# Utilizing Load Shifting for Optimal Compressor Sequencing in Industrial
Refrigeration

Rohit Konda, Vikas Chandan, Jesse Crossno, Blake Pollard, Dan Walsh, Rick
Bohonek, and Jason R. Marden R. Konda<EMAIL_ADDRESS>and J. R. Marden
(jrmarden@ece.ucsb.edu)are with the Department of Electrical and Computer
Engineering at the University of California, Santa Barbara, CA. Vikas Chandan
<EMAIL_ADDRESS>Jesse Crossno<EMAIL_ADDRESS>Blake Pollard
<EMAIL_ADDRESS>and Dan Walsh<EMAIL_ADDRESS>are with
CrossnoKaye®. Rick Bohonek<EMAIL_ADDRESS>is with Butterball LLC®.
This work is supported by funding from CrossnoKaye.

###### Abstract

The ubiquity and energy needs of industrial refrigeration has prompted several
research studies investigating various control opportunities for reducing
energy demand. This work focuses on one such opportunity, termed _compressor
sequencing_ , which entails intelligently selecting the operational state of
the compressors to service the required refrigeration load with the least
possible work. We first study the static compressor sequencing problem and
observe that deriving the optimal compressor operational state is
computationally challenging and can vary dramatically based on the
refrigeration load. Thus we introduce load shifting in conjunction with
compressor sequencing, which entails strategically precooling the facility to
allow for more efficient compressor operation. Interestingly, we show that
load shifting not only provides benefits in computing the optimal compressor
operational state, but also can lead to significant energy savings. Our
results are based on and compared to real-world sensor data from an operating
industrial refrigeration site of Butterball LLC®located in Huntsville, AR,
which demonstrated that without load shifting, even optimal compressor
operation results in compressors often running at intermediate capacity
levels, which can lead to inefficiencies. Through collected data, we
demonstrate that a load shifting approach for compressor sequencing has the
potential to reduce energy use of the compressors up to $20\%$ compared to
optimal sequencing without load shifting.

##  I Introduction

Industrial refrigeration systems are present in a multitude of sectors, not
limited to food processing, plastics, electronics, and chemical processing [1,
2, 3]. Altogether, industrial refrigeration accounts for approximately $8.4\%$
of total energy usage in the U.S [4]. As such, there are tremendous energy
saving opportunities available in industrial refrigeration, not only through
updating hardware components, but also increasing the sophistication of the
implemented control algorithms. Algorithmic improvements are potentially more
enticing, as they can realize significant energy savings with minimal capital
expenditures to retrofit the system.

The four central components of a prototypical refrigeration system include the
evaporators, compressors, condensers, and the expansion valve, with
interconnections as illustrated in Figure 1. We study a common industrial
refrigeration process, where ammonia refrigerant is circulated in a closed
loop in a vapor compression cycle to move heat against the thermal gradient of
the system. Informally, the main thermodynamic steps in an ideal vapor
compression cycle are summarized as follows111The actualized refrigeration
process deviates from this idealization considerably, but we simplify for the
purpose of presentation.:

$(1\to 2)$ The refrigerant vapor flows through a compressor, where it is
compressed from a low pressure, referred to as suction pressure, to a high
pressure, referred to as discharge pressure. A consequence of this compression
is an increase in the temperature of the refrigerant vapor, which now takes
the form of a super-heated vapor.

$(2\to 3)$ The refrigerant super-heated vapor is then fed to the condenser,
where constant pressure heat rejection occurs, and heat is released to the
ambient environment, resulting in condensation of the ammonia. A consequence
of this heat rejection is that the refrigerant transitions from a super-heated
vapor to a saturated liquid.

$(3\to 4)$ The refrigerant is then expanded adiabatically across an expansion
valve, reducing the temperature and pressure and resulting in a vapor-liquid
mixture.

$(4\to 1)$ Cooled refrigerant liquid flows through the evaporator, where heat
absorption from the system via evaporation of the refrigerant occurs, and
super-heated vapor is fed back to the compressor, completing the cycle.

We direct the interested reader to [3] for a comprehensive review of
refrigeration systems.

Figure 1: A simplified diagram of the refrigeration components are depicted
showing the flow of ammonia through the vapor compression process. Figure 2:
This figure highlights the cumulative distribution functions for the slide
valve position for four compressors operating at the Butterball facility
during the month of June, 2023. Here, the slide valve position is associated
with compressor capacity, where $100\%$ means that the compressor is running
at full capacity. We also highlight the percentages in which each compressor
is operating at full capacity (where the slide valve sensor is measured above
$99\%$) or trim as well as the percentage of time the compressor is turned on
and off. Note that the compressors are often operating in trim, suggesting
that there are potential opportunities to save energy by operating the
compressors at full capacity more often.

The configuration and operation of the entire refrigeration system can have
significant impacts on the cost of operation; this can either be measured
through total power, electric cost, carbon emissions, etc. Infrastructural
retrofits of the refrigeration system, including changing the choice of
refrigerant, hardware specifications of components, or general system layout,
can be typically costly to implement. Accordingly, a more viable way to reduce
costs is to strategically adjust the control policies of the refrigeration
components to meet the required heat extraction while minimizing the
operational cost. For example, _thermal load shifting_ has received
significant attention as a methodology to preemptively cool a facility in
order to take financial advantage of dynamic energy cost-rate structures [5,
6, 7, 8, 9, 10, 11]. Additionally, another approach is _set point
optimization_ , where the set points for suction and discharge pressure are
dynamically adjusted to drive the refrigeration system towards an energy
optimal operating state while maintaining the desired achievable cooling
demands [12, 13, 14]. For these domains, standard control and optimization
techniques, such as model predictive control and dynamic programming, can be
implemented for attaining near-optimal control strategies.

While most of the existing control approaches for energy optimization focus on
the evaporators, e.g., thermal load shifting, it is important to highlight
that the compressors represent the dominant energy expenditure in most
refrigeration systems, typically accounting for more than $85\%$ of the total
energy usage of the refrigeration process. For example, Figure 3 highlights
the approximate breakdown of the energy demand associated with an industrial
refrigeration site of Butterball, which is large poultry processing facility,
during the month of June, 2023. It is widely known that compressors are
operated most efficiently when running at full capacity [15, 16]; however, the
typical control objective for the compressors is suction pressure
stabilization. In this way, the operational state of the compressors is
directly dependent on the state of the evaporators and this can ultimately
lead to the compressors operating in an inefficient manner, i.e., at partial
capacity. Figure 2 confirms this phenomena, directly highlighting the
cumulative distribution functions for the slide valve position of the four
compressors operating at the Butterball facility during this time
period.222This data was acquired through direct partnership with CrossnoKaye
(see crossnokaye.com), which focuses on the derivation and implementation of
intelligent control systems for industrial refrigeration systems in the cold
food and beverage domain. CrossnoKaye has been monitoring and controlling the
refrigeration system at the Butterball facility since April 2023 Here, the
slide valve position can be viewed in the same light as capacity, where
$100\%$ means that the compressor is running at full capacity. Note that the
slide valve positions are often significantly below $100\%$, suggesting that
there are potential energy saving opportunities in algorithmic improvements
for compressor scheduling and control.

Figure 3: Approximate power breakdown for a Butterball facility at Huntsville.

‘

This paper shifts the control focus from the evaporators directly to the
compressors, where the goal is to optimize the operational state of the
compressors to serve the required refrigeration load. We formalize this
optimization problem as the _compressor sequencing problem_ [15]. We begin by
characterizing the optimal solution to the _static_ compressor sequencing
problem, which focuses on satisfying a given refrigeration load at a single
time instance. Interestingly, we show that the optimal solution can be derived
via a simple water-filling algorithm. However, this water-filling algorithm
must be executed with the correct compressor sequence, which is dependent on
the refrigeration load and can be difficult to compute, to recover the optimal
solution. We note that the optimal compressor sequence may also vary
drastically as a function of the refrigeration load, hence creating
significant issues from both an implementation and computation perspective.

Given these limitations, our second result shifts attention from the static
compressor sequencing problem to the _dynamic_ compressor sequencing problem.
Here, we are provided with a given time-dependent profile of the refrigeration
load that we need to serve over a given horizon. See forthcoming Figure 7 as
an illustration of a typical refrigeration load over a month long horizon at
the Butterball facility. Unlike the static problem, this dynamic formulation
gives us the flexibility to exploit load shifting, where one preemptively
cools the facility (i.e., services future loads), so that the compressors can
operate in a more efficient fashion, i.e., more often at full capacity. Our
main results characterize the optimal solution of this dynamic compressor
sequencing problem. Not only does load shifting provide substantial potential
for energy savings, interestingly, the resulting optimal solution is more
implementable and tractable than the optimal solution of the static problem.
In particular, the optimal solution is the result of water-filling algorithm
at each instance, where now the order of the compressors is fixed and can be
set a priori. This ensures that the resulting behavior will be far more
predictable and stable than just implementing the solution of the optimal
static compressor sequencing each period, thus making it more amenable to real
world implementation.

In order to practically realize the energy saving opportunities associated
with compressor sequencing, we implement a numerical case study on the
operation of the Butterball facility in Section III. The basis of this
numerical study was from collected time-series data on compressor
configurations and refrigeration load estimates. Using these refrigeration
load estimates, our initial results suggest that the potential energy savings
could be significant, with upwards of $20\%$ reduction in total energy
expenditure when comparing optimal compressor sequencing with load shifting to
optimal compressor sequencing without load shifting. Furthermore, this paper
provides a number of supporting results characterizing properties of the
optimal and near-optimal online load shifting algorithms.

While we introduce load shifting as a novel mechanism to address the
compressor sequencing problem, practical implementations of the proposed
algorithms would require additional practical considerations. Increasing or
decreasing the capacities of the compressors carelessly can result in
undesirable swings in suction and discharge pressure and lead to system
instability. Classically, evaporators handle thermal regulation while the
compressors handle pressure regulation. One possible way to implement load
shifting is to directly control compressor operation while having the
evaporators handle pressure regulation, either by turning additional
evaporators on/off or adjusting the fan speed. Nevertheless, this work is
merely gauged at assessing the potential benefits of load shifting for
compressor sequencing with the realization of practical control strategies to
be relegated to future work.

##  II Optimal Compressor Sequencing

In this section, we formalize the control problem for optimal compressor
sequencing. Here, the operational state of the compressors (e.g. the on/off
status as well as the slide valve position) is chosen such that the thermal
demands are met with the least cost, which we measure in terms of energy
usage. We discuss potential opportunities for algorithmic improvements in this
section.

###  II-A Preliminaries

Many large scale refrigeration systems employ algorithms for intelligently
choosing the operational state of the compressors. In refrigeration systems
with multiple compressors, one must decide the operational state of these
compressors that is necessary to service the underlying refrigeration load.
More formally, let $\mathcal{C}$ denote a finite set of compressors (for
Butterball, $\mathcal{C}=\\{C_{1},C_{2},C_{3},C_{4}\\}$), where each
compressor $c\in\mathcal{C}$ is associated with a minimum and maximum heat
capacity, $q_{c}^{-}$ and $q_{c}^{+}$ respectively, as well as a power-heat
curve $P_{c}:Q_{c}\to\mathbb{R}_{\geq 0}$ where
$Q_{c}=0\cup[q_{c}^{-},q_{c}^{+}]$ designates the viable refrigeration loads
on compressor $c$, with $0$ indicating the compressor is turned off. Here,
$P_{c}(q_{c})\geq 0$ is the power required to serve heating load $q_{c}\in
Q_{c}$ through compressor $c$. We assume that $P_{c}(0)=0$ and $P_{c}$ is
concave and increasing over the interval $[q_{c}^{-},q_{c}^{+}]$, which
implies that compressors operate more efficiently at higher capacities.
Specifically for the compressors in operation at Butterball, we assume an
affine structure for the power-heat curves, where for any compressor
$c\in{\cal C}$ and thermal load $q_{c}\in[q_{c}^{-},q_{c}^{+}]$ we have

$P_{c}(q_{c})=P_{c}(q_{c}^{-})+\left(\frac{q_{c}-q_{c}^{-}}{q_{c}^{+}-q_{c}^{-}}\right)\left(P_{c}(q_{c}^{+})-P_{c}(q_{c}^{-})\right).$

We validate the affine models for the power-heat curves against collected data
on estimated refrigeration load and compressor power, which is shown in Figure
4. The extreme points of these power-heat curves is summarized in Table I.

Compressor | C1 | C2 | C3 | C4
---|---|---|---|---
Model | Screw | Screw | Screw | Screw
$q_{c}^{-}$ (kW) | 220 | 239 | 165 | 284
$q_{c}^{+}$ (kW) | 3000 | 2126 | 1760 | 2351
$P(q_{c}^{-})$ (kW) | 124 | 173 | 142 | 181
$P(q_{c}^{+})$ (kW) | 262 | 427 | 356 | 494
TABLE I: Compressor Characteristics

Note that for simplicity, we removed the dependence on slide valve position to
provide a direct relationship between thermal load and power.

Figure 4: For each of the compressors, the estimated power and heat capacity
for each minute in the month of June was recorded for a Butterball facility.
We depict the resulting spread in the given figure and notice a fairly affine
relationship, which we denote in red. This is also supported from
manufacturing simulation software for the compressors.

The problem of compressor sequencing centers on the goal of meeting the
incoming refrigeration load, which we denote by
$q^{\mathrm{in}}\in\mathbb{R}_{\geq 0}$, with the least possible energy
expenditure. More formally, the goal is to identify compressor loads
$\\{q_{c}\\}_{c\in\mathcal{C}}$ that satisfy the incoming refrigeration load,
i.e., $\sum_{c\in\mathcal{C}}q_{c}\geq q^{\mathrm{in}}$, and minimize the
total work expenditure as measured by the total power usage by the
compressors, i.e., $\sum_{c\in{\cal C}}P_{c}(q_{c})$. We denote this
compressor assignment by the policy $\pi:\mathbb{R}_{\geq
0}\rightarrow\prod_{c\in{\cal C}}Q_{c}$, where
$\pi(q^{\mathrm{in}})=\\{q_{c}\\}_{c\in{\cal C}}$ designates the refrigeration
loads for each compressor $c\in\mathcal{C}$. The cost of a policy $\pi$ for a
given $q^{\mathrm{in}}$ is defined by
$J_{\pi}(q^{\mathrm{in}})=\sum_{c\in\mathcal{C}}P_{c}(\pi_{c}(q^{\mathrm{in}}))$.
We will henceforth remove the dependence on the compressor set, i.e., denote
$\\{\cdot\\}_{c\in{\cal C}}$ as merely $\\{\cdot\\}$, for notational
simplicity.

###  II-B Fixed Order Compressor Sequencing

The industrial standard for compressor operation is to meet a given
refrigeration load $q^{\mathrm{in}}$ through a water filling algorithm with a
pre-determined order of compressors $\mathcal{C}$. For ease of presentation,
let the set of compressors $\mathcal{C}=\\{c_{1},c_{2},\dots,c_{m}\\}$
naturally denote the order of the compressors, i.e., $c_{1}$ first, $c_{2}$
second, etc. Then the operation of the compressors according to this policy,
represented by $\pi^{\mathrm{FO}}$, is given by Algorithm 1.

Algorithm 1 Water Filling Algorithm

$\mathcal{O}$, $q^{\mathrm{in}}$, $q^{\mathrm{tot}}\leftarrow 0$,
$q_{c}\leftarrow 0$ for all $c\in\mathcal{C}$

for $c$ in $\mathcal{O}$ do

if $q^{\mathrm{in}}>q^{\mathrm{tot}}$ then

$q_{c}\leftarrow q_{c}^{+}$

$q^{\mathrm{tot}}\leftarrow q^{\mathrm{tot}}+q_{c}^{+}$

end if

end for

for $c$ in $\mathrm{reverse}(\mathcal{O})$ do

if $q^{\mathrm{in}}\leq q^{\mathrm{tot}}$ and $q_{c}\neq 0$ then

$d\leftarrow\min\\{q_{c}^{+}-q_{c}^{-},q^{\mathrm{tot}}-q^{\mathrm{in}}\\}$

$q_{c}\leftarrow q_{c}-d$

$q^{\mathrm{tot}}\leftarrow q^{\mathrm{tot}}-d$

end if

end for

return $\\{q_{c}\\}_{c\in\mathcal{C}}$

We will denote the fixed order policy as
$\pi^{\mathrm{FO}}(q^{\mathrm{in}};\mathcal{O})$, where $\mathcal{O}$
describes a specific ordering of the compressor set $\mathcal{C}$. Note that
Algorithm 1 returns a thermal load profile $\\{q_{c}\\}$ that is guaranteed to
satisfy the inequality $\sum_{c\in\mathcal{C}}q_{c}\geq q^{\mathrm{in}}$
provided that we assume that $\min_{c}q_{c}^{-}\leq
q^{\mathrm{in}}\leq\sum_{c\in\mathcal{C}}q_{c}^{+}$. When $q_{c}=q_{c}^{+}$,
we say that the compressor is operating at full capacity. Alternatively, when
$q_{c}^{+}>q_{c}\geq q_{c}^{-}$, we say that the compressor is operating in
trim. A numerical example of Algorithm 1 is shown below.

######  Example 1.

Consider the set of compressors $\mathcal{C}$ associated with Butterball with
order and characteristics following from Table I. Let Algorithm 1 be run for
an incoming refrigeration load $q^{\mathrm{in}}=3100$. Since
$q^{\mathrm{in}}>q_{1}^{+}=3000$, we first set $q_{1}\leftarrow q_{1}^{+}$ and
$q_{2}\leftarrow q_{2}^{+}=2126$ according to the first loop in Algorithm 1.
Since $3000+2126>3100$, we run the second loop to scale back the compressor
loads. First we set $q_{2}\leftarrow q_{2}^{-}=239$, resulting in a total load
of $3239>3100$. Lastly, to match $q^{\mathrm{in}}$, we scale back
$q_{1}\leftarrow 3000-139$ to result in compressor loads $q_{1}=2861$,
$q_{2}=239$, $q_{3}=0$, and $q_{4}=0$.

The central tuning parameter of the policy
$\pi^{\mathrm{FO}}(q^{\mathrm{in}};\mathcal{O})$ is the order $\mathcal{O}$
that the water filling algorithm is run under. There may be significant
differences in energy usage for different orders, especially if compressors
vary in the efficiency. This phenomenon is displayed in Figure 5, where we see
huge gap in energy usage between the best and worst compressor sequence when
considering the four compressors in operation at Butterball.

Figure 5: In this figure, for each possible refrigeration load, we calculate
the power usage between water filling using the best and worst order of the
four compressors in operation at Butterball. Notice that there can be
potentially a large gap between power usage between the different orders.

Note that for intermediate thermal loads, the efficiency loss according to a
fixed-order water filling algorithm could be in upwards of $49\%$ (see Table
II).

###  II-C Optimal Compressor Sequencing

While the standard water-filling algorithms provide a straightforward approach
to compressor sequencing, moving away from a fixed order scheme may lead to
more efficient compressor operation. Hence, we consider the problem of optimal
compressor sequencing in this section, where the goal is to determine the
compressor state that meets the refrigeration load with the least possible
energy expenditure. More formally, the operation of the compressors would be
determined by the solution of the following non-convex optimization problem.

$\displaystyle J^{*}(q^{\mathrm{in}})$ $\displaystyle=\min_{q_{c}\in
Q_{c}}\sum_{c\in\mathcal{C}}P_{c}(q_{c})$ (1) $\displaystyle\text{s.t.
}\sum_{c\in\mathcal{C}}q_{c}\geq q^{\mathrm{in}}$

The following proposition characterizes the structure of optimal solution to
the compressor sequencing problem in Eq. (1). In fact, regardless of the
refrigeration load $q^{\mathrm{in}}$, the optimal compressor state can be
realized by a water-filling algorithm with a specific compressor order that
depends on $q^{\mathrm{in}}$.

######  Proposition 1.

Let $q^{\mathrm{in}}$ be the incoming refrigeration load. The optimal
compressor state, as given by the solution of Eq. (1), can be realized by the
water filling algorithm given in Algorithm 1 with a specific order
$\mathcal{O}$ that depends on $q^{\mathrm{in}}$.

###### Proof.

To show this statement, we show equivalently that for all the compressors,
only one compressor $c\in\mathcal{C}$ has $q_{c}^{+}>q^{*}_{c}>q_{c}^{-}$ in
the optimal solution to Eq. (1). Note that if this is true, the optimal order
$\mathcal{O}$ coincides to when the compressors are ordered decreasing in
their capacities $\\{q^{*}_{c}\\}$. Given this order, Algorithm 1 will produce
the equivalent capacity $\\{q^{\mathrm{alg}}_{c}\\}=\\{q^{*}_{c}\\}$ that
match the optimal solution.

We show the claim that at most one
$q^{*}_{c}\notin\\{q_{c}^{-},q_{c}^{+},0\\}$ is not at the endpoints through
contradiction. Let $i$ and $j$ be the compressors at partial capacity in the
optimal solution. Notice that the function $P_{i}(q_{i}+d)+P_{j}(q_{j}-d)$ is
a concave function of $d$, by assumption of concavity of $P_{i}$ and $P_{j}$
and preservation of concavity under affine transformations. For any feasible
$d$, note that the constraint in Eq. (1) is always satisfied if we change
$q_{i}$ to $q_{i}+d$ and $q_{j}$ to $q_{j}-d$. Additionally, since
$P_{i}(q_{i}+d)+P_{j}(q_{j}-d)$ is concave, the optimal value for $d$ must
occur at either endpoints of the feasible interval. Thus in the optimal
solution, $q_{i}$ or $q_{j}$ must be either at $q_{c}^{-}$ or $q_{c}^{+}$ or
$0$, ensuing in contradiction. ∎

This structural result demonstrates that the optimal control algorithm for
compressor sequencing could be realized from the perspective of a partitioning
process where the admissible refrigeration loads are partitioned into various
regions, and each region is associated with a distinct ordering as shown in
Figure 6. However, this approach to compressor sequencing has significant
problems from both a computation and implementation perspective. First,
solving the optimization problem in Eq. (1) represents a mixed integer
optimization that grows exponentially in complexity in the size of the
compressor set $\mathcal{C}$. Furthermore, observe that the optimal order can
change drastically as a function of the refrigeration load as highlighted in
Figure 6. This means that the state of the compressors could shift wildly
during operation, which may be infeasible due to delays in changing compressor
capacities and causing unnecessary variability in compressor operation. Thus,
we look to load shifting as a medium to smooth out the compressor sequencing
problem - we discuss this in the next section.

Figure 6: We depict the optimal compressor sequence for the four compressors
in Butterball. For each given refrigeration load $q^{\mathrm{in}}$, we
pictorially describe the optimal order where each compressor is associated
with a unique color and the height determines its placement in the order.

###  II-D Optimal Compressor Sequencing with Load Shifting

Load shifting is common practice in refrigeration systems for reducing
operational costs. Load shifting involves the process of preemptively cooling
a facility, thereby using the product within the facility as a thermal battery
to save on future cooling demands. Accordingly, for this setting we will think
about cooling needs over a given discrete horizon $[0,1,\dots,T]$ where the
refrigeration load at each stage $k$ is given by $q^{\mathrm{in}}(k)$ and
$\mathbf{q^{\mathrm{in}}}=\\{q^{\mathrm{in}}(k)\\}_{0\leq k\leq T}$. Here, we
will assume that there is complete knowledge of the refrigeration load over
the horizon at the initial stage $k=0$. This assumption will allow us to
hypothetically assess the potential opportunities associated with load
shifting for compressor sequencing on realistic refrigeration load profiles as
provided in forthcoming Figure 7, which highlights the refrigeration load over
the month of June, 2023.

The goal of optimal compressor sequencing is to establish a new shifted
thermal load demand trajectory
$\mathbf{q^{\mathrm{sh}}}=\\{q^{\mathrm{sh}}(0),\dots,q^{\mathrm{sh}}(T)\\}$
and dynamic compressor states
$\mathbf{q}_{c}=\\{q_{c}(0),\dots,q_{c}(T)\\}\\}$ that minimize the cumulative
energy expenditure. Here, we require the shifted load demand trajectory
satisfies

$\sum_{k=0}^{\tau}q^{\mathrm{sh}}(k)\geq\sum_{k=0}^{\tau}q^{\mathrm{in}}(k),\
\forall\tau\in[0,T],$

where the provided cooling exceeds the refrigeration load required for any
horizon $[0,\tau]$ with $\tau\in\\{0,\dots,T\\}.$ Accordingly, our new
optimization takes on the following form:

$\displaystyle J^{*}(\mathbf{q^{\mathrm{in}}})$
$\displaystyle=\min_{\mathbf{q^{\mathrm{sh}}},\\{\mathbf{q}_{c}\\}}\ \
\frac{1}{T}\sum_{k=0}^{T}\sum_{c\in\mathcal{C}}P_{c}(q_{c}(k))$ (2)
$\displaystyle\text{s.t. }q_{c}(k)$ $\displaystyle\in Q_{c}\ \text{ for all
}c\in\mathcal{C},k\in[0,T],$
$\displaystyle\sum_{k=0}^{\tau}q^{\mathrm{sh}}(k)$
$\displaystyle\geq\sum_{k=0}^{\tau}q^{\mathrm{in}}(k)\ \text{ for all
}\tau\in[0,T],$ $\displaystyle\sum_{c\in\mathcal{C}}q_{c}(k)$
$\displaystyle\geq q^{\mathrm{sh}}(k)\ \text{ for all }k\in[0,T].$

This optimization has two sets of decision variables: the shifted thermal load
trajectory $\mathbf{q^{\mathrm{sh}}}$ and the dynamic compressor loads
$\\{\mathbf{q}_{c}\\}$. The first constraint ensures that the compressors
loads are viable for every stage $k$. The second constraint dictates that that
the shifted load trajectory $\mathbf{q^{\mathrm{sh}}}$ needs to deliver at
least as much cooling as any nominal thermal profile
$\mathbf{q^{\mathrm{in}}}$ for any horizon $\tau\in[0,T]$. The last constraint
ensures that the total compressor load needs to match the shifted thermal load
$\mathbf{q^{\mathrm{sh}}}$ at each stage $k$. The optimization problem in Eq.
(2) is determined through $\\{\mathbf{q}_{c}\\}$ and
$\mathbf{q^{\mathrm{sh}}}$, whereas the static compressor sequencing problem
in Eq. (1) fixes $q^{\mathrm{sh}}(k)=q^{\mathrm{in}}(k)$ for all $k$. While
this results in a seemingly more complex optimization problem, the following
proposition demonstrates that the optimal solution is actually attained by a
fixed order water-filling algorithm.

######  Proposition 2.

Let $\mathbf{q^{\mathrm{in}}}$ be the dynamic refrigeration load. Furthermore,
consider a set of compressors $\mathcal{C}=\\{c_{1},\dots,c_{m}\\}$ that are
ordered in terms of increasing marginal costs of cooling at full capacity,
i.e., if compressor $i$ comes before compressor $j$ in the order
$\mathcal{O}^{\mathrm{sh}}$, we have that

$\frac{q_{i}^{+}}{P_{i}(q_{i}^{+})}\leq\frac{q_{j}^{+}}{P_{j}(q_{j}^{+})}.$
(3)

Then the following optimization problem yields the same optimal cost in Eq.
(2) when $T\to\infty$:

$\displaystyle J^{*}(\mathbf{q^{\mathrm{in}}})$
$\displaystyle=\min_{\mathbf{q^{\mathrm{sh}}}}\ \
\frac{1}{T}\sum_{k=0}^{T}\sum_{c\in\mathcal{C}}P_{c}(q_{c}(k))$ (4)
$\displaystyle\text{ s.t. }q_{c}(k)$
$\displaystyle=\pi_{c}^{\mathrm{FO}}(q^{\mathrm{sh}}(k))\text{ for all
}c\in\mathcal{C},k\in[0,T],$
$\displaystyle\sum_{k=0}^{\tau}q^{\mathrm{sh}}(k)$
$\displaystyle\geq\sum_{k=0}^{\tau}q^{\mathrm{in}}(k),\ \text{ for all
}\tau\in[0,T]$

where $\pi_{c}^{\mathrm{FO}}(\cdot)$ comes from the water filling algorithm in
Algorithm 1 with the above ordering $\mathcal{O}^{\mathrm{sh}}$.

###### Proof.

Let $\mathbf{\hat{q^{\mathrm{sh}}}},\\{\mathbf{\hat{q}}_{c}\\}$ be the optimal
solution for Eq. (2). As $P_{c}$ is monotonic, we note that
$\sum_{c\in\mathcal{C}}\hat{q}_{c}(k)=\hat{q^{\mathrm{sh}}}(k)$ must be hold
with equality for all $k$ in the optimal solution. Then, for a given
$\hat{q^{\mathrm{sh}}}(k)$, the optimal compressor loads
$\\{\hat{q}_{c}(k)\\}$ can be given through the water filling algorithm in
Algorithm 1 for each $k$ for some order $\mathcal{O}(k)$. This can be shown
with arguments similar to the proof of Proposition 1.

Now we show that $\mathcal{O}(k)=\mathcal{O}^{\mathrm{sh}}$ for all $k$. Let
$i$ and $j$ be compressors such that $i$ comes before $j$ in
$\mathcal{O}^{\mathrm{sh}}$ but $j$ comes before $i$ in $\mathcal{O}(k)$. We
claim that this is only possible a finite number of times. If not, there are
an infinite number of times where $\hat{q}_{i}(k)=0$, but $\hat{q}_{j}(k)>0$.
However, there exist a load shift and time points
$\\{k_{\ell}\\}_{1\leq\ell\leq N}$ time steps in which $q_{i}(k_{\ell})\to
q\leq q_{i}^{+}$ for $1\leq\ell\leq M<N$ and $q_{j}(k_{\ell})\to 0$ for
$M<\ell\leq N$ which produces a more efficient solution, since the marginal
cost of cooling for compressor $i$ is less than compressor $j$. Thus the water
filling algorithm with order $\mathcal{O}^{\mathrm{sh}}$ recovers an optimal
solution to Eq. (2) when $T\to\infty$. ∎

From Proposition 2, we see that a simple fixed order water-filling algorithm
achieves the optimal solution to Eq. (2) given the correct shifted load
$\mathbf{q^{\mathrm{sh}}}$. We can also evaluate the potential cost benefits
between compressor sequencing with and without load shifting. We characterize
the greatest possible difference in cost in the next proposition. For
notational ease, we define the ratios
$R_{\max}=\max_{c}P_{c}(q_{c}^{-})/q_{c}^{-}$ and
$R_{\min}=\min_{c}P_{c}(q_{c}^{+})/q_{c}^{+}$.

######  Proposition 3.

For a given dynamic refrigeration load $\mathbf{q^{\mathrm{in}}}$, let
$J^{\mathrm{cs}}(\mathbf{q^{\mathrm{in}}})$ be the trajectory cost in Eq. (2)
associated with compressor sequencing without load shifting and let
$J^{*}(\mathbf{q^{\mathrm{in}}})$ be the optimal trajectory cost with load
shifting with $T\to\infty$. The fractional difference between the costs is
upper and lower bounded by

$0\leq\frac{J^{\mathrm{cs}}(\mathbf{q^{\mathrm{in}}})-J^{*}(\mathbf{q^{\mathrm{in}}})}{J^{*}(\mathbf{q^{\mathrm{in}}})}\leq\frac{R_{\max}-R_{\min}}{R_{\min}}$
(5)

###### Proof.

We first note that in Eq. (2), we recover the optimization without load
shifting (or Eq. (1)) if we impose the constraint
$\mathbf{q^{\mathrm{sh}}}=\mathbf{q^{\mathrm{in}}}$ directly. Therefore, we
have that $J^{*}(\mathbf{q^{\mathrm{in}}})\leq
J^{\mathrm{cs}}(\mathbf{q^{\mathrm{in}}})$ necessarily, and the ratio is
always non-negative. We first show the upper bound when considering one
compressor; the ratios simplify to $R_{\min}=P_{c}(q_{c}^{-})/q_{c}^{-}$ and
$R_{\max}=P_{c}(q_{c}^{+})/q_{c}^{+}$. Consider two thermal load profiles
$\mathbf{q}^{1}$ and $\mathbf{q}^{2}$ of length $T\gg 0$, where
$q^{1}(k)=q_{c}^{-}$ for $T-\mathrm{round}(D\cdot q_{c}^{+})\leq k<T$ and $0$
elsewhere and $q^{2}(k)=q_{c}^{+}$ for $0\leq k<\mathrm{round}(D\cdot
q_{c}^{-})$ and $0$ elsewhere. We choose $D$ and $T$ large enough that
$\sum_{k}q^{1}(k)$ is approximately equal to $\sum_{k}q^{2}(k)$ and
$\mathrm{round}(D\cdot q_{c}^{-})<T-\mathrm{round}(D\cdot q_{c}^{+})$. By our
definitions, $\mathbf{q}^{2}$ is a load shifted version of $\mathbf{q}^{1}$,
in that it satisfies $\sum^{\tau}_{k=0}q^{2}(k)\geq\sum^{\tau}_{k=0}q^{1}(k)$
for all $\tau\in[0,T]$. Thus without load shifting for $\mathbf{q}^{1}$, the
power usage is approximately $P_{c}(q_{c}^{-})\times D\cdot q_{c}^{+}$ and
with load shifting to $\mathbf{q}^{2}$, the power usage is approximately
$P_{c}(q_{c}^{+})\times D\cdot q_{c}^{-}$. Thus the ratio of power use matches
the upper bound $(R_{\max}-R_{\min})/R_{\min}$. It can be easily verified that
this example attains the worst case ratio via convexity of $P_{c}$. Extension
to multiple compressors follows an analogous argument. A similar construction
can be assembled, where in $\mathbf{q}^{2}$, the most efficient compressor
satisfies the required load at full capacity and in $\mathbf{q}^{1}$, the
least efficient compressor satisfies the required load at the minimum
capacity. ∎

Proposition 3 provides a possible range of energy savings when utilizing load
shifting. By the worst-case constructions of $\mathbf{q^{\mathrm{in}}}$, we
see that if the profile $\mathbf{q^{\mathrm{in}}}$ fluctuates significantly,
implementing load shifting can generate the most energy savings; however, if
$q^{\mathrm{in}}$ is relatively constant, the energy savings may be less. For
the compressors operating in Butterball, the fractional difference is between
$0\leq\frac{J^{\mathrm{cs}}(\mathbf{q^{\mathrm{in}}})-J^{*}(\mathbf{q^{\mathrm{in}}})}{J^{*}(\mathbf{q^{\mathrm{in}}})}\leq
8.85$, suggesting significant energy savings through load shifting. We
validate this in the next section.

Figure 7: Over a month, we display the estimated refrigeration load serviced
by the compressors, where each data point is associated with every minute in
June 2023. In this figure, we also apply an average filter of 20 min. to
smooth out the data. Data points for refrigeration load were calculated
through estimates of compressor power and COP through sensor readings for each
time point.

##  III The Butterball Facility: A Case Study on Opportunities in Compressor
Sequencing

In this section, we evaluate the potential energy savings possible through
compressor sequencing and load shifting on a case study of the Butterball
facility. Here, we use the predicted refrigeration load profile from the
Butterball facility during the month of June 2023, depicted in Figure 7, to
serve as a prototypical example for our case study. This profile was estimated
using direct measurements of power usage of the compressors, as well derived
coefficient of performance or COP (sometimes CP or CoP) of the refrigeration
system, which is defined as the ratio of useful cooling to work done by the
compressors. We observe that the thermal profile is relatively cyclic, where
the peaks correspond to working hours during the work week (when new product
requiring cooling typically enters during standard operating hours), and the
lower plateaus represent weekends (representing off hours of the facility).

In the Butterball facility, all compressors in operation are screw
compressors, which use a screw thread to trap and compress a volume of gas.
For screw compressors, the main control parameter to modulate the capacity is
a continuous slide valve control.333It is also sometimes possible to modulate
the capacity of a screw compressor through speed control with a variable speed
drive. While this paper does not explicitly consider this method, our analysis
extends to this scenario as well. Increasing the slide valve will expose more
of the screw thread, increasing the volume of gas to be compressed and
increases the capacity of the compressor. As highlighted previously,
compressors operate at their highest efficiency when running at full capacity.
The slide valve being at the minimum position corresponds to the minimum heat
capacity $q_{c}^{-}$ (and similarly for $q_{c}^{+}$) for compressor $c$.

The main energy savings studied in this paper is summarized in Table II.
First, we see that there are significant energy savings if good fixed orders
are used. When using only optimal compressor sequencing, the possible energy
savings are up to only $5\%$ as compared to the best fixed order. However,
when utilizing compressor sequencing with load shifting, we see energy savings
for up to $20\%$. We also compare the performance of a simple, online version
of a compressor sequencing algorithm with load shifting, given in Algorithm 2.
In this online algorithm, the compressors service either the mean
refrigeration or the required refrigeration load at full capacity. We see that
this simple implementation achieves similar energy savings to the optimal
compressor sequencing with load shifting. Thus we validate load shifting as a
viable mechanism for garnering energy savings with regards to compressor
sequencing. In future work, we will extend these results to construct
realizable control algorithms to offer actualized energy savings.

Algorithm 2 Online Load-Shifting Algorithm

$\mathbf{q^{\mathrm{in}}}$,
$(k,q^{\mathrm{k}},q^{\mathrm{tot}})\leftarrow(0,0,0)$,
$\mathbf{q}_{c}\leftarrow\mathbf{0}$ for all $c\in\mathcal{C}$

for $k\leq T$ do

$q^{\mathrm{k}}\leftarrow q^{\mathrm{k}}+q^{\mathrm{in}}(k)$

$q^{\mathrm{tot}}\leftarrow\max\\{q^{\mathrm{k}},\mathrm{mean}(\mathbf{q^{\mathrm{in}}})\\}$

for $c$ in $\mathcal{O}^{\mathrm{sh}}$ do

if $q^{\mathrm{tot}}>0$ then

$q_{c}\leftarrow q_{c}^{+}$

$q^{\mathrm{tot}}\leftarrow q^{\mathrm{tot}}-q_{c}^{+}$

$q^{\mathrm{k}}\leftarrow q^{\mathrm{k}}-q_{c}^{+}$

end if

end for

$k\leftarrow k+1$

end for

Methodology | Average Power
---|---
Worst Fixed Order |  $856.7$ kW
Best Fixed Order |  $562.3$ kW
Compressor Sequencing (C.S) |  $551.0$ kW
Online C.S with Load Shifting |  $444.3$ kW
C.S with Load Shifting |  $443.5$ kW
TABLE II: Cost of Algorithms

##  IV Conclusion

Improving the energy efficiency of industrial refrigeration systems has the
potential to have significant impacts on energy usage. This work focuses on
optimizing the efficiency of the compressors of the refrigeration cycle, as
the compressors comprise the largest energy demand of the system.
Specifically, we focus on the problem of compressor sequencing, where the
operating conditions of the compressors are optimized given a thermal load
requirement. We first study the compressor sequencing problem and characterize
the structure of the optimal solution as a variant of a water-filling
algorithm. However, deriving the optimal solution is computationally hard in
general. Thus we utilize the mechanism of load shifting, where we utilize pre-
cooling, to derive more efficient compressor sequencing algorithms. We observe
that load shifting promotes straight-forward compressor sequencing algorithms
and allows for additional energy savings. We validate our results on an
operational Butterball facility. Through sensor data, we see that load
shifting can potentially reduce energy use to up to $20\%$. In the future, we
will also explore possible extensions, such as incorporating more realistic
heat flow models, tackling load shifting through the lens of inventory
control, exploiting dynamic energy prices to inform precooling, and derive
stable implementations to be employed on facilities.

## References

  * [1] Z. Sun et al., “Comprehensive performance analysis of cascade refrigeration system with two-stage compression for industrial refrigeration,” Case Studies in Thermal Engineering, vol. 39, p. 102400, 2022.
  * [2] F. Fabrega, J. Rossi, and J. d’Angelo, “Exergetic analysis of the refrigeration system in ethylene and propylene production process,” Energy, vol. 35, no. 3, pp. 1224–1231, 2010.
  * [3] W. F. Stoecker, Industrial refrigeration handbook.  McGraw-Hill Education, 1998.
  * [4] “Manufacturing energy consumption survey,” tech. rep., US Energy Information Administration, 2018.
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  * [8] Y. Yao and D. K. Shekhar, “State of the art review on model predictive control (mpc) in heating ventilation and air-conditioning (hvac) field,” Building and Environment, vol. 200, p. 107952, 2021.
  * [9] R. Pattison, C. Touretzky, T. Johansson, I. Harjunkoski, and M. Baldea, “Optimal process operations in fast-changing electricity markets: framework for scheduling with low-order dynamic models and an air separation application,” Industrial & Engineering Chemistry Research, vol. 55, pp. 4562–4584, 2016.
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  * [11] A. Vishwanath, V. Chandan, and K. Saurav, “An iot-based data driven precooling solution for electricity cost savings in commercial buildings,” IEEE Internet of Things Journal, vol. 6, no. 5, pp. 7337–7347, 2019.
  * [12] L. S. Larsen, C. Thybo, J. Stoustrup, and H. Rasmussen, “Control methods utilizing energy optimizing schemes in refrigeration systems,” in 2003 European Control Conference (ECC), pp. 1973–1977, IEEE, 2003.
  * [13] K. A. Manske, “Performance optimization of industrial refrigeration systems,” 2000.
  * [14] L. F. S. Larsen, Model based control of refrigeration systems.  Department of Control Engineering, Aalborg University, 2006.
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  * [16] K. Manske, D. Reindl, and S. Klein, “Evaporative condenser control in industrial refrigeration systems,” International journal of refrigeration, vol. 24, no. 7, pp. 676–691, 2001.


REDISCUSSION OF ECLIPSING BINARIES. PAPER XIII.

THE F-TYPE TWIN SYSTEM IT CASSIOPEIAE

By John Southworth

Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK

IT Cas is a detached eclipsing binary system containing two F3 V stars in an
orbit of period 3.90 d and eccentricity 0.089. Light curves are available from
three sectors of observations from the Transiting Exoplanet Survey Satellite
(TESS), and extensive radial velocity measurements have been published by Lacy
et al. 1. We model these data using the jktebop code to determine the physical
properties of the system. We find masses of $1.324\pm 0.009$ and $1.322\pm
0.008$$~{}{\rm M}_{\odot}$, and radii of $1.555\pm 0.004$ and $1.551\pm
0.005$$~{}{\rm R}_{\odot}$. The two stars are identical to within the
uncertainties, and the depths of the primary and secondary eclipses are also
indistinguishable. Using the effective temperature of $6740\pm 105$ K from
Lacy et al. (for both stars) gives a distance to the system of $505.5\pm 8.3$
pc, in good agreement with the value of $515.0\pm 4.4$ pc from the Gaia DR3
parallax. The properties of the stars are consistent with theoretical
predictions for a solar chemical composition and an age of 2 Gyr. No
pulsations are apparent in the TESS photometry.

## Introduction

We are currently pursuing a project to determine precise and accurate masses
and radii for detached eclipsing binaries (dEBs) for which high-quality
spectroscopic orbits and new light curves from the Transiting Exoplanet Survey
Satellite (TESS) are available 2. The immediate aims of the project are to use
the new TESS data to improve the measurements of the properties of dEBs that
are already in the Detached Eclipsing Binary Catalogue
(DEBCat***https://www.astro.keele.ac.uk/jkt/debcat/, ref. 3), and to add new
dEBs to this catalogue. The longer-term aims are to use dEBs to improve our
understanding of stellar physics and to help calibrate theoretical
evolutionary models 4; 5.

The availability of high-quality light curves from space missions such as
CoRoT 6, Kepler 7 and TESS 8 has revolutionised many areas of stellar physics
9; 10. Their effect has been keenly felt for binary stars 11; 12; 13, with
light curves of a quality unachievable from the ground now available for
thousands of dEBs 14; 15; 16 of which many have an extensive observational
history.

One of these number is IT Cassiopeiae (Table I), an F-type dEB which has been
observed using TESS and for which high-quality radial velocity (RV)
measurements have been published. IT Cas was discovered by Fadeeva using
photographic plates from Moscow 17; 18. Photometric analyses have been carried
out by several authors since 18; 19; 20; 21; 22; 23. The system has a small
orbital eccentricity and exhibits apsidal motion 18; 23; 24; 25 with a period
of $U=877\pm 78$ yr 25.

Lacy 26 obtained high-resolution spectra which showed the system to be double-
lined, with both components having narrow spectral lines indicative of low
rotational velocity. Lacy et al. 1 (hereafter L97) subsequently presented
extensive new spectroscopy and photometry from which they determined the
physical properties of the system; these include the only mass and radius
measurements published so far. The available light curves had poor coverage of
the first and last contact points of the eclipses. In this work we use a
space-based light curve and the radial velocities (RVs) of L97 to obtain
improved measurements of the physical properties of IT Cas.

Khaliullin & Kozyreva 18 detected periodic variability in their light curve of
a secondary eclipse, and deduced that the primary component was a $\delta$
Scuti pulsator. Holmgren & Wolf 23 detected periodic variability in their
light curve of a _primary_ eclipse, and deduced that the _secondary_ component
was a $\delta$ Scuti pulsator. Lacy et al. 1 did not confirm either variation
and suggested that they were erroneous. Our own analysis below finds no
evidence for short-period variability. Kozyreva et al. 24 found slow
variations in the brightness of IT Cas with a timescale of about one month.
They found them to occur independent of the choice of comparison star so
attributed them to the dEB.

Table I: Basic information on IT Cas.  Property | Value | Reference
---|---|---
Right ascension (J2000) | 23:42:01.38 | 27
Declination (J2000) | +51:44:36.8 | 27
Tycho designation | TYC 3650-959-1 | 28
Gaia DR3 designation | 1944868020357285504 | 27
Gaia DR3 parallax |  $1.9419\pm 0.0165$ mas | 27
TESS Input Catalog designation | TIC 26801525 | 29
$U$ magnitude | $11.631\pm 0.020$ | 30
$B$ magnitude | $11.640\pm 0.013$ | 30
$V$ magnitude | $11.152\pm 0.010$ | 30
$J$ magnitude | $10.212\pm 0.020$ | 31
$H$ magnitude | $9.957\pm 0.021$ | 31
$K_{s}$ magnitude | $9.915\pm 0.016$ | 31
Spectral type | F3 V + F3 V | This work

## Observational material

FIG. 1:  TESS short-cadence SAP photometry of IT Cas from sectors 17 (top), 24
(middle) and 57 (bottom). The flux measurements have been converted to
magnitude units then rectified to zero magnitude by the subtraction of
quadratic functions.

The TESS mission8 observed IT Cas in sectors 17 (2019/10/07 to 2019/11/02), 24
(2020/04/16 to 2020/05/13) and 57 (2022/09/30 to 2022/10/29). All three
sectors were observed in short cadence mode with a 120 s sampling rate. The
simple aperture photometry (SAP) and pre-search data conditioning SAP (PDCSAP)
data 32 are almost indistinguishable, so we used the SAP data in our analysis
for consistency with previous papers in this series. The eclipses in the PDC
light curves are approximately 0.05 mag deeper than in the SAP light curves,
indicating that the PDC data have been corrected for contaminating light. We
prefer to model the SAP data and fit for third light, on the grounds that it
is better to adjust the model to match the data than adjust the data to match
the model.

The reduced data were downloaded from the MAST archive†††Mikulski Archive for
Space Telescopes,
https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html and converted
from flux units to relative magnitude. We required a QUALITY flag of zero,
which yielded 12 942 of the 18 012 datapoints from sector 17, 16 309 of 19 074
from sector 24, and 17 990 of 20 712 from sector 57. We further trimmed the
data from sector 24 to remove parts of the light curve associated with
incompletely-observed eclipses, leaving 14 459 datapoints for further analysis
(Fig. 1). We did not use the errors provided with the datapoints as they are
too small. We preferred instead to determine the precision of the photometry
from the scatter around the best-fitting model.

FIG. 2:  Best fit to the TESS sector 17 light curve of IT Cas using jktebop
for the primary (left) and secondary (right) eclipses. The residuals are shown
on an enlarged scale in the lower panels.

We queried the Gaia DR3 database‡‡‡https://vizier.cds.unistra.fr/viz-
bin/VizieR-3?-source=I/355/gaiadr3 in the region of IT Cas. A total of 146
additional sources are listed within 2 arcmin – the constellation of
Cassiopeiae is close to the galactic plane so has a relatively high surface
density of point sources. The brightest of these is fainter than IT Cas by
4.31 mag in the $G_{\rm RP}$ passband (a light ratio of 1.9%). This implies
that there is a small but significant amount of contaminating light in the
TESS data which must be accounted for in the light curve analysis.

## Light curve analysis

Inspection of the SAP light curves showed that the eclipse depths vary between
the three TESS sectors. The primary eclipses are approximately 0.685 mag deep
in sectors 17 and 57, and 0.705 mag deep in sector 24. The secondary eclipses
show exactly the same behaviour. Apsidal motion is unlikely to cause this
because the apsidal period is too long and the eclipse depths do not vary in
antiphase. The probable explanation is that the different spacecraft
orientations in the three sectors, combined with the relatively large 21′′
pixel size, means that the amount of contaminating light changes. The three
light curves should therefore be fitted separately.

FIG. 3:  Same as Fig. 2 but for the TESS data from sector 24.

The light curves from the three sectors were each modelled using version 43 of
the jktebop§§§http://www.astro.keele.ac.uk/jkt/codes/jktebop.html code 33; 34.
The parameters of the fit included the fractional radii of the stars,
expressed as their sum ($r_{\rm A}+r_{\rm B}$) and ratio ($k={r_{\rm
B}}/{r_{\rm A}}$), the orbital inclination ($i$), the central surface
brightness ratio ($J$), the amount of contaminating light ($L_{3}$) and the
coefficients of the reflection effect. The orbital eccentricity ($e$) and
argument of periastron ($\omega$) were included using the Poincaré elements
($e\cos\omega$ and $e\sin\omega$). The secondary eclipse was found to occur at
orbital phase 0.552.

Following the results from Southworth 35 we included limb darkening using the
power-2 law 36 and the re-parameterisation into $h_{1}$ and $h_{2}$ given by
Maxted 37. The two stars are almost identical so we forced them to have the
same limb darkening coefficients, and fitted for both coefficients.

IT Cas exhibits slow apsidal motion, but a full analysis is beyond the scope
of the current work. We therefore determined an orbital ephemeris separately
for each TESS sector and did not interpret them further. The primary and
secondary eclipses are of practically indistinguishable depth and we cannot
confidently decide which is which. This contrasts with a similar situation we
found for ZZ Boo 38 where the TESS data definitively determined – for the
first time – which of the two types of eclipses was deeper and thus by
definition the primary. In the case of IT Cas the eclipse depths are even more
similar and also will change over the apsidal period as $\omega$ cycles round.
We therefore adopted the same convention for eclipse identifications as L97.
We refer to the star eclipsed at phase 0.0 as star A and to its companion as
star B.

FIG. 4:  Same as Fig. 2 but for the TESS data from sector 57.

The best fits to the eclipse light curves are shown in Figs. 2, 3 and 4. Their
parameters are given in Table II. Uncertainties in the parameters were
determined using both Monte Carlo and residual-permutation simulations as
implemented in jktebop 39; 40, the two alternatives being in close agreement
for all parameters. The consistency between sectors is high, with the values
for all but two parameters being in good agreement. Given this and the very
small errorbars, we adopted the straight mean of the values for each
parameter. We did the same for the errorbars, foregoing the division by
$\sqrt{3}$ to convert to standard error. The values of $L_{3}$ are not in good
agreement, as was expected given the change in eclipse depths between sectors.
The disagreement in $e\cos\omega$ is stronger (reduced $\chi^{2}=49$) and
remains unexplained: we have multiplied the final errorbar in this quantity by
a factor of seven to account for the discrepancy.

Table II:  Adopted parameters of IT Cas measured from the TESS light curves using the jktebop code. The uncertainties are 1$\sigma$ and were determined using Monte Carlo and residual-permutation simulations. Parameter | Sector 17 | Sector 24 | Sector 57 | Adopted
---|---|---|---|---
Fitted parameters: |  |  |  |
Time of primary eclipse (BJDTDB) | $58778.308750\pm 0.000013$ | $58973.141188\pm 0.000017$ | $59869.370277\pm 0.000015$ |
Orbital period (d) | $3.896644\pm 0.000005$ | $3.896638\pm 0.000008$ | $3.896646\pm 0.000005$ |
Orbital inclination (∘) | $89.714\pm 0.015$ | $89.679\pm 0.022$ | $89.727\pm 0.017$ | $89.707\pm 0.018$
Sum of the fractional radii | $0.21568\pm 0.00014$ | $0.21535\pm 0.00023$ | $0.21558\pm 0.00024$ | $0.21554\pm 0.00020$
Ratio of the radii | $0.9986\pm 0.0021$ | $0.9944\pm 0.0035$ | $0.9942\pm 0.0050$ | $0.9957\pm 0.0035$
$e\cos\omega$ | $0.080838\pm 0.000007$ | $0.080877\pm 0.000010$ | $0.081059\pm 0.000033$ | $0.08092\pm 0.00012$
$e\sin\omega$ | $-0.03644\pm 0.00032$ | $-0.03662\pm 0.00039$ | $-0.03624\pm 0.00099$ | $-0.03643\pm 0.00057$
Central surface brightness ratio | $0.99963\pm 0.00054$ | $0.99995\pm 0.00083$ | $1.00112\pm 0.00086$ | $1.00023\pm 0.00074$
Third light | $0.0481\pm 0.0030$ | $0.0222\pm 0.0044$ | $0.0536\pm 0.0096$ |
LD coefficient $c$ | $0.602\pm 0.075$ | $0.540\pm 0.038$ | $0.595\pm 0.020$ | $0.579\pm 0.044$
LD coefficient $\alpha$ | 0.50 (fixed) | 0.50 (fixed) | 0.50 (fixed) |
Derived parameters: |  |  |  |
Fractional radius of star A | $0.10791\pm 0.00013$ | $0.10798\pm 0.00020$ | $0.10782\pm 0.00030$ | $0.10790\pm 0.00021$
Fractional radius of star B | $0.10777\pm 0.00014$ | $0.10737\pm 0.00024$ | $0.10776\pm 0.00029$ | $0.10763\pm 0.00022$
Eccentricity | $0.08867\pm 0.00013$ | $0.08878\pm 0.00015$ | $0.08879\pm 0.00040$ | $0.08875\pm 0.00023$
Argument of periastron (∘) | $335.74\pm 0.19$ | $335.64\pm 0.23$ | $335.91\pm 0.60$ | $335.76\pm 0.34$
Light ratio $\ell_{\rm B}/\ell_{\rm A}$ | $0.9969\pm 0.0038$ | $0.9887\pm 0.0062$ | $1.0000\pm 0.0085$ | $0.9952\pm 0.0062$

## Radial velocities

FIG. 5:  RVs of IT Cas from L97 (filled circles for star A and open circles
for star B) compared to the best-fitting spectroscopic orbits from our own
analysis using jktebop (solid curves). The residuals are given in the lower
panels separately for the two components.

We are aware of only one published spectroscopic study of IT Cas: that of L97.
We copied the RVs from that work and fitted them ourselves to confirm the
results. Errorbars were assigned to the two different sources of RVs (“CHSL”
and “CfA”) according to the standard errors given in table 10 of L97, and were
subsequently adjusted by a small amount to force $\chi_{\nu}^{\,2}=1.0$. A
solution to the RVs was obtained with jktebop, fitting for the velocity
amplitudes ($K_{\rm A}$ and $K_{\rm B}$) and systemic velocities
($V_{\rm\gamma,A}$ and $V_{\rm\gamma,B}$) of the two stars, and the reference
time of mid-eclipse. The quantities $e\cos\omega$ and $e\sin\omega$ were fixed
to the values given in Table II – an alternative solution with these
parameters fitted returned values of $K_{\rm A}$ and $K_{\rm B}$ larger by an
insignificant 0.05 km s-1.

Uncertainties were determined using Monte Carlo simulations 41 and found to be
slightly larger than the formal errors from the covariance matrix. Our results
are: $K_{\rm A}=93.85\pm 0.24$ km s-1, $K_{\rm B}=94.03\pm 0.31$ km s-1,
$V_{\rm\gamma,A}=-38.16\pm 0.18$ km s-1 and $V_{\rm\gamma,B}=-37.94\pm 0.24$
km s-1. These agree very well with the value from L97, the main difference in
the two analyses being our use of newer and more precise $e\cos\omega$ and
$e\sin\omega$ values. A plot of the spectroscopic orbits is given in Fig. 5.

## Search for pulsations

Previous studies of IT Cas have claimed the detection of $\delta$ Scuti
pulsations in either star A 18 or star B 23, but their presence was not
confirmed (L97). Pulsations in dEBs are widespread 42; 43; 44 and are
important laboratories for stellar physics 45; 46; 47. The components of IT
Cas have $T_{\rm eff}$s and masses within the lower half of the $\delta$ Scuti
instability strip 48; 49. We therefore searched for pulsations in the TESS
light curve of this system.

This was done by performing a frequency analysis on the residuals of the best
fit to the full light curves from the three TESS sectors individually,
covering frequencies from 0 to 100 d-1. The sectors were not combined as this
would have led to strong aliasing effects. We find no evidence for pulsations
within the frequency range considered, to a 3$\sigma$ upper limit of 0.10
mmag. Although $\delta$ Scuti stars do show variations in pulsation amplitude,
the likely explanation is that the features seen in the older light curves of
IT Cas are red noise rather than of astrophysical origin.

Kozyreva et al. 24 found slow variations in the brightness of IT Cas on a
monthly timescale. The TESS data are not well suited to the detection of
periodicity on such long timescales, so we did not investigate this
possibility further.

## Physical properties of IT Cas

Table III: Physical properties of IT Cas defined using the nominal solar units given by IAU 2015 Resolution B3 (ref. 50).  Parameter | Star A | Star B
---|---|---
Mass ratio $M_{\rm B}/M_{\rm A}$ | $0.9981\pm 0.0042$
Semimajor axis of relative orbit ($\mathcal{R}^{\rm N}_{\odot}$) | $14.414\pm 0.030$
Mass ($\mathcal{M}^{\rm N}_{\odot}$) | 1.3244 $\pm$ | 0.0094 | 1.3218 $\pm$ | 0.0080
Radius ($\mathcal{R}^{\rm N}_{\odot}$) | 1.5552 $\pm$ | 0.0044 | 1.5513 $\pm$ | 0.0045
Surface gravity ($\log$[cgs]) | 4.1765 $\pm$ | 0.0022 | 4.1778 $\pm$ | 0.0021
Density ($~{}\rho_{\odot}$) | 0.3521 $\pm$ | 0.0022 | 0.3540 $\pm$ | 0.0023
Synchronous rotational velocity ( km s-1) | 20.19 $\pm$ | 0.06 | 20.14 $\pm$ | 0.06
Effective temperature (K) | 6740 $\pm$ | 105 | 6740 $\pm$ | 105
Luminosity $\log(L/\hbox{$\mathcal{L}^{\rm N}_{\odot}$})$ | 0.653 $\pm$ | 0.027 | 0.651 $\pm$ | 0.027
$M_{\rm bol}$ (mag) | 3.108 $\pm$ | 0.068 | 3.113 $\pm$ | 0.068
Distance (pc) | $505.5\pm 8.3$

The physical properties of IT Cas were determined using the jktabsdim code 51,
the measured values of $r_{\rm A}$, $r_{\rm B}$, $i$, $P$ and $e$ from Table
II, and the $K_{\rm A}$ and $K_{\rm B}$ from above. The results are given in
Table III, where the errorbars have been propagated from all input parameters
using a perturbation approach. The uncertainties on the radii of the stars are
0.3%, which is slightly greater than the lower limit of 0.2% to which results
are expected to be reliable 52. The largest source of uncertainty for both the
masses _and radii_ of the stars is the uncertainty in the velocity amplitudes,
thanks to the high quality of the TESS light curve. The two stars are
identical to within the errorbars, with differences in mass of $0.3\pm 1.2$%
and in radius of $0.5\pm 0.6$%. The agreement between our results and those of
L97 is good but slightly less than expected given the errorbars.

L97 gave a $T_{\rm eff}$ value for both stars of $6740\pm 105$ K, somewhat
larger than the values of $6579\pm 135$ K in the TESS Input Catalog (TICv8 29)
and $6330\pm 8$ K in Gaia DR3. To check this we determined the distance to the
system using the jktabsdim code, the $UBV$ magnitudes from Lacy 30, the
$JHK_{s}$ magnitudes from 2MASS 31 converted to the Johnson system using the
transformations from Carpenter 53, an interstellar reddening of
$E(B-V)=0.083\pm 0.056$ mag from the stilism¶¶¶https://stilism.obspm.fr online
tool 54; 55, and the surface brightness versus $T_{\rm eff}$ relations from
Kervella et al. 56. This yielded $505.5\pm 8.3$ pc, which compares well with
the $515.0\pm 4.4$ pc from simple inversion of the parallax of the system from
Gaia DR3. The lower $T_{\rm eff}$ from TICv8 is ruled out to 2$\sigma$, and
the Gaia DR3 $T_{\rm eff}$ to higher confidence. Based on this, we accept the
$T_{\rm eff}$ from L97 as suitable for both stars.

The adopted $T_{\rm eff}$ value corresponds to a spectral class of F3 on the
scale of Pecaut & Mamajek 57. We therefore infer a spectral type of the system
of F3 V + F3 V. This is somewhat earlier than the F5 mentioned by L97 and the
F6 given by Simbad∥∥∥http://simbad.u-strasbg.fr/simbad/sim-
id?Ident=it+cas&submit=submit+id (without reference). This spectral type was
arrived at from the $T_{\rm eff}$ and evolutionary stage of the stars, and is
not a true spectral classification.

## Summary and conclusions

IT Cas is a dEB containing two F3 V stars on a 3.90 d orbit with a small
orbital eccentricity. The two stars are identical in mass, radius and $T_{\rm
eff}$ to within the uncertainties. TESS observed the system during three
sectors covering approximately 3 yr, giving light curves of very high quality.
We have modelled these data using the jktebop code to determine the
photometric properties of the system. We also analysed the RVs of IT Cas
published by L97, finding results in good agreement with that work. From the
measured parameters we have calculated the physical properties of the system
(Table II) to precisions of 0.6% in mass and 0.3% in radius. The $T_{\rm
eff}$s found by L97 yield a distance measurement in full agreement with the
parallax from Gaia DR3. We searched for and found no evidence for pulsations
in the light curve.

As a sanity check we have compared the masses, radii and $T_{\rm eff}$s of the
stars to predictions from the parsec stellar evolutionary models 58. A
fractional metal abundance by mass of $Z=0.014$ and an age of $2.0\pm 0.1$ Gyr
provide a good match to the measured properties. IT Cas is now a well-
understood dEB, but would benefit from high-resolution spectroscopy for the
measurement of its photospheric chemical composition and more precise $T_{\rm
eff}$ values. The system shows apsidal motion, and the measured apsidal period
would be a useful addition to a detailed comparison between the properties of
IT Cas and theoretical predictions.

## Acknowledgements

We are grateful to Steve Overall and an anonymous referee for useful comments
on a draft of this work. This paper includes data collected by the TESS
mission and obtained from the MAST data archive at the Space Telescope Science
Institute (STScI). Funding for the TESS mission is provided by the NASA’s
Science Mission Directorate. STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
The following resources were used in the course of this work: the NASA
Astrophysics Data System; the SIMBAD database operated at CDS, Strasbourg,
France; and the ar$\chi$iv scientific paper preprint service operated by
Cornell University.

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