On the Influence of “red leak” of Light Filters on the Brightness Estimates of
Stars of Late Spectral Types Illustrated by the Observations of Rapid
Variability of Symbiotic Stars

G. E. Nikishev1, N. A. Maslennikova1a, A. M. Tatarnikov1, K.Yu. Parusov1, A.A.
Belinski1

1 M.V.Lomonosov Moscow State University, Moscow 119191, Russia.
E-mail<EMAIL_ADDRESS>
Received: 22.09.2023/ Accepted: 17.10.2023.

The results of modeling the dependence of the “red leak” of photometric
filters on various factors (color index $V-R$, luminosity class, interstellar
reddening, air mass and PWV) during observations of stars are presented. The
error arising from not taking into account the “red leak” in the case of
filters used on the 0.6-m telescope of the CMO SAI can amount to
$0.6^{m}-0.8^{m}$ for late stars. Algorithms for reducing observational data
are presented for filters $U$ and $B$. The results of observations of the
rapid variability of two symbiotic stars CH Cyg and SU Lyn with cold
components of very late spectral types are presented. For CH Cyg, rapid
variability was detected on both observation dates. Taking into account the
“red leak” effect, the amplitude in the $B$ band was 0.10m on November 6, 2019
and 0.19m on December 15, 2022, with a characteristic variability time of
about 20 minutes. For SU Lyn, no rapid brightness variability was detected in
the $B$ band on February 2, 2023 (with an accuracy of 0.003m).

Keywords: photometry, photometric systems, symbiotic stars.

PACS: 95.75.De

## Introduction

Symbiotic stars are binary systems, in the spectra of which the spectrum of
cool stars — a red giant - and emission lines with a high ionization
potential, typical, for example, for the spectra of planetary nebulae, are
simultaneously visible [1]. Thus, one component of the binary system is a red
giant with a temperature of 3500 K (sometimes a red supergiant or yellow
giant), and the second is a hot subdwarf (a star similar to the central stars
of planetary nebulae) with a temperature $T_{hot}>30000$ К (in some systems
the hot component is a white dwarf or neutron star). The maximum radiation of
the hot component occurs in the far UV band, so it ionizes the substance lost
by the cold component. As a result, an extended ionized shell is formed, in
which emission lines such as H I, He I, He II, [O III], etc. are formed. An
accretion disk may exist around the hot component of a symbiotic star (for
example, [2], [3], [4], [5]). This combination of objects of different types
in one system leads to the fact that symbiotic stars exhibit manifestations of
activity of different scales (from thousandths of a magnitude to several
magnitudes) and at different time periods — from several minutes to tens and
hundreds of years.

One of the types of variability of symbiotic stars is rapid brightness
variability, similar in its parameters to the flicker effect of cataclysmic
binaries [6] — variability with a characteristic amplitude of up to tenths of
a magnitude at times from several minutes to an hour. At the same time, the
variability can be either periodic (for example, in BF Cyg [7] or Z And [8]),
or irregular — CH Cyg [9], T CrB [10], CSS 1102 [11] etc. The origin of
variability in the first case is a rotation of the hot component, and in the
second, apparently, transient phenomena during the accretion.

The search for and study of the rapid variability of symbiotic stars is
currently being carried out photometrically with CCD cameras. Observations are
carried out in the spectral range in which the maximum contribution of the
variable source is expected. The cold component of the system has a maximum
emission in the red or near-infrared region of the spectrum, its emission at
characteristic monitoring times (hundreds of minutes) can be considered
constant and its contribution drops rapidly with decreasing wavelength.
Therefore, observations are usually made in the $U$ or $B$ bands (center
wavelength 360 nm or 440 nm, respectively). However, modern CCD detectors have
a very wide sensitivity bandwidth, extending from 300-350 nm to 1100-1200 nm.
If the used shortwave filter has transmission not only in the working area,
but also in the red part of the spectrum — the so-called “red leak”, then
during observations not only photons in the band, for example, $B$, but also
several from the long-wave part will be recorded. When observing very red
objects (e.g., late M-stars), the effect can be very large even for small
(fractions of a percent) amounts of red leak. The effects of red leak were
affected, for example, by
HST111https://www.stsci.edu/instruments/wfpc2/Wfpc2_hand/HTML/W2_30.html UV
filters and the $u$ filter of the SDSS
survey222https://www.sdss4.org/dr17/imaging/caveats/. The necessity of taking
into account the red leak in symbiotic star observations [12] is also known.
To some extent, the influence of this effect can be reduced by selecting
comparison stars of similar spectral types. However, this is not always
possible - firstly, such stars may have their own variability, and secondly,
stars with the required color indices may simply not be in the field of view
of the photometer.

In this paper, we study the impact of the “red leak” of the CCD photometer
filters of the RC600 telescope of the Caucasus Mountain Observatory of SAI MSU
(CMO) on the results of red stars photometry and determine a method for taking
this effect into account using the example of processing rapid variability
observations of the symbiotic star CH Cyg and variable star SU Lyn, suspected
of belonging to the class of symbiotic stars, the cool components of which
have late spectral types.

##  1\. Observations

At the SAI CMO [13], the 60-cm RC600 telescope manufactured by ASA (Austria)
has been actively operating since 2019. It is used for a large volume of
photometric observations. For this purpose, a CCD photometer [14] based on the
Andor iKON-L CCD matrix (image scale 0.67 arcsec/pixel) is used. The
photometer is equipped with a filter wheel containing $UBVR_{c}I_{c}$ filters
of the Johnson-Cousins photometric system. The transmission curves of all
filters are available on the CMO website333https://arca.sai.msu.ru/filters.
The $R_{c}$ and $I_{c}$ bands used in this work will be referred to as $R$ and
$I$ in the text. The sensitivity curves of the CCD matrix detector and the
transmission curves of the short-wave $U$ and $B$ filters, used since 2019,
registered on the Avantes avaspec 2048 spectrophotometer, are shown in Fig. 1.
Also installed on the RC600 are the $g^{\prime}$, $r^{\prime}$, $i^{\prime}$,
and $z^{\prime}$ filters of the SDSS photometric system. The transmission
curve of the short-wave $g^{\prime}$ filter outside the main band is very
similar to the $B$ curve, so our results of the ”red leak” research in the $B$
band are also applicable to observations in this filter.

Fig. 1: Transmission curves of the $U$ (blue line) and $B$ (red line) filters,
atmosphere (black line), camera entrance window (purple line) and bandwidth of
the Andor iKON-L CCD camera (green line). Dashed lines show the response
curves of the corresponding instrumental bands (product of filter bandwidth,
entrance window and camera sensitivity)

The photometric observations discussed below were carried out at RC600. The
symbiotic star CH Cyg was observed on 06.11.2019 and 15.12.2022 in the $B$
band. The variable SU Lyn suspected of belonging to symbiotic stars [15] was
observed on 25.02.2023 in the $U$ band. The photometric observations were
processed in the standard way (accounting for the bias, dark and flat field,
and aperture photometry). The errors were estimated for field stars whose
brightness is close to the magnitudes of the objects. For CH Cyg, the single
measurement errors on 06.11.2019 were $0.006^{m}$, on 15.12.2022 —
$0.005^{m}$, and for SU Lyn — $0.003^{m}$.

##  2\. Synthetic photometry

As can be seen from Fig. 1, the $U$ and $B$ filters currently used on the
RC600 telescope have significant red leak. For the $U$ filter it is
represented as a narrow peak at 700 nm with a maximum transmittance of about
$0.2\%$, while for the $B$ filter it consists of two regions — red (in the
wavelength range of $550-700$ nm with an average transmittance of $<0.1\%$)
and extensive infrared with an average transmittance of about $1\%$.

The final value of red leak is also influenced by the transmission of the
Earth’s atmosphere and the entrance window of the camera, and the sensitivity
of the detector. Fig. 1 shows the Earth’s atmospheric transmittance curve for
a high-altitude observatory (CMO altitude is about 2100 m above sea level). It
was derived from the ESO
SkyCalc444https://www.eso.org/observing/etc/bin/gen/form?INS.MODE=swspectr+INS.NAME=SKYCALC
online calculator for an altitude of 2400 m and water content PWV=2.5 mm, and
then adjusted to the altitude and pressure typical of CMO according to [16].
It can be seen that the influence of the atmosphere on the leakage value can
be practically neglected. The sensitivity curve of the detector has slopes in
both the UV and IR parts of the spectrum, and affects the considered filters
differently. The UV slope of the curve reduces the efficiency of the $U$
filter in its main bandwidth (thereby increasing the relative contribution of
red leak). The IR slump of the curve significantly affects the value of the IR
part of the $B$ filter leak (see Fig. 1).

To determine the magnitude of the red leakage contribution to the signal in
the short-wavelength instrumental bands, we performed synthetic photometry on
the spectra of stars of different spectral types (from O5 to M10) presented in
the [17] spectra library. The synthetic signal for a star of each spectral
type was calculated by integrating the convolution of the spectrum
$I(\lambda)$ with the atmospheric transmission curve $A(\lambda)$ and the
instrumental band response curves $F(\lambda)$ shown in Fig. 1 (the multiplier
$\lambda$ under the integral is used to convert the energy into the number of
photons):

$N_{ph}\sim\int^{\lambda_{2}}_{\lambda_{1}}\lambda
I(\lambda)A(\lambda)F(\lambda)d\lambda$ (1)

Then, the value of the red leak (in stellar magnitudes) can be calculated by
the formula:

$\delta{\rm mag}=2.5\lg\frac{\int^{1100}_{300}\lambda
I(\lambda)A(\lambda)F(\lambda)d\lambda}{\int^{\lambda_{2}}_{\lambda_{1}}\lambda
I(\lambda)A(\lambda)F(\lambda)d\lambda}$ (2)

In the numerator, integration is carried out over the entire spectral band,
and in the denominator – between wavelengths that limit the working band of
the filter (for the $U$ filter $\lambda_{1}=315$ nm, $\lambda_{2}=470$ nm, for
the $B$ filter $\lambda_{1}=365$ nm, $\lambda_{2}=575$ nm).

The color indices $V-R$ for each spectrum were also calculated by synthetic
photometry. The spectrum of the star A0V was used as a standard, and the
filters $V$ and $R$ were considered free of red leakage.

Fig. 2: Dependence of the red leak in the $U$ band on color for stars of
different luminosity classes (circles — I and II classes, triangles — III
class, black dots — IV and V classes; the corresponding lines show smoothed
dependences)

Fig. 2 shows the results of calculations of the red leak value for the $U$
filter. It can be seen that for stars of late spectral types (later than about
G0), the single dependence of the leakage value on the color index splits into
two — for I – III classes and IV – V classes. In the region $(V-R)\approx 0.8$
(K stars), the curves split once more — for red giants, the leakage value
decreases with increasing color index. Thus, to correctly account for the read
leak in the $U$ band, it is necessary to know not only the color of the star
(or spectral types), but also the luminosity class.

Fig. 3: Dependence of the red leak value in the $U$ band on color for the case
of interstellar reddening with color excess $E(B-V)=1$ for stars of different
luminosity classes

As the calculations show, the magnitude of the red leak is affected not only
by the color index but also by the value of the interstellar reddening. Fig. 3
shows the leakage in the $U$ band for the case of $E(B-V)=1$. It can be seen
that it is still possible to identify separate color dependences of the
leakage for late-type stars of different luminosity classes. The leakage grows
rapidly with increasing $V-R$ and reaches $2^{m}$ (i.e., the leakage
contribution to the total signal is 6 times larger than the useful signal).

In the $B$ band (Fig. 4), there is practically no separation of the dependence
by luminosity classes — the leakage value slowly increases from 0 for hot
stars to $0.05^{m}$ for stars of spectral type $K0-K1$, after which it begins
its sharp increase, and at $V-R\sim 1.5$ it can exceed $0.5^{m}$. Interstellar
reddening, just as in the case of the $U$ band, significantly increases the
leakage value for stars of the same spectral types.

Fig. 4: Dependence of the red leak value in the $B$ band on color for stars of
luminosity classes I – V in the case of no interstellar reddening (dots and
solid line) and in the case of excess color $E(B-V)=1$ under the normal
reddening law (triangles and dashed line)  Fig. 5: Dependence of the red leak
value in the $U$ band on the color of stars for different values of airmass at
$PWV=2.5$ mm: $M_{z}=1$ — black triangles, $M_{z}=1.15$ — red squares,
$M_{z}=2$ — blue squares. The lines show the positions of the sequences
highlighted in Fig. 2 Fig. 6: Dependence of the red leak value in the $U$ band
on the color of the stars for different values of $PWV$ at $Mz=1.15$:
$PWV=2.5$ mm — black triangles, $PWV=10$ mm — red squares, $PWV=30$ mm — blue
squares. The lines show the positions of the sequences highlighted in Fig. 2
Fig. 7: Dependence of the red leak value in the $B$ band on the color of stars
for different values of airmass at $PWV=2.5$ mm. The designations are the same
as in Fig.5 Fig. 8: Dependence of the red leak value in the $B$ band on the
color indices for different values of $PWV$ at $Mz=1.15$. The notations are
the same as in Fig. 6

There are at least two other factors affecting the observed energy
distribution in the spectra of stars — the change in the absorption of
radiation by the Earth’s atmosphere due to changes in the object’s altitude
above the horizon and the change in the amount of water vapor. The effect of
these factors on the red leak value is shown in Fig. 5 and Fig. 6 for the $U$
band and in Fig. 7 and Fig. 8 for the $B$ band. It can be seen that in the $U$
band the influence of PWV is practically not noticeable, while in the $B$ band
this influence is present and can reach $\sim 0.1^{m}$ for red stars.

##  3\. Photometric monitoring of symbiotic stars

Photometric monitoring surveys to search for and study the rapid variability
of symbiotic stars in the CMO are periodically carried out with the RC600
telescope. They usually take 2-4 hours (with a typical variability time of
10-60 min, see, e.g., [11], [18]), during which CCD-frames are taken
continuously (with an interval for the readout time, which for the Andor
iKON-L matrix is 4 s) of the sky area around the star under study. Brightness
estimates are made by aperture photometry by comparing the signals accumulated
in each frame from the variable star and from the comparison stars. It is
considered that the comparison stars should have a spectral type close to that
of the star under study — this avoids some problems in reducing the
observations. In the case of observations of symbiotic stars, this requirement
is difficult to fulfill because one of the components of a symbiotic star is
usually a red giant, and such stars may not be in close proximity to the
variable. Therefore, all stars in the field with a sufficient signal-to-noise
ratio serve as comparison stars.

The fast brightness variability of a symbiotic star or flickering — a chaotic
change in the star’s luminosity associated with an accretion disk or hot
component whose emission contributes noticeably to the total flux from the
system in the $U$ and $B$ bands and is almost invisible in the longer
wavelength bands dominated by the emission from the red giant. The flux from
the red giant can be considered constant on characteristic flicker effect
timescales, so the additive signal from the red leakage remains constant in
observations of fast light variability. However, many symbiotic stars have
active and quiescent states in which the accretion rate is different, i.e.,
the contribution of the accretion disk to the system’s light in the short-
wavelength bands changes. Thus, in the blue region, the spectrum of a
symbiotic star is very different from that of normal stars, so the plots
presented above, which are intended to correct observations of normal stars,
cannot be used directly to account for red leakage.

In general, to calculate the red leak value, it is necessary to know the
energy distribution in the red region (spectral type or color index), the
magnitude of $E(B-V)$, the brightness in the $R_{c}$ or $I_{c}$ band, and the
altitude of the object at the time of the observation. Using these data and
the bandwidth of the filter, camera entrance window, and detector, the number
of red leak photons for a given object is calculated. The object’s brightness
measured during the observations in the $U$ or $B$ bands is converted to a
photon flux from which the calculated red leak is subtracted. The flux is then
converted back to stellar magnitudes.

For the case where the object is a normal star in the red part of the spectrum
(e.g., a red giant in a symbiotic system), we can use the data in Fig. 2 – 8
with a correction for the difference in the brightness of the system in the
$U$ (or $B$) band from the spectral class expected for a normal star,
coinciding with that of the red giant.

Fig. 9: Light curves of CH Cyg in the $B$ band (observed — blue dots,
corrected for red leak — green dots) and control star (red dots) obtained on
06.11.2019 (left) and 15.12.2022 (right). The control stars are $\sim 3^{m}$
fainter than the variable  Fig. 10: Light curves of SU Lyn in the $U$ band
(observed curve —blue dots, corrected for red leakage —green dots) and
comparison stars with color indices $V-R=0.75$ (purple dots), $V-R=0.63$
—brown dots, and $V-R<0.3$ — red and yellow dots. The curves of the comparison
stars are arbitrarily shifted along the vertical axis, with fainter stars
showing a larger dispersion of the brightness estimates

Fig. 9 shows the results of observations of the fast variability of CH Cyg in
the $B$ band made on November 6, 2019 and December 15, 2022. The duration of
the first monitoring was 2h 12m, the second — about 45 m. On both light
curves, irregular brightness fluctuations are clearly visible. The maximum
amplitude was $~{}0.07^{m}$ and $0.17^{m}$, respectively, with the RMS of the
control star brightness being $0.006^{m}$ and $0.005^{m}$. It is difficult to
distinguish the characteristic time of variability in the 2019 curve, but
several short-term episodes of brightness change can be noted. On the curve of
December 15, 2022, the same short-term light rises with an amplitude of $\sim
0.1^{m}$, separated by a time interval of $~{}20$ minutes, and low-amplitude
flicker noise between them are clearly visible.

To account for the red leak in the CH Cyg light curves, the following system
parameters were taken: spectral class of the cold component CH Cyg — M7III
[19], $E(B-V)=0$ [20]. The brightness of CH Cyg in the $R$ band according to
the [21] was $\sim 7.2^{m}$ on 11/06/2019, and $\sim 7.2^{m}$ on 12/15/2022 —
$\sim 5.3^{m}$. The object’s altitude during the observations varied in the
range 51-25∘.

The corrected light curves are shown in Fig. 9. Although the cool component of
CH Cyg was $2^{m}$ fainter in 2019 than in 2022, the average red leak on
11/06/2019 was higher —it was about $0.4^{m}$, and on 12/15/2022 — $0.17^{m}$.
This is because the contribution of the hot component in the $B$ band in 2019
was much lower (by $\sim 3^{m}$), resulting in a large relative leakage
contribution. After accounting for the leakage, the amplitude of the fast
brightness variability increased on 06.11.2019 by 30% (from $0.07^{m}$ to
$0.10^{m}$), and on 15.12.2022 by about 10% (from $0.17^{m}$ to $0.19^{m}$).
The brightness of CH Cyg in the $B$ band became fainter ($10.66^{m}$ and
$7.83^{m}$, respectively). From Fig. 9 (left panel) we can see that the
leakage correction allowed us to identify a dip in the light curve at $\Delta
JD>0.28$, blurred by the additional red leakage associated with the increase
in $M_{z}$ during the monitoring.

The light curves of CH Cyg are very different (Fig. 9). On 12/15/2022,
compared to 11/6/2019, the $B$-band brightness increased by $\sim 3^{m}$ and
the amplitude of the flickering increased by a factor of 2. The $B-V$ color
index decreased from $1.8^{m}$ to $1.3^{m}$ [21], indicating an increased
contribution of the accretion disk to the system’s brightness. Despite the
fact that CH Cyg is a double system with an orbital period of 15.6 years and a
red giant pulsation period of 750 days [22], we cannot explain these changes
by the eclipse of the hot component, since on 06.11.2019 the orbital phase was
0.19, and on 15.12. 2022 — 0.39. In the quiescent state, the spectrum of CH
Cyg is dominated by the emission of the cool component — the red giant; the
contribution of the accretion disk and nebula in the blue region of the
spectrum becomes noticeable only when the accretion rate increases [23].
Therefore, we can assume that the accretion disk had a small luminosity on
06.11.2019, and the observations on 15.12.2022 occurred during a period of
significantly increased accretion rate, so the contribution of the accretion
disk to the system’s luminosity in the $B$ band increased and the amplitude of
the flickering increased.

Fig. 10 shows the result of SU Lyn brightness measurements for more than 3
hours. It is well seen that there are no oscillations on the light curve, but
there is an almost linear trend — over 200 minutes the light has increased by
$0.15^{m}$. A trend can also be seen in comparison stars with a large color
index: for the star with $V-R=0.75$, the light changes were $0.10^{m}$, for
the star with $V-R=0.63$ — $0.09^{m}$. The brightness of the comparison stars
with small $V-R$ color indices did not show any trend. During the observation
period, the star’s altitude changed from $57^{\circ}$ to $30^{\circ}$.

The SU Lyn spectrum matches well with the spectrum of the red giant M5III [24]
even in the blue region. Thus, the contribution of the accretion disk, if the
disk is present in the system, can be neglected in the $U$ and $B$ bands.
Therefore, the plots shown in Figs. 2 and 5 were used to correct the red
leakage. The magnitude of the red leak for SU Lyn ranged from $0.88^{m}$ to
$1.04^{m}$ depending on the height of the object. After taking into account
the red leak, the light curve trend disappeared (see Fig. 10), which may
indicate the correctness of the correction.

##  4\. Conclusion

The paper considers the manifestation of the "red leak" effect of the short-
wavelength filters $U$ and $B$ in photometric observations with the CCD camera
of the RC600 telescope of the CMO of SAI. The presence of transmission outside
the main band of the filters leads to a significant contribution of this
effect — up to $0.6-0.8^{m}$ for normal stars of late spectral types. For
stars of early spectral types, the effect is small for the $B$ band, but can
be noticeable in the $U$ band, where it is noticeable at the level of a few
percent starting with A0-stars.

Fig. 2 shows that the red leak in the $U$ band appears differently for stars
having the same color indices but belonging to different luminosity classes.
No such separation is observed in the $B$ band. This difference may be due to
the shape of the filter transmission curves outside the working band — in the
$U$ filter, part of the red leak signal is collected in a narrow transmission
peak in the 700 nm region, whereas in the $B$ band, the entire spurious signal
is collected in the IR region of the spectrum (see Fig. 1). When processing
photometric observations and selecting comparison stars, we usually deal only
with color indices, paying no attention to either the actual spectral type or
the luminosity class of the comparison stars used in photometry. However, when
working in the $U$ band, this can lead to significant errors when correcting
for the effect of the red leak of the filter (the magnitude of the effect
differs by a factor of 2 for stars of different luminosity classes).

The red leak contribution to the final brightness of the system is influenced
by several effects that change the ratio of the fluxes in the short-wavelength
bands relative to the long-wavelength bands. This can be interstellar
reddening, which leads to the weakening of the object in the $U$ and $B$ bands
and sharply increases the red leak contribution, which for normal stars at
$E(B-V)=1$ can reach $2^{m}$, i.e., the spurious signal is several times
larger than the useful one (see Figs. 3 and 4).

The dependence of the magnitude of the "red leak" in the bands $U$ and $B$ on
the interstellar reddening is explained by the strong fall of the radiation
flux in the working transmission region of short-wavelength light filters
relative to the flux in the bands $V$, $R$, and $I$ due to the action of
absorption. For example, for the normal law of interstellar reddening at
$E(B-V)=1$, the absorption will be $~{}2^{m}$ larger in the $U$ band than in
the $R$ band, from which most of the leakage signal is collected. Therefore,
the absolute magnitude of the leak will become smaller (since there is also
absorption in the red region, although smaller in magnitude), with a
significant increase in its relative contribution.

The value of the "red leak" is also affected by the absorption of radiation in
the Earth’s atmosphere. In this work, synthetic photometry was carried out for
several characteristic values of the factors that produce this absorption —
the change in PWV and the change in the airmass $M_{z}$. The first factor has
little or no effect in the $U$ band, but is noticeable in the $B$ band. The
second factor has a stronger effect in the $U$ band. The influence of the
airmass is confirmed by observations of the red variable SU Lyn (spectral type
M5III) — the observed effect is in good agreement with the model effect (Fig.
10).

In this paper, we propose methods to account for the red leak for stars whose
spectrum is close to that of normal stars or differs in the blue region. In
the first case, the accounting can be done directly using the data of Fig. 2 –
8. In the second case, the correction algorithm is more complicated and is
described in the section "Photometric monitoring".

In this paper, we present the results of the search for fast variability in
two objects — the known pecular symbiotic star CH Cyg and the recently
discovered symbiotic star SU Lyn. No flicker effect was detected in SU Lyn.
When the red leak is taken into account, the trend in its light curve
disappeared, which confirms the correctness of the correction. The importance
of accounting for the red leak is also confirmed by the observations of the
flickering in CH Cyg. The amplitude is an important parameter characterizing
the rapid variability of the brightness. After the red leak correction, it
increased to $0.10^{m}$ on 06.11.2019 and $0.19^{m}$ on 15.12.2022. After the
correction, the star’s brightness became lower: on 06.11.2019 in the $B$ band,
the brightness decrease was $\sim 0.4^{m}$, and on 15.12.2022 the brightness
dropped by $\sim 0.2^{m}$. The different magnitude of the correction (with an
almost unchanged spectral type of the cool component) is due to the different
contribution to the total brightness of the accretion disk system on these
dates. It is also interesting to note that, despite the later spectral type of
the cool component CH Cyg compared to SU Lyn, the correction for it is
smaller. This is due to the presence in CH Cyg of additional emission in the
short-wavelength region, which is absent in SU Lyn.

The work was performed using equipment purchased under the Lomonosov Moscow
State University Development Program. The work of A. M. Tatarnikov (analysis
of observational results and synthetic photometry) was supported by the
Russian Science Foundation (grant 23-12-00092). N. A. Maslenikova
(observations of symbiotic stars, processing of observations of rapid
variability) is grateful for support from the Fund for the Development of
Theoretical Physics and Mathematics <<BAZIS>> (grant 22-2-10-21-1).

## References

  * 1. Boyarchuk A.A. // Soviet Astronomy. 1968. 11, 818.
  * 2. Bath G.T. and Pringle J.E. // MNRAS. 1982. 201, 345.
  * 3. Kafatos M. and Michalitsianos A.G.  // NASA Conference Publication. 1982. 2238, 452.
  * 4. Viotti R., Altamore A., Baratta G.B. et al. // NASA Conference Publication. 1982. 2238, 446.
  * 5. Bollimpalli D.A., Hameury J.M. and Lasota J.P. // MNRAS. 2018. 481, 5422.
  * 6. Bruch A. // A&A. 2000. 359, 998.
  * 7. Formiggini L. and Leibowitz E.M. // MNRAS. 2009. 396, 1507.
  * 8. Sokoloski J.L. and Bildsten L. // ApJ. 1999. 517, 919.
  * 9. Stoyanov K.A., Marti J., Zamanov R. et al. // Bulgarian Astronomical Journal. 2018. 28, 42.
  * 10. Dobrotka A., Hric L., Casares J.  et al. // MNRAS. 2010. 402, 2567.
  * 11. Maslennikova N.A., Tatarnikova A.A., Tatarnikov A.M., Ikonnikova N.P. and Dodin A.V. // Astron. Lett. 2022. 48, 38.
  * 12. Munari U. and Moretti S. // Baltic Astronomy. 2012. 21, 22.
  * 13. Shatsky N., Belinski A., Dodin A. et al. // Ground-Based Astronomy in Russia. Proc. 2020. 1. P. 127.
  * 14. Berdnikov L.N., Belinskii A.A., Shatskii N.I. et al. // Astr. Rep. 2020. 64, 310.
  * 15. Iłkiewicz K., Mikołajewska J., Scaringi S. et al. // MNRAS. 2022. 510, 2707.
  * 16. Noll S., Kausch W., Barden M. et al. // A&A. 2012. 543, A92.
  * 17. Pickles A.J. // PASP. 1998. 110, 863.
  * 18. Maslennikova N.A., Tatarnikov A.M. and Tatarnikova A.A. // Astr. Bull. 2023. 78, 325.
  * 19. Shenavrin V.I., Taranova O.G. and Nadzhip A.E. // Astr. Rep. 2023. 78, 325.
  * 20. Green G.M., Schlafly E., Zucker C. et al. // ApJ. 2019. 887, 93.
  * 21. Kloppenborg B.K. // Observations from the AAVSO International Database. https://www.aavso.org
  * 22. Hinkle K.H., Fekel F.C. and Joyce R.R. // ApJ. 2009. 692, 1360.
  * 23. Tarasova T.N. and Skopal A. // Astr. Lett. 2021. 47, 235.
  * 24. Mukai K., Luna G.J.M., Cusumano G. et al. // MNRAS. 2016. 461, L1.


* [12] R. L. Workman and Others.  “Review of Particle Physics”.  _PTEP_ , 2022, 083C01 (2022).  doi:10.1093/ptep/ptac097.
  * [13] A. Hook.  “Solving the Hierarchy Problem Discretely”.  _Phys. Rev. Lett._ , 120(26), 261802 (2018).  doi:10.1103/PhysRevLett.120.261802.  eprint 1802.10093.
  * [14] L. Di Luzio _et al._ “An even lighter QCD axion”.  _JHEP_ , 2021(05), 184 (2021).  doi:10.1007/JHEP05(2021)184.  eprint 2102.00012.
  * [15] L. Di Luzio _et al._ “Dark matter from an even lighter QCD axion: trapped misalignment”.  _JCAP_ , 2021(10), 001 (2021).  doi:10.1088/1475-7516/2021/10/001.  eprint 2102.01082.
  * [16] P. Sikivie.  “Invisible Axion Search Methods”.  _Rev. Mod. Phys._ , 93(1), 015004 (2021).  doi:10.1103/RevModPhys.93.015004.  eprint 2003.02206.
  * [17] P. W. Graham and S. Rajendran.  “Axion dark matter detection with cold molecules”.  _Phys. Rev. D_ , 84, 055013 (2011).  doi:10.1103/PhysRevD.84.055013.
  * [18] P. W. Graham and S. Rajendran.  “New observables for direct detection of axion dark matter”.  _Phys. Rev. D_ , 88, 035023 (2013).  doi:10.1103/PhysRevD.88.035023.
  * [19] Y. V. Stadnik and V. V. Flambaum.  “Axion-induced effects in atoms, molecules, and nuclei: Parity nonconservation, anapole moments, electric dipole moments, and spin-gravity and spin-axion momentum couplings”.  _Phys. Rev. D_ , 89(4), 043522 (2014).  doi:10.1103/PhysRevD.89.043522.  eprint 1312.6667.
  * [20] D. Budker _et al._ “Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr)”.  _Phys. Rev. X_ , 4(2), 021030 (2014).  doi:10.1103/PhysRevX.4.021030.  eprint 1306.6089.
  * [21] P. Sikivie.  “Experimental Tests of the Invisible Axion”.  _Phys. Rev. Lett._ , 51, 1415 (1983).  doi:10.1103/PhysRevLett.51.1415.  [Erratum: Phys.Rev.Lett. 52, 695 (1984)].
  * [22] P. Sikivie.  “Detection Rates for ’Invisible’ Axion Searches”.  _Phys. Rev. D_ , 32, 2988 (1985).  doi:10.1103/PhysRevD.36.974.  [Erratum: Phys.Rev.D 36, 974 (1987)].
  * [23] P. W. Graham _et al._ “Experimental Searches for the Axion and Axion-Like Particles”.  _Ann. Rev. Nucl. Part. Sci._ , 65, 485 (2015).  doi:10.1146/annurev-nucl-102014-022120.  eprint 1602.00039.
  * [24] W. Bernreuther.  “The Electric dipole moment of the muon”.  _Z. Phys. C_ , 56, S97 (1992).  doi:10.1007/BF02426781.
  * [25] C. Abel _et al._ “Search for axionlike dark matter through nuclear spin precession in electric and magnetic fields”.  _Physical Review X_ , 7, 041034 (2017).  ISSN 2160-3308.  doi:10.1103/PhysRevX.7.041034.
  * [26] T. S. Roussy _et al._ “Experimental Constraint on Axionlike Particles over Seven Orders of Magnitude in Mass”.  _Phys. Rev. Lett._ , 126, 171301 (2021).  doi:10.1103/PhysRevLett.126.171301.  eprint 2006.15787.
  * [27] S. P. Chang _et al._ “Axion dark matter search using the storage ring EDM method”.  _PoS_ , PSTP2017, 036 (2018).  doi:10.22323/1.324.0036.
  * [28] S. P. Chang _et al._ “Axionlike dark matter search using the storage ring EDM method”.  _Phys. Rev. D_ , 99, 083002 (2019).  doi:10.1103/PhysRevD.99.083002.
  * [29] F. Abusaif _et al._ _Storage ring to search for electric dipole moments of charged particles : feasibility study_.  CERN Yellow Reports. Monographs, ISSN 2519-8068, eISSN 2519-8076; 3. CERN, Geneva (2021).  ISBN 978-92-9083-607-0.  doi:10.23731/CYRM-2021-003.
  * [30] O. Kim and Y. K. Semertzidis.  “New method of probing an oscillating EDM induced by axionlike dark matter using an RF Wien filter in storage rings”.  _Phys. Rev. D_ , 104(9), 096006 (2021).  doi:10.1103/PhysRevD.104.096006.  eprint 2105.06655.
  * [31] L. Krauss _et al._ “Calculations for Cosmic Axion Detection”.  _Phys. Rev. Lett._ , 55, 1797 (1985).  doi:10.1103/PhysRevLett.55.1797.
  * [32] H. Georgi, D. B. Kaplan, and L. Randall.  “Manifesting the Invisible Axion at Low-energies”.  _Phys. Lett. B_ , 169, 73 (1986).  doi:10.1016/0370-2693(86)90688-X.
  * [33] G. Raffelt and D. Seckel.  “Bounds on Exotic Particle Interactions from SN 1987a”.  _Phys. Rev. Lett._ , 60, 1793 (1988).  doi:10.1103/PhysRevLett.60.1793.
  * [34] K. Choi, K. Kang, and J. E. Kim.  “Invisible Axion Emissions From SN1987A”.  _Phys. Rev. Lett._ , 62, 849 (1989).  doi:10.1103/PhysRevLett.62.849.
  * [35] M. Carena and R. D. Peccei.  “The Effective Lagrangian for Axion Emission From SN1987A”.  _Phys. Rev. D_ , 40, 652 (1989).  doi:10.1103/PhysRevD.40.652.
  * [36] R. Barbieri _et al._ “Axion to magnon conversion: a scheme for the detection of galactic axions”.  _Phys. Lett. B_ , 226, 357 (1989).  doi:10.1016/0370-2693(89)91209-4.
  * [37] P. V. Vorob’ev, I. V. Kolokolov, and V. F. Fogel.  “Ferromagnetic detector of (pseudo)Goldstone bosons”.  _JETP Lett._ , 50, 65 (1989).  http://jetpletters.ru/ps/1125/article_17041.shtml.
  * [38] A. I. Kakhidze and I. V. Kolokolov.  “Antiferromagnetic axions detector”.  _Sov. Phys. JETP_ , 72, 598 (1991).
  * [39] P. V. Vorob’ev, A. I. Kakhidze, and I. V. Kolokolov.  “Axion wind: A Search for cosmological axion condensate”.  _Phys. Atom. Nucl._ , 58, 959 (1995).
  * [40] P. W. Graham _et al._ “Spin Precession Experiments for Light Axionic Dark Matter”.  _Phys. Rev. D_ , 97(5), 055006 (2018).  doi:10.1103/PhysRevD.97.055006.  eprint 1709.07852.
  * [41] Y. Stadnik.  _Manifestations of Dark Matter and Variations of the Fundamental Constants of Nature in Atoms and Astrophysical Phenomena_.  Ph.D. thesis, New South Wales U. (2017).  doi:10.1007/978-3-319-63417-3.
  * [42] C. Smorra _et al._ “Direct limits on the interaction of antiprotons with axion-like dark matter”.  _Nature_ , 575(7782), 310 (2019).  doi:10.1038/s41586-019-1727-9.  eprint 2006.00255.
  * [43] P. W. Graham _et al._ “Storage ring probes of dark matter and dark energy”.  _Phys. Rev. D_ , 103(5), 055010 (2021).  doi:10.1103/PhysRevD.103.055010.  eprint 2005.11867.
  * [44] A. J. Silenko.  “Relativistic spin dynamics conditioned by dark matter axions”.  _Eur. Phys. J. C_ , 82(10), 856 (2022).  doi:10.1140/epjc/s10052-022-10827-7.  eprint 2109.05576.
  * [45] N. N. Nikolaev.  “Spin of protons in NICA and PTR storage rings as an axion antenna”.  _Pisma Zh. Eksp. Teor. Fiz._ , 115(11), 683 (2022).  doi:10.1134/S0021364022600653.
  * [46] A. J. Silenko.  “High precision description and new properties of a spin-1 particle in a magnetic field”.  _Phys. Rev. D_ , 89(12), 121701 (2014).  doi:10.1103/PhysRevD.89.121701.  eprint 1404.4953.
  * [47] A. J. Silenko.  “General description of spin motion in storage rings in the presence of oscillating horizontal fields”.  _EPL_ , 118(6), 61003 (2017).  doi:10.1209/0295-5075/118/61003.  eprint 1706.02065.
  * [48] V. Bargmann, L. Michel, and V. L. Telegdi.  “Precession of the polarization of particles moving in a homogeneous electromagnetic field”.  _Phys. Rev. Lett._ , 2, 435 (1959).  doi:10.1103/PhysRevLett.2.435.
  * [49] T. Fukuyama and A. J. Silenko.  “Derivation of Generalized Thomas-Bargmann-Michel-Telegdi Equation for a Particle with Electric Dipole Moment”.  _Int. J. Mod. Phys._ , A28, 1350147 (2013).  doi:10.1142/S0217751X13501479.  eprint 1308.1580.
  * [50] E. Stephenson.  “A Search for Axion-like Particles with a Horizontally Polarized Beam In a Storage Ring”.  _PoS_ , PSTP2019, 018 (2020).  doi:10.22323/1.379.0018.
  * [51] G. Guidoboni _et al._ “How to reach a thousand-second in-plane polarization lifetime with $0.97\text{$-$}\mathrm{GeV}/c$ deuterons in a storage ring”.  _Phys. Rev. Lett._ , 117, 054801 (2016).  doi:10.1103/PhysRevLett.117.054801.
  * [52] D. Eversmann _et al._ “New method for a continuous determination of the spin tune in storage rings and implications for precision experiments”.  _Phys. Rev. Lett._ , 115, 094801 (2015).  doi:10.1103/PhysRevLett.115.094801.
  * [53] R. Maier.  “Cooler synchrotron COSY: Performance and perspectives”.  _Nucl. Instrum. Meth. A_ , 390, 1 (1997).  doi:10.1016/S0168-9002(97)00324-0.
  * [54] W. Haeberli.  “Sources of polarized ions”.  _Annual Review of Nuclear Science_ , 17(1), 373 (1967).  doi:10.1146/annurev.ns.17.120167.002105.
  * [55] D. Chiladze _et al._ “Determination of deuteron beam polarizations at COSY”.  _Phys. Rev. ST Accel. Beams_ , 9, 050101 (2006).  doi:10.1103/PhysRevSTAB.9.050101.
  * [56] M. Tanifuji.  _Polarization Phenomena in Physics: Applications to Nuclear Reactions_.  World Scientific, Singapore (2018).  ISBN 987-981-3230-88-0.  doi:10.1142/10731.
  * [57] H. H. Barschall and W. Haeberli (eds.).  _The Madison Convention, Polarization Phenomena in Nuclear Reactions_.  University of Wisconsin Press, Madison, WI (1971).  https://www.osti.gov/biblio/4726823.
  * [58] F. Müller _et al._ “Measurement of deuteron carbon vector analyzing powers in the kinetic energy range 170–380 MeV”.  _Eur. Phys. J. A_ , 56, 1 (2020).  ISSN 1434-601X.  doi:10.1140/EPJA/S10050-020-00215-8.
  * [59] Z. Bagdasarian _et al._ “Measuring the polarization of a rapidly precessing deuteron beam”.  _Phys. Rev. Accel. Beams_ , 17, 052803 (2014).  doi:10.1103/PhysRevSTAB.17.052803.
  * [60] P. Benati _et al._ “Synchrotron oscillation effects on an rf-solenoid spin resonance”.  _Phys. Rev. Accel. Beams_ , 15, 124202 (2012).  doi:10.1103/PhysRevSTAB.15.124202.
  * [61] P. Benati _et al._ “Erratum: Synchrotron oscillation effects on an rf-solenoid spin resonance [Phys. Rev. ST Accel. Beams 15, 124202 (2012)]”.  _Phys. Rev. Accel. Beams_ , 16, 049901 (2013).  doi:10.1103/PhysRevSTAB.16.049901.
  * [62] G. Guidoboni _et al._ “Connection between zero chromaticity and long in-plane polarization lifetime in a magnetic storage ring”.  _Phys. Rev. Accel. Beams_ , 21, 024201 (2018).  doi:10.1103/PhysRevAccelBeams.21.024201.
  * [63] J. Slim _et al._ “Electromagnetic Simulation and Design of a Novel Waveguide RF Wien Filter for Electric Dipole Moment Measurements of Protons and Deuterons”.  _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ , 828, 116 (2016).  doi:10.1016/j.nima.2016.05.012.
  * [64] J. Slim _et al._ “The driving circuit of the waveguide RF Wien filter for the deuteron EDM precursor experiment at COSY”.  _Journal of Instrumentation_ , 15(03), P03021 (2020).  doi:10.1088/1748-0221/15/03/p03021.
  * [65] F. Rathmann, N. N. Nikolaev, and J. Slim.  “Spin dynamics investigations for the electric dipole moment experiment”.  _Phys. Rev. Accel. Beams_ , 23, 024601 (2020).  doi:10.1103/PhysRevAccelBeams.23.024601.
  * [66] J. Slim _et al._ “First detection of collective oscillations of a stored deuteron beam with an amplitude close to the quantum limit”.  _Phys. Rev. Accel. Beams_ , 24(12), 124601 (2021).  doi:10.1103/PhysRevAccelBeams.24.124601.  eprint 2101.07582.
  * [67] JEDI Collaboration.  “Replication Data for: First Search for Axion-Like Particles in a Storage Ring Using a Polarized Deuteron Beam” (2022).  doi:10.26165/JUELICH-DATA/HHNVQ3.
  * [68] G. J. Feldman and R. D. Cousins.  “Unified approach to the classical statistical analysis of small signals”.  _Phys. Rev. D_ , 57, 3873 (1998).  doi:10.1103/PhysRevD.57.3873.
  * [69] S. P. Chang.  _Studies on Axion-EDM experiment using the storage ring method_.  Ph.D. dissertation, https://library.kaist.ac.kr/search/ctlgSearch/posesn/view.do?bibctrlno=996474&se=t0&ty=B&_csrf=cdfcdff0-c27d-4b60-9bef-cbb0967f7d6d, The Korea Advanced Institute of Science and Technology (KAIST), Daedeok Innopolis, Daejeon, South Korea (2022).
  * [70] S. Plaszczynski _et al._ “A novel estimator of the polarization amplitude from normally distributed Stokes parameters”.  _Monthly Notices of the Royal Astronomical Society_ , 439(4), 4048 (2014).  ISSN 0035-8711.  doi:10.1093/mnras/stu270.
  * [71] D. Eversmann, J. Pretz, and M. Rosenthal.  “Amplitude estimation of a sine function based on confidence intervals and Bayes' theorem”.  _Journal of Instrumentation_ , 2016(11), P05003 (2016).  doi:10.1088/1748-0221/11/05/p05003.
  * [72] D. Aybas _et al._ “Search for axionlike dark matter using solid-state nuclear magnetic resonance”.  _Phys. Rev. Lett._ , 126, 141802 (2021).  doi:10.1103/PhysRevLett.126.141802.
  * [73] I. Schulthess _et al._ “New Limit on Axionlike Dark Matter Using Cold Neutrons”.  _Phys. Rev. Lett._ , 129(19), 191801 (2022).  doi:10.1103/PhysRevLett.129.191801.  eprint 2204.01454.
  * [74] C. O’Hare.  “cajohare/axionlimits: Axionlimits”.  https://cajohare.github.io/AxionLimits/ (2020).  doi:10.5281/zenodo.3932430.
  * [75] N. Bar, K. Blum, and G. D’Amico.  “Is there a supernova bound on axions?”  _Phys. Rev. D_ , 101(12), 123025 (2020).  doi:10.1103/PhysRevD.101.123025.  eprint 1907.05020.
  * [76] L. Caloni _et al._ “Novel cosmological bounds on thermally-produced axion-like particles”.  _Journal of Cosmology and Astroparticle Physics_ , 2022(09), 021 (2022).  doi:10.1088/1475-7516/2022/09/021.
  * [77] M. Pospelov and A. Ritz.  “Theta induced electric dipole moment of the neutron via QCD sum rules”.  _Phys. Rev. Lett._ , 83, 2526 (1999).  doi:10.1103/PhysRevLett.83.2526.  eprint hep-ph/9904483.
  * [78] R. J. Crewther _et al._ “Chiral Estimate of the Electric Dipole Moment of the Neutron in Quantum Chromodynamics”.  _Phys. Lett. B_ , 88, 123 (1979).  doi:10.1016/0370-2693(79)90128-X.  [Erratum: Phys.Lett.B 91, 487 (1980)].
  * [79] V. Baluni.  “CP-nonconserving effects in quantum chromodynamics”.  _Phys. Rev. D_ , 19, 2227 (1979).  doi:10.1103/PhysRevD.19.2227.
  * [80] K. Ottnad _et al._ “New insights into the neutron electric dipole moment”.  _Phys. Lett. B_ , 687, 42 (2010).  doi:10.1016/j.physletb.2010.03.005.  eprint 0911.3981.
  * [81] N. Yamanaka and E. Hiyama.  “Enhancement of the CP-odd effect in the nuclear electric dipole moment of 6Li”.  _Phys. Rev. C_ , 91(5), 054005 (2015).  doi:10.1103/PhysRevC.91.054005.  eprint 1503.04446.
  * [82] J. Bsaisou _et al._ “Nuclear Electric Dipole Moments in Chiral Effective Field Theory”.  _JHEP_ , 2015(03), 104 (2015).  doi:10.1007/JHEP03(2015)104.  [Erratum: JHEP 05, 083 (2015).], eprint 1411.5804.
  * [83] I. B. Khriplovich and R. A. Korkin.  “P and T odd electromagnetic moments of deuteron in chiral limit”.  _Nucl. Phys. A_ , 665, 365 (2000).  doi:10.1016/S0375-9474(99)00403-0.  eprint nucl-th/9904081.
  * [84] O. Lebedev _et al._ “Probing CP violation with the deuteron electric dipole moment”.  _Phys. Rev. D_ , 70, 016003 (2004).  doi:10.1103/PhysRevD.70.016003.  eprint hep-ph/0402023.
  * [85] C. P. Liu and R. G. E. Timmermans.  “P- and T-odd two-nucleon interaction and the deuteron electric dipole moment”.  _Phys. Rev. C_ , 70, 055501 (2004).  doi:10.1103/PhysRevC.70.055501.  eprint nucl-th/0408060.
  * [86] I. R. Afnan and B. F. Gibson.  “Model Dependence of the 2H Electric Dipole Moment”.  _Phys. Rev. C_ , 82, 064002 (2010).  doi:10.1103/PhysRevC.82.064002.  eprint 1011.4968.
  * [87] J. de Vries _et al._ “Parity- and Time-Reversal-Violating Form Factors of the Deuteron”.  _Phys. Rev. Lett._ , 107, 091804 (2011).  doi:10.1103/PhysRevLett.107.091804.  eprint 1102.4068.
  * [88] J. Bsaisou _et al._ “The electric dipole moment of the deuteron from the QCD $\theta$-term”.  _Eur. Phys. J. A_ , 49, 31 (2013).  doi:10.1140/epja/i2013-13031-x.  eprint 1209.6306.
  * [89] J. Bsaisou _et al._ “P- and T-Violating Lagrangians in Chiral Effective Field Theory and Nuclear Electric Dipole Moments”.  _Annals Phys._ , 359, 317 (2015).  doi:10.1016/j.aop.2015.04.031.  eprint 1412.5471.
  * [90] A. Wirzba, J. Bsaisou, and A. Nogga.  “Permanent Electric Dipole Moments of Single-, Two-, and Three-Nucleon Systems”.  _Int. J. Mod. Phys. E_ , 26(01n02), 1740031 (2017).  doi:10.1142/S0218301317400316.  eprint 1610.00794.
  * [91] A. Hook and J. Huang.  “Probing axions with neutron star inspirals and other stellar processes”.  _JHEP_ , 2018(06), 036 (2018).  doi:10.1007/JHEP06(2018)036.  eprint 1708.08464.
  * [92] K. Blum _et al._ “Constraining Axion Dark Matter with Big Bang Nucleosynthesis”.  _Phys. Lett. B_ , 737, 30 (2014).  doi:10.1016/j.physletb.2014.07.059.  eprint 1401.6460.
  * [93] G. G. Raffelt.  “Astrophysical axion bounds”.  _Lect. Notes Phys._ , 741, 51 (2008).  doi:10.1007/978-3-540-73518-2_3.  eprint hep-ph/0611350.
  * [94] P. Carenza _et al._ “Improved axion emissivity from a supernova via nucleon-nucleon bremsstrahlung”.  _JCAP_ , 2019(10), 016 (2019).  doi:10.1088/1475-7516/2019/10/016.  [Erratum: JCAP 05, E01 (2020)], eprint 1906.11844.
  * [95] I. M. Bloch _et al._ “Axion-like Relics: New Constraints from Old Comagnetometer Data”.  _JHEP_ , 2020(01), 167 (2020).  doi:10.1007/JHEP01(2020)167.  eprint 1907.03767.
  * [96] I. M. Bloch _et al._ “New constraints on axion-like dark matter using a Floquet quantum detector”.  _Sci. Adv._ , 8(5), abl8919 (2022).  doi:10.1126/sciadv.abl8919.  eprint 2105.04603.
  * [97] D. F. Jackson Kimball _et al._ “Overview of the Cosmic Axion Spin Precession Experiment (CASPEr)”.  _Springer Proc. Phys._ , 245, 105 (2020).  doi:10.1007/978-3-030-43761-9_13.  eprint 1711.08999.
  * [98] A. Garcon _et al._ “Constraints on bosonic dark matter from ultralow-field nuclear magnetic resonance”.  _Sci. Adv._ , 5(10), eaax4539 (2019).  doi:10.1126/sciadv.aax4539.  eprint 1902.04644.
  * [99] E. G. Adelberger _et al._ “Particle Physics Implications of a Recent Test of the Gravitational Inverse Sqaure Law”.  _Phys. Rev. Lett._ , 98, 131104 (2007).  doi:10.1103/PhysRevLett.98.131104.  eprint hep-ph/0611223.
  * [100] G. Vasilakis _et al._ “Limits on New Long Range Nuclear Spin-Dependent Forces Set with a K-He3 Comagnetometer”.  _Phys. Rev. Lett._ , 103(26), 261801 (2009).  doi:10.1103/PhysRevLett.103.261801.  eprint 0809.4700.
  * [101] A. Bhusal, N. Houston, and T. Li.  “Searching for Solar Axions Using Data from the Sudbury Neutrino Observatory”.  _Phys. Rev. Lett._ , 126(9), 091601 (2021).  doi:10.1103/PhysRevLett.126.091601.  eprint 2004.02733.
  * [102] M. Buschmann _et al._ “Upper Limit on the QCD Axion Mass from Isolated Neutron Star Cooling”.  _Phys. Rev. Lett._ , 128(9), 091102 (2022).  doi:10.1103/PhysRevLett.128.091102.  eprint 2111.09892.
  * [103] J. Pretz _et al._ “Statistical sensitivity estimates for oscillating electric dipole moment measurements in storage rings”.  _Eur. Phys. J. C_ , 80(2), 107 (2020).  doi:10.1140/epjc/s10052-020-7664-9.  eprint 1908.09678.


# Deep Recurrent Electricity Theft Detection in AMI Networks with Random
Tuning of Hyper-parameters

Mahmoud Nabil1, Muhammad Ismail2, Mohamed Mahmoud1, Mostafa Shahin2, Khalid
Qaraqe2, and Erchin Serpedin2

1Department of Electrical and Computer Engineering, Tennessee Tech.
University, TN, USA  2 Department of Electrical and Computer Engineering,
Texas A&M University at Qatar, Doha, Qatar

###### Abstract

Modern smart grids rely on advanced metering infrastructure (AMI) networks for
monitoring and billing purposes. However, such an approach suffers from
electricity theft cyberattacks. Different from the existing research that
utilizes shallow, static, and customer-specific-based electricity theft
detectors, this paper proposes a generalized deep recurrent neural network
(RNN)-based electricity theft detector that can effectively thwart these
cyberattacks. The proposed model exploits the time series nature of the
customers’ electricity consumption to implement a gated recurrent unit
(GRU)-RNN, hence, improving the detection performance. In addition, the
proposed RNN-based detector adopts a random search analysis in its learning
stage to appropriately fine-tune its hyper-parameters. Extensive test studies
are carried out to investigate the detector’s performance using publicly
available real data of 107,200 energy consumption days from 200 customers.
Simulation results demonstrate the superior performance of the proposed
detector compared with state-of-the-art electricity theft detectors.

###### Index Terms:

Electricity theft detection, cyberattacks, AMI networks, deep machine
learning.

##  I Introduction

Electricity theft results in high financial losses for several countries such
as the United States ($\$6$ billion/year) and India ($\$17$ billion/year) [1],
[2]. Other developing countries lose almost $50\%$ of their electricity
revenue due to theft [3]. Recently, advanced metering infrastructure (AMI)
networks are utilized within smart grids for monitoring, asset management, and
billing purposes. The AMI networks rely on smart meters located in the
customers’ premises to regularly report their energy consumption. This
approach has the potential to hinder traditional physical electricity theft
attacks including line hooking or meter tampering [4]. Nevertheless, a new
category of electricity theft attacks that target the AMI networks has
appeared, namely electricity theft cyberattacks. These attacks jeopardize the
integrity of the customer’s energy consumption data as malicious customers
hack into their smart meters to manipulate their own energy consumption
values. Despite the difficulties associated with detecting such cyberattacks,
the customers’ fine-grained energy consumption data is considered to be a
promising tool that can be used to boost automated electricity theft detection
mechanisms.

Several automated theft detection techniques have been proposed in literature
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. These detection techniques
can be categorized into three main groups: machine learning, state estimation,
and game theory-based techniques. Compared with these techniques, machine
learning-based techniques present the most mature and effective mechanism to
tackle this problem due to their superior detection rate on real datasets [1].
However, the existing techniques suffer from several limitations. First, the
existing detectors employ shallow machine learning architectures such as
support vector machines (SVMs), while deep learning architectures can better
capture the behavior of the input data resulting in a higher detection rate.
In addition, the existing detectors rely on static input data [1, 13],
however, exploiting the temporal correlation within the time series energy
consumption data of the customers can further enhance the detection
performance. Moreover, some of the existing works [1] adopt customer-specific
models, where a detector’s model is developed for each customer using only its
energy consumption data in order to detect any future thefts. However,
customer-specific models are not practical due to several reasons. On one
hand, it would be impossible to train new detector models for new customers
joining the system as these new customers do not have any historical energy
consumption records. Furthermore, customer-specific detectors are not robust
against contamination attacks, where the customer data that is used to train
the detector is already malicious yet falsely labeled as honest data to
confuse the detector.

In this work, our objective is to develop a generalized deep recurrent neural
network (RNN)-based electricity theft detector. In specific, RNNs offer a
considerable leverage while working with time series data since it is
represented as a feedback network where the output at the current state is the
input for the next state. To the best of our knowledge, only [14] investigates
the application of deep neural networks for electricity theft detection. While
a long-short-term-memory (LSTM) architecture of RNNs is investigated in [14]
to tackle the electricity theft detection problem, a comprehensive study is
needed to assess the performance of RNN-based detectors on real energy
consumption data sets. In addition, further investigation should be considered
in tuning the network hyper-parameters to further improve the training time
and the model’s performance.

The contributions of this paper are summarized as follows:

  * •

We propose a generalized RNN-based electricity theft detector using hidden
gated recurrent unit (GRU) layers. The generality of the model stems from its
training on the energy consumption load profiles of different residential
customers’ patterns.

  * •

We carry out hyper-parameter tuning based on a random search to further
improve the performance. Random search is proven to outperform grid search by
finding a good hyper-parameter set in faster execution time [18].

  * •

We evaluate the performance of the proposed detector using publicly available
real data of $107,200$ energy consumption days from $200$ customers against a
set of different types of cyber attacks. Simulation results demonstrate the
robustness of the proposed model against existing detectors.

The remainder of this paper is organized as follows. Section II reviews the
related work. Energy consumption data is presented in Section III. The
proposed detection approach is explained in Section IV. Simulation results and
discussions are presented in Section V. Finally, conclusions are drawn in
Section VI.

##  II Related Work

Several studies have addressed the problem of detecting electricity theft in
the literature. Machine learning studies are the most pertinent to the
proposed work. In [5], the authors compared the performance of two
classification algorithms based on single feedforward neural network, namely,
extreme learning machine (ELM), and online sequential extreme learning machine
(OSELM), against SVM. The proposed approach was able to achieve $70\%$
classification rate with ELM. Shallow SVM based detector has also been
employed in [19] and achieved a detection accuracy up to $60\%$ on real data
set. The performance of this detector was further improved in [6] to reach an
accuracy of $72\%$ using a fuzzy inference system as a post-processing stage.
A graph-based approach is proposed in [7] that uses optimum path forest for
electricity theft detection and outperforms SVM with an accuracy of $89\%$. To
the best of our knowledge, the best reported performance on a publicly
available data set was achieved by [1], which employs an SVM classifier, with
an average detection rate of $94\%$ and a false acceptance rate of $11\%$.

Deep learning techniques for electricity theft detection are studied only in
[14], where the author presents a comparison between different deep learning
architectures such as convolutional neural networks (CNN), LSTM-RNNs, and
stacked auto-encoders. Nevertheless, only two types of cyberattacks are
presented and synthetic data is used to evaluate the performance of the
proposed models, which cannot be compared reliably against models trained on
real consumption data. Moreover, the effect of fine-tuning the models hyper-
parameters is not studied in [14].

TABLE I: Cyber Attacks in [1].  Cyber Attack
---
$f_{1}(E_{c}(d,t))=\alpha E_{c}(d,t)$
$f_{2}(E_{c}(d,t))=\beta(d,t)E_{c}(d,t)$
$f_{3}(E_{c}(d,t))=\begin{cases}0&\forall
t\in[t_{\text{\scriptsize{i}}}(d),t_{\text{\scriptsize{f}}}(d)]\\\
E_{c}(d,t)&\forall
t\notin[t_{\text{\scriptsize{i}}}(d),t_{\text{\scriptsize{f}}}(d)]\end{cases}$
$f_{4}(E_{c}(d,t))=\mathbb{{E}}[E_{c}(d)]$
$f_{5}(E_{c}(d,t))=\beta(d,t)\mathbb{{E}}[E_{c}(d)]$
$f_{6}(E_{c}(d,t))=E_{c}(d,T-t+1)$

##  III Energy Consumption Data

Define a set of customers $\mathbb{\mathcal{C}}$, a set of days
$\mathbb{\mathcal{{D}}}=\\{1,\dots,D\\}$, and a set of periods
$\mathbb{\mathcal{{T}}}=\\{1,\dots,T\\}$ of equal duration within each day. A
smart meter is installed for each customer to regularly report the energy
consumption for load monitoring and billing. For each customer $c$, the energy
consumption value at a specific day $d$ and time $t$ is denoted as
$E_{c}(d,t)$. Hence, each customer has an energy consumption record matrix
$E_{c}$ of actual consumptions, where the rows of $E_{c}$ span the days in
$\mathbb{\mathcal{{D}}}$, while the columns span the periods in
$\mathcal{{T}}$. The smart meter reports the consumption to the utility for
customer $c$ on day $d$ and period $t$ as $R_{c}(d,t)$. An honest customer
reports the actual energy consumed at the end of the consumption period, and
hence, $R_{c}(d,t)=E_{c}(d,t)$. On contrary, a malicious customer aims to
manipulate $E_{c}(d,t)$ to lower its electricity bill by modifying the smart
meter readings. Hence, the cyberattack is formally defined as
$R_{c}(d,t)=f(E_{c}(d,t))$, where $f(\cdot)$ denotes a cyberattack function
that results in a reduced version for the malicious customer electricity bill.
Although it is easy to collect honest samples for a customer $c$ by monitoring
its reports to the utility, malicious reports might not be easy to collect.
This is because, the customer reports may all be honest or it has not been
detected previously as a malicious customer. To tackle this problem, a set of
cyberattack functions were defined in [1] as indicated in Table I. Each
function $f(\cdot)$ creates a different attack scenario that aims to reduce
the customers’ energy consumption $E_{c}(d,t)$. The first attack
$f_{1}(\cdot)$ aims to reduce $E_{c}(d,t)$ by some fraction, where $\alpha$
denotes a flat reduction percentage. On the other hand, attack $f_{2}(\cdot)$
dynamically reduces $E_{c}(d,t)$ by a value controlled by the time function
$\beta(d,t)$. The third attack $f_{3}(\cdot)$ represents a selective time
filtering function, where the malicious customer reports zero reading during
the interval $[t_{\text{\scriptsize{i}}}(d),t_{\text{\scriptsize{f}}}(d)]$,
otherwise, the customer reports the actual consumption $E_{c}(d,t)$. Here,
$t_{\text{\scriptsize{i}}}(d)$ and $t_{\text{\scriptsize{f}}}(d)$ denotes the
initial and the final periods of the interval. The next two attacks
$f_{4}(\cdot)$ and $f_{5}(\cdot)$ are based on the expected value of the
energy consumption for the malicious customer for a given day denoted by
$\mathbb{{E}}[E_{c}(d)]$. In $f_{4}(\cdot)$ , the attacker reports a flat
value during the day, while in $f_{5}(\cdot)$ the attacker reduces
$\mathbb{{E}}[E_{c}(d)]$ dynamically from time to time using function
$\beta(d,t)$. The last attack $f_{6}(\cdot)$ is a reverse function that
reorders the energy consumption reports during the day so that higher reports
are assigned to low tariff periods.

Figure 1: Architecture of general RNN-based electricity theft detector.

Applying the aforementioned cyber attacks on the energy consumption record
matrix $E_{c}$ results in six matrices $M_{c,i}$, where $i\in\\{1,\dots,6\\}$,
for each attack $f_{i}(\cdot)$. The complete data set for a customer is
denoted as $\hat{X}_{c}$, which is a concatenation for $E_{c}$ and all
malicious matrices $M_{c,i}$. In addition, each row $X_{c}(d,\cdot)$
represents either an honest day sample (i.e, $E_{c}$ samples) labeled with 0
or a malicious sample (i.e., $M_{c,i}$ samples) labeled with 1. Since
$\hat{X}_{c}$ is an imbalanced dataset where the ratio of the honest samples
to the malicious samples is 1:6, the detector may be biased towards the
dominant class category (i.e., malicious samples). To mitigate the bias
effect, the adaptive synthetic sampling approach (ADASYN) [20] is used to
oversample the minor class so that the ratio between the honest and the
malicious samples is almost the same. Each data set $\hat{X}_{c}$ is
partitioned into two disjoints data sets for model evaluation, specifically, a
training set $\hat{X}_{c,\text{\sc{tr}}}$ and atesting set
$\hat{X}_{c,\text{\sc{tst}}}$ with ratio 3:2. The training sets of all
customers in $\mathbb{\mathcal{C}}$ are merged together to form
$\hat{X}_{\text{\sc{tr}}}$. Similarly, the test sets of all customers are
merged together to form $\hat{X}_{\text{\sc{tst}}}$. Hence, all customers have
the same opportunity in training and evaluating the general model. As
different customers have different ranges of energy consumption values during
different report periods $\mathbb{\mathcal{{T}}}$, different consumption
periods have different opportunities to influence the optimization algorithm
for the final model. Hence, in order to enforce equal opportunities for
different consumption periods, feature scaling is applied to
$\hat{X}_{\text{\sc{tr}}}$. The mean and the variance of
$\hat{X}_{\text{\sc{tr}}}$ are calculated and used to scale
$\hat{X}_{\text{\sc{tr}}}$ so that each consumption period would have zero
mean and unit standard deviation. The output of the feature scaling stage is
denoted as $X_{\text{\sc{tr}}}$. The same scaling scores are further applied
to the test set to get $X_{\text{\sc{tst}}}$.

To evaluate the performance of the detector, three metrics are used as
follows. The detection rate (DR) measures the percentage of the correctly
detected malicious attacks. The false acceptance (FA) rate measures the
percentage of the honest samples that are falsely identified as malicious. The
highest difference (HD) measures the difference between DR and FA [1] . The
detection latency is ignored as electricity theft attacks do not result in an
immediate loss to the utility. Moreover, malicious customers can be fined
later for any detected thefts.

Data: $X_{\text{\sc{tr}}}$

Result: Optimal parameters $U_{(.)}^{l}$,$W_{(.)}^{l}$,$V_{(.)}^{l}$, and
$b_{(.)}^{l}$ $\forall l\in[1,\ldots,L]$

1

2Initialization: Weights $U_{(.)}^{l}$,$W_{(.)}^{l}$,$V_{(.)}^{l}$, and
$b_{(.)}^{l}$ $\forall l\in[1,\ldots,L]$, $i=1$

3while  _$i\neq I$_ do

4 Initialize: $m=1$

5 while  _$m\neq M$_ do

6

7 for  _each training example $x$ in mini-batch $m$_ do

8

9 Feed Forward:

10 for  _each recurrent layer $l$_ do

11 for  _each time step $t$_ do

12

13 $z_{t}^{l}=\sigma(o_{t}^{l-1}U_{z}^{l}+s_{t-1}^{l}W_{z}^{l}+b_{z}^{l})$

14 $r_{t}^{l}=\sigma(o_{t}^{l-1}U_{r}^{l}+s_{t-1}^{l}W_{r}^{l}+b_{r}^{l})$

15 $h_{t}^{l}=\tanh(o_{t}^{l-1}U_{h}^{l}+(s_{t-1}^{l}\odot
r^{l})W_{h}^{l}+b_{h}^{l})$

16 $s_{t}^{l}=(1-z^{l})\odot h^{l}+z^{l}\odot s_{t-1}^{l}$

17 $o_{t}^{l}=\text{softmax}(V^{l}s_{t}^{l}+b_{o}^{l})$

18 end for

19

20 end for

21

22 Back propagation: Compute $\nabla_{U_{(.)}^{l}}C(x)$,
$\nabla_{V_{(.)}^{l}}C(x)$, $\nabla_{W_{(.)}^{l}}C(x)$, and
$\nabla_{b_{(.)}^{l}}C(x)$

23 end for

24

25 Weight and bias update:

26 $U_{(.)}^{l}=U_{(.)}^{l}-\frac{\eta}{K}\sum_{x}\nabla_{U_{(.)}^{l}}C(x)$

27 $V_{(.)}^{l}=V_{(.)}^{l}-\frac{\eta}{K}\sum_{x}\nabla_{V_{(.)}^{l}}C(x)$

28 $W_{(.)}^{l}=W_{(.)}^{l}-\frac{\eta}{K}\sum_{x}\nabla_{W_{(.)}^{l}}C(x)$

29 $b_{(.)}^{l}=b_{(.)}^{l}-\frac{\eta}{K}\sum_{x}\nabla_{b_{(.)}^{l}}C(x)$

30 end while

31

32 end while

33

Algorithm 1 Electricity Theft Detector Training

##  IV Design of Electricity Theft Detector

This section describes the two stages used to develop the RNN-based general
detector. The first is a training stage that describes the detector
architecture and parameters, and the second is the hyper-parameter
optimization stage that uses a random search-based approach for fine-tuning.

###  IV-A Training Stage

Data: $X_{\text{\sc{tr}}}$

Result: Optimal hyper parameters $L^{*}$,$N^{*}$,$O^{*}$, $A_{H}^{*}$, and
$A_{O}^{*}$

1

2Initialization: Weights $U_{(.)}^{l}$,$W_{(.)}^{l}$,$V_{(.)}^{l}$, and
$b_{(.)}^{l}$ $\forall l\in[1,\ldots,L]$, $i=1$

3while  _$i\neq I$_ do

4

5 $L[i]\leftarrow\mathcal{L}$ , $N[i]\leftarrow\mathcal{N}$
,$O[i]\leftarrow\mathcal{O}$, $A_{H}[i]\leftarrow\mathcal{A}_{H}$,
$A_{O}[i]\leftarrow\mathcal{A}_{O}$

6 for  _each $\bar{X}_{\text{\sc{tr}}}$ , $\bar{X}_{\text{\sc{tst}}}$ in
k-folds$(X_{\text{\sc{tr}}})$_ do

7

8 Apply Algorithm 1 using sampled hyper-parameters and record DR, FA and
accuracy.

9 end for

10

11 Record average values across folds DR[$i$], FA[$i$] and accuracy[$i$].

12 end while

13

14Report the hyperparameters of the top three models.

Algorithm 2 Random Search Hyper Parameters Optimization

The architecture of the RNN-based general detector is shown in Fig. 1. The
input layer is denoted by vector $x_{c}(d)$, which represents the consumption
vector of customer $c$ during day $d$ at different reporting periods
$t\in\mathcal{{T}}$. The RNN architecture consists of $L$ hidden GRU layers
each with $N$ neurons. Each GRU layer except the last one accepts a sequence
vector as input and produces a sequence output vector. Unlike traditional
feedforward layers, recurrent layers are more efficient in exploiting patterns
from sequential information. The output layer has two neurons that define the
malicious and the honest classes, where the true label can be represented by
one-hot vector $y(x_{c}(d))=(0\,\,1)^{T}$ for honest customers, and
$y(x_{c}(d))=(1\,\,0)^{T}$ for malicious customers.

The output vector from each layer $l\in[1,\ldots,L]$ is denoted as $o^{l}$,
where $o^{1}=x(d)$. Each hidden GRU layer $l\in[2,\ldots,L-1]$ has the
following parameters:

  * •

The input at time step $t$ is $o_{t}^{l-1}$, which results from the previous
layer $l-1$.

  * •

The hidden state $s_{t}^{l}$ at time step $t$ that represents the memory
computed based on the previous hidden state $s_{t-1}^{l}$ of the same layer.

  * •

The update gate
$z_{t}^{l}=\sigma(o_{t}^{l-1}U_{z}^{l}+s_{t-1}^{l}W_{z}^{l}+b_{z}^{l})$, which
determines a combination between the new input $o_{t}^{l-1}$ and the previous
memory $s_{t-1}^{l}$. Here, $U_{z}^{l}$ and $W_{z}^{l}$ are learnable weights,
$\sigma(.)$ is the activation function, and $b_{z}^{l}$ is the bias vector.

  * •

The reset gate
$r_{t}^{l}=\sigma(o_{t}^{l-1}U_{r}^{l}+s_{t-1}^{l}W_{r}^{l}+b_{r}^{l})$, which
determines the amount of the previous memory $s_{t-1}^{l}$ that will
contribute to the next state using the equation
$h_{t}^{l}=\tanh(o_{t}^{l-1}U_{h}^{l}+(s_{t-1}^{l}\odot
r^{l})W_{h}^{l}+b_{h}^{l})$ . Here, $U_{r}^{l}$, $W_{r}^{l}$, $U_{h}^{l}$, and
$W_{h}^{l}$ are learnable weight matrices, $b_{r}^{l}$ and $b_{h}^{l}$ are the
bias vectors, and $\odot$ is the Hadamard product.

  * •

The next state is simply $s_{t+1}^{l}=(1-z^{l})\odot h^{l}+z^{l}\odot
s_{t}^{l}$ , and the output at time $t+1$ is
$o_{t+1}^{l}=\text{softmax}(V^{l}s_{t+1}^{l}+b_{o}^{l})$, where $V^{l}$ is a
learnable weight matrix.

The detector training stage involves learning the parameters
$U_{(.)}^{l}$,$W_{(.)}^{l}$,$V_{(.)}^{l}$, and $b_{(.)}^{l}$ for all layers,
which results in the desired output vector $y(x_{c}(d))$ for any input
$x_{c}(d)$. The optimization objective to find these weights can be achieved
by minimizing the cross-entropy cost function $C$ defined as

$\displaystyle\underset{\Theta}{\min}\frac{-1}{S}\sum_{X_{\text{\sc{tr}}}}$
$\displaystyle\\{y^{\text{\sc{t}}}(x_{c}(d))\ln(o^{L})+$ (1)
$\displaystyle(1-y^{\text{\sc{t}}}(x_{c}(d)))\ln(1-o^{L})\\},$

Where $\Theta$ denotes the RNN parameters
$U_{(.)}^{l}$,$W_{(.)}^{l}$,$V_{(.)}^{l}$, and $b_{(.)}^{l}$ for all layers
and $S$ represents the total number of the training samples for all customers,
which equals to the number of rows in $X_{\text{\sc{tr}}}$. Solving (1) for
the optimal parameters $\Theta$ is done using an iterative gradient descent-
based optimization algorithm. This is achieved by partitioning
$X_{\text{\sc{tr}}}$ into $M$ mini-batches of equal size and executing
Algorithm 1 for $I$ total iterations. Each iteration $i$ has two main stages,
namely, feed forward and back-propagation. In the feed forward stage, the
training samples in the mini-batch are passed through all the network layers
to calculate the predicted output vectors. On the other hand, the back
propagation stage uses the mini-batches to calculate the gradient of the cost
function in (1) with respect to the network weights [21]. Thereafter, the
computed gradients are used to modify the weights and biases for each
iteration. Note that, the back propagation within each layer is basically a
back propagation through time (BPTT) as each layer is a GRU. In addition,
$\nabla_{a}$ in Algorithm 1 denotes the partial derivative with respect to
$a$. Algorithm 1 uses stochastic gradient descent (SGD) for optimization with
learning rate $\eta$.

###  IV-B Hyper-parameters Optimization Stage

TABLE II: Sampled hyper-parameters results of Algorithm 2 using k-fold cross
validation.

$L$ | $N$ | $A_{H}$ | $A_{O}$ | $O$ |  | DR | FA | HD | Accuracy
---|---|---|---|---|---|---|---|---|---
2 | 201 | Hard Sigmoid | Sigmoid | Adam |  | 0.930 | 0.029 | 0.900 | 0.950
384 | Sigmoid | Sigmoid | SGD |  | 0.707 | 0.313 | 0.394 | 0.697
428 | Sigmoid | Softmax | Adamax |  | 0.946 | 0.036 | 0.910 | 0.955
270 | Relu | Sigmoid | Nadam |  | 0 | 0 | 0 | 0.499
306 | Hard Sigmoid | Softmax | SGD |  | 0.727 | 0.324 | 0.403 | 0.701
258 | Hard Sigmoid | Sigmoid | SGD |  | 0.713 | 0.311 | 0.402 | 0.701
154 | Hard Sigmoid | Softmax | Adagrad |  | 0.880 | 0.067 | 0.812 | 0.906
250 | Tanh | Sigmoid | Adadelta |  | 0.916 | 0.051 | 0.865 | 0.932
351 | Sigmoid | Softmax | Adamax |  | 0.941 | 0.035 | 0.905 | 0.952
284 | Hard Sigmoid | Softmax | SGD |  | 0.739 | 0.340 | 0.398 | 0.699
105 | Tanh | Sigmoid | Adagrad |  | 0.877 | 0.083 | 0.793 | 0.896
3 | 385 | Tanh | Sigmoid | Adadelta |  | 0.928 | 0.037 | 0.890 | 0.945
307 | Sigmoid | Sigmoid | RMSprop |  | 0.923 | 0.031 | 0.891 | 0.945
239 | Tanh | Sigmoid | Adagrad |  | 0.609 | 0.328 | 0.281 | 0.640
272 | Hard Sigmoid | Sigmoid | Adadelta |  | 0.885 | 0.071 | 0.813 | 0.906
110 | Sigmoid | Softmax | Nadam |  | 0.928 | 0.036 | 0.891 | 0.945
251 | Tanh | Sigmoid | Nadam |  | 0 | 0 | 0 | 0.499
310 | Hard Sigmoid | Softmax | Adam |  | 0.939 | 0.033 | 0.905 | 0.952
407 | Hard Sigmoid | Sigmoid | Adagrad |  | 0.867 | 0.075 | 0.791 | 0.895
193 | Sigmoid | Sigmoid | Adadelta |  | 0.875 | 0.068 | 0.807 | 0.903
4 | 492 | Relu | Sigmoid | SGD |  | 0 | 0 | 0 | 0.499
198 | Tanh | Softmax | RMSprop |  | 0.419 | 0.186 | 0.232 | 0.615
352 | Sigmoid | Softmax | Adamax |  | 0.931 | 0.026 | 0.905 | 0.952
215 | Tanh | Softmax | Adamax |  | 0.942 | 0.028 | 0.914 | 0.957
275 | Tanh | Softmax | Adam |  | 0 | 0 | 0 | 0.499
202 | Relu | Sigmoid | Adagrad |  | 0 | 0 | 0 | 0.499
393 | Sigmoid | Sigmoid | RMSprop |  | 0.939 | 0.044 | 0.894 | 0.947
165 | Sigmoid | Softmax | Adagrad |  | 0.898 | 0.073 | 0.825 | 0.912
390 | Hard Sigmoid | Sigmoid | Adadelta |  | 0.608 | 0.359 | 0.249 | 0.624
310 | Tanh | Sigmoid | RMSprop |  | 0.187 | 0.126 | 0.061 | 0.530

Tuning the hyper-parameters of the detector is a challenging and a time-
consuming task, however, optimal parameters improve the detector performance.
Since an exhaustive search on hyper-parameters is practically infeasible due
to the incurred high computational complexity, random search is used to find a
sub-optimal but an efficient solution in a reduced time compared with the
exhaustive grid search [18]. In this paper, random search is used to tune the
following hyper-parameters: the network architecture in terms of the number of
hidden layers and the number of neurons within each hidden layer, the type of
activation functions used at the RNN layers, the type of activation functions
used at the output layer, and the type of gradient descent-based optimization
algorithm used in finding the optimal RNN parameters (weights and biases)
[21]. Denote $\mathcal{L}$ and $\mathcal{N}$ as two uniform distributions that
represent the number of hidden layers and neurons to be sampled by the random
search algorithm. In addition, Let $\mathcal{O}$ represent a uniform
distribution of optimization algorithms that can include Adam, SGD, Adamax,
etc. [21]. Moreover, two uniform distributions are defined to represent the
set of activation functions used by the hidden and output layers, namely,
$\mathcal{A}_{H}$ and $\mathcal{A}_{O}$, respectively. Algorithm 2 shows the
random search over $X_{\text{\sc{tr}}}$ for the selection of the optimal
hyper-parameters defined as $P^{*}$. The maximum number of search iterations
is denoted as $I$, where at each iteration all the distributions are sampled
uniformly and the sampled model is evaluated using K-fold cross-validation.

##  V Numerical Results and Discussion

To evaluate the performance of the proposed RNN-based general electricity
theft detector, real smart meter data from the Irish Smart Energy Trials is
used [22]. The data set was published by the Electric Ireland and Sustainable
Energy Authority of Ireland in January 2012. It contains the energy
consumption readings for $5000$ customers over $536$ days between $2009-2010$.
The customers report their readings every 30 minutes (i.e., the RNN input size
is $48$), and hence, the total number of reports per customer is $25,728$.
Daily reports from $200$ customers with $107,200$ days are considered as
honest energy consumption records, which are then used to launch the
cyberattacks given in Table I for each customer. For cyberattacks
$f_{1}(\cdot)$, $f_{2}(\cdot)$, and $f_{5}(\cdot)$, $\alpha$ and $\beta(d,t)$
are random variables that are uniformly distributed over the interval
$[0.1,0.8]$ [1]. For attack $f_{3}(\cdot)$, $t_{\text{\scriptsize{i}}}(d)$ is
a uniform random variable in $[0,42]$, and the duration of the attack, i.e.,
$t_{\text{\scriptsize{f}}}(d)-t_{\text{\scriptsize{i}}}(d)$, is a uniform
random variable in $[8,48]$, and hence, the maximum value of $t_{f}(d)=48$.
Applying all cyberattacks for each customer results in $\hat{X}_{c}$ data set
which contains $536$ honest samples (days) and $3,216$ malicious samples. Each
sample in $\hat{X}_{c}$ has $48$ energy consumption values. For each
$\hat{X}_{c}$ data set, over-sampling is performed to balance the size of
honest and malicious classes. Consequently, the total size of each
$\hat{X}_{c}$ data set contains a total of $6,432$ honest and malicious
samples. The total number of samples for all $c\in\mathcal{C}$ is $1,286,400$.
Each $\hat{X}_{c}$ data set is partitioned into training set
$\hat{X}_{c,\text{\sc{tr}}}$ and testing set $\hat{X}_{c,\text{\sc{tst}}}$
with ratio 3:2. The training sets for all customers are merged together to
form $\hat{X}_{\text{\sc{tr}}}$. Similarly, the test sets for all customers
are merged together to form $\hat{X}_{\text{\sc{tst}}}$. Feature scaling on
training data produces $X_{\text{\sc{tr}}}$ and applying the same scores on
the test data produces $X_{\text{\sc{tst}}}$.

In Algorithm 1, the total number of epochs $I=10$ and the batch size is $350$.
For Algorithm 2, the options used for the random search are the following:
$\mathcal{L}=\\{2,3,4\\}$, $\mathcal{N}=[100,500]$,
$\mathbb{\mathbb{\mathcal{O}=}}${SGD, Adadelta, Adagrad , Adam, Adamax, Nadam,
RMSprop}, $\mathcal{{A}}_{H}=${Sigmoid, Relu, Hard Sigmoid, Tanh}, and
$\mathcal{{A}}_{O}=${Sigmoid, Softmax}. In addition, the total number of
iterations is $I=30$, the drop out rate is set to $0.2$ and the initialization
of the network neurons is set to Glorot Uniform [21]. All the experiments were
performed on the high-performance cluster (HPC) of the Tennessee Tech
University using two NVIDIA Tesla K80 GPUs. Tensorflow and Python Keras
Library are used for the implementation.

Table II gives the random search evaluation results obtained by applying
Algorithm 2 on the sampled hyper-parameters. The results give the average
detection performance over K-fold cross-validation (with $K=3$) on
$X_{\text{\sc{tr}}}$ in terms of DR, FA, and HD. As indicated in the table,
the proposed general electricity theft detector benefits from the recurrent
deep architecture as the HD can achieve $91.4\%$. It can also be noticed that
all the networks that have Relu samples failed to optimize properly, which is
attributed to the finding discussed in [23] in which deep sigmoidal networks
can outperform deep Relu networks if initialized properly. Moreover, all the
networks with SGD optimizer struggle to achieve good results, which can be
attributed to the small number of epochs used for the training. Finally, all
the promising models have Softmax activation for the output layer. The top
three models with respect to HD for each number of layers $l\in\\{2,3,4\\}$
are highlighted in Table II and further used for performance evaluation using
the test set $X_{\text{\sc{tst}}}$.

TABLE III: Average Performance of the Theft Detector.

| MD1 | MD2 | MD3 | [1]
---|---|---|---|---
DR (%) | 93.4 | 92.5 | 92.7 | 94
FA (%) | 6.9 | 5.0 | 5.3 | 11
HD (%) | 86.4 | 87.4 | 87.4 | 83

Table III gives a comparison between the top three models highlighted in Table
II and the model proposed in [1], which was tested by the same data used in
this paper. The first three columns in Table III give the average performance
results when using the best hyper-parameters highlighted in Table II, such
that the models with $l=2$, $l=3$ , and $l=4$ are represented by MD1, MD2, and
MD3, respectively. The last column gives the results from [1]. While the
detector in [1] is trained on the same dataset used in this paper, it is a
customer-specific electricity theft detector based on a shallow machine
learning architecture (SVM). As indicated in Table II, the best HD is achieved
when MD2 or MD3 hyper-parameters are applied, which outperforms the proposed
detector in [1] by a $4\%$ increase. Moreover, the FA is improved by almost
$55\%$. The results indicate that an improvement in the electricity theft
detection performance can be achieved using RNN architecture. In addition, our
proposed detector is a general model that does not rely on a specific customer
data. Hence, it is more robust against contamination attacks compared with the
proposed detector in [1].

##  VI Conclusion

A general RNN-based electricity theft detector is proposed. The proposed
detector exploits the time series nature of the customers’ energy consumption
to implement GRU deep hidden layers that can learn complex patterns of
customers’ consumption yielding better detection performance. Random search is
employed to tune the detector hyper-parameters and further improve the
performance. Load profiles comprising $107,200$ days for different customers
over two years are the basis for the experiments. The proposed RNN-based
detector achieves a detection rate up to $93\%$ and a false acceptance rate as
low as $5\%$, which presents a $55\%$ reduction in false acceptance, and $4\%$
improvement in the highest difference rate compared with a benchmark detector
in the literature. Moreover, the proposed detector is more robust against
contamination attacks as it does not rely on a specific customer data.

##  VII Acknowledgment

This publication was made possible by NPRP grant # NPRP9-055-2-022 from the
Qatar National Research Fund (a member of Qatar Foundation). The statements
made herein are solely the responsibility of the authors.

## References

  * [1] P. Jokar, N. Arianpoo, and V. C. Leung, “Electricity theft detection in AMI using customers’ consumption patterns,” _IEEE Transactions on Smart Grid_ , vol. 7, no. 1, pp. 216–226, 2016.
  * [2] S. K. Singh, R. Bose, and A. Joshi, “Entropy-based electricity theft detection in AMI network,” _IET Cyber-Physical Systems: Theory & Applications_, 2017.
  * [3] P. Antmann, “Reducing technical and non-technical losses in the power sector,” _World Bank, Washington, DC_ , 2009.
  * [4] P. Pickering, “E-Meters Offer Multiple Ways to Combat Electricity Theft and Tampering,” Last accesed: Nov 2017. [Online]. Available: http://www.electronicdesign.com/meters/e-meters-offer-multiple-ways-combat-electricity-theft-and-tampering
  * [5] A. Nizar, Z. Dong, and Y. Wang, “Power utility nontechnical loss analysis with extreme learning machine method,” _IEEE Transactions on Power Systems_ , vol. 23, no. 3, pp. 946–955, 2008.
  * [6] J. Nagi, K. S. Yap, S. K. Tiong, S. K. Ahmed, and F. Nagi, “Improving SVM-based nontechnical loss detection in power utility using the fuzzy inference system,” _IEEE Transactions on power delivery_ , vol. 26, no. 2, pp. 1284–1285, 2011.
  * [7] C. Ramos, A. de Sousa, J. Papa, and X. Falcao, “A new approach for nontechnical losses detection based on optimum-path forest,” _IEEE Transactions on Power Systems_ , vol. 26, no. 1, pp. 181–189, 2011.
  * [8] E. Angelos, O. Saavedra, C. Cortés, and A. de Souza, “Detection and identification of abnormalities in customer consumptions in power distribution systems,” _IEEE Transactions on Power Delivery_ , vol. 26, no. 4, pp. 2436–2442, 2011.
  * [9] C.-H. Lin, S.-J. Chen, C.-L. Kuo, and J.-L. Chen, “Non-cooperative game model applied to an advanced metering infrastructure for non-technical loss screening in micro-distribution systems,” _IEEE Transactions on Smart Grid_ , vol. 5, no. 5, pp. 2468–2469, 2014.
  * [10] S. Amin, G. A. Schwartz, A. A. Cardenas, and S. S. Sastry, “Game-theoretic models of electricity theft detection in smart utility networks: Providing new capabilities with advanced metering infrastructure,” _IEEE Control Systems_ , vol. 35, no. 1, pp. 66–81, 2015.
  * [11] Y. Zhou, X. Chen, A. Y. Zomaya, L. Wang, and S. Hu, “A dynamic programming algorithm for leveraging probabilistic detection of energy theft in smart home,” _IEEE Transactions on Emerging Topics in Computing_ , vol. 3, no. 4, pp. 502–513, 2015.
  * [12] Y. Liu and S. Hu, “Cyberthreat analysis and detection for energy theft in social networking of smart homes,” _IEEE Transactions on Computational Social Systems_ , vol. 2, no. 4, pp. 148–158, 2015.
  * [13] A. Jindal, A. Dua, K. Kaur, M. Singh, N. Kumar, and S. Mishra, “Decision tree and SVM-based data analytics for theft detection in smart grid,” _IEEE Transactions on Industrial Informatics_ , vol. 12, no. 3, pp. 1005–1016, 2016.
  * [14] R. R. Bhat, R. D. Trevizan, R. Sengupta, X. Li, and A. Bretas, “Identifying nontechnical power loss via spatial and temporal deep learning,” in _Proc. of 15th IEEE International Conference on Machine Learning and Applications (ICMLA)_ , 2016, pp. 272–279.
  * [15] T.-S. Zhan, S.-J. Chen, C.-C. Kao, C.-L. Kuo, J.-L. Chen, and C.-H. Lin, “Non-technical loss and power blackout detection under advanced metering infrastructure using a cooperative game based inference mechanism,” _IET Generation, Transmission & Distribution_, vol. 10, no. 4, pp. 873–882, 2016.
  * [16] M. Tariq and H. V. Poor, “Electricity theft detection and localization in grid-tied microgrids,” _IEEE Transactions on Smart Grid_ , 2016.
  * [17] F. Xiao and Q. Ai, “Electricity theft detection in smart grid using random matrix theory,” _IET Generation, Transmission & Distribution_, 2017.
  * [18] J. Bergstra and Y. Bengio, “Random search for hyper-parameter optimization,” _Journal of Machine Learning Research_ , vol. 13, no. Feb, pp. 281–305, 2012\.
  * [19] J. Nagi, K. S. Yap, S. K. Tiong, S. K. Ahmed, and M. Mohamad, “Nontechnical loss detection for metered customers in power utility using support vector machines,” _IEEE transactions on Power Delivery_ , vol. 25, no. 2, pp. 1162–1171, 2010.
  * [20] H. He, Y. Bai, E. A. Garcia, and S. Li, “Adasyn: Adaptive synthetic sampling approach for imbalanced learning,” in _Proc. of IEEE International Joint Conference on Computational Intelligence_ , 2008, pp. 1322–1328.
  * [21] I. Goodfellow, Y. Bengio, and A. Courville, _Deep learning_. MIT press, 2016.
  * [22] “Irish Social Science Data Archive,” Last accesed: Nov 2017. [Online]. Available: http://www.ucd.ie/issda/data/commissionforenergyregulationcer/
  * [23] J. Pennington, S. Schoenholz, and S. Ganguli, “Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice,” in _Advances in neural information processing systems_ , 2017, pp. 4788–4798.