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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2018-147 LHCb-PAPER-2018-016 September 27, 2018

Measurement of ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ production in proton-
proton collisions at $\sqrt{s}$ = 8$\mathrm{\,Te\kern-2.07413ptV}$

LHCb collaboration†††Authors are listed at the end of this paper.

A measurement of ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ production cross-
section is presented using data, corresponding to an integrated luminosity of
2$\mbox{\,fb}^{-1}$, from $pp$ collisions at $\sqrt{s}$ =
8$\mathrm{\,Te\kern-1.00006ptV}$ collected by the LHCb experiment. The
$\tau^{+}\tau^{-}$ candidates are reconstructed in final states with the first
tau lepton decaying leptonically, and the second decaying either leptonically
or to one or three charged hadrons. The production cross-section is measured
for $Z$ bosons with invariant mass between 60 and
120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, which decay to tau leptons with
transverse momenta greater than 20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
pseudorapidities between 2.0 and 4.5. The cross-section is determined to be
$\sigma_{pp\\!\rightarrow{Z}\\!\rightarrow{\tau^{+}\tau^{-}}}=95.8\pm 2.1\pm
4.6\pm 0.2\pm 1.1\mathrm{\,pb}$, where the first uncertainty is statistical,
the second is systematic, the third is due to the LHC beam energy uncertainty,
and the fourth to the integrated luminosity uncertainty. This result is
compatible with NNLO Standard model predictions. The ratio of the cross-
sections for ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ to
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ (${Z}\\!\rightarrow{e}^{+}{e}^{-}$),
determined to be $1.01\pm 0.05$ ($1.02\pm 0.06$), is consistent with the
lepton-universality hypothesis in $Z$ decays.

Published in JHEP 09 (2018) 159

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

##  1 Introduction

The measurement of the production cross-section for a $Z$ boson111$Z$ refers
to $Z$/$\gamma$⋆, i.e. includes contributions from the virtual photon
production and interference. using different decay modes in proton-proton
($pp$) collisions, $\sigma_{pp\\!\rightarrow Z\\!\rightarrow f\bar{f}}$, is an
important verification of Standard Model (SM) predictions. The ratio of the
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ production cross-sections to other
leptonic decay modes provides a test of lepton universality (LU). The LEP
experiments have performed high accuracy tests of LU at the $Z$ pole, with a
precision better than 1% [1]. Consequently, the observation in proton-proton
collisions of any apparent deviation from LU in $Z$ decays would be an
evidence of new phenomena producing final-state leptons, like in the
theoretical context of mSUGRA [2], constrained NMSSM [3], Randall-Sundrum
models [4, 5], or lepton-violating decays of Higgs-like bosons [6, 7, 8, 9,
10].

This analysis extends the LHCb results obtained with $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ [11] to
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$. The cross-section is measured for
leptons from the $Z$ decay with transverse momentum ($p_{\mathrm{T}}$) above
20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and a $Z$ invariant mass between 60
and 120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, as for the previously
published ${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ and
${Z}\\!\rightarrow{e}^{+}{e}^{-}$ cross-sections [12, 13]. The cross-section
measurements in the pseudorapidity range $2.0<\eta<4.5$ covered by the LHCb
experiment are complementary to those with the central detectors ATLAS [14]
and CMS [15].

In the present analysis, the reconstruction of the tau-pair candidates is
performed in both leptonic and hadronic decay modes of the tau, requiring at
least one leptonic mode for the tau-pair candidate. The reconstruction of
high-$p_{\mathrm{T}}$ tau leptons in the 3-prong decay mode is performed for
the first time in LHCb.

##  2 Detector and datasets

The LHCb detector [16, 17] is a single-arm forward spectrometer designed for
the study of particles containing $b$ or $c$ quarks. The detector includes a
high-precision tracking system consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of
$4{\mathrm{\,Tm}}$, and three stations of silicon-strip detectors and straw
drift tubes placed downstream of the magnet. The tracking system provides a
measurement of momentum of charged particles with a relative uncertainty that
varies from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/p_{\mathrm{T}}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Photons, electrons and hadrons are
identified by a calorimeter system consisting of scintillating-pad (SPD) and
preshower detectors (PS), an electromagnetic calorimeter (ECAL) and a hadronic
calorimeter (HCAL). Muons are identified by a system composed of five stations
of alternating layers of iron and multiwire proportional chambers.

The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. The
hardware trigger imposes a global event cut (GEC) requiring the hit
multiplicity in the SPD to be less than 600, to prevent events with high
occupancy from dominating the processing time in the software trigger.

This analysis uses $pp$ collisions at
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ corresponding to a total integrated
luminosity of $\mathcal{L}=(1976\pm 23)\mbox{\,pb}^{-1}$ [18]. Simulated data
samples are used to study the event selection, determine efficiencies, and
estimate systematic uncertainties. In the simulation, $pp$ collisions are
generated using Pythia 8 [19, *Sjostrand:2006za] with a specific LHCb
configuration [21], and parton density functions taken from CTEQ6L [22].
Decays of hadronic particles are described by EvtGen [23], in which final-
state radiation is generated using Photos [24]. The interaction of the
generated particles with the detector, and its response, are implemented using
the Geant4 toolkit [25, *Agostinelli:2002hh] as described in Ref. [27].

##  3 Event selection

The $Z$ boson is reconstructed from $\tau$ particles decaying into leptonic
(muons or electrons) or hadronic (one or three charged hadrons) final states.
Charged tracks are reconstructed by the tracking system and matched with
clusters of ECAL/HCAL cells and hits in the muon detector. Muon candidates are
identified by matching tracks to hits in the muon stations downstream of the
calorimeters. They are required to leave hits in at least three muon stations,
or four muon stations if they have
$p_{\mathrm{T}}>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Electron candidates
must fail the muon identification criteria and fall within the acceptance of
the PS, ECAL, and HCAL sub-detectors. On average, 30% of a material radiation
length is crossed by a particle before the bending magnet, causing a
considerable energy loss by bremsstrahlung for electrons and positrons. Hence,
the electron or positron candidate momentum is corrected using a
bremsstrahlung photon recovery technique [28]. However, since the ECAL is
designed to register particles from heavy-flavour hadron decays, calorimeter
cells with transverse energy above about 10$\mathrm{\,Ge\kern-1.00006ptV}$
saturate the electronics, and lead to incomplete electron bremsstrahlung
recovery. A large energy deposit in the PS, ECAL, but not in HCAL is required,
satisfying $E_{\text{PS}}>50\mathrm{\,Me\kern-1.00006ptV}$,
$E_{\text{ECAL}}/p>0.1$, and $E_{\text{HCAL}}/p<0.05$, where $p$ is the
reconstructed momentum of the electron candidate, after applying the
bremsstrahlung photon recovery. Charged hadrons are required to be within the
HCAL acceptance, deposit an energy of $E_{\text{HCAL}}/p>0.05$, and must fail
the muon identification criteria. The pion mass is assigned to all charged
hadrons.

The analysis is divided into seven “streams”, labelled as
$\tau_{\mu}\tau_{\mu}$, $\tau_{\mu}\tau_{e}$, $\tau_{\mu}\tau_{h1}$,
$\tau_{\mu}\tau_{h3}$, $\tau_{e}\tau_{e}$, $\tau_{e}\tau_{h1}$, and
$\tau_{e}\tau_{h3}$, where the subscript denotes the final state
reconstructed. Charge-conjugate processes are implied throughout. The streams
are chosen such that at least one $\tau$ lepton decays leptonically. The tau-
pair candidates are selected by triggers requiring muons or electrons with a
minimum transverse momentum of 15${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
trigger efficiency is between 70% and 85%, depending on the number of leptons
in the stream. The final states presented in this analysis account for 58% of
all ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ decays. In the following, a $\tau$
candidate corresponds to a single particle for the $\tau_{e}$, $\tau_{\mu}$,
and $\tau_{h1}$ decay channels, or a combination of the three hadrons in the
case of $\tau_{h3}$. A pair of $\tau$ candidates must be associated to the
same PV. In case where multiple PVs are presented in the event, the associated
PV is defined as that with a smallest change in vertex-fit $\chi^{2}$ when it
is reconstructed with and without the $\tau$ candidate.

The dominating backgrounds are of QCD origin with one or several jets (call
“QCD events” in the following), as well as electroweak processes, mainly
$W$/$Z$ +jets (“$V\\!j$”). The following requirements on the transverse
momentum of $\tau$ decay products are used to reduce these backgrounds. For
all the streams the triggering lepton must have
$p_{\mathrm{T}}>20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. For the processes
$\tau_{\mu}\tau_{\mu}$, $\tau_{e}\tau_{e}$, and $\tau_{\mu}\tau_{e}$ the
second lepton $p_{\mathrm{T}}$ threshold is
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The hadron of the $\tau_{h1}$
candidates is required to have
$p_{\mathrm{T}}>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. For the $\tau_{h3}$
decay channel, each of the three charged hadrons are selected with
$p_{\mathrm{T}}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and at least one must
be above 6${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In addition, the $\tau_{h3}$
candidates must have a total $p_{\mathrm{T}}$ in excess of
12${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an invariant mass in the range
0.7 to 1.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. This leads to the
$\tau_{h3}$ identification efficiency of about 30%, comparable with the value
of 35% found in the context of the
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{\tau^{+}}{{\nu}_{\tau}}$ analysis [29]. For
all streams, the reconstructed direction of the $\tau$ candidate must be in
the fiducial geometrical acceptance $2.0<\eta<4.5$.

Additional selection criteria are needed to suppress background processes due
to semileptonic $c$\- or $b$-hadron decays, misidentification of hadrons as
leptons, or, especially in the $\tau_{h3}$ stream, combinations of unrelated
particles.

Signal candidates tend to have back-to-back tracks in the plane transverse to
the beam axis, and a higher invariant mass than the background. Hence, the
tau-pair is required to have an invariant mass above
20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, or
30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the stream containing
$\tau_{h1}$,$\tau_{h3}$ candidates. Additionally, for the dilepton streams
$\tau_{\mu}\tau_{\mu}$, $\tau_{e}\tau_{e}$, the selected mass range is below
60${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, to avoid the on-shell
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ and ${Z}\\!\rightarrow{e}^{+}{e}^{-}$
regions. The absolute difference in azimuthal angle of the two $\tau$
candidates is required to be greater than $2.7$ radians. The above selections
are found to be 70 to 80% efficient, depending on the analysis stream.

Charged particles in QCD events tend to be associated with jet activity, in
contrast to signal candidates where they are isolated. An isolation variable,
$\hat{I}_{p_{\rm T}}$, is defined as the $p_{\mathrm{T}}$ of the candidate
divided by the transverse component of the vectorial sum of all track momenta
in a cone surrounding the candidate of radius $R_{\eta\phi}=0.5$, defined in
the pseudorapidity-azimuthal angle $(\eta-\phi)$ space. A fully isolated
candidate has $\hat{I}_{p_{\rm T}}=1$, while lower values indicate the
presence of jet activity. The selection $\hat{I}_{p_{\rm T}}>0.9$ is applied
to all $\tau$ candidates, with an efficiency of more than 64% for the tau-pair
signal and rejecting about 98% of QCD events.

The lifetime of the $\tau$ lepton is used to separate the signal from prompt
background. For the $\tau$ decay channels with a single charged particle, it
is not possible to reconstruct a secondary vertex and a selection on the
particle IP to the associated PV is applied. The efficiency on the signal from
these criteria is in the range 71 to 79%.

In the $\tau_{h3}$ case, a vertex reconstruction is possible: the maximum
distance between the three tracks in the $\eta-\phi$ space is required to be
less than $0.005\cdot p_{\mathrm{T}}$ where $p_{\mathrm{T}}$ is the transverse
momentum of $\tau_{h3}$ in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The proper
decay time is subsequently estimated from the distance of the reconstructed
vertex to the associated PV, and the momentum of the candidate, taken as an
approximation of the $\tau$ momentum. A minimum of 60$\mathrm{\,fs}$ is
imposed for this variable, efficiently discarding the prompt background whilst
keeping about 77% of the signal. For the $\tau_{h3}$ decay, a correction to
the mass is also possible by exploiting the direction of flight, recovering
part of the momentum lost due to undetected particles. The corrected mass is
defined as

$m_{\text{corr}}\equiv\sqrt{m^{2}+p^{2}\sin^{2}\theta}+p\sin\theta\,,$ (1)

where $m$ and $p$ are the invariant mass and momentum computed from the three
tracks and $\theta$ is the angle between the momentum and flight direction of
the candidate. The requirement
$m_{\text{corr}}<3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ reduces the QCD
background by about 50% and the $V\\!j$ background by about 60%, retaining 80%
of the signal. Fig. 1 shows the mass distributions of $\tau_{h3}$ candidates
before and after correction for data, compared to the distributions of
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ decays and of the $V\\!j$ background
from simulation.

Figure 1:  Distributions of invariant (dashed line) and corrected (full line,
shaded) mass of $\tau_{h3}$ candidates from the $\tau_{\mu}\tau_{h3}$ channel.
The yields are normalised to the integrated luminosity of the data. The
results from data are represented by the black points. The error bars
represent the statistical uncertainty only. The distributions are compared to
the signal distributions from simulated ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$
(blue) events and the $V\\!j$ (red) background.

In the $\tau_{e}\tau_{e}$ and $\tau_{\mu}\tau_{\mu}$ streams an additional
background component arises from ${Z}\\!\rightarrow l^{+}l^{-}$ decays. This
process produces two muons or two electrons with similar $p_{\mathrm{T}}$
values, in contrast to signal which tends to have unbalanced $p_{\mathrm{T}}$
due to the missing momentum from unreconstructed neutrinos and neutral
hadrons. The $p_{\mathrm{T}}$ asymmetry, $A_{p_{\rm T}}$, is defined as the
absolute $p_{\mathrm{T}}$ difference of the two candidates divided by their
sum. For the two leptonic streams $A_{p_{\rm T}}$ is required to be greater
than $0.1$. A particular case is the $\tau_{\mu}\tau_{e}$ stream, where
background from $V\\!j$ processes arises, with one lepton coming from the jet
causing a relatively large $p_{\mathrm{T}}$ imbalance with respect to the
lepton from the ${W}/{Z}$ boson. A suppression by a factor of two of this
source of background, with a loss of 10% of the signal is obtained imposing a
maximal $A_{p_{\rm T}}$ value of 0.6. For $\tau_{h1}$ and $\tau_{h3}$ the
$A_{p_{\rm T}}$ criterion has been found inefficient for background rejection,
hence no such a constraint is imposed to these two decay modes.

##  4 Signal and background estimation

Figure 2:  Invariant-mass distributions for (a) $\tau_{\mu}\tau_{\mu}$, (b)
$\tau_{e}\tau_{e}$, (c) $\tau_{\mu}\tau_{h1}$, (d) $\tau_{e}\tau_{h1}$, (e)
$\tau_{\mu}\tau_{h3}$, (f) $\tau_{e}\tau_{h3}$, (g) $\tau_{\mu}\tau_{e}$
candidates with the excluded mass ranges indicated by the gray areas. The
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ simulation (red) is normalised to the
observed signal. The $Z$ (blue), QCD (brown), and electroweak (magenta)
backgrounds are estimated from data. The ${t}{\overline{t}}$, $VV$ backgrounds
and cross-feed (green) are estimated from simulation (see text) and generally
not visible.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

After the selections described in the previous Section, a maximum of one
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ candidate per event is found. The number
of signal candidates is determined from the number of observed candidates in
data subtracted by the total number of estimated backgrounds. The results are
summarized in Table 1. The invariant-mass distributions for such candidates
are shown in Fig. 2, for the seven analysis streams.

A data-driven approach is used to estimate the amount of background from QCD
and $V\\!j$ processes. Same-sign (SS) tau-pair candidates are selected with
identical criteria as the signal, but requiring the tau candidates to have
identical electric charge. From simulation, the SS candidates yield is found
to originate mainly from QCD and $V\\!j$ processes, while the mis-
reconstructed ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ process contributes less
than 1%:
$N^{\text{SS}}_{\text{}}=N^{\text{SS}}_{\text{QCD}}+N^{\text{SS}}_{\text{{$V\\!j$}}}+N^{\text{SS}}_{\text{${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$}}$.
The last term originates, for instance, from an electron either from a
${\pi}^{0}$ decay or pair-production, a single hadron from partially-
reconstructed $\tau_{h3}$, 3-prong from false combinatorics, or a muon from a
misidentified hadron. The amount of QCD and $V\\!j$ events in the SS dataset
is determined by a fit to the
$p_{\mathrm{T}}{(\tau_{1})}-p_{\mathrm{T}}{(\tau_{2})}$ distribution, for each
analysis stream [11]. In the fit, the QCD distribution templates are taken
from an SS QCD-enriched dataset, obtained by the anti-isolation requirement
$\hat{I}_{p_{\rm T}}<0.6$; the distributions templates for the two $V\\!j$
processes ($W$+jet, $Z$+jet) are obtained from simulation and are found to be
statistically consistent. Subsequently, the number of QCD and $V\\!j$
background candidates is computed as $N_{\text{QCD}}=r_{\text{QCD}}\cdot
N^{\text{SS}}_{\text{QCD}}$, and $N_{{V\\!j}}=r_{{V\\!j}}\cdot
N^{\text{SS}}_{\text{{$V\\!j$}}}$. The value of $r_{{V\\!j}}$ is obtained from
simulation, considering both $W$ and $Z$ contributions, and varies from
$1.05\pm 0.08$ for the $\tau_{e}\tau_{e}$ up to $2.37\pm 0.30$ for the
$\tau_{\mu}\tau_{h1}$. The same-sign and opposite-sign QCD-enriched datasets
provide the $r_{\text{QCD}}$ values, which are all close to unity, with the
exception of $1.30\pm 0.05$, obtained for $\tau_{\mu}\tau_{\mu}$.

The ${Z}\\!\rightarrow l^{+}l^{-}$ decays ($l=e,\mu$) are a background for all
the streams, except for $\tau_{\mu}\tau_{h3}$ and $\tau_{e}\tau_{h3}$. The
number of ${Z}\\!\rightarrow l^{+}l^{-}$ decays contaminating the
$\tau_{\mu}\tau_{\mu}$ stream is determined by applying all selection criteria
except for the requirement on the dimuon mass: this produces a sample with a
clear peak at the $Z$ mass, as well as an off-shell contribution at lower
mass, as shown in Fig. 2. A template distribution obtained from simulation is
normalised to the data in the 80–100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
mass interval. The fraction of genuine ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$
candidates in the normalisation region is found to be negligible from
simulation. The contribution from ${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ decays
to the background in the signal region is inferred from the normalised
distribution. A similar procedure is applied to estimate the
$\tau_{e}\tau_{e}$ background from ${Z}\\!\rightarrow{e}^{+}{e}^{-}$ decays,
but with the normalisation performed in the
70–100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ interval to account for the
electron momentum resolution degraded by an incomplete electron bremsstrahlung
recovery. For this process, 1% of non-$Z$ background candidates are subtracted
from the normalisation region, as estimated from SS dilepton events.

The process ${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ can be observed as a fake
$\tau_{\mu}\tau_{h1}$ candidate when one of the muons is misidentified as a
charged hadron. This background is evaluated by applying the
$\tau_{\mu}\tau_{h1}$ selection but requiring a second identified muon rather
than a hadron, and scaling by the probability for a muon to be misidentified
as a hadron. The misidentification probability, obtained from simulation and
cross-checked using a tag-and-probe method applied to
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ data (requiring an identified muon as a
tag, and an oppositely-charged track as a probe), is of the order of $10^{-3}$
for muons with $p_{\mathrm{T}}<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and
$10^{-4}$–$10^{-5}$ at larger $p_{\mathrm{T}}$ values. The uncertainty on the
estimation of this background is obtained from the lepton misidentification
probability uncertainty combined with the statistical uncertainty of the
dimuon candidates sample. A similar procedure allows the estimation of
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$, ${Z}\\!\rightarrow{e}^{+}{e}^{-}$
backgrounds in $\tau_{\mu}\tau_{e}$, $\tau_{e}\tau_{h1}$ streams.

Other background processes are due to diboson decays, ${t}{\overline{t}}$
events, and $Z$ decays into $b$ hadrons. Their contributions are relatively
small and obtained from simulation.

Some of the selected tau-pair candidates may not originate from the stream
under study. For instance, a $\tau_{h1}$ candidate may be selected from a
partially reconstructed $\tau_{h3}$ candidate. The fraction of cross-feed
candidates is obtained from the ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$
simulated sample. The statistical uncertainty is 1 to 3%, to which a small
contribution from the uncertainties on the branching fractions of the
contaminating streams is added.

Table 1:  Expected backgrounds yields and total number of candidates observed.
In the last row the uncertainties are the statistical and systematic
contributions combined.

|  000$\tau_{\mu}\tau_{\mu}$ |  000$\tau_{\mu}\tau_{h1}$ |  000$\tau_{\mu}\tau_{h3}$ |  000$\tau_{e}\tau_{e}$ |  000$\tau_{e}\tau_{h1}$ |  000$\tau_{e}\tau_{h3}$ |  000$\tau_{\mu}\tau_{e}$
---|---|---|---|---|---|---|---
${Z}\\!\rightarrow l^{+}l^{-}$ |  $249.7$ p m 8.8 |  $1.2$ p m 0.5 |  000— |  $420.8$ p m 25.3 |  $16.1$ p m 2.2 |  000— |  $25.3$ p m 5.4
QCD |  $50.9$ p m 10.2 |  $235.8$ p m 19.3 |  $21.2$ p m 5.3 |  $42.7$ p m 8.8 |  $330.8$ p m 22.8 |  $19.4$ p m 5.1 |  $160.0$ p m 16.9
$V\\!j$ |  $12.7$ p m 7.4 |  $144.2$ p m 43.0 |  $5.1$ p m 3.4 |  $5.8$ p m 2.7 |  $68.3$ p m 19.7 |  $10.1$ p m 5.8 |  $65.3$ p m 25.7
$VV$ |  $0.2$ p m 0.1 |  $1.2$ p m 0.2 |  $0.2$ p m 0.1 |  $0.2$ p m 0.1 |  $0.8$ p m 0.1 |  $0.2$ p m 0.1 |  $10.0$ p m 0.5
${t}{\overline{t}}$ |  $1.0$ p m 0.2 |  $2.2$ p m 0.2 |  $0.6$ p m 0.1 |  $0.2$ p m 0.0 |  $0.7$ p m 0.1 |  $0.1$ p m 0.0 |  $5.5$ p m 0.2
${Z}\\!\rightarrow{{b}{\overline{b}}}$ |  $0.8$ p m 0.4 |  $0.3$ p m 0.2 |  $0.1$ p m 0.1 |  $0.1$ p m 0.1 |  $0.3$ p m 0.2 |  $0.1$ p m 0.1 |  $0.3$ p m 0.2
Cross-feed |  $4.5$ p m 1.1 |  $22.2$ p m 2.5 |  $13.9$ p m 2.0 |  $13.0$ p m 3.9 |  $16.5$ p m 2.4 |  $7.3$ p m 1.7 |  $52.5$ p m 4.2
Total bkg. |  $319.9$ p m 12.7 |  $407.1$ p m 37.5 |  $41.1$ p m 5.3 |  $482.7$ p m 24.2 |  $433.5$ p m 22.0 |  $37.2$ p m 5.8 |  $318.9$ p m 23.6
Observed | $696$ | $1373$ | $205$ | $610$ | $861$ | $110$ | $1322$
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ |  $376.1$ p m 29.0 |  $965.9$ p m 52.1 |  $163.9$ p m 14.2 |  $127.3$ p m 32.9 |  $427.5$ p m 35.8 |  $72.8$ p m 11.1 |  $1003.1$ p m 41.8

##  5 Cross-section measurement

The production cross-section of $Z$ boson to tau-pair is measured for each
analysis stream using

$\sigma_{pp\\!\rightarrow{Z}\\!\rightarrow{\tau^{+}\tau^{-}}}=\frac{N_{\text{obs}}/\varepsilon_{\text{rec}}^{\text{obs}}-\sum_{\rm
k}N_{\text{bkg,k}}/\varepsilon_{\text{rec}}^{\text{bkg,k}}}{\mathcal{L}\;{\mathcal{B}}\;\mathcal{A}\;\varepsilon_{\text{sel}}},$
(2)

where $N_{\text{obs}}$ is the number of observed $Z$ bosons and
$N_{\text{bkg,k}}$ is the estimated background from source k.

The total integrated luminosity is denoted by $\mathcal{L}$, and $\mathcal{B}$
is the product of the branching fractions of the tau lepton pair to decay to
the given final state, with values and uncertainties taken from the world
averages [30]. The acceptance factor, $\mathcal{A}$, is needed to normalise
the results of each analysis stream to the kinematical region
$60<M_{\tau\tau}<120~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
$2.0<\eta^{\tau}<4.5$, and
$p_{\mathrm{T}}^{\tau}>20~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, which
allows the comparison with the ${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$,
${Z}\\!\rightarrow{e}^{+}{e}^{-}$ decay measurements in LHCb [12, 13]. This
factor is the fraction of ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ events where
the generated $\tau$ satisfy the chosen kinematical selections, which also
fulfill the fiducial acceptance selection. The value of $\mathcal{A}$ for each
stream is obtained from simulation, using the POWHEG-BOX [31, *Nason:2004rx,
*Frixione:2007vw, *Alioli:2010xd] at next-to-leading order with PDF
MSTW08NLO90cl [35], and Pythia 8.175 [19, *Sjostrand:2006za]. The uncertainty
on $\mathcal{A}$ from the choice of PDF is estimated following the procedure
explained in Ref. [36].

The event reconstruction and selection efficiencies,
$\varepsilon_{\text{rec}}$ and $\varepsilon_{\text{sel}}$, as well as their
uncertainties, are estimated from simulation and calibrated using a data-
driven method (where applicable) derived from the method described in Refs.
[11, 12, 13]. The term $\varepsilon_{\text{rec}}$ is the product of the GEC,
trigger, tracking and particle identification efficiencies. The smallest value
of $\varepsilon_{\text{rec}}$ is found to be 9% in the $\tau_{e}\tau_{h3}$
stream, while the largest value is 65% for $\tau_{\mu}\tau_{\mu}$. The GEC
efficiency is determined from ${Z}\\!\rightarrow l^{+}l^{-}$ decays in data
collected with a relaxed requirement. The muon and electron trigger
efficiencies are evaluated as a function of $\eta$ and $p_{\mathrm{T}}$ using
a tag-and-probe method applied on ${Z}\\!\rightarrow l^{+}l^{-}$ decays. The
tracking efficiency for muons uses a tag-and-probe method from
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ decays in data, whereas for electrons
and charged hadrons simulated samples are used. The particle identification
efficiency is also obtained by a tag-and-probe procedure. In order to cover
the signal $p_{\mathrm{T}}$ spectrum, different data samples are selected:
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\;{\mu^{+}\mu^{-}}$
decays for muons, ${Z}\\!\rightarrow{e}^{+}{e}^{-}$ and
${{{B}^{+}}}\rightarrow\;{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{(\rightarrow\;{e^{+}e^{-}})}{{K}^{+}}$ decays for electrons, and
${{D}^{*+}}\rightarrow\;{{D}^{0}}{(\rightarrow\;{{K}^{-}}{{\pi}^{+}})}{{\pi}^{+}}$
decays for charged hadrons.

The efficiency of the selection ranges between 20% for $\tau_{e}\tau_{e}$ and
50% for $\tau_{\mu}\tau_{e}$. The values are obtained from the simulation.
Corrections at the level of 1% are inferred by the comparison of the
selection-variable distributions for ${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$
decays in data and simulated samples, which are also added to the systematic
uncertainty.

Table 2:  Relative uncertainties of the various contributions affecting the cross-section measurement, given in percent. The uncertainties are correlated between streams, except in rows denoted with †. | $\tau_{\mu}\tau_{\mu}$ | $\tau_{\mu}\tau_{h1}$ | $\tau_{\mu}\tau_{h3}$ | $\tau_{e}\tau_{e}$ | $\tau_{e}\tau_{h1}$ | $\tau_{e}\tau_{h3}$ | $\tau_{\mu}\tau_{e}$
---|---|---|---|---|---|---|---
Tau branching fractions product | 0.5 | 0.3 | 0.5 | 0.5 | 0.3 | 0.5 | 0.3
PDF, acceptance, FSR | 1.3 | 1.9 | 1.5 | 1.3 | 1.9 | 1.5 | 1.3
Reconstruction | 2.1 | 3.1 | 5.6 | 4.5 | 5.4 | 7.0 | 2.7
Selection | 5.0 | 3.5 | 4.7 | 5.7 | 3.5 | 5.1 | 3.9
Background estimation† | 3.4 | 3.9 | 3.2 | 19.0 | 5.2 | 8.0 | 2.4
Systematic | 6.4 | 6.2 | 8.0 | 20.3 | 8.4 | 11.8 | 5.2
Statistical† | 6.9 | 3.8 | 8.1 | 17.6 | 6.6 | 13.1 | 3.4
Beam energy | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2
Luminosity | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2
Total | 9.6 | 7.5 | 11.5 | 27.0 | 10.8 | 17.7 | 6.5

A summary of uncertainties is given in Table 2, with the statistical
uncertainty from $N_{\text{obs}}$ obtained assuming Poissonian statistics. The
contribution of the LHC beam energy uncertainty [37] is of 0.2% as studied
with the Dynnlo generator [38]. The integrated luminosity is measured using
van der Meer scans [39] and beam-gas imaging method [40], giving a combined
uncertainty of 1.2% [18].

The cross-section results compared with the previous
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ and ${Z}\\!\rightarrow{e}^{+}{e}^{-}$
measurements inside the same acceptance region at
8$\mathrm{\,Te\kern-1.00006ptV}$ [13, 12], are presented in Fig. 3, where the
region is defined for $Z$ bosons with an invariant mass between 60 and
120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ decaying to leptons with
$p_{\mathrm{T}}>20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<\eta<4.5$.
The predictions from theoretical models are calculated with the Fewz [41, 42]
generator at NNLO for the PDF sets ABM12 [43], CT10 [44], CT14 [45], HERA15
[46], MSTW08 [35], MMHT14 [47], and NNPDF30 [48]. A best linear unbiased
estimator is used to combine the measurements from all streams taking into
account their correlations, giving a $\chi^{2}$ per degree of freedom of 0.69
($p$-value of 0.658). The combined cross-section is

$\sigma_{pp\\!\rightarrow{Z}\\!\rightarrow{\tau^{+}\tau^{-}}}=95.8\pm 2.1\pm
4.6\pm 0.2\pm 1.1\mathrm{\,pb}\,,$

where the uncertainties are statistical, systematic, due to the LHC beam
energy uncertainty, and to the integrated luminosity uncertainty,
respectively.

Lepton universality is tested from the cross-section ratios [12, 13]

$\frac{\sigma^{8\mathrm{\,Te\kern-0.70004ptV}}_{pp\\!\rightarrow{Z}\\!\rightarrow{\tau^{+}\tau^{-}}}}{\sigma^{8\mathrm{\,Te\kern-0.70004ptV}}_{pp\\!\rightarrow{Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}}}=1.01\pm
0.05\quad,\quad\frac{\sigma^{8\mathrm{\,Te\kern-0.70004ptV}}_{pp\\!\rightarrow{Z}\\!\rightarrow{\tau^{+}\tau^{-}}}}{\sigma^{8\mathrm{\,Te\kern-0.70004ptV}}_{pp\\!\rightarrow{Z}\\!\rightarrow{e}^{+}{e}^{-}}}=1.02\pm
0.06\,,$

where the uncertainties due to the LHC beam energy and to the integrated
luminosity are assumed to be fully correlated as the analyses share the same
dataset, whilst the statistical and systematic uncertainties are assumed to be
uncorrelated.

Figure 3:  Summary of the measurements of ${Z}\\!\rightarrow l^{+}l^{-}$
production cross-section inside the LHCb acceptance region from $pp$
collisions at 8$\mathrm{\,Te\kern-1.00006ptV}$. The error bar represents the
total uncertainty. The dotted inner error bar corresponds to the statistical
contribution. The coloured band corresponds to the combined measurement of
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ from this analysis. The last 7 rows
represent the NNLO predictions with different parameterizations of the PDFs.

##  6 Conclusion

A measurement of ${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ production cross-
section in $pp$ collisions at $\sqrt{s}$ = 8$\mathrm{\,Te\kern-1.00006ptV}$
inside LHCb fiducial acceptance region is reported, where the region is
defined as a tau-pair of invariant mass between 60 and
120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, with the tau leptons having a
transverse momentum greater than 20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and
pseudorapidity between 2.0 and 4.5.

The reconstruction of tau-pair candidates is performed in both leptonic and
hadronic decay modes of the tau lepton, requiring at least one leptonic mode
for the tau-pair combination. The backgrounds to
${Z}\\!\rightarrow{\tau^{+}\tau^{-}}$ are mainly from QCD and $W$/$Z$ +jets
and are estimated with a data-driven method.

The production cross-section with all uncertainties summed in quadrature
yields $95.8\pm 5.2$$\mathrm{\,pb}$, in agreement with the SM prediction. The
results are consistent with the ${Z}\\!\rightarrow{e}^{+}{e}^{-}$ and
${Z}\\!\rightarrow{\mu}^{+}{\mu}^{-}$ cross-sections measured at LHCb. They
are compatible with LU at the level of 6%.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open-source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France), Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China), RFBR, RSF
and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund,
the Royal Society, the English-Speaking Union and the Leverhulme Trust (United
Kingdom).

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LHCb collaboration

R. Aaij27, B. Adeva41, M. Adinolfi48, C.A. Aidala73, Z. Ajaltouni5, S. Akar59,
P. Albicocco18, J. Albrecht10, F. Alessio42, M. Alexander53, A. Alfonso
Albero40, S. Ali27, G. Alkhazov33, P. Alvarez Cartelle55, A.A. Alves Jr59, S.
Amato2, S. Amerio23, Y. Amhis7, L. An3, L. Anderlini17, G. Andreassi43, M.
Andreotti16,g, J.E. Andrews60, R.B. Appleby56, F. Archilli27, P. d’Argent12,
J. Arnau Romeu6, A. Artamonov39, M. Artuso61, K. Arzymatov37, E. Aslanides6,
M. Atzeni44, S. Bachmann12, J.J. Back50, S. Baker55, V. Balagura7,b, W.
Baldini16, A. Baranov37, R.J. Barlow56, S. Barsuk7, W. Barter56, F.
Baryshnikov70, V. Batozskaya31, B. Batsukh61, V. Battista43, A. Bay43, J.
Beddow53, F. Bedeschi24, I. Bediaga1, A. Beiter61, L.J. Bel27, N. Beliy63, V.
Bellee43, N. Belloli20,i, K. Belous39, I. Belyaev34,42, E. Ben-Haim8, G.
Bencivenni18, S. Benson27, S. Beranek9, A. Berezhnoy35, R. Bernet44, D.
Berninghoff12, E. Bertholet8, A. Bertolin23, C. Betancourt44, F. Betti15,42,
M.O. Bettler49, M. van Beuzekom27, Ia. Bezshyiko44, S. Bhasin48, J. Bhom29, S.
Bifani47, P. Billoir8, A. Birnkraut10, A. Bizzeti17,u, M. Bjørn57, M.P.
Blago42, T. Blake50, F. Blanc43, S. Blusk61, D. Bobulska53, V. Bocci26, O.
Boente Garcia41, T. Boettcher58, A. Bondar38,w, N. Bondar33, S. Borghi56,42,
M. Borisyak37, M. Borsato41,42, F. Bossu7, M. Boubdir9, T.J.V. Bowcock54, C.
Bozzi16,42, S. Braun12, M. Brodski42, J. Brodzicka29, D. Brundu22, E.
Buchanan48, A. Buonaura44, C. Burr56, A. Bursche22, J. Buytaert42, W.
Byczynski42, S. Cadeddu22, H. Cai64, R. Calabrese16,g, R. Calladine47, M.
Calvi20,i, M. Calvo Gomez40,m, A. Camboni40,m, P. Campana18, D.H. Campora
Perez42, L. Capriotti56, A. Carbone15,e, G. Carboni25, R. Cardinale19,h, A.
Cardini22, P. Carniti20,i, L. Carson52, K. Carvalho Akiba2, G. Casse54, L.
Cassina20, M. Cattaneo42, G. Cavallero19,h, R. Cenci24,p, D. Chamont7, M.G.
Chapman48, M. Charles8, Ph. Charpentier42, G. Chatzikonstantinidis47, M.
Chefdeville4, V. Chekalina37, C. Chen3, S. Chen22, S.-G. Chitic42, V.
Chobanova41, M. Chrzaszcz42, A. Chubykin33, P. Ciambrone18, X. Cid Vidal41, G.
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A. Comerma-Montells12, A. Contu22, G. Coombs42, S. Coquereau40, G. Corti42, M.
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Filippov36, R.A. Fini14, M. Fiorini16,g, M. Firlej30, C. Fitzpatrick43, T.
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Féo Pereira Rivello Carvalho27, E. Gabriel52, A. Gallas Torreira41, D.
Galli15,e, S. Gallorini23, S. Gambetta52, M. Gandelman2, P. Gandini21, Y.
Gao3, L.M. Garcia Martin72, B. Garcia Plana41, J. García Pardiñas44, J. Garra
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Gerick12, E. Gersabeck56, M. Gersabeck56, T. Gershon50, D. Gerstel6, Ph.
Ghez4, S. Gianì43, V. Gibson49, O.G. Girard43, L. Giubega32, K. Gizdov52, V.V.
Gligorov8, D. Golubkov34, A. Golutvin55,70, A. Gomes1,a, I.V. Gorelov35, C.
Gotti20,i, E. Govorkova27, J.P. Grabowski12, R. Graciani Diaz40, L.A. Granado
Cardoso42, E. Graugés40, E. Graverini44, G. Graziani17, A. Grecu32, R.
Greim27, P. Griffith22, L. Grillo56, L. Gruber42, B.R. Gruberg Cazon57, O.
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Johnson42, C.R. Jones49, C. Joram42, B. Jost42, N. Jurik57, S. Kandybei45, M.
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Keizer49, M. Kelsey61, M. Kenzie49, T. Ketel28, E. Khairullin37, B. Khanji12,
C. Khurewathanakul43, K.E. Kim61, T. Kirn9, S. Klaver18, K. Klimaszewski31, T.
Klimkovich11, S. Koliiev46, M. Kolpin12, R. Kopecna12, P. Koppenburg27, I.
Kostiuk27, S. Kotriakhova33, M. Kozeiha5, L. Kravchuk36, M. Kreps50, F.
Kress55, P. Krokovny38,w, W. Krupa30, W. Krzemien31, W. Kucewicz29,l, M.
Kucharczyk29, V. Kudryavtsev38,w, A.K. Kuonen43, T. Kvaratskheliya34,42, D.
Lacarrere42, G. Lafferty56, A. Lai22, D. Lancierini44, G. Lanfranchi18, C.
Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, A. Leflat35, J.
Lefrançois7, R. Lefèvre5, F. Lemaitre42, O. Leroy6, T. Lesiak29, B.
Leverington12, P.-R. Li63, T. Li3, Z. Li61, X. Liang61, T. Likhomanenko69, R.
Lindner42, F. Lionetto44, V. Lisovskyi7, X. Liu3, D. Loh50, A. Loi22, I.
Longstaff53, J.H. Lopes2, G.H. Lovell49, D. Lucchesi23,o, M. Lucio Martinez41,
A. Lupato23, E. Luppi16,g, O. Lupton42, A. Lusiani24, X. Lyu63, F. Machefert7,
F. Maciuc32, V. Macko43, P. Mackowiak10, S. Maddrell-Mander48, O. Maev33,42,
K. Maguire56, D. Maisuzenko33, M.W. Majewski30, S. Malde57, B. Malecki29, A.
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Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin Benito40, M. Marinangeli43,
P. Marino43, J. Marks12, G. Martellotti26, M. Martin6, M. Martinelli42, D.
Martinez Santos41, F. Martinez Vidal72, A. Massafferri1, R. Matev42, A.
Mathad50, Z. Mathe42, C. Matteuzzi20, A. Mauri44, E. Maurice7,b, B. Maurin43,
A. Mazurov47, M. McCann55,42, A. McNab56, R. McNulty13, J.V. Mead54, B.
Meadows59, C. Meaux6, F. Meier10, N. Meinert67, D. Melnychuk31, M. Merk27, A.
Merli21,q, E. Michielin23, D.A. Milanes66, E. Millard50, M.-N. Minard4, L.
Minzoni16,g, D.S. Mitzel12, A. Mogini8, J. Molina Rodriguez1,z, T.
Mombächer10, I.A. Monroy66, S. Monteil5, M. Morandin23, G. Morello18, M.J.
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K. Müller44, V. Müller10, P. Naik48, T. Nakada43, R. Nandakumar51, A. Nandi57,
T. Nanut43, I. Nasteva2, M. Needham52, N. Neri21, S. Neubert12, N. Neufeld42,
M. Neuner12, T.D. Nguyen43, C. Nguyen-Mau43,n, S. Nieswand9, R. Niet10, N.
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S. Ogilvy18, R. Oldeman22,f, C.J.G. Onderwater68, A. Ossowska29, J.M. Otalora
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Patel55, C. Patrignani15,e, A. Pearce42, A. Pellegrino27, G. Penso26, M. Pepe
Altarelli42, S. Perazzini42, D. Pereima34, P. Perret5, L. Pescatore43, K.
Petridis48, A. Petrolini19,h, A. Petrov69, S. Petrucci52, M. Petruzzo21,q, B.
Pietrzyk4, G. Pietrzyk43, M. Pikies29, M. Pili57, D. Pinci26, J. Pinzino42, F.
Pisani42, A. Piucci12, V. Placinta32, S. Playfer52, J. Plews47, M. Plo
Casasus41, F. Polci8, M. Poli Lener18, A. Poluektov50, N. Polukhina70,c, I.
Polyakov61, E. Polycarpo2, G.J. Pomery48, S. Ponce42, A. Popov39, D.
Popov47,11, S. Poslavskii39, C. Potterat2, E. Price48, J. Prisciandaro41, C.
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Rademacker48, M. Rama24, M. Ramos Pernas41, M.S. Rangel2, F. Ratnikov37,x, G.
Raven28, M. Ravonel Salzgeber42, M. Reboud4, F. Redi43, S. Reichert10, A.C.
dos Reis1, F. Reiss8, C. Remon Alepuz72, Z. Ren3, V. Renaudin7, S.
Ricciardi51, S. Richards48, K. Rinnert54, P. Robbe7, A. Robert8, A.B.
Rodrigues43, E. Rodrigues59, J.A. Rodriguez Lopez66, M. Roehrken42, A.
Rogozhnikov37, S. Roiser42, A. Rollings57, V. Romanovskiy39, A. Romero
Vidal41, M. Rotondo18, M.S. Rudolph61, T. Ruf42, J. Ruiz Vidal72, J.J.
Saborido Silva41, N. Sagidova33, B. Saitta22,f, V. Salustino Guimaraes62, C.
Sanchez Gras27, C. Sanchez Mayordomo72, B. Sanmartin Sedes41, R.
Santacesaria26, C. Santamarina Rios41, M. Santimaria18, E. Santovetti25,j, G.
Sarpis56, A. Sarti18,k, C. Satriano26,s, A. Satta25, M. Saur63, D.
Savrina34,35, S. Schael9, M. Schellenberg10, M. Schiller53, H. Schindler42, M.
Schmelling11, T. Schmelzer10, B. Schmidt42, O. Schneider43, A. Schopper42,
H.F. Schreiner59, M. Schubiger43, M.H. Schune7, R. Schwemmer42, B. Sciascia18,
A. Sciubba26,k, A. Semennikov34, E.S. Sepulveda8, A. Sergi47,42, N. Serra44,
J. Serrano6, L. Sestini23, P. Seyfert42, M. Shapkin39, Y. Shcheglov33,†, T.
Shears54, L. Shekhtman38,w, V. Shevchenko69, E. Shmanin70, B.G. Siddi16, R.
Silva Coutinho44, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, N.
Skidmore12, T. Skwarnicki61, J.G. Smeaton49, E. Smith9, I.T. Smith52, M.
Smith55, M. Soares15, l. Soares Lavra1, M.D. Sokoloff59, F.J.P. Soler53, B.
Souza De Paula2, B. Spaan10, P. Spradlin53, F. Stagni42, M. Stahl12, S.
Stahl42, P. Stefko43, S. Stefkova55, O. Steinkamp44, S. Stemmle12, O.
Stenyakin39, M. Stepanova33, H. Stevens10, S. Stone61, B. Storaci44, S.
Stracka24,p, M.E. Stramaglia43, M. Straticiuc32, U. Straumann44, S. Strokov71,
J. Sun3, L. Sun64, K. Swientek30, V. Syropoulos28, T. Szumlak30, M.
Szymanski63, S. T’Jampens4, Z. Tang3, A. Tayduganov6, T. Tekampe10, G.
Tellarini16, F. Teubert42, E. Thomas42, J. van Tilburg27, M.J. Tilley55, V.
Tisserand5, S. Tolk42, L. Tomassetti16,g, D. Tonelli24, D.Y. Tou8, R. Tourinho
Jadallah Aoude1, E. Tournefier4, S. Tourneur43, M. Traill53, M.T. Tran43, A.
Trisovic49, A. Tsaregorodtsev6, A. Tully49, N. Tuning27,42, A. Ukleja31, A.
Usachov7, A. Ustyuzhanin37, U. Uwer12, C. Vacca22,f, A. Vagner71, V.
Vagnoni15, A. Valassi42, S. Valat42, G. Valenti15, R. Vazquez Gomez42, P.
Vazquez Regueiro41, S. Vecchi16, M. van Veghel27, J.J. Velthuis48, M.
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Viemann67, X. Vilasis-Cardona40,m, A. Vitkovskiy27, M. Vitti49, V. Volkov35,
A. Vollhardt44, B. Voneki42, A. Vorobyev33, V. Vorobyev38,w, J.A. de Vries27,
C. Vázquez Sierra27, R. Waldi67, J. Walsh24, J. Wang61, M. Wang3, Y. Wang65,
Z. Wang44, D.R. Ward49, H.M. Wark54, N.K. Watson47, D. Websdale55, A.
Weiden44, C. Weisser58, M. Whitehead9, J. Wicht50, G. Wilkinson57, M.
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F.F. Wilson51,42, J. Wimberley60, M. Winn7, J. Wishahi10, W. Wislicki31, M.
Witek29, G. Wormser7, S.A. Wotton49, K. Wyllie42, D. Xiao65, Y. Xie65, A. Xu3,
M. Xu65, Q. Xu63, Z. Xu3, Z. Xu4, Z. Yang3, Z. Yang60, Y. Yao61, L.E.
Yeomans54, H. Yin65, J. Yu65,ab, X. Yuan61, O. Yushchenko39, K.A. Zarebski47,
M. Zavertyaev11,c, D. Zhang65, L. Zhang3, W.C. Zhang3,aa, Y. Zhang7, A.
Zhelezov12, Y. Zheng63, X. Zhu3, V. Zhukov9,35, J.B. Zonneveld52, S.
Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
8LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14INFN Sezione di Bari, Bari, Italy
15INFN Sezione di Bologna, Bologna, Italy
16INFN Sezione di Ferrara, Ferrara, Italy
17INFN Sezione di Firenze, Firenze, Italy
18INFN Laboratori Nazionali di Frascati, Frascati, Italy
19INFN Sezione di Genova, Genova, Italy
20INFN Sezione di Milano-Bicocca, Milano, Italy
21INFN Sezione di Milano, Milano, Italy
22INFN Sezione di Cagliari, Monserrato, Italy
23INFN Sezione di Padova, Padova, Italy
24INFN Sezione di Pisa, Pisa, Italy
25INFN Sezione di Roma Tor Vergata, Roma, Italy
26INFN Sezione di Roma La Sapienza, Roma, Italy
27Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
28Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
29Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
30AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
31National Center for Nuclear Research (NCBJ), Warsaw, Poland
32Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
33Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
34Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
35Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
36Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
37Yandex School of Data Analysis, Moscow, Russia
38Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
39Institute for High Energy Physics (IHEP), Protvino, Russia
40ICCUB, Universitat de Barcelona, Barcelona, Spain
41Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
42European Organization for Nuclear Research (CERN), Geneva, Switzerland
43Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
44Physik-Institut, Universität Zürich, Zürich, Switzerland
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 27
69National Research Centre Kurchatov Institute, Moscow, Russia, associated to
34
70National University of Science and Technology ”MISIS”, Moscow, Russia,
associated to 34
71National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
34
72Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 40
73University of Michigan, Ann Arbor, United States, associated to 61
74Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xNational Research University Higher School of Economics, Moscow, Russia
ySezione INFN di Trieste, Trieste, Italy
zEscuela Agrícola Panamericana, San Antonio de Oriente, Honduras
aaSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
abPhysics and Micro Electronic College, Hunan University, Changsha City, China
†Deceased


# Short-term prediction of photovoltaic power generation using Gaussian
process regression

Yahya Al Lawati*, Jack Kelly+ and Dan Stowell  *School of Electronic
Engineering and Computer Science, Queen Mary University of London, London, UK
+Open Climate Fix, London, UK

###### Abstract

Photovoltaic (PV) power is affected by weather conditions, making the power
generated from the PV systems uncertain. Solving this problem would help
improve the reliability and cost effectiveness of the grid, and could help
reduce reliance on fossil fuel plants. The present paper focuses on evaluating
predictions of the energy generated by PV systems in the United Kingdom
Gaussian process regression (GPR). Gaussian process regression is a Bayesian
non-parametric model that can provide predictions along with the uncertainty
in the predicted value, which can be very useful in applications with a high
degree of uncertainty. The model is evaluated for short-term forecasts of 48
hours against three main factors – training period, sky area coverage and
kernel model selection – and for very short-term forecasts of four hours
against sky area. We also compare very short-term forecasts in terms of cloud
coverage within the prediction period and only initial cloud coverage as a
predictor.

##  I Introduction

Fossil fuels are the primary source of energy worldwide, accounting for 84% of
primary energy use [1]. However, the world seeks alternatives, and renewable
energy has gained interest: its consumption share has grown strongly to over
40% (excluding hydroelectricity) of the global growth in primary energy in
2019 compared to 2018, with solar and wind power being the greatest
beneficiaries [1]. The chief concern that accompanies solar energy is its
uncertainty, which is mainly affected by weather conditions [2]. Being able to
predict power generation mere hours ahead would help control the amount that
must be generated from fossil fuels, reducing the amount of carbon dioxide
(CO2) emissions produced. There are two main forecasting methods: model-based
approaches and data-driven approaches. An example of model-based approaches is
numerical weather prediction (NWP) which employs a set of equations to
describe the flow of fluids. Model-based approaches can be complicated and
computationally costly. Data-driven approaches, on the other hand, do not use
any physical model. They are easier to implement and require no prior
knowledge about weather forecasts. Forecasting techniques can be deterministic
or probabilistic [2][3]. Probabilistic approaches can predict a range of
values with probabilities, and thus are more appropriate for forecasting
applications.

In this work we evaluate the accuracy of short-term predictions of the amount
of energy generated from photovoltaic (PV) systems using Gaussian process
regression (GPR), trained using historical data of power output from PV
systems together with cloud coverage obtained from high-resolution visible
(HRV) satellite images.

Integrating PV systems into the grid cannot be efficient unless there is
reliable information about power forecast [7]. This is because, whilst it is
possible to handle small energy drops by generating slightly energy more than
demand, significant power drops must be replaced within seconds. Fossil fuel
plants can take hours to spin from start, and storage battery devices are
costly and inefficient for large quantities of photovoltaic panels [8, 9].
Short-term energy prediction can help to determine how much spinning reserve
to schedule [8]. Spinning reserve is the unused power capacity which can be
activated on the authority of the system operator to serve additional demand
[10]. Numerical weather prediction models are useful for predicting elements
such as temperature and wind speed, especially in the short term. However,
predicting solar irradiance can be inaccurate and very short-term predictions
might be inapplicable, for several reasons [8, 11]:

  * •

Numerical weather prediction models require at least an hour to execute. This
is a problem, especially for cloud forecasting, as it can change rapidly.

  * •

There is no high demand for solar irradiance.

  * •

Solar irradiance is computationally expensive to calculate.

The gap between the necessity and difficulty of predictions which use
conventional forecasts opens the door for machine learning. Different solar
forecasting models have been developed throughout the years, some of which
have achieved great success in solar forecasting. Artificial neural networks
[12, 13] and support vector machines [3, 14] are examples of popular models.
Artificial Neural networks are known for their ability to model complex and
non-linear relationships. For time-series applications Gaussian processes
(GPs) appear to be recommended for their ability to handle complex
relationships [10]. Our appendix explains GPs in more detail; we next discuss
research conducted on GPs for PV modelling.

Weighted Gaussian Process Regression (WGPR) [2] assigns weights to each data
sample, such that high outliers will have lower weights. The forecasting
process involves multiple levels, and the model considers eight attributes:
solar radiation, photosynthetic active radiation (PAR), ambient temperature,
wind speed, gust speed, precipitation, wind direction, and humidity. The model
has a training data period of approximately 33 days and a validation period of
five consecutive days with five-minute intervals. Solar radiation and
photosynthetic active radiation (PAR) were the most important predictors
amongst the eight considered.

Gaussian process quantile regression (GPQR) model is discussed in [10, 14].
Quantile regression aims to estimate the quantiles of the conditional
distribution of predicted values given predictor values, which makes the model
more robust against outlier predictions. In addition, it captures the
relationship between input and output variables. The GP model is trained with
data from 20 days to predict the power load for the following seven days. The
results show that GPQR performed better than GPR by 0.26% on average (using
mean absolute percentage error, MAPE) for one-hour predictions and 0.49% on
average for two-hour predictions.

Other papers have discussed models based on GPs [16, 17, 18]. Both the models
we discussed show better results when compared to GPR in their own
comparisons. However, the accuracy difference is not large, and the added
sophistication may imply heavier computational demands.

According to [14], clouds are one of the factors that have a significant
influence on solar radiation. It is essential to note that, other than the
issues mentioned regarding NWP, cloud coverage data are parameterised, whilst
depending on satellite images can provide more accurate readings [8].
Parameterisation means that the information is converted to an abstract value
that presents general knowledge. For example, instead of resolving clouds,
general information is provided, such as whether the day was sunny or cloudy.

The question raised in this paper is: “Can a simple Gaussian Process
Regression model accuracy enhance by depending on satellite images instead of
Numerical Weather Predictions?” This paper’s contributions can be summarised
as:

  * •

Exploring different factors that can enhance short-term forecasts (48 hours)
according to satellite images.

  * •

Exploring different factors that can enhance very short-term forecasts (four
hours) according to satellite images at each point of the forecast and initial
satellite images at the moment of the forecast.

  * •

Comparing forecasts depending on cloud coverage at each point and the initial
cloud coverage for very short-term forecasts (four hours).

##  II Methodology

The model was trained based on historical data and cloud coverage. The
experiments were divided into two sets. The first set of experiments examined
three factors in order to obtain the most accurate solar power forecasts for
the following 48 hours: training period, sky coverage area and kernel
structure. The second set of experiments examined very short-term predictions
for the subsequent four hours in terms of sky coverage area. In addition, a
comparison was made between forecasts with consideration of cloud coverage
within the prediction period and forecasts with only initial cloud coverage
provided. In addition, a third set of experiments was conducted to test the
influence of the periodic kernel on model accuracy. However, the model failed
to follow the trend of the actual power generated. Hence, this set of
experiments is not discussed in the paper.
The results of the first and second sets evaluated amongst four different PV
systems in the United Kingdom. The paper assumes that the cloud forecast data
will be provided either by cloud forecasting models or by prediction
algorithms in case applied as a real-world application. This section describes
two aspects: data preparation and model accuracy evaluation.

###  II-A Data Preparation

The model uses historical data from given PV systems and cloud coverage to
predict the amount of energy in the following four and 48 hours. It uses three
data sources, including historical data on power generated from different
systems and the metadata of those systems, such as location, system capacity
and other system details. The third source is satellite images in five-minute
intervals. The first step was to identify the geospatial boundary in
transverse Mercator projection (metres) of the United Kingdom. The next step
in the data preparation was to load PV system metadata and convert latitude
and longitude to transverse Mercator projection. Then, systems outside the
United Kingdom were identified and removed from the data. After that, data on
energy generated from systems over time were loaded, and the period was
specified. Data was then aligned with the metadata, in addition to removing
systems with missing metadata and corrupt systems that generated power
overnight. Next, the satellite data were loaded. The time series rearranged
from date and time to integer sequence of numbers where the distance between
two numbers equals to five minutes and mapped with the data. Cloud coverage
was measured through average visible HRV level in the area above the PV
system. Both cloud coverage and time series were merged with the power
generated by the PV system to form the training and testing set for the model.

###  II-B Model Accuracy Evaluation

This subsection explores different factors that affect model accuracy. There
are several well-known methods for evaluating the performance of a prediction
model. Mean square error, root mean square error (RMSE) and MAPE are examples
of error-based measurement. In this paper, MAE is used to evaluate the
accuracy of the model. It is defined as follows:

$MAE(Y,Y*)=\frac{1}{N}\sum_{i=1}^{N}\left|{y_{i}-y*_{i}}\right|$ (1)

Mean absolute error measures how much error expected on average from a PV
system; in this case, it is measured in watts (W). If the model has a MAE of
100 W trained against a PV system, then the predicted value can differ from
the actual value by 100 W on average. The reason why MAE was chosen for this
study is that it helps quantify the actual difference in power load generated
on average. Four PV systems around the United Kingdom were tested in each
trial. The tested PV systems had the numbers 709, 1556, 1627 and 1872, which
were randomly selected from a pool of PV systems, and had a capacity of 2,460
W; 3,870 W; 2,820 W and 3,960 W, respectively. The experiments were divided
into two sets. The first set of experiments predicted the power generated from
PV systems over the next 48 hours by testing different factors, given the
cloud coverage at each point. The first factor to examine from the first set
is the training period. The drawback with the Gaussian process is that it does
not scale well by increasing the number of observations, in which the training
complexity can reach $O(N^{3})$. Consequently, selecting the features that
will be considered in the model and the number of observations that will be
trained are crucial for the training process. In addition, a long training
period adds seasonal uncertainty and efficiency reduction over time, which may
affect accuracy. Four periods are examined to compare training periods and the
accuracy of the model. The periods are one week, two weeks, three weeks and
one month. The second factor examined is the average HRV value of the sky area
above the PV system. If the sky area considered is too large, then the HRV
values will not be relevant and will affect the accuracy of the model. If the
coverage area is too small, then the solar irradiance captured by a PV system
will also be inaccurate. Each pixel on the map represents 1,000 m on the
ground. The tested sky areas were 2 by 2 pixels, 6 by 6 pixels and 12 by 12
pixels. The third factor examined is the kernel. The kernel represents the
prior knowledge about the data and how the data are correlated. The kernel has
a significant effect on the model’s accuracy; therefore, it should be
carefully selected.
Before discussing different kernel choices, it is essential to note that the
white noise kernel is added to the main kernel in all experiments conducted in
this paper. This kernel represents the noise in the data by adding uncertainty
to the observed data, and it does not change the prediction. The white noise
kernel represented by the following:

$K_{WN}(x_{i},x_{j})=\sigma^{2}\delta(i,j)$ (2)

where $\delta$ is the Kronecker delta, and $\sigma^{2}$ is the variance
parameter.
Due to the sun’s movement, power generated from PV systems is periodic in
nature. Accordingly, the main kernel in the model is the periodic kernel.
However, the latter is a wrap kernel and must have a base kernel that inherits
from a stationary kernel. In other words, the periodic kernel is applied in a
domain of a given stationary kernel. The periodic kernel with squared
exponential stationary kernel is given in (3). The squared exponential (SE)
kernel (4), rational quadratic (RQ) kernel (5) and Matern kernel (6) are
examined in this paper:

$\begin{split}K_{per-SE}(x_{j},x_{j};h,w,T)=\\\
h^{2}exp\left(-\frac{1}{2w^{2}}\sin^{2}\left(\pi\left|\frac{x_{i}-x_{j}}{\lambda}\right|\right)\right)\end{split}$
(3)
$K_{SE}=h^{2}exp\left[-\left(\frac{x_{i}-x_{j}}{\lambda}\right)^{2}\right]$
(4)
$K_{RQ}(x_{i},x_{j})=h^{2}\left(1+\frac{(x_{i}-x_{j})^{2}}{\alpha\lambda^{2}}\right)^{-\alpha}$
(5)
$\begin{split}K_{M}(x_{i},x_{j})=h^{2}\frac{1}{\Gamma(v)2^{v-1}}\left(2\sqrt{v}\frac{|x_{i}-x_{j}|}{\lambda}\right)\\\
\mathbb{B}_{v}\left(2\sqrt{v}\frac{|x_{i}-x_{j}|}{\lambda}\right)\end{split}$
(6)

where $h$ is amplitude, $\lambda$ is the input scale, $T$ is the period, $w$
is the roughness (similar to role of $\lambda$ in stationary covariances),
$\Gamma()$ is s standard Gamma function, $\mathbb{B}()$ is a modified Bessel
function of second order, $\alpha$ is known as the index, and $v$ controls the
degree of differentiability of the resultant functions [24] The first two
kernels provide a high degree of freedom with relatively few hyperparameters,
whilst the third kernel is better for rougher and less smooth variations.

The second set of experiments predicted the energy generated over the next
four hours by fixing the training period to three weeks and the periodic
kernel to Matern12 as a base kernel. The experiments compared the accuracy of
the predictions against a sky coverage of 6 by 6 pixels and 12 by 12 pixels
with and without cloud coverage input during the specified period. When cloud
coverage was not considered, the last observed cloud coverage was used to
predict the following four hours.

##  III Results

Results for the first set of experiments (Table I) demonstrate that MAE was
lowest with one week of training and the highest with one month of training.
However, one week of training may not be enough to cover different weather
situations; therefore, three weeks of training was considered as a period in
the next set of tests, as it has the second lowest MAE. Moreover, it is worth
mentioning that there was a significant difference in training time between
each training period, since many GP tasks have complexity of $O(N^{3})$.

TABLE I: Set one Experiments result Training Period |  Sky Coverage |  Kernel Structure |  System 709 |  System 1556 |  System 1627 |  System 1872 |  Average (MAE)
---|---|---|---|---|---|---|---
1 week |  2X2 pixels |  Matern12 |  112.07 |  233.14 |  199.46 |  143.73 |  $\boldsymbol{172.1}$
2 weeks |  |  |  161.9 |  418.8 |  162.1 |  142.15 |  221.24
3 weeks |  |  |  231.54 |  237.17 |  161.1 |  161.08 |  197.72
1 month |  |  |  139.38 |  423.79 |  177 |  288.52 |  257.17
3 weeks |  2X2 |  Matern12 |  231.54 |  237.17 |  161.1 |  161.08 |  197.7225
6X6 |  |  105.94 |  322.23 |  162.5 |  154.79 |  186.365
12X12 |  |  109.59 |  227.2 |  165.89 |  146.6 |  $\boldsymbol{162.32}$
3 weeks |  2X2 |  Squared Exponential |  113.705 |  236.67 |  177.54 |  152.13 |  170.01125
|  Rational Quadratic |  124.1 |  223.95 |  709.615 |  147.52 |  301.29625
|  Matern12 |  109.59 |  227.2 |  165.89 |  146.6 |  $\boldsymbol{162.32}$

Figure 1: Energy prediction vs. actual generation for a PV system with initial
cloud input on a cloudy day with scattered clouds, over 4 hours. MAE: 327.41.

On sky coverage, the model performed better with 12 by 12 pixel coverage for a
48-hour forecast, with average MAE of 162.32. A kernel selection test found
the Matern12 kernel to perform best on average. The conducted experiments
showed that a three-week training period, a 12 by 12 pixel sky coverage and a
periodic kernel with Matern12 base kernel yielded the best results for 48-hour
predictions. As the model made use of cloud cover data, the results may be
affected by the accuracy of cloud coverage prediction. Figure 3 shows MAE for
the recommended settings per tested days.

Figure 2: MAE box plot for 48-hours forecasts presented by testing periods TABLE II: Set two Experiments result System Number | With Cloud Coverage | Without Cloud Coverage
---|---|---
6X6 | 12X12 | 6X6 | 12X12
709 | 145.45 | 285.99 | 322.98 | 327.67
1556 | 457.81 | 842.47 | 527.75 | 837.665
1627 | 180.74 | 192.12 | 196.57 | 202.56
1872 | 240.52 | 224.62 | 317.97 | 320.91
Average | 256.04 | 386.3 | 341.31 | 422.2

We also tested the difference in accuracy with and without cloud cover
information, with a much shorter forecasting period of four hours (Table II,
Figure 3). We analysed four selected days, each day featuring different
weather conditions. The lowest MAE was in a clear sky and highest with
scattered clouds. The model was able to follow the general trend in all cases,
but cloud cover improved predictions. Note that for PV system 1556, errors
were generally larger than for others.

Figure 3: MAE box plot with outliers for four-hour forecasts. Upper: with
consideration of cloud coverage presented to the PV system. Lower: without.

##  IV Conclusion

This paper evaluated three different factors which affect the GPR model when
predicting power generated by PV systems: the training period, sky coverage
area and kernel. The main advantage of a GP model is that not only can it
predict the amount of power generated, but it also captures the uncertainty of
the predicted value at each point. The analysis shows that the Matern12 kernel
is the best one amongst the tested kernels, as it was more flexible and could
capture the uncertainty presented in the generated power. Furthermore, a
training period of three weeks and a sky coverage area of 12 by 12 pixels were
the best choices for 48-hour forecasts. In comparison, a sky coverage area of
6 by 6 pixels was preferable for four-hour forecasts.

The results show that the model was able to accurately predict power generated
by PV systems when cloud coverage was stable. However, the model suffered from
tracking very rapid changes in cloud coverage when the clouds were scattered.
The results also confirmed that the main factor for short-term predictions of
solar power is solar irradiance and that cloud coverage has a significant
effect on power production.

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##  Appendix A Gaussian process

Gaussian process (GP) is a probability distribution over possible functions
that fit a set of points. Having a joint distribution of these random
variables generates a multivariate Gaussian distribution. A univariate
Gaussian distribution can be described by mean and variance.

$\mathcal{N}\sim(\mu,\sigma^{2})$ (7)

where $\mathcal{N}$ represents Gaussian distribution with mean $\mu$ and
variance $\sigma^{2}$.
The joint univariate Gaussian distribution is described instead by mean and a
covariance matrix.

$\mathcal{N}\sim(\mu,\Sigma)$ (8)

where $\mathcal{N}$ represents Gaussian distribution with mean $\mu$ and
covariance $\Sigma$.
The covariance matrix will contains the variance of each random variable,
which expressed diagonally in the matrix, and the covariance between the
random variables. The covariance between variables represents how two
variables are correlated. Assuming that we have two-variable Gaussian
distribution with variables $y_{1},y_{2}$ having $\mu=0$ and covariance matrix
$\begin{bmatrix}1&0.9\\\ 0.9&1\end{bmatrix}$. Figure 6 shows the possible
values that $y_{2}$ can have given $y_{1}$, described by the red Gaussian
distribution as $y_{1}$ and $y_{2}$ are correlated.

Figure 4: Possible values for $y_{2}$ given $y_{1}$

Figure 7 shows an example of a ten-dimensional posterior distribution, with
observations on locations 2, 6, and 8. The black line represents the mean and
the grey area represents the standard deviation.

Figure 5: 10 dimensional posterior distribution with observations on locations
2, 6, and 8 [24]

Thus far, what has been presented is a discrete set of predictions and
observations points. However, in order to model observations with real values,
we require arbitrarily large sets and must calculate its covariance or
otherwise have the covariance generated by the covariance kernel function,
which will provide the covariance element between any arbitrary samples.
Choosing the covariance kernel function insufficient on its own. Other
properties, such as the right length scale that represents the data more
accurately, can still be vague. Such properties are called hyperparameters,
and they define the shape of the covariance kernel function. Finding the best
values for these hyperparameters that represent the given data is called
’training the model’.
For a set of random variables, such as $X=\\{x_{1},x_{2},...,x_{n}\\}$, then
we can define the covariance matrix can be defined as

$\boldsymbol{K}(X,X)=\begin{pmatrix}k(x_{1},x_{1})&k(x_{1},x_{2})&\dotsm&k(x_{1},x_{n})\\\
k(x_{2},x_{1})&k(x_{2},x_{2})&\dotsm&k(x_{2},x_{n})\\\
\vdots&\vdots&\vdots&\vdots\\\
k(x_{n},x_{1})&k(x_{n},x_{2})&\dotsm&k(x_{n},x_{n})\end{pmatrix}$ (9)

The entire function evaluation associated with points in $X$ is a draw from
Multivariate Gaussian distribution.

$p(Y(X))=\mathcal{N}(\mu(X),K(X,X))$ (10)

where $Y=\\{y_{1},y_{2},\dotsm,y_{n}\\}$ are dependent function values,
evaluated with random variables $X=\\{x_{1},x_{2},...,x_{n}\\}$.
Using the Multivariate Gaussian distribution we can obtain posterior
distribution over $\boldsymbol{y*}$ for test datum $\boldsymbol{x*}$ given by

$p(\boldsymbol{y*})=\mathcal{N}(\boldsymbol{m*},\boldsymbol{C*})$ (11)

where,

$\boldsymbol{m*}=\boldsymbol{\mu}(X*)+\boldsymbol{K}(X*,X)\boldsymbol{K}(X,X)^{-1}(Y(X)-\mu(X)),$
(12)
$\boldsymbol{C*}=\boldsymbol{K}(X*,X*)-\boldsymbol{K}(X*,X)\boldsymbol{K}(X,X)^{-1}\boldsymbol{K}(X*,X)^{T}.$
(13)

Figure 8 is an example of Gaussian process plot in which the red points
represent observations and the blue line represents the function that best
fits the observations. The blue area represents the uncertainty of the output
[6][21][22][24].

Figure 6: Example of gaussian Process output [23]