# Entanglement-breaking channels are a quantum memory resource

Lucas B. Vieira<EMAIL_ADDRESS>Institute for Quantum Optics and Quantum
Information (IQOQI), Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Faculty of Physics, University of
Vienna, Boltzmanngasse 5, 1090 Vienna, Austria  Huan-Yu Ku
<EMAIL_ADDRESS>Department of Physics, National Taiwan Normal University,
Taipei 11677, Taiwan  Costantino Budroni<EMAIL_ADDRESS>Department of Physics “E. Fermi” University of Pisa, Largo B. Pontecorvo 3,
56127 Pisa, Italy  Faculty of Physics, University of Vienna, Boltzmanngasse 5,
1090 Vienna, Austria  Institute for Quantum Optics and Quantum Information
(IQOQI), Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria

###### Abstract

Entanglement-breaking channels (equivalently, measure-and-prepare channels)
are an important class of quantum operations noted for their ability to
destroy multipartite _spatial_ quantum correlations. Inspired by this
property, they have also been employed in defining notions of “classical
memory”, under the assumption that such channels effectively act as a
classical resource. We show that, in a single-system multi-time scenario,
entanglement-breaking channels are still a quantum memory resource: a qudit
going through an entanglement-breaking channel cannot be simulated by a
classical system of same dimension. We provide explicit examples of memory-
based output generation tasks where entanglement-breaking channels outperform
classical memories of the same size. Our results imply that entanglement-
breaking channels cannot be generally employed to characterize classical
memory effects in temporal scenarios without additional assumptions.

_Introduction.—_ The notion of classical memory, in the sense of storage of
information, is familiar to us from our everyday life. When we talk about the
number of bits (or, more commonly, one of its multiples: bytes, kilobytes,
etc.) a device has, we refer the logarithm of the number of distinguishable
internal states, where information can be stored and retrieved. Needless to
say, memory is an essential resource for information processing, and a finite
amount of memory imposes fundamental limitations on which tasks can be
achieved. With the advent of quantum information and quantum computation,
distinguishing between classical and quantum memory effects is of utmost
importance for identifying scenarios where quantum systems offer an advantage
over classical ones [1, 2, 3, 4]. An approach towards this goal is by
establishing bounds on information processing tasks achievable with classical
memory alone. A violation of such bounds then provides a certificate of non-
classicality, often also providing a quantifiable metric for the _quantumness_
of the underlying memory resource [5, 6, 7, 8, 9, 10, 11, 12].

Any information processing task consists of manipulating information stored in
a memory over time. This is achieved by sequential operations performed on the
memory, with classical and quantum systems offering vastly different
capabilities on what memory states and their transformations are possible. In
recent years, the formalism of quantum supermaps [13, 14, 15], also known as
process tensors or quantum combs, has enabled a thorough investigation of
temporal quantum protocols, as it is naturally well-suited for the study of
multi-time statistics [7, 16, 17]. Within this framework, many quantum-
inspired notions have been employed as conflicting _definitions_ of “classical
memory” [1, 16, 17, 4, 18]; see Ref. [17] for a detailed characterization.

A typical approach involves entanglement-breaking (EB) channels, which are
known to be exactly equivalent to measure-and-prepare (M&P) operations [19].
As the input quantum state is destroyed by the measurement operation and
replaced by a classical label (albeit encoded in the prepared output quantum
state), one may be inclined to claim that such channels are essentially
equivalent to classical channels [1, 9, 10, 11, 16], i.e., that they can only
establish classical correlations between their inputs and outputs. EB channels
have been used in this manner, e.g., to develop resource theories of quantum
memories [1], the benchmarking and application of quantum devices [9],
characterizing non-classical memory effects in quantum processes [4, 20, 16]
and bosonic Gaussian channels [21, 11], and in establishing a hierarchy
between competing notions of classical memory [17]. A potential issue with
such entanglement-inspired definitions is that they still operate within the
quantum formalism, and therefore, may not truly characterize the limitations
inherent to classical systems. Clearly, a precise distinction between
classical and quantum memory effects must emerge from explicit constructions
within each theory.

Here, we show that the previous interpretation and applications of EB channels
can be misleading in the case of temporal scenarios involving sequential
measurements. We address the following question: is the memory of a “$d$-it”
(a classical system with $d$ states) equivalent to that of a _qudit_
($d$-level quantum system) passing through an EB channel? We answer it in the
negative by presenting explicit examples where a qudit passing through an EB
channel outperforms a $d$-it of classical memory in the task of generating
correlations in time. We conclude that EB channels cannot be used to define or
characterize classical memories in the most general settings.

Figure 1: Temporal correlations can be investigated by a classical or quantum
system with $d$ distinguishable internal states, in the initial state $s_{0}$,
which is repeatedly measured by the same instrument at multiple times. The
measurement of state $s_{t}$ produces the outcome $a_{t+1}$ and changes the
state to $s_{t+1}$.

_Preliminary notions.—_ A channel $\mathcal{E}_{A}$ is said to be entanglement
breaking if $(\mathcal{E}_{A}\otimes\text{id}_{B})(\rho_{AB})$ is separable
for any bipartite state $\rho_{AB}$. A well-known result establishes that any
EB channel $\mathcal{E}$ can be written as a measure-and-prepare (M&P) channel
in the _Holevo form_ [22, 19]

$\mathcal{E}(\rho)=\sum_{i=0}^{m}\operatorname{tr}\left[\rho
E_{i}\right]\sigma_{i},$ (1)

for $\\{E_{i}\\}_{i=1}^{m}$ a Positive Operator-Valued Measure (POVM), i.e.
$E_{i}\geq 0$ and $\sum_{i=1}^{m}E_{i}=\mathbbm{1}$, and $\sigma_{i}$ quantum
states. Here, $m$ can be interpreted as the number of labels—in principle,
arbitrarily large—required in the channel’s intermediate classical memory in
order to describe its operation [19, 1]. EB channels can be fully
characterized [19] by the fact their Choi matrix [23, 24],
$C_{\mathcal{E}}:=(\mathcal{E}\otimes\text{id})(\left|\Phi\right\rangle\left\langle\Phi\right|)$
with $\left|\Phi\right\rangle$ a maximally entangled state, is separable and
with rank $r\geq d$, which also implies any such channel can be rewritten with
at most $m=d^{2}$ operators $E_{i}$ and $\sigma_{i}$. These channels can be
understood in terms of a quantum-classical-quantum operation: (1) The input
quantum state $\rho$ is first measured, producing a classical outcome label
$i$, out of a total $m$, with probability $\operatorname{tr}\left[\rho
E_{i}\right]$, (2) Conditioned on $i$, the corresponding quantum state
$\sigma_{i}$ is passed forward as output; see Fig. 2b.

To properly distinguish between classical and quantum memories, we investigate
their capabilities when generating temporal correlations in the simplest
scenario, i.e., that of a single, finite-dimensional system subject to the
same dichotomous measurement at multiple times, as shown in Fig. 1. Such a
system can be described within the formalism of finite-state machines [6, 25].
In the classical case, the most general description of its behavior is by an
initial stochastic vector $\pi$ over its $d$ distinct states (i.e., a $d$-it
of memory), and a pair of (row) sub-stochastic transition matrices
$T=(T_{0},T_{1})$ describing its dynamics upon each outcome $a\in\\{0,1\\}$,
with $T_{0}+T_{1}$ row stochastic. The probability of a length-$L$ sequence of
outcomes $\bm{a}=a_{1}a_{2}\cdots a_{L}$ is then given by

$p(\bm{a}|T,d)=\pi T_{a_{1}}\cdots T_{a_{L}}\eta,$ (2)

with $\eta=(1,\dots,1)^{\text{T}}$. The quantum case is defined analogously,
with a single $d$-dimensional quantum system (a qudit), initially at state
$\rho_{0}$, being repeatedly measured by a quantum instrument
$\mathcal{I}=(\mathcal{I}_{0},\mathcal{I}_{1})$, with effects
$\mathcal{I}_{a}$ being completely positive (CP) trace non-increasing maps
such that $\mathcal{I}_{0}+\mathcal{I}_{1}$ is completely positive trace-
preserving (CPTP). The probability of a sequence $\bm{a}$ is given by

$p(\bm{a}|\mathcal{I},d)=\operatorname{tr}\left[\mathcal{I}_{a_{L}}\circ\cdots\circ\mathcal{I}_{a_{1}}(\rho_{0})\right].$
(3)

In both cases, the system’s state is the sole resource available to generate
the probability distributions, and any dynamics is entirely governed by the
effects associated with an outcome $a_{t}$. In this way, classical and quantum
memories are placed on equal footing as a memory resource. Crucially, memory
is the central resource for this task, as any arbitrary temporal correlation
can be generated by a classical system with enough memory, i.e., enough
internal states [5, 26, 27]. Consequently, the system’s dimension establishes
fundamental limitations on the set of achievable temporal correlations [28, 6,
27, 8, 25], providing a clear notion for distinguishing between _manifestly_
classical and quantum memories. Note that we do not allow for external storage
of information (e.g., by conditioning on past outcomes, or invoking an
external reference clock), so that the total amount of memory can be
quantified by the number of states $d$ of the system.

One possible approach to distinguish between classical and quantum memory
effects is by establishing upper bounds on the maximum probability for
individual sequences to be produced within classical and quantum mechanics [6,
8, 25]. The notion of _deterministic complexity_ of a sequence $\bm{a}$,
denoted by $\operatorname{DC}(\bm{a})$, was introduced in Ref. [25] as the
minimum dimension $d$ for a system (classical or quantum) to be capable of
generating $\bm{a}$ with probability one. Thus, if
$d<\operatorname{DC}(\bm{a})$, then _no_ classical or quantum system can
produce the sequence $\bm{a}$ deterministically, implying both classical and
quantum theory must obey non-trivial upper bounds for this maximum
probability. Since $d=2$ is the smallest non-trivial amount of memory, these
observations also highlight the importance of studying temporal correlations
over three or more time steps, as every sequence of length two can be produced
deterministically with either a bit or qubit.

As the underlying memory resource can be explicitly confined within each
theory, such non-trivial upper bounds provide a sharp demarcation between
classical and quantum memories and their capabilities, but they are generally
difficult to obtain precisely [25, 29, 30]. In Ref. [25], a universal upper
bound of $1/\mathrm{e}$ was conjectured for all sub-deterministic classical
memory scenarios, whereas no non-trivial universal quantum bound seems to
exist.

Figure 2: (a) The sequential measurement protocol considered in this work. An
isolated quantum system in an initial state $\rho_{0}$ is repeatedly passed
through an entanglement-breaking channel $\mathcal{E}$ before being measured
by a quantum instrument $\mathcal{I}$, obtaining a sequence of outcomes
$\bm{a}=a_{1}a_{2}\dots a_{L}$. (b) The entanglement-breaking channel
$\mathcal{E}$ understood as a measure-and-prepare operation on a
$d$-dimensional quantum system, with a $m$-dimensional intermediate classical
space. (c) The $m\times m$ transition matrix of the effective classical model.

_The sequential measurement protocol.—_ We investigate the non-classicality of
entanglement-breaking channels by inserting an EB channel $\mathcal{E}$, as in
Eq. 1, before each quantum measurement in Eq. 3 (depicted in Fig. 2a),
obtaining

$\displaystyle\begin{split}&p(\bm{a}|\mathcal{I},\mathcal{E},d)=\operatorname{tr}\left[\mathcal{I}_{a_{L}}\circ\mathcal{E}\circ\cdots\circ\mathcal{I}_{a_{2}}\circ\mathcal{E}\circ\mathcal{I}_{a_{1}}\circ\mathcal{E}(\rho_{0})\right]\\\
&=\sum_{i_{1}\dots
i_{L}=1}^{m}\operatorname{tr}\left[\rho_{0}E_{i_{1}}\right]\operatorname{tr}\left[\mathcal{I}_{a_{1}}(\sigma_{i_{1}})E_{i_{2}}\right]\cdots\operatorname{tr}\left[\mathcal{I}_{a_{L}}(\sigma_{i_{L}})\mathbbm{1}\right]\\\
&=\sum_{i_{1}\dots
i_{L+1}=1}^{m}\pi_{i_{1}}[T_{a_{1}}]_{i_{1}i_{2}}\cdots[T_{a_{L}}]_{i_{L}i_{L+1}}\eta_{i_{L+1}},\end{split}$
(4)

where we use
$[T_{a}]_{ij}:=\operatorname{tr}\left[\mathcal{I}_{a}(\sigma_{i})E_{j}\right]$,
$\pi_{i}:=\operatorname{tr}\left[\rho_{0}E_{i}\right]$, and
$\eta=(1,\dots,1)^{\text{T}}$ as in Eq. 2. Note that, since the $E_{i}$ are
POVM elements, each trace term denoted a probability. We have thus collected
all such probabilities in the matrices $T_{a}$ such that

$\sum_{a,j}[T_{a}]_{ij}=\operatorname{tr}\left[\textstyle{\sum_{a}}\mathcal{I}_{a}(\sigma_{i})\cdot\textstyle{\sum_{j}}E_{j}\right]=\operatorname{tr}\left[\hat{\sigma}_{i}\mathbbm{1}\right]=1,$
(5)

where we use the fact $E_{i}$ form a POVM and $\sum_{a}\mathcal{I}_{a}$ is
CPTP, thus $\hat{\sigma}_{i}=\sum_{a}\mathcal{I}_{a}(\sigma_{i})$ has unit
trace. Furthermore, we can define an initial probability vector $\pi$ arising
from the first application of the channel, with $\sum_{i}\pi_{i}=1$. The above
results imply we can model the protocol classically, by recognizing $T_{a}$ as
$m\times m$ sub-stochastic transition matrices and $\pi$ as an initial
stochastic vector (Fig. 2c). Eq. 4 can thus be equivalently written as in Eq.
2, with $m$ instead of $d$ states.

We are led to conclude that the repeated inclusion of an arbitrary
entanglement-breaking channel before the quantum instrument allows the
resulting temporal correlations to be, indeed, described classically. This
observation seems to justify the association of EB channels with classical
memories. Nevertheless, as we already discussed, any temporal correlation can
be obtained via a classical system if enough memory is available, such that a
meaningful comparison must be made between systems of _bounded memory_ , i.e.,
bounded dimension.

A central observation is that the effective classical model thus obtained is
over $m$ “virtual” classical states (i.e., the number of outcomes in the
measure-and-prepare description of the channel), not $d$ as in the dimension
of the underlying Hilbert space. This suggests that a qudit passing through an
EB channel may be able to generate stronger correlations, i.e., carry more
memory, than a classical $d$-it. Clearly, the exception is the case $m\leq d$,
where any quantum model can be converted into a classical one of the same
dimension (and vice versa) by choosing $E_{i},\sigma_{i}$, and $\mathcal{I}$
acting diagonally on the same basis; see App. A for the explicit construction.

Figure 3: (Color online). Probabilities of one-tick sequences $\bm{a}^{L}_{\text{ot}}$ for classical one-way models and the quantum construction using ETFs, both using $d=L-1$. One-way models are conjectured to be optimal for all $L=d-1$, and are known to achieve the exact upper bounds $\Omega_{C}(\bm{a}^{L}_{\text{ot}},d)$ for $L=3,4$, and $5$. The bounds for $L=4$ and $5$ are violated by this explicit quantum construction using ETFs, revealing non-classical memory effects of EB channels. L | d |  Classical ($\Omega_{C}$) |  Quantum ($\Omega_{Q}$) | $\Omega_{Q}/\Omega_{C}$
---|---|---|---|---
3 | 2 | $(2/3)^{3}=0.\overline{296}$ | $\overset{?}{=}0.296296$ | $\overset{?}{=}1$
4 | 3 | $(3/4)^{4}=0.31640625$ | $\geq 0.359523$ | $\geq 1.136270$
5 | 4 | $(4/5)^{5}=0.32768$ | $\geq 0.368445$ | $\geq 1.124405$
Table 1: Results from numerical optimization for various one-tick sequences of
length $L$, and Hilbert space dimension $d=L-1$. The quantum models found
numerically for $L=4$ and $5$ outperform the previous ETF construction, and
are included in App. D. No violations were found for $d=2$.

_Quantum advantages with EB channels.—_ Here, we show that EB channels with
$m>d$ are able to generate non-classical memory effects. This can be
investigated by choosing a sequence $\bm{a}$ with
$d<\operatorname{DC}(\bm{a})$, then computing the maximum probability
achievable over all classical models with $d$ states:

$\displaystyle\Omega_{C}(\bm{a},d)$ $\displaystyle=\max_{T}\;p(\bm{a}|T,d).$
(6)

Analogously, using a $d$-dimensional quantum system, we look for instruments
$\mathcal{I}$ and EB channels $\mathcal{E}$ with $m>d$ such that

$\displaystyle p(\bm{a}|\mathcal{I},\mathcal{E},d)>\Omega_{C}(\bm{a},d).$ (7)

Crucially, the maxima $\Omega_{C}(\bm{a},d)$ must be found explicitly within
classical theory, either by exact computation or by obtaining an upper bound,
whereas for the quantum value of $p(\bm{a}|\mathcal{I},\mathcal{E},d)$ an
explicit model $(\mathcal{I},\mathcal{E})$ must be found. If the classical
memory bound has been violated even in the presence of an EB channel
$\mathcal{E}$, as in Eq. 7, then the system still acts as a quantum memory
resource for the task of generating temporal correlations.

To achieve this goal, we consider the _one-tick sequences_ [8, 25] of length
$L$, defined as $\bm{a}_{\text{ot}}^{L}:=0\dots 01$ and having
$\operatorname{DC}(\bm{a}^{L}_{\text{ot}})=L$, under the scenario $d=L-1$. As
the case of $L=2$ is trivial, we consider the smallest non-trivial cases
$L\geq 3$. For the classical model, analytical upper bounds can be computed
via semidefinite programming (SDP) [29]. The case $L=3$ and $4$ were already
solved in [29], and here we computed explicitly the case $L=5$ with the same
method. In all these cases, the upper bounds computed via SDP coincide, up to
the numerical precision, with the values obtained via the _one-way model_ [8],
which is conjectured to be optimal for all $d=L-1$ scenarios [8, 25]. We are
therefore sure to have obtained the exact bounds for the cases $L=3,4,$ and
$5$. An explicit quantum model violating these classical bounds can be
obtained with the following construction. Consider the deterministic classical
model on $m$ states given by $[T_{0}]_{i,i+1}=1$ for $i=1,\dots,m-1$,
$[T_{1}]_{m,1}=1$, zeros for all other entries, and an initial state
$\pi=(1,0,\dots,0)$. The idea is to approximate this model by emulating
$m=d+1$ nearly-orthogonal virtual states with the available $d$-dimensional
quantum system.

With $\zeta=\mathrm{e}^{2\pi\mathrm{i}/(m-1)}$, we define $\mathcal{E}$ using

$\left|\psi_{n}\right\rangle=\frac{1}{\sqrt{d-1}}\sum_{k=1}^{d-1}\zeta^{nk}\left|k\right\rangle,\quad
n=1,\dots,m-1,$ (8)

then let $\sigma_{n}=\left|\psi_{n}\right\rangle\left\langle\psi_{n}\right|$
and $E_{n+1}=\tfrac{d-1}{d}\sigma_{n}$ for $n=1,\dots,m-1$, and finally
setting $\rho_{0}=\sigma_{m}=E_{1}=\left|0\right\rangle\left\langle 0\right|$.
The instrument is defined as $\mathcal{I}_{a}(\rho)=K_{a}\rho K^{\dagger}_{a}$
with $K_{0}=\mathrm{diag}(1,\cdots,1,0)$ and $K_{1}=\mathbbm{1}-K_{0}$. This
quantum model corresponds to isolating the $(d-1)$-dimensional subspace
orthogonal to $\left|0\right\rangle$, assembling a harmonic complex
_equiangular tight frame_ (ETF) [31] of size $m-1$ within this subspace, then
using it for the virtual states in the transitions of $T_{0}$. ETFs are useful
for this task as they provide a generalization of orthonormal bases,
saturating the Welch bounds for minimum overlaps [32]. As seen in Fig. 3, this
quantum ETF construction clearly violates the exact classical bounds for $L=4$
and $5$, conclusively showing that general EB channels cannot be applied to
characterize classical memory effects.

The quantum model presented above provides an explicit violation of the
classical bound. This is, however, not necessarily the maximum one can achieve
in the quantum case. To investigate the quantum maximum with EB channels, we
have also performed numerical optimization via gradient descent methods,
obtaining lower bounds for the maxima $\Omega_{Q}(\bm{a},d)$. Results are
summarized in Table 1. The quantum models found numerically slightly
outperform the previous ETF construction. See App. C for more details
regarding the optimization and App. D for the explicit models found.

_Conclusion and discussion.—_ Entanglement breaking channels are universally
recognized as classical memories [1, 2, 3, 9, 16, 4, 10, 17, 12, 11], and,
indeed they are classical under many points of view. Quite surprisingly,
however, we found that there exist scenarios where they still act as genuine
quantum memories. More precisely, we answer in the negative the following
question: is a qudit passing through an entanglement-breaking channel
equivalent to a classical $d$-it?

To address this question, we have investigated a task where memory is the
central resource, namely, the generation of correlations in time. Indeed, it
has been shown that any temporal correlation can be reproduced by a classical
system if enough memory is available [5, 26, 27], such that differences
between classical and quantum temporal correlations only emerge if memory is
bounded [6, 8, 25]. As classical memory is understood in terms of the number
of internal states $d$ of the system, a fair comparison between classical and
quantum theory must involve a quantum system of the same size. Differences
between quantum and classical memories can be found by investigating the most
general set of memory operations allowed in each theory, then establishing
bounds on their capabilities.

The violations of classical bounds we have found showcase the importance of
investigating classical memory effects in terms of manifestly classical
descriptions, instead of relying on quantum-inspired analogies, as otherwise
unexpected quantum effects may persist. As we have shown, the common
presumption that EB channels act as a classical resource carries additional
assumptions, which could be rather strong; see discussion on App. B.
Furthermore, our results emphasize that, in the context of temporal
correlations, it is essential to look beyond qubits and two-time scenarios, as
otherwise important distinctions between classical and quantum systems are not
apparent, as also recognized in Ref. [17]; see also App. D.

Our findings inspire several future research directions. Understanding the
minimal requirements for an EB channel to truly act as a classical memory
could offer insights into the origins of the quantum advantage in the context
of temporal correlations, allowing a more definitive understanding of
classical and quantum memory effects in non-Markovian processes; see also App.
B. Furthermore, since the quantum advantage seen in this work stems from the
non-orthogonality of the states $\sigma_{i}$, an investigation of the quantum
upper bounds in the presence of EB channels could lead to new insights into
the optimal encoding of classical information within quantum systems for
various sequential tasks, with potential applications to, e.g., quantum
random-access codes [33, 34, 35, 36] or quantum compression of classical
predictive models for stochastic processes [37, 38, 39].

Naturally, there are many distinct features one could wish from a “quantum
memory”, e.g., preserving coherence or entanglement [1, 9, 10], or enabling
non-classical multi-time statistics [7, 17]. If the quantum memory is required
to preserve entanglement, then EB channels are obviously useless as a quantum
resource. However, quantum effects are not restricted to spatial correlations,
and in considering the temporal character inherent to any information
processing task, we are required to also consider the fundamental differences
between classical and quantum temporal correlations. Our results indicate a
more careful and detailed investigation of quantum memory resources is
warranted, such that their precise origin and advantage over classical
memories can be properly understood, quantified, and successfully applied in
emergent quantum technologies.

_Acknowledgements.—_ We thank Simon Milz and Philip Taranto for valuable
discussions. This work is supported by the Austrian Science Fund (FWF) through
projects ZK 3 (Zukunftskolleg) and F7113 (BeyondC). This work is supported by
the Ministry of Science and Technology, Taiwan, (Grants No. MOST
111-2917-I-564-005, 112-2112-M-003-020-MY3), and Higher Education Sprout
Project of National Taiwan Normal University (NTNU) and the Ministry of
Education (MOE) in Taiwan.

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##  Appendix A Direct correspondence between classical and quantum models for
$m\leq d$

Using $\mathcal{A}$ for the set of outcomes, let $T=(T_{a})_{a\in\mathcal{A}}$
and $\\{\left|i\right\rangle\\}_{i=1}^{d}$ denote an arbitrary classical model
and orthonormal basis, respectively. We consider a generic quantum instrument
written in its Kraus decomposition

$\mathcal{I}_{a}(\rho)=\sum_{k=1}^{n_{a}}K_{ak}\rho
K_{ak}^{\dagger},\quad\text{with}\quad\sum_{a\in\mathcal{A}}\sum_{k=1}^{n_{a}}K_{ak}^{\dagger}K_{ak}=\mathbbm{1},$
(9)

such that $\sum_{a}\mathcal{I}_{a}$ is trace preserving. We can define a Kraus
operator for each non-zero $[T_{a}]_{ij}$, via the action
$K_{aij}\left|i\right\rangle=\sqrt{[T_{a}]_{ij}}\left|j\right\rangle$, such
that $\mathcal{I}$ acts diagonally in this basis. Now, by choosing
$E_{i}=\sigma_{i}=\left|i\right\rangle\left\langle i\right|$, we have a
completely dephasing channel $\mathcal{E}_{\text{CD}}$ in the same basis,
$\mathcal{E}_{\text{CD}}(\rho):=\sum_{i=1}^{m}\left\langle
i\right|\rho\left|i\right\rangle\left|i\right\rangle\left\langle i\right|$.
Finally, by using the initial state
$\rho_{0}=\sum_{i}\pi_{i}\left|i\right\rangle\left\langle i\right|$, we
conclude

$\operatorname{tr}\left[\mathcal{I}_{a_{L}}\circ\mathcal{E}\circ\cdots\circ\mathcal{I}_{a_{1}}\circ\mathcal{E}(\rho_{0})\right]=\pi
T_{a_{1}}\cdots T_{a_{L}}\eta.$ (10)

Together with the results from Eqs. 4 and 5, this establishes the
correspondence between classical and quantum scenarios for $m\leq d$, and
motivates the investigation of the $m>d$ case.

##  Appendix B Conditions for classicality

Since generic EB channels can violate bounds for classical memories, it is
desirable to know the conditions under which non-classical memory effects are
actually erased by their application, as often assumed, i.e., such that no
choice of quantum instrument $\mathcal{I}$ can lead to non-classical temporal
correlations.

A full investigation of these conditions is beyond the scope of this paper.
Nevertheless, understanding some sufficient conditions offers insights on the
strength of the implicit assumptions being made when applying EB channels as
means of characterizing classical memory effects. We present two independent
sufficient conditions: either the states $\\{\sigma_{i}\\}_{i=1}^{m}$ commute,
and/or the POVM elements $\\{E_{i}\\}_{i=1}^{m}$ commute.

If $[\sigma_{i},\sigma_{j}]=0$, $\forall i,j$, then a common eigenbasis exists
in which all operators can be written as
$\sigma_{i}=\mathrm{diag}(s^{0}_{i},\dots,s^{d-1}_{i})$. Without loss of
generality, we can express the channel in this basis as

$\begin{split}\mathcal{E}(\rho)=\sum_{i=1}^{m}\operatorname{tr}\left[E_{i}\rho\right]\sigma_{i}=\sum_{i=1}^{m}\sum_{\ell=0}^{d-1}\operatorname{tr}\left[E_{i}\rho\right]s_{i}^{\ell}\left|\ell\right\rangle\left\langle\ell\right|\\\
=\sum_{\ell=0}^{d-1}\operatorname{tr}\left[\left(\sum_{i=1}^{m}s_{i}^{\ell}E_{i}\right)\rho\right]\left|\ell\right\rangle\left\langle\ell\right|=\sum_{\ell=0}^{d-1}\operatorname{tr}\left[F_{\ell}\rho\right]\left|\ell\right\rangle\left\langle\ell\right|,\end{split}$
(11)

where $F_{\ell}:=\sum_{i}s_{i}^{\ell}E_{i}$ and $\\{F_{\ell}\\}_{\ell=1}^{d}$
is a valid POVM. In fact, $s_{i}^{\ell}\geq 0$ since $\sigma_{i}\geq 0$ and
$\sum_{\ell}s_{i}^{\ell}=1$ since
$\operatorname{tr}\left[\sigma_{i}\right]=1$, which implies $F_{\ell}\geq 0$
and
$\sum_{\ell}F_{\ell}=\sum_{i,\ell}s_{i}^{\ell}E_{i}=\sum_{i}E_{i}=\openone$.
We thus have a new definition of the channel $\mathcal{E}$ which uses only $d$
outcomes and states. But from App. A, this implies there exists an equivalent
classical model with $d$ states. The case for $[E_{i},E_{j}]=0$ is analogous.

##  Appendix C Numerical optimization

For $\bm{a}^{L}_{\text{ot}}=0\dots 01$ and $d=L-1$, the best classical bounds
known are achieved by the so-called one-way models [6, 25]

$[T_{0}]_{i,i}=1/L,\quad[T_{0}]_{i,i+1}=[T_{1}]_{d,1}=1-1/L,$ (12)

with $p(\bm{a}^{L}_{\text{ot}}|T,d)=(1-\nicefrac{{1}}{{L}})^{L}$, which are
conjectured to give the optimal probability for a given $d$ in any sub-
deterministic classical scenario (i.e., $d<\operatorname{DC}(\bm{a})$). For
$d=2,3$, and $4$, these upper bounds are known to be exact, and can be
computed with semidefinite programming techniques [29].

In the quantum case, numerical optimization was performed via gradient descent
techniques using the Adam algorithm [40], with an additional exponentially
decreasing learning rate from $0.07$ to $10^{-12}$ over 50 000 iterations, and
many trials with random initial values. We have found sufficient to optimize
for rank-1 $E_{i},\sigma_{i}$, with no further assumptions. Each
$\mathcal{I}_{a}$ was defined with a single Kraus operator, as no advantage
was observed in using more. As the probability in Eq. 4 is convex for
$\rho_{0}$, and we maximize for a single sequence, this initial state can be
assumed pure, and w.l.o.g., set to $\rho_{0}=\left|0\right\rangle\left\langle
0\right|$. We have the following constrained optimization problem:

$\displaystyle\begin{split}\textbf{Find:}&\quad\max_{\mathcal{E},\mathcal{I}}p(\bm{a}|\mathcal{E},\mathcal{I},d)\\\
\textbf{Such that:}&\quad E_{i},\sigma_{i}\geq 0\;\forall i,\\\
&\quad\sum_{i}E_{i}=\mathbbm{1},\;\operatorname{tr}\left[\sigma_{i}\right]=1,\;\forall
i,\\\ &\quad\sum_{a}K_{a}^{\dagger}K_{a}=\mathbbm{1},\end{split}$ (13)

with $\mathcal{E}(\rho)=\sum_{i=1}^{m}\operatorname{tr}\left[\rho
E_{i}\right]\sigma_{i}$ and $\mathcal{I}_{a}(\rho)=K_{a}\rho K_{a}^{\dagger}$.
This can be relaxed into an unconstrained optimization by using arbitrary
$d\times d$ complex matrices $A_{i},B_{i},C_{a}$, then defining
$\sigma_{i}:=A_{i}^{\dagger}A_{i}/\mathrm{tr}[A_{i}^{\dagger}A_{i}]$ for the
states. For the POVM, we first define $\tilde{B}_{i}:=B_{i}^{\dagger}B_{i}$
and compute
$\lambda^{E}_{\text{max}}:=\max\text{eigs}(\sum_{i}\tilde{B}_{i})$, such that
letting $E_{i}:=\tilde{B}_{i}/\lambda^{E}_{\text{max}}$ gives
$\sum_{i}E_{i}\leq\mathbbm{1}$. For the instrument we compute
$\lambda^{\mathcal{I}}_{\text{max}}=\max\text{eigs}(\sum_{a}C_{a}^{\dagger}C_{a})$
and define $K_{a}:=C_{a}/\sqrt{\lambda^{\mathcal{I}}_{\text{max}}}$, giving
$\sum_{a}K_{a}^{\dagger}K_{a}\leq\mathbbm{1}$. In this way, the gradient of
the objective function will naturally favor solutions satisfying
$\sum_{i}E_{i}=\mathbbm{1}$ and $\sum_{a}K_{a}^{\dagger}K_{a}=\mathbbm{1}$.
The above construction is general, but can be easily adapted for rank-1
$\sigma_{i}$ and $E_{i}$. The best quantum models found for $L=4$ and $5$ are
included in App. D.

##  Appendix D Numerical examples violating the classical the bound

The following are the best quantum models found for $L=4$ and $5$ using
gradient descent techniques, as described in App. C. These optimal values for
a given dimension and sequence length required high accuracy numerical
optimization to be found, and rely on several significant figures in order to
adequately satisfy the constraints in Eq. 13. Despite considerable efforts, no
closed-form description of these models was found. Since all optimal $E_{i}$
and $\sigma_{i}$ found were rank-1, we include the vectors such that
$E_{i}=\left|e_{i}\right\rangle\left\langle e_{i}\right|$ and
$\sigma_{i}=\left|\phi_{i}\right\rangle\left\langle\phi_{i}\right|$. We
provide only 4 significant digits of precision, for simplicity, but
convergence was obtained up to 10 significant digits. Both cases assume
$\rho_{0}=\left|0\right\rangle\left\langle 0\right|$ and
$\mathcal{I}_{a}(\rho)=K_{a}\rho K_{a}^{\dagger}$.

Model for $L=4$:

$\displaystyle\begin{split}&\left|e_{1}\right\rangle=\begin{bmatrix}1\\\ 0\\\
0\end{bmatrix},\quad\left|e_{2}\right\rangle=\begin{bmatrix}0\\\
-0.09692-0.41924i\\\
0.74404\end{bmatrix},\quad\left|e_{3}\right\rangle=\begin{bmatrix}0\\\
0.07184\,+0.30898i\\\
0.65475\end{bmatrix},\quad\left|e_{4}\right\rangle=\begin{bmatrix}0\\\
-0.18847-0.82383i\\\ -0.13307\end{bmatrix},\\\
&\left|\phi_{1}\right\rangle=\begin{bmatrix}0\\\ -0.08293-0.35949i\\\
0.92946\end{bmatrix},\quad\left|\phi_{2}\right\rangle=\begin{bmatrix}0\\\
0.09971\,+0.4288i\\\
0.89788\end{bmatrix},\quad\left|\phi_{3}\right\rangle=\begin{bmatrix}0\\\
0.96462\\\ 0.05913\,-0.25695i\end{bmatrix},\quad\left|\phi_{4}\right\rangle=\
\begin{bmatrix}1\\\ 0\\\ 0\end{bmatrix},\\\ &K_{0}=\mathrm{diag}(0,1,1),\qquad
K_{1}=\mathrm{diag}(1,0,0).\end{split}$ (14)

Model for $L=5$:

$\displaystyle\begin{split}&\left|e_{1}\right\rangle=\begin{bmatrix}1\\\ 0\\\
0\\\ 0\end{bmatrix},\quad\left|e_{2}\right\rangle=\begin{bmatrix}0\\\ \
-0.18168-0.03287i\\\ -0.33582-0.01918i\\\
0.83859\end{bmatrix},\quad\left|e_{3}\right\rangle=\ \begin{bmatrix}0\\\
0.06342\,+0.62645i\\\ -0.36075+0.0699i\\\ -0.32475\end{bmatrix},\quad\
\left|e_{4}\right\rangle=\begin{bmatrix}0\\\ -0.40607-0.42989i\\\
-0.28345-0.24945i\\\ -0.4251\end{bmatrix}\
,\quad\left|e_{5}\right\rangle=\begin{bmatrix}0\\\ -0.3906+0.2592i\\\
0.78055\\\ 0.05633\,+0.08616i\ \end{bmatrix},\\\
&\left|\phi_{1}\right\rangle=\begin{bmatrix}0\\\ -0.25627-0.10917i\\\
-0.29951-0.06935i\\\ 0.90988\
\end{bmatrix},\quad\left|\phi_{2}\right\rangle=\begin{bmatrix}0\\\ 0.78968\\\
0.17386\,+0.37575i\\\ \
-0.02481+0.45209i\end{bmatrix},\quad\left|\phi_{3}\right\rangle=\begin{bmatrix}0\\\
0.73235\\\ 0.46844\,\ -0.13164i\\\
0.35015\,-0.32293i\end{bmatrix},\quad\left|\phi_{4}\right\rangle=\begin{bmatrix}0\\\
\ -0.34778+0.40375i\\\ 0.82555\\\
0.14552\,+0.11544i\end{bmatrix},\quad\left|\phi_{5}\right\rangle=\
\begin{bmatrix}1\\\ 0\\\ 0\\\ 0\end{bmatrix},\\\
&K_{0}=\mathrm{diag}(0,1,1,1),\qquad K_{1}=\mathrm{diag}(1,0,0,0).\end{split}$
(15)

For $L=3$ and $d=2$, we were unable to find any violation of the classical
bound, strongly suggesting EB-based temporal non-classicality only emerges
beyond two-level systems. Furthermore, since $\operatorname{DC}(\bm{a})\leq L$
for any sequence of length $L$, either a qubit or a bit are sufficient to
produce any correlations on two time steps. It is therefore essential to also
look beyond two-time steps in order to reveal fundamental differences between
classical and quantum systems in this sequential measurement scenario.


# Electrification of Clay Calcination: A First Look into Dynamic Modeling and
Energy Management for Integration with Sustainable Power Grids
††thanks: EcoClay (Project Agreement No. 64021-7009) is partially funded by
the Danish Energy Technology Development and Demonstration Program (EUDP)
under the Danish Energy Agency.

Bruno Laurini  DTU Wind
Technical University of Denmark
Kongens Lyngby, Denmark
<EMAIL_ADDRESS>Nicola Cantisani  Wilson R. Leal da Silva  DTU Compute
Technical University of Denmark
Kongens Lyngby, Denmark
<EMAIL_ADDRESS>Green Innovation
FLSmidth Cement
Copenhagen, Denmark
<EMAIL_ADDRESS>Yi Zong  DTU Wind
Technical University of Denmark
Kongens Lyngby, Denmark
<EMAIL_ADDRESS>John Bagterp Jørgensen  DTU Compute
Technical University of Denmark
Kongens Lyngby, Denmark
<EMAIL_ADDRESS>
###### Abstract

This article explores the electrification in clay calcination, proposing a
dynamic model and energy management strategy for the integration of
electrified calcination plants into sustainable power grids. A theoretical
dynamic modeling of the electrified calcination process is introduced, aiming
at outlining temperature profiles and energy usage - thus exploring the
feasibility of electrification. The model serves as a tool for optimizing
parameters, estimating system behavior, and enabling model-based process
control. An innovative energy management model is also presented, ensuring
efficient assimilation of electrified calcination plants into the power grid.
It encapsulates demand-supply balancing and optimizes renewable energy usage.
In essence, we provide an insightful pathway to a more sustainable cement
production, underlining the value of renewable energy sources and effective
energy management in the context of clay calcination.

Keywords: calcined clay; cement; electrification; energy management;
sustainability.

##  I Introduction

The cement industry is a hard-to-abate sector that contributes to around 8.0%
of the world’s CO2 emission [1]. In the production process of Ordinary
Portland Cement, OPC, (i.e. CEM I [2]), the emission of CO2 originates from
limestone calcination and fuel combustion, with a total of 0.80 $t$ CO2 / $t$
cement, assuming a coal-fired system. The cement industry has been
experimenting with complementary technologies to help reduce its CO2
emissions. The current industrial effort to deliver a more sustainable cement
is towards using more Supplementary Cementitious Materials (SCM) in composite
cements, while reducing emissions from fossil fuels utilisation via
electrification. To that end, calcined clay has gained popularity as SCM due
to its worldwide availability, excellent pozzolanic properties, and low cost
[3]. The main industrial solutions for clay calcination rely on the use of
fossil fuels. Hence, the replacement of fuel combustion with full
electrification can further reduce the CO2 footprint from composite cements.
In particular, the use of electricity in a flash calciners requires a
electrical hot gas generator to heat the exhaust gas from the preheating
cyclones to the desired temperature necessary to calcine clays - usually in
the range of 750 - 850∘C depending on the clay type.

While there exist many positive aspects in electrifying clay calcination
process - a comprehensive discussion on technical, economic, and environmental
potentials are provided in [4] and [5] \- the integration of such electrified
process with the power grid presents technical challenges. The large increase
in electrical demand due to electrification might require investments in both
the plant’s electrical infrastructure and power transmission system [6]. The
latter is dependent on the system’s actual power transfer limits, where power
demand fluctuations of such large industrial load are to have a considerable
impact on the power system’s operation. Hence, coordination with system
operators and adoption of smart power management strategies are crucial for
guaranteeing stability.

The adoption of demand-side management measures in electrified plants
represents an opportunity for innovation, helping deliver a more sustainable
product at a lower operation cost because: a) the use of energy management
systems to control dispatchable units (i.e. flexible loads, storage and on-
site generation) can help reduce emissions and operational costs, and b) the
employment of demand response (DR) strategies by the cement plant can help
system operators to handle the intermittence of Renewable Energy Sources
(RES), supporting the integration of such resources into the power grid. In
view of that, we propose a theoretical dynamic model and energy management
strategy for the integration of electrified clay calcination plants into
sustainable energy grids.

##  II Theoretical model

In this section, we describe the mathematical and physical foundation of the
cement plant’s Energy Management System (EMS) and a dynamic model of the clay
calcination process.

###  II-A Energy Management System (EMS)

An EMS aims to optimize the energy generation and usage inside an energy
system. It can be applied to systems of different scales, such as residential
and commercial buildings, micro-grids, power plants, and industrial plants.
For instance, a residential building can be equipped with photovoltaic
modules, lithium-ion batteries, and a HVAC system. In this case, a Building
Energy Management System (BEMS) could be used to optimize the power dispatch
of flexible technologies for given thermal comfort limits and dynamic
electricity contracts (e.g. Time-of-use rates, Real-time pricing) with the
utility [7].

The benefit of using smart energy management strategies is rather minimal in
small-scale energy systems. Nonetheless, in an industrial context, they can
lead to large costs and CO2 savings. The complexity of EMS increases alongside
the scale of the system due to factors such as: distinct processes taking
place simultaneously, large critical loads, power distribution complexity, and
potential participation in power markets.

From a cement plant outlook, we propose an industrial EMS based on an Optimal
Power Flow (OPF) algorithm. Most power distribution networks are based on
radial topology, and the power flow within them can be modeled using Branch
Flow Model (BFM), which focuses on describing the physical relationship
between branch variables such as Branch current and power flows. Assuming a
radial distribution topology, the directed graph that represents the network
can be described by $G=(N,E)$ . From Ohm’s law, the voltage drop between two
buses reads:

$V_{i}-V_{j}=z_{ij}I_{ij}\quad\forall(i,j)\in E,$ (1)

where $(i,j)$ denotes a network link - connecting bus $i$ and $j$ \- and $E$
represents the set of buses. The branch power flow $S_{i}j$ definition reads:

$S_{ij}=V_{i}I_{ij}^{*}\quad\forall(i,j)\in E.$ (2)

Let $N_{p}(j)\subseteq N$ be the bus set containing the parent of $j$,
$N_{c}(j)\subseteq N$ the bus set of all the children of $j$, and
$N_{ss}\subseteq N$ be the set of substation buses; then, the power balance in
the branch power flow depicted in Fig. 1(a) can be written as:

$\sum_{i\in
N_{\mathrm{p}}(j)}\left(S_{ij,t}-z_{ij}\left|I_{ij}\right|^{2}\right)+S_{j,t}=\sum_{k\in
N_{\mathrm{c}}(j)}S_{jk,t}\quad\forall j\in N\backslash N_{\mathrm{ss}}.$ (3)

(a) Branch power flow

(b) Conceptual topology of the plant’s power distribution

Figure 1: Single line diagrams

Together, (1) to (3) constitute the branch flow equations and can be used for
power flow analysis of a given network. After some manipulation of (1) and
(2), we can arrive in a voltage drop formulation that do not contain complex
numbers, only magnitudes, i.e.:

$U_{j}=U_{i}-2\left(P_{ij}r_{ij}+Q_{ij}x_{ij}\right)+\left(r_{ij}^{2}+x_{ij}^{2}\right)l_{ij}\quad\forall(i,j)\in
E,$ (4)

where $U$ and $l$ are squared voltage and current magnitudes. The complex
power balance in (3) can be decomposed into its real and imaginary
counterparts by:

$\sum_{i\in N_{\mathrm{p}}(j)}\left(P_{ij}-l_{ij}r_{ij}\right)-\sum_{k\in
N_{\mathrm{c}}(j)}P_{jk}=P_{j}\quad\forall j\in N\backslash N_{\mathrm{ss}}$
(5)

and

$\sum_{i\in N_{\mathrm{p}}(j)}\left(Q_{ij}-l_{ij}x_{ij}\right)-\sum_{k\in
N_{\mathrm{c}}(j)}Q_{jk}=Q_{j}\quad\forall j\in N\backslash N_{\mathrm{ss}}$
(6)

Furthermore, the squared branch current magnitude is related to the voltage
drop, and active and reactive power flow through the branch, according to the
apparent power definition, i.e.:

$l_{ij}=\frac{P_{ij}^{2}+Q_{ij}^{2}}{U_{i}}\quad\forall(i,j)\in E.$ (7)

Along with the power generation/absorption constraints, the aforementioned
equations can be cast in a standard optimization form. Such problem represents
an OPF model, where a given objective function is minimized given a set of
equalities and inequalities constraints. For example, a Distribution System
Operator (DSO) could run an OPF aiming at power loss minimization and
Conservation of Voltage Reduction (CVR), in a context of high penetration of
Distributed Energy Resources (DERs) [8]. Our proposed EMS for a cement plant
is designed in a similar context, with the main difference being that a DSO
cannot control all the loads and generators that are connected to the
distribution grid. In the industrial case, the cement plant owns both the
network and the loads/generators, i.e. there are many control actions that can
be set to achieve the desired objective.

Figure 1(b) shows a conceptual topology of a cement plant’s power
distribution, where on-site renewable generation (e.g. a PV and a Wind power
plant), Battery Energy Storage System (BESS) and the main grid are used to
feed both flexible and non-flexible loads. A cement plant is typically divided
into functional operating departments according to the different processes.
Usually, separate electrical feeder circuits are used to guarantee electrical
power supply for each important department depending on their size. The loads
of each department require different amounts of active and reactive power that
are supplied by the upstream buses.

Moreover, depending on the nature of the process, a load can be flexible or
not. In other words, it can modulate its power absorption within a specific
range that depends on the process itself. A clay calcination department -
containing an electrified clay calcination process - can be a source of power
flexibility, especially when coupled with energy storage solutions (e.g.
thermal storage). When looking at the plant’s power supply part, the main
substation represents the dominant source of supply. In this area,
Photovoltaic, Wind turbines, and Battery Energy Storage Systems are connected
to the AC network through power converters. Such technologies can help
increase the plant’s power flexibility and reduce both Scope 2 emissions costs
and costs of buying power from the utility. Note that, while many industrial
plants have fixed electricity contracts with the utility company, there is
also an opportunity for the plant to participate in power markets directly, in
case it has sufficient power capacity and smart energy management strategies.

###  II-B Dynamic model

The establishment of a dynamic model of the clay calcination process is the
key to assess and simulate the plant’s performance under changing conditions.
Such a model is a proxy to evaluate and optimize controllers for the process.
Figure 2 shows the process diagram of the clay calcination in a loop with gas
re-circulation. The process starts by introducing ground clay in the first
cyclone. The clay is pre-heated by two cyclones before entering the calciner.
Next, the calcination reaction is completed in the calciner and the mixture of
gas-solid undergoes separation in the last cyclone. Finally, the hot gas is
recirculated in a loop. Our approach towards modelling this dynamic process is
presented as follows.

The plant-wide dynamic model consists of connected units. The main units are
the calciner and cyclones; and this model results in the following system of
differential algebraic equations (DAEs):

$\displaystyle\dot{x}(t)=f(x,y,u,d,p),$ (8a) $\displaystyle 0=g(x,y,u,d,p),$
(8b)

where $x$ and $y$ are the differential and algebraic variables. $u,d$ and $p$
inputs, disturbances and model parameters, respectively. The properties and
reaction of the materials and substances are evaluated by an independent
chemical and thermo-physical model that is shared by all the units. The
chemical model comprise stoichiometry and kinetics of the dehydroxylation
reaction of clay, i.e.:

$\mathrm{Al_{2}O_{2}\cdot 2SiO_{2}\cdot
2H_{2}O(s)\rightarrow}\mathrm{Al_{2}O_{2}\cdot
2SiO_{2}(s)}\mathrm{+2H_{2}O(g).}$ (9)

This reaction is modeled as a third-order reaction as in [9]. The material’s
thermo-physical properties are evaluated by building a thermodynamic library,
which can compute the material’s enthalpy $H$, volume $V$, and internal energy
$U$ as a function of temperature $T$, pressure $P$, and number of models $n$
by means of:

$\displaystyle V$ $\displaystyle=V(T,P,n),$ (10a) $\displaystyle H$
$\displaystyle=H(T,P,n),$ (10b) $\displaystyle U$ $\displaystyle=H-PV.$ (10c)

The dynamic behaviour of each unit is modeled by using first principles, i.e.
mass and energy balances. Next, the calciner is modeled as plug flow reactor -
as fully described in [10]. The mass and energy balances are expressed as
partial differential equations in length $z$ and time $t$, i.e.:

$\displaystyle\partial_{t}c$ $\displaystyle=-\partial_{z}N+R,$ (11a)
$\displaystyle\partial_{t}\hat{u}_{s}$
$\displaystyle=-\partial_{z}\tilde{H}_{s}+\hat{J}_{sg}+\hat{Q}_{amb,s},$ (11b)
$\displaystyle\partial_{t}\hat{u}_{g}$
$\displaystyle=-\partial_{z}\tilde{H}_{g}-\hat{J}_{sg}+\hat{Q}_{amb,g}.$ (11c)

where $c,N,R,\hat{u},\tilde{H}$ are the concentrations, material fluxes,
chemical production rates, volumetric internal energy, and enthalphy flux,
respectively. The subscripts $s$ and $g$ indicate solid and gas.
$\hat{J}_{sg}$ and $\hat{Q}_{amb}$ are the solid to gas heat transfer and heat
transfer to the ambient, respectively.

The velocity of the material stream is modeled explicitly as a function of the
pressure drop (by means of the Darcy-Weisbach equation). After spatial
discretization of (11), the model is completed by adding the algebraic
equations:

$\displaystyle V(T_{s},P,c_{s})+V(T_{g},P,c_{g})-1=0,$ (12a) $\displaystyle
U(T_{s},P,c_{s})-\hat{u}_{s}=0,$ (12b) $\displaystyle
U(T_{g},P,c_{g})-\hat{u}_{g}=0.$ (12c)

The cyclones are modeled as a single cell without spatial discretization. This
model consists of mass and energy balances for solid and gas, formulated as
DAEs in the same fashion as the calciner. The material fluxes depend on the
cyclone separation efficiency, which is modeled explicitly as in [11]. The
velocities at the inlet and outlet are modeled as a function of the pressure
drop between the input and output streams.

Other minor process units (e.g. electric hot gas generator - EHGG, circulating
fan and particle filter) are modeled using static relationships and
incorporated in the model as algebraic equations. In addition, the connection
between the units is realised by lumping the flow resistance in the connecting
tube as a pressure node [12]. The flow rate $F$ between 2 units (1 and 2)
reads:

$F=C\sqrt{\frac{(P_{1}-P_{2})(P_{1}+P_{2})}{T_{1}}},$ (13)

where $C$ represents the flow resistance.

Figure 2: Clay calcination process diagram.

The inputs $u$ (variables) of the plant-wide model are the clay feed, pressure
rise after the circulating fan, and fresh air intake. The disturbances $d$ are
the electrical power to the hot gas generator and the ambient temperature.
Notice that the model may be easily extended to a system of stochastic
differential algebraic equations to account for different sources of
uncertainty and model/plant mismatch. Nonetheless, such extension is beyond
the scope of this publication.

##  III Model integration

The integration of the EMS and Dynamic calcination models provides a link
between the electricity supply of the cement plant and the thermodynamics of
the clay calcination process. Such a link allows for emulating the physical
constraints of both systems and investigating how the plant can operate close
to optimal conditions from costs, emissions, and technical standpoint. While
both models are linked through the power absorbed by the EHGG, they have
different temporal/spatial scales, following a similar modelling approach to
the hierarchical algorithm for integrated scheduling and control proposed in
[13]. The EMS is based on a systemic approach, where the focus is on modeling
the electrical interaction of different processes through the power
distribution network and scheduling dispatchable units.

The scheduling problem is directly related to daily production plan of the
cement plant and to wholesale electricity markets. Therefore, a model with
relatively large time-horizon (e.g. 24 hours) and low frequency (e.g. 1 hour)
is used. Conversely, the dynamic model of the clay calcination plant enables a
detailed simulation of the process on the temporal scale of seconds, capturing
transient states and detailed process dynamics.

The dynamic model keeps track of crucial variables such as temperature in the
calciner, calcined clay at the outlet, calcination degree in every step of the
process, etc. A local plant controller is implemented to control the process.
As mention in Section II.A, the EMS aims to optimize the energy utilization in
the cement plant considering techno-economic and emission-wise aspects, i.e.
the objective function reads:

$\displaystyle\min\quad$
$\displaystyle\phi=\phi_{c}+\phi_{CO_{2}}+\phi_{u}-\phi_{cc}$ (14a) s.t.
$\displaystyle\text{Network constraints},$ (14b)
$\displaystyle\text{Operational/process constraints}.$ (14c)

The terms in the objective function are operational cost including electricity
purchase ($\phi_{c}$), CO2 emissions ($\phi_{CO_{2}}$), voltage deviation
($\phi_{u}$), and calcined clay production ($\phi_{cc}$).

Figure 3: Integration framework of the EMS and dynamic calcination model.

Solving the optimization problem (14) will provide scheduling decisions for
the chosen time horizon. These may include power set-points for generation
units and storage, or the operational status of voltage controlling devices.
The use of different objective terms characterizes a multi-objective
optimization and allows for the EMS to make scheduling decisions based on a
combination of different objectives. Weights may need to be used in the
objective function to tune the performance of the optimizer. Amongst the
decision variables, $\widetilde{P}^{EHGG}$ represents the power set-point sent
to the clay calcination plant controller.

Note that the EMS sets the active power consumed by the EHGG based on a
higher-level model: as such, it does not capture the dynamics of the clay
calcination process. The actual power used by the EHGG in real-time will
actually depend on the process dynamics and the plant controller. Fig. 3 shows
the proposed model integration framework.

##  IV Conclusion

This paper presents a theoretical methodology aimed at optimizing energy
management in cement plants, with a focus on an electrified clay calcination
process. We propose an EMS model alongside a dynamic model for the clay
calcination. Also, we discuss an integration framework to link the power
management of the plant with the electricity requirements of the electrified
calcination process. This framework is designed to reduce operational costs
associated with electricity procurement, while mitigating indirect carbon
dioxide emissions from the power grid. By addressing technical and production-
related constraints, our approach aims to achieve these objectives while
ensuring the fulfillment of production targets and technical requirements.
Future work should be carried out to extend the EMS and dynamic models
including a market bidding strategy that encompasses participation in Day-
ahead and balancing electricity markets and advanced control strategies based
on model predictive control.

##  V Acknowledgments

The EcoClay project (Project Agreement No. 64021-7009) - which is partially
funded by the Danish Energy Technology Development and Demonstration Program
(EUDP) under the Danish Energy Agency.

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