tCO 2 e of PGE 3 , respectively. They share responsibility for LOSS 3 proportionally, 4/6 for PGE 4and 2/6 for PROD 4 . So PGE 4 is responsible for 2 tCO 2 e of LOSS 3 and PROD 4 is responsible for 1 tCO 2 e of LOSS 3 . Put these together and PGE 4 is responsible for 6 tCO 2 e of PGE 2 and PROD 4 is responsible for 3 tCO 2 e of PGE 2 . These are also the quantities they are responsible for of PGE 1 , because the only difference between PGE1 and PGE

2 is the 1 tCO 2 e that PROD 2 is responsible for. So PGE 4 is responsible for 6/10 of PGE 1 . As shown in the calculation above in Equation F.22, this confirms that P = 6/10. Note that because LOSS 1 occurs before all products, responsibility for it is shared in the same proportion as responsibility for PGE PR-1 , so it does not need to be accounted for in P

. 5 Although Equation F.21 is a more general representa tion of NBE than the previous examples, it does not cover all possible scenarios. If the carbo n chain is more complex, and there are multiple losses after the first product, then the ∑ .345 B @ B76 term in the numerator of the fraction that shares out responsibility for LOSS iwill not account for PROD k ’s share of any subsequent losses, instead assigning full responsibility for those subsequent losses to PGE s . Instead of demonstrating this with another example, the same effect can be illustrated by calculating P with respect to PGE 0 instead of PGE PR-1 . This is not a more general way of calculating P, but simply an equivalent expression. With this change, one needs to account for LOSS 1 in P , resulting in the following equations:

 = + 	
 N*OP  ∑ Y

" \ ∑ 	
  J ∑ 
 P 	
   P   P#"J   #"  " ] Z  "#

  (EQ. F.23)

 = 

  
      

   ^ _ _ _ `
+ 	
 N*OP  ∑ Y

" \ ∑ 	
  J ∑ 
 P 	
   P   P#"J   #"  " ] Z  "#

  a b b b c

. (EQ. F.24)

Again assuming j =1 and adding numerical values gives the following calculation of NBE: =   %

'   '   ^ _ _ _ `
+  &%      \ Q JQ'J% Q' $ 

 ]   %   Q'

$ 

' a

b b b c (EQ. F.25)

5 Note that while LOSS 1 does not show up in the P term in this example, it does not drop out of the NBE calculation entirely as it is accounted for in the L term.November 2014 F-13

 = % K − %  J%  % % 
' L = % 9 − %( ' : = % 9 − [

' : = % 9 '

' : = . [. (EQ. F.26)
As shown in this numerical example, this expanded e quation still generates the same answer for NBE in this example. More complex supply chain carbon trails will requir e continually more complex expressions for P. Comparing the two equivalent expressions above for P in Equations F.20 and F.23, defining P in relation to PGE PR-1 required an additional nested summation. Adding lo sses and products to the example will similarly require additional nested su mmations to fully assign responsibility for subsequent losses to the products and by-products i n the supply chain. An important note about the additional complexity required to fully account for a more complex supply chain, is that using the more complex version of the equation (e.g., Equ ation F.21 instead of Equation F.19) will always assign more responsibility for the losses to produc ts and by-products, thus lowering the calculated value of P

, lowering the calculated NBE , and lowering the ultimate BAF value, so it is in the interest of the facility to use the more complex expression. For implementation purposes though, it may be that in some cases supply chains are uniform enough that default values of P could be calculated, so that facilities would not need to perform these com plex calculations. The examples contained in this appendix show how to calculate NBE in a way that meticulously “follows the tons of biogenic CO 2 ” through a series of hypothetical stationary sourc e production processes. In cases where the supply chain is long and complex, this quickly becomes a very complicated exercise. In application, it is unlikel y that all of the relevant carbon masses will be measurable and known. As was described above for L , it may be necessary to estimate P without actually performing the full calculations described in these examples. Nonetheless, it is useful to think through this idealized application to better understand the NBE equation and how it is adaptable across different production processes and points of assessment.