#### 2.1. Conceptual Framework

This study uses a partial budgeting framework to maximize the return to fertilizer costs subject to other costs held constant both at the grower and refiner levels. Other aspects required to grow the feedstock remain unchanged and therefore will not affect decisions surrounding input application. A similar assumption is made for the biorefinery, except transportation costs and feedstock quality are also considered.

The farmer is assumed to maximize net returns from switchgrass production. The partial net returns for the farm (

$N{R}_{Farm})$ can be expressed as follows in Equation (1):

where

P is price received for switchgrass feedstock per metric ton,

S is switchgrass yield per hectare with

P *

S equaling gross revenues per hectare,

C represents the variable application costs of nitrogen, and

$O{C}_{Farm}$ are other costs per hectare.

P,

S, and

C could be in the case of

P and are in the case of

S and

C functions of the quantity of applied nitrogen (

N_{a}). If the marketplace incorporates the effects of

N on ash content and ultimately conversion efficiency, then price of switchgrass,

P, could be discounted according to the amount of

N applied. The costs of applying

${N}_{a}$ are not impacted by the rate since applying some

N is required, and therefore it is included in

$O{C}_{Farm}$.

As with the farmer, the bio-refinery is assumed to maximize net returns,

$N{R}_{Refinery}$, and can be characterized using Equation (2):

where

FP is the expected price received for the biofuel product in

$/liter;

FY is the per metric ton conversion rate of feedstock to fuel which is impacted by switchgrass quantity (

S) as previously defined and ash content (

Ash), which are, in turn, both functions of

${N}_{a}$ application;

Q is the annual quantity of biomass converted to fuel (metric tons);

FC are the delivered feedstock costs; and

OC are other conversion facility costs. The renewable identification numbers (

RIN) are a second output of the facility and are attached to the fuel output.

RINs are credits that the US EPA uses to track and enforce renewable fuels mandate compliance in the US.

RINs are attached to renewable fuels as they are produced and detached when the renewable fuel is mixed with fossil fuels. Essentially,

RINs are records of individual batches of renewable fuel being blended into the US gasoline, aviation and diesel fuels, and take on a value as companies supplying fossil fuels attempt to meet compliance requirements. Pathways are approved by the EPA. A cellulosic pathway (Pathway L) using switchgrass as a feedstock receives a D code of 7. The non-ester renewable fuels with a heating value of at least 123,500 has an equivalence value (EV) of 1.7 [

28].

In Ou et al. [

12], the analysis is concerned with the total ash, moisture, and carbon content of switchgrass. For the purpose of this study, moisture and carbon are held constant as these variables are not believed to be impacted by applied

N. Ash content was allowed to vary as ash content was identified as having significant impact on conversion yield.

#### 2.1.1. Switchgrass Yield and Nitrogen Application

For purposes of model specification selection for switchgrass yield as a function of

N, four different yield functions are estimated and tested against each other including a quadratic response function (QRF), a mixed quadratic response function (MQRF), a linear response plateau (LRP), and a mixed linear response plateau (MLRP). As in Boyer et al. [

16], techniques such as the log likelihood ratio (LLR) test, as well as the Akaike information criterion (AIC), adjusted AIC (AICC), and Bayesian information criterion (BIC) fit statistics are used in model specification selection and testing. Using these aforementioned model fit testing criteria, the mixed quadratic response function (MQRF) is selected to further explain the relationship between switchgrass yield and

N application. As in [

16], the MQRF displayed in Equation (3) shows the relationship between

N fertilizer application and switchgrass yield as:

where

${\beta}_{0},{\beta}_{1},\mathrm{and}{\beta}_{2}$ are the yield response parameters estimated using observed yields; ε is the random error term accounting for variation in yields from unexplained factors;

$v$ is included to capture the yield variability from year-to-year due to changes in weather, harvest timing, and other situational effects which are assumed to be unrelated to the independent variables and independent of the error term;

${N}_{a}$ is

N-fertilizer application rate in kilograms per hectare;

${N}_{a}$^{2} represents a quadratic

N term; and all other variables have been previously defined. The results of the yield response function provide an estimated quantity of switchgrass at the farm-gate level in dry metric tons per hectare for a given fertilization rate.

#### 2.1.2. Ash Content and N Fertilizer Application

Similar to the switchgrass yield response function, an MQRF is estimated for the effect of

N fertilizer application on ash content in switchgrass. Chemical composition and yield variability from year-to-year result from changes in weather, harvest timing, and other situational effects justifies including random effects in the function which are assumed to be independent of the error term. The ash content as a function of

N is written in Equation (4):

where

${\delta}_{0},{\delta}_{1},\text{}\mathrm{and}\text{}{\delta}_{2}$ are the percent ash response parameters estimated using observed applied nitrogen levels; ε is the random error term accounting for variation in ash from unexplained factors;

$v$ is included to capture the ash variability from year-to-year due to changes in weather, harvest timing, and other situational effects which are assumed to be unrelated to the independent variables and independent of the error term; and all variables have been previously defined.

The biorefinery assumed in this study is replicated from Ou et al. [

12] for facility capacity as well as controls for ash content, moisture, and carbon. To estimate biofuel yield in liters, a quadratic response function (QRF) is used. The response function estimating biofuel yield is not a mixed model showing year random effects as there is no information on year-to-year variation. Equation (5) represents the biorefinery yield equation and is a function of feedstock characteristics:

where

FY represents the fuel yield in gasoline equivalent liters per dry metric ton;

ϕ_{i} represents the six parameter estimates; and

Ash, Moist, and

Carbon are components of switchgrass used as a feedstock to produce fuel expressed in percentage terms.

As in Ou et al. [

12], a feedstock capacity of 2000 metric tons of feedstock per day is used for this analysis. Assuming the facility runs at 90% efficiency, the total feedstock processing capacity is 656,000 metric tons per year. In this study, both moisture and carbon content are held constant at 35% and 46%, respectively, when predicting biofuel yield. When varying ash content from zero to five percent, biofuel yield ranges from 137.78 to 108.32 liters per dry metric ton. The fuel price paid to the biorefinery for their final output is assumed to be

$0.71 per liter [

29].

Several model specifications were tested including the quadratic response function (QRF), MQRF, linear response plateau (LRP), and the mixed linear response plateau (MLRP). As in [

30], techniques such as the log likelihood ratio (LLR) test, as well as the Akaike information criterion (AIC), adjusted AIC (AICC), and Bayesian information criterion (BIC) fit statistics, are used in model specification selection and testing. The MQRF was selected for both Equations (3) and (4), and the QRF was selected for Equation (5). The results indicated that the mixed effect models were stronger when considering year random effects. With respect to biofuel yield, however, no year random effects are present. Thus, only the LRP and QRF models were estimated. Based on the results of AIC, BIC, and LLR testing, the QRF function proved to be the best fit at predicting biofuel yield.

#### 2.1.3. Optimizing N Fertilizer for Farms and Refinery

The base case of the switchgrass farmer in this analysis is considered “naïve” toward the effects of ash content. In the base case, the producer does not receive a higher price for quality-considered product. Therefore, to maximize net returns, the optimal

N fertilizer application is only a function of quantity or yield (S). Expanding the NR function given in Equation (1), substituting in the deterministic portion of Equation (3), the equation for the farmer can be written as follows:

where the price paid to producers (

P_{S}) is frequently referred to as the farmgate feedstock price or just feedstock price and is multiplied by Equation (3), showing the impact of

N fertilizer on switchgrass yield (S), and is less than both the fertilizer price (

P_{N}) multiplied by the fertilizer quantity (

${N}_{a}$) and the farm’s other costs in growing switchgrass

$(O{C}_{Farm}$). In this calculation it is assumed that yield will not impact

OC_{Farm}. This equation provides the net returns level associated with a variable quantity of

N fertilizer application.

The optimal farm

N rate (

${N}_{a}^{\ast}$) is obtained by equating the first order condition of Equation (6) to 0 and solving for

${N}_{a}^{*}$ [

31]:

The result of the first order condition indicates the global maximum on the net returns curve estimated in Equation (6) and determines the optimal N_{a} application rate for the farmer.

With respect to the biorefinery, net returns will be maximized when ash is minimized. The nitrogen level where ash is minimized is determined by taking the first order derivative of Equation (4) with respect to nitrogen, setting it equal to 0, and solving for nitrogen (Equation (8)).

where

${N}_{min}^{**}$ is the nitrogen level that minimizes ash in the biomass feedstock, and the

${\delta}_{i}$ are parameter estimates previously defined in Equation (4) [

31]. At this level of nitrogen, ash will be minimized and the biorefinery rate of conversion of biomass will be maximized with respect to ash content. The biorefinery’s profit function is dependent on the conversion of biomass to fuel and takes the form:

where

$N{R}_{Refinery}$ is the net returns of the biorefinery,

$F{P}_{f}$ is the biorefinery fuel price in

$/liter for fuel type f,

RIN is the value of the renewable identification number provided by the EPA in

$.liter, FC are the delivered feedstock costs in

$,

OC_{Refinery} is the biorefinery’s other costs of production in

$,

$F{Y}_{f}$ is the biorefinery fuel yield in liters per dry metric ton of switchgrass for fuel type

f, $E{V}_{f}$ is the equivalence value for fuel type

f, and

Q is the annual quantity of switchgrass purchased for conversion in metric tons.

#### 2.2. Data

Three equations are estimated in this analysis and measure the impact nitrogen has on yield and ash content of the feedstock, as well as the impact of ash content on the biorefinery yield. Once a year, harvest was conducted and samples drawn. The harvest occurred post senescence and the collected samples were tested for moisture content in a forced air oven. They were then placed in storage. The switchgrass yield data and its quality information resulted from experiments that were conducted at the University of Tennessee’s Research and Education Center in Milan, Tennessee (35°56/N, 88°43/W) on Alamo switchgrass from 2004–2014 [

30]. Yield estimates were determined from this experiment. The experiment was conducted on three different landscapes, with four nitrogen application rates, and five initial seeding rates.

For chemical composition, there were 160 samples tested from a subsample of the experiment for chemical composition. A subset of the data utilized in this analysis was analyzed for switchgrass chemical composition including waste minerals or ash content as a percentage of switchgrass feedstock. Samples from harvests in 2006, 2008, 2010, 2012, and 2013 were used in the analysis. Other years of data were available, but because of resource availability, quality data measurements were conducted for these select years. The initial two years were not chosen as switchgrass maturity is not reached until the third growing season. The plots utilized for the study are abbreviated by field N21 and 212. Field 212 is characterized with a well-drained Grenada silt loam soil type, with no slope in an upland position of site. Field N21 has a well-drained Vicksburg silt loam soil type and no slope, and positioned in a flood plain. Both fields were planted in switchgrass with plots identified within the field. Nitrogen fertilizer was applied to sub plots with rates of application of 0, 67, 134, and 201 kg/ha. The source of the

N was ammonium nitrate (NH

_{4}NO

_{3}). Each of these treatments was applied annually. According to Mooney et al. [

32], switchgrass requires very little P and K; however, 89 kilograms of P

_{2}O

_{5} and K

_{2}O were applied per hectare annually to ensure that nutrients removed from the fields during harvest were replaced.

Samples were ground up using a Thomas Scientific (Swedesboro, NJ, USA) Model 4 Wiley mill using a 40 mesh (0.425 mm) screen. Since ground biomass’s moisture content can rapidly change when exposed to air, the ground samples sat for three weeks at ambient conditions (approximately 23 °C, 63% relative humidity). Total solids measurements were completed using a sub-sample dried in a 105 °C convection oven for a minimum of 4 h to determine the percent of total solid prior to compositional analyses. Each sample was first combusted at 575 °C for 24 h and weighed for measurement of total ash content. A total of eight primary components were quantified as a mass percentage of the oven dried biomass (on % dry basis): extractives, cellulose, hemicellulose (combined values for xylan, galactan, arabinan, and mannan), lignin, and total ash. For further details on the process used to conduct chemical composition analysis, see [

29].