When Strings Surprise

Nissan Itzhaki1,2,3 and Uri Peleg2 1 Department of Physics, Princeton
University, Princeton, NJ 08544, USA

2 School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, 69978,
Israel

3School of Natural Sciences, Institute for Advanced Study

1 Einstein Drive, Princeton, NJ 08540, USA

###### Abstract

We argue that on-shell excitations with large negative energies are created
rapidly when the string coupling increases with time. This does not indicate
an inconsistency in string theory since the negative energy on-shell
excitation is always entangled with an on-shell excitation with a positive
energy. The total energy of this energy-EPR state vanishes. We discuss the
reason the energy-EPR states appear in string theory and the role they might
play in black hole physics.

It is fair to say that thirty years after Joe Polchinski posed the question
[1] there is still confusion about what string theory is. This short note aims
to add to the confusion.

We claim that in soft backgrounds ($\alpha^{\prime}R\ll 1$ and
$g_{s}=e^{\Phi}\ll 1$) in which the string coupling grows slowly with time,
on-shell excitations with large negative energies are rapidly created. These
excitations do not imply inconsistency since they do not appear by themselves.
An on-shell excitation with large negative energy is always created together
with another on-shell excitation with positive energy such that the total
energy vanishes. Schematically, the state takes the form

$|\Psi\rangle=\int dE_{1}|E_{1}\rangle\otimes|E_{2}=-E_{1}\rangle.$ (1)

Namely, we argue that string theory admits energy-EPR states where the total
energy (rather than total spin in [2, 3]) vanishes. The vacuum in quantum
field theory is filled with pairs of this form. The difference is that here
the excitations are on-shell. This means, in particular, that an observer can
take her time in making a precise measurement of the energy of, say, the first
excitation and obtain a negative value, $E_{1}<0$.

We start by considering the simplest background in which $\partial_{\mu}\Phi$
is time-like and points to the future: a time-like linear dilaton background

$ds^{2}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}+\mbox{Compact},\quad\Phi=Qt.$ (2)

We work with $\alpha^{\prime}=1$ and take $Q>0$, so that the string coupling
constant, $g_{s}=e^{Qt}$, blows up in the future. We consider $Q\ll 1$ which
means that, at least naively, a low-energy effective description is expected
to be valid. As we shall see the creation of the energy-EPR states occurs long
before approaching the strong coupling region at large $t$. To illustrate the
existence of states like (1) we need only one non-compact space-like
direction. We take the number of non-compact space-like directions to be three
to emphasize that this could happen in our world.111As we shall see the
process that leads to (1) is a local process that requires only that locally
$\partial_{t}\Phi>0$. Hence measuring an on-shell excitation with a large
negative energy does not mean we are heading towards a singularity in the
future.

For any positive $Q$, no matter how small, this background admits classical
string solutions to the equation of motions and Virasoro constraints that are
absent for $Q=0$ [4]

$t(\sigma,\tau)=t_{0}+Q\ln\left(\frac{1}{2}\cosh\left(\frac{\sigma}{Q}\right)+\frac{1}{2}\cosh\left(\frac{\tau}{Q}\right)\right),$
(3)

and $x=x_{0}+\sigma,~{}y=y_{0},~{}z=z_{0}$ with $-\infty<\sigma,\tau,<\infty.$
This solution describes a closed folded string that is created at
$t=t_{0},~{}x=x_{0},~{}y=y_{0}$, and $z=z_{0}$. The string expands rapidly and
asymptotically the fold travels at the speed of light. See figure 1.

At the fold, $\tau=0$, we have $\partial_{\tau}t=0$. Therefore, the same
solution in the upper half-plane ($-\infty<\sigma<\infty,~{}0\leq\tau<\infty$)
with a Neumann boundary condition at $\tau=0$ describes an Instant Open String
(IOS)222We thank I. Klebanov for this observation.. Like the fold of the IFS,
the endpoints of the IOS travel faster than light.

Figure 1: The IFS solution. The green arrows represent the null energy flux at
the fold, pointing backward in time to indicate negative energy. The flux
becomes more negative over time. This feeds energy into the bulk of the IFS,
allowing it to grow and become macroscopic.

The origin of the discontinuity in $Q$ – the fact that a solution exists for
any $Q>0$ and does not exist for $Q=0$ – is the following. By definition,
$\alpha^{\prime}$ corrections associated with the target space curvature, $H$
field, and second derivatives cannot dominate, in soft backgrounds, the
leading Virasoro constraints,

$g_{\mu\nu}\partial_{\pm}X^{\mu}\partial_{\pm}X^{\nu}.$ (4)

The dilaton gradient, however, is different. It is formally subleading in the
$\alpha^{\prime}$ expansion, but since it contributes to the Virasoro
constraints a linear term in $X^{\mu}$,

$\alpha^{\prime}\partial_{\mu}\Phi\partial_{\pm}^{2}X^{\mu},$ (5)

it can dominate (4) for a short (world sheet) time, even if
$\partial_{\mu}\Phi$ is small. This is what allows the string to fold. Near
the fold (5) dominates (4), and away from the fold (4) dominates. There is a
sense in which a time-like dilaton gradient violates the equivalence
principle. It triggers the creation of light strings that are simply absent in
its absence.333Space-like dilaton gradient also leads to solutions for any
$Q\neq 0$ that are absent when $Q=0$ [7]. The difference is that these
solutions have an infinite energy so they do not affect the low-energy
dynamics.

One might argue that the existence of instant strings (IFSs and/or IOSs) is
not that surprising. The background in which they are created is time-
dependent, and it is often the case that time dependence leads to the creation
of states from the vacuum. Hence it is natural to wonder if this is yet
another stringy version of the Schwinger mechanism [5] in which the role of
the electric field is played by $Q$.444The open string Schwinger mechanism was
discussed in [8], and a closed string analog, in which the $H$ field plays the
role of the electric field, was discussed in [9].  There are, in fact, some
crucial differences between instant string creation and the Schwinger
mechanism. We find it useful to discuss these differences as they emphasize
the unique features of instant strings and why they decay into (1).

The first difference is the creation scale. In the Schwinger mechanism, the
creation scale grows as the electric field is decreased. In particular, the
creation scale blows up as the electric field vanishes. Consequently
calculating basic quantities, such as the production rate, is challenging for
a varying electric field. For instant strings, the situation is the opposite.
As is clear from (3) the creation scale is $Q$. Hence as we decrease $Q$ the
instant string creation process becomes more and more local. Thus as long as
the curvature and the second derivatives of $\Phi$ are small they cannot
affect local properties of the IFS. For example, in the time-like linear
dilaton background (2) the IFS production rate is [6]

$\Gamma_{IFS}\sim\frac{Q^{2}}{g_{s}^{2}},$ (6)

where $g_{s}$ is the string coupling at $t_{0}$.555In the appendix we present
further evidence for this equation. The locality argument above implies that
in a more general background, with a time-like $\partial_{\mu}\Phi$ that
points to the future, it is

$\Gamma_{IFS}\sim\frac{(\partial_{\mu}\Phi)^{2}}{g_{s}^{2}}.$ (7)

Moreover, the instant string solution is expected to be well approximated by
(3) at distances shorter than the curvature scale.

This locality argument also implies that the instant strings are not related
to the perturbative tachyons that usually appear in (2) since the length scale
associated with the creation of these tachyons, $1/Q$, is much larger than the
scale associated with the creation of instant strings, $Q$. Indeed a small
second derivative of $\Phi$ can render the tachyon massless (or massive), but,
as discussed above, it will have little effect on the instant strings.

The second difference is that in the Schwinger effect, the role of the
electric field is twofold. It triggers the pair creation, and it also feeds
the pair with energy after the creation. It accelerates the electron, say, to
the left while accelerating the positron to the right. In the instant string
case, $Q$ is only the trigger for its creation, but it does not feed it with
energy after the creation. The instant string feeds itself. Namely, the
instant string is created from the vacuum with zero energy (and momentum),
and, as apparent from the existence of the zero modes $t_{0}$ and $x_{0}$, the
total energy and momentum remain zero at later times. This was verified in a
direct calculation [10]. As expected, away from the fold the only non-
vanishing component of the energy-momentum tensor is $T_{uv}$, (with $u=t+x$
and $v=t-x$) which in the small $Q$ limit takes a particular simple form666The
exact energy-momentum tensor can be found in [10].

$T_{uv}=\frac{1}{2\pi\alpha^{\prime}}\Theta(u)\Theta(v),$ (8)

associated with the tension of the folded string. Energy momentum conservation
fixes $T_{uu}$ and $T_{vv}$ at the fold to be

$T_{uu}=-\frac{1}{2\pi\alpha^{\prime}}\Theta(v)v\delta(u),\quad
T_{vv}=-\frac{1}{2\pi\alpha^{\prime}}\Theta(u)u\delta(v),$ (9)

which implies that at the fold there is a negative null flux. Evidently, the
way the instant string feeds itself is by transferring energy from the folds
toward the bulk of the string allowing it to grow with time.

The instant string solution is quite unusual. On the one hand, it describes a
light state with $E=P=0$. On the other hand, even from afar, it does not look
at all like a particle. In particular, it becomes macroscopic at late times,
and so, at finite string coupling, it can split. The IFS splits into two
folded strings (see Figure 2), and the IOS splits into two open strings.

Figure 2: IFS decay, the offset of the breaking point from the center, $\Delta
x$, determines the distribution of the bulk energy between the two components,
leading to $E_{1}=-E_{2}=\frac{\Delta x}{2\pi\alpha^{\prime}}$. The total
momentum of each component is due to the folds $P_{2}=-P_{1}=\frac{\Delta
t}{2\pi\alpha^{\prime}}$. The widths of each component’s bulk are respectively
$L_{1}=\Delta t+\Delta x$ and $L_{2}=\Delta t-\Delta x$. The blue and green
arrows at the folds in each time slice represent the energy and momentum
associated with the inner and outer folds.

Since the total momentum and energy of an instant string vanish we have

$E_{2}=-E_{1},~{}~{}~{}~{}P_{2}=-P_{1}.$ (10)

If the splitting takes place right in the middle of the instant string then
$E_{1}=E_{2}=0$. If the splitting occurs at some point to the right (left) of
the middle point then $E_{2}=-E_{1}>(<)0$. The splitting of an instant string
is a local process that does not depend on the location of the splitting (as
long as it is away from the fold). Hence the wave function of the two strings
associated with an instant string that splits at time $\Delta t$ after its
creation is well approximated by

$|\Psi(\Delta t)\rangle\sim\int_{-\frac{\Delta
t}{2\pi\alpha^{\prime}}}^{\frac{\Delta
t}{2\pi\alpha^{\prime}}}dE_{1}\left|E_{1}=\Delta
x/2\pi\alpha^{\prime},P_{1}=-\Delta
t/2\pi\alpha^{\prime}\right\rangle\otimes\left|E_{2}=-E_{1},P_{2}=-P_{1}\right\rangle,$
(11)

which can be viewed as an energy-EPR state. Note that one of the strings
always has a negative energy and that this negative energy is typically quite
large. It is of the order of $\Delta t/\alpha^{\prime}$. Using [11] we can
estimated that $\Delta t\sim l_{s}/g_{s}$ and so $E_{1}\sim M_{s}/g_{s}$.

Although the state $\left|E_{1},P_{1}\right\rangle$ has fixed quantum numbers,
and, unlike the IFS, from afar it does look like a particle, it cannot be
described by a $(1,1)$ vertex operator (even if $E_{1}>0$). The reason is that
while the total energy and momentum associated with this state do not vary in
time there is quite a bit of dynamic involved in the time evolution of
$\left|E_{1},P_{1}\right\rangle$. The simplest way to see this is to consider
the energy-momentum tensor associated with $\left|E_{1},P_{1}\right\rangle$.
This state has two folds (or ends, in the case of the IOS). One inherited from
the instant string and another due to the splitting. Causality implies that
the fold that was inherited from the instant string is not aware of the
splitting. Hence the negative null flux at the fold is still, say,

$T_{uu}=-\frac{1}{2\pi\alpha^{\prime}}\Theta(v)v\delta(u),$ (12)

and, in particular, it decreases with time (it becomes more negative). The
bulk of the string still contributes $T_{uv}=\frac{1}{2\pi\alpha^{\prime}}$,
and so by energy-momentum conservation at the new fold there is a positive
null flux

$T_{uu}=\frac{1}{2\pi\alpha^{\prime}}(v-L_{2})\Theta(v-L_{2})\delta(u-L_{1}),$
(13)

with $L_{1}=\Delta t+\Delta x$ and $L_{2}=\Delta t-\Delta x,$ that grows with
time (see figure (2)). The mechanism by which $\left|E_{1},P_{1}\right\rangle$
evolves in time is by transferring energy from the fold inherited from the IFS
to the new fold through the bulk of the string.

For $L_{1}\gg l_{s}$, we expect $\left|E_{1},P_{1}\right\rangle$ to split
further. It is natural to expect the splitting to stop when $L_{1}\sim l_{s}$.
This suggests that the final state associated with the decay of an instant
string involves two states, inherited from the fold of the IFS, with negative
energy of the order of $-M_{s}/g_{s}$. They point in opposite directions and
are entangled with many soft modes with positive energy. The total energy of
this energy-EPR state vanishes.

We would like to end with some questions:
$\bullet$ What are the possible imprints of the energy-EPR states? In
Cosmological scenarios that involve a time-dependent dilaton, these states are
created but they do not contribute to the average time evolution. The main
contribution in Cosmology is due to the IFSs (before they decay) that induce
negative pressure at no energy density cost [12]. The implication of this will
be discussed elsewhere [13]. The energy-EPR states do appear to be relevant
for fluctuations. It should be interesting to study the differences between
the fluctuations associated with the energy-EPR states and standard
cosmological fluctuations and see if there is a sharp prediction that can be
made.

Another possibility is a direct detection of an on-shell excitation with
negative energy. Unfortunately, at least in the IFS decay case, the negative
energy excitation appears to couple only via gravity to the standard model
fields which makes detection unrealistic. Note that since the energy is
negative, the gravitational shock wave produced by such an excitation induces
time advance which could lead to causality violation. To violate causality we
need, however, to have several such excitations and control their production
location and momenta. This does not appear to be an easy task given the way
they are produced.

$\bullet$ Why do these energy-EPR states appear in string theory? A possible
answer is related to the fact that the dilaton determines the amount of
classical or coarse-grained entropy via $G_{N}^{-1}\sim e^{-2\Phi}.$ As a
result when the dilaton varies in time so does the classical entropy. This
appears to be the source of the radiation of quantum or fine-grained entropy
in the form of energy-EPR states. The IFS appears to play the role of a
convertor as it converts the coarse-grained entropy into a fine-grained
entropy.

If we define $\Psi=e^{-\Phi},$ we have that the coarse-grained entropy scales
like $\Psi^{2}$ and (7) implies that $(\partial\Psi)^{2}$ determines the fine-
grained entropy production. It is, therefore, natural to dub $\Psi$ an
entropon. The term ”entropon” appears in the condensed matter literature in
the context of active solids [14], which are solids that involve self-
propelled excitations. Amusingly, there seems to be some analogy between
active matter and string theory with time-dependent dilaton. Standard closed
string modes are the analog of phonons. Both are excitations that are present
even when the solid is inactive. When the dilaton grows with time string
theory becomes active. It includes new self-propelled excitations: the instant
strings that grow by feeding themselves. Their decay products, the energy-EPR
states, dominate the entropy production.

$\bullet$ Are there implications to Black Holes? The BH in which it is easiest
to address this is the near extremal NS5-branes [15], i.e. the 2D BH [16, 17,
18, 19]. The region behind the horizon of such a BH includes a time-like
$\partial_{\mu}\Phi$ that points to the future. In [6] it was shown that the
production rate (7) implies that the number of IFSs an infalling observer
encounters on the way to the singularity is of the order of the BH entropy.
Here we claim that an IFS decays into an energy-EPR state, which means that
the Bekenstein-Hawking entropy associated with near extremal NS5-branes is of
the order of the fine-grained entropy associated with the energy-EPR states
that are created inside the BH. Combining this with [20], assuming that the
energy-EPR state also forms a wormhole, we seem to conclude that near extremal
NS5-branes are filled with tiny wormholes. In the context of JT gravity, a
related claim was made in [21]. Since the energy-EPR states involve
excitations with negative energy, the nature of these wormholes, if they
exist, is likely to be nonstandard.

Acknowledgements: We thank D. Gross, A. Hashimoto, G. Horowitz, V. Hubeny, J.
Minahan, I. Klebanov, H. Ooguri, M. Rangamani, and A. Sen for discussions.
Work supported in part by the ISF (grant number 256/22). This research was
supported in part by grant NSF PHY-2309135 to the Kavli Institute for
Theoretical Physics (KITP).

##  Appendix A Evidence for (6)

A production rate that scales like $1/g_{s}^{2}$ should leave its mark on the
sphere partition function. In particular, if the initial state at $t=t_{i}$ is
the vacuum, $|0(t_{i})\rangle$, then the amplitude to remain in the vacuum at
a later time $t_{f}$ is exponentially suppressed due to IFS production

$\langle
0(t_{i})|0(t_{f})\rangle=\exp\left(-V\int_{t_{i}}^{t_{f}}\Gamma_{IFS}(t)~{}dt\right)\sim\exp\left(-VQe^{-2Qt_{i}}\right),$
(14)

where $V$ is the volume and we used (6). We assume here that $t_{f}-t_{i}\gg
Q$, which implies that the integral is dominated by $t_{i}$. We also take
$Q\ll 1$ which means that the dilute IFS-gas approximation is valid and that
interactions among the IFSs can be neglected. In this case (14) should be
compared with the time-like linear dilaton sphere partition function.

In string theory, however, $t_{i}$ and $t_{f}$ cannot be finite. We must take
$t_{i}=-\infty$ and $t_{f}=\infty$, which gives $0$. A way to put an effective
cutoff at $t_{i}$ in string theory is to add a Liouville potential. In space-
like linear dilaton both options for the Liouville wall ($\exp(b\phi)$ and
$\exp(\phi/b)$) cut off the strong coupling region. In the time-like case, we
can either cut the strong coupling region (in the future, in our case) or the
weak coupling region. For this particular calculation, we have to cut off the
weak coupling region since there $\Gamma_{IFS}$ blows up, and assume that the
singularity at the future does not matter for this calculation since there
$\Gamma_{IFS}$ vanishes.

As usual (see e.g. [22]) the full partition function $Z_{vac}$ is related to
the single string partition function, $Z_{1}$, via

$Z_{vac}=\exp(Z_{1}).$ (15)

Thus, for $Q\ll 1$, the time-like Liouville $Z_{1}$ should be compared with

$Z_{1}=-Qe^{-2Qt_{i}}.$ (16)

It appears that an agreement with IFS consideration requires that $Z_{1}$ is
real and negative. This is not standard for the partition function in a time-
like direction, which usually is imaginary.

Time-like Liouville theory, however, is not a standard theory. In particular,
its relation to space-like Liouville, via an analytic continuation of $b$, is
rather subtle [23, 24]). Luckily, using the Coulomb gas approach, $Z_{1}$ was
calculated in time-like Liouville by Giribet [25], who found

$Z_{1}=\frac{(1+b^{2})(\pi\Lambda\gamma(-b^{2}))^{Q/b}}{\pi^{3}Q\gamma(-b^{2})\gamma(-b^{-2})}.$
(17)

KPZ scaling [26] relates the $(\pi\Lambda\gamma(-b^{2}))^{Q/b}$ with the
$e^{-2Qt_{i}}$ in (16). The comparison we are left with is between the $-Q$ in
(16) and the factor of $(1+b^{2})/Q\gamma(-b^{2})\gamma(-b^{-2})$ in (17),
which, for $Q\ll 1$, indeed agree, up to a numerical factor that we cannot
determine at the moment. To check the numerical factor one needs to calculate
$\Gamma_{IFS}$ in the presence of the Liouville wall.

## References

  * [1] J. Polchinski, “What is string theory?,” [arXiv:hep-th/9411028 [hep-th]].
  * [2] D. Bohm, “Quantum Theory, ”Prentice Hall, Englewood Cliffs, 1951.
  * [3] J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics Physique Fizika 1, 195-200 (1964) doi:10.1103/Physics Physique Fizika.1.195
  * [4] N. Itzhaki, “Stringy instability inside the black hole,” JHEP1810, 145 (2018) doi:10.1007/JHEP10(2018)145[arXiv:1808.02259[hep-th]].
  * [5] J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev. 82, 664-679 (1951) doi:10.1103/PhysRev.82.664
  * [6] A. Hashimoto, N. Itzhaki and U. Peleg, “A Worldsheet Description of Instant Folded Strings,” . J. High Energ. Phys. 2023, 88 (2023). doi:10.1007/JHEP02(2023)088 [arXiv:2209.04988 [hep-th]].
  * [7] J. M. Maldacena, “Long strings in two dimensional string theory and non-singlets in the matrix model,” JHEP 09, 078 (2005) doi:10.1088/1126-6708/2005/09/078 [arXiv:hep-th/0503112 [hep-th]].
  * [8] C. Bachas and M. Porrati, “Pair creation of open strings in an electric field,” Phys. Lett. B 296, 77-84 (1992) doi:10.1016/0370-2693(92)90806-F [arXiv:hep-th/9209032 [hep-th]].
  * [9] F. Dowker, J. P. Gauntlett, G. W. Gibbons and G. T. Horowitz, “Nucleation of p-branes and fundamental strings,” Phys. Rev. D 53, 7115-7128 (1996) doi:10.1103/PhysRevD.53.7115 [arXiv:hep-th/9512154 [hep-th]].
  * [10] K. Attali and N. Itzhaki, “The Averaged Null Energy Condition and the Black Hole Interior in String Theory,” Nucl. Phys. B 943, 114631 (2019) doi:10.1016/j.nuclphysb.2019.114631 [arXiv:1811.12117 [hep-th]].
  * [11] J. Dai and J. Polchinski, “The Decay of Macroscopic Fundamental Strings,” Phys. Lett. B 220, 387-390 (1989) doi:10.1016/0370-2693(89)90892-7
  * [12] N. Itzhaki, “String Theory and The Arrow of Time,” JHEP 03, 192 (2021) doi:10.1007/JHEP03(2021)192 [arXiv:2101.10142 [hep-th]].
  * [13] N. Itzhaki and U. Peleg, to appear
  * [14] L. Caprini, U. Marini Bettolo Marconi, A. Puglisi, and H. Lowen “Entropons as collective excitations in active solids,” J. Chem. Phys. 159, 041102 (2023)
  * [15] J. M. Maldacena and A. Strominger, “Semiclassical decay of near extremal five-branes,” JHEP 12, 008 (1997) doi:10.1088/1126-6708/1997/12/008 [arXiv:hep-th/9710014 [hep-th]].
  * [16] E. Witten, “On string theory and black holes,” Phys. Rev. D 44, 314-324 (1991) doi:10.1103/PhysRevD.44.314
  * [17] G. Mandal, A. M. Sengupta and S. R. Wadia, “Classical solutions of two-dimensional string theory,” Mod. Phys. Lett. A 6, 1685-1692 (1991) doi:10.1142/S0217732391001822
  * [18] S. Elitzur, A. Forge and E. Rabinovici, “Some global aspects of string compactifications,” Nucl. Phys. B 359, 581-610 (1991) doi:10.1016/0550-3213(91)90073-7
  * [19] R. Dijkgraaf, H. L. Verlinde and E. P. Verlinde, “String propagation in a black hole geometry,” Nucl. Phys. B 371, 269-314 (1992) doi:10.1016/0550-3213(92)90237-6
  * [20] J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” Fortsch. Phys. 61, 781-811 (2013) doi:10.1002/prop.201300020 [arXiv:1306.0533 [hep-th]].
  * [21] D. Stanford and Z. Yang, “Firewalls from wormholes,” [arXiv:2208.01625 [hep-th]].
  * [22] Polchinski, J. (1998) String Theory. Vol. 1, “An Introduction to the Bosonic String”.
  * [23] A. B. Zamolodchikov, “Three-point function in the minimal Liouville gravity,” Theor. Math. Phys. 142, 183-196 (2005) doi:10.1007/s11232-005-0003-3 [arXiv:hep-th/0505063 [hep-th]].
  * [24] D. Harlow, J. Maltz and E. Witten, “Analytic Continuation of Liouville Theory,” JHEP 12, 071 (2011) doi:10.1007/JHEP12(2011)071 [arXiv:1108.4417 [hep-th]].
  * [25] G. Giribet, “On the timelike Liouville three-point function,” Phys. Rev. D 85, 086009 (2012) doi:10.1103/PhysRevD.85.086009 [arXiv:1110.6118 [hep-th]].
  * [26] V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov, “Fractal Structure of 2D Quantum Gravity,” Mod. Phys. Lett. A 3, 819 (1988) doi:10.1142/S0217732388000982


# Explainable Incipient Fault Detection Systems for Photovoltaic Panels

Seshapalli Sairam, Seshadhri Srinivasan , Giancarlo Marafioti , B. Subathra,
Geir Mathisen , Korkut Bekiroglu Seshapalli Sairam is with Dept. of
Instrumentation and Control Engineering, Kalasalingam Academy for Research and
Education, Krishnankoil, Srivilliputtur, India. e-mail :
<EMAIL_ADDRESS>Srinivasan is with GE Corporate Research Center,
Bangalore-560068. e-mail<EMAIL_ADDRESS>Marafioti is
with SINTEF, Cybernetics. e-mail<EMAIL_ADDRESS>Subathra is
with Dept. of Instrumentation and Control Engineering, Kalasalingam Academy
for Research and Education, Krishnankoil, Srivilliputtur, India. e-mail :
<EMAIL_ADDRESS>Mathisen is with Norwegian University of Science and
Technology Faculty of Engineering, Cybernetics. e-mail :
<EMAIL_ADDRESS>Bekiroglu is with the College of Engineering, SUNY
Polytechnic Institute, Utica 13503, NY, USA. e-mail :
<EMAIL_ADDRESS>
###### Abstract

This paper presents an eXplainable Fault Detection and Diagnosis System
(XFDDS) for incipient faults in PV panels. The XFDDS is a hybrid approach that
combines the model-based and data-driven framework. Model-based FDD for PV
panels lacks high fidelity models at low irradiance conditions for detecting
incipient faults. To overcome this, a novel irradiance based three diode model
(IB3DM) is proposed. It is a nine parameter model that provides higher
accuracy even at low irradiance conditions, an important aspect for detecting
incipient faults from noise. To exploit PV data, an extreme gradient boosting
(XGBoost) is used due to its ability to detecting incipient faults. Lack of
explainability, feature variability for sample instances, and false alarms are
challenges with data-driven FDD methods. These shortcomings are overcome by
hybridization of XGBoost and IB3DM, and using eXplainable Artificial
Intelligence (XAI) techniques. To combine the XGBoost and IB3DM, a fault-
signature metric is proposed that helps reducing false alarms and also trigger
explanation on detecting incipient faults. To provide explainability, an
eXplainable Artificial Intelligence (XAI) application is developed. It uses
the local interpretable model-agnostic explanations (LIME) framework and
provides explanations on classifier outputs for data instance. These
explanations help field engineers/technicians for performing troubleshooting
and maintenance operations. The proposed XFDDS is illustrated using
experiments on different PV technologies and our results demonstrate the
perceived benefits.

###### Index Terms:

Explainable Artificial Intelligence (XAI) incipient fault, eXplainable Fault
Detection and Diagnosis System (XFDDS), eXtreme Gradient Boosting (XGBoost).

##  I Introduction

###  I-A Motivation

Exponential growth in photovoltaic (PV) deployments has raised interest in its
reliable operation [1]. As PV panels are installed in harsh environments and
subjected to varying weather conditions, they are prone to diverse faults
(permanent, incipient, and intermittent) with different severity levels [2].
Such faults could diminish energy production, accelerate aging, and even cause
fire hazards [3]. Therefore, detecting and locating faults early (at the
incipient stage) is pivotal for the PV panel’s reliable operations [4].
Detecting incipient faults challenging as the signatures are less evident due
to low magnitude, and the problem accentuates at low irradiance conditions.
Also, incipient faults are quite intermittent and show up for a short
duration. Therefore, detecting them becomes more challenging. Nevertheless,
incipient faults could develop as a severe fault in the long-run if left
undetected/unattended, leading to costly replacements and maintenance
operations [5]. Consequently, detecting incipient faults has gained
significant traction recently [6]. While fault-detection and diagnosis (FDD)
systems are proven to improve PV system reliability [7], there are few
challenges that need to be addressed for detecting incipient faults: (i) high
fidelity PV models providing good accuracy at low irradiance conditions are
required, (ii) existing FDD methods cannot reason their decisions to the field
engineers/technicians, and (iii) difficult distinguishing between false alarms
and incipient faults. Our objective in this paper is to propose an eXplainable
Fault Detection and Diagnosis Systems (XFDDS) for incipient faults in PV
panels that address challenges with FDD systems.

###  I-B Literature Review

The FDD methods in the literature can be broadly discerned as being— model-
based (MB), signal-based (SB), and data-driven (DD) [8]. The MB methods use PV
panel models (set of nonlinear equations) followed by signal-analysis (e.g.,
correlation analysis) on the input-output data from the model to detect faults
[9]. The single-diode model (SDM) [10], double-diode model (DDM) [11], and
three-diode models (TDM) [12, 13] are widely used in FDD systems. While
existing models provide good accuracy at high irradiance, their accuracy is
less at low irradiance conditions. The SB methods use fault signatures from
sensor data to detect faults [14]. Widely used SB methods are: statistical
signal processing [15], I-V (current-voltage) characteristics analysis [16],
power loss analysis [17] and, voltage and current measurements [18]. More
recently, SB-FDD methods using two-stage support vector machines [19], multi-
signal decomposition, and fuzzy inference systems [20] have also been
proposed. The DD methods using labeled fault-data and artificial intelligence
(AI) techniques have shown promise in improving detection accuracy due to
their powerful model representation capabilities. The DD methods leverage
historical labeled data and powerful models from AI techniques to perform
multi-class regression or classification, which are quite important for
detecting faults [21]. In the literature, FDD methods using AI models such as
random forest [22], collaborative filtering [23], extreme gradient boosting
[24], and such techniques have been proposing (see,[25] and references
therein). Despite these advances, detecting incipient faults addressing the
fundamental challenges of accuracy at low irradiance conditions, lack of
explainability on decisions to field engineers/technicians, and distinguishing
false alarms from incipient faults is rather unexplored in the literature to
our best knowledge.

###  I-C Contributions

This paper proposes an XFDSS for PV panels addressing challenges with FDD
system. It is a hybrid method that combines model-based and data-driven
approaches. To overcome accuracy challenges under low irradiance conditions
and detect incipient faults, an irradiance based three diode model (IB3DM)
proposed. The model uses irradiance and temperature in parameter computations
inherently, thereby increasing its accuracy even at low irradiance conditions.
For fault explainability and distinguishing false alarms from incipient
faults, the IB3DM is combined with data-based approaches that perform multi-
label classification. This paper uses the extreme gradient boosting (XGBoost)
based multi-label classifier due to its suitability to detect incipient
faults. As XGBoost cannot explain its decisions to the field
technician/engineer, recently developed eXplainable AI (XAI) techniques are
used. The XAI extends the capabilities of the AI techniques by providing
explanations on decisions on individual data-instances [26], a key aspect in
incipient fault detection. We show that these explanations are very useful for
field engineers/technicians to understand the fault-causes and fault-type. The
local interpretable model-agnostic explanations (LIME) approach is used [27]
to provide the explanations. The main idea is to perturb the features and
compute the importance and variable thresholds for being classified as faults
on individual samples. Main contributions are:

  1. 1.

A novel three diode model called the Irradiance Based Three Diode Model
(IB3DM) which inherently captures the influences of solar irradiance and
ambient temperature;

  2. 2.

Design an XFDS leveraging the accuracy of IB3DM, XGBoost, and LIME;

  3. 3.

Illustrate the IB3DM and XFDS using experiments and simulations on different
PV technologies.

The paper is organized into five sections. The components of XFDDS is
explained in Section II. The IB3DM for PV panel and its parameter computation
is explained in Section III. The XFDDS methodology is presented in Section IV.
Results are presented in Section V and conclusions are presented in Section
VI.

##  II Explainable Fault Detection and Diagnosis System

The main challenges with existing FDD techniques are:

  * (C1)

Lack of high fidelity models capturing PV panel performance at low irradiance
conditions;

  * (C2)

Existing FDD methods lack explanations to field engineers/technicians on why a
particular sample was classified as faults and the variable thresholds on
which this decision on a fault is made;

  * (C3)

Data-based models compute feature importance for a particular fault on the
global data, whereas incipient faults are intermittent and there are
inconsistencies within data instances as well;

  * (C4)

Data-based models cause false alarms due to mis-classification.

Figure 1: Explainable fault detection system

The XFDDS proposed in our work addresses the challenges (C1)-(C4) and its
schematic is shown in Fig. 1. Its main components are: climate service, IB3DM,
fault-signature metric, the XGBoost classifier, XAI triggers, sample store and
XAI application. The IB3DM uses the climate service (a web-application) to
obtain solar irradiance and temperature for predicting the PV outputs
(voltage, current, and power). Exploiting IB3DMs’ model accuracy, a fault-
signature metric is defined (see Section IV-B) which serves as a trigger for
obtaining explanations from XAI application and such samples are stored in
sample store, a local cache. The XGBoost based classifier is a combination of
multiple classifiers, and regression tree (CART) ensemble created using
boosting techniques [24]. The XGBoost is selected as the data-based model in
our application, as it naturally fits the incipient fault-detection framework
as detailed later. Two challenges with XGBoost are lack of explainability and
false alarms [28]. Moreover, the feature importances for a particular fault
are computed based on global data, contrary to this incipient faults are
intermittent with data varying among fault samples (feature inconsistency
problem). The particular of XAI application generate these explanations on a
data-instances, which is very important for incipient faults. These
explanations help the user to identify the fault types and variable thresholds
based on which the fault was detected. In what follows the IB3DM model is
first proposed and then the XFDDS approach is illustrated.

##  III Novel Irradiance-based three diode model

Our model parameters depend on irradiance and module temperature and it
addresses the challenge (C1). Therefore, we call our model irradiance-based
three diode model (IB3DM). The IB3DM is an extension of the TDM proposed in
[12, 13] wherein $I_{P}$, the light generated current depends on irradiance
and module temperature. Further, in IB3DM, the ideality factors are not fixed;
rather, they are obtained as a solution to an optimization problem by
specifying bounds ([0, 2]). This is a deviation from existing works in three
diode models where higher ideality factors are used leading to low fill-factor
that could be achieved only in industrial-grade panels. This makes existing
TDM unsuitable for residential PV panels.

###  III-A Equivalent Circuit and Model Parameters

Our idea is to propose a three diode model that accurately captures the PV
cells’ performance even under low irradiance conditions.

######  Remark 1.

In IB3DM, the source current is modelled as a dependent current source that is
a function of irradiance and module temperature, denoted by
$I_{p}(G,\leavevmode\nobreak\ T)$. The photo-generated current has two parts;
the first part which is a premultiplier is linearly dependent on the
irradiance and acts as a scaling factor for the second part that depends on
panel temperature.

Suppose the nominal phase current is denoted by $I_{p,n}$, then the dependence
on phase current on the irradiance is given by,

$I_{P}(G,T)=\Bigg{(}\frac{G}{G_{STC}}\Bigg{)}\Big{(}I_{p,n}+K_{I}\times(T-T_{r})\Big{)}$
(1)

where $K_{I}$ is a constant computed from data-sheets. The source current is
in parallel to three diodes with a series resistance $R_{s}$ and shunt
resistance $R_{sh}$, as shown in Fig. 2. With this modification, the current
source is a function of irradiance and module temperature. From the equivalent
circuit, the output current is given by

$I=I_{P}-I_{l1}-I_{l2}-I_{l3}-I_{sh}$ (2)

Similarly, current through the shunt resistance $R_{sh}$ in Fig. 2 is,

$I_{sh}=\frac{V+IR_{s}}{R_{sh}}$ (3)

The diode saturation currents can be calculated as:

$I_{01}=\frac{I_{S}+K_{I}\times(T-T_{r})}{exp\big{(}\frac{V_{OC}+K_{V}\times(T-T_{r})}{{V_{t}\times
n_{01}}}\big{)}-1},$ (4)

$I_{02}=\frac{I_{S}+K_{I}\times(T-T_{r})}{exp\big{(}\frac{V_{OC}+K_{V}\times(T-T_{r})}{{V_{t}\times
n_{02}}}\big{)}-1},$ (5)

$I_{03}=\frac{I_{S}+K_{I}\times(T-T_{r})}{exp\big{(}\frac{V_{OC}+K_{V}\times(T-T_{r})}{{V_{t}\times
n_{03}}}\big{)}-1},$ (6)

The saturation currents strongly depends on the temperature as indicated by
equations (4)-(6). Note that the coefficient $K_{V}$ is from the
manufacturers’ data-sheet and used to compute I-V curve for different
temperatures as seen the equations (4)-(6). The junction thermal voltage is
given by,

$V_{t}=\frac{N_{s}kT}{q}.$ (7)

Combining equations (1)-(7), one can obtain the equations relating the output
current, output voltage, and model parameters for the IB3DM:

$\begin{split}I&=I_{P}-I_{01}\Bigg{(}exp\Big{(}{\frac{V+IR_{s}}{n_{01}V_{t}}}\Big{)}-1\Bigg{)}\\\
&-I_{02}\Bigg{(}exp\Big{(}{\frac{V+IR_{s}}{n_{02}V_{t}}}\Big{)}-1\Bigg{)}\\\
&-I_{03}\Bigg{(}exp\Big{(}{\frac{V+IR_{s}}{n_{03}V_{t}}}\Big{)}-1\Bigg{)}-\frac{V+IR_{s}}{R_{sh}},\end{split}$
(8)

where $n_{01}$, $n_{02}$, and $n_{03}$ are diode ideality factors.
Consequently, the IB3DM has nine parameters given by
$\mathcal{P}_{IB3DM}=\\{I_{P},I_{l1},I_{l2},I_{I3},R_{S},R_{sh},n_{01},n_{02},n_{03}\\}$
which should be obtained from I-V curve data. Next step is to compute the
model parameters and an optimization based approach is proposed as detailed in
the next section.

Figure 2: Equivalent circuit of IB3DM.

###  III-B The IB3DM Parameter Computation

To compute the IB3DMs’ model parameters, first the $I_{P}$ is calculated using
$G$ and $T$ from equation (1). Next, we define the objective function as the
root mean square error between the experimental V-I and the estimated model is
given by,

$\begin{split}f_{m}(V,I,\mathcal{P})&=I-I_{P}+I_{01}\Bigg{(}exp\Big{(}{\frac{V+IR_{s}}{n_{01}V_{t}}}\Big{)}-1\Bigg{)}\\\
&+I_{02}\Bigg{(}exp\Big{(}{\frac{V+IR_{s}}{n_{02}V_{t}}}\Big{)}-1\Bigg{)}\\\
&+I_{03}\Bigg{(}exp\Big{(}{\frac{V+IR_{s}}{n_{03}V_{t}}}\Big{)}-1\Bigg{)}+\frac{V+IR_{s}}{R_{sh}}\end{split}$
(9)

It measures the difference between experimental I-V curve data and the one
calculated using the model, i.e.,
${f_{m}(V,I,\mathcal{P})}=I_{measured}-I_{calculated}$ with $V$ varying over
the operating range. To compute the model parameters, we utilized the root
mean square error (RMSE) as a metric denoted by,
$\mathcal{J}=\sqrt{\frac{1}{N_{e}}\sum_{i=1}^{N_{e}}f_{m}(V,I,\mathcal{P})^{2}}$.
The parameter computation problem is modelled as an optimization problem given
by,

$\displaystyle\underset{\mathcal{P}_{IB3DM}}{\operatorname{min}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathcal{J}$ (10) $\displaystyle
s.t.$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\eqref{eq:6}-\eqref{eq:10}$

Clearly, (10) is nonlinear and non-convex that is computationally complex to
solve using conventional optimization techniques. Usually meta-heuristic
approaches are used for computations for this type of optimization problems
[29]. Our analysis utilizes five different meta-heuristic algorithms that are
explained in results section.

##  IV Explainable Fault Detection and Diagnosis Methodology

###  IV-A Extreme Gradient Boosting for Fault Classification

Gradient boosting decision tree is a powerful machine learning algorithm
wherein multiple weak learner ensemble form a strong learner [30]. The XGBoost
main idea is that the training set in the current instance is related to the
learning results from previous learning, and the weights on data-samples are
adjusted on each sample in each iteration. These features naturally fits the
incipient fault-detection scenario as labeled fault-data are available
sequentially, and weight adaptation in each sample increases model accuracy.
Furthermore, in XGBoost the current decision tree (leaf) is fitted based on
the residuals from the previous trees and it uses the gradient boosting
principle wherein new decision trees are constructed to correlate to the
negative gradient of the loss-function. A detailed description of XGBoost
could be found in [31, 32].

_This investigation uses XGBoost for detecting two incipient faults:(i) line-
to-line (LL) fault which denotes short-circuit within the string or on
multiple PV strings and (ii) partial shading._ Two different classification
cases are considered: binary and multi-class classification. In binary
classification, labeled data about whether a given sample is faulty or healthy
operating condition is predicted without considering the fault-type (LL or
partial shading), whereas in multi-class fault classification, the fault-type
are also included as additional class.

Figure 3: Confusion Matrix for Binary Classification

In binary classification, voltage, current, irradiance, and power are the
inputs and a binary variable modelling the sample to be faulty/healthy is the
output. Dichotomous search optimization was used for tuning hyper-parameters
that resulted in 100 learners. During training, the classifier showed 93-95%
accuracy, and validation had 86-93% accuracy. The confusion matrix for the
binary classifier is shown in Fig. 3. One can see that the classifier performs
extremely well for binary classification.

Figure 4: Confusion Matrix for Multi-Class Classification

In multi-class classification, additional labels on the fault type (LL), or
partial shading) is added to the data-set. Then the XGBoost classifier is
trained to identify the fault label. The XGBoost hyper parameters was
optimized using dichotomous search and it had 130 learners. During the
training, the classifier showed 88-92% accuracy, and 78-82% during validation.
Confusion matrix for the multi-class classification during validation is shown
in Fig. 4. While its accuracy is reasonable, the challenges (C2)-(C4) are not
addressed by XGBoost.

###  IV-B Explainable Fault Detection

To provide local explanations and address challenges (C2)-(C4), the data-
driven approach is first fused with model-based approach by proposing a fault-
signature metric (FSM) given by,

$\sigma(G,P)=\gamma(G)\leavevmode\nobreak\
exp\big{(}{\frac{-\|G-G_{s}\|_{\ell_{2}}}{\Sigma_{G}}}\big{)}^{-1}\times
exp\big{(}{\frac{-\|P-\hat{P}\|_{\ell_{2}}}{\Sigma_{P}}}\big{)}^{-1},$ (11)

where $\gamma(G),\Sigma_{G}$, and $\Sigma_{P}$ are the scaling factor as a
function of irradiance, variances in the actual solar irradiance, and power
generated by the solar panel, respectively. Also, $\|\leavevmode\nobreak\
\|_{\ell_{2}}$ represents $\ell_{2}$ norm. $\hat{P}$ is the estimated power
computed by the IB3DM, and the signature computes the deviations in power. The
fault-signature metric serves two purposes: (i) it weights low variability at
low irradiance conditions higher than at high irradiance conditions helps
overcoming noise and preventing false alarms and (ii) triggers for
explanations on detecting an incipient fault by using thresholds on fault-
signature metric.

The XFDDS uses FSM to eliminate false alarms by comparing it with results of
the XGBoost. Second, event triggers for fault explanations are generated using
FSM. Once the FSM exceeds a known threshold, explanations are asked from the
XAI application and such samples are stored in sample store.

On receiving triggers, the XAI application is activated that uses the local
interpretable model-agnostic explanations (LIME) [27] framework. It utilizes
surrogate modelling wherein the model is considered a black-box, and the
features are perturbed to find feature importance on a particular sample. The
data instance is perturbed, and samples are generated from the data-set
distribution, which is weighted based on their distances from the current
point. Then feature selection is applied to keep the relevant variables, and
the linear model is trained on the weighted data-set automatically within the
algorithm. Once trained, the model explains to the user about the variables
and their thresholds that made the model decide a particular sample instance
as a fault/normal operation. This is quite useful in detecting incipient
faults as they are intermittent and occur for a few fault-samples.

_While the core of XFDS is still the XGBoost, the XAI extracts explanations on
why a particular sample was classified as being faulty/healthy._ Further,
thresholds on variables that helped make these decisions are also provided,
which is quite useful in detecting even the fault type. For example, a LL
fault is characterized by high voltage but a lower current and power. As power
drops due to circulating currents are very hard to predict. Moreover,
intermittent LL faults are difficult to catch FDDS. However, with XAI could
reason out incipient LL faults.

Figure 5: Experimental setup using a PV module.

##  V Results

The proposed XFDDS could be implemented on simple hardware as illustrated in
this section. In our experiments, the fault-detection is implemented by
interfacing embedded hardware with computer. However, the computer could be
replaced with any system-on-chip (SoC) with limited computing power. The
computer interfaces to the web application directly and obtains the weather
data. The weather station recordings are ported as a comma-separated variable.
The ATmega 328P processor is used as the data-acquisition unit, which
interfaces to the computer through the software application and receives the
measurements from sensors. Measurements were obtained from the current sensor
(INA169), temperature sensor (DS18B20), and voltage sensor (F031-06),
respectively. The data was transmitted to the computer using the UART
(Universal Asynchronous Receiver/Transmitter), a serial communication mode to
PLX-DAQ and the measurements are stored in a data-base for further processing.
Temperature sensor Maxim IC DS 18B20 is a 1-wire digital temperature sensor
that reports temperature in Celsius with 9-12 bit precision and has a working
range of -55 to 125$\mathrm{\SIUnitSymbolCelsius}$. A rheostat is used as the
load, and the experiments are conducted at International Research Center,
Kalasalingam University, India. Experiments with PV panel, sensors interfaced
with computers, and Python was used to implement the fault-detection scheme
(see Fig. 5).

###  V-A The IB3DM Model Parameter Estimation

As stated earlier, the IB3DM parameter computation requires solving a
nonlinear and non-convex optimization problem in (10). To overcome
computational difficulties, five different meta-heuristic algorithms are used:
(i) firefly, (ii) particle swarm optimization (PSO), (iii) teaching-learning
based optimization (TLBO), (iv) biogeography based optimization (BBO), and (v)
shuffled frog leaping algorithm (SFLA) to estimate the model parameters of the
IB3DM.

The parameter is computed for two different PV technologies: monocrystalline
(STM5-20/36) at 1000 $\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$ at
33$\mathrm{\SIUnitSymbolDegree}$ and polycrystalline (Solartech SPM-020P-R)
with 1000 $\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$ at
45$\mathrm{\SIUnitSymbolDegree}$ with each panel having 36 cells in series.

The IB3DM parameters computed for a monocrystalline panel for the SDM, DDM,
CD3DM, and IB3DM with the five meta-heuristic algorithms is shown in Tab. I.
The RMSE value ($\mathcal{J}$) are shown in Tab. I. One can observe that IB3DM
offers better accuracy than existing models as evinced by their low RMSE
values. In addition, fire-fly algorithm provides better estimates of the I-V
curves.

Figure 6: (a) V-I Curve of the SDM, DDM, C3DM versus IB3DM, (b) P-V Curve of
SDM, DDM, C3DM, and IB3DM (Monocrystalline).

Figure 7: (a) V-I Curve of the SDM, DDM, C3DM versus IB3DM, (b) P-V Curve of SDM, DDM, C3DM, and IB3DM (Polycrystalline PV panel) Table I: Comparison of meta-heuristic algorithms for computing the model parameters of the Monocrystalline panels Model | Algorithm | $I_{P}(A)$ | $I_{01}(A)$ | $I_{02}(A)$ | $I_{03}(A)$ | $n_{1}$ | $n_{2}$ | $n_{3}$ | $R_{s}(\Omega)$ | $R_{sh}(\Omega)$ | $K$ | $RMSE$
---|---|---|---|---|---|---|---|---|---|---|---|---
| Firefly | 1.6644 | 1.55E-06 | - | - | 1.94 | - | - | 0.176 | 752.29 | - | 0.012245
| PSO | 1.6626 | 2.87E-06 | - | - | 1.87 | - | - | 0.3917 | 599.78 | - | 0.060392
SDM | TLBO | 1.6634 | 2.86E-06 | - | - | 1.67 | - | - | 0.4255 | 598.55 | - | 0.042516
| BBO | 1.6605 | 9.08E-07 | - | - | 1.99 | - | - | 0.6372 | 642.66 | - | 0.065356
| SFLA | 1.663 | 6.02E-06 | - | - | 1.96 | - | - | 0.2385 | 690.27 | - | 0.083196
| Firefly | 1.6645 | 1.71E-06 | 3.01E-12 | - | 1.85 | 1.72 | - | 0.2396 | 739.49 | - | 0.010945
| PSO | 1.7029 | 3.08E-05 | 6.24E-05 | - | 1.64 | 1.55 | - | 0.1263 | 606.28 | - | 0.052871
DDM | TLBO | 1.6638 | 1.14E-10 | 5.21E-06 | - | 1.52 | 1.91 | - | 0.5215 | 695.23 | - | 0.042925
| BBO | 1.7029 | 3.08E-05 | 6.24E-05 | - | 1.94 | 1.55 | - | 0.3263 | 606.28 | - | 0.085258
| SFLA | 1.6613 | 5.66E-06 | 2.24E-08 | - | 1.76 | 1.89 | - | 0.2199 | 673.52 | - | 0.092876
| Firefly | 1.6645 | 2.71E-06 | 3.01E-12 | 1.08E-05 | 1.72 | 1.57 | 1.47 | 0.296 | 739.49 | 0.0092 | 0.004852
| PSO | 1.7029 | 1.08E-05 | 2.24E-05 | 1.72E-05 | 1.52 | 1.22 | 1.42 | 0.2263 | 656.28 | 0.0272 | 0.018471
CD3DM | TLBO | 1.6638 | 2.24E-10 | 3.31E-06 | 4.24E-08 | 1.49 | 1.19 | 1.24 | 0.3215 | 595.23 | 0.0231 | 0.009229
| BBO | 1.7029 | 1.98E-05 | 2.41E-05 | 3.23E-08 | 1.76 | 1.28 | 1.62 | 0.2363 | 596.28 | 0.0185 | 0.024528
| SFLA | 1.6613 | 2.66E-06 | 5.24E-08 | 4.76E-06 | 1.62 | 1.24 | 1.32 | 0.2699 | 473.52 | 0.0289 | 0.050476
| Firefly | 1.6633 | 2.93E-06 | 5.10E-15 | 1.54E-07 | 1.35 | 1.46 | 1.24 | 0.0917 | 804.43 | - | 0.005463
| PSO | 1.7133 | 6.88E-04 | 1.80E-10 | 1.63E-09 | 1.02 | 1.09 | 1.14 | 0.3618 | 477.24 | - | 0.007824
IB3DM | TLBO | 1.6622 | 1.89E-08 | 8.67E-08 | 1.19E-05 | 1.08 | 1.06 | 1.15 | 0.3917 | 761.51 | - | 0.005936
| BBO | 1.6683 | 8.52E-06 | 4.13E-06 | 2.85E-04 | 1.03 | 1.12 | 1.39 | 0.2511 | 570.46 | - | 0.008193
| SFLA | 1.6683 | 8.52E-06 | 4.13E-06 | 2.85E-04 | 1.13 | 1.18 | 1.29 | 0.3511 | 570.46 | - | 0.008262
Table II:  Comparison of meta-heuristic algorithms for computing the model parameters of the Polycrystalline panel Model | Algorithm | $I_{P}(A)$ | $I_{01}(A)$ | $I_{02}(A)$ | $I_{03}(A)$ | $n_{1}$ | $n_{2}$ | $n_{3}$ | $R_{s}(\Omega)$ | $R_{sh}(\Omega)$ | $K$ | $RMSE$
---|---|---|---|---|---|---|---|---|---|---|---|---
| Firefly | 1.047 | 4.43E-05 | - | - | 1.85 | - | - | 1.7763 | 612.53 | - | 0.030324
| PSO | 1.0458 | 5.86E-05 | - | - | 1.96 | - | - | 1.7286 | 680.97 | - | 0.549642
SDM | TLBO | 1.0345 | 1.91E-05 | - | - | 1.86 | - | - | 1.9521 | 684.96 | - | 0.495359
| BBO | 1.0343 | 1.43E-04 | - | - | 1.97 | - | - | 1.4589 | 653.48 | - | 0.576291
| SFLA | 1.0466 | 1.31E-04 | - | - | 1.98 | - | - | 1.5583 | 699.79 | - | 0.762183
| Firefly | 1.0435 | 5.15E-05 | 6.03E-13 | - | 1.83 | 1.65 | - | 1.7621 | 611.84 | - | 0.027368
| PSO | 1.0676 | 7.88E-05 | 6.03E-08 | - | 1.91 | 1.71 | - | 1.7272 | 591.83 | - | 0.389554
DDM | TLBO | 1.0323 | 2.14E-05 | 5.37E-06 | - | 1.79 | 1.69 | - | 1.1532 | 692.98 | - | 0.069885
| BBO | 1.0881 | 1.55E-05 | 6.62E-04 | - | 1.93 | 1.79 | - | 1.0243 | 641.61 | - | 0.401265
| SFLA | 1.0971 | 1.67E-04 | 3.53E-05 | - | 1.95 | 1.82 | - | 1.0004 | 358.15 | - | 0.495239
| Firefly | 1.0745 | 1.62E-06 | 1.92E-12 | 3.28E-05 | 1.62 | 1.48 | 1.28 | 1.0235 | 798.25 | 0.0052 | 0.011619
| PSO | 1.0929 | 6.54E-05 | 1.54E-05 | 1.92E-05 | 1.85 | 1.62 | 1.59 | 1.0563 | 696.28 | 0.0237 | 0.387132
CD3DM | TLBO | 1.6538 | 1.52E-10 | 2.26E-06 | 5.42E-08 | 1.68 | 1.34 | 1.45 | 1.5235 | 495.23 | 0.0259 | 0.065293
| BBO | 1.0029 | 1.98E-05 | 2.41E-05 | 3.23E-08 | 1.89 | 1.67 | 1.62 | 1.0063 | 606.28 | 0.0262 | 0.537698
| SFLA | 1.0513 | 3.66E-08 | 3.24E-06 | 3.76E-09 | 1.91 | 1.73 | 1.71 | 1.0039 | 623.52 | 0.0325 | 0.553676
| Firefly | 1.047 | 2.27E-10 | 2.39E-04 | 2.03E-12 | 1.42 | 1.37 | 1.06 | 1.0954 | 729.53 | - | 0.004856
| PSO | 1.049 | 8.33E-06 | 4.64E-04 | 3.70E-12 | 1.66 | 1.54 | 1.23 | 1.9449 | 693.99 | - | 0.006806
IB3DM | TLBO | 1.042 | 3.32E-06 | 1.06E-04 | 1.17E-05 | 1.53 | 1.45 | 1.03 | 1.2107 | 671.01 | - | 0.005077
| BBO | 1.039 | 1.43E-05 | 4.11E-05 | 4.67E-05 | 1.72 | 1.62 | 1.35 | 1.5124 | 697.74 | - | 0.007493
| SFLA | 1.045 | 2.63E-05 | 2.92E-07 | 2.26E-05 | 1.79 | 1.67 | 1.49 | 1.9119 | 691.77 | - | 0.007994

A comparison of the I-V and P-V curves with model parameters computed by the
firefly algorithm for the different PV models: SDM, DDM, CD3DM, and IB3DM is
shown in Fig. 6. The accuracy provided by IB3DM at MPP is also shown in the
zoomed portion. Similarly, accuracy is also high at low and high irradiance
conditions as well.

The IB3DM model parameters and its comparison with other PV models for poly-
crystalline panel (SPM-020P-R) is shown in Tab. II. The low RMSE values are
indicative of the accuracy provided by IB3DM. Further, firefly algorithm
provides the best model parameters compared with other meta-heuristic
techniques. Similarly, the I-V and P-V curves for the different diode model is
shown in Fig. 7. This results shows the model accuracy across irradiance
levels. These results demonstrate the IB3DMs’ ability to provide model
accuracy across different irradiance levels and PV technologies.

###  V-B Explainable Incipient Fault Detection

Having computed the model parameters, the next step is to fuse IB3DM with XAI
to implement the XFDDS. First, we show the ability of IB3DM to detect faults
and then extensions to providing explanations are presented. Two studies used
to illustrate XFDDS capabilities:

  1. (i)

Single-cell in a module is partially shaded in a PV panel consisting of 36
cells in series;

  2. (ii)

Most cells in a array are partially shaded (8 among 36) in different
proportions (10-90%).

These two cases cover most scenarios envisaged during the PV panel operations
and the incipient fault was created artificially for the study.

###  V-C Case Study 1

This study considers fault in polycrystalline panel (Solartech SPM-020P-R)
using IB3DM. To illustrate IB3DM ability to detect fault, the V-I and P-V
curves are computed at different irradiance levels (0-1000
$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$, 800
$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$, and 450
$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$). The I-V and P-V curves are shown in
Fig.9 (a) and (b), respectively. One can see that the IB3DM predictions
exactly coincide with experimental V-I and P-V curves, whereas model errors
are high with other models. Furthermore, slope of the PV curves at lower
voltages are indicative of the incipient fault and this is illustrated from
fault signatures at low irradiance conditions (see, Fig. 8). Using this fault-
signature metric, the XFDDS could avoids false alarms (false positives). This
results illustrates the ability of IB3DM to detect partial shading in single
cell and avoid false alarms raised by XGBoost classifier. Moreover, detecting
incipient faults is challenging with IB3DM alone.

Figure 8: Fault signature for different irradiance.

Figure 9: (a) V-I Curve for different irradiance, and (b) P-V Curve for
different irradiance (polycrystalline).

###  V-D Case Study 2

The IB3DMs’ ability to detect faults in _monocrystalline array_ with partial
shading of different magnitudes across PV panel is illustrated. Variations in
power curves with the SDM, DDM, CD3DM, and IB3DM for irradiance levels (0-1000
$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$) are considered. Due to partial
shading, the voltage reduction is not severe, whereas the current reduction is
quite high which gets reflected in the power curve as well. This is
illustrated in Fig. 10 and is indicative of a fault. This is observed in
fault-signature metric as well. These findings when combined with XGBoost
based classifier could reduce mis-classification and false alarms.

Figure 10: Power curve for Partial shading condition for PV panel

There are two shortcomings with just IB3DM based detection. It requires enough
significant sample averages to detect incipient faults which is seldom
possible due to their intermittent nature. Next, they cannot identify fault
types nor provide explanations.

###  V-E XFDDS for Incipient Faults

The XFDDS is implemented on two different XGBoost implementations, binary and
multi-class classification. The XGBoost leverages data and averts false alarms
by using IB3DM fault-signature metric. In this study, the monocrystalline
panel string consisting of 56 panels at the International Research Center,
Kalasalingam University, is used as the pilot to demonstrate the XFDDS. In
binary classification, the XGBoost classifies whether a given test sample is
faulty/healthy. This is compared with the fault-signature metric outcomes.
Using a threshold on FSM and based on XGBoost classifier outcomes,
explanations are triggered.

In our study, the partial shading fault was created artificially by covering
panels with papers. A sampling time of 1 $\mathrm{s}$ and the fault is created
at 2400 samples for a duration of 32 minutes. These samples are passed to the
IB3DM, which detects the presence of incipient fault through its fault
signature. This triggers the XAI application to generate explanations. The
explanations for partial shading faults are shown in Fig. 11. The explanations
are very intuitive for detecting incipient partial shading fault. As the
explanation shows that the irradiance (G $\geq$ 896) and the voltage (V $\geq$
141.29 $\mathrm{V}$) , but the current ($\leq$ 9.77 $\mathrm{A}$ ) and the
power magnitudes ($\leq$ 1354 $\mathrm{W}$) are low. This is indicative of a
partial shading fault for the field engineer.

Figure 11: Partial shading fault

Similarly, the LL fault was created by shorting out the lines within a single
string for a short duration. The IB3DM generates fault-signature of the power
curve, which triggered the explanations that are shown in Fig. 12 which is
explaining that current values are less than 7.44$\mathrm{A}$ and power values
lesser than 1112.52 $\mathrm{W}$ while the voltage is greater than 141.29
$\mathrm{V}$ helps the XFDDS decide that there is a LL fault.

Figure 12: Line-to-Line Fault

Explanations on a sample which was false positive detected by XGBoost and
explanations provided by the XAI is shown in Fig.13. Here the decision was
based on the current value, which was less than 7.84 $\mathrm{A}$. However,
the power level greater than 1112.89 $\mathrm{W}$ shows that this is not a
fault. This way, false positives could be avoided by explanations, and this
denotes an intermittent behaviour of the load to the field engineer.

Figure 13: Line-to-Line Fault Explanation Figure 14: Explanations for LL-fault
Multi-Label Classification

Next, the XFDDS was applied to multi-class classification problem. In this
case, the fault-labels were given as input as well. The IB3DM was used to
trigger explanations using thresholds on fault-signature metric. The
explanations for the incipient LL fault is shown in Fig. 14. In this case the
XGBoost identifies the LL fault and the causes are illustrated by current (
less than 7.84 $\mathrm{A}$) and power (less than 1122.38 $\mathrm{W}$).

Figure 15: Explanations for Partial Shading Faults

The explanations for partial shading faults are shown in Fig. 15. The
explanations are very intuitive as the current and power are less than a
threshold, whereas the irradiance is greater than 896
$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$. The fault could be easily understood
by the field technicians. Furthermore, the threshold values are explained to
the field engineers on current, power, irradiance, and voltage. This result
also shows that the explanations and the thresholds are oblivious to the
classification label, i.e., binary or multi-class.

##  VI Conclusions

This paper presented an explainable fault-detection and diagnosis system
(XFDDS) for detecting incipient faults in PV panels. Its main components were:
irradiance based three diode model and eXplainable artificial intelligence
(XAI) application. The IB3DM used irradiance and temperature to compute its
model parameters, thereby increasing its accuracy even at low irradiation
conditions. The model parameters were computed solving a non-convex
constrained optimization problem with five different meta-heuristic
approaches. Our results demonstrated the model fidelity of the IB3DM even at
low irradiance conditions. To exploit the aggregated data from PV panels, an
extreme gradient boosting (XGBoost) based classifier was used and it had two
shortcomings: false alarms and lack of explainability. False alarms were
reduced by combining IB3DM with XGBoost classifier by proposing a fault-
signature metric (FSM). Explanations were provided by extending the XGBoost
with local interpretable model-agnostic explanations (LIME) in the XAI
application. The explanations are quite useful in identifying faults and
planning maintenance operations. The proposed XFDDS implementation, deployment
and demonstration on multiple PV technologies was used to illustrate the
capabilities. Extending XFDDS to include multiple faults and translating the
approach to an edge are the future course of this investigation.

## References

  * [1] T. Rajesh, K. Tamilselvan, A. Vijayalakshmi, C. N. Kumar, and K. A. Reddy, “Design and implementation of an automatic solar tracking system for a monocrystalline silicon material panel using mppt algorithm,” _Materials Today: Proceedings_ , 2020.
  * [2] B. K. Karmakar and A. K. Pradhan, “Detection and classification of faults in solar pv array using thevenin equivalent resistance,” _IEEE Journal of Photovoltaics_ , vol. 10, no. 2, pp. 644–654, 2020.
  * [3] Y. Zhao, R. Ball, J. Mosesian, J.-F. de Palma, and B. Lehman, “Graph-based semi-supervised learning for fault detection and classification in solar photovoltaic arrays,” _IEEE Transactions on Power Electronics_ , vol. 30, no. 5, pp. 2848–2858, 2014.
  * [4] D. S. Pillai, F. Blaabjerg, and N. Rajasekar, “A comparative evaluation of advanced fault detection approaches for pv systems,” _IEEE Journal of Photovoltaics_ , vol. 9, no. 2, pp. 513–527, 2019.
  * [5] B. Jin, D. Li, S. Srinivasan, S.-K. Ng, K. Poolla, and A. Sangiovanni-Vincentelli, “Detecting and diagnosing incipient building faults using uncertainty information from deep neural networks,” in _2019 IEEE International Conference on Prognostics and Health Management (ICPHM)_. IEEE, 2019, pp. 1–8.
  * [6] E. Garoudja, F. Harrou, Y. Sun, K. Kara, A. Chouder, and S. Silvestre, “Statistical fault detection in photovoltaic systems,” _Solar Energy_ , vol. 150, pp. 485–499, 2017.
  * [7] A. Mellit, G. M. Tina, and S. A. Kalogirou, “Fault detection and diagnosis methods for photovoltaic systems: A review,” _Renewable and Sustainable Energy Reviews_ , vol. 91, pp. 1–17, 2018.
  * [8] Z. Gao, C. Cecati, and S. X. Ding, “A survey of fault diagnosis and fault-tolerant techniques—part i: Fault diagnosis with model-based and signal-based approaches,” _IEEE Transactions on Industrial Electronics_ , vol. 62, no. 6, pp. 3757–3767, 2015.
  * [9] Y. Chaibi, M. Malvoni, A. Chouder, M. Boussetta, and M. Salhi, “Simple and efficient approach to detect and diagnose electrical faults and partial shading in photovoltaic systems,” _Energy Conversion and Management_ , vol. 196, pp. 330–343, 2019.
  * [10] S. Shongwe and M. Hanif, “Comparative analysis of different single-diode pv modeling methods,” _IEEE Journal of photovoltaics_ , vol. 5, no. 3, pp. 938–946, 2015.
  * [11] F. Bradaschia, M. C. Cavalcanti, A. J. do Nascimento, E. A. da Silva, and G. M. de Souza Azevedo, “Parameter identification for pv modules based on an environment-dependent double-diode model,” _IEEE Journal of Photovoltaics_ , vol. 9, no. 5, pp. 1388–1397, 2019.
  * [12] V. Khanna, B. Das, D. Bisht, P. Singh _et al._ , “A three diode model for industrial solar cells and estimation of solar cell parameters using pso algorithm,” _Renewable Energy_ , vol. 78, pp. 105–113, 2015.
  * [13] M. H. Qais, H. M. Hasanien, and S. Alghuwainem, “Identification of electrical parameters for three-diode photovoltaic model using analytical and sunflower optimization algorithm,” _Applied Energy_ , vol. 250, pp. 109–117, 2019\.
  * [14] A. Triki-Lahiani, A. B.-B. Abdelghani, and I. Slama-Belkhodja, “Fault detection and monitoring systems for photovoltaic installations: A review,” _Renewable and Sustainable Energy Reviews_ , vol. 82, pp. 2680–2692, 2018\.
  * [15] M. Davarifar, A. Rabhi, A. El-Hajjaji, and M. Dahmane, “Real-time model base fault diagnosis of pv panels using statistical signal processing,” in _2013 International Conference on Renewable Energy Research and Applications (ICRERA)_. IEEE, 2013, pp. 599–604.
  * [16] S. Fadhel, C. Delpha, D. Diallo, I. Bahri, A. Migan, M. Trabelsi, and M. Mimouni, “Pv shading fault detection and classification based on iv curve using principal component analysis: Application to isolated pv system,” _Solar Energy_ , vol. 179, pp. 1–10, 2019.
  * [17] M. H. Ali, A. Rabhi, A. El Hajjaji, and G. M. Tina, “Real time fault detection in photovoltaic systems,” _Energy Procedia_ , vol. 111, pp. 914–923, 2017\.
  * [18] L. Chen and X. Wang, “Adaptive fault localization in photovoltaic systems,” _IEEE Transactions on Smart Grid_ , vol. 9, no. 6, pp. 6752–6763, 2017.
  * [19] Z. Yi and A. H. Etemadi, “Line-to-line fault detection for photovoltaic arrays based on multiresolution signal decomposition and two-stage support vector machine,” _IEEE Transactions on Industrial Electronics_ , vol. 64, no. 11, pp. 8546–8556, 2017.
  * [20] ——, “Fault detection for photovoltaic systems based on multi-resolution signal decomposition and fuzzy inference systems,” _IEEE Transactions on Smart Grid_ , vol. 8, no. 3, pp. 1274–1283, 2016.
  * [21] Y. Zhao, T. Li, X. Zhang, and C. Zhang, “Artificial intelligence-based fault detection and diagnosis methods for building energy systems: Advantages, challenges and the future,” _Renewable and Sustainable Energy Reviews_ , vol. 109, pp. 85–101, 2019.
  * [22] K. Dhibi, R. Fezai, M. Mansouri, M. Trabelsi, A. Kouadri, K. Bouzara, H. Nounou, and M. Nounou, “Reduced kernel random forest technique for fault detection and classification in grid-tied pv systems,” _IEEE Journal of Photovoltaics_ , 2020.
  * [23] Y. Zhao, D. Li, T. Lu, Q. Lv, N. Gu, and L. Shang, “Collaborative fault detection for large-scale photovoltaic systems,” _IEEE Transactions on Sustainable Energy_ , 2020.
  * [24] Y. Gan, Z. Chen, L. Wu, C. Long, S. Cheng, and P. Lin, “A novel fault diagnosis method for pv arrays using extreme gradient boosting classifier,” 2019\.
  * [25] A. Ebrahimifakhar, A. Kabirikopaei, and D. Yuill, “Data-driven fault detection and diagnosis for packaged rooftop units using statistical machine learning classification methods,” _Energy and Buildings_ , vol. 225, p. 110318, 2020\.
  * [26] X.-H. Li, C. C. Cao, Y. Shi, W. Bai, H. Gao, L. Qiu, C. Wang, Y. Gao, S. Zhang, X. Xue _et al._ , “A survey of data-driven and knowledge-aware explainable ai,” _IEEE Transactions on Knowledge and Data Engineering_ , 2020\.
  * [27] S. Bramhall, H. Horn, M. Tieu, and N. Lohia, “Qlime-a quadratic local interpretable model-agnostic explanation approach,” _SMU Data Science Review_ , vol. 3, no. 1, p. 4, 2020.
  * [28] A. Torres-Barrán, Á. Alonso, and J. R. Dorronsoro, “Regression tree ensembles for wind energy and solar radiation prediction,” _Neurocomputing_ , vol. 326, pp. 151–160, 2019.
  * [29] V. J. Chin, Z. Salam, and K. Ishaque, “Cell modelling and model parameters estimation techniques for photovoltaic simulator application: A review,” _Applied Energy_ , vol. 154, pp. 500–519, 2015.
  * [30] R. Sun, G. Wang, W. Zhang, L.-T. Hsu, and W. Y. Ochieng, “A gradient boosting decision tree based gps signal reception classification algorithm,” _Applied Soft Computing_ , vol. 86, p. 105942, 2020.
  * [31] J. Fan, X. Wang, L. Wu, H. Zhou, F. Zhang, X. Yu, X. Lu, and Y. Xiang, “Comparison of support vector machine and extreme gradient boosting for predicting daily global solar radiation using temperature and precipitation in humid subtropical climates: A case study in china,” _Energy Conversion and Management_ , vol. 164, pp. 102–111, 2018.
  * [32] N. Sapountzoglou, J. Lago, and B. Raison, “Fault diagnosis in low voltage smart distribution grids using gradient boosting trees,” _Electric Power Systems Research_ , vol. 182, p. 106254, 2020.