# Generalised Killing Spinors on
Three-Dimensional Lie Groups

Diego Artacho∗ Department of Mathematics, Imperial College London, London, SW7
2AZ, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

For each $n\in\mathbb{N}$, we construct a Riemannian spin manifold carrying a
generalised Killing spinor with $n$ distinct eigenvalues, providing the first
known examples of such objects for $n>4$. For each left-invariant metric on a
$3$-dimensional Lie group, we explicitly find all invariant generalised
Killing spinors. We find many new examples with three distinct eigenvalues,
and we prove that, in some cases, this is the generic situation.

$\displaystyle\small{}^{*}\textit{Imperial College London}$

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Differential forms approach to the spin representation
    2. 2.2 Invariant connections and Nomizu maps
  3. 3 The Heisenberg groups
  4. 4 Three-dimensional Lie groups
  5. A Explicit matrices
    1. A.1 $L(3,-1)$
    2. A.2 $L(3,1)$
    3. A.3 $L(3,2,x)$, $0<\absolutevalue{x}\leq 1$
    4. A.4 $L(3,3)$
    5. A.5 $L(3,4,x)$, $x\geq 0$
    6. A.6 $L(3,5)$
    7. A.7 $L(3,6)$

##  1\. Introduction

Let $(M,g)$ be a Riemannian manifold equipped with a spin structure. Let
$\Sigma M$ be the associated spinor bundle. A spinor
$\psi\in\Gamma\left(\Sigma M\right)$ is said to be a generalised Killing
spinor if there exists $A\in\Gamma\left(\operatorname{End}(TM)\right)$ a
$g$-symmetric endomorphism field such that, for every vector field
$X\in\Gamma\left(TM\right)$,

(1.1) $\widetilde{\nabla}^{g}\psi=A(X)\cdot\psi\,,$

where $\widetilde{\nabla}^{g}$ is the spin connection and $\cdot$ denotes
Clifford multiplication. Such a spinor is an eigenspinor of the Dirac operator
of eigenvalue $\tr(A)$. These spinors encode geometric properties of the
manifold, and they arise naturally in the study of hypersurfaces [1, 2],
special geometries and $G$-structures [3].

Such an endomorphism field $A$ diagonalises in a local orthonormal frame, and
it has real eigenvalues, which are smooth functions $M\to\mathbb{R}$. If $A$
is identically zero, $\psi$ is said to be parallel. If $A$ is diagonal and
non-zero, $\psi$ is said to be Killing.

We say that a generalised Killing spinor $\psi$ has $r$ distinct eigenvalues
if the endomorphism $A$ does. If $r=1$, $\psi$ is a Killing spinor. Examples
with two distinct eigenvalues are relatively common, see [4, 5]. In [6], the
authors mention that generalised Killing spinors with three eigenvalues are
rare, and that the only example known to them of a space carrying one of these
is the quaternionic Heisenberg group. The first examples with four eigenvalues
were given in [5]. No examples with a higher number of eigenvalues are known.

The aim of this paper is two-fold. In Section 3 we explicitly build, for each
$n\in\mathbb{N}$, a left-invariant metric on the $n$-th real Heisenberg group
$H_{2n+1}$ which carries a generalised Killing spinor with $n+1$ distinct
eigenvalues –see Theorem 3.3. These are the first known examples of such
objects for $n>3$.

In Section 4, we answer the following question:

###### Question.

Let $G$ be a $3$-dimensional Lie group and $g$ a left-invariant metric on $G$.
Does $g$ carry invariant generalised Killing spinors? If so, what is the
maximum number of distinct eigenvalues?

The answer to this question is given in Theorem 4.2. We summarise this result
in Table 1.

$\mathfrak{g}$ | $\dim\left(\Sigma_{\text{inv}}^{GK}\right)$ |  $r$
---|---|---
$L(3,-1)$ | $0$ |  $-$
$L(3,1)$ | $2$ |  $2$
$L(3,2,x)$ |  $2$, if $x=-1$ |  $3$
$0$, if $x\neq-1$ |  $-$
$L(3,3)$ | $0$ |  $-$
$L(3,4,x)$ |  $2$, if $x=0$ |  $3$
$0$, if $x\neq 0$ |  $-$
$L(3,5)$ | $2$ |  $3$
$L(3,6)$ | $2$ |  $3$
Table 1. Maximum (and generic) number of distinct eigenvalues, denoted as $r$,
of invariant generalized Killing spinors on a Lie group with Lie algebra
$\mathfrak{g}$ equipped with any left-invariant metric.
$\Sigma_{\text{inv}}^{GK}$ denotes the complex dimension of the vector space
of invariant generalised Killing spinors.

##  2\. Preliminaries

###  2.1. Differential forms approach to the spin representation

For a more general introduction to this approach to the spin representation,
see [5]. Let $n\in\mathbb{N}$, and let $G$ be a $(2n+1)$-dimensional Lie
group. Left-invariant Riemannian metrics on $G$ are in bijective
correspondence with inner products on its Lie algebra $\mathfrak{g}$. Let $B$
be such an inner product, and let $g$ be the corresponding metric on $G$. Let
$(e_{1},\dots,e_{2n+1})$ be a $B$-orthonormal basis of $\mathfrak{g}$.

First, consider the complexification $\mathfrak{g}^{\mathbb{C}}$ of the real
vector space $\mathfrak{g}$. It is clear that

$\mathfrak{g}^{\mathbb{C}}=\operatorname{span}_{\mathbb{C}}\left\\{e_{1}\right\\}\oplus
L\oplus L^{\prime}\,,$

where

$L=\operatorname{span}_{\mathbb{C}}\left\\{x_{p}=\frac{1}{\sqrt{2}}\left(e_{2p}-ie_{2p+1}\right)\right\\}_{p=1,\dots,n}\,,\qquad
L^{\prime}=\operatorname{span}_{\mathbb{C}}\left\\{y_{p}=\frac{1}{\sqrt{2}}\left(e_{2p}+ie_{2p+1}\right)\right\\}_{p=1,\dots,n}\,.$

Let

$\Sigma=\Lambda^{\bullet}L^{\prime}\,.$

Note that we can let $\mathfrak{g}^{\mathbb{C}}$ act on $\Sigma$ by extending
the following: for each $\eta\in\Sigma$ and $1\leq p\leq n$,

$e_{1}\cdot\eta=i\left(\eta_{\text{even}}-\eta_{\text{odd}}\right)\,,\qquad
x_{p}\cdot\eta=i\sqrt{2}x_{p}\lrcorner\eta\,,\qquad
y_{p}\cdot\eta=i\sqrt{2}y_{p}\wedge\eta\,.$

This action of $\mathbb{C}^{n}\cong\mathfrak{g}^{\mathbb{C}}$ on $\Sigma$
extends to a non-trivial action of the complex Clifford algebra
$\mathbb{C}\rm{l}(2n+1)$ on $\Sigma$. And the dimension of $\Sigma$ is
$2^{n}$. Hence, the restriction of this action to $\operatorname{Spin}(2n+1)$
is isomorphic to the spin representation. Note that, for each $1\leq p\leq n$
and $\eta\in\Sigma$,

$\qquad
e_{1}\cdot\eta=i\left(\eta_{\text{even}}-\eta_{\text{odd}}\right)\,,\qquad
e_{2p}\cdot\eta=i\left(x_{p}\lrcorner\eta+y_{p}\wedge\eta\right)\,,\qquad
e_{2p+1}\cdot\eta=\left(y_{p}\wedge\eta-x_{p}\lrcorner\eta\right)\,.$

$G$ has a unique $G$-invariant spin structure in the sense of [7]. Its
associated spinor bundle is given by the trivial bundle

$G\times\Sigma\,,$

and sections of this bundle correspond to smooth maps $G\to\Sigma$. $G$ acts
on the space of spinors as follows: for $x\in G$ and a spinor $\psi$,

$(x\cdot\psi)(y)=\psi(x^{-1}\cdot y).$

A spinor $\psi$ is said to be invariant if it is stabilized by all elements of
$G$, i.e., if $\psi$ is a constant map. Hence, elements of $\Sigma$ correspond
to invariant spinors.

###  2.2. Invariant connections and Nomizu maps

For an extensive treatment of this topic, see [8]. We will only need the
following result:

######  Lemma 2.1.

Let $m\in\mathbb{N}$ and let $G$ be an $m$-dimensional Lie group. Let $B$ be
an inner product on its Lie algebra $\mathfrak{g}$, let $g$ be the
corresponding left-invariant metric on $G$, and let $\nabla^{g}$ be the Levi-
Civita connection of $g$. Define the Nomizu map of $\nabla^{g}$ to be the
unique linear map $\Uplambda\colon\mathfrak{g}\to\mathfrak{so}(m)$ satisfying

$\Uplambda(X)(Y)=\frac{1}{2}\left[X,Y\right]+U(X,Y)\,,$

where

$B\left(U(X,Y),W\right)=\frac{1}{2}\left(B\left(\left[W,X\right],Y\right)+B\left(X,\left[W,Y\right]\right)\right)\,.$

Let $\widetilde{\Uplambda}=f\circ\Uplambda$, where

$\displaystyle f\colon$
$\displaystyle\mathfrak{so}(m)\overset{\cong}{\longrightarrow}\mathfrak{spin}(m)$
$\displaystyle e_{i}\wedge e_{j}\mapsto\frac{1}{2}e_{i}\cdot e_{j}\,.$

Let $\psi\in\Sigma$ be an invariant spinor. Then, for each $X\in\mathfrak{g}$,

$\widetilde{\nabla}^{g}_{X}\psi=\widetilde{\Uplambda}(X)\cdot\psi\,,$

where $\cdot$ denotes the differential of the spin representation introduced
in 2.1. ∎

##  3\. The Heisenberg groups

In this section we exhibit the first known examples of generalised Killing
spinors with an arbitrary number of distinct eigenvalues. For each
$n\in\mathbb{N}$, we will find a left-invariant metric on the $n$-th
Heisenberg group $H_{2n+1}$ which carries an invariant generalised Killing
spinor with $(n+1)$ distinct eigenvalues.

We start by recalling the definition of the Heisenberg groups.

######  Definition 3.1.

Let $n\in\mathbb{N}$. The $n$-th (real) Heisenberg group $H_{2n+1}$ is defined
as

$H_{2n+1}=\left\\{\begin{pmatrix}1&v^{T}&x\\\ 0&\operatorname{Id}_{n}&w\\\
0&0&1\end{pmatrix}\in\operatorname{M}_{n+2}\left(\mathbb{R}\right)\colon
v,w\in\mathbb{R}^{n},\,x\in\mathbb{R}\right\\}\,,$

where we think of elements $v\in\mathbb{R}^{n}$ as column vectors and $v^{T}$
denotes the transpose of $v$.

Note that, with the obvious topology and smooth structure, $H_{2n+1}$ is a Lie
group which is diffeomorphic to $\mathbb{R}^{2n+1}$. The Lie algebra of
$H_{2n+1}$ is

$\mathfrak{h}_{2n+1}=\operatorname{span}_{\mathbb{R}}\left\\{\hat{Z},\,\hat{E}_{p},\,\hat{F}_{p}\right\\}_{p=1,\dots,n}\,,$

where

$\hat{Z}=\begin{pmatrix}1&0^{T}&1\\\ 0&\operatorname{Id}_{n}&0\\\
0&0&1\end{pmatrix}\,,\qquad\hat{E}_{p}=\begin{pmatrix}1&e_{p}^{T}&0\\\
0&\operatorname{Id}_{n}&0\\\
0&0&1\end{pmatrix}\,,\qquad\hat{F}_{p}=\begin{pmatrix}1&0^{T}&0\\\
0&\operatorname{Id}_{n}&e_{p}\\\ 0&0&1\end{pmatrix}\,$

and $e_{p}$ denotes the $p$-th vector of the standard basis of
$\mathbb{R}^{n}$. These generators of $\mathfrak{h}_{2n+1}$ satisfy, for
$1\leq p,q\leq n$,

$\left[\hat{E}_{p},\hat{F}_{q}\right]=\delta_{p,q}\hat{Z}\,,\qquad\left[\hat{E}_{p},\hat{Z}\right]=\left[\hat{F}_{p},\hat{Z}\right]=\left[\hat{E}_{p},\hat{E}_{q}\right]=\left[\hat{F}_{p},\hat{F}_{q}\right]=0\,,$

where $\delta$ is the Kronecker delta.

Fix $a,b\in\mathbb{R}_{>0}^{p}$ and $c\in\mathbb{R}_{>0}$. Define $B$ to be
the unique inner product on $\mathfrak{h}_{2n+1}$ which makes the basis

$\left\\{Z=\frac{1}{\sqrt{c}}\hat{Z}\,,E_{p}=\frac{1}{\sqrt{a_{p}}}\hat{E}_{p},\,F_{p}=\frac{1}{\sqrt{b_{p}}}\hat{F}_{p}\right\\}_{p=1,\dots,n}$

orthonormal. Note that $B$ naturally induces a left-invariant metric on
$H_{2n+1}$, which we will denote by $g$.

######  Lemma 3.2.

The Nomizu map
$\Uplambda\colon\mathfrak{h}_{2n+1}\to\mathfrak{so}(\mathfrak{h}_{2n+1})$ of
the Levi-Civita connection of $g$ is given by

$\displaystyle\Uplambda\left(Z\right)=-\frac{1}{2}\sum_{p=1}^{n}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\wedge
F_{p}\,,$
$\displaystyle\Uplambda\left(F_{p}\right)=-\frac{1}{2}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\wedge
Z\,,$
$\displaystyle\Uplambda\left(E_{p}\right)=\frac{1}{2}\sqrt{\frac{c}{a_{p}b_{p}}}F_{p}\wedge
Z\,.$

###### Proof.

Recall from [9, 8]111There is a sign mistake in [9], which is fixed in [8].
that the Nomizu map $\Uplambda$ of the Levi-Civita connection of the invariant
metric $g$ on $H_{2n+1}$ induced by the inner product $B$ on
$\mathfrak{h}_{2n+1}$ satisfies, for every $X,Y\in\mathfrak{h}_{2n+1}$,

$\Uplambda(X)(Y)=\frac{1}{2}\left[X,Y\right]+U(X,Y)\,,$

where

$B\left(U(X,Y),W\right)=\frac{1}{2}\left(B\left(\left[W,X\right],Y\right)+B\left(X,\left[W,Y\right]\right)\right)\,.$

Fix $1\leq p\leq n$. Let us first compute $\Uplambda\left(E_{p}\right)$. Note
that, for $1\leq q\leq n$,

$\displaystyle\Uplambda\left(E_{p}\right)(E_{q})$
$\displaystyle=\frac{1}{2}\left[E_{p},E_{q}\right]+U(E_{p},E_{q})=0\,,$
$\displaystyle\Uplambda\left(E_{p}\right)(F_{q})$
$\displaystyle=\frac{1}{2}\left[E_{p},F_{q}\right]+U(E_{p},F_{q})=\frac{1}{2}\left[E_{p},F_{q}\right]=$
$\displaystyle=\frac{1}{2}\frac{1}{\sqrt{a_{p}b_{q}}}\left[\hat{E}_{p},\hat{F}_{q}\right]=\delta_{p,q}\frac{1}{2}\frac{1}{\sqrt{a_{p}b_{q}}}\hat{Z}=\delta_{p,q}\frac{1}{2}\sqrt{\frac{c}{a_{p}b_{q}}}Z\,,$
$\displaystyle\Uplambda\left(E_{p}\right)(Z)$
$\displaystyle=\frac{1}{2}\left[E_{p},Z\right]+U(E_{p},Z)=U(E_{p},Z)=$
$\displaystyle=\frac{1}{2}B\left(\left[F_{p},E_{p}\right],Z\right)F_{p}=-\frac{1}{2}\sqrt{\frac{c}{a_{p}b_{p}}}F_{p}\,.$

This proves that $\Uplambda\left(E_{p}\right)$ is as claimed. Let us now
compute $\Uplambda\left(F_{p}\right)$. Similarly to the previous computations,
for each $1\leq q\leq n$ we have that

$\displaystyle\Uplambda\left(F_{p}\right)(E_{q})$
$\displaystyle=\frac{1}{2}\left[F_{p},E_{q}\right]+U(F_{p},E_{q})=\frac{1}{2}\left[F_{p},E_{q}\right]=-\delta_{p,q}\frac{1}{2}\sqrt{\frac{c}{b_{p}a_{q}}}Z\,,$
$\displaystyle\Uplambda\left(F_{p}\right)(F_{q})$
$\displaystyle=\frac{1}{2}\left[F_{p},F_{q}\right]+U(F_{p},F_{q})=0\,,$
$\displaystyle\Uplambda\left(F_{p}\right)(Z)$
$\displaystyle=\frac{1}{2}\left[F_{p},Z\right]+U(F_{p},Z)=U(F_{p},Z)=$
$\displaystyle=\frac{1}{2}B\left(\left[E_{p},F_{p}\right],Z\right)E_{p}=\frac{1}{2}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\,.$

This proves that $\Uplambda\left(F_{p}\right)$ is as claimed. Finally, let us
compute $\Uplambda\left(Z\right)$. For each $1\leq p\leq n$, we have that

$\displaystyle\Uplambda\left(Z\right)(E_{p})$
$\displaystyle=\frac{1}{2}\left[Z,E_{p}\right]+U(Z,E_{p})=U(Z,E_{p})=$
$\displaystyle=\frac{1}{2}B\left(Z,\left[F_{p},E_{p}\right]\right)F_{p}=-\frac{1}{2}\sqrt{\frac{c}{a_{p}b_{p}}}F_{p}\,,$
$\displaystyle\Uplambda\left(Z\right)(F_{p})$
$\displaystyle=\frac{1}{2}\left[Z,F_{p}\right]+U(Z,F_{p})=U(Z,F_{p})=$
$\displaystyle=\frac{1}{2}B\left(Z,\left[E_{p},F_{p}\right]\right)E_{p}=\frac{1}{2}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\,,$
$\displaystyle\Uplambda\left(Z\right)(Z)$
$\displaystyle=\frac{1}{2}\left[Z,Z\right]+U(Z,Z)=0\,.$

This finishes the proof. ∎

Recalling the notation of Section 2.1, we can now state our main result.

######  Theorem 3.3.

The invariant spinor on $H_{2n+1}$ determined by $1\in\Sigma$ is a generalised
Killing spinor with the following set of eigenvalues:

$\left\\{\lambda_{p}=\frac{1}{4}\sqrt{\frac{c}{a_{p}b_{p}}}\,,\,\mu=-\sum_{q=1}^{n}\lambda_{q}\right\\}_{p=1,\dots,n}\,.$

In particular, choosing $b_{p}=c=1$ and $a_{p}=p^{2}$, there are $n+1$
distinct eigenvalues.

###### Proof.

For $X\in\mathfrak{h}_{2n+1}$, let
$\widetilde{\Uplambda}(X)\in\mathfrak{spin}(2n+1)$ be the image of
$\Uplambda(X)\in\mathfrak{so}(2n+1)$ under the isomorphism

$e_{i}\wedge e_{j}\mapsto\frac{1}{2}e_{i}\cdot e_{j}\,.$

We will check that, for all $1\leq p\leq n$,

$\displaystyle\widetilde{\Uplambda}\left(E_{p}\right)\cdot
1=\lambda_{p}E_{p}\cdot 1\,,$
$\displaystyle\widetilde{\Uplambda}\left(F_{p}\right)\cdot
1=\lambda_{p}F_{p}\cdot 1\,,$
$\displaystyle\widetilde{\Uplambda}\left(Z\right)\cdot 1=\mu Z\cdot 1\,.$

From Lemma 3.2, we get

$\displaystyle\widetilde{\Uplambda}\left(E_{p}\right)=\frac{1}{4}\sqrt{\frac{c}{a_{p}b_{p}}}F_{p}\cdot
Z\,,$
$\displaystyle\widetilde{\Uplambda}\left(F_{p}\right)=-\frac{1}{4}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\cdot
Z\,,$
$\displaystyle\widetilde{\Uplambda}\left(Z\right)=-\frac{1}{4}\sum_{p=1}^{n}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\cdot
F_{p}\,.$

Hence,

$\displaystyle\widetilde{\Uplambda}\left(E_{p}\right)\cdot
1=\frac{1}{4}\sqrt{\frac{c}{a_{p}b_{p}}}F_{p}\cdot Z\cdot
1=\frac{i}{4}\sqrt{\frac{c}{a_{p}b_{p}}}F_{p}\cdot
1=\frac{i}{4}\sqrt{\frac{c}{a_{p}b_{p}}}y_{p}\,,$
$\displaystyle\widetilde{\Uplambda}\left(F_{p}\right)\cdot
1=-\frac{1}{4}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\cdot Z\cdot
1=-\frac{i}{4}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\cdot
1=\frac{1}{4}\sqrt{\frac{c}{a_{p}b_{p}}}y_{p}\,,$
$\displaystyle\widetilde{\Uplambda}\left(Z\right)\cdot
1=-\frac{1}{4}\sum_{p=1}^{n}\sqrt{\frac{c}{a_{p}b_{p}}}E_{p}\cdot F_{p}\cdot
1=-\frac{i}{4}\sum_{p=1}^{n}\sqrt{\frac{c}{a_{p}b_{p}}}\cdot 1\,.$

And because

$E_{p}\cdot 1=iy_{p}\,,\qquad F_{p}\cdot 1=y_{p}\,,\qquad Z\cdot 1=i1\,,$

we conclude. ∎

##  4\. Three-dimensional Lie groups

In this section we study the existence of left-invariant metrics on
$3$-dimensional Lie groups which admit invariant generalised Killing spinors.
We use the classification of $3$-dimensional real Lie algebras in [10] –see
Appendix A.

First, we prove a surprising fact:

######  Lemma 4.1.

Let $G$ be a $3$-dimensional Lie group with Lie algebra $\mathfrak{g}$, and
let $g$ be a left-invariant metric on $G$ carrying a non-zero invariant
generalised Killing spinor with associated endomorphism $A$. Then, every
invariant spinor is generalised Killing with endomorphism $A$.

###### Proof.

Let $\left(e_{1},e_{2},e_{3}\right)$ be a $g$-orthonormal basis of
$\mathfrak{g}$. Let $\widetilde{\Uplambda}$ be the lift to
$\mathfrak{spin}(3)$ of the Nomizu map of the Levi-Civita connection of $g$.
Then, there exist linear forms

$\lambda,\mu,\nu\colon\mathfrak{g}\to\mathbb{R}$

such that, for every $X\in\mathfrak{g}$,

$\widetilde{\Uplambda}(X)=\lambda(X)e_{1}e_{2}+\mu(X)e_{1}e_{3}+\nu(X)e_{2}e_{3}\,,$

and hence

$\displaystyle\widetilde{\Uplambda}(X)\cdot 1$
$\displaystyle=\lambda(X)e_{1}e_{2}\cdot 1+\mu(X)e_{1}e_{3}\cdot
1+\nu(X)e_{2}e_{3}\cdot 1$
$\displaystyle=i\lambda(X)e_{1}\cdot\left(y_{1}\right)+\mu(X)e_{1}\cdot\left(y_{1}\right)+\nu(X)e_{2}\cdot\left(y_{1}\right)$
$\displaystyle=\lambda(X)y_{1}-i\mu(X)y_{1}+i\nu(X)1$
$\displaystyle=\lambda(X)e_{3}\cdot 1-\mu(X)e_{2}\cdot 1+\nu(X)e_{1}\cdot 1$
$\displaystyle=\left(\lambda(X)e_{3}-\mu(X)e_{2}+\nu(X)e_{1}\right)\cdot 1\,.$

Similarly,

$\displaystyle\widetilde{\Uplambda}(X)\cdot y_{1}$
$\displaystyle=\lambda(X)e_{1}e_{2}\cdot y_{1}+\mu(X)e_{1}e_{3}\cdot
y_{1}+\nu(X)e_{2}e_{3}\cdot y_{1}$
$\displaystyle=i\lambda(X)e_{1}\cdot\left(1\right)-\mu(X)e_{1}\cdot\left(1\right)-\nu(X)e_{2}\cdot\left(1\right)$
$\displaystyle=-\lambda(X)1-i\mu(X)1-i\nu(X)y_{1}$
$\displaystyle=\lambda(X)e_{3}\cdot y_{1}-\mu(X)e_{2}\cdot
y_{1}+\nu(X)e_{1}\cdot y_{1}$
$\displaystyle=\left(\lambda(X)e_{3}-\mu(X)e_{2}+\nu(X)e_{1}\right)\cdot
y_{1}\,.$

Define $B\in\operatorname{End}(\mathfrak{g})$ by

$X\mapsto\lambda(X)e_{3}-\mu(X)e_{2}+\nu(X)e_{1}\,.$

It is clear that, for each $X\in\mathfrak{g}$, and every invariant spinor
$\Phi\in\Sigma$,

(4.1) $\widetilde{\Uplambda}(X)\cdot\Phi=B(X)\cdot\Phi\,.$

But, by hypothesis, there exists a non-zero invariant generalised Killing
spinor $\Psi$ with associated symmetric endomorphism
$A\in\operatorname{End}(\mathfrak{g})$. Hence, for every $X\in\mathfrak{g}$,

$A(X)\cdot\Psi=\widetilde{\Uplambda}(X)\cdot\Psi=B(X)\cdot\Psi\,.$

As $\Psi\neq 0$, this implies that $A=B$, and hence $B$ is symmetric. So, by
(4.1), every invariant generalised Killing spinor is generalised Killing with
endomorphism $A$. ∎

We can now state the main result of this section.

######  Theorem 4.2.

Let $G$ be a $3$-dimensional Lie group with Lie algebra $\mathfrak{g}$.

  1. (1)

If $\mathfrak{g}=L(3,-1)$, $L(3,2,x)$ for some $0<\absolutevalue{x}\leq 1$ and
$x\neq-1$, $L(3,3)$ or $L(3,4,y)$ for some $y>0$, then no left-invariant
metric on $G$ carries non-trivial invariant generalised Killing spinors.

  2. (2)

If $\mathfrak{g}=L(3,1)$, then every left-invariant metric on $G$ carries a
$2$-dimensional space of invariant generalised Killing spinors. Moreover, the
set of left-invariant metrics for which the number of distinct eigenvalues of
these spinors is strictly less than $2$ has measure zero in the set of left-
invariant metrics on $G$.

  3. (3)

If $\mathfrak{g}=L(3,2,-1)$, $L(3,4,0)$, $L(3,5)$ or $L(3,6)$, then every
left-invariant metric on $G$ carries a $2$-dimensional space of invariant
generalised Killing spinors. Moreover, the set of left-invariant metrics for
which the number of distinct eigenvalues of these spinors is strictly less
than $3$ has measure zero in the set of all left-invariant metrics on $G$.

###### Proof.

The strategy in every case will be the following. We start by fixing a basis
$(f_{1},f_{2},f_{3})$ of $\mathfrak{g}$ for which the structure constants take
the form given in the classification [10] –see Appendix A. Then, we consider a
general invertible matrix

$P=\begin{pmatrix}\alpha&\beta&\gamma\\\ \delta&\varepsilon&\zeta\\\
\eta&\theta&\iota\\\ \end{pmatrix}\in\operatorname{GL}(3,\mathbb{R})\,,$

and define $B_{P}$ to be the unique inner product on $\mathfrak{g}$ such that
the basis $\left(e_{1},e_{2},e_{3}\right)$ defined by

$\displaystyle e_{1}=\alpha f_{1}+\delta f_{2}+\eta f_{3}\,,$ $\displaystyle
e_{2}=\beta f_{1}+\varepsilon f_{2}+\theta f_{3}\,,$ $\displaystyle
e_{3}=\gamma f_{1}+\zeta f_{2}+\iota f_{3}\,,$

is $B_{P}$-orthonormal.

Then, we compute the endomorphism $A$ of $\mathfrak{g}$ such that, for every
$X\in\mathfrak{g}$,

(4.2) $\widetilde{\Uplambda}(X)\cdot 1=A(X)\cdot 1\,.$

This exists by the proof of Lemma 4.1. This is an easy but long and tedious
computation which is analogous to the one in Section 3. The explicit results
are collected in Appendix A.

Finally, by Lemma 4.1, there are only two cases. If $A$ is symmetric, then all
invariant spinors are generalised Killing with endomorphism $A$. If not, the
only invariant generalised Killing spinor is $0$. In Table 2, we show the
value of $A-A^{T}$ in each case.

$\mathfrak{g}$ | $A-A^{T}$
---|---
$L(3,-1)$ | $\frac{1}{2}\begin{pmatrix}0&-\zeta&\varepsilon\\\ \zeta&0&-\delta\\\ -\varepsilon&\delta&0\\\ \end{pmatrix}$
$L(3,1)$ | $\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$
$L(3,2,x)$ | $\frac{x+1}{2}\begin{pmatrix}0&-\iota&\theta\\\ \iota&0&-\eta\\\ -\theta&\eta&0\\\ \end{pmatrix}$
$L(3,3)$ | $\begin{pmatrix}0&-\iota&\theta\\\ \iota&0&-\eta\\\ -\theta&\eta&0\\\ \end{pmatrix}$
$L(3,4,x)$ | $x\begin{pmatrix}0&-\iota&\theta\\\ \iota&0&-\eta\\\ -\theta&\eta&0\\\ \end{pmatrix}$
$L(3,5)$ | $\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$
$L(3,6)$ | $\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$
Table 2. For each Lie algebra $\mathfrak{g}$, value of $A-A^{T}$.

∎

Hitchin considered in [11] a family of left-invariant deformations of the
round metric on $S^{3}\cong\operatorname{SU}(2)$, and computed the spectrum of
the Dirac operator. Here we have considered the whole family of left-invariant
metrics.

######  Remark 4.3.

It is very surprising that the symmetry of $A$ is completely independent of
the choice of left-invariant metric – see Table 2.

##  Appendix A Explicit matrices

Let $G$ be a $3$-dimensional Lie group with Lie algebra $\mathfrak{g}$.

$\mathfrak{g}=\operatorname{span}_{\mathbb{R}}\\{f_{1},f_{2},f_{3}\\}\,.$

Left-invariant metrics on $G$ are in bijective correspondence with inner
products on $\mathfrak{g}$, which in turn are in bijective correspondence with
invertible matrices

$P=\begin{pmatrix}\alpha&\beta&\gamma\\\ \delta&\varepsilon&\zeta\\\
\eta&\theta&\iota\\\ \end{pmatrix}\in\operatorname{GL}(3,\mathbb{R})\,.$

Namely, to the matrix $P$ corresponds the unique inner product $B_{P}$ on
$\mathfrak{g}$ such that the basis

$\displaystyle e_{1}=\alpha f_{1}+\delta f_{2}+\eta f_{3}\,,$ $\displaystyle
e_{2}=\beta f_{1}+\varepsilon f_{2}+\theta f_{3}\,,$ $\displaystyle
e_{3}=\gamma f_{1}+\zeta f_{2}+\iota f_{3}\,,$

is $B_{P}$-orthonormal. Let $g_{P}$ be the Riemannian metric on $G$
corresponding to $B_{P}$. We saw in the proof of Lemma 4.1 that there exists
$A\in\operatorname{End}(\mathfrak{g})$ such that, for all invariant spinors
$\phi\in\Sigma$ and all $X\in\mathfrak{g}$,

$\widetilde{\Uplambda}(X)\cdot\phi=A(X)\cdot\phi\,.$

For each of the non-abelian $3$-dimensional Lie algebras [10], we give an
explicit expression of the matrix of $A$ in the basis $(e_{1},e_{2},e_{3})$.
These matrices are obtained using Wolfram Mathematica by following the same
procedure illustrated in Section 3.

###  A.1. $L(3,-1)$

There Lie bracket is given by

| $f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---|---
$f_{1}$ | $0$ | $f_{1}$ | $0$
$f_{2}$ | $-f_{1}$ | $0$ | $0$
$f_{3}$ | $0$ | $0$ | $0$

This is a direct sum

$L(3,-1)=L(2,1)\oplus L(1,1)\,.$

Then,

$A=\frac{1}{4\det(P)}\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{pmatrix}\,,$

where

$\displaystyle
a_{11}=\delta(\alpha\iota\zeta+\alpha\theta\varepsilon+\beta\eta\varepsilon+\eta\gamma\zeta)-\alpha\eta(\varepsilon^{2}+\zeta^{2})-\delta^{2}(\beta\theta+\gamma\iota)+(\gamma\varepsilon-\beta\zeta)(\iota\varepsilon-\theta\zeta)\,,$
$\displaystyle
a_{12}=2(\gamma\varepsilon-\beta\zeta)(\eta\zeta-\delta\iota)\,,$
$\displaystyle
a_{13}=2(\gamma\varepsilon-\beta\zeta)(\delta\theta-\eta\varepsilon)\,,$
$\displaystyle
a_{21}=2(\delta\gamma-\alpha\zeta)(\theta\zeta-\iota\varepsilon)\,,$
$\displaystyle
a_{22}=-\delta(\alpha\iota\zeta-\alpha\theta\varepsilon-\beta\eta\varepsilon+\eta\gamma\zeta)-\alpha\eta(\varepsilon-\zeta)(\varepsilon+\zeta)-\delta^{2}(\beta\theta-\gamma\iota)-(\gamma\varepsilon-\beta\zeta)(\iota\varepsilon-\theta\zeta)\,,$
$\displaystyle
a_{23}=-2(\delta\gamma-\alpha\zeta)(\delta\theta-\eta\varepsilon)\,,$
$\displaystyle
a_{31}=2(\beta\delta-\alpha\varepsilon)(\iota\varepsilon-\theta\zeta)\,,$
$\displaystyle
a_{32}=-2(\beta\delta-\alpha\varepsilon)(\delta\iota-\eta\zeta)\,,$
$\displaystyle
a_{33}=-\delta(-\zeta(\alpha\iota+\eta\gamma)+\alpha\theta\varepsilon+\beta\eta\varepsilon)-\alpha\eta(\zeta^{2}-\varepsilon^{2})-\delta^{2}(\gamma\iota-\beta\theta)-(\gamma\varepsilon-\beta\zeta)(\iota\varepsilon-\theta\zeta)\,.$

###  A.2. $L(3,1)$

There Lie bracket is given by

| $f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---|---
$f_{1}$ | $0$ | $0$ | $0$
$f_{2}$ | $0$ | $0$ | $f_{1}$
$f_{3}$ | $0$ | $-f_{1}$ | $0$

This is the $3$-dimensional Heisenberg algebra $\mathfrak{h}_{3}$. In this
case,

$A=\frac{1}{4\det(P)}\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{pmatrix}\,,$

where

$\displaystyle
a_{11}=\delta^{2}(\iota^{2}+\theta^{2})-2\delta\eta(\iota\zeta+\theta\varepsilon)+\eta^{2}(\varepsilon^{2}+\zeta^{2})-(\iota\varepsilon-\theta\zeta)^{2}\,,$
$\displaystyle
a_{12}=2(\delta\iota-\eta\zeta)(\iota\varepsilon-\theta\zeta)\,,$
$\displaystyle
a_{13}=2(\delta\theta-\eta\varepsilon)(\theta\zeta-\iota\varepsilon)\,,$
$\displaystyle
a_{21}=2(\delta\iota-\eta\zeta)(\iota\varepsilon-\theta\zeta)\,,$
$\displaystyle
a_{22}=\delta^{2}(\theta^{2}-\iota^{2})+\delta\eta(2\iota\zeta-2\theta\varepsilon)+\eta^{2}(\varepsilon^{2}-\zeta^{2})+(\iota\varepsilon-\theta\zeta)^{2}\,,$
$\displaystyle
a_{23}=2(\delta\iota-\eta\zeta)(\delta\theta-\eta\varepsilon)\,,$
$\displaystyle
a_{31}=2(\delta\theta-\eta\varepsilon)(\theta\zeta-\iota\varepsilon)\,,$
$\displaystyle
a_{32}=2(\delta\iota-\eta\zeta)(\delta\theta-\eta\varepsilon)\,,$
$\displaystyle
a_{33}=\delta^{2}(\iota^{2}-\theta^{2})+2\delta\eta(\theta\varepsilon-\iota\zeta)+\eta^{2}(\zeta^{2}-\varepsilon^{2})+(\iota\varepsilon-\theta\zeta)^{2}\,.$

###  A.3. $L(3,2,x)$, $0<\absolutevalue{x}\leq 1$

For $0<\absolutevalue{x}\leq 1$, we denote by $L(3,2,x)$ the Lie algebra with
Lie bracket given by

| $f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---|---
$f_{1}$ | $0$ | $0$ | $f_{1}$
$f_{2}$ | $0$ | $0$ | $xf_{2}$
$f_{3}$ | $-f_{1}$ | $-xf_{2}$ | $0$

Note that, for $x=-1$, this is the Poincaré algebra $\mathfrak{p}(1,1)$, i.e.,
the Lie algebra of the Poincaré group $P(1,1)$ –see [12, Section 1.2.5]. For
general $x$,

$A=\frac{1}{2\det(P)}\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{pmatrix}\,,$

where

$\displaystyle a_{11}=$
$\displaystyle-\delta\eta(\gamma\iota+\beta\theta)+\alpha\delta(\iota^{2}+\theta^{2})+\beta(\eta-\iota)(\eta+\iota)+\delta^{2}(\iota^{2}+\theta^{2})+\eta^{2}(\gamma^{2}+\varepsilon^{2}+\zeta^{2})$
$\displaystyle-\gamma^{2}\theta^{2}-\iota^{2}\varepsilon^{2}+2\iota\theta\varepsilon\zeta-\theta^{2}\zeta^{2}\,,$
$\displaystyle a_{12}=$
$\displaystyle-\gamma\iota(\alpha\theta+\beta\eta+\delta\theta
x-\eta\varepsilon x)+\iota\zeta(\alpha\theta x+\eta(\beta x+\varepsilon
x))-\iota^{2}(\alpha(\beta-\varepsilon x)+\delta(\beta x+\varepsilon x))$
$\displaystyle+\eta\gamma^{2}\theta+\eta\theta\zeta^{2}\,,$ $\displaystyle
a_{13}=$ $\displaystyle-\beta\theta(\iota(\alpha+\delta
x)+\eta(\gamma-x\zeta))+\alpha\theta(\gamma\theta+\iota\varepsilon x-\theta
x\zeta)+\beta^{2}\eta\iota+(\delta\theta-\eta\varepsilon)(\gamma\theta
x-\iota\varepsilon+\theta\zeta)\,,$ $\displaystyle a_{21}=$
$\displaystyle-\gamma\iota(\alpha\theta+\beta\eta-\delta\theta
x+\eta\varepsilon x)+\iota\zeta(\alpha\theta x+\eta(\varepsilon-\beta
x)+\delta\theta)-\iota^{2}(\alpha(\beta+\varepsilon
x)+\delta(\varepsilon-\beta x))$
$\displaystyle+\eta\gamma^{2}\theta+\eta\theta\zeta^{2}\,,$ $\displaystyle
a_{22}=$
$\displaystyle-\delta\eta(\gamma\iota+\beta\theta)-\iota^{2}(\alpha^{2}+\delta^{2}-\varepsilon^{2})+\alpha^{2}\theta^{2}-2\beta\theta(\alpha\eta+\gamma\iota)+2\alpha\eta\gamma\iota+\beta^{2}(\eta^{2}+\iota^{2})+\delta^{2}\theta^{2}$
$\displaystyle+2\iota\zeta(\delta\eta-\theta\varepsilon)-2\delta\eta\theta\varepsilon+\gamma^{2}(\theta^{2}-\eta^{2})+\eta^{2}\varepsilon^{2}-\eta^{2}\zeta^{2}+\theta^{2}\zeta^{2}\,,$
$\displaystyle a_{23}=$
$\displaystyle\alpha^{2}\iota\theta+\beta\eta(-\alpha\iota+\delta\iota
x+\eta(\gamma-x\zeta))-\alpha\eta(\gamma\theta+\iota\varepsilon x-\theta
x\zeta)+(\delta\theta-\eta\varepsilon)(\delta\iota-\eta(\gamma x+\zeta))\,,$
$\displaystyle a_{31}=$ $\displaystyle-\beta\theta(\iota(\alpha-\delta
x)+\eta(\gamma+x\zeta))+\alpha\theta(\gamma\theta-\iota\varepsilon x+\theta
x\zeta)+\beta^{2}\eta\iota-(\delta\theta-\eta\varepsilon)(\gamma\theta
x+\iota\varepsilon-\theta\zeta)\,,$ $\displaystyle a_{32}=$
$\displaystyle\alpha^{2}\iota\theta+\beta\eta(\eta(\gamma+x\zeta)-\iota(\alpha+\delta
x))-\alpha\eta(\gamma\theta-\iota\varepsilon x+\theta
x\zeta)+(\delta\theta-\eta\varepsilon)(\delta\iota+\eta\gamma x-\eta\zeta)\,,$
$\displaystyle a_{33}=$
$\displaystyle\alpha^{2}\iota^{2}-\alpha^{2}\theta^{2}+2\beta\theta(\alpha\eta-\gamma\iota)-2\eta(\alpha\gamma\iota+\delta\iota\zeta-\delta\theta\varepsilon)+\beta^{2}(\iota^{2}-\eta^{2})+\delta^{2}\iota^{2}-\delta^{2}\theta^{2}$
$\displaystyle+\eta^{2}(\gamma^{2}-\varepsilon^{2}+\zeta^{2})+\gamma^{2}\theta^{2}+\iota^{2}\varepsilon^{2}-2\iota\theta\varepsilon\zeta+\theta^{2}\zeta^{2}\,.$

###  A.4. $L(3,3)$

The Lie bracket is given by

| $f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---|---
$f_{1}$ | $0$ | $0$ | $f_{1}$
$f_{2}$ | $0$ | $0$ | $f_{1}+f_{2}$
$f_{3}$ | $-f_{1}$ | $-f_{1}-f_{2}$ | $0$

In this case,

$A=\frac{1}{4\det(P)}\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{pmatrix}\,,$

where

$\displaystyle
a_{11}=\delta^{2}(\iota^{2}+\theta^{2})-2\delta\eta(\iota\zeta+\theta\varepsilon)+\eta^{2}(\varepsilon^{2}+\zeta^{2})-(\iota\varepsilon-\theta\zeta)^{2}\,,$
$\displaystyle
a_{12}=2(\iota(-\alpha\iota\varepsilon+\delta\iota(\beta+\varepsilon)+\gamma(\eta\varepsilon-\delta\theta))-\iota\zeta(\theta(\delta-\alpha)+\eta(\beta+\varepsilon))+\eta\theta\zeta^{2})\,,$
$\displaystyle
a_{13}=2((\iota\varepsilon-\theta\zeta)(\alpha\theta-\delta\theta+\eta\varepsilon)+\beta\theta(\eta\zeta-\delta\iota)+\gamma\theta(\delta\theta-\eta\varepsilon))\,,$
$\displaystyle
a_{21}=2(\iota(\iota\varepsilon(\alpha+\delta)-\beta\delta\iota+\gamma(\delta\theta-\eta\varepsilon))+\iota\zeta(-\theta(\alpha+\delta)+\beta\eta-\eta\varepsilon)+\eta\theta\zeta^{2})\,,$
$\displaystyle
a_{22}=\delta^{2}(\theta^{2}-\iota^{2})+2\delta\eta(\iota\zeta-\theta\varepsilon)+\eta^{2}(\varepsilon-\zeta)(\varepsilon+\zeta)+(\iota\varepsilon-\theta\zeta)^{2}\,,$
$\displaystyle
a_{23}=2(\eta(-\alpha\iota\varepsilon+\alpha\theta\zeta+\eta\varepsilon(\gamma+\zeta))+\beta\eta(\delta\iota-\eta\zeta)+\delta^{2}\iota\theta-\delta\eta(\theta(\gamma+\zeta)+\iota\varepsilon))\,,$
$\displaystyle
a_{31}=2((\iota\varepsilon-\theta\zeta)(\eta\varepsilon-\theta(\alpha+\delta))+\beta\theta(\delta\iota-\eta\zeta)+\gamma\theta(\eta\varepsilon-\delta\theta))\,,$
$\displaystyle
a_{32}=2(\eta\zeta(\eta\varepsilon-\theta(\alpha+\delta))+\alpha\eta\iota\varepsilon+\beta\eta(\eta\zeta-\delta\iota)+\delta^{2}\iota\theta+\delta\eta\gamma\theta-\delta\eta\iota\varepsilon-\eta^{2}\gamma\varepsilon)\,,$
$\displaystyle
a_{33}=\delta^{2}(\iota-\theta)(\iota+\theta)+2\delta\eta(\theta\varepsilon-\iota\zeta)+\eta^{2}(\zeta^{2}-\varepsilon^{2})+(\iota\varepsilon-\theta\zeta)^{2}\,.$

###  A.5. $L(3,4,x)$, $x\geq 0$

For $x\geq 0$, the Lie bracket is given by

| $f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---|---
$f_{1}$ | $0$ | $0$ | $xf_{1}-f_{2}$
$f_{2}$ | $0$ | $0$ | $f_{1}+xf_{2}$
$f_{3}$ | $-xf_{1}+f_{2}$ | $-f_{1}-xf_{2}$ | $0$

For $x=0$, this is the Euclidean algebra $\mathfrak{e}(2)$, i.e., the Lie
algebra of the Lie group $E(2)$ –see [12, Section 1.2.5]. For general $x$,

Then,

$A=\frac{1}{4\det(P)}\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{pmatrix}\,,$

where

$\displaystyle a_{11}=$
$\displaystyle\alpha^{2}\iota^{2}+\alpha^{2}\theta^{2}+2\beta\theta(\gamma\iota-\alpha\eta)-2\eta(\alpha\gamma\iota+\delta\iota\zeta+\delta\theta\varepsilon)+\beta^{2}(\eta-\iota)(\eta+\iota)+\delta^{2}\iota^{2}+\delta^{2}\theta^{2}$
$\displaystyle+\eta^{2}(\gamma^{2}+\varepsilon^{2}+\zeta^{2})-\gamma^{2}\theta^{2}-\iota^{2}\varepsilon^{2}+2\iota\theta\varepsilon\zeta-\theta^{2}\zeta^{2}\,,$
$\displaystyle a_{12}=$ $\displaystyle
2\left(-\gamma\iota(\alpha\theta+\beta\eta+\delta\theta x-\eta\varepsilon
x)-\iota\zeta(-\alpha\theta x+\eta(\beta x+\varepsilon
x))+\iota^{2}(\alpha(\beta-\varepsilon x)+\delta(\beta x+\varepsilon
x))\right.$
$\displaystyle\left.+\eta\gamma^{2}\theta+\eta\theta\zeta^{2}\right)\,,$
$\displaystyle a_{13}=$ $\displaystyle 2\left(-\beta\theta(\iota(\alpha+\delta
x)+\eta(\gamma-x\zeta))+\alpha\theta(\gamma\theta+\iota\varepsilon x-\theta
x\zeta)+\beta^{2}\eta\iota+(\delta\theta-\eta\varepsilon)(\gamma\theta
x-\iota\varepsilon+\theta\zeta)\right)\,,$ $\displaystyle a_{21}=$
$\displaystyle 2\left(-\gamma\iota(\alpha\theta+\beta\eta-\delta\theta
x+\eta\varepsilon x)-\iota\zeta(\alpha\theta x+\eta(\varepsilon-\beta
x)+\delta\theta)+\iota^{2}(\alpha(\beta+\varepsilon
x)+\delta(\varepsilon-\beta x))\right.$
$\displaystyle\left.+\eta\gamma^{2}\theta+\eta\theta\zeta^{2}\right)\,,$
$\displaystyle a_{22}=$
$\displaystyle-\iota^{2}(\alpha^{2}+\delta^{2}-\varepsilon^{2})+\alpha^{2}\theta^{2}-2\beta\theta(\alpha\eta+\gamma\iota)+2\alpha\eta\gamma\iota+\beta^{2}(\eta^{2}+\iota^{2})+\delta^{2}\theta^{2}+2\iota\zeta(\delta\eta-\theta\varepsilon)$
$\displaystyle-2\delta\eta\theta\varepsilon+\gamma^{2}(\theta^{2}-\eta^{2})+\eta^{2}\varepsilon^{2}-\eta^{2}\zeta^{2}+\theta^{2}\zeta^{2}\,,$
$\displaystyle a_{23}=$ $\displaystyle
2\left(\alpha^{2}\iota\theta+\beta\eta(-\alpha\iota+\delta\iota
x+\eta(\gamma-x\zeta))-\alpha\eta(\gamma\theta+\iota\varepsilon x-\theta
x\zeta)+(\delta\theta-\eta\varepsilon)(\delta\iota-\eta(\gamma
x+\zeta))\right)\,,$ $\displaystyle a_{31}=$ $\displaystyle
2\left(-\beta\theta(\iota(\alpha-\delta
x)+\eta(\gamma+x\zeta))+\alpha\theta(\gamma\theta-\iota\varepsilon x+\theta
x\zeta)+\beta^{2}\eta\iota-(\delta\theta-\eta\varepsilon)(\gamma\theta
x+\iota\varepsilon-\theta\zeta)\right)\,,$ $\displaystyle a_{32}=$
$\displaystyle
2\left(\alpha^{2}\iota\theta+\beta\eta(\eta(\gamma+x\zeta)-\iota(\alpha+\delta
x))-\alpha\eta(\gamma\theta-\iota\varepsilon x+\theta
x\zeta)+(\delta\theta-\eta\varepsilon)(\delta\iota+\eta\gamma
x-\eta\zeta)\right)\,,$ $\displaystyle a_{33}=$
$\displaystyle\alpha^{2}\iota^{2}-\alpha^{2}\theta^{2}+2\beta\theta(\alpha\eta-\gamma\iota)-2\eta(\alpha\gamma\iota+\delta\iota\zeta-\delta\theta\varepsilon)+\beta^{2}(\iota^{2}-\eta^{2})+\delta^{2}\iota^{2}-\delta^{2}\theta^{2}$
$\displaystyle+\eta^{2}(\gamma^{2}-\varepsilon^{2}+\zeta^{2})+\gamma^{2}\theta^{2}+\iota^{2}\varepsilon^{2}-2\iota\theta\varepsilon\zeta+\theta^{2}\zeta^{2}\,.$

###  A.6. $L(3,5)$

The Lie bracket is given by

| $f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---|---
$f_{1}$ | $0$ | $f_{1}$ | $-2f_{2}$
$f_{2}$ | $-f_{1}$ | $0$ | $f_{3}$
$f_{3}$ | $2f_{2}$ | $-f_{3}$ | $0$

This is the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$. In this case,

$A=\frac{1}{2\det(P)}\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{pmatrix}\,,$

where

$\displaystyle a_{11}=$
$\displaystyle\alpha^{2}(\iota^{2}+\theta^{2})+\beta\left(-2\alpha\eta\theta-\delta^{2}\theta+\delta\eta\varepsilon+2\gamma\iota\theta-\iota\varepsilon\zeta+\theta\zeta^{2}\right)+\alpha\left(\delta\iota\zeta+\delta\theta\varepsilon-\eta\left(2\gamma\iota+\varepsilon^{2}+\zeta^{2}\right)\right)$
$\displaystyle+\beta^{2}(\eta-\iota)(\eta+\iota)+\gamma\left(\delta^{2}(-\iota)+\delta\eta\zeta+\eta^{2}\gamma-\theta(\gamma\theta+\varepsilon\zeta)+\iota\varepsilon^{2}\right)\,,$
$\displaystyle a_{12}=$
$\displaystyle\beta\iota(2\alpha\iota+\delta\zeta)+\zeta(\alpha\iota\varepsilon+\delta\gamma\theta)-2\gamma\iota(\alpha\theta+\delta\varepsilon)-\alpha\theta\zeta^{2}-\beta\eta\left(2\gamma\iota+\zeta^{2}\right)+\eta\gamma(2\gamma\theta+\varepsilon\zeta)\,,$
$\displaystyle a_{13}=$
$\displaystyle\beta(-2\alpha\iota\theta+\delta\iota\varepsilon-2\delta\theta\zeta-2\eta\gamma\theta+\eta\varepsilon\zeta)+\alpha\left(2\gamma\theta^{2}-\iota\varepsilon^{2}+\theta\varepsilon\zeta\right)+2\beta^{2}\eta\iota+\gamma\varepsilon(\delta\theta-\eta\varepsilon)\,,$
$\displaystyle a_{21}=$
$\displaystyle\beta\iota(2\alpha\iota+\delta\zeta)+\zeta(\alpha\iota\varepsilon+\delta\gamma\theta)-2\gamma\iota(\alpha\theta+\delta\varepsilon)-\alpha\theta\zeta^{2}-\beta\eta\left(2\gamma\iota+\zeta^{2}\right)+\eta\gamma(2\gamma\theta+\varepsilon\zeta)\,,$
$\displaystyle a_{22}=$
$\displaystyle\alpha^{2}(\theta-\iota)(\iota+\theta)-\beta\left(2\alpha\eta\theta+\delta^{2}\theta-\delta\eta\varepsilon+2\gamma\iota\theta-\iota\varepsilon\zeta+\theta\zeta^{2}\right)+\alpha\delta(\theta\varepsilon-\iota\zeta)$
$\displaystyle+\alpha\eta\left(2\gamma\iota-\varepsilon^{2}+\zeta^{2}\right)+\beta^{2}(\eta^{2}+\iota^{2})+\gamma\left(-\delta\eta\zeta+\iota(\delta-\varepsilon)(\delta+\varepsilon)-\eta^{2}\gamma+\theta(\gamma\theta+\varepsilon\zeta)\right)\,,$
$\displaystyle a_{23}=$ $\displaystyle
2\alpha^{2}\iota\theta+\beta\eta(-2\alpha\iota+\delta\zeta+2\eta\gamma)+\delta(\alpha\iota\varepsilon+\alpha\theta\zeta+\eta\gamma\varepsilon)-2\alpha\eta(\gamma\theta+\varepsilon\zeta)-\beta\delta^{2}\iota-\delta^{2}\gamma\theta\,,$
$\displaystyle a_{31}=$
$\displaystyle\beta(-2\alpha\iota\theta+\delta\iota\varepsilon-2\delta\theta\zeta-2\eta\gamma\theta+\eta\varepsilon\zeta)+\alpha\left(2\gamma\theta^{2}-\iota\varepsilon^{2}+\theta\varepsilon\zeta\right)+2\beta^{2}\eta\iota+\gamma\varepsilon(\delta\theta-\eta\varepsilon)\,,$
$\displaystyle a_{32}=$ $\displaystyle
2\alpha^{2}\iota\theta+\beta\eta(-2\alpha\iota+\delta\zeta+2\eta\gamma)+\delta(\alpha\iota\varepsilon+\alpha\theta\zeta+\eta\gamma\varepsilon)-2\alpha\eta(\gamma\theta+\varepsilon\zeta)-\beta\delta^{2}\iota-\delta^{2}\gamma\theta\,,$
$\displaystyle a_{33}=$
$\displaystyle\alpha^{2}\iota^{2}-\alpha^{2}\theta^{2}+\beta\left(2\alpha\eta\theta+\delta^{2}\theta-\delta\eta\varepsilon+2\gamma\iota\theta-\iota\varepsilon\zeta+\theta\zeta^{2}\right)+\alpha\delta\iota\zeta-\alpha\delta\theta\varepsilon$
$\displaystyle+\alpha\eta\left(-2\gamma\iota+\varepsilon^{2}-\zeta^{2}\right)+\beta^{2}\left(\iota^{2}-\eta^{2}\right)-\delta^{2}\gamma\iota+\delta\eta\gamma\zeta+\eta^{2}\gamma^{2}+\gamma^{2}\theta^{2}-\gamma\iota\varepsilon^{2}+\gamma\theta\varepsilon\zeta\,.$

###  A.7. $L(3,6)$

The Lie bracket is given by

| $f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---|---
$f_{1}$ | $0$ | $f_{3}$ | $-f_{2}$
$f_{2}$ | $-f_{3}$ | $0$ | $f_{1}$
$f_{3}$ | $f_{2}$ | $-f_{1}$ | $0$

This is the Lie algebra
$\mathfrak{sp}(1)\cong\mathfrak{su}(2)\cong\mathfrak{so}(3)$. In this case,

$A=\frac{1}{4\det(P)}\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{pmatrix}\,,$

where

$\displaystyle a_{11}=$
$\displaystyle\iota^{2}(\alpha^{2}+\delta^{2}-\varepsilon^{2})+\alpha^{2}\theta^{2}+\alpha^{2}\varepsilon^{2}+\alpha^{2}\zeta^{2}-2\alpha\beta(\delta\varepsilon+\eta\theta)-2\alpha\gamma(\delta\zeta+\eta\iota)$
$\displaystyle+\beta^{2}(\delta^{2}+\eta^{2}-\iota^{2}-\zeta^{2})+2\beta\gamma(\iota\theta+\varepsilon\zeta)+\gamma^{2}(\delta^{2}+\eta^{2}-\theta^{2}-\varepsilon^{2})+\delta^{2}\theta^{2}$
$\displaystyle+2\iota\zeta(\theta\varepsilon-\delta\eta)-2\delta\eta\theta\varepsilon+\eta^{2}\varepsilon^{2}+\eta^{2}\zeta^{2}-\theta^{2}\zeta^{2}\,,$
$\displaystyle a_{12}=$ $\displaystyle
2(-\gamma(\alpha\iota\theta+\alpha\varepsilon\zeta+\beta\delta\zeta+\beta\eta\iota)+\alpha\beta(\iota^{2}+\zeta^{2})+\gamma^{2}(\delta\varepsilon+\eta\theta)+(\delta\iota-\eta\zeta)(\iota\varepsilon-\theta\zeta))\,,$
$\displaystyle a_{13}=$ $\displaystyle
2(-\beta(\alpha\iota\theta+\alpha\varepsilon\zeta+\delta\gamma\varepsilon+\eta\gamma\theta)+\alpha\gamma(\theta^{2}+\varepsilon^{2})+\beta^{2}(\delta\zeta+\eta\iota)+(\delta\theta-\eta\varepsilon)(\theta\zeta-\iota\varepsilon))\,,$
$\displaystyle a_{21}=$ $\displaystyle
2(-\gamma(\alpha\iota\theta+\alpha\varepsilon\zeta+\beta\delta\zeta+\beta\eta\iota)+\alpha\beta(\iota^{2}+\zeta^{2})+\gamma^{2}(\delta\varepsilon+\eta\theta)+(\delta\iota-\eta\zeta)(\iota\varepsilon-\theta\zeta))\,,$
$\displaystyle a_{22}=$
$\displaystyle-\zeta^{2}(\alpha^{2}+\eta^{2}-\theta^{2})-\alpha^{2}\iota^{2}+\alpha^{2}\theta^{2}+\alpha^{2}\varepsilon^{2}-2\beta(\alpha\delta\varepsilon+\alpha\eta\theta+\gamma\iota\theta+\gamma\varepsilon\zeta)$
$\displaystyle+2\delta(\alpha\gamma\zeta+\eta\iota\zeta-\eta\theta\varepsilon)+2\alpha\eta\gamma\iota+\beta^{2}(\delta^{2}+\eta^{2}+\iota^{2}+\zeta^{2})-\delta^{2}(\gamma^{2}+\iota^{2}-\theta^{2})-\eta^{2}\gamma^{2}$
$\displaystyle+\eta^{2}\varepsilon^{2}+\gamma^{2}\theta^{2}+\gamma^{2}\varepsilon^{2}+\iota^{2}\varepsilon^{2}-2\iota\theta\varepsilon\zeta\,,$
$\displaystyle a_{23}=$ $\displaystyle
2(\alpha^{2}(\iota\theta+\varepsilon\zeta)-\alpha\beta(\delta\zeta+\eta\iota)-\alpha\gamma(\delta\varepsilon+\eta\theta)+\beta\gamma(\delta^{2}+\eta^{2})+(\delta\iota-\eta\zeta)(\delta\theta-\eta\varepsilon))\,,$
$\displaystyle a_{31}=$ $\displaystyle
2(-\beta(\alpha\iota\theta+\alpha\varepsilon\zeta+\delta\gamma\varepsilon+\eta\gamma\theta)+\alpha\gamma(\theta^{2}+\varepsilon^{2})+\beta^{2}(\delta\zeta+\eta\iota)+(\delta\theta-\eta\varepsilon)(\theta\zeta-\iota\varepsilon))\,,$
$\displaystyle a_{32}=$ $\displaystyle
2(\alpha^{2}(\iota\theta+\varepsilon\zeta)-\alpha\beta(\delta\zeta+\eta\iota)-\alpha\gamma(\delta\varepsilon+\eta\theta)+\beta\gamma(\delta^{2}+\eta^{2})+(\delta\iota-\eta\zeta)(\delta\theta-\eta\varepsilon))\,,$
$\displaystyle a_{33}=$
$\displaystyle\zeta^{2}(\alpha^{2}+\eta^{2}+\theta^{2})+\alpha^{2}\iota^{2}-\alpha^{2}\theta^{2}-\alpha^{2}\varepsilon^{2}+2\alpha\beta(\delta\varepsilon+\eta\theta)-2\delta\zeta(\alpha\gamma+\eta\iota)-2\alpha\eta\gamma\iota$
$\displaystyle+\beta^{2}(-\delta^{2}-\eta^{2}+\iota^{2}+\zeta^{2})-2\beta\gamma(\iota\theta+\varepsilon\zeta)+\delta^{2}(\gamma^{2}+\iota^{2}-\theta^{2})+2\delta\eta\theta\varepsilon+\eta^{2}\gamma^{2}-\eta^{2}\varepsilon^{2}$
$\displaystyle+\gamma^{2}\theta^{2}+\gamma^{2}\varepsilon^{2}+\iota^{2}\varepsilon^{2}-2\iota\theta\varepsilon\zeta\,.$

## Acknowledgements

The author would like to thank his supervisor, Marie-Amélie Lawn, and Ilka
Agricola for their comments on the first draft of this paper.

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# Regression Based Anomaly Detection in Electric Vehicle State of Charge
Fluctuations Through Analysis of EVCI Data

Sagar Babu Mitikiria, Yash Tiwarib, Vedantham Lakshmi Srininvasc, Mayukha
Pal∗d (Corresponding author: ∗Mayukha Pal)aMitikiri Sagar Babu is a Data
Science Research Intern at ABB Ability Innovation Center, Hyderabad 500084,
India and also a Ph.D. Research Scholar at the Department of Electrical
Engineering, IIT (ISM) Dhanbad, Dhanbad, 826004, India.bYash Tiwari is with
ABB Ability Innovation Center, Bangalore-560048, IN, working as a Data
ScientistcDr. Vedantham Lakshmi Srinivas is an Asst. Professor in the
Department of Electrical Engineering, IIT (ISM) Dhanbad, Dhanbad, 826004,
India.dDr. Mayukha Pal is with ABB Ability Innovation Center,
Hyderabad-500084, IN, working as Global R&D Leader – Cloud & Analytics
(e-mail: mayukha.pal@in.abb.com).

###### Abstract

With the increase in the number of electric vehicles (EV), there is a need for
the development of the EV charging infrastructure (EVCI) to facilitate fast
charging, thereby mitigating the EV congestion at charging stations. The role
of the public charging station depot is to charge the vehicle, prioritizing
the achievement of the desired state of charge (SoC) value for the EV battery
or charging till the departure of the EV, whichever occurs first. The
integration of cyber and physical components within EVCI defines it as a cyber
physical power system (CPPS), increasing its vulnerability to diverse cyber
attacks. When an EV interfaces with the EVCI, mutual exchange of data takes
place via various communication protocols like the Open Charge Point Protocol
(OCPP), and IEC 61850. Unauthorized access to this data by intruders leads to
cyber attacks, potentially resulting in consequences like energy theft, and
revenue loss. These scenarios may cause the EVCI to incur higher charges than
the actual energy consumed or the EV owners to remit payments that do not
correspond adequately to the amount of energy they have consumed. This article
proposes an EVCI architecture connected to the utility grid and uses the EVCI
data to identify the anomalies or outliers present in the EV transmitted data,
particularly focusing on SoC irregularities. The proposed methodology involves
utilizing a ridge regression based machine learning (ML) model for predicting
changes in the SoC. The adversaries have the capability of spoofing these
change in SoC values, consequently making the EVCI incapable of achieving the
desired task. Three distinct spoofing techniques namely, decimal shifting,
incremental array spoofing, and random spoofing are implemented on the data
and subsequently tested with the proposed methodology. The results show that
the proposed methodology detects the anomaly accurately and also classifies
the type of spoofing that causes the anomaly.

###### Index Terms:

EVCI, SoC, fast charging, spoofing, anomaly detection, Ridge regression,

##  I Introduction

The global electric vehicle (EV) market sales have exhibited robust growth
reaching 2 million units sold in the first quarter of 2022, which is a 75%
increase in sales compared to 2021 [1]. Due to this rising demand, the EV
charging infrastructure (EVCI) needs to be developed at the same rate. To
reduce the EV charging stations (EVCS) complications in penetration with the
grid [2] and in charging such as charging speed, compatibility, range anxiety,
grid power availability, and congestion of EVs at charging stations, etc.,
These charging stations are located at parking areas of corporate offices,
commercial malls, and residential apartments, where there is probability for
the vehicle staying in these locations, for the dwell times [3]. Due to the
intermittent power available from the utility grid, charging stations are
integrated with energy storage systems like battery energy storage systems
(BESS), or renewable energy systems (RES) to meet it’s power demand. In
addition to their primary function of EV charging, these stations offer
auxiliary grid-related services, including the stabilization of frequency and
voltage levels, provision of reactive power support, etc., These services
require bi-directional power flow that enables the flow of electricity from
the grid-to-vehicle (G2V mode) and also from the vehicle-to-grid (V2G mode),
particularly during the instances when the demand for the loads in the grid
exceeds the generation [4]. Furthermore, the charging infrastructure has
exhibited consistent augmentation in charging capacity, such as ultra-fast
direct current (DC) [5] and extreme fast charging (XFC) [6] stations equipped
with multiple modules. These high-speed charging stations could deliver up to
350 kW of rated power, enabling rapid charging cycles within 10 minutes
timeframe, thereby establishing a comparable refueling experience to that of
internal combustion engine vehicles (ICEVs).

Figure 1: Proposed architecture of EVCI

As a consequence of rapid fast charging, the grid perceives the EVCI
aggregated with EVs as a bulk power system load (during G2V mode) with
inherent energy storage capacity such as BESS. When all EVs are simultaneously
charged, the grid experiences various voltage and frequency stress,
particularly during peak demand periods. These conditions necessitate load
shedding actions, thereby compromising the overall reliability of the grid
[7]. To sustain the balance between load and generation, the EVCI actively
participates in demand response (DR), entailing the charging of the EVs during
the off-peak periods, and the charged EVs along with the BESS function as
sources during the peak periods [8, 9]. The integration of the EVCI with
functions other than charging, such as load balancing, demand response,
frequency response, etc., requires a communication network with internet-of-
things (IoT) between various operators like the distribution system operator
(DSO), EVCI administrators, and the EV users. These communication requirements
across different elements of the EVCI play a significant role in defining it
as a cyber physical power system (CPPS) [10]. The CPPS is a physically
interconnected system merged with cyber components encompassing computing,
control, and communication functions allowing bi-directional flow of power and
information thereby enabling the implementation of advanced smart grid
technologies [11].

Due to the substantial dependence of the CPPS on the cyber systems, their
communication capabilities are highly vulnerable to malicious cyber intrusions
[12]. The vulnerabilities in the EVCI [13] may be internal and external
vulnerabilities such as EV-X or EVCS-X interface vulnerabilities. The former
consists of CAN (control area network) bus [14], tyre pressure monitoring
system (TPMS) vulnerabilities [15]. In the latter one, the X indicates various
physically accessible ports, Internet Service Protocols (ISP), vendors,
roadside infrastructures, and vehicular vulnerabilities. Adversaries exploit
these vulnerabilities, to infiltrate the system thereby initiating a cyber
attack. These attacks result in anomalies within the data, where the data
exchange occurs between the diverse physical elements by way of the cyber
components. The anomalies in the data of EVCI due to the cyber attacks cause
service disruptions, charging delays, financial loss, safety risks, resource
drain, grid instability, reputation damage, etc., [16, 17, 18]. So, these
anomalies need to be detected and a proper defense mechanism has to be
implemented to mitigate or to prevent the effects of cyber-attacks.

##  II related work

Many researchers have addressed the cybersecurity of power grids, but the
cybersecurity of EVCI has not gained much attention. The ways of cyberattacks,
targeting the vital infrastructure are changing with diversified patterns in
the energy distribution sector through remote communications, virtual private
network (VPN) links, and corporate networks. The physical parts of the CPPS
could be damaged by breaching its information and communication technology
(ICT) infrastructure and obtaining more precise access to the supervisory
control and data acquisition (SCADA) system which controls and monitors the
components of the power grid without the need of physical attack [19]. After
the cyberattacks on the Ukrainian power grid, particularly in the years 2015
and 2016 showed the necessity of cyber-physical security for substations and
SCADA systems, U.S. officials from the Department of Energy (DoE), the
Department of Homeland Security (DHS), the Federal Bureau of Investigation
(FBI), and the North American Electric Reliability Corporation (NERC)
increased their efforts realizing it, as their opportunity to examine the
strategies and tactics used by the aggressor to predict the likelihood
occurrence of the cyberattacks and it,s type in future [20, 21]. In both
years, the attacker’s motive is to successfully compromise the industrial
control system (ICS) causing potential infrastructure threats, equipment
failure, and power outages. Therefore, to improve the resilience, reliability,
safety, and, security of power systems, it is essential to strengthen the
cybersecurity of industrial automation and control systems in the power grid.

In [13], various vulnerabilities are addressed that arise due to the growth of
EVs from the power grid perspective. It characterizes the vulnerabilities
susceptible to exploitation, imposing potential risks to EVs and EVCI
equipment, the power grid, or a combination of both. The cyber security issues
associated with electric chargers and the potential consequences of cyber
attacks are examined in [22]. It extensively explores the conceivable cyber
attacks on utility operations, power systems, interconnected systems, and
billing procedures. Additionally, it deliberates on recommendations and
optimal strategies for enhancing cybersecurity within EVCI systems. The
attacks on EV charging stations and their users were simulated demonstrating
practical impacts on the power grid leading to service disruptions and grid
failures are presented in [23]. The findings emphasize the need for mitigation
strategies and uncover concerns regarding the insufficient security measures
implemented in deployed charging stations.

##  III Contributions and Paper Organization

The primary contribution of this article is detecting the anomalies in the
$\Delta SoC$ values of EVs when connected to the charging stations. Since
there is collinearity between the SoC and the charging current values [24],
the charging current behavior is reflected in the SoC being a relative
parameter thereby, $\Delta SoC$. A ridge regression based ML model is used for
predicting the $\Delta SoC$ values. With the help of the predicted $\Delta
SoC$ values, the EVCI administrator is able to detect the type of vehicle,
calculate the charging time, and also in estimating the battery capacities.
This work predicts the $\Delta SoC$ values from the obtained EVCI data and
does not rely on the EV data. So, if there is any communication failure or
incorrect estimations in the SoC of the EVs, the model won’t be affected. The
proposed ML model considers the current values of utility, storage system, and
charging ports as the inputs of the model and $\Delta SoC$ values as the
output of the model. Therefore, predicting the $\Delta$SoC values with the
current values at various ports and sensors data and the ML models helps in
finding the charging behaviors of the EV, the type of the EVs arriving at the
charging station, and preventing energy theft by estimating the amount of
energy transferred to the EV battery with the SoC values given the battery
capacity is known.

The predicted SoC ($\Delta SoC_{predicted}$) values are compared with the EV
communicated SoC values ($\Delta SoC_{actual}$) which are assumed to be
spoofed later. The absolute difference is calculated between these two data
points and compared with a predefined threshold iteratively. Furthermore, the
overall contributions of this article are as follows:

  * •

A grid connected EVCI system is proposed and simulated to obtain the data.

  * •

The proposed EVCI model consists of both cyber and physical components
including both wired and wireless communications by utilizing various
protocols such as OCPP and IEC 61850 (GOOSE).

  * •

A ridge regression based ML model is implemented to predict the SoC value of
the EV’s battery.

  * •

This work detects the anomalies in the $\Delta SoC$ of the EV’s battery using
the EVCI data.

  * •

The anomaly is detected based on the absolute differences between the
communicated and predicted $\Delta SoC$ and by observing the nature of these
differences continuously.

  * •

Various types of spoofing scenarios are simulated on the testing data for
producing the anomaly data.

  * •

In addition to anomaly detection, the proposed methodology also considers the
spoofing classification.

  * •

The proposed methodology is tested and validated for different cases.

This article structure commences with an introduction to the CPPS, EVCI
infrastructure types, and their basic operation in Section I. The
comprehensive review of literature on the cybersecurity aspects of EVCI when
interfaced with the utility grid is presented in Section II. Then Section III
describes the author’s contribution to the work. The subsequent Section IV
outlines the modeling framework for the proposed EVCI system. Section V
describes the parameters of the data acquired for both the training and
testing data. It also provides the pre-processing of the data for using it as
an input to the ML model. Within Section VI, the methodology and ML model
employed for the prediction of $\Delta SoC$ are explained. The metrics used
for evaluating the model performance are explained in Section VII.
Additionally, various case studies are also provided to validate the proposed
anomaly detection framework and spoofing classification. Finally, Section VIII
concludes with the achievements of this study while mentioning the potential
areas for future research.

##  IV Modelling of the System

The modeling of the proposed EVCI system is categorized into two parts: 1.
Physical system modeling (representing the physical components like the
charging board (CB), utility grid, BESS, etc.,) and 2. Cyber system modeling
(consisting of communications, and protocols like OCPP). Fig. 1 illustrates
the comprehensive architecture of the EVCI comprising these two systems.

###  IV-A Physical System Modelling

The physical system modeling of an EVCI comprises multiple units pertaining to
the power flow and energy storage. The key components in the physical system
of the EVCI include the point of common coupling (PCC) linking the EVCI to the
utility grid through a transformer, BESS, CBs, and charging ports. Fig. 1
illustrates the architectural components within the physical system of EVCI,
responsible for storing the root mean square (RMS) values of the various
measured electrical parameters such as active power (P), reactive power (Q),
voltage (V), and current (I).

The proposed system utilizes a step-down type transformer of rating
13.5kV$/$480V placed between the PCC and the EVCI as shown in Fig. 1. The main
function of this transformer is to step down the grid level medium voltage
(MV) to the low voltage (LV). The BESS and CBs are connected to the PCC
through this transformer. This article considers four CBs namely CB-0, CB-1,
CB-2, and CB-3. To facilitate EV charging, the charging ports are connected to
the CBs. The detailed representation of these charging ports and their current
notations are provided in Table I. Two types of charging ports are considered
in this work namely, Terra HP charger that supports from 175-350 kW fast
charging and Terra 53 charger supporting up to 50 kW power [25].

TABLE I: Charger currents representation Charging board | Charging ports | Notation
---|---|---
CB-0 | EV0_Terra53 | IEV0
CB-1 | EV1_TerraHP-Cord_a | IEV1a
EV1_TerraHP-Cord_b | IEV1b
CB-2 | EV2_TerraHP-Cord_a | IEV2a
EV2_TerraHP-Cord_b | IEV2b
CB-3 | EV3_TerraHP | IEV3

Since the main aim of this work is detecting the anomalies, the modeling of
EVs is done considering only two types of vehicles whose battery types are BEV
300 and BEV 150 where BEV stands for battery electric vehicle. The BEV 300
battery type EVs are charged with a Terra HP charger that can charge up to 300
kW and the BEV 150 battery type EVs are connected to Terra 53 chargers for
charging the power up to 300 kW. For all the other types of EVs, the nominal
charging power is considered 50 kW and connected with Terra 53 chargers.

It is crucial to note that any vehicle upon connecting to the EVCS transmits
some amount of data. This data encompasses details such as charging protocol,
EV type, arriving time, departure time, initial SoC, and target SoC [26]. The
target SoC signifies the user-demanded SoC, representing the desired charge
level the EV should attain before the scheduled departure time. The EV also
continuously transmits the SoC values to the EVCI through chargers. The EVCI
charges the EVs by supplying the grid power or the BESS power depending on
availability.

####  IV-A1 BESS modeling

A simple model of the XFC station depot connected with a BESS is represented
in Fig. 1. The parameters and the specifications of the BESS used in this work
are provided in Table II. This paper focuses on the modeling of public EVCS,
with the assumption of uncontrollable arrival rates of EVs and unpredictable
power flow to the EVSE. Since the grid power is also not controllable,
Therefore the only power (or current) in the EVCI is the BESS.

TABLE II: BESS parameters and specifications Parameter |  Value
---|---
Maximum charging power (kW) |  500
Maximum Discharging power (kW) |  500
Energy capacity (kWh) |  250
Charging efficiency (%) |  95
Discharging efficiency (%) |  95
Maximum SoC (%) |  90
Minimum SoC (%) |  20
Initial SoC (%) |  50
Maximum power import (kW) |  1000
Maximum power export (kW) |  1000

###  IV-B Cyber System Modelling

The cyber system consists of XMS (extreme fast charging managing system)
operator and XMS cloud. Both the wired and wireless communications are
employed in this system. The EVSE, BESS, and transformer parameter measurement
circuits are communicated to the XMS cloud in a wired mode through the IEC
61850 messaging protocol, and the charging ports and the EVs communicate to
the XMS operator and the XMS cloud system in a wireless mode through OCPP
(Open Charge Point Protocol). Fig. 1 shows different modes of communication,
connecting the various parts of the proposed system.

The IEC 61850 is a standard messaging protocol used for multicast messages
such as generic object-oriented substation events (GOOSE). It recommends only
message integrity and authenticity. Generally, in substation automation
systems, GOOSE messages are used to carry the breaker open or close commands
[27]. Another protocol used here is OCPP, which is an open source to
facilitate the communication between the charging ports and the backend
systems. This was proposed by the Dutch foundation ElaadNL and should be
standardized and certified by OCA (Open Charge Alliance) to ensure uniformity
and cross-vendor compatibility [28]. The electrical parameters data is
transmitted from the physical parts of the system through these protocols.
Besides, the charging ports also transmit another parameter known as charge
status (CS) which is a binary that returns 1 when the EV is charging and 0
otherwise. Both the physical and cyber systems simulations are performed in
real-time on the OPAL-RT simulation platform using both MATLAB and Python
environments.

Figure 2: Heatmap of correlation among various parameters related to charging
port EV0$\\_$Terra53

##  V Data acquisition and preprocessing

The data required for developing the proposed methodology is acquired and
stored for both training and testing separately. The training data is
collected for four days at a rate of one sample per second and the testing
data is collected for one day at the same rate. The acquired data consists of
variables PPC current (IPCC), BESS current (IBESS), charging port current
(IEVn), active power (PEVn) and reactive power (QEVn) of charging port, charge
status ($CS$) where $n$ is the representation of the EV charging port. Since
it is required for the EVCI to know the SoC of the connected EV, The SoC of
the connected EV is continuously transmitted to the EVCI through OCPP as
explained later. This data is also collected and stored in the variable SoCEVn
at an equivalent sampling rate.

Since the main aim of this article is detecting the anomalies in $\Delta SoC$,
it is calculated by subtracting the present instant SoC value from its
previous instant. This $\Delta SoC$ calculation is performed for all the EVs
connected to various charging ports and stored in variables labeled as $\Delta
SoC_{EVn}$. It is the actual or calculated SoC, termed as $\Delta
SoC_{actual}$. This data variable is the output of the proposed ML model where
various types of spoofing techniques are assumed to be done in the form of a
cyber attack. This study explores three distinct types of spoofing techniques:
decimal shifting, incremental array spoofing, and random spoofing. Table III
presents the numerical levels associated with each type of spoofing technique.
These values are chosen to simulate stealthy attack scenarios [29] and
implemented individually on the testing data at different instances.

TABLE III: Spoofing techniques: value ranges and types Noise type |  Range (or) Type
---|---
Random |  Uniform distribution (in range of 10-2)
Shifting in decimals |  $-0.009$ to $+0.009$
Incremental array |  Arithmetic progression

Fig. 4 shows the communicated and predicted $\Delta SoC$ values where the
spikes in communicated $\Delta SoC$ values signify the large variation in it
that occurs due to the arrival or departure of the EVs at the respective
charging port of the EVCI. Since these data points are not considered
anomalies, the spoofing techniques are simulated to the particular windows of
the obtained time series data with lengths more than the length of these spike
durations. The random spoofing replaces the $\Delta SoC_{actual}$ values with
random values within the specified range by following a uniform distribution.
Similarly, the incremental array spoofing technique also replaces a series of
$\Delta SoC_{actual}$ values, with these series of values should be in
arithmetic progression. The scaling and shifting in decimals type of spoofing
is performing mathematical calculations to the actual output values for
simulating the spoofing scenario. The actual output of the time series data is
replaced with the spoofed time series output of the testing data and later
compared with the output of the ML model ie., $\Delta SoC_{predicted}$ for
anomaly detection.

For any ML model, the input data has to be preprocessed. Due to the spikes and
high variations in $\Delta SoC$ values, the min-max normalization fails
because of the irregular distribution of data. The standard scaling method is
used to preprocess the data as required by the Ml model.

##  VI Methodology

###  VI-A Multicollinearity

Multicollinearity is defined as the phenomenon of the existence of a strong
correlation between various variables in a regression model [30]. This
phenomenon arises from diverse circumstances, including the inherent physical
interpretations of variables, such as metrics or measurements utilized within
software defect prediction models. The interdependence among these variables
could be influenced by the intrinsic nature of their meanings, contributing to
the multicollinearity. Another reason for multicollinearity is the large
collection of metrics. For constructing a better model we tend to collect
metrics fitting to the defects as many as possible resulting in serious
multicollinearity issues in datasets [31].

The multicollinearity in the acquired datasets exists between the battery
current of the EV (IEVn) and its estimated SoC values, where n indicates the
charging port to which the EV is connected. Since coulomb counting [32] is the
most adopted method, it estimates the SoC as a relative measure of charging or
discharging EV current and integrates it over time and is given as follows:

$\textit{SoC}(t)=\textit{SoC}(t-1)+\frac{I(t)}{Q_{n}}\Delta\textit{t}$ (1)

Where $SoC(t)$ is the SoC to be estimated at present instant, $SoC(t-1)$ is
the SoC of the battery at the previous time instant. $I(t)$ is current at
present time instant, Qn is the capacity of the battery pack and $\Delta t$ is
the discrete sampling period of the battery management systems (BMS). Fig. 2
demonstrates the correlations with the help of heat-map between the various
variables showing the high collinearity between the current and SoC values.

There also exists another strong collinearity between the multiple input
variables of the ML model such as IPCC, IBESS, and IEVn (currents flowing
through all the EVs). This collinearity is easily verified by Kirchoff’s
current law (KCL) and is given as follows:

$\textit{I${}_{PCC}$}=\textit{I${}_{BESS}$}+\sum_{k=1}^{N}\textit{I${}_{CBk}$}$
(2)

Where ICBk is the EVCI’s CB current whose values are equal to the sum of all
charging ports current connected to the particular CB. N is the total number
of CBs present in the charging station.

Figure 3: Proposed Anomaly detection methodology flowchart

###  VI-B Ridge Regression

In the context of multiple linear regression applications, ridge regression is
the most commonly used parameter forecasting technique particularly used to
overcome the drawbacks associated with collinearity that arise frequently in
other regression scenarios [33]. It helps to mitigate the negative impact of
collinearity on the least squares (LS) estimator. The main feature of this
ridge regression is addressing the problem of collinearity without removing
any data variable. With the other regression estimators, there is a
probability of coefficients becoming larger in absolute values and sometimes
may also have a wrong sign. The likelihood of encountering these challenges
increases the deviation in prediction vectors from orthogonality. As discussed
in the previous sections, due to the collinearity present between the multiple
input variables in this work, considering the multiple linear regression
standard model:

$\textbf{y}=\textbf{x}\beta+\varepsilon$ (3)

Where $\mathit{E}(\varepsilon)=0$,
$\mathit{E}(\varepsilon\varepsilon^{\prime})=\sigma^{2}I_{n}$, and x is a full
rank matrix of n$\times$p dimensions. The bold letter symbols denote vectors
and matrices. It is assumed that the variables are standardized to the
correlation form (x’x) and the response variable with each of the explanatory
variables is given by the vector $\gamma$ $\equiv$ x’y. Let $\hat{\beta}$ be
the LS estimate of the variable $\beta$ and is given by

$\hat{\beta}=(\textbf{x}^{\prime}\textbf{x})^{-1}\textbf{x}^{-1}\textbf{y}$
(4)

The average distance between the $\beta$ and $\hat{\beta}$ directly
contributes to the challenges in this standard estimation. It is achieved by
the standard squared distance (L2) between $\beta$ and $\hat{\beta}$ with the
following properties:

$\displaystyle L^{2}=(\hat{\beta}-\beta)^{\prime}(\hat{\beta}-\beta)$ (5)
$\displaystyle\mathit{E}(L^{2})=\sigma^{2}\text{trace}(\textbf{x}^{\prime}\textbf{x})^{-1}$
(6)
$\displaystyle\mathit{E}(\hat{\beta}^{\prime}\hat{\beta})=\beta^{\prime}\beta+\sigma^{2}\text{trace}(\textbf{x}^{\prime}\textbf{x})^{-1}$
(7)

Figure 4: Comparison of actual and predicted $\Delta SoC$ values of the EVs
connected to charging port EV0$\\_$Terra53

Traditional linear regression methods, when dealing with high collinear data
suffer from coefficient stability as the input matrix (x) becomes singular due
to correlated independent variables. This paper adopted ridge regression to
provide a realistic solution by penalizing excessive coefficients shrinking
toward null values, by incorporating the L2 normalization penalty term into
the loss function ($\lambda\left\|\beta\right\|^{2}_{2}$). This penalty term
makes the model coefficients smaller and more stable. The parameters $\lambda$
and $\alpha$ govern the trade-off between achieving a good fit of the model
and restricting the coefficients to small values respectively. The final
solution for the ridge regression is given as follows:

$\hat{\beta}_{ridge}=\text{argmin}_{\beta}\left\\{\left\|y-X\beta\right\|^{2}_{2}+\alpha\left\|\beta\right\|^{2}_{2}\right\\}$
(8)

Grid search and cross-validation (GS-CV) method is employed to rigorously
evaluate a set of variables [34]. This involves assessing the model’s
performance across each fold of cross-validation. To ensure the model’s
generalization ability beyond the training data, the optimal configuration was
selected based on the fold-averaged performance metric. The choice of specific
parameters influences the extent of coefficient shrinkage subsequently
impacting the stability and accuracy of the model. Through GS-CV, the value of
the parameter achieves a good trade-off between variance reduction and minimum
bias [35].

This basic strategy reduces multicollinearity by providing three critical
benefits: reduced variance (ensuring stable estimates), improved bias-variance
trade-off (favoring generalizability), and higher model stability (reducing
noise sensitivity). As a result, ridge regression enables researchers to build
robust and interpretable models even when confronted with tough collinear
data, improving the quality of scientific discourse and encouraging more
robust conclusions.

The performance of the six regression models for all the six EV charging ports
is evaluated individually with suitable metrics. Table V depicts the various
metrics such as mean squared error (MSE) and its values are also provided to
evaluate the model.

Algorithm 1 Anomaly Detection

Initialize the value threshold and window length as max$\\_$iter.

Record the values $I_{PCC}$, $I_{BESS}$, $I_{EVX}$, $CS$, and $\Delta
SoC_{actual}$.

for $i=1,2,\ldots$max$\\_$iter do

Predict $\Delta SoC_{predicted}$ using ridge regression ML model.

Compute the absolute difference (DM) between the values $\Delta SoC_{actual}$
and $\Delta SoC_{predicted}$.

if DM $\geq$ threshold then

x $\leftarrow D_{M}(i)$, y $\leftarrow\mid D_{M}(i+1)-D_{M}(i)\mid$.

Go to Algorithm-2, Algorithm-3, and Algorithm-4.

end if

end for

The comparison between actual and predicted $\Delta SoC$ values of the EVs
connected to the charging port EV0$\\_$Terra53 are shown in Fig. 4. Since
there are large deviations in the actual $\Delta SoC$ values, these scenarios
occur during the arrival and departure of the EVs to the EVCI as depicted in
Fig. 4, it is due to the reason that during the EV arrival to the EVCI, SoC
changes abruptly from 0 to a greater value and also during departure, the SoC
drops from a higher value to 0. So, the proposed methodology considers a time
window of different lengths for the different charging ports. As a result,
whenever the DM exceeds the threshold value, the methodology starts the
iteration count and checks for the larger thresholds in the consecutive data
samples throughout the window. If the DM exceeds the threshold throughout the
window then the methodology detects it as an anomaly. The Algorithm 1 provides
the pseudocode for the anomaly detection.

Algorithm 2 Shifting in Decimals Spoofing

1:$a\leftarrow D_{M}(i)$

2:for $j=1,2,\ldots$max$\\_$iter do

3: if $a=D_{M}(i)$ then

4: $j\leftarrow j+1$

5: if j = max$\\_$iter then

6: Display Shifting in decimals Spoofing detected

7: end if

8: end if

9:end for

Algorithm 3 Incremental Spoofing

1:$a\leftarrow\mid D_{M}(i+1)-D_{M}(i)\mid$

2:for $j=1,2,\ldots$max$\\_$iter do

3: if $D_{M}(i+j)=(a*j)+D_{M}(i+j+1)$ then

4: $j\leftarrow J+1$

5: if j = max$\\_$iter then

6: Display Incremental array spoofing detected

7: end if

8: end if

9:end for

In addition to anomaly detection, the proposed methodology also categorizes
the type of spoofing that causes anomaly in the target data i.e., $\Delta
SoC$. Since this work considers the three types of spoofing techniques namely,
shifting in decimals, incremental array spoofing, and random spoofing, the
proposed framework classifies in parallel to the anomaly detection. The
Algorithm 2 describes the procedure for detecting the shifting in decimals
type of spoofing which records the absolute difference once it reaches the
threshold and compares it with the other succeeding values for the entire
window duration if all the values are equal then the resulting spoofing is of
shifting in decimals type.

The incremental array spoofing is detected by observing the series of absolute
differences between the predicted and communicated $\Delta SoC$ for the window
if the initial distance reaches the threshold. If any progression is observed
the proposed methodology returns the type of progression present in the data.
The flow diagram for this type of spoofing detection is provided in Algorithm
3. If the spoofing does not belong to the any of aforementioned categories,
then it is termed as random spoofing. It compares the absolute differences
with the threshold iteratively only if the conditions of the shifting in
decimals and incremental spoofing aren’t achieved as explained in Algorithm 4.
The overall framework of the proposed methodology combining all the
pseudocodes is depicted in Fig. 3. Furthermore, the step-wise procedure of the
proposed methodology is as follows:

  1. 1

Initialize the values iterative count (i), window length, and threshold.

  2. 2

Predict the output of the model ($\Delta SoC_{predicted}$).

  3. 2

Compute DM = abs($\Delta SoC_{predicted}$ \- $\Delta SoC_{actual}$).

  4. 4

if DM $\leq$ threshold go to step 2. else, go to next step.

  5. 5

x = D${}_{M}(i)$, y = $\mid D_{M}(i+1)-D_{M}(i)\mid$ and j = 1.

  6. 6

If $D_{M}(i+j)$ = a, increment j to j+1
else, go to step 8.

  7. 7

If j = 10, display shifting in decimal spoofing detected and break.

  8. 8

If y = $\mid D_{M}(i+1)-D_{M}(i)\mid$, increment j to j+1
else, go to step 10.

  9. 9

If j = 10, display Incremental array spoofing detection and break.

  10. 10

If i $<$ 10, increment i to i+1
else, display random spoofing detected.

Algorithm 4 Random Spoofing

1:if D${}_{M}(i)\geq threshold$ then

2: $i\leftarrow i+1$

3: if i = max$\\_$iter then

4: Display Random type Spoofing detected

5: end if

6:end if

##  VII Results & Case Study

Various ML and neural network (NN) models are tested for the acquired data by
preprocessing with standard scaling. The parameters and the performance metric
values used in this model are shown in Table V. The MSE values of the
different model shows that the ridge regression model is the best fit for
predicting $\Delta SoC$ values using the EVCI data. The ride regression model
was implemented using scikit-learn in Python environment, and robust model
assessment was ensured by conducting the grid search with k-fold cross-
validation strategy with k = 20. The optimal hyperparameter $\alpha$ is
determined to be 10.05. This ridge regression model is used for predicting the
$\Delta SoC$, and the performance of the model is also evaluated by metrics
such as mean squared error (MSE). Table V depicts the values of the MSE metric
of all the ML models used for each charging port. The testing data is used to
validate the proposed methodology by spoofing it for various cases are shown
with the obtained results.

TABLE IV: performance metrics comparison of various models for predicting $\Delta SoC$ Model |  Parameters |  MSE
---|---|---
Linear regression |  fit$\\_$intercept = false |  1.771121117
Multi-layer perceptron |  Adam optimizer, Dropout = 0.2 |  1.77119094
Support vector regression (SVR) |  C = 1.5, epsilon = 0.18 |  1.77747376
Random Forest (RF) |  number of estimators = 230 |  2.01821433
Ridge regression |  alpha = 10 |  0.000194
TABLE V: Proposed regression model performance across various charging ports charging Port | MSE
---|---
EV0$\\_$Terra53 | 0.000194
EV1$\\_$TerraHP-Cord$\\_$a | 0.000217
EV1$\\_$TerraHP-Cord$\\_$b | 0.000180
EV2$\\_$TerraHP-Cord$\\_$a | 0.000324
EV2$\\_$TerraHP-Cord$\\_$b | 0.000129
EV3$\\_$TerraHP | 0.000356

The different spoofing types considered in this work are shifting by decimals,
incremental array shifting, and random spoofing. These are implemented on the
charging port EV0$\\_$Terra53 and the proposed anomaly detection methodology
is tested for all types of spoofing. since the results are shown for only one
charging port the proposed framework, could be scaled to the other remaining
charging ports also. The results obtained for detecting the anomaly caused by
the different spoofing types on the charging port EV0$\\_$Terra53 are further
discussed as different case studies.

Figure 5: Shifting in decimals case study indicating (a) communicated and
predicted $\Delta SoC$ values highlighting the spoofed points, anomaly
detection for the spoofed windows for the instances starting with (b) 409 and
(c) 19481 each window of size 10 samples Figure 6: Incremental array shifting
case study indicating (a) communicated and predicted $\Delta SoC$ values
highlighting the spoofed points, anomaly detection for the spoofed windows for
the instances starting with (b) 35687 and (c) 53880 each window of size 10
samples Figure 7: Random array spoofing case study indicating (a) communicated
and predicted $\Delta SoC$ values highlighting the spoofed points, anomaly
detection for the spoofed windows for the instances starting with (b) 190 and
(c) 81908 each window of size 10 samples

###  VII-A Case Study: Sifting in decimals

This case consists of shifting the actual or communicated $\Delta SoC$ values
of the connected EVs. Since this work considers shifting the decimals for a
smaller window size, it does not have any impact on the EVCI. However,
shifting the communicated $\Delta SoC$ values continuously for longer
durations may affect the EVCI system and also the EV battery. A few sample
windows of size 10 samples each are considered at random instances on the
testing data for spoofing. Fig. 5(a) shows the actual communicated $\Delta
SoC$ (spoofed) and the predicted $\Delta SoC$ values along with spoofed data
points marked distinctly, for the charging port EV0$\\_$Terra53 of the testing
data. The shifting of $\Delta SoC$ values is done by adding a constant number
to the communicated $\Delta SoC$ values. If this spoofing increases the actual
value then EVCI assumes that EV is charged at higher speeds and the charging
process is commuted without achieving the desired task. In cases of spoofing
with lower values, the charging process is not commuted even after achieving
the desired task. hence, these issues are prevented by detecting this type of
spoofing. A threshold is predefined for this anomaly detection based on the
type of EV connected to the respective charging port. Fig. 5(b) and Fig. 5(c)
depicts the anomaly detection for this type of spoofing for different time
windows at time instances 190 and 81908 respectively along with the threshold
where the spoofed values are the communicated $\Delta SoC$ values. The results
show that the proposed anomaly detection methodology works fine and it also
classifies this shifting type spoofing with an accuracy of 99.31$\%$.

###  VII-B Case Study: Incremental array shifting

The incremental shifting of the $\Delta SoC$ consists of the shifting of the
communicated $\Delta SoC$ values to a significant amount. The earlier shifting
in decimal consists of adding or replacing the actual values with a constant
value, whereas this shifting type adds or replaces the actual values with a
series of progression values. This type of spoofing plays a crucial role
during the EVs arrival and departure duration where the $\Delta SoC$ values
are in higher magnitudes. The communicated (spoofed) values and the predicted
values of the $\Delta SoC$ are shown in Fig. 6(a) with the spoofed data points
marked distinctly for the EVs arriving at the charging port EV0$\\_$Terra53 of
the testing data. This spoofing makes the EVCI assume that the EV is charging
at faster or slower rates than desired which results in disturbing the
charging and discharging rates (C-rates) of the EVs. To mitigate these issues,
this type of spoofing is detected by employing a predefined threshold. Since
this progression based shifting may be incremental or decremental, the
proposed methodology is verified for the incremental array shifting but it
could be applied to both types of spoofing. This work considers shifting the
$\Delta SoC$ values by adding a series of the arithmetic progression to the
existing values thus making the SoC increase at a constant faster rate. Fig.
6(b) and Fig. 6(c) show the anomaly detection window for this type of spoofing
considered at a faster rate in the testing data for the different instances.
These results indicate that the proposed methodology has 90.84$\%$ accuracy in
detecting the anomalies attributed to spoofing, specifically related to the
array shifting of the SoC values with a sequence of arithmetic progression
values.

###  VII-C Case Study: Random Spoofing

The type of spoofing techniques that do not belong to any of the
aforementioned categories are classified as random spoofing. Random spoofing
consists of replacing the actual$\Delta SoC$ values with random numbers. A few
sample windows of size 10 instances each are spoofed and these time windows
are chosen randomly. The communicated (spoofed) and the predicted $\Delta SoC$
values are shown in Fig. 7(a) along with the spoofed points marked distinctly,
for the charging port EV0$\\_$Terra53 of the testing data. The effect of
random spoofing is irregular variations in $\Delta SoC$ values resulting in
the EVCI not achieving the desired tasks. This work considers replacing the
actual $\Delta SoC$ value with a series of random numbers within the specified
range as presented in Table III. Fig. 7(b) and Fig. 7(c) show the anomaly
detection where the spoofing is of random type. The results show that the
proposed methodology has 93$\%$ accuracy in detecting the random spoofing and
classifying it.

##  VIII Conclusions and Future Work

An EVCI consisting of four CBs and six charging ports is simulated as shown in
Fig. 1. The communications of the EVCI system are also simulated with suitable
protocols like OCPP and GOOSE. The required data is obtained from the
simulation and segregated into training and testing data as needed. A data-
driven approach is used to predict the $\Delta SoC$, which adopts a ridge
regression based ML model due to the multicollinearity between the data
variables. This work assumes that anomalies are caused due to spoofing the
communicated $\Delta SoC$ values. Absolute differences between the predicted
$\Delta SoC$ and the communicated $\Delta SoC$ are computed and these
differences are compared with the predefined threshold. The proposed
methodology iteratively compares the absolute differences with the threshold
to detect the anomalies. This work differentiates the anomalies with the EV’s
arrival and departure scenarios.

Three types of spoofing techniques are implemented on the testing data and the
proposed algorithm is validated. various case studies are evaluated to
validate the proposed algorithm for the spoofing classification. The spoofing
techniques are classified by observing the patterns in the calculated absolute
differences. Spoofing $\Delta SoC$ values results in the irregular and
improper charging of the EVs leading to energy theft, and financial loss to
the EVCI administrators and the EV users. The possible future works are as
follows:

  * •

Load forecasting of the EVCI as the manipulation of the SoC leads to
performance degradation of batteries and causes irregular loading behavior on
EVCI.

  * •

Study of the grid instabilities due to large scale cyber attacks targeting the
multiple EV’s SoC impacting the power grid, leading to fluctuations and
potential instabilities in the grid.

  * •

Congestion management and handling the unplanned charging sessions that may
arise due to false or manipulated SoC values.

  * •

Estimating the revenue loss incurred by the energy theft due to the cyber
attacks on the EVCI.

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