# Magnetic Order in Honeycomb Layered U2Pt6Ga15 Studied by Resonant X-ray and
Neutron Scatterings

Chihiro Tabata  Materials Sciences Research Center, Japan Atomic Energy
Agency, Tokai 319-1195, Japan  Advanced Science Research Center, Japan Atomic
Energy Agency, Tokai 319-1195, Japan  Fusako Kon  Graduate School of Science,
Hokkaido University, Sapporo 060-0810, Japan  Kyugo Ota  Graduate School of
Science and Engineering for Education, University of Toyama, Toyama 930-8555,
Japan  Ruo Hibino  Graduate School of Science, Hokkaido University, Sapporo
060-0810, Japan  Yuji Matsumoto  Graduate School of Science and Engineering,
University of Toyama, Toyama 930-8555, Japan  Hiroshi Amitsuka  Graduate
School of Science, Hokkaido University, Sapporo 060-0810, Japan  Hironori
Nakao  Photon Factory, Institute of Materials Structure Science, High Energy
Accelerator Organization, Tsukuba 305-0801, Japan  Condensed Matter Research
Center, Institute of Materials Structure Science, High Energy Accelerator
Organization, Tsukuba 305-0801, Japan  Yoshinori Haga  Advanced Science
Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan  Koji
Kaneko  Materials Sciences Research Center, Japan Atomic Energy Agency, Tokai
319-1195, Japan  Advanced Science Research Center, Japan Atomic Energy Agency,
Tokai 319-1195, Japan

###### Abstract

Antiferromagnetic (AF) order of U2Pt6Ga15 with the ordering temperature
$T_{\rm N}$ = 26 K was investigated by resonant X-ray scattering and neutron
diffraction on single crystals. This compound possesses a unique crystal
structure in which uranium ions form honeycomb layers and then stacks along
the $c$-axis with slight offset, which gives rise to a stacking disorder. The
AF order can be described with the propagation vector of $\bm{q}=(1/6,1/6,0)$
in the hexagonal notation. The ordered magnetic moments orient perpendicular
to the honeycomb layers, indicating a collinear spin structure consistent with
Ising-like anisotropy. The magnetic reflections are found to be broadened
along $c^{*}$ indicating that the stacking disorder results in anisotropic
correlation lengths. The semi-quantitative analysis of neutron diffraction
intensity, combined with group theory considerations based on the
crystallographic symmetry, suggests a zig-zag type magnetic structure for the
AF ground state, in which the AF coupling runs perpendicular to the stacking
offset, characterized as $\bm{q}=(1,0,0)_{\rm orth}$. The realization of the
zig-zag magnetic structure implies the presence of frustrating intralayer
exchange interactions involving both ferromagnetic (FM) first-neighbor and AF
second and third-neighbor interactions in this compound.

##  I Introduction

Frustrated magnets have been a promising research field to accommodate diverse
and intriguing phenomena, including topological magnetic order [1], spin-
liquid states [2], quantum Hall effect [3, 4], and unconventional
superconductivity [5, 6]. When only the nearest neighbor antiferromagnetic
(AF) interaction, $J_{1}$, is effective, magnetic frustration arises in
triangular lattice and their derivatives such as Kagomé and pyrochlore
lattices. In contrast, a honeycomb network does not have such frustrations
since an AF $J_{1}$ interaction results in a Néel type AF order. However,
magnetic frustrations can emerge when the next-nearest AF interaction,
$J_{2}$, becomes active. Theoretical studies on the $J_{1}$-$J_{2}$ honeycomb
lattice predict the formation of a highly degenerate ground state, which can
lead to a spin liquid state [7] or exotic multiple-$\bm{q}$ topological
magnetic order[8]. Recently particular focus is on Kitaev quantum spin liquid,
which serves as a solvable theoretical model for a two-dimensional (2D)
honeycomb layer with Ising interactions, as represented by ${\alpha}$-RuCl3
and iridates[9, 10, 11]. However, finding an ideal Kitaev compound is
challenging due to small distortions from the honeycomb lattice and/or non-
negligible interlayer interactions in actual compounds, which can lead to the
formation of specific orders. Considerable efforts have been undertaken to
investigate ideal 2D honeycomb lattice compounds.

In this context, $R_{2}$Pt${}_{6}X_{15}$($R$=rare earths, $X$=Ga, Al) could
provide an interesting playground. $R_{2}$Pt${}_{6}X_{15}$ crystallizes in the
hexagonal Sc0.6Fe2Si4.9-type structure with the space group $P6_{3}/mmc$ (no.
194, $D_{6}^{h4}$) [12, 13], illustrated in Fig. 1(a). One remarkable feature
of this structure is that $R$ and $X$ atoms have a regular vacancies of 1/3
and 2/3, respectively. These systematic vacancies result in the formation of a
honeycomb network for the $R$ atoms [14, 15, 16]. The unit cell contains two
well-separated honeycomb layers of $\sim$8 Å along the $c$-axis. Note that
this stacking has a slight shift along [01$\bar{1}$0]. As there exists three
equivalent [01$\bar{1}$0], there appears three domains in terms of stacking as
displayed in Fig. 1 (b). Consequently, while the symmetry of the local unit
cell is orthorhombic, an average structure with three domains results in a
hexagonal symmetry. To avoid confusion, subscripts “hex” and “orth” are used
to represent averaged hexagonal and orthorhombic crystal systems,
respectively. The relationship between the hexagonal and orthorhombic cells is
depicted in Fig. 1 (b). The presence of the three domains indicates intrinsic
stacking disorder of this structure. Together with the long separation between
honeycomb layers, weak interlayer interaction is expected, in other words,
$R_{2}$Pt${}_{6}X_{15}$ appears likely a host of quasi-2D magnetic behavior.
Indeed, diffuse scattering along the $l$ direction around (1/3, 1/3, $l$)hex
reflections was confirmed in the X-ray diffraction experiments, indicating
short range correlations along the $c$-axis[13, 15].

Figure 1:  Crystal structure of $R_{2}$Pt${}_{6}X_{15}$ drawn by VESTA [17].
(a) Average hexagonal structure. (b) Orthorhombic structure with three domains
rotated by 120∘ each around $c_{\rm hex}$ ($c_{\rm orth}$). Pt and $X$ atoms
are not shown.

Here we focus on the new 5$f$ variant U2Pt6Ga15. U2Pt6Ga15 also has a stacking
disorder with lattice constants of $a_{\rm hex}~{}=~{}4.304$ Å and $c_{\rm
hex}~{}=~{}16.30$ Å[15]. Magnetization measurements reveal strong uniaxial
nature of this compound with the easy axis along $c_{\rm hex}$ [15]. Despite a
lack of the structural long-range correlation along the $c$-axis, a bulk
crystal exhibits a fairly sharp transition at $T_{\rm N}$ = 26 K as observed
in magnetization, specific heat, and electrical resistivity measurements. This
sharp transition implies a dominant in-plane and relatively weak interlayer
exchange interactions, as pointed out by the previous study of bulk physical
properties [15]. To our best knowledge, an uranium compound with a 2D
honeycomb structure is rarely known. U2Pt6Ga15 may provide interesting
ordering phenomena under frustration between 5$f$ electrons with strong spin-
orbit interaction. In order to get microscopic insights into magnetic order in
U2Pt6Ga15, resonant X-ray scattering (RXS) and neutron diffraction experiments
were carried out on single crystalline samples.

##  II Experiment

Single crystalline U2Pt6Ga15 was grown by the Ga-self flux method, of which
details were reported in Ref. 15. Rod-like crystals with a typical dimension
of 1 $\times$ 1 $\times$ 3 mm3 ($\sim$5 mg) were used in both RXS and neutron
diffraction experiments. These crystals also display similar (1/3, 1/3,
$l$)hex diffuse scatterings, consistent with those reported in the literature
[13, 15]. The RXS experiments were performed at BL-11B of Photon Factory,
using a two-circle diffractometer for soft X-ray installed inside a high-
vacuum chamber [18].The typical vacuum level in the chamber was $10^{-5}$ Pa
throughout the measurements. An incident X-ray energy was set to around 3.72
keV to fulfill a resonance condition of the U $M_{4}$ edge. The two samples
with different orientations were attached to Cu blocks by using silver paste,
and the Cu blocks were mounted onto a cold finger of a liquid 4He flow
cryostat with base temperature of about 6 K. A horizontal scattering geometry
with the incident $\pi$-polarization, in which the electric field vector lying
in the scattering plane, was employed. The polarization of the scattered X-ray
was analyzed using an Al (111) single crystal.

The neutron diffraction experiments were carried out on the thermal triple-
axis spectrometer TAS-1 installed at the reactor hall of the research reactor
JRR-3 of Japan Atomic Energy Agency in Tokai, Japan. A vertical focusing PG
monochromator and analyzer were used. The spectrometer was fixed to the
elastic condition in a triple-axis mode with a wavelength of 2.40 Å. The
collimation set of open-80’-40’-80’ was employed, and higher order
contamination was removed by a PG filter placed before the sample. The single
crystalline sample was fixed to an Al plate to have a ($hhl$) horizontal
scattering plane. The sample was sealed in an Al can filled with a He exchange
gas, and was attached to a cold finger of a closed-cycle refrigerator and
cooled down to 3 K.

##  III Results

###  III.1 Resonant X-ray scattering

Superlattice reflections originating from magnetic ordering were searched by
scans along the high symmetry reciprocal lattice axes, [100]${}^{*}_{\rm
hex}$, [001]${}^{*}_{\rm hex}$, and [110]${}^{*}_{\rm hex}$ at the resonant
condition of the U $M_{4}$ absorption edge at 11 K. Here, the experimental
data are represented in the hexagonal notation due to the intrinsic difficulty
in distinguishing individual orthorhombic domains in diffraction experiments.
Figure 2 shows a scan profile along [110]hex. Satellite peaks were found at
(5/6, 5/6, 0)hex and (7/6, 7/6, 0)hex besides a fundamental peak at (1, 1,
0)hex. Additional scans confirmed that superlattice reflections form a six-
fold symmetry around (1, 1, 0)hex in the $a^{*}_{\rm hex}$-$b^{*}_{\rm hex}$
plane, reflecting the average hexagonal symmetry, as illustrated by the inset
of the figure. This result indicates that the propagation vector of the AF
state can be described by $\bm{q}=(1/6,1/6,0)_{\rm hex}$. Figure 3 shows the
peak intensity of the typical satellite reflection (7/6, 1/6, 0)hex as a
function of incident X-ray energy. The well-defined peak around 3.722 keV
corresponds to the U $M_{4}$ absorption edge, indicating that the satellite
originates from a resonant scattering involving 5$f$ electrons. The
temperature variation of the satellite peaks is depicted in Fig. 4. Upon
cooling, the RXS signal intensity gradually increases below $T_{\rm N}$,
indicative of order parameter development. This increase in intensity is not
accompanied by a peak shift, as shown in the right panel in Fig. 4 (b).
Namely, the ordering vector $\bm{q}$ remains commensurate with
$\bm{q}=(1/6,1/6,0)_{\rm hex}$ across the entire temperature range below
$T_{\rm N}$. The peak width remains constant along the $h$ direction
throughout the temperature variation, while it slightly broadens with
increasing temperature in the $l$ direction.

Figure 2: The line scan profile along [110]hex of U2Pt6Ga15 measured under the
resonant condition of the U M4 edge. The inset illustrates the observed
diffraction pattern around the 110 fundamental reflection and the scan
trajectory. Figure 3: Energy dependence of the resonant X-ray magnetic
scattering at $(7/6,1/6,0)_{\rm hex}$ (closed circles), shown with the
fluorescence background (open circles). Figure 4: Temperature dependence of
(a) peak profiles and (b) integrated intensity, peak positions and full width
at half maximum (FWHM) of the resonant X-ray magnetic scattering at
$(5/6,5/6,0)_{\rm hex}$.

The observed magnetic reflections have a highly anisotropic peak shape in
reciprocal space as represented by the ($h,0,l$) intensity map for the (7/6,
1/6, 0)hex reflection in Fig. 5 (a). The peak has a streak along the $l$
direction at 6 K, indicating that short-range correlations remain along the
$c_{\rm hex}$-axis well below $T_{\rm N}$. Figure 5 (b) displays the peak
profiles along the [100]${}^{*}_{\rm hex}$ and [001]${}^{*}_{\rm hex}$
directions, together with those of the (1, 1, 0)hex fundamental reflection.
The magnetic reflection is broader than the fundamental reflection for both
directions. Corresponding to the streak in the color map, the profile along
the $l$-direction is substantially broadened with a Lorentzian form. The
correlation lengths of the magnetic order, estimated from the full width at
half maximum (FWHM), are approximately 70 unit cells (260 Å) along the $a_{\rm
hex}$-axis and 7 unit cells (100 Å) along the $c_{\rm hex}$-axes. Given the
high resolution of the synchrotron X-rays, the negligible impact of its
resolution on the observed peak width suggests that the width predominantly
reflects the intrinsic nature of the sample, revealing the 2D nature of
magnetic correlations in U2Pt6Ga15.

Figure 5: (a) A color map of the RXS peak of U2Pt6Ga15 in the $(h,0,l)_{\rm
hex}$ reciprocal lattice plane. (b) Line profiles of the RXS peak and the
fundamental reflection peak along $(h,0,0)_{\rm hex}$ and $(0,0,l)_{\rm hex}$
lines.

To investigate orientation of the ordered moment, the polarization of the
scattered X-ray was analyzed in two configurations: $c_{\rm hex}$ (a) within
and (b) normal to the horizontal scattering plane. Figure 6 shows the RXS
intensity as a function of the analyzer crystal angle, $\phi_{\rm A}$, for the
two scattering geometries depicted in Fig. 6(a) and (b). Polarization
components $\pi^{\prime}$ and $\sigma^{\prime}$ of the scattered X-ray were
obtained for $\phi_{\rm A}$ = 90∘ and 0∘, respectively. As shown in the
figure, the two configurations give contrasting dependencies on $\phi_{\rm
A}$. In configuration (a), the maximum intensity was observed at $\phi_{\rm
A}$=0∘, corresponding to $\sigma^{\prime}$ polarization, while in
configuration (b), it was observed at $\phi_{\rm A}$=90∘, corresponding to
$\pi^{\prime}$ polarization.

The observed polarization dependences in both configurations consistently
demonstrate the magnetic moments parallel to the $c_{\rm hex}$ axis. First,
the strong RXS intensity in the $\pi$-$\sigma^{\prime}$ process in
configuration (a) indicates that a rotation of the polarization vector of the
X-ray occurs. This cannot be caused by the Thomson’s scattering process, and
necessitates magnetic dipoles or even higher-rank multipoles. Strong intensity
in neutron diffraction indicates magnetic dipole origin, as detailed in the
following section. The finite intensity in the $\pi$-$\pi^{\prime}$ channel is
a background signal which is not intrinsic to the magnetic order, as evidenced
by the absence of peak structure in the rocking curve.

Next, $\phi_{\rm A}$ dependence in configuration (b) is discussed based on the
orientation of the magnetic dipole moment relative to the polarization. The
RXS amplitude arising from magnetic dipole is given by
$-i(\bm{\varepsilon}^{\prime}\times\bm{\varepsilon})\cdot\bm{u}$ [19], where
${\bm{\varepsilon}}$ and $\bm{{\varepsilon}}^{\prime}$ are polarization
vectors of the incident and scattered X-ray, respectively, and $\bm{u}$ is a
unit vector parallel to the magnetic moment.The absence of detectable signal
in the $\pi$-$\sigma^{\prime}$ strongly suggests that the ordered moments are
pointing to the $c_{\rm hex}$ axis. Indeed, the Ising-like moments along to
the $c_{\rm hex}$ axis, suggested by the bulk measurements[15], well reproduce
the observed $\phi_{\rm A}$ dependences as illustrated by the dotted lines in
Figs. 6(a) and (b).

Figure 6: Polarization analyses of the RXS signal for the magnetic scatterings
measured at 10 K. The schematic view of geometric configurations of the sample
and optical system are also shown for two configurations with the $c_{\rm
hex}$-axis (a) parallel and (b) perpendicular to the scattering plane. The
vectors of $\bm{k}$, $\bm{k^{\prime}}$, and $\bm{\kappa}$ are wave vectors of
the incident and scattered X-ray, and the scattering vector, respectively.

###  III.2 Neutron scattering

The RXS experiment succeeded to determine the propagation vector $\bm{q}$ and
the orientation of the ordered moment. On the other hand, a long wavelength of
X-ray for the U M4 edge limits an accessible reciprocal space. A complete
magnetic structure, namely the spin arrangement within the unit cell, could
not be determined owing to this restriction. To determine complete spin
structure, the neutron diffraction experiment was performed on a single
crystal.

Figure 7(a) shows peak profiles of the magnetic Bragg reflection along $l$
through (5/6, 5/6, 0). The magnetic reflection with $\bm{q}=(1/6,1/6,0)_{\rm
hex}$ which develops below $T_{\rm N}$ was confirmed in the neutron scattering
as well. The peak profile is reproduced by a broad Lorentzian.The integrated
intensity as a function of temperature in Fig. 7(b) reveals gradual evolution
of the magnetic peak below $T_{\rm N}$ = 26 K. This increase is slightly
steeper in neutron than RXS, which may reflect different penetration length of
two probes. On the other hand, $T_{\rm N}$ and the broad nature of the
magnetic peak are consistent between two probes, indicating that the observed
short range correlation stems not from surface effect, but from bulk.

Figure 7: Temperature dependence of (a) the peak profile and (b) integrated
intensity of the neutron magnetic reflection at $\bm{q}=(5/6,5/6,0)_{\rm hex}$
of U2Pt6Ga15.

To determine the magnetic structure, magnetic peaks in wide $\bm{Q}$-space
were measured at the base temperature. Figure 8 shows line scan profiles along
$l$ for (1/6, 1/6, $l$)hex, (5/6, 5/6, $l$)hex, and (7/6, 7/6, $l$)hex at 3 K.
In all three scans, magnetic peak intensities decrease with increasing $l$,
and is more pronounced for (1/6, 1/6, $l$)hex. This feature is consistent with
the Ising-type magnetic structure as magnetic neutron diffraction intensity is
proportional to the square of the magnetic moment perpendicular to the
scattering vector $\bm{Q}$.

One remarkable feature is a systematic extinction; intense magnetic peaks were
observed only at $l$ = even for all three scans. The systematic absence of the
magnetic peak at $l$=odd could reflect characteristic structure along the
$c$-axis. U ions in the unit cell can be regarded to form a double-layered
structure along $c$, though it does not fulfill precise translational symmetry
with (0, 0, 1/2)hex and have a slight displacement on stacking. The observed
systematic absence with $l$=odd suggests that the two layers in the unit cell
have a ferromagnetic coupling. Details of a magnetic structure model will be
discussed in the next section.

Figure 8: Line profiles along [001] in the reciprocal lattice space measured
at the base temperature, 3.8 K. The top, middle, and bottom panels show (1/6,
1/6, $l$), (5/6, 5/6, $l$), and (7/6, 7/6, $l$) lines, respectively.

##  IV Discussion

First, a possible magnetic structure of U2Pt6Ga15 is discussed in this
section. Typically, a straightforward quantitative analysis of magnetic Bragg
intensities from a neutron diffraction measurement gives possible magnetic
structure models. There exists two difficult factors, however, in the present
case. One is the elongation of the magnetic Bragg peaks along $l$, that makes
it hard to deduce reliable absolute intensities by conventional scans. Another
difficulties arise from domains in terms of stacking displacement of the U
layer along $c_{\rm hex}$ ($c_{\rm orth}$). As mentioned before, the honeycomb
layers do not stack up perfectly along $c_{\rm hex}$ ($c_{\rm orth}$), but are
shifted along $b_{\rm orth}$, resulting in the formation of three domains
rotated by 120∘ with each other (see Fig. 1 (b)). Because the present neutron
experiment covered only first quadrant of the 2D scattering plane in the
reciprocal space, effects of domain could not be taken into account.

To overcome these intrinsic difficulties, we utilize a group theoretical
approach. Whereas the domain structure prevents quantitative analysis,
relationship between the propagation vector and the stacking offset could be
determined. Hereafter, the orthorhombic notation is employed in the following
discussion to handle the stacking offset properly. The present RXS and neutron
diffraction experiments revealed following characteristics of the magnetic
order of U2Pt6Ga15:

  1. (i)

The propagation vector of the magnetic order is $\bm{q}=(1/6,1/6,0)_{\rm
hex}$.

  2. (ii)

The ordered magnetic moments orient parallel to $c_{\rm hex}$ ($c_{\rm
orth}$).

  3. (iii)

The neighboring honeycomb layers in the unit cell couple ferromagnetically.

Owing to a choice on coupling between the stacking displacement and the
magnetic propagation, there exists three possibilities to describe
$\bm{q}=(1/6,1/6,0)_{\rm hex}$ in the orthorhombic notation,
$\bm{q}=(1,0,0)_{\rm orth}$, $(1/2,-1/2,0)_{\rm orth}$, and $(1/2,1/2,0)_{\rm
orth}$, as displayed in Fig. 9. By applying the group theory for the result
(i), 16 magnetic space groups are allowed, and the additional condition (ii)
excludes half of the candidates[20]. The condition (iii) narrows down to two
models both described by the propagation vectors $\bm{q}=(1,0,0)_{\rm orth}$,
the zig-zag structure $P_{B}mmn$ (#59.414) and the stripy structure $P_{B}nma$
(#62.454), as displayed in Fig.10.

Figure 9: Schematic of the relationship between the local orthorhombic lattice
system and the average hexagonal lattice system for U2Pt6Ga15. The cross
symbols indicate the positions of the observed magnetic scattering. The closed
circles represent the squared structure factor calculated for the orthorhombic
structure.

Note that the two structures give identical intensity ratio for the observed
magnetic reflections in (1/6, 1/6, $l$)hex, (5/6, 5/6, $l$)hex, and (7/6, 7/6,
$l$)hex. On the other hand, there appears a clear contrast for another high
symmetry point, such as $(1/2,1/2,0)_{\rm hex}$, which corresponds to
$(3,0,0)_{\rm orth}$. A magnetic peak is not expected at $(1/2,1/2,0)_{\rm
hex}$ for the zig-zag structure, whereas the stripy structure gives more than
3 times stronger intensity for $(1/2,1/2,0)_{\rm hex}$ than that for
$(1/6,1/6,0)_{\rm hex}$. Absence of a magnetic peak at $(1/2,1/2,0)_{\rm hex}$
in both neutron and RXS experiments indicates that the zig-zag structure is
favorable as the magnetic structure in the ground state of U2Pt6Ga15. The
observed and calculated neutron diffraction intensities are summarized in
Table 1.

Figure 10: Magnetic structure candidates for U2Pt6Ga15 suggested by the RXS and neutron diffraction experiments. The black circles and light grey circles indicate down spins and up spins, respectively. Four ($2\times 2$) magnetic unit cells are shown for each. Table 1: Observed and calculated values of intensity ratios of magnetic reflections at (1/6, 1/6, 0)hex and (1/2, 1/2, 0)hex. | $I_{\rm calc}$ | $I_{\rm obs}$
---|---|---
Zig-zag | Stripy
$I_{\frac{1}{2}\frac{1}{2}0}/I_{\frac{1}{6}\frac{1}{6}0}$ | 0.00 | 3.46 | 0.00(1)

Having determined the spin configuration, we now discuss the results in terms
of the honeycomb lattice layers. The zig-zag order in honeycomb layered
systems has been found in several transition metal oxides such as ZnMnO3 [21],
Na2IrO3 [22], and Na3Co2SbO6 [23]. In these cases, ferromagnetic (FM) first-
nearest neighbors ($J_{1}$) and AF second-nearest neighbors interactions
($J_{2}$) are considered to be essential. The AF $J_{2}$ comparative to the FM
$J_{1}$ brings frustration to a honeycomb network, preventing system from
ordering in the Néel-type AF state. Such a situation in an Ising spin system
was theoretically analyzed within classical spin models[24], pointing out that
an introduction of a finite AF interaction between third-nearest neighbors,
$J_{3}$, stabilizes the zig-zag order out of another degenerating phase with a
doubled magnetic unit cell. Thus, the zig-zag type magnetic structure in the
ground state in U2Pt6Ga15 can be basically understood within a classical Ising
spin system, that is characterized by 2D nature with frustration. The negative
in-plane Weiss temperature, $\Theta_{W}{\perp}=-204$ K may be a consequence of
the considerable AF interactions of $J_{2}$ and $J_{3}$, although the
crystalline electric field interaction should be taken into account.

It is worth mentioning the magnetic correlations between honeycomb layers and
the effect of the stacking disorder in U2Pt6Ga15. The magnetic ordering itself
is robust against the disorder. This is demonstrated by the sharp transition
at $T_{\rm N}$ in the bulk properties[15] and the consistency of $T_{\rm N}$
across samples in our RXS and neutron scattering results. The key factor of
the robustness may be the 2D nature in the magnetic interaction, namely, the
order is primarily driven by the intralayer magnetic interactions. This
assumption finds support in the theoretical model that indicates the zig-zag
order does not necessitate interlayer interactions[24]. As suggested by the
positive Weiss temperature along the $c$-axis ($\Theta_{\parallel}=+58$ K)
[15], an interlayer coupling could be FM. Although it is not considered
significant, the FM interlayer interaction can play a supportive role when the
system favors the zig-zag structure as this structure accommodates the highest
number of interlayer FM spin pairs, which maximizes energy gains in the
interlayer coupling. To elucidate overall spin exchange interactions in
details, inelastic neutron scattering experiments to investigate spin
excitation are in progress.

##  V Conclusion

Magnetic order in a honeycomb layered antiferromagnet U2Pt6Ga15 was
investigated using resonant X-ray and neutron scatterings on single
crystalline samples. Magnetic Bragg reflections described by a propagation
vector of $\bm{q}=(1/6,1/6,0)_{\rm hex}$, which is $\bm{q}=(1,0,0)_{\rm orth}$
in the orthorhombic system, were observed in the both RXS and neutron
scattering experiments. The observed magnetic reflections revealed anisotropic
magnetic correlation lengths with relatively short interlayer correlation
reflecting the characteristic crystal structure with a systematic stacking
disorder. The RXS experiment provided evidence of U 5$f$ electron ordering, as
strong resonant scattering at the U $M_{4}$ absorption edge was detected. The
polarization dependence of the RXS signal explicitly indicates the ordered
moments perpendicular to the honeycomb layer, consistent with the Ising-type
anisotropy in the bulk magnetization. The neutron scattering experiments,
followed by group theoretical analysis, successfully determined that the spin
structure in the ground state is of the zig-zag type. The zig-zag structure
and the quasi-two-dimensional correlation length suggest competing in-plane
spin exchange interactions, involving both FM $J_{1}$, and AF $J_{2}$ and
$J_{3}$ interactions, along with relatively weak interlayer coupling in this
compound.

###### Acknowledgements.

We acknowledge valuable discussions with Andrew D. Christianson. We would like
to thank to Yoshinori Kitajima for assistance in using BL-11B of Photon
Factory. This work was supported by JSPS KAKENHI Grants Nos. JP23H04867,
JP23H04871, JP21K14644, JP20H01864, JP21H01027, JP21H04987, JP23H01132, and
JP23K03332. The synchrotron radiation experiments were performed at Photon
Factory with the approval of Photon Factory Program Advisory Committee
(Proposal No. 2020G072).

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# Renewable Energy Management in Smart Home Environment via Forecast Embedded
Scheduling based on Recurrent Trend Predictive Neural Network

Mert Nakıp[cor1]<EMAIL_ADDRESS>Institute of Theoretical and Applied
Informatics, Polish Academy of Sciences (PAN), 44–100 Gliwice, Poland  Onur
Çopur<EMAIL_ADDRESS>Prime Vision, 2600 JA, Delft,Netherlands  Emrah
Biyik<EMAIL_ADDRESS>Department of Energy Systems Engineering,
Yaşar University, 35100, Izmir, Turkey  Cüneyt Güzeliş
<EMAIL_ADDRESS>Department of Electrical and Electronics
Engineering, Yaşar University, 35100, Izmir, Turkey

###### Abstract

Smart home energy management systems help the distribution grid operate more
efficiently and reliably, and enable effective penetration of distributed
renewable energy sources. These systems rely on robust forecasting,
optimization, and control/scheduling algorithms that can handle the uncertain
nature of demand and renewable generation. This paper proposes an advanced ML
algorithm, called Recurrent Trend Predictive Neural Network based Forecast
Embedded Scheduling (rTPNN-FES), to provide efficient residential demand
control. rTPNN-FES is a novel neural network architecture that simultaneously
forecasts renewable energy generation and schedules household appliances. By
its embedded structure, rTPNN-FES eliminates the utilization of separate
algorithms for forecasting and scheduling and generates a schedule that is
robust against forecasting errors. This paper also evaluates the performance
of the proposed algorithm for an IoT-enabled smart home. The evaluation
results reveal that rTPNN-FES provides near-optimal scheduling $37.5$ times
faster than the optimization while outperforming state-of-the-art forecasting
techniques.

###### keywords:

energy management, forecasting, scheduling, neural networks, recurrent trend
predictive neural network

††journal: Applied Energy

##  1 Introduction

Residential loads account for a significant portion of the demand on the power
system. Therefore, intelligent control and scheduling of these loads enable a
more flexible, robust, and economical power system operation. Moreover, the
distributed nature of the local residential load controllers increases system
scalability. On the distribution level, the smart grid benefits from the
increased adoption of residential demand and generation control systems,
because they improve system flexibility, help to achieve a better demand-
supply balance, and enable increased penetration of renewable energy sources.
Increasing flexibility of the building energy demand depends on multiple
developments, including accurate forecasting and effective scheduling of the
loads, incorporation of renewable energy sources such as solar and wind power,
and integration of suitable energy storage technologies (e.g. batteries and/or
electric vehicle charging) into the building energy management system.
Advanced control, optimization and forecasting approaches are necessary to
operate these complex systems seamlessly.

In this paper, in order to address this problem, we propose a novel embedded
neural network architecture, called Recurrent Trend Predictive Neural Network
based Forecast Embedded Scheduling (rTPNN-FES), which simultaneously forecasts
the renewable energy generation and schedules the household appliances
(loads). rTPNN-FES is a unique neural network architecture that enables both
accurate forecasting and heuristic scheduling in a single neural network. This
architecture is comprised of two main layers: 1) the Forecasting Layer which
consists of replicated Recurrent Trend Predictive Neural Networks (rTPNN) with
weight-sharing properties, and 2) the Scheduling Layer which contains parallel
softmax layers with customized inputs each of which is assigned to a single
load. In this paper, we also develop a 2-Stage Training algorithm that trains
rTPNN-FES to learn the optimal scheduling along with the forecasting. However,
the proposed rTPNN-FES architecture does not depend on the particular training
algorithm, and the main contributions and advantages are provided by the
architectural design. Note that the rTPNN model was originally proposed by
Nakıp et al. [1] for multivariate time series prediction, and its superior
performance compared to other ML models was demonstrated when making
predictions based on multiple time series features in the case of multi-sensor
fire detection. On the other hand, rTPNN has not yet been used in an energy
management system and for forecasting renewable energy generation.

Furthermore, the advantages of using rTPNN-FES instead of a separate
forecaster and scheduler are in three folds:

  1. 1.

rTPNN-FES learns how to construct a schedule adapted to forecast energy
generation by emulating (mimicking) optimal scheduling. Thus, the scheduling
via rTPNN-FES is highly robust against forecasting errors.

  2. 2.

The requirements of rTPNN-FES for the memory space and computation time are
significantly lower compared to the combination of a forecaster and an optimal
scheduler.

  3. 3.

rTPNN-FES proposes a considerably high scalability for the systems in which
the set of loads varies over time, e.g. adding new devices into a smart home
Internet of Things (IoT) network.

We numerically evaluate the performance of the proposed rTPNN-FES architecture
against 7 different well-known ML algorithms combined with optimal scheduling.
To this end, publicly available datasets [2, 3] are utilized for a smart home
environment with 12 distinct appliances. Our results reveal that the proposed
rTPNN-FES architecture achieves significantly high forecasting accuracy while
generating a close-to-optimal schedule over a period of one year. It also
outperforms existing techniques in both forecasting and scheduling tasks.

The remainder of this paper is organized as follows: Section 2 reviews the
differences between this paper and the state-of-the-art. Section 3 presents
the system set-up and initiates the optimization problem. Section 4 presents
the rTPNN-FES architecture and the 2-Stage Training algorithm which is used to
learn and emulate the optimal scheduling. Section 5 presents the performance
evaluation and comparison. Finally, Section 6 summarizes the main
contributions of this paper.

##  2 Related Works

In this section, we present the comparison of this paper with the-state-of-the
art works in three categories: 1) The works in the first category develop an
optimization-based energy management system without interacting with ML. 2)
The works in the second category focus on forecasting renewable energy
generation using either statistical or deep learning techniques. 3) The works
in the last category develop energy management systems using ML algorithms.

###  2.1 Optimization-based Energy Management Systems

We first review the recent works which developed optimization-based energy
management systems. In [4], Shareef et al. gave a comprehensive summary of
heuristic optimization techniques used for home energy management systems. In
[5], Nezhad et al. presented a model predictive controller for a home energy
management system with loads, photovoltaic (PV) and battery electric storage.
They formulated the MPC as a mixed-integer programming problem and evaluated
its economic performance under different energy pricing schemes. In [6],
Albogamy et al. utilized Lyapunov-based optimization to regulate HVAC loads in
a home with battery energy storage and renewable generation. In [7], S. Ali et
al. considered heuristic optimization techniques to develop a demand response
scheduler for smart homes with renewable energy sources, energy storage, and
electric and thermal loads. In [8], G. Belli et al. resorted to mixed integer
linear programming for optimal scheduling of thermal and electrical appliances
in homes within a demand response framework. They utilized a cloud service
provider to compute and share aggregate data in a distributed fashion. In [9],
variants of several heuristic optimization methods (optimal stopping rule,
particle swarm optimization, and grey wolf optimization) were applied to the
scheduling of home appliances under a virtual power plant framework for the
distribution grid. Then, their performance was compared for three types of
homes with different demand levels and profiles.

There is a wealth of research on optimization and model predictive controller-
based scheduling of residential loads. In this literature, usually, prediction
of the load demand and generation (if available) are pursued independently
from the scheduling algorithm and are merely used as a constraint parameter in
the optimization problem. The discrepancy in predicted and observed demand and
generation may lead to poor performance and robustness issues. The proposed
rTPPN-FES in this paper handles forecast and scheduling in a unified way and,
therefore, provides robustness in the presence of forecasting errors.

###  2.2 Forecasting of Renewable Energy Generation

We now briefly review the related works on forecasting renewable energy
generation, which have also been reviewed in more detail in the literature,
i.e. [10, 11].

The earlier research in this category forecast energy generation using
statistical methods. For example, in [12], Kushwaha et al. use the well-known
seasonal autoregressive integrated moving average technique to forecast the PV
generation in 20-minute intervals. In [13], Rogier et al. evaluated the
performance of a nonlinear autoregressive neural network on forecasting the PV
generation data collected through a LoRa-based IoT network. In [14], Fentis et
al. used Feed Forward Neural Network and Least Square Support Vector
Regression with exogenous inputs to perform short-term forecasting of PV
generation. In [15] analyzed the performances of (Autoregressive Integrated
Moving Average) ARIMA and Artificial Neural Network (ANN) for forecasting the
PV energy generation. In [16], Atique et al. used ARIMA with parameter
selection based on Akaike information criterion and the sum of the squared
estimate to forecast PV generation. In [17], Erdem and Shi analyzed the
performance of autoregressive moving averages to forecast wind speed and
direction in four different approaches such as decomposing the lateral and
longitudinal components of the speed. In [18], Cadenas et al. performed a
comparative study between ARIMA and nonlinear autoregressive exogenous
artificial neural network on the forecasting wind speed.

The recent trend of research focuses on the development of ML and (neural
network-based) deep learning techniques. In [19], Pawar et al. combined ANN
and Support Vector Regressor (SVR) to predict renewable energy generated via
PV. In [20], Corizzo et al. forecast renewable energy using a regression tree
with an adopted Tucker tensor decomposition. In [21] forecast the PV
generation based on the historical data of some features such as irradiance,
temperature and relative humidity. In [22], Shi et al. proposed a pooling-
based deep recurrent neural network technique to prevent overfitting for
household load forecast. In [23], Zheng et al. developed an adaptive neuro-
fuzzy system that forecasts the generation of wind turbines in conjunction
with the forecast of weather features such as wind speed. In [24], Vandeventer
et al. used a genetic algorithm to select the parameters of SVM to forecast
residential PV generation. In [25], van der Meer et al. performed a
probabilistic forecast of solar power using quantile regression and dynamic
Gaussian process. In [26], He and Li have combined quantile regression with
kernel density estimation to predict wind power density. In [27], Alessandrini
et al. used an analogue ensemble method to problematically forecast wind
power. In [28], Cervone et al. combined ANN with the analogue ensemble method
to forecast the PV generations in both deterministic and probabilistic ways.
Recently in [29], Guo et al. proposed a combined load forecasting method for a
Multi Energy Systems (MES) based on Bi-directional Long Short-Term Memory
(BiLSTM). The combined load forecasting framework is trained with a multi-
tasking approach for sharing the coupling information among the loads.

Although there is a significantly large number of studies to forecast
renewable energy generation and/or other factors related to generation, this
paper differs sharply from the existing literature as it proposes an embedded
neural network architecture called rTPNN-FES that performs both forecasting
and scheduling simultaneously.

###  2.3 Machine Learning Enabled Energy Management Systems

In this category, we review the recent studies that aim to develop energy
management systems enabled by ML, especially for residential buildings.

The first group of works in this category used scheduling (based on either
optimization or heuristic) using the forecasts provided by an ML algorithm. In
[30], Elkazaz et al. developed a heuristic energy management algorithm for
hybrid systems using an autoregressive ML for forecasting and optimization for
parameter settings. In [31], Zaouali et al. developed an auto-configurable
middle-ware using Long-Short Term Memory (LSTM) based forecasting of renewable
energy generated via PV. In [32], Shakir et al. developed a home energy
management system using LSTM for forecasting and Genetic Algorithm for
optimization. In [33], Manue et al. used LSTM to forecast the load for battery
utilization in a solar system in a smart home system. In [34] developed a
hybrid system of renewable and grid-supplied energy via exponential weighted
moving average-based forecasting and a heuristic load control algorithm. In
[35], Aurangzeb et al. developed an energy management system which uses a
convolutional neural network to forecast renewable energy generation. Finally,
in [36], in order to distribute the load and decrease the costs, Sarker et al.
developed a home energy management system based on heuristic scheduling.

The second group of works in this category developed energy management systems
based on reinforcement learning. In [37], Ren et al. developed a model-free
Dueling-double deep Q-learning neural network for home energy management
systems. In [38], Lissa et al. used ANN-based deep reinforcement learning to
minimize energy consumption by adjusting the hot water temperature in the PV-
enabled home energy management system. In [39], Yu et al. developed an energy
management system using a deep deterministic policy gradient algorithm. In
[40], Wan et al. used a deep reinforcement learning algorithm to learn the
energy management strategy for a residential building. In [41], Mathew et al.
developed a reinforcement learning-based energy management system to reduce
both the peak load and the electricity cost. In [42], Liu et al. developed a
home energy management system using deep and double deep Q-learning techniques
for scheduling home appliances. In [43], Lu et al. developed an energy
management system with hybrid CNN-LSTM based forecasting and rolling horizon
scheduling. In [44], Ji et al. developed a microgrid energy management system
using the Markov decision process for modelling and ANN-based deep
reinforcement learning for determining actions.

Deep learning-based control systems are also very popular for off-grid
scenarios, as off-grid energy management systems are gaining increasing
attention to provide sustainable and reliable energy services. In References
[45] and [46], the authors developed algorithms based on deep reinforcement to
deal with the uncertain and stochastic nature of renewable energy sources.

All of these works have used ML techniques, especially deep learning and
reinforcement learning, to build energy management systems. Moreover, in a
recent work [47], Nakıp et al. mimicked the scheduling via ANN and developed
an energy management system using this ANN-based scheduling. However, in
contrast with rTPNN-FES proposed in this paper, none of them has used ANN to
generate scheduling or combined forecasting and scheduling in a single neural
network architecture.

##  3 System Setup and Optimization Problem

Figure 1: The illustration of the system considered by rTPNN-FES

In this section, we present the assumptions, mathematical definitions and the
optimization problem related to the system setup which is used for embedded
forecasting scheduling via rTPNN-FES and shown in Figure 1. During this paper,
rTPNN-FES is assumed to perform at the beginning of a scheduling window that
consists of equal-length $S$ slots and has a total duration of $H$ in actual
time (i.e. the horizon length). In addition, the length of each slot $s$
equals $H/S$, and the actual time instance at which the slot $s$ starts is
denoted by $m_{s}$. Then, we let $g^{m_{s}}$ denote the power generation by
the renewable energy source within slot $s$. Also, $\hat{g}^{m_{s}}$ denotes
the forecast of $g^{m_{s}}$.

We let $\mathcal{N}$ be the set of devices that need to be scheduled until $H$
(in other words until the end of slot $S$), and $N$ denote the total number of
devices, i.e. $|\mathcal{N}|=N$. Each device $n\in\mathcal{N}$ has a constant
power consumption per slot denoted by $E_{n}$. In addition, $n$ should be
active uninterruptedly for $a_{n}$ successive slots. That is, when $n$ is
started, it consumes $a_{n}E_{n}$ until it stops. Moreover, we assume that the
considered renewable energy system contains a battery with a capacity of
$B_{max}$, where the stored energy in this battery is used via an inverter
with a supply limit of $\Theta$. We assume that there is enough energy in
total (the sum of the stored energy in the battery and total generation) to
supply all devices within $[0,H]$.

At the beginning of the scheduling window, we forecast the renewable energy
generation and schedule the devices accordingly. To this end, as the main
contribution of this paper, we combine the forecaster and scheduler in a
single neural network architecture, called rTPNN-FES, which shall be presented
in Section 4.

Optimization Problem: We now define the optimization problem for the non-
preemptive scheduling of the starting slots of devices to minimize _user
dissatisfaction_. In other words, this optimization problem aims to distribute
the energy consumption over slots prioritizing “user satisfaction”, assuming
that the operation of each device is uninterruptible. In this article, we
consider a completely off-grid system –which utilizes only renewable energy
sources– where it is crucial to achieve near-optimal scheduling to use limited
available resources. Recall that this optimization problem is re-solved at the
beginning of each scheduling window for the available set of devices
$\mathcal{N}$ using the forecast generation $\hat{g}^{m_{s}}$ over the
scheduling window in Figure 1.

Moreover, for each $n\in\mathcal{N}$, there is a predefined cost of user
dissatisfaction, denoted by $c_{(n,s)}$, for scheduling the start of $n$ at
slot $s$. This cost can take value in the range of $[0,+\infty)$, and
$c_{(n,s)}$ set to $+\infty$ if the user does not want slot $s$ to be reserved
for device $n$. As we shall explain in more detail in Section 5, we determine
the user dissatisfaction cost $c_{(n,s)}$ as the increasing function of the
distance between $s$ and the desired start time of the considered device $n$.
We should note that the definition of the user dissatisfaction cost only
affects the numerical results since the proposed rTPNN-FES methodology does
not depend on its definition.

Then, we let $x_{(n,s)}$ denote a binary schedule for the start of the
activity of device $n$ at slot $s$. That is, $x_{(n,s)}=1$ if device $n$ is
scheduled to start at the beginning of slot $s$, and $x_{(n,s)}=0$ otherwise.
In addition, in our optimization program, we let $x^{*}_{(n,s)}$ be a binary
decision variable and denote the optimal value of $x_{(n,s)}$. Accordingly, we
define the optimization problem as follows:

$min~{}~{}\sum_{n\in\mathcal{N}}\sum_{s=1}^{S}x^{*}_{(n,s)}c_{(n,s)}$ (1)

subject to

$\displaystyle\sum_{s=1}^{S-(a_{n}-1)}x^{*}_{(n,s)}=1,\qquad\forall
n\in\mathcal{N}$ (2)
$\displaystyle\sum_{n\in\mathcal{N}}\sum_{s^{\prime}=[s-(a_{n}-1)]^{+}}^{s}E_{n}x^{*}_{(n,s^{\prime})}\leq\Theta,\quad\forall
s\in\\{1,\dots,S\\}$ (3)
$\displaystyle\sum_{n\in\mathcal{N}_{i}}\sum_{s^{\prime}=[s-(a_{n}-1)]^{+}}^{s}E_{n}x^{*}_{(n,s^{\prime})}\leq\hat{g}^{m_{s}}+B_{max},$
(4) $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\forall
s\in\\{1,\dots,S\\}$
$\displaystyle\sum_{n\in\mathcal{N}}\sum_{s^{\prime}=1}^{s}\sum_{s^{\prime\prime}=[s^{\prime}-(a_{n}-1)]^{+}}^{s^{\prime}}E_{n}x^{*}_{(n,s^{\prime\prime})}\leq
B+\sum_{s^{\prime}=1}^{s}\hat{g}^{m_{s^{\prime}}},$ (5)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\forall s\in\\{1,\dots,S\\}$

where $[\Xi]^{+}=\Xi$ if $\Xi\geq 1$; otherwise, $[\Xi]^{+}=1$. The objective
function (1) minimizes the total user dissatisfaction cost over all devices as
($\sum_{n\in\mathcal{N}}\sum_{s=1}^{S}x^{*}_{(n,s)}c_{(n,s)}$). While
minimizing user dissatisfaction, the optimization problem also considers the
following constraints:

  * •

Uniqueness and Operation constraint in (2) ensures that each device $n$ is
scheduled to start exactly at a single slot between $1$-st and
$[S-(a_{n}-1)]$-th slot. The upper limit for the starting of the operation of
device $n$ is set to $[S-(a_{n}-1)]$ because $n$ must operate for successive
$a_{n}$ slots before the end of the last slot $S$.

  * •

Inverter Limitation constraint in (3) limits total power consumption at each
slot $s$ to the maximum power of $\Theta$ that can be provided by the
inverter. Note that the term
$\sum_{s^{\prime}=s-(a_{n}-1)}^{s}x^{*}_{(n,s^{\prime})}$ is a convolution
which equals $1$ if device $n$ is scheduled to be active at slot $s$ (i.e. $n$
is scheduled to start between $s-(a_{n}-1)$ and $s$).

  * •

Maximum Storage constraint in (4) ensures that the scheduled consumption at
each slot $s$ does not exceed the sum of the predicted generation
($\hat{g}^{m_{s}}$) at this slot and the maximum energy ($B_{max}$) that can
be stored in the battery.

  * •

Total Consumption constraint in (5) ensures that the scheduled total power
consumption until each slot $s$ is not greater than the summation of the
stored energy, $B$, at the beginning of the scheduling window and the total
generation until $s$. This constraint is used as we are considering a
completely off-grid system.

##  4 Recurrent Trend Predictive Neural Network based Forecast Embedded
Scheduling (rTPNN-FES)

Figure 2: Recurrent Trend Predictive Neural Network based Forecast Embedded
Scheduling (rTPNN-FES)

In this section, we present our rTPNN-FES neural network architecture. Figure
2 displays the architectural design of rTPNN-FES which aims to generate
scheduling for the considered window while forecasting the power generation
through this window automatically and simultaneously. To this end, rTPNN-FES
is comprised of two main layers of “Forecasting Layer” and “Scheduling Layer”,
and it is trained using the “2-Stage Training Procedure”.

We let $\mathcal{F}$ be the set of features and
$\mathcal{F}\equiv\\{1,\dots,F\\}$. In addition, $z_{f}^{m_{s}}$ denotes the
value of input feature $f$ in slot $s$ which starts at $m_{s}$, where this
feature can be considered as any external data , such as weather predictions,
that are directly or indirectly related to power generation $g^{m_{s}}$. We
also let $\tau_{f}$ be a duration of time when the system developer has
observed that the feature $f$ has periodicity; $\tau_{0}$ represents the
periodicity duration for $g^{m_{s}}$. Note that we do not assume that the
features will have a periodic nature. If there is no observed periodicity,
$\tau_{f}$ can be set to $H$.

As shown in Figure 2, the inputs of rTPNN-FES are
$\\{g^{m_{s}-2\tau_{0}},g^{m_{s}-\tau_{0}}\\}$ and
$\\{z^{m_{s}-2\tau_{f}}_{f},z^{m_{s}-\tau_{f}}_{f}\\}$ for $f\in\mathcal{F}$,
and the output of that is
$\\{x_{n,s}\\}_{n\in\\{1,\dots,N\\}}^{s\in\\{1,\dots,S\\}}$.

###  4.1 Forecasting Layer

Forecasting Layer is responsible for forecasting the power generation within
the architecture of rTPNN-FES. For each slot $s$ in the scheduling window,
rTPNN-FES forecasts the renewable energy generation $\hat{g}^{m_{s}}$ based on
the collection of the past feature values for two periods,
$\\{z^{m_{s}-2\tau_{f}}_{f},z^{m_{s}-\tau_{f}}_{f}\\}_{f\in\mathcal{F}}$, as
well as the past generation for two periods
$\\{g^{m_{s}-2\tau_{0}},g^{m_{s}-\tau_{0}}\\}$. To this end, this layer
consists of $S$ parallel rTPNN models that share the same parameter set
(connection weights and biases). That is, in this layer, there are $S$
replicas of a single trained rTPNN; in other words, one may say that a single
rTPNN is used with different inputs to forecast the traffic generation for
each slot $s$. Therefore, all but one of the Trained rTPNN blocks are shown as
transparent in Figure 2.

The weight sharing among rTPNN models (i.e. using replicated rTPNNs) has the
following advantages:

  * •

The number of parameters in the Forecasting Layer decreases by a factor of
$S$; thus reducing both time and space complexity.

  * •

By avoiding rTPNN training repeated $S$ times, the training time is also
reduced by a factor of $S$.

  * •

Because a single rTPNN is trained on the data collected over $S$ different
slots, the rTPNN can now capture recurrent trends and relationships with
higher generalization ability.

####  4.1.1 Structure of rTPNN

Figure 3: The structure of rTPNN used in rTPNN-FES

We now briefly explain the structure of rTPNN, which has been originally
proposed in [1], for our rTPNN-FES neural network architecture. As shown in
Figure 3 displaying the structure of rTPNN, for any $s$, the inputs of rTPNN
are $\\{g^{m_{s}-2\tau_{0}},g^{m_{s}-\tau_{0}}\\}$ and
$\\{z^{m_{s}-2\tau_{f}}_{f},z^{m_{s}-\tau_{f}}_{f}\\}$ for $f\in\mathcal{F}$,
and the output is $\hat{g}^{m_{s}}$. In addition, the rTPNN architecture
consists of $(F+1)$ Data Processing (DP) units and $L$ fully connected layers,
including the output layer.

####  4.1.2 DP units

In the architecture of rTPNN, there is one DP unit either for the past values
of energy generation, denoted by $\textrm{DP}_{0}$ or for each time series
feature $f$, denoted by $\textrm{DP}_{f}$. That is, $\textrm{DP}_{f}$ for any
feature $f$ (including $f=0$) has the same structure but its corresponding
input is different for each $f$. For example, the input of $\textrm{DP}_{f}$
is $\\{z_{f}^{m_{s}-2\tau_{f}},z_{f}^{m_{s}-\tau_{f}}\\}$ corresponding to any
time series feature $f\in\\{1,\dots,F\\}$ while the input of $\textrm{DP}_{0}$
is the past values of energy generation
$\\{g^{m_{s}-2\tau_{0}},g^{m_{s}-\tau_{0}}\\}$. Thus, one may notice that
$\textrm{DP}_{0}$ is the only unit with a special input.

During the explanation of the DP unit, we focus on a particular instance
$\textrm{DP}_{f}$, which is also shown in detail in Figure 3. Using
$\\{z_{f}^{m_{s}-2\tau_{f}},z_{f}^{m_{s}-\tau_{f}}\\}$ input pair,
$\textrm{DP}_{f}$ aims to learn the relationship between this pair and each of
the predicted trend $t_{f}^{s}$ and the predicted level $l_{f}^{s}$. To this
end, $\textrm{DP}_{f}$ consists of Trend Predictor and Level Predictor sub-
units each of which is a linear recurrent neuron.

As shown in Figure 3, Trend Predictor of $\textrm{DP}_{f}$ computes the
weighted sum of the change in the value of feature $f$ from $m_{s}-2\tau_{f}$
to $m_{s}-\tau_{f}$ and the previous value of the predicted trend. That is,
$\textrm{DP}_{f}$ calculates the sum of the difference between
$(z_{f}^{m_{s}-\tau_{f}}-z_{f}^{m_{s}-2\tau_{f}})$ with connection weight of
$\alpha^{1}_{f}$ and the previous value of the predicted trend $t_{f}^{s-1}$
with the connection weight of $\alpha^{2}_{f}$ as

$t_{f}^{s}=\alpha^{1}_{f}\,(z_{f}^{m_{s}-\tau_{f}}-z_{f}^{m_{s}-2\tau_{f}})+\alpha^{2}_{f}\,t_{f}^{s-1}$
(6)

By calculating the trend of a feature and learning the parameters in (6),
rTPNN is able to capture behavioural changes over time, particularly those
related to the forecasting of $\hat{g}^{m_{s}}$.

Level Predictor sub-unit of $\textrm{DP}_{f}$ predicts the level of feature
value, which is the smoothed version of the value of feature $f$, using only
$z_{f}^{m_{s}-\tau_{f}}$ and the previous state of the predicted level
$l_{f}^{s-1}$. To this end, it computes the sum of the
$z_{f}^{m_{s}-\tau_{f}}$ and $l_{f}^{s-1}$ with weights of $\beta^{1}_{f}$ and
$\beta^{2}_{f}$ respectively as

$l_{f}^{s}=\beta^{1}_{f}\,z_{f}^{m_{s}-\tau_{f}}+\beta^{2}_{f}\,l_{f}^{s-1}$
(7)

By predicting the level, we can reduce the effects on the forecasting of any
anomalous instantaneous changes in the measurement of any other feature $f$.

Note that parameters $\alpha^{1}_{f}$, $\alpha^{2}_{f}$, $\beta_{f}^{1}$ and
$\beta_{f}^{2}$ of Trend Predictor and Level Predictor sub-units are learned
during the rTPNN training like all other parameters (i.e. connection weights).

####  4.1.3 Feed-forward of rTPNN

We now describe the calculations performed during the execution of the rTPNN;
that is, when making a prediction via rTPNN. To this end, first, let
$\mathbf{W}_{l}$ denote the connection weight matrix for the inputs of hidden
layer $l$, and $\mathbf{b}_{l}$ denote the vector of biases of $l$. Thus, for
each $s$, the forward pass of rTPNN is as follows:

  1. 1.

Trend Predictors of $\textrm{DP}_{0}$-$\textrm{DP}_{F}$:

$\displaystyle
t_{0}^{s}=\alpha^{1}_{0}(g^{m_{s}-\tau_{0}}-g^{m_{s}-2\tau_{0}})+\alpha^{2}_{0}t_{0}^{s-1},$
$\displaystyle
t_{f}^{s}=\alpha^{1}_{f}(z_{f}^{m_{s}-\tau_{f}}-z_{f}^{m_{s}-2\tau_{f}})+\alpha^{2}_{f}t_{f}^{s-1},\quad\forall
f\in\mathcal{F}$ (8)

  2. 2.

Level Predictors of $\textrm{DP}_{0}$-$\textrm{DP}_{F}$:

$\displaystyle
l_{0}^{s}=\beta^{1}_{0}g^{m_{s}-\tau_{0}}+\beta^{2}_{0}l_{0}^{s-1},$
$\displaystyle
l_{f}^{s}=\beta^{1}_{f}z_{f}^{m_{s}-\tau_{0}}+\beta^{2}_{f}l_{f}^{s-1},\qquad\forall
f\in\mathcal{F}$ (9)

  3. 3.

Concatenation of the outputs of $\textrm{DP}_{0}$-$\textrm{DP}_{F}$ to feed to
the hidden layers:

$\displaystyle\mathbf{z}^{s}=[t_{0}^{s},l_{0}^{s},g^{m_{s}-\tau_{0}},\dots,t_{F}^{s},l_{F}^{s},z_{F}^{m_{s}-\tau_{F}}]$
(10)

  4. 4.

Hidden Layers from $l=1$ to $l=L$:

$\displaystyle\mathbf{O}^{s}_{1}=\Psi(\mathbf{W}_{1}{(\mathbf{z}^{s})}^{T}+\mathbf{b}_{1}),$
(11)
$\displaystyle\mathbf{O}^{s}_{l}=\Psi(\mathbf{W}_{l}\mathbf{O}^{s}_{l-1}+\mathbf{b}_{l}),\quad\forall
l\in\\{2,\dots,L-1\\}$ (12)
$\displaystyle\hat{g}^{m_{s}}=\Psi(\mathbf{W}_{L}\mathbf{O}^{s}_{L-1}+\mathbf{b}_{L}),$
(13)

where ${(\mathbf{z}^{s})}^{T}$ is the transpose of the input vector
$\mathbf{z}^{s}$, $\mathbf{O}^{s}_{l}$ is the output vector of hidden layer
$l$, and $\Psi(\cdot)$ denotes the activation function as an element-wise
operator.

###  4.2 Scheduling Layer

The Scheduling Layer consists of $N$ parallel softmax layers, each responsible
for generating a schedule for a single device’s start time. A single softmax
layer for device $n$ is shown in Figure 4. Since this layer is cascaded behind
the Forecasting Layer, each device $n$ is scheduled to be started at each slot
$s$ based on the output of the Forecasting Layer $\hat{g}^{m_{s}}$ as well as
the system parameters $c_{(n,s)}$, $E_{n}$, $B$, $B_{max}$ and $\Theta$ for
this device $n$ and this slot $s$.

Figure 4: The structure of Scheduling Layer

In Figure 4, each arrow represents a connection weight. Accordingly, for
device $n$ for slot $s$ in a softmax layer of the Scheduling Layer, a neuron
first calculates the weighted sum of the inputs as

$\displaystyle\alpha_{(n,s)}$ $\displaystyle=$ $\displaystyle
w^{g}_{(n,s)}g^{m_{s}}+w^{B}_{(n,s)}\frac{B}{S}-w^{c}_{(n,s)}c_{(n,s)}$
$\displaystyle-w^{E}_{(n,s)}E_{n}-w^{\Theta}_{(n,s)}\Theta-w^{B_{max}}_{(n,s)}B_{max}$

where all connection weights of $w^{g}_{(n,s)}$, $w^{B}_{(n,s)}$,
$w^{c}_{(n,s)}$, $w^{E}_{(n,s)}$, $w^{\Theta}_{(n,s)}$, and
$w^{B_{max}}_{(n,s)}$ are _strictly positive_. In addition, the signs of the
terms are determined considering the intuitive effect of the parameter on the
schedule decision for device $n$ at slot $s$. For example, the higher
$g^{m_{s}}$ makes slot $s$ a better candidate to schedule $n$, while the
higher user dissatisfaction cost $c_{(n,s)}$ makes slot $s$ a worse candidate.
In addition, a softmax activation is applied at the output of this neuron:

$x_{(n,s)}~{}=~{}\Phi(\alpha_{(n,s)})~{}=~{}\frac{e^{\alpha_{(n,s)}}}{\sum_{s=1}^{S}\alpha_{(n,s)}}$
(15)

###  4.3 2-Stage Training Procedure

We train our rTPNN-FES architecture to learn the optimal scheduling of devices
as well as the forecasting of energy generation in a single neural network. To
this end, we first assume that there is a collected dataset comprised of the
actual values of $g^{m_{s}}$ and $\\{z_{f}^{m_{s}}\\}_{f\in\mathcal{F}}$ for
$s\in\\{1,\dots,S\\}$ for multiple scheduling windows. Note that rTPNN-FES
does not depend on the developed 2-stage training procedure, so it can be used
with any training algorithm. For each window in this dataset, the 2-stage
procedure works as follows:

####  4.3.1 Stage 1 - Training of rTPNN Separately for Forecasting

In this first stage of training, in order to create a forecaster, the rTPNN
model (Figure 3) is trained separately from the rTPNN-FES architecture (Figure
2). To this end, the deviation of $\hat{g}^{m_{s}}$ from $g^{m_{s}}$ for
$s\in\\{1,\dots,S\\}$, i.e. the forecasting error of rTPNN, is measured via
Mean Squared Error as

$MSE_{\textrm{forecast}}\equiv\frac{1}{S}\sum_{s=1}^{S}(g^{m_{s}}-\hat{g}^{m_{s}})^{2}$
(16)

We update the parameters (connection weights and biases) of rTPNN via back-
propagation with gradient descent, in particular the Adam algorithm, to
minimize $MSE_{\textrm{forecast}}$, where the initial parameters are set to
parameters found in previous training. We repeat updating parameters as many
epochs as required without over-fitting to the training samples.

When Stage 1 is completed, the parameters of “Trained rTPNN” in Figure 2 are
replaced by the resulting parameters found in this stage. Then, the parameters
of Trained rTPNN are frozen to continue further training of rTPNN-FES in Stage
2. That is, the parameters of Trained rTPNN are not updated in Stage 2.

####  4.3.2 Stage 2 - Training of rTPNN-FES for Scheduling

In Stage 2 of training, in order to create a scheduler emulating optimization,
the rTPNN-FES architecture (Figure 2) is trained following the steps shown in
Figure 5.

Figure 5: The steps in Stage 2 training of rTPNN-FES to learn to schedule

The steps in Stage 2 shown in Figure 5 are as follows:

  1. 1

The optimal schedule,
$\\{x_{n,s}^{*}\\}_{n\in\\{1,\dots,N\\}}^{s\in\\{1,\dots,S\\}}$ is computed by
solving the optimization problem given in Section 3 in (1)-(5).

  2. 2

The feed-forward output of rTPNN-FES,
$\\{x_{n,s}\\}_{n\in\\{1,\dots,N\\}}^{s\in\\{1,\dots,S\\}}$, which is the
estimation of scheduling, is computed through (6)-(15) using the architecture
in Figure 2.

  3. 3

The performance of rTPNN-FES for scheduling, i.e. total estimation error of
rTPNN-FES, is measured via Categorical Cross-Entropy as

$CCE_{\textrm{schedule}}\equiv-\sum_{n=1}^{N}\sum_{s=1}^{S}x_{n,s}^{*}\log(x_{n,s})$
(17)

  4. 4

The parameters (connection weights and biases) in the “Scheduling Layers” of
rTPNN-FES are updated via back-propagation with gradient decent (using Adam
optimization algorithm) to minimize $CCE_{\textrm{schedule}}$.

As soon as this training procedure is completed, i.e. during real-time
operation, rTPNN-FES generates both forecasts of renewable energy generations,
$\\{\hat{g}^{m_{s}}\\}_{s\in\\{1,\dots,S\\}}$ and a schedule
$\\{x_{n,s}\\}_{{n\in\\{1,\dots,N\\}}}^{s\in\\{1,\dots,S\\}}$ that emulates
the optimization.

##  5 Results

In this section, we aim to evaluate the performance of our rTPNN-FES. To this
end, during this section, we first present the considered datasets and hyper-
parameter settings. We also perform a brief time-series data analysis aiming
to determine the most important features for the forecasting of PV energy
generation. Then, we numerically evaluate the performance of our technique and
compare that with some existing techniques.

###  5.1 Methodology of Experiments

####  5.1.1 Datasets

For the performance evaluation of the proposed rTPNN-FES, we combine two
publicly available datasets [2] and [3]. The first dataset [2] consists of
hourly solar power generation (kW) of various residential buildings in
Konstanz, Germany between 22-05-2015 and 12-03-2017. Within this dataset, we
consider only the residential building called “freq_DE_KN_residential1_pv”
which corresponds to 15864 samples in total. The second dataset contains
weather-related information which is scraped with World Weather Online (WWO)
API [3]. This API provides 19 features related to temperature, precipitation,
illumination and wind.

####  5.1.2 Experimental Set-up

Table 1: Household Appliances in the Smart Home Environment Appliance Name | Power Consumption (kW) | Active Duration | Desired Start Time
---|---|---|---
Washing Machine (warm wash) | 2.3 | 2 | 14
Dryer (avg. load) | 3 | 2 | 16 (earliest 15)
Robot Vacuum Cleaner | 0.007 | 2 | 15
Iron | 1.08 | 2 | 8
TV | 0.15 | 3 | 20
Refrigerator | 0.083 | 24 | non-stop
Oven | 2.3 | 1 | 18
Dishwasher | 2 | 2 | 21
Electric Water Heater | 0.7 | 1 | 6, 17
Central AC | 3 | 2 | 6, 18
Pool Filter Pump | 1.12 | 8 | 10
Electric Vehicle Charger | 7.7 | 8 | 21 (earliest 18 latest 23)

Considering the limitations of the available dataset, we perform our
experiments on a virtual residential building which is, each year, actively
used between May and September. It is assumed that there are 12 different
smart home appliances in active months. These appliances are shown in Table 1,
where each appliance should operate at least once a day. Note that Electric
Water Heater and Central AC operate twice a day, where the desired start times
are 6:00 and 17:00 for the heater, and 6:00 and 18:00 for the AC. In order to
produce sufficient energy for the operation of these appliances, this building
has its own PV system which consists of the following elements: 1) PV panels
for which the generations are taken from the dataset [2] explained above, 2)
three batteries with $13.5$ kWh capacity of each, and 3) inverter with a power
rate of 10kW.

Furthermore, during our experimental work, we set $H=24~{}h$, and we define
the user dissatisfaction cost $c_{(n,s)}$ for each device $n$ at each slot $s$
based on the “Desired Start Time”, which is given in Table 1, as

$c_{(n,s)}=1-\frac{1}{\sigma_{n}\sqrt{2\pi}}\,\exp\Bigg{(}-\frac{1}{2}\,\bigg{(}\frac{s-\mu_{n}}{\sigma_{n}}\bigg{)}^{2}\Bigg{)}$
(18)

where $\mu_{n}$ is the desired start time of $n$, and $\sigma_{n}$ is the
acceptable variance for the start of $n$. The value of $\sigma_{n}$ is $1$ for
Iron and Electric Water Heater, $2$ for TV, Oven, Dishwasher and AC, $3$ for
Washing Machine and Dryer, and $5$ for Robot Vacuum Cleaner. Also, the value
of $c_{(n,s)}$ is set to infinity for $s$ lower than the earliest start time
and for that greater than the latest start time.

Recall that the Water Heater and AC, which are activated twice a day, are
modelled as two separate devices.

####  5.1.3 Implementation and Hyper-Parameter Settings for rTPNN-FES

We implemented rTPNN-FES by using Keras API on Python 3.7.13. The experiments
are executed on the Google Colab platform with an operating system of Linux
5.4.144 and a 2.2GHz processor with 13 GB RAM.

Forecasting Layer is trained on this platform via the adam optimizer for 40
epochs with $10^{-3}$ initial learning rate. In order to exploit the PV
generation trend on daily basis, the batch size is fixed at $24$. Moreover, an
$L_{2}$ regularization term is injected into Trend and Level Predictors in the
rTPNN layer in order to avoid gradient vanishing. Finally, we used fully
connected layers of rTPNN which are respectively comprised of $F+1$ and
$\lceil(F+1)/2\rceil$ neurons with sigmoid activation. Scheduling Layer of
each device is trained on the same platform also using the adam optimizer for
20 epochs with a batch size of $1$ and initial learning rate of $10^{-3}$.
Note that setting the batch size to $1$ is due to the particular
implementation of rTPNN-FES which uses the Keras library. In addition, the
infinity values of $c_{(n,s)}$ are set to $100$ at the inputs of the
scheduling layer in order to be able to calculate the neuron activation. We
also set the periodicity $\tau_{0}$ of $g^{m_{s}}$ as $24~{}h$.

Furthermore, the source codes of the rTPNN-FES and experiments in this paper
are shared in [48] in addition to the repository of the original rTPNN.

####  5.1.4 Genetic Algorithm-based Scheduling for Comparison

Genetic algorithms (GAs) have been widely used in scheduling tasks due to
their ability to effectively solve complex optimization problems. GAs are able
to incorporate various constraints and prior knowledge into the optimization
process, making them well-suited for scheduling tasks with many constraints.
GAs are also able to efficiently search through a vast search space to find
near-optimal solutions, even for problems with a large number of variables
[49]. These characteristics make GAs powerful tools for finding high-quality
solutions in our experimental setup and good candidates to compare against
rTPNN-FES.

The experiments are executed on the Google Colab platform with the same
hardware configurations of rTPNN-FES. In this experimental setting, a
chromosome is a daily schedule matrix. The cross-over is made by swapping
device schedules by selecting a random cross-over point out of the total
number of devices and mutation is introduced by changing the scheduled time of
a single device randomly with probability $0.1$. The GA application starts
with sampling feasible solutions out of 5000 random solutions as an initial
population. After that, 1000 new generations are simulated while the
population size is fixed to 200 by making selections in an elitist style.

###  5.2 Forecasting Performance of rTPNN-FES

We now compare the forecasting performance of rTPNN with the performances of
LSTM, MLP, Linear Regression, Lasso, Ridge, ElasticNet, Random Forest as well
as 1-Day Naive Forecast.1111-Day Naive Forecast equals to the original time
series with 1-day lag. Recall that in recent literature, References [31, 32,
33] used LSTM, and Reference [14, 15, 19, 38] used MLP.

During our experimental work, the dataset is partitioned into training and
test sets with the first 300 days (corresponding to 7200 samples) and the rest
361 days (corresponding to 8664 samples) respectively.

First, Table 2 presents the performances of all models on both training and
test sets with respect to Mean Squared Error (MSE), Mean Absolute Error (MAE),
Mean Absolute Percentage Error (MAPE) and Symmetric Mean Absolute Percentage
Error (SMAPE) metrics, which are calculated as

$\displaystyle MSE$ $\displaystyle=$
$\displaystyle\frac{1}{S}\sum_{s=1}^{S}(g^{m_{s}}-\hat{g}^{m_{s}})^{2}$ (19)
$\displaystyle MAE$ $\displaystyle=$
$\displaystyle\frac{1}{S}\sum_{s=1}^{S}\Big{|}g^{m_{s}}-\hat{g}^{m_{s}}\Big{|}$
(20) $\displaystyle MAPE$ $\displaystyle=$
$\displaystyle\frac{100\%}{S}\sum_{s=1}^{S}\Big{|}\frac{g^{m_{s}}-\hat{g}^{m_{s}}}{g^{m_{s}}}\Big{|}$
(21) $\displaystyle SMAPE$ $\displaystyle=$
$\displaystyle\frac{100\%}{S}\sum_{s=1}^{S}\frac{\big{|}g^{m_{s}}-\hat{g}^{m_{s}}\big{|}}{(|g^{m_{s}}|+|\hat{g}^{m_{s}}|)/2}$
(22)

In Table 2, the results on the test set show that rTPNN outperforms all of the
other forecasters for the majority of the error metrics while some forecasters
may perform better in individual metrics. However, observations on an
individual error metric (without considering the other metrics) may be
misleading due to its properties. For example, the MAPE of Ridge Regression is
significantly low but MSE, MAE and SMAPE of that are high. The reason is that
Ridge is more accurate in forecasting samples with high energy generation than
forecasting those with low generations. Moreover, rTPNN is shown to have high
generalization ability since it performs well for both training and test sets
with regard to all metrics. Also, only rTPNN and LSTM are able to achieve
better performances than the benchmark performance of the 1-Day Naive Forecast
with respect to MSE, MAE and SMAPE.

We also see that SMAPE yields significantly larger values than those of other
metrics (including MAPE) because SMAPE takes values in $[0,200]$ and has a
scaling effect as a result of the denominator in (22). In particular, the
absolute deviation of forecast values from the actual values is divided by the
sum of those. Therefore, under- and over-forecasting have different effects on
SMAPE, where under-forecasting results in higher SMAPE.

Table 2: Comparison of the forecasting performance of rTPNN with that of state-of-the-art forecasters with respect to MSE, MAE, MAPE, and SMAPE excluding nights Forecasting Methods | Training Set | Test Set
---|---|---
MSE | MAE | MAPE | SMAPE | MSE | MAE | MAPE | SMAPE
rTPNN | 2.23 | 1.13 | 3.72 | 51.84 | 2.58 | 1.21 | 10.67 | 54.42
LSTM | 2.18 | 1.18 | 4.95 | 54.83 | 2.56 | 1.26 | 13.59 | 57.98
MLP | 2.77 | 1.35 | 6.33 | 60.57 | 3.09 | 1.42 | 14.25 | 63.06
Linear Regression | 2.78 | 1.28 | 4.92 | 57.71 | 3.16 | 1.35 | 6.08 | 60.38
Lasso Regression | 8.61 | 2.12 | 4.06 | 88.68 | 8.7 | 2.14 | 11.16 | 90.68
Ridge Regression | 2.78 | 1.29 | 4.93 | 57.74 | 3.16 | 1.36 | 6.11 | 60.41
ElasticNet Regression | 8.61 | 2.12 | 4.06 | 88.68 | 8.7 | 2.14 | 11.16 | 90.69
RandomForestRegressor | 0.3 | 0.41 | 1.5 | 24.75 | 3.18 | 1.36 | 6.82 | 60
1-Day Naive Forecast | 3.68 | 1.25 | 2.76 | 56.63 | 4.25 | 1.37 | 1.26 | 58.29
Figure 6: Forecasting results of the three most competitive models (rTPNN,
LSTM and MLP) with respect to results in Table 2 for the time between fifth
and seventh days in the test set

Next, in Figure 6, we present the actual energy generation between the fifth
and the seventh days of the test set as well as those forecast by the best
three techniques (rTPNN, LSTM and MLP). Our results show that the predictions
of rTPNN are the closest to the actual generation within the predictions of
these three techniques. In addition, we see that rTPNN can successfully
capture both increases and decreases in energy generation while LSTM and MLP
struggle to predict sharp increases and decreases.

Figure 7: Histogram of the forecasting error in kW measured as
$(\hat{g}^{m_{s}}-g^{m_{s}})$ for each $m_{s}$ in the test set

Finally, Figure 7 displays the histogram of the forecasting error that is
realized by each of rTPNN, LSTM, and MLP on the test set. Our results in this
figure show that the forecasting error of rTPNN is around zero for the
significantly large number of samples (around 5000 out of 8664 samples). We
also see that the absolute error is smaller than $2$ for $93\%$ of the
samples. We also see that the overall forecasting error is lower for rTPNN
than both LSTM and MLP.

###  5.3 Scheduling Performance of rTPNN-FES

We now evaluate the scheduling performance of rTPNN-FES for the considered
smart home energy management system. To this end, we compare the schedule
generated by rTPNN-FES with that by optimization (solving (1)-(5)) using
actual energy generations as well as the GA-based scheduling (presented in
Section 5.1.4). Note that although the schedule generated by the optimization
using actual generations is the best achievable schedule, it is practically
not available due to the lack of future information about the actual
generations.

Figure 8: Comparison of rTPNN-FES against the optimal scheduling and GA-based
scheduling with respect to the scheduling cost (top) for the days of the test
set and (bottom) as the boxplot of the cost difference.

Figure 8 (top) displays the comparison of rTPNN-FES against the optimal
scheduling and the GA-based scheduling regarding the cost value for the days
of the test set. In this figure, we see that rTPNN-FES significantly
outperforms GA-based scheduling achieving close-to-optimal cost. In other
words, the user dissatisfaction cost – which is defined in (1) – of rTPNN-FES
is significantly lower than the cost of GA-based scheduling, and it is
slightly higher than that of optimal scheduling. The average cost difference
between rTPNN-FES and optimal scheduling is $1.3\%$ and the maximum difference
is about $3.48\%$.

Furthermore, Figure 8 (bottom) displays the summary of the statistics for the
cost difference between rTPNN-FES and the optimal scheduling as well as the
difference between GA-based and optimal scheduling as a boxplot. In Figure 8
(bottom), we first see that the cost difference is significantly lower for
rTPNN-FES, where even the upper quartile of rTPNN-FES is smaller than the
lower quartile of GA-based scheduling. We also see that the median of the cost
difference between rTPNN-FES and optimal scheduling is $0.13$ and the upper
quartile of that is about $0.146$. That is, the cost difference is less than
$0.146$ for $75\%$ of the days in the test set. In addition, we see that there
are only $7$ outlier days for which the cost is between $0.19$ and $0.3$.
According to the results presented in Figure 8, rTPNN-FES can be considered as
a successful heuristic with a low increase in cost.

###  5.4 Evaluation of the Computation Time

In Table 3, we present measurements on the training and execution times of
each forecasting model. Our results first show that the execution time of
rTPNN ($0.17~{}ms$) is comparable with the execution time of LSTM and highly
acceptable for real-time applications. On the other hand, the training time
measurements show that the training of rTPNN takes longer than that of other
forecasting models. Accordingly, one may say that there is a trade-off between
training time and the forecasting performance of rTPNN.

Table 3: Training and Execution Times for Forecasting | Forecasting
---
Methods
| Training Time
---
(seconds)
| Execution Time
---
(milliseconds)
rTPNN | 210 | 0.17
LSTM | 70 | 0.14
MLP | 47 | 0.08
| Random
---
Forest
11.8 | 0.12
| Linear
---
Regression
0.004 | 0.0025
| Lasso
---
Regression
0.005 | 0.0012
| Ridge
---
Regression
0.004 | 0.0012
| Elastic Net
---
Regression
0.007 | 0.0012

Figure 9 displays the computation time of rTPNN-FES and that of optimization
combined with LSTM (the second-best forecaster after rTPNN) in seconds. Note
that we do not present the computation of GA-based scheduling in this figure
since it takes $4.61$ seconds on average – which is approximately 3 orders of
magnitude higher than the computation time of rTPNN-FES and 1 order of
magnitude higher than that of optimization – to find a schedule for a single
window. Our results in this figure show that rTPNN-FES requires significantly
lower computation time than optimization to generate a daily schedule of
household appliances. The average computation time of rTPNN-FES is about
$4~{}ms$ while that of optimization with LSTM is $150~{}ms$. That is, rTPNN-
FES is $37.5$ times faster than optimization with LSTM to simultaneously
forecast and schedule. Although the absolute computation time difference seems
insignificant for a small use case (as in this paper), it would have important
effects on the operation of large renewable energy networks with a high number
of sources and devices.

Figure 9: Computation time (in seconds) comparison between rTPNN-FES and
optimal scheduling under LSTM forecaster

##  6 Conclusion

We have proposed a novel neural network architecture, called Recurrent Trend
Predictive Neural Network based Forecast Embedded Scheduling (namely rTPNN-
FES), for smart home energy management systems. The rTPNN-FES architecture
forecasts renewable energy generation and schedules household appliances to
use renewable energy efficiently and to minimize user dissatisfaction. As the
main contribution of rTPNN-FES, it performs both forecasting and scheduling in
a single architecture. Thus, it 1) provides a schedule that is robust against
forecasting and measurement errors, 2) requires significantly low computation
time and memory space by eliminating the use of two separate algorithms for
forecasting and scheduling, and 3) offers high scalability to grow the load
set (i.e. adding devices) over time.

We have evaluated the performance of rTPNN-FES for both forecasting renewable
energy generation and scheduling household appliances using two publicly
available datasets. During the performance evaluation, rTPNN-FES is compared
against 8 different techniques for forecasting and against the optimization
and genetic algorithm for scheduling. Our experimental results have drawn the
following conclusions:

  * •

The forecasting layer of rTPNN-FES outperforms all of the other forecasters
for the majority of MSE, MAE, MAPE, and SMAPE metrics.

  * •

rTPNN-FES achieves a highly successful schedule which is very close to the
optimal schedule with only $1.3\%$ of the cost difference.

  * •

rTPNN-FES requires a much shorter time than both optimal and GA-based
scheduling to generate embedded forecasts and scheduling, although the
forecasting time alone is slightly higher than other forecasters.

Future work shall improve the training of rTPNN-FES by directly minimizing the
cost of user dissatisfaction (or other scheduling costs) to eliminate the
collection of optimal schedules for training. In addition, the integration of
a predictive dynamic thermal model into the rTPNN-FES framework shall be
pursued in future studies. (Such integration is required to utilize more
advanced HVAC scheduling/control system designs.) It would also be interesting
to observe the performance of rTPNN-FES for large-scale renewable energy
networks. Furthermore, since the architecture of rTPNN-FES is not dependent on
the particular optimization problem formulated in this paper, rTPNN-FES shall
be applied for other forecasting/scheduling problems such as optimal dispatch
in microgrids, flow control in networks, and smart energy distribution in
future work.

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