# Optimal Control to Handle Variations in Biomass Feedstock Characteristics
and Reactor In-Feed Rate

Fikri Kucuksayacigil  Mohammad Roni 111Corresponding author:
<EMAIL_ADDRESS>Sandra D. Eksioglu  Qiushi Chen  Energy and Environment
Science & Technology Department, Idaho National Laboratory, Idaho Falls, ID
Industrial Engineering Department, University of Arkansas, Fayetteville, AR
Glenn Department of Civil Engineering, Clemson University, Clemson, SC

###### Abstract

The variations in feedstock characteristics such as moisture and particle size
distribution lead to an inconsistent flow of feedstock from the biomass pre-
processing system to the reactor in-feed system. These inconsistencies result
in low on-stream times at the reactor in-feed equipment. This research
develops an optimal process control method for a biomass pre-processing system
comprised of milling and densification operations to provide the consistent
flow of feedstock to a reactor’s throat. This method uses a mixed-integer
optimization model to identify optimal bale sequencing, equipment in-feed
rate, and buffer location and size in the biomass pre-processing system. This
method, referred to as the hybrid process control (HPC), aims to maximize
throughput over time. We compare HPC with a baseline feed forward process
control. Our case study based on switchgrass finds that HPC reduces the
variation of a reactor’s feeding rate by 100% without increasing the operating
cost of the biomass pre-processing system for biomass with moisture ranging
from 10 to 25%. A biorefinery can adapt HPC to achieve its design capacity.

###### keywords:

biomass , process control , optimization , feedstock , biochemical

††journal: Energy

##  1 Introduction

Biomass characteristics vary by moisture, composition, particle size, shape,
and density. These variabilities often cause poor flowability of materials at
pre-processing systems that lead to clogging of equipment [1]. For example,
variable moisture content causes variability in particle sizes. Whereas small
particles may plug weep holes, larger particles may plug bins. Poor
flowability often stems from arching, bridging, and rat-holing in the system
[1]. Solid materials have unanticipated physical interactions with equipment,
which often cause interruptions in the flow. These interruptions negatively
affect down-steam processes.

The variability of biomass characteristics impacts biorefineries in different
ways. Variations of particle size and distribution, moisture, ash, and heat
contents negatively affect the integration of biomass feeding systems and
conversion processes and result in low or unreliable on-stream time and long
start-up times. For these reasons a number of biorefineries, which were
planning to ramp-up production, had to delay these plans. For example, Bell
[2] and Merrow [3] show that feeding systems reached only 77% of their
designed production amount at the end of the first year of operation (see [2]
for findings related to general particulate processes and [3] for initial
efforts to reveal and communicate drawbacks of particulate processes).
Bioenergy experts report that capital expenditures and operating expenses have
risen due to operational difficulties of handling biomass. These findings are
proven empirically [4] and via engineering perspectives [5]. The challenges
outlined here emerge as significant barriers for biomass to become a
sustainable source of energy. As a consequence, major commercial biorefineries
(e.g., DuPont) have discontinued their production in the U.S. [6].
Furthermore, investors are becoming reluctant to invest in the biofuel
industry due to the high risks stemming from variability of biomass
characteristics, market, and supply chain [7]. Sustainable operations of a
biorefinery under biomass variations remains a challenge [8].

A number of efforts have been made to mitigate the impacts of variations in
biomass characteristics and to improve the design throughput of a biofuel
conversion process. First, new designs of the equipment in the pre-possessing
system have been proposed to improve flowability. There is rich literature
examining physical interactions of biomass and equipment design to gain
insight towards improving flowability. As an example, Dooley et al. [9]
proposed a rotary shear mechanism to chop biomass efficiently. Engineers
alternatively prefer improving existing equipment, usually supported by
modeling tools such as the discrete element method (DEM). For example,
Ketterhagen et al. [10] used DEM to identify the correct geometry of a hopper
to ensure uninterrupted flow of particles. Crawford et al. [11] investigated
the effects of chemical additives, moisture content, particle interaction, and
wall friction forces on the flowability of biomass materials. Wu et al. [12]
measured physical characteristics of biomass and introduce designs of handling
and storage equipment that align with these measurements.

Second, process control via advanced modeling and simulation tools has shown
to reduce the negative impacts that variations of biomass characteristics have
on achieving the design throughput of a conversion process. Process control
shows a prominent potential to identify robust process designs that handle
variations in feedstock characteristics and equipment down-time. There are
abundant studies in the literature focusing on process control that model
inflow and outflow for equipment in consideration. Marino et al. [13, 14]
developed a data-driven decision support systems, which inform operators
regarding the performance of grinders given the moisture content of bales, in-
feed rate, and screen size. Jaramillo and Sanchez [15] proposed a mathematical
formulation to model outflow from a pretreatment continuous tubular reactor.
Numbi and Xia [16] adopted a system optimization perspective to model optimal
processing of high-pressure grinding rolls. Hartley et al. [17] emphasized the
effect of biomass physical characteristics on the performance of a pre-
processing system by modeling biomass flow via a discrete-event simulation.

Third, taking preventative measures reduces equipment down-time due to
variations of biomass characteristics. For instance, processing high-moisture
bales at a lower rate minimizes the risk of clogging. We regard preventative
measures as practical applications that improve the reliability of biomass
pre-processing systems. One area in reliability is concerned with minimizing
variability in production systems by adjusting control variables (e.g., feed
at a lower rate). Strong mathematical assumptions, however, limit the
applicability of the models in this area. Das et al. [18] analyzed a
production system where random failures of equipment occur. They provide a
closed-form expression of the economic production lot-size. Li and Meerkov
[19] suggested bounds on throughput variances so that a proper serial line of
production can be designed. Assaf et al. [20] and Colledani et al. [21]
analyzed the variability of output from unreliable equipment in single-stage
or simplified multi-stage systems.

Last, but not least, changing the underlying structure of feeding systems by
relocating equipment and storage areas has shown to improve the design
throughput of a conversion process. For example, adding buffers to the system
ensures a smooth flow. Indeed, improving production systems via investigating
the effectiveness of buffer size and location is a rich area of research. A
number of studies make probabilistic assumptions concerning processing time,
biomass supply, etc. These models, however, are generally limited due to
strong assumptions. Tempelmeier [22] investigated (continuous) flow of
materials in production systems and discussed different algorithms to
determine optimal buffer sizes and their locations. Tan and Gershwin [23]
analyzed a two-stage continuous-flow system by assuming that transitions among
states are Markovian and one buffer with finite capacity is present between
stages. Similarly, Turki et al. [24] determined an optimal buffer size for a
single-product machine with random failures. In Hosseini and Tan [25], each
stage in a two-stage system has different states of processing with general
distribution of transition times. The authors developed a mixed-integer linear
programming model to determine optimal level of buffer by maximizing profit.

All efforts mentioned above have some limitations. Equipment design and
process control are usually tied to a single equipment. In other words, these
efforts tend to ignore the impacts of biomass flow to the performance of an
integrated feeding and pre-processing system. Nevertheless, it is crucial that
the design of a hopper or the control of a grinder are informed by the
performance of predecessor and successor equipment. This is of significance to
make system-wise optimal decisions. Moreover, studies concerning the
reliability and buffer analysis are usually coupled with strong and
unrealistic mathematical assumptions. Being tied to single equipment and
strong mathematical assumptions makes the above-mentioned approaches
inappropriate to model and analyze biomass feeding systems.

In our work, we mitigate the limitations of the above-mentioned approaches.
This paper considers the whole biomass feeding system from beginning to end by
tracking the flow dynamics and alterations in the physical characteristics of
materials throughout the process. We develop and analyze process control
strategies to handle the variation in feedstock characteristics and reactor
in-feed rate, and do not restrict ourselves to strong modeling assumptions. As
a result, our proposed model designs a reliable biomass feeding system for an
integrated biorefinery.

We point out a few characteristics of our approach that distinguish it from
other works in the literature. First, our approach considers sequencing of
bales. Processing high-moisture bales often takes longer than usual due to the
possibility of clogging in grinders. Such a practice becomes a bottleneck to
reaching higher target rates. One way to overcome this challenge is to process
low-moisture bales first to build-up inventory, and then, consume this
inventory while processing high-moisture bales. Through this research, we
analyze the impact that different sequences of bales have on achieving the
maximum feeding rate, buffer size, and cost of operating the system. To the
best of our knowledge, there is no study in the literature that have analyzed
the effect of bale sequencing on the performance of feeding systems.

Second, our approach controls buffer size and location as the means for
managing the variability of biomass characteristics and maintaining a
continuous feeding of feedstock to a reactor at the maximum possible feeding
rate. Finally, our work is concerned with the comparison of two different
control approaches, the hybrid process control (HPC), which aims to maximize
throughput over time, and a state-of-the-art baseline feed forward process
control (BFFPC). Through this research, we provide cost/benefit analysis for
BFFPC and HPC under different settings of a biomass feeding system. In
summary, our study is uniquely positioned to devise a control approach for
biomass feeding systems.

This research makes the following contributions to the literature. First, we
propose HPC for biomass feeding systems that devises bale sequencing, as well
as buffer location and size. We compare HPC with a state-of-the-art BFFPC for
biomass pre-prossessing systems. Second, we exhibit how to formulate optimal
control approaches using linear and mixed-integer linear programmings. These
formulations are general to a large extent and can be applied to several
biomass feeding systems. Third, we communicate a number of significant
managerial insights. We convey that HPC is a potential solution to reduce
variation of reactor feeding rate without increasing cost.

##  2 Material and Methods

###  2.1 Feedstock Pre-processing Description

Feedstock pre-processing in this study includes two-stage size reduction and
densification. The first-stage size reduction uses a grinder to break the bale
into smaller particles. The second-stage size reduction further reduces the
particle size via a hammer mill. Densification converts biomass to pellets.
Figures 1(a) and 1(b) illustrate the system with equipment. We refer the
interested readers to a study of biomass feedstock supply system design and
analysis by [26] for additional information about equipment and feedstock pre-
processing systems. A supplemental file presents figures of several equipment
and process demonstration unit (PDU). Table LABEL:supp-tab:eqspecs and
LABEL:supp-tab:costdata list some specifications for equipment.

(a) Without fractional milling

(b) With fractional milling

Figure 1: A biomass pre-processing system

The feedstock pre-processing system that we utilize in this study has two
different designs based on whether a separation unit is utilized between
first-stage and second-stage grinding. Process without fractional milling does
not have a separation process. After being processed in the first-stage
grinder, materials go through the second-stage grinder for further size
reduction. Such a practice produces fines that are prohibitive for the
successful operation of the system. In process with fractional milling,
following first-stage grinding, a separation unit separates small-sized
particles and pushes them to a storage unit (we call the percentage of small-
sized particles the “bypass ratio”). Such a practice results in tighter
particle size distribution, a minimum amount of fines, and reduced energy
consumption [27].

###  2.2 Description of Proposed Control Method in the Feedstock Pre-
processing System

Primary control variables in the feedstock pre-processing system are: (i)
screen sizes in grinders; (ii) in-feed rate (amount of material processed per
unit of time); (iii) mill speed in grinders; (iv) discharge rates from storage
units; (v) feeding rate of pellet mill; and (vi) amount of moisture to add to
pellet mix. Operators usually adjust these variables to ensure uniformity of
particle sizes and increase pellet quality. Since screen size and moisture
requirements can be determined in advance based on pellet particle size and
quality requirements, we fix the value of the following control variables to
predetermined values – screen size associated with first- and second-stage
size reduction and moisture.

The proposed control method (e.g., HPC) is an adaption of the feedback loop
control and incorporates additional biomass process specific controls,
examples of which are given in studies of refrigerators/cooling systems [28]
and demand-side management systems [29]. Thus, HPC is a control approach whose
primary goal is to meet a desire output response by adjusting bale sequencing
decisions, biomass in-feed rate of different equipment, and buffer location
and size. Consider the feedstock pre-processing design with fractional
milling. Processing of bales with different moisture leads to additional
operational constraints. For example, in-feed rate of second-stage grinding is
constrained by moisture content of biomass since high moisture biomass is fed
at a slow rate to prevent clogging in the system. Low moisture biomass is
processed faster. This variation in moisture causes a fluctuation in the
feedstock processing and reactor feeding rates.

The fluctuation of the reactor feeding rate can be handled in the proposed HPC
method in the following way. First, the operator characterizes the bale based
on its moisture range (e.g., high-, medium-, and low-moisture ranges). Second,
the operator decides when to process low-, medium-, or high-moisture bales, as
well as when to build-up/use the buffer to maintain a continuous feeding of
the reactor. By simultaneous optimization of bale sequencing, in-feed rate,
and buffer location and size, the HPC determines an optimal trajectory that
maintains the continuous feeding of a reactor. If the current process buffer
size (e.g., metering bin) is not sufficient, then optimization also decides by
how much to increase the buffer size (see Figure LABEL:supp-fig:hpc_control).

The proposed HPC method is compared with the state-of-the-art BFFPC to
demonstrate the differences and benefits of the proposed method. BFFPC faces
the same operational constraints caused by moisture. This BFFPC method
maximizes the total throughput over time but does not necessarily maintain a
continuous feeding of the reactor. The BFFPC operates as follows. First, the
operator characterizes bales based on moisture range, but feeds bales in an
unsystematic way. That means, bales are not arranged in a sequence according
to their moisture contents. We will henceforth use the term “random pattern”
to refer to this arrangement. Second, in contrast to the HPC method, BFFPC
finds an optimal trajectory that maximizes throughput by optimizing in-feed
rate, but it does not control buffer location and size (see Figure LABEL:supp-
fig:bffpc_control).

We summarize the commonalities and differences of BFFPC and HPC in Table
LABEL:supp-tab:control_summary.

###  2.3 Optimization Model Development

The optimization model identifies the minimum time to process a fixed number
of bales using HPC and BFFPC under various operational constraints. The
optimal trajectory to process a fixed number of bales is found via an
algorithm as shown in Figure LABEL:supp-fig:flowchart. The goal is to shorten
the modeling horizon iteratively until no better solution is found. In each
iteration of this algorithm, an optimization model is solved. The
implementation of this algorithm is described in supplementary material.

####  2.3.1 MILP model for HPC

We use the following notation to model HPC. Throughout the text, we use dry
Mega gram (dry Mg) as mass unit.

Sets:

  * 1.

$\mathcal{I}$: Equipment used in a biomass feeding system

  * 2.

$\mathcal{I_{G}}$: Grinders

  * 3.

$\mathcal{I_{M}}$: Storage units

  * 4.

$\mathcal{I}_{i}$: Predecessor of equipment $i$

  * 5.

$\mathcal{T}$: Periods in modeling horizon

  * 6.

$\mathcal{L}$: Biomass moisture levels considered in model

  * 7.

$\mathcal{C}$: Potential expansion options for storage units,
$\mathcal{C}\coloneqq\\{0,0.1,0.2,\ldots,1\\}$ where 0.1 means capacity is
expanded by 10%.

Indices:

  * 1.

$i\in\mathcal{I}$: Equipment locations in the system

  * 2.

$t\in\mathcal{T}$: Periods

  * 3.

$s\in\mathcal{L}$: Moisture level

  * 4.

$k\in\mathcal{C}$: Potential expansion option.

Parameters:

  * 1.

$\alpha$: Penalty for adjustment of reactor feeding ($/dry Mg/unit of time)

  * 2.

$\eta$: Unit sale price of pellets ($/dry Mg)

  * 3.

$c_{is}$: Unit cost of equipment $i$ based on moisture level $s$ ($/unit of
time)

  * 4.

$c_{isk}$: Cost of operating equipment $i\in\mathcal{I_{M}}$ with processing
capacity $k$ for biomass with moisture level $s$ ($)

  * 5.

$w$, $h$: Width and height of a bale (meters)

  * 6.

$j_{st}$: An indicator parameter which takes the value 1 if biomass with
moisture level $s$ is processed in period $t$, and takes the value 0 otherwise

  * 7.

$d_{s}$: Density of biomass bale with moisture level of $s$ (dry Mg/cubic
meter)

  * 8.

$\bar{d}_{is}$: Average density of biomass with moisture level $s$ in
equipment $i\in\mathcal{I_{M}}$ (dry Mg/cubic meter)

  * 9.

$\overline{x}_{is}$: Capacity of equipment $i$ when processing materials with
moisture level of $s$ (dry Mg/unit of time)

  * 10.

$n_{s}$: The number of bales with moisture level $s$

  * 11.

$q_{s}$: The mass of a bale with moisture level $s$ (dry Mg)

  * 12.

$\mu_{i}$: Dry matter loss in equipment $i\in\mathcal{I_{G}}$ (%)

  * 13.

$\theta_{s}$: Bypass ratio when processing biomass with moisture level of $s$
(%)

  * 14.

$\overline{m}_{i}$: Mass capacity of equipment $i\in\mathcal{I_{M}}$ (dry Mg)

  * 15.

$\overline{v}_{i}$: Volume capacity of equipment $i\in\mathcal{I_{M}}$ (cubic
meter)

  * 16.

$\gamma_{t}\triangleq wh\sum_{s\in\mathcal{L}}d_{s}j_{st}$: Amount of biomass
per meter of bale conveyor at period $t$ (dry Mg/meter).

Decision variables:

_Binary variables_

  * 1.

$J_{k}$: takes the value 1 if the capacity of an equipment is expanded by
$k$%, takes the value 0 otherwise.

_Continuous variables_

  * 1.

$X_{t}$: In-feed rate of the system at period $t$ (dry Mg/unit of time)

  * 2.

$X_{it}$: Outflow from equipment $i$ at period $t$ (dry Mg/unit of time)

  * 3.

$\mathbb{X}_{ist}$: Outflow of material with moisture level $s$ from equipment
$i\in\mathcal{I_{M}}$ at period $t$ (dry Mg/unit of time)

  * 4.

$\mathbb{M}_{ist}$: Inventory level of biomass with moisture level $s$ in
equipment $i\in\mathcal{I_{M}}$ at period $t$ (dry Mg)

  * 5.

$Y_{t}$: Conveyor speed at period $t$ (meters/unit of time)

  * 6.

$\beta_{t}^{+}$, $\beta_{t}^{-}$: Decision variables to control reactor
feeding rate deviations (dry Mg/unit of time).

The optimization model we make reference to in Figure LABEL:supp-fig:flowchart
is defined by Equations (1)-(21) and corresponds to HPC. The objective
function (1) minimizes the variations of reactor’s feeding, maximizes the
throughput of the system, and determines optimal location and size of the
buffer. The first term in the objective maximizes the throughput of the system
by maximizing $\eta X_{|\mathcal{I}|t}$, the hypothetical revenue gained by
producing/selling one unit of pellet at period $t$. The second term in the
objective minimizes the volatility of the feeding rate of the reactor to
ensure a continuous and smooth feeding of the reactor, a key success indicator
for a biomass feeding system. The last term in the objective minimizes the
cost of expanding the capacity of storage units.

$\displaystyle\max\sum_{t\in\mathcal{T}}\left[\eta
X_{|\mathcal{I}|t}-\alpha\left(\beta_{t}^{+}+\beta_{t}^{-}\right)\right]-\sum_{i\in\mathcal{I_{M}}}\sum_{k\in\mathcal{C}}\sum_{s\in\mathcal{L}}\sum_{t\in\mathcal{T}}c_{isk}j_{st}J_{k}$
(1)

We use Equations (1)-(21) to obtain optimal trajectories of system in-feed
rate, reactor feeding rate, and inventory in storage units. Operating cost of
equipment however has unit of $/hour. Thus, after obtaining the optimal
trajectories, we calculate the corresponding cost of operating the system via

$\displaystyle\sum_{i\in\left(\mathcal{I}-\mathcal{I_{M}}\right)}\sum_{s\in\mathcal{L}}\sum_{t\in\mathcal{T}}c_{is}+\sum_{i\in\mathcal{I_{M}}}\sum_{k\in\mathcal{C}}\sum_{s\in\mathcal{L}}\sum_{t\in\mathcal{T}}c_{isk}j_{st}J_{k}$
(2)

given the optimal value of $J_{k}$. Therefore, the optimal cost we present in
our numerical analyses considers only system operating costs.

Capacity constraints: Each equipment has processing capacities, which we model
via

$\displaystyle X_{it}$
$\displaystyle\leq\sum_{s\in\mathcal{L}}j_{st}\overline{x}_{is}$
$\displaystyle\forall i\in\mathcal{I},\forall t\in\mathcal{T}$ (3)
$\displaystyle\sum_{s\in\mathcal{L}}\mathbb{M}_{ist}$
$\displaystyle\leq\overline{m}_{i}\left(1+\sum_{k\in\mathcal{C}}kJ_{k}\right)$
$\displaystyle\forall i\in\mathcal{I_{M}},\forall t\in\mathcal{T}$ (4)
$\displaystyle\sum_{s\in\mathcal{L}}\frac{\mathbb{M}_{ist}}{\bar{d}_{is}}$
$\displaystyle\leq\overline{v}_{i}\left(1+\sum_{k\in\mathcal{C}}kJ_{k}\right)$
$\displaystyle\forall i\in\mathcal{I_{M}},\forall t\in\mathcal{T}$ (5)
$\displaystyle 1$ $\displaystyle\geq\sum_{k\in\mathcal{C}}J_{k}.$ (6)

Constraint (3) indicate that the outflow of biomass from each equipment should
not exceed its pre-determined processing capacity, which is driven by the
moisture level of the biomass processed at each period. Constraint (4) limit
the mass of biomass inventoried and constraint (5) limit the volume of biomass
inventoried in a storage unit. This consideration is necessary because for
loose biomass (low density), the volume capacity is restrictive, whereas mass
capacity turns out to be restrictive in the case when biomass is highly dense.
These constraints also consider the impact that extensions of the storage
capacity have on the mass and volume of the inventory. We calculate the
average density of biomass $\bar{d}_{is}$ as the weighted average of biomass
densities conveyed via the two conveyors connected to a metering bin. The
corresponding weights are $\theta_{s}$ and $1-\theta_{s}$. Constraint (6)
ensures that only one capacity expansion is chosen. If we desire to run HPC
without the capacity expansion option, we simply adjust constraint (6) by
replacing its left-hand side with 0.

Operational constraints: The system under consideration has operational
capacities related to system in-feed. The in-feed rate of the system refers to
the amount of biomass fed to the system in a period. Constraint (7) calculates
the in-feed rate based on the physical dimensions of a bale, its density, and
conveyor speed. Constraint (8) ensures that every bale is processed.
$\beta_{t}^{+}$ and $\beta_{t}^{-}$ calculate the change of reactor feeding
rate from one period to the next. If
$X_{|\mathcal{I}|t}-X_{|\mathcal{I}|t-1}>0$, $\beta_{t}^{+}>0$ and
$\beta_{t}^{-}=0$, otherwise, if $X_{|\mathcal{I}|t}-X_{|\mathcal{I}|t-1}<0$,
$\beta_{t}^{+}=0$ and $\beta_{t}^{-}>0$. In the case when the feeding of
reactor does not change in successive time periods,
$\beta_{t}^{+}=\beta_{t}^{-}=0$.

$\displaystyle X_{t}$ $\displaystyle=\gamma_{t}Y_{t}$ $\displaystyle\forall
t\in\mathcal{T}$ (7) $\displaystyle\sum_{t\in\mathcal{T}}j_{st}X_{t}$
$\displaystyle=q_{s}n_{s}$ $\displaystyle\forall s\in\mathcal{L}$ (8)
$\displaystyle\beta_{t}^{+}-\beta_{t}^{-}$
$\displaystyle=X_{|\mathcal{I}|t}-X_{|\mathcal{I}|t-1}$ $\displaystyle\forall
t\in\mathcal{T}$ (9)

Flow balance constraints: Feeding system is essentially a flow system that
needs to satisfy flow balance constraints. Constraint (10) computes the
outflow from the grinders. The outflow is reduced by the dry matter losses in
the grinders. Constraint (11) shows that the amount of biomass that does not
bypass will continue flowing on drag chain conveyor 5 (d.c.5). The biomass
that bypasses the separation unit flows on screw conveyor 6 (s.c.6) that feeds
the metering bin, as indicated by constraint (12). Constraint (13) represents
that flow balance (flow in equals flow out) for other equipment.

$\displaystyle X_{it}$
$\displaystyle=\sum_{p\in\mathcal{I}_{i}}(1-\mu_{i})X_{pt}$
$\displaystyle\forall i\in\mathcal{I_{G}},\forall t\in\mathcal{T}$ (10)
$\displaystyle X_{it}$
$\displaystyle=\sum_{p\in\mathcal{I}_{i}}\left(1-\sum_{s\in\mathcal{L}}j_{st}\theta_{s}\right)X_{pt}$
$\displaystyle i=\text{d.c.5},\forall t\in\mathcal{T}$ (11) $\displaystyle
X_{it}$
$\displaystyle=\sum_{p\in\mathcal{I}_{i}}\left(\sum_{s\in\mathcal{L}}j_{st}\theta_{s}\right)X_{pt}$
$\displaystyle i=\text{s.c.6},\forall t\in\mathcal{T}$ (12) $\displaystyle
X_{it}$ $\displaystyle=\sum_{p\in\mathcal{I}_{i}}X_{pt}$ $\displaystyle\forall
i\in\mathcal{I}-\\{\text{d.c.5,
s.c.6},\mathcal{I_{M}},\mathcal{I_{G}}\\},\forall t\in\mathcal{T}$ (13)

Inventory balance constraints: Constraint (14) is the mass balance constraint
for storage units. Constraint (15) finds the total outflow from each storage
unit at period $t$. We assume that the system does not carry initial inventory
in storage units, presented by constraint (16).

$\displaystyle\mathbb{M}_{ist}$
$\displaystyle=\mathbb{M}_{ist-1}+\sum_{p\in\mathcal{I}_{i}}j_{st}X_{pt}-\mathbb{X}_{ist}$
$\displaystyle\forall i\in\mathcal{I_{M}},\forall s\in\mathcal{L},\forall
t\in\mathcal{T}$ (14) $\displaystyle X_{it}$
$\displaystyle=\sum_{s\in\mathcal{L}}\mathbb{X}_{ist}$ $\displaystyle\forall
i\in\mathcal{I_{M}},\forall t\in\mathcal{T}$ (15)
$\displaystyle\mathbb{M}_{is0}$ $\displaystyle=0$ $\displaystyle\forall
i\in\mathcal{I_{M}},\forall s\in\mathcal{L}$ (16)

Non-negativity constraints: The following are the non-negativity constraints.

$\displaystyle 0$ $\displaystyle\leq X_{t}$ $\displaystyle\forall
t\in\mathcal{T}$ (17) $\displaystyle 0$ $\displaystyle\leq X_{it}$
$\displaystyle\forall i\in\mathcal{I},\forall t\in\mathcal{T}$ (18)
$\displaystyle 0$ $\displaystyle\leq Y_{t}$ $\displaystyle\forall
t\in\mathcal{T}$ (19) $\displaystyle 0$
$\displaystyle\leq\mathbb{X}_{ist},\mathbb{M}_{ist}$ $\displaystyle\forall
i\in\mathcal{I_{M}},\forall s\in\mathcal{L},\forall t\in\mathcal{T}$ (20)
$\displaystyle 0$ $\displaystyle\leq\beta_{t}^{+},\beta_{t}^{-}$
$\displaystyle\forall t\in\mathcal{T}$ (21)

We refer to the MILP formulation above ((1), (3) to (21)) as Problem H for
HPC.

Notice that this version of HPC includes fractional milling. If we desire to
use HPC without fractional milling, we make the following adjustments in the
above model. We set $\theta_{s}=0,\forall s\in\mathcal{L}$, which impacts
constraints (11) and (12). Fixing the value of $\theta_{s}=0$ also impacts
constraint (5) since $\theta_{s}$ and $1-\theta_{s}$ are used as weights in
calculations of $\bar{d}_{is}$. Since screw conveyor 6 does not exist in this
problem setting, we deduct the cost of screw conveyor 6 from the cost
expression (2).

####  2.3.2 LP model for BFFPC

We develop an LP formulation of BFFPC by updating the model introduced above
for HPC. In this section, we only list the modifications we make.

Objective function: The objective function of BFFPC focuses on maximizing the
amount of biomass processed within the time horizon.

$\displaystyle\max\sum_{t\in\mathcal{T}}\left[\eta X_{|\mathcal{I}|t}\right]$
(22)

We compute the cost of operating the system via

$\displaystyle\sum_{i\in\mathcal{I}}\sum_{s\in\mathcal{L}}\sum_{t\in\mathcal{T}}c_{is}$
(23)

Capacity constraints of storage units: Since BFFPC does not consider expansion
of storage units capacities, we replace constraints (4) and (5) of HPC with
constraints (24) and (25), as shown below.

$\displaystyle\sum_{s\in\mathcal{L}}\mathbb{M}_{ist}$
$\displaystyle\leq\overline{m}_{i}$ $\displaystyle\forall
i\in\mathcal{I_{M}},\forall t\in\mathcal{T}$ (24)
$\displaystyle\sum_{s\in\mathcal{L}}\frac{\mathbb{M}_{ist}}{\bar{d}_{is}}$
$\displaystyle\leq\overline{v}_{i}$ $\displaystyle\forall
i\in\mathcal{I_{M}},\forall t\in\mathcal{T}$ (25)

Notice that BFFPC does not use variables $J_{k}$; thus, constraint (6) is not
considered. Since BFFPC does not aim to control the variations of a reactor’s
feeding rate, variables $\beta_{t}^{+}$, $\beta_{t}^{-}$ and constraints (9)
and (21) are not considered as well.

The remaining of the constraints are the same in both HPC and BFFPC model
formulations. We refer to the LP formulation presented above (e.g., (22), (3),
(7), (8), (10)-(20), (24), and (25)) as Problem B for BFFPC.

###  2.4 Solution Approach

We implement the LP and MILP models in Julia [30], utilizing JuMP modeling
language [31], and use the Gurobi solver [32]. We execute the models on a
computer that has 32 GB RAM and an Intel(R) Xeon(R) CPU E5-1620 v2 CPU running
at 3.7 GHz. We use a 1-minute granularity in the optimization. We set a
30-minute time limit and 1% optimality gap for Gurobi. The minimum, average,
and maximum CPU time to solve a single LP (Problem B) are 0.282, 0.405, and
0.642 seconds. The corresponding minimum, average, and maximum CPU time to
solve the MILP Problem H are 0.318, 0.721, and 1.774 seconds. The running time
of the algorithm presented in Figure LABEL:supp-fig:flowchart varies from
18.87 seconds to 66.59 seconds. This variation results from different patterns
of feeding bales to the system.

##  3 Case Study

###  3.1 Technical Process Data

We utilize technical process data from the PDU at Idaho National Laboratory
(INL). Table 1 summarizes the data collected by analyzing historical grinding
and densification experiments of switchgrass. The data is grouped based on the
moisture level of biomass and based on processes. Notice that grinders consume
more energy when biomass has a high moisture level [33]. Due to grinding,
biomass loses moisture. Wet material loss is the largest for biomass with high
moisture levels [34]. Since particle sizes alter during grinding, material
density changes. Bulk density of biomass diminishes during first-stage
grinding as biomass in bale form is tapped. In contrast, density increases
during the second-stage grinding as the particles get smaller. As noted
before, wet biomass is processed slow to prevent clogging. Hence, the maximum
in-feed rate of grinders decreases for high-moisture materials, which
negatively impacts the processing capacity of the system. There are several
options to select screen size combinations for coupled grinders. Experiments
at PDU show that it is the best practice to use screens with a diameter of
76.2-mm and 11.11-mm in first-stage and second-stage grinders, respectively
[35]. Utilizing these screens minimizes the amount of fine particles.

|  |  | Moisture Levels
---|---|---|---
|  |  | High | Medium | Low
Bale | Moisture (%) | 25 | 17.50 | 10
Dry bulk density (dry Mg/cubic meter) | 0.144 | 0.144 | 0.144
First-stage | Operating conditions | Screen size (mm) | 76.2 | 76.2 | 76.2
Dry bulk density (dry Mg/cubic meter) | 0.144 | 0.144 | 0.144
Moisture (%) | 25 | 17.5 | 10
Process performance | Energy consumption (kWh/dry Mg) | 25.68 | 21.95 | 7.10
Grinder | Moisture loss (%) | 4.77 | 3.00 | 0.5
Dry matter loss (%) | 1.50 | 1.50 | 1.50
Bulk density change (dry Mg/cubic meter) | -0.09 | -0.10 | -0.11
Maximum in-feed rate (dry Mg/hour) | 2.20 | 4.53 | 5.23
Separation | Operating conditions | Screen size (mm) | 6.35 | 6.35 | 6.35
Dry bulk density (dry Mg/cubic meter) | 0.053 | 0.041 | 0.039
Moisture (%) | 20.23 | 14.50 | 9.50
Unit | Process performance | Moisture loss (%) | 0.71 | 0.50 | 0.00
Bypass ratio (%) | 40.48 | 44.98 | 49.98
Second-stage | Operating conditions | Screen size (mm) | 11.11 | 11.11 | 11.11
Dry bulk density (dry Mg/cubic meter) | 0.053 | 0.041 | 0.039
Moisture (%) | 19.52 | 13.99 | 9.50
Process performance | Energy consumption (kWh/dry Mg) | 49.71 | 16.58 | 14.67
Grinder | Moisture loss (%) | 4.00 | 3.00 | 0.7
Dry matter loss (%) | 0.50 | 0.50 | 0.50
Bulk density change (dry Mg/cubic meter) | 0.066 | 0.082 | 0.090
Maximum in-feed rate (dry Mg/hour) | 1.59 | 2.80 | 5.23
Pelleting | Operating conditions | Dry bulk density (dry Mg/cubic meter) | 0.119 | 0.123 | 0.129
Moisture (%) | 15.52 | 10.99 | 8.80
Process performance | Energy consumption (kWh/dry Mg) | 90.39 | 60.63 | 55.12
Equipment | Moisture loss (%) | 3.92 | 1.50 | 0.00
Bulk density change (dry Mg/cubic meter) | 0.547 | 0.542 | 0.537
Maximum in-feed rate (dry Mg/hour) | 3.33 | 3.81 | 4.76
Table 1: Technical process-related data used as input for BFFPC and HPC

Table 1 also presents the technical process data for the separation unit. The
bypass ratio is smallest in the case when biomass is highly moist. When
materials are wet, they tend to be cohesive, cling to one another, and end up
creating aggregates. This reduces the amount of materials bypassing the
separation unit. Materials lose moisture at this stage as well. Wet materials
lose the largest amounts of moisture during separation. Table 1 also lists the
technical process data for densification equipment (via pelleting mechanism).
Pelleting equipment consumes the highest amount of energy in a biomass feeding
system [36]. When material is wet, energy consumption rises. Moisture loss
occurs due to heating during densification. Wet materials lose the highest
amounts of moisture during pelletization. Operators feed pelleting equipment
at a lower rate when material is highly moist to ensure the production of
high-quality pellets.

All equipment have a processing capacity of 13.61 dry Mg/hour with the
exception of the first-stage grinder, second-stage grinder, and pelleting.
Metering and storage bins have a storage capacity of 4.54 dry Mg. The volume
capacity of these storage units is 24.07 cubic meter. Width, height, and
length of a bale are 1.22, 0.91, and 2.44 meters, respectively. The mass of a
bale is 0.392 dry Mg. We assume that the sale price of pellets is $77.16/dry
Mg. We assume that unit cost of adjustment of reactor feeding is $5.51/(dry
Mg/minute). We conduct a sensitivity analysis with different values of sale
price of pellets and unit cost of adjustment of reactor feeding. We observe
that solutions are not sensitive to these parameters.

###  3.2 Techno-economic Data

We use the Biomass Logistics Model (BLM) to estimate input cost data, such as
the unit operation cost of pre-processing, handling, and storing used in the
model [37]. BLM is an integrated framework that quantifies the impacts of
feedstock supply system design on costs, greenhouse gas emissions, and energy
consumption. It uses a system dynamics simulation model to simulate the flow
of biomass through the whole supply chain, track the change in biomass
physical characteristics during operations, and adjust cost/benefit of the
engineered system resulting from these operations. BLM estimates the cost of
operating a feeding system in $/hour (see Table LABEL:supp-tab:costdata).

In the case when the capacity of the storage units expands, the corresponding
operation cost per minute increases. We use the following scaling rule [38] to
estimate the new costs.

$\displaystyle\frac{c_{is}}{c_{isk}}$
$\displaystyle=\left(\frac{\overline{m}_{i}}{(1+k)\overline{m}_{i}}\right)^{0.6}$
$\displaystyle\forall i\in\mathcal{I_{M}},\forall s\in\mathcal{L},\forall
k\in\mathcal{C}$ (26)

where 0.6 represents economies of scale.

##  4 Results

In this section, we present the results from implementing the models developed
for process design, namely BFFPC and HPC, with and without fractional milling.
We compare the performance of BFFPC and HPC by investigating: (i) the optimal
trajectories; (ii) the average reactor feeding rate and its variability; (iii)
the effect of buffer size expansion and location on system’s performance; (iv)
the cost of operating the system; and ($v$) the minimum time to process all
bales.

We assume that a total of 200 bales are to be processed in the feeding system.
Of these, 60 are low-moisture bales, 100 are medium-moisture bales, and 40 are
high-moisture bales. These bales are processed in HPC following certain
patterns, which are to process six low-moisture bales, then to process ten
medium-moisture bales, and finally, to process four high-moisture bales. This
pattern is repeated until all 200 bales are processed. This pattern is known
as 6L,10M,4H X10. BFFPC processes the same number of bales with same moisture
distribution. However, bales are not sequenced according to their moisture
level. The feeding pattern is determined randomly and known as “random
pattern” in this section.

The following is a list of performance metrics we use to compare HPC and
BFFPC:

  * 1.

Maximum inventory (Max. Inv.) in storage units is
$\max_{t}\sum_{s\in\mathcal{L}}\mathbb{M}_{ist},\forall i\in\mathcal{I_{M}}$.

  * 2.

Average reactor feeding (Av. Feed.) is
$\frac{\sum_{t\in\mathcal{T}}X_{|\mathcal{I}|t}}{|\mathcal{T}|}$.

  * 3.

Variability of reactor’s feeding (Coef. Var.) is measured via the coefficient
of variation of $X_{|\mathcal{I}|t}$, which is
$\frac{\text{std}\left(X_{|\mathcal{I}|t}\right)}{\left(\frac{\sum_{t\in\mathcal{T}}X_{|\mathcal{I}|t}}{|\mathcal{T}|}\right)}$
where std is standard deviation operator.

  * 4.

Cost of operating systems (Cost) for HPC and BFFPC is calculated via (2) and
(23), respectively.

  * 5.

Minimum time to process 200 bales (Min. Time).

There are other studies that use the coefficient of variation to measure the
variability in production systems, examples of which are given in systems
reliability literature [39, 19]. Coefficient of variation shows the extent of
deviation from average performance in a process. In our case, it indicates how
much the feeding rate of a reactor differs from one period to the next.

###  4.1 Optimal trajectories of feeding rate of reactor, in-feed rate, and
inventory

In this section, we illustrate the optimal trajectories of a reactor’s feeding
rate, a system’s in-feed rate, and the inventory level. Figure 2 (a)
illustrates the optimal trajectories of reactor’s feeding rates for BFFPC and
HPC for the process design without fractional milling. Reactor’s feeding rate
in BFFPC varies from 0 to 5 dry Mg/hour, whereas it is constant in HPC. In the
process with fractional milling, BFFPC feeds the reactor at a rate ranging
from 2 dry Mg/hour to 5 dry Mg/hour, whereas the HPC feeds the reactor at a
constant rate (see Figure 2 (b)). These results indicate that HPC reduces the
variation of a reactor’s feeding rate by 100%, regardless of whether
fractional milling is utilized. We further observe that processing times of
HPC are slightly shorter than those of BFFPC. It is observed that BFFPC
without fractional milling exhibits more variability as compared to BFFPC with
fractional milling.

(a) Without fractional milling

(b) With fractional milling

Figure 2: Optimal trajectories of reactor feeding rates for BFFPC (with
feeding rate “random pattern”) and HPC (with feeding rate 6L,10M,4H X10)

Figure 3 compares the trajectories of a systems’ in-feed rate. We observe that
both BFFPC and HPC adjust bale in-feed rates differently. Although the maximum
in-feed rate for both pre-processing systems is the same, the sequences of
bale fed are different. HPC’s in-feed trajectory has a more consistent pattern
(same repeated adjustments). If we compare in-feed rate trajectories with and
without fractional milling, it is observed that the range of the in-feed rate
lies in an interval of 0 dry Mg/hour and 5 dry Mg/hour without fractional
milling, this range is narrower with fractional milling. For instance, we
rarely see in-feed rate drops to 0 dry Mg/hour with fractional milling.

(a) Without fractional milling

(b) With fractional milling

Figure 3: Optimal trajectories of bale in-feed rates for BFFPC (with feeding
rate “random pattern”) and HPC (with feeding rate 6L,10M,4H X10)

Figure 4 reveals optimal trajectories of the total inventory level in metering
and storage bins. We observe that the build up of inventory in storage units
for BFFPC is irregular. On the other hand, inventory build up for HPC follows
a pattern. Biomass accumulates in storage units while processing low- and
medium-moisture bales, and biomass is discharged while processing high-
moisture bales. The breakpoints observed for HPC correspond to the times when
operators switch between bales with different moisture levels. This approach
to inventory control minimizes inventory level. The results of Figure 4 show
that HPC tends to keep more inventory than BFFPC. The higher inventory level
leads to a shorter processing time, as well as continuous and smooth feeding
of the reactor.

For the feeding patterns presented in Figure 4, we calculate the total
inventory carried over the entire modeling horizon. It is observed that BFFPC
with fractional milling keeps more inventory than BFFPC without fractional
milling. However, HPC without fractional milling carries more inventory than
HPC with fractional milling. The same conclusions are reached when analyzing
HPC and BFFPC with other feeding patterns.

(a) Without fractional milling

(b) With fractional milling

Figure 4: Optimal trajectories of total inventory in metering and storage bins
for BFFPC (with feeding rate “random pattern”) and HPC (with feeding rate
6L,10M,4H X10)

###  4.2 Average feeding rate of reactor and variability in feeding

In this section, we compare the average feeding rate of the reactor and the
variability in feeding. If BFFPC is replaced by HPC in the process design
without fractional milling, a reduction in the coefficient of variation goes
from 5.48% to 100%. The average feeding rates increases from 2.713 dry Mg/hour
to 2.723 dry Mg/hour, as shown in Figure 5(a). If BFFPC is replaced by HPC in
the process design with fractional milling, similar improvements are observed
in the average feeding rate of reactor and variability in feeding. HPC
increases the average feeding rate from 3.696 dry Mg/hour to 3.780 dry
Mg/hour, as shown in Figure 5(b). The reduction in the coefficient of
variation in BFFPC and HPC are 62.38% and 100%, respectively.

(a) Without fractional milling

(b) With fractional milling

Figure 5: Average feeding rate of reactor and reduction in coefficient of
variation of reactor feeding rate for BFFPC (with feeding rate “random
pattern”) and HPC (with feeding rate 6L,10M,4H X10)

###  4.3 Optimal size of buffers to improve the reactor feeding rate

In this section, we evaluate the impact of buffer size and location to the
performance of the feeding system. We remind the reader that the proposed MILP
formulation allows capacity expansions of storage units, if economical. We
experiment with different feeding patterns in which 60 low-moisture bales, 100
medium-moisture bales, and 40 high-moisture bales are fed to the system in
this order (we denote this pattern with 60L,100M,40H). When using the feeding
pattern 6L, 10M, 4H X10, the system does not inventory in large amounts; thus,
the storage capacity is often not violated. The storage capacity impacts the
performance of the system when processing this new pattern.

In Figures 6 and 7, “optimized buffer size capacity” refers to the model that
allows expansion (as needed) of the capacity of storage units, and “fixed
capacity” refers to the model that does not allow the expansion of the
capacity of storage units. We note that the system without fractional milling
does not need expansion in storage units; and thus optimized buffer size
capacity equals original capacities. In an optimal solution of the system with
fractional milling, buffer size capacities are expanded by 100%.

The results of Figures 6(b) and Figure 7(b) show that the HPC system improves
its performance when processing the feeding pattern 60L, 100M, 40H and
additional storage capacity is available. In this case, the process throughput
is 3.649 dry Mg/hour, the reduction in the coefficient of variation is 100%,
the average unit cost is $37.15/dry Mg, and the processing time is 21.07
hours. If the existing storage capacity (of 4.54 dry Mg and 24.07 cubic meter)
in the process design did not increase, the average throughput would be lower
(3.564 dry Mg/hour), the reduction in coefficient of variation would be lower
(83.06%), the average unit cost would be lower ($35.97/dry Mg), and the
process time would be higher (21.57 hours). The additional investment cost is
the main reason why the average unit cost is higher in a system that uses
additional storage capacity than a system that does not. These additional
costs outweigh the reductions of processing costs due to shorter processing
times. The impact of buffer size is minimal to the process design without
fractional milling. This is because system feeding is relatively slower and it
does not require extra storage capacity.

(a) Without fractional milling

(b) With fractional milling

Figure 6: Average feeding rate of reactor and coefficient of variation in
reactor feeding rate with fixed and optimized capacities of storage units

(a) Without fractional milling

(b) With fractional milling

Figure 7: Cost of operating system and minimum time to process all bales with
fixed and optimized capacities of storage units

###  4.4 Cost of operating the system

Table 2 summarizes the estimated costs of operating the system and the minimum
time required to process 200 bales for BFFPC and HPC. We observe that HPC
reduces costs by 0.36% and processing time by 0.35% over BFFPC in a process
without fractional milling. When fractional milling is utilized, HPC’s
economical advantages over BFFPC increase. HPC saves 2.22% in costs and 2.24%
in processing time. These results also indicate improvements of processing
time and costs due to fractional milling.

Structure | Control | Cost ($/dry Mg) | Min. Time (hours)
---|---|---|---
Without fractional milling | BFFPC | 42.32 | 28.27
HPC | 42.16 | 28.17
With fractional milling | BFFPC | 34.71 | 20.80
HPC | 33.94 | 20.33
Table 2: Cost of operating the system and minimum time to process 200 bales
for BFFPC (with feeding rate “random pattern”) and HPC (with feeding rate
6L,10M,4H X10)

##  5 Conclusion and Future Research Directions

The variations in feedstock characteristics, such as feedstock moisture and
particle size distribution lead to an inconsistent flow of feedstock from the
biomass pre-processing system to a rector in-feed system. In this research, we
developed an optimal process control method for a biomass pre-processing
system comprised of milling and densification operations to provide consistent
flow of feedstock to a reactor’s throat. Our case study based on switchgrass
finds that the proposed optimal process control reduces the variation of
reactor feeding rate by 100% without increasing the cost of a biomass pre-
processing system operating if the biomass moisture ranges from 10 to 25%.
This variability reduction is possible via identification of optimal bale
sequencing, equipment in-feed rate in the process, and process buffer location
and size in the biomass pre-processing system. A biorefinery can adapt
proposed control to achieve its design capacity.

Future research will focus on integrating the outcomes of Discrete Element
Method (DEM) models with optimization models. DEM models are typically used to
simulate the mechanical and flow behavior of feedstocks in equipment for given
feedstock characteristics and equipment design. The results from these
simulations can be used to develop functional relationships between feedstock
characteristics (e.g., type, moisture content, particle size distribution,
density), equipment setting, and biomass flow. These functional relationships
can be incorporated into the optimization model to control the flow of biomass
to the reactor. For example, biomass type, moisture level and mill speed in
grinders impact biomass density. The density of biomass impacts its volume,
and as a consequence, the storage capacity of the metering bin. Thus, outcomes
of DEM models can be used to develop functional relationships between biomass
type, biomass moisture level, mill speed in grinders and density to be
included in optimization. There are a few DEM models for biomass feedstocks,
such as, Guo et al. [40] which focuses on switchgrass and Xia et al. [41]
which focuses on pinewood. We will incorporate the outcomes of similar models
to optimization to improve the accuracy of our results.

##  6 Credit author statement

Fikri Kucuksayacigil: Develop and implementation of model utilizing
cplex/Gurobi, prepare results, writing original draft. Mohammad Roni:
Conceptualization, develop methodology, scoping the study, data collection,
writing: prepare final manuscript Sandra D. Eksioglu: Develop methodology,
supervision, writing- review and editing. Qiushi Chen:Writing - review &
editing.

##  7 Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  8 Acknowledgment

The funding of this work was provided by the U.S. Department of Energy, Office
of Energy Efficiency and Renewable Energy, Bioenergy Technologies Office under
award Number DE- EE0008255 and Department of Energy Idaho Operations Office
Contract No. DE-AC07-05ID14517. The authors greatly thank INL staff in the DOE
Biomass Feedstock National User Facility for providing technical data for this
study. Specifically, we acknowledge Neal A. Yancey and Jaya S. Tumuluru, who
are the individual INL contributors to this research who provided helpful
comments, data, and other forms of support for this analysis. The views
expressed herein are those of the author only, and do not necessarily
represent the views of DOE or the U.S. Government.

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