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 Assessment at the Boiler/Fermenter Mouth

 ................................................................................................................................................................................. F-3

2.2. Example 2: Changing the Point of Assessment to Stack Emissions ............................................. F-6

2.3. Example 3: Changing the Point of Assessment to Forest/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 CO2 Trail—Toward Fully Generalizing 

the NBE Equation ........................................................................................................................................... F-11

List of Figures 

Figure F-1: Conceptual Diagram Illustrating the Calculation of Potential Gross Emissions at 

Different Points of Assessment. ..................................................................................................................... F-3

Figure F-2: Stack Emissions as the Point of Assessment. .................................................................................... 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 appendix are intended to illustrate the various 

carbon flows that contribute to the calculation of net atmospheric contribution of biogenic CO2

emissions associated with the production, processing, 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 CO2 emissions for a stationary source could be calculated at different points of assessment, 

depending on the specific policy or programmatic context, 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 Biogenic Emissions (NBE) from stationary source 

biomass consumption equation can be presented as the 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 measurable 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 CO2

contributions associated with different stages of biogenic 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 used by a specific entity or generally 

consumed). This is a quantity that could be measured or estimated at different points of 

assessment (e.g., at the boiler mouth, stationary source gate, feedstock production site, or at 

the stack: wherever the point of assessment needs to be. Thus, this term can have different 

values indicated by subscripts, representing different points along the supply chain). 

• L is a unitless adjustment factor greater than or equal 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 production 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 one, 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 products, 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 unitless factors that relate the total biogenic 

carbon content of the feedstock grown at the feedstock production site, i.e. (PGE)*(L), to 

related landscape biogenic carbon pools. The details of these terms are discussed elsewhere 

in this framework. For the purposes of this appendix, we can think of the terms (GROW + 

AVOIDEMIT + SITETNC + LEAK)*(PGE)*(L) as the estimated net contribution to the 

atmosphere associated with growing, harvesting, producing, processing, and using the 

feedstock that was measured at the point of assessment. This amount is then multiplied by 

P to determine the share that is actually emitted and is the responsibility of a particular 

entity. 

The framework itself is designed to be flexible enough 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 interpreted 
November 2014 F-3 

as biogenic emissions at the boiler mouth, the stack, or total potential biogenic emissions at the 

farm gate once biomass has been harvested. Additionally, 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 supply chain, we need to understand how the L 

and P terms are calculated and what exactly they represent. 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 subsequent examples. 

2.1. Example 1: Simple Carbon Flow with Point of Assessment 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 representation uses atmospheric 

accounting methods, so terms that represent an emission (or potential emission) to the atmosphere 

(e.g., LOSS, PGE) are positive, and terms that represent sequestration (e.g., PROD) are negative. 

Figure F-1: Conceptual Diagram Illustrating the Calculation 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 to 

be applied with limited data requirements. The example here is designed as a thought experiment 

to “follow the tons of biogenic CO2,” 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). This can be generalized to PGEi in order cover any 

number of measurement points, where i indexes over all the points of measurement. The red 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/fermenter mouth. 

The terms LOSS and PROD represent the actual tons of carbon lost in transportation 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., LOSS1 occurs before PGE1, and PROD2 occurs before PGE2. 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 actual emissions, and PGE2 represents actual 

emissions out the stack (note that at this point the emissions are no longer “potential,” but for 

notational reasons we still refer to these stack emissions as PGE). The middle row is for potential 

emissions at various assessment points. These values 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 products 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 value. 

• PGE0: Represents PGE at the forest/farm gate. This is the total harvested biogenic CO2 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 stationary source facility. 

• LOSS1: Represents the biogenic CO2 lost in transportation or storage between the 

forest/farm gate (PGE0) and the boiler/fermenter mouth (PGE1). 

• PGE1: Represents PGE at the boiler/fermenter mouth. 

• PROD2: Represents biogenic CO2 stored in long-term product pools (including lumber, 

ethanol, or other purely marketable products produced with a portion of the harvested 

biomass) or other nonmarketable industrial by-products (including ash). 

• PGE2: Represents emissions at the stack. 

Thus, the equations relating these variables are: 

PGE1 = PGE0 – LOSS1 = 10 – 4 = 6 tCO2e. (EQ. F.3)

PGE2 = PGE1 + PROD2 = 6 + (-2) = 4 tCO2e. (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 that 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 PGEj, 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 detailed discussions of how they can be 

calculated in practice. It is important to note that GROW, AVOIDEMIT, SITETNC, and LEAK are all 

unitless.1 For the purposes of this appendix, we can define our 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 specific 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 overall BAF value. However, we can also think of each of 

these elements as being relative to the total amount of biomass that is harvested at the forest or 

farm (PGE0). G, A, S, and Lk are variables that represent the actual tons of landscape net emissions 

resulting from producing PGE0 tons of biomass.2 In that sense, (G+A+S+Lk) represents actual net 

emissions on the landscape caused by a harvest of PGE0. The landscape-level emissions are 

normalized by PGE0 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 biomass 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 (PGE0). 

L is defined as the ratio of harvested potential biogenic emissions (PGE0) and PGE evaluated at 

assessment point j. 

 = 



 (EQ. F.6)

Note that this fraction will result in a positive number greater than or equal to one. Also note that 

although LOSS does not appear in this equation, we could substitute 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 transformation of PGEj to PGE0, implicitly 

capturing any losses and products3 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 PGEj. That is, while landscape-level terms implicitly capture the 

emissions impact of total biogenic feedstock harvests (PGE0), L provides the necessary adjustment 

for total potential biogenic emissions within the stationary 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 combustion by exiting the stationary source through 

forms other than as stack emissions. P is a unitless term between zero and one that scales PGE

down so that the portion of biogenic feedstock carbon embodied in products or byproducts is not 

included in the final results when calculating the assessment factor for biogenic CO2 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 the forest/farm, 

so L only needs to account for these losses. However, if any products or by-products are produced between the 

forest/farm and the point of assessment (as in the example presented later in section 1.5), then in order to scale PGEi

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 version 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 represents the stack. In this example, PROD2 is 

the only product, so this summation would simply be equal to PROD2. Because PRODi is always 

negative and the absolute value of the sum of all PRODi must be less than PGEj, the sum of all PRODi

divided by PGEj will be a fraction between negative one and zero, and P will be a positive number 

between zero and one. 

P is one minus the share of PGEj 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 plus a negative number.) In 

effect, we are taking away the portion of carbon the products or by-products are responsible for 

and attributing the remainder to the facility. Although this is a relatively simple example, the 

mathematical representation of this term will change 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 PGE0 in numerator of the L term cancels with the PGE0 in the ratio of (G+A+S+Lk)/ PGE0, and the 

PGEj in the denominator of the L term cancels with the first PGEj 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 landscape caused by a harvest of PGE0. All of this 

canceling is important, because G, A, S, and Lk are not observable, and even PGE0 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 reference point or future anticipated baseline modeling 

methods) and calculate net emissions while only needing to observe PGEj. 

Now, applying numerical values from Figure F-1 and assuming that (G+A+S+Lk) = 3 tCO2e (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 Stack Emissions 

Continuing through the following examples, necessary modifications are made to the BAF equation 

for alternative points of assessment. The biogenic CO2 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 of 
November 2014 F-7 

assessment. This provides a consistency check because this exercise merely alters the point of 

assessment but not the biogenic carbon trail. 

The second example also relies on the simple hypothetical carbon trail from Figure F-1, with the 

only difference being that the new point of assessment is stack emissions (represented by PGE2), 

i.e., “j=2.” 

Figure F-2: Stack Emissions as the Point of Assessment. 

In the first example, all losses occurred before the point of assessment, and all products were 

produced after the point of assessment. In this example, 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. Equation 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 PGE0 to PGEj. Although the expression for L has 

not changed, the numerical value and the interpretation will be different, because PGEj is now stack 

emissions. In our original interpretation, L was a unitless adjustment factor greater than or equal to 

one, that scales PGE as it was measured at the point of assessment up to 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 products and losses before the point of assessment, 

L scales PGE as it was measured at the point of assessment up to account for any losses during 

transportation or storage between the forest/farm and the point of assessment and to account for 

any products produced between the forest/farm and the point of assessment. In the general form, L

does more than just account for losses, it serves as a general scaling factor to relate the carbon 

content of the biomass feedstock at the point of assessment (PGEj) to the carbon content of biomass 

that was grown and harvested at the forest/farm and that was required to generate that feedstock 

(PGE0), 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 PGE0. The remaining job for the P term is to determine what share of these net landscape 

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 ratio of all products produced to 

PGEj, this new fraction deducts contributions from products that occur prior to the point of 

assessment from PGEj in the denominator (./01 − ∑ .3456

1

678 ). Thinking back to Equation F.4, 

PGEj = PGEj-1 + PRODj, rearranging this expression and generalizing, the new denominator is simply 

the expression for PGEj-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 point. Because all that has changed in this example 

is the point of assessment, the value calculated for 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 tCO2e, NBE

is calculated in equation 0. 

 = (  %

   (

 9 + &'

(&&': = % 9 − '$

: = % (

$

 = '	;*
'+ (EQ. F.12)

Note that although the point of assessment has changed, 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 Forest/Farm Gate 

The third example (Figure F-3) also relies on the simple hypothetical carbon trail from Figures F-1 

and F-2, with the only difference being that the new point of assessment is the forest/farm 

(represented by PGE0), i.e., j=0. 

Figure F-3: Forest/Farm as the Point of Assessment. 

In this example, because the point of assessment is at PGE0, the L term is equal to one. The L term is 

still theoretically scaling between the point of assessment and the forest/farm level, but as they are
November 2014 F-9 

the same, the L term has a value of one. PGE0 is directly measured as it is PGEj in this instance, so 

PGEj does not need to be scaled up, and L is simply equal to one. 

In this example, losses (LOSS1) do not enter into the L term, but because the mass balance has not 

changed, losses still are occurring, but they are occurring after the point of assessment, so the 

question is how are losses accounted for now? In all examples, (PGEj)*(L) = PGE0. The equation then 

assumes that all of PGE0 is emitted, unless the P term specifies that a share of PGE0 is sequestered or 

contained in products. This means that once PGEj is transformed to PGE0, unless there are products, 

the calculation does not change regardless of whether the emissions come from a loss or from the 

stack. Consider for a moment an example without products (i.e., P=1); here <=0 =

./01 

>?@AB

C>DE

 C>DE

C>DF

, 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, although 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 LOSS1 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 is produced. Similar to the previous example, in 

the denominator of the fraction in the P term, losses are deducted from PGEj in the denominator 

(./01 − ∑ G4HH6

CI

6718 , and as demonstrated in Equation 0F.13, this is essentially the expression 

for PGEj+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 PRODi > 0). 

In effect, as the P term ensures that the facility is not held responsible 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 shares 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 tCO2e, 

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 before any products are produced, 

both forms of the equation as established in examples 2 and 3 must be combined. Doing so results 

in Equation 0F.17. This generalized form can be applied to all the previous examples of biogenic CO2

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 PGES 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 (LOSSi) occur before the first product (PRODi) is generated. 

As explained in the previous examples, the denominator in the fraction in the P term uses the 

relationships in Equations F.3 and F.4 to transform PGEj 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 forest/farm. PGE begins at 0. 

• S: Last point on the carbon trail; represents the stack. PGES is stack emissions. 

• PR: Index number for the first product produced on the carbon trail. PRODPR is the first 

product. 

The indexing for points along the carbon trail begins 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 next 

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 indistinguishable from stack emissions PGEs in these equations. 

To simplify the equation, it is assumed that LOSS is not specified after the last product is produced 

in the carbon trail, and instead any such losses are rolled into the calculation of PGEs. 

With this notation in mind, Equations F.16 and F.17 are rewritten as: 

 = + 	
N*OP

	Q 

(EQ. F.18)

 

4

 If LOSSi and PRODi occurred at the same stage i, it would be ambiguous which came first and whether that PRODi

should share responsibility for LOSSi or not. For purposes of specifying the full theoretical carbon trail, a PGE 

would need to be inserted between LOSSi and a PRODi. 
November 2014 F-11 

 =  






 



 9 + 	
N*OP

	Q 

:, (EQ. F.19)

Where we also define a new variable, .345RSTUV = ∑ .3456

@

678 , or the sum of all products 

produced along the carbon supply chain. PGEPR-1 is potential gross emissions at the point in the 

carbon supply chain before the last product is produced. In examples 1 through 3 above, PGEPR-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 before any products are produced in the carbon 

supply chain. If a product is produced at a point in the carbon chain before a loss occurs, then that 

product should not be held responsible for the subsequent loss. Example 5 below shows how the 

NBE equation can be fully generalized to cover this situation. 

2.5. Example 5: Extending to a More Complex Biogenic CO2 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 that this example includes four potential points 

of assessment for PGEj, and multiple points where products are produced and 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 + PRODTotal / PGE1 = 1+(−3/10) = 7/10. This would be incorrect 

though, because it assigns the facility full responsibility for the LOSS3 term and thus does not 

account for the portion of LOSS3 that PROD4 is accountable for. To properly account for how 

responsibility for LOSS3 is shared, one should apply the following equations 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 that (G+A+S+Lk) = 3 tCO2e, gives the 

following calculation of NBE: 

 =   %

' '  K + &%%

Q'

$



  L = % K − %% 

%



  L = % 9 − ( : = % 9

$

 : = . [ (EQ. F.22)

This example is more complicated than the earlier examples but can still be thought of intuitively. 

At the end of the carbon trail, PGE4 and PROD4 are responsible for 4 tCO2e and 2 tCO2e of PGE3,

respectively. They share responsibility for LOSS3 proportionally, 4/6 for PGE4 and 2/6 for PROD4. So 

PGE4 is responsible for 2 tCO2e of LOSS3 and PROD4 is responsible for 1 tCO2e of LOSS3. Put these 

together and PGE4 is responsible for 6 tCO2e of PGE2 and PROD4 is responsible for 3 tCO2e of PGE2. 

These are also the quantities they are responsible for of PGE1, because the only difference between 

PGE1 and PGE2 is the 1 tCO2e that PROD2 is responsible for. So PGE4 is responsible for 6/10 of PGE1. 

As shown in the calculation above in Equation F.22, this confirms that P = 6/10. Note that because 

LOSS1 occurs before all products, responsibility for it is shared in the same proportion as 

responsibility for PGEPR-1, so it does not need to be accounted for in P.

5

Although Equation F.21 is a more general representation of NBE than the previous examples, it 

does not cover all possible scenarios. If the carbon chain is more complex, and there are multiple 

losses after the first product, then the ∑ .345B

@

B76 term in the numerator of the fraction that shares 

out responsibility for LOSSi will not account for PRODk’s share of any subsequent losses, instead 

assigning full responsibility for those subsequent losses to PGEs. Instead of demonstrating this with 

another example, the same effect can be illustrated by calculating P with respect to PGE0 instead of 

PGEPR-1. This is not a more general way of calculating P, but simply an equivalent expression. With 

this change, one needs to account for LOSS1 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 LOSS1 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 equation still generates the same answer for 

NBE in this example. 

More complex supply chain carbon trails will require 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 PGEPR-1 required an additional nested summation. Adding losses and products to the 

example will similarly require additional nested summations to fully assign responsibility for 

subsequent losses to the products and by-products in 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., Equation F.21 instead of Equation F.19) will always 

assign more responsibility for the losses to products 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 complex calculations. 

The examples contained in this appendix show how to calculate NBE in a way that meticulously 

“follows the tons of biogenic CO2” through a series of hypothetical stationary source production 

processes. In cases where the supply chain is long and complex, this quickly becomes a very 

complicated exercise. In application, it is unlikely 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.