aainstitutetext:  College of Physics and Electronic Engineering, Qilu Normal
University
2 Wenbo Road, Jinan 250300, Chinabbinstitutetext: College of Physics and Space
Science, China West Normal University
1 Shida Road, Nanchong 637000 China

# An Improvement for Quantum Tunneling Radiation of Fermions in a Stationary
Kerr-Newman Black Hole Spacetime

J. Zhang,111Corresponding author. a M. Q. Liu a Z. E. Liu b and S. Z. Yang
<EMAIL_ADDRESS>
###### Abstract

By introducing a specific etheric-like vector in the Dirac equation with
Lorentz Invariance Violation (LIV) in the curved spacetime, an improved method
for quantum tunneling radiation of fermions is proposed. As an example, we
apply this new method to a charged axisymmetric Kerr-Newman black hole.
Firstly, considering LIV theory, we derive a modified dynamical equation of
fermion with spin 1/2 in the Kerr-Newman black hole spacetime. Then we solve
the equation and find the increase or decrease of black hole’s Hawking
temperature and entropy are related to constants $a$ and $c$ of the Dirac
equation with LIV in the curved spacetime. As $c$ is positive, the new Hawking
temperature is about $\frac{\sqrt{1+2a+2cmk_{r}^{2}}}{\sqrt{1+2a}}$ times
higher than that without modification, but the entropy will decrease. We also
make a brief discussion for the case of high spin fermions.

##  1 Introduction

Black hole is a mysterious cosmic body with extremely intense gravity. With
the detection of LIGO and Virgo, more and more activities of black holes are
found, so current theoretical researches for black holes become more and more
significant. Black holes can be divided into three types: static black hole,
stationary black hole and dynamic black hole. Hawking firstly proved that
black holes have thermal radiation in theory by studying the quantum effect
near the horizon of black holes haw74 ; haw75 . Hawking radiation effectively
links gravitational theory, quantum theory and thermodynamic statistics
physics, and inspired other researchers to study the thermodynamical evolution
of black holes dam76 ; san88 ; zha91 ; yan95 . Parikh et al. pointed that the
behavior of Hawking radiation can be regarded as a quantum tunneling par00 ;
par06 . In their assumption, the event horizon of black hole is a potential
barrier, virtual particles yielded inside the horizon have a certain
probability to escape from this barrier and be converted to real particles
radiating out from the black hole. Refs.liu07 ; akh06 ; sri99 ; sha02 ; med02
; iso06 ; zha06 ; yan05 ; liu19  adopted the quantum tunneling method to
investigate Hawking radiation for different types of black holes. Srinivasan
et al. derived the Hamilton-Jacobi equation in curved spacetime from a scalar
field equation sri99 ; sha02 . Kerner and Mann et al. studied the tunneling
radiation of Dirac particles using a semi-classical theory ker08a ; ker08b ;
cri08 . Lin and Yang proposed a new method to study the quantum tunneling
radiation of fermions lin09b ; lin09a ; lin11 ; yan10 . Their method can also
be used to study the quantum tunneling radiation of bosons. The results
obtained in Refs.lin09b ; lin09a ; lin11 ; yan10  show that the Hamilton-
Jacobi equation in curved spacetime is a basic equation of particle dynamic,
which reflects the inherent consistency between Lorentz symmetry theory and
the Hamilton-Jacobi equation. Recently, we considered a light dispersion
relationship derived from string theory to research the modified quantum
tunneling rates for spherical symmetry and axisymmetryic black holeszha20a ;
zha20b .

General relativity is a gravitational theory that cannot be renormalized, so
several modified gravitational theories have been proposed. Since the Lorentz
Invariation Violation (LIV) may exist at high energy cases, various gravity
models based on LIV have been proposed hor09 ; jac01 ; lin14 . In principle,
LIV theory can solve the problem of renormalization of gravitational theory.
In addition, some studies on LIV suggest that the dark matter may be just one
of the effects of LIV theoretical models muk10 . In string theory,
electrodynamics and non-abel theory, LIV has attracted extensive attention
col07 ; jac99 ; kos89 . In the recent years, Dirac equation with LIV term in
the flat spacetime has been studied by introducing etheric-like field terms
cas11 ; nas15 . In this theory, Lorentz symmetry disappears due to the
existence of the ether-like field. Therefore, some special properties which
are inconsistent with Lorentz symmetry theory will emerge at the high energy
case. This is an interesting topic which needs to be explored in depth. On the
other hand, the dynamics of fermions with LIV in the curved spacetime is also
an attractive subject worthy further study, which will influence the
correction to quantum tunneling radiation of black hole. At present, the
quantum tunneling radiation of Dirac particles with etheric-like field terms
has been investigated in spherically symmetric black holes pu19 . We
investigate the influence of different etheric-like vector $u^{\alpha}$ on the
solution of the modified Hamilton-Jacobi equation by using both the semi-
classical approximation and beyond the semi-classical approximationzha20c .

In this paper, the quantum tunneling radiation of fermions is modified in the
axisymmetric charged Kerr-Newman black hole by considering a specific ether-
like field vector term. Our paper is organized as follows: In Sec.2,
considering LIV theory, the dynamical equation of fermions with spin 1/2 is
derived for Kerr-Newman black hole. In Sec.3, we solve this dynamical equation
and obtain the corrected physical quantities such as Hawking temperature and
tunneling rate of the black hole. We make some discussions in Sec.4.

##  2 Lorentz Invariance Violation Theory and Dirac-Hamilton-Jacobi Equation

In the Ref.nas15 , Nascimento et al. researched the particle’s action which
includes LIV in the flat spacetime. Transferring the normal derivative to
covariant derivative, and extending the commutation relation of gamma matrices
$\bar{\gamma}^{\mu}$ and $\bar{\gamma}^{\nu}$ in the flat spacetime to that in
the curved sapcetime, we obtain Dirac equation of fermion, spin of which is
1/2, with LIV in the curved spacetime as:

$\\{\gamma^{\mu}D_{\mu}[1+\hbar^{2}\frac{a}{m^{2}}(\gamma^{\mu}D_{\mu})^{2}]+\frac{b}{\hbar}\gamma^{5}+c\hbar(u^{\alpha}D_{\alpha})^{2}-\frac{m}{\hbar})\\}\Psi=0,$
(1)

where $m$ is the mass of fermion, $a$, $b$ and $c$ is small constants. Gamma
matrices $\gamma^{\mu}$ satisfy the condition:

$\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}I,$ (2)
$\gamma^{5}\gamma^{\mu}+\gamma^{\mu}\gamma^{5}=0,$ (3)

where $g^{\mu\nu}$ is the inverse metric tensor, and $I$ is the unit matrix.
In the flat spacetime, Eq.(2) reduces to
$\bar{\gamma}^{\mu}\bar{\gamma}^{\nu}+\bar{\gamma}^{\nu}\bar{\gamma}^{\mu}=2\delta^{\mu\nu}I$,
and Eq.(3) changes to
$\bar{\gamma}^{5}\bar{\gamma}^{\mu}+\bar{\gamma}^{\mu}\bar{\gamma}^{5}=0$. In
Eq.(1)

$D_{\mu}=\partial_{\mu}+\frac{i}{2}\Gamma^{\alpha\beta}_{\mu}\pi_{\alpha\beta}-\frac{i}{\hbar}qA_{\mu},$
(4)

where $q$ is the charge of fermion, and $A_{\mu}$ is the electromagnetic
potential of black hole. The second term at the right side of Eq.(4) is spin
connection which is a very small term in the dynamical equation and thus can
be ignored. $u^{\alpha}$ is an etheric-like vector, which satisfies:

$u^{\alpha}u_{\alpha}={\rm const}.$ (5)

In order to solve the Dirac equation for fermions with spin 1/2, we assume its
wave function is

$\Psi=\psi_{AB}{\rm e}^{\frac{{\rm i}}{\hbar}S}=\left(\begin{array}[]{c}A\\\
B\end{array}\right){\rm e}^{\frac{{\rm i}}{\hbar}S},$ (6)

where $A$ and $B$ are matrix elements in the column matrix, $S$ is the
Hamilton principal function. Substituting Eq.(4) and Eq.(6) into Eq.(1), we
get

$\displaystyle\\{i\gamma^{\mu}(\partial_{\mu}S-qA_{\mu})[1-\frac{a}{m^{2}}\gamma^{\alpha}\gamma^{\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})]$
$\displaystyle-
cu^{\alpha}u^{\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})+b\gamma^{5}-m\\}\Psi=0.$
(7)

Using Eq.(2), we have

$\gamma^{\alpha}\gamma^{\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})=g^{\alpha\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta}).$
(8)

Combining Eq.(8) and Eq.(2), they yields

$\displaystyle i\gamma^{\mu}(\partial_{\mu}S-qA_{\mu})\Psi$
$\displaystyle=[1-\frac{a}{m^{2}}\gamma^{\alpha}\gamma^{\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})]^{-1}$
(9)
$\displaystyle\\{cu^{\alpha}u^{\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})+b\gamma^{5}+m\\}\Psi$
$\displaystyle=[1+\frac{a}{m^{2}}\gamma^{\alpha}\gamma^{\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})+\mathcal{O}(a^{2})]$
$\displaystyle\\{cu^{\alpha}u^{\beta}(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})+b\gamma^{5}+m\\}\Psi$
$\displaystyle=[1+(\frac{c}{m}u^{\alpha}u^{\beta}+\frac{a}{m^{2}}g^{\alpha\beta})(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})$
$\displaystyle-\frac{b}{m}\gamma^{5}]m\Psi.$

Since $b\ll m$, so $\frac{b}{m}\gamma^{5}$ is very small. Multiplying
$i\gamma^{\nu}(\partial_{\nu}S-qA_{\nu})$ at both sides of Eq.(9), we get

$\displaystyle-\gamma^{\mu}\gamma^{\nu}(\partial_{\mu}S-qA_{\mu})(\partial_{\nu}S-qA_{\nu})\Psi=m^{2}+2(cmu^{\alpha}u^{\beta}+ag^{\alpha\beta})$
$\displaystyle(\partial_{\alpha}S-qA_{\alpha})(\partial_{\beta}S-qA_{\beta})\Psi+\mathcal{O},$
(10)

i. e.,

$\displaystyle[g^{\mu\nu}(\partial_{\mu}S-qA_{\mu})(\partial_{\nu}S-qA_{\nu})$
$\displaystyle+2(cmu^{\mu}u^{\nu}+ag^{\mu\nu})(\partial_{\mu}S-qA_{\mu})(\partial_{\nu}S-qA_{\nu})+m^{2}]\Psi=0.$
(11)

This is a matrix equation. The condition for this matrix equation to have
nontrivial solutions is that the value of determinant of the wave function’s
coefficient matrix is zero, i. e.,

$(g^{\mu\nu}+2cmu^{\mu}u^{\nu}+2ag^{\mu\nu})(\partial_{\mu}S-qA_{\mu})(\partial_{\nu}S-qA_{\nu})+m^{2}=0,$
(12)

or

$(g^{\mu\nu}+\frac{2cmu^{\mu}u^{\nu}}{1+2a})(\partial_{\mu}S-qA_{\mu})(\partial_{\nu}S-qA_{\nu})+m^{2}/(1+2a)=0.$
(13)

Considering $a$ is a very small constant and adopting Taylor expansion for the
last term in the Eq.(13), we get

$(g^{\mu\nu}+\frac{2cmu^{\mu}u^{\nu}}{1+2a})(\partial_{\mu}S-qA_{\mu})(\partial_{\nu}S-qA_{\nu})+m^{2}(1-2a)=0.$
(14)

From Eq.(1) to Eq.(14), we get a new dynamical equation for Dirac particles.
Comparing with the normal Hamilton-Jacobi equation, one can find that, as
$2cmu^{\mu}u^{\nu}=a=0$, Eq.(14) returns to the normal Hamilton-Jacobi
equation. We call this deformed equation as Dirac-Hamilton-Jacobi equation.
The choice of the Hamilton principal function $S$ depends on the selected
coordinate and line element. In a stationary spacetime, it can be generally
expressed as $S=S(t,r,\theta,\varphi)$.

##  3 Correction to Tunneling radiation of Fermions in the Kerr-Newman
Spacetime

In the Boyer-Lindquist coordinate, the line element of Kerr-Newman black hole
is written as wal84

$ds^{2}=\rho^{2}(\frac{dr^{2}}{\Delta}+d\theta^{2})+\frac{\sin^{2}\theta}{\rho^{2}}[(r^{2}+a_{kn}^{2})d\varphi-
a_{kn}dt]^{2}-\frac{\Delta}{\rho^{2}}[dt-a_{kn}\sin^{2}\theta d\varphi]^{2},$
(15)

where

$\displaystyle\Delta=r^{2}+a_{kn}^{2}-2Mr+Q^{2},$
$\displaystyle\rho^{2}=r^{2}+a_{kn}^{2}\cos^{2}\theta,$ (16)

where $M$ and $a$ are the mass and angular momentum of unit mass of black
hole. From Eq.(15) and Eq.(3), one can get the components of non-zero
covariant metric tensors

$\displaystyle g_{tt}=\frac{1}{\rho^{2}}(a_{kn}^{2}\sin^{2}\theta-\Delta),$
$\displaystyle g_{rr}=\frac{\rho^{2}}{\Delta},\qquad\qquad$ $\displaystyle
g_{\theta\theta}=\rho^{2},\qquad\qquad\qquad\quad$ $\displaystyle
g_{\varphi\varphi}=\frac{\sin^{2}\theta}{\rho^{2}}[(r^{2}+a_{kn}^{2})-\Delta
a_{kn}^{2}\sin^{2}\theta],$ $\displaystyle
g_{t\varphi}=\frac{a_{kn}\sin^{2}\theta}{\rho^{2}}(Q^{2}-2Mr),\,$ (17)

and inverse metric tensors

$\displaystyle
g^{tt}=\frac{1}{\rho^{2}}(a_{kn}^{2}\sin^{2}\theta-\frac{r^{2}+a_{kn}^{2}}{\Delta}),$
$\displaystyle g^{rr}=\frac{\Delta}{\rho^{2}},$ $\displaystyle
g^{\theta\theta}=\frac{1}{\rho^{2}},$ $\displaystyle
g^{\varphi\varphi}=\frac{1}{\rho^{2}}(\frac{1}{\sin^{2}\theta}-\frac{a_{kn}^{2}}{\Delta}),$
$\displaystyle g^{t\varphi}=\frac{1}{\rho^{2}}(\frac{Q^{2}-2Mr}{\Delta}).$
(18)

According to null super-surface equation

$g^{\mu\nu}\frac{\partial f}{\partial x^{\mu}}\frac{\partial f}{\partial
x^{\nu}}=0,$ (19)

the event horizon of black hole $r_{H}$ satisfies following equation

$\Delta|_{r=r_{H}}=r^{2}_{H}-2Mr_{H}+a_{kn}^{2}+Q^{2}=0.$ (20)

The electromagnetic potential of the Kerr-Newman black hole

$\displaystyle A_{\mu}$ $\displaystyle=(A_{t},0,0,A_{\varphi}),$ (21)
$\displaystyle A_{t}=-\frac{Qr}{\rho^{2}},$ $\displaystyle
A_{\varphi}=\frac{a_{kn}Qr\sin^{2}\theta}{\rho^{2}}.$

Substituting Eq.(3) and (3) into Eq.(14), we get the dynamical equation of
spin 1/2 fermions with mass $m$ and charge $q$ in the curved spacetime,

$\displaystyle
g^{tt}(\partial_{t}S-qA_{t})^{2}+2g^{t\varphi}(\partial_{t}S-qA_{t})(\partial_{\varphi}S-qA_{\varphi})+g^{rr}(\partial_{r}S)^{2}+g^{\theta\theta}(\partial_{\theta}S)^{2}$
$\displaystyle+g^{\varphi\varphi}(\partial_{\varphi}S-qA_{\varphi})^{2}+(1-2a)m^{2}+\frac{2mc}{1+2a}[u^{t}u^{t}(\partial_{t}S-qA_{t})^{2}+u^{r}u^{r}(\partial_{r}S)^{2}$
$\displaystyle+u^{\theta}u^{\theta}(\partial_{\theta}S)^{2}+u^{\varphi}u^{\varphi}(\partial_{\varphi}S-qA_{\varphi})^{2}+2u^{t}u^{\varphi}(\partial_{t}S-qA_{t})(\partial_{\varphi}S-qA_{\varphi})$
$\displaystyle+2u^{t}u^{r}(\partial_{t}S-qA_{t})\partial_{r}S+2u^{t}u^{\theta}(\partial_{t}S-qA_{t})\partial_{\theta}S$
$\displaystyle+2u^{r}u^{\varphi}\partial_{r}S(\partial_{\varphi}S-qA_{\varphi})+2u^{\theta}u^{\varphi}\partial_{\theta}S(\partial_{\varphi}S-qA_{\varphi})]=0.$
(22)

After substituting the components of $g^{\mu\nu}$ into Eq.(3), multiplying
$\rho^{2}$ on both sides of the equation and merging the similar terms, the
dynamical equation are as follows:

$\displaystyle\Delta(\frac{\partial S}{\partial
r})^{2}-\frac{1}{\Delta}[(r^{2}+a_{kn}^{2})\frac{\partial S}{\partial
t}+a_{kn}\frac{\partial
S}{\partial\varphi}+eQr]^{2}+r^{2}m^{2}(1-2a)+(\frac{\partial
S}{\partial\theta})^{2}$ $\displaystyle+(\frac{1}{\sin\theta}\frac{\partial
S}{\partial\varphi}+a_{kn}\sin\theta\frac{\partial S}{\partial
t})^{2}+\rho^{2}m^{2}(1-2a)\cos^{2}\theta+\frac{2mc\rho^{2}}{1+2a}[u^{t}u^{t}(\frac{\partial
S}{\partial t}-qA_{t})^{2}$ $\displaystyle+u^{r}u^{r}(\frac{\partial
S}{\partial r})^{2}+u^{\theta}u^{\theta}(\frac{\partial
S}{\partial\theta})^{2}+u^{\varphi}u^{\varphi}(\frac{\partial
S}{\partial_{\varphi}}-qA_{\varphi})^{2}$
$\displaystyle+2u^{t}u^{\varphi}(\frac{\partial S}{\partial
t}-qA_{t})(\frac{\partial
S}{\partial_{\varphi}}-qA_{\varphi})+2u^{t}u^{r}(\partial_{t}S-qA_{t})\partial_{r}S$
$\displaystyle+2u^{t}u^{\theta}(\partial_{t}S-qA_{t})\partial_{\theta}S+2u^{r}u^{\varphi}\partial_{r}S(\partial_{\varphi}S-qA_{\varphi})$
$\displaystyle+2u^{\theta}u^{\varphi}\partial_{\theta}S(\partial_{\varphi}S-qA_{\varphi})]=0.$
(23)

To solve the above equation, we must choose special $u^{t}$, $u^{r}$,
$u^{\theta}$ and $u^{\varphi}$ which must satisfy Eq.(5). According to
Eqs.(15)-(3) and the metric tensor of Kerr-Newman black hole, we choose the
following $u^{\alpha}$:

$\displaystyle
u^{t}=\frac{k_{t}\rho}{(a_{kn}^{2}\sin^{2}\theta-\Delta)^{1/2}},$
$\displaystyle u^{r}=\frac{k_{r}\Delta^{1/2}}{\rho},$ $\displaystyle
u^{\theta}=\frac{k_{\theta}}{\rho},$ $\displaystyle
u^{\varphi}=\frac{k_{\varphi}\rho}{\sin\theta a_{kn}(Q^{2}-2Mr)},$ (24)

where $k_{t},\ k_{r},\ k_{\theta},\ k_{\varphi}$ are constants, which
satisfies

$u^{\alpha}u_{\alpha}={k^{2}_{\alpha}},$ (25)

where $\alpha$ denotes $t,\ r,\ \theta,\ \varphi$. Substituting Eq.(3) and
Eq.(20) into Eq.(3), and considering the limit at the event horizon of black
hole, we can get the dynamical equation of spin $1/2$ fermions at the event
horizon of black hole, that is,

$\Delta^{2}\left|{}_{r\rightarrow
r_{H}}\right.(1+\frac{2cmk^{2}_{r}}{1+2a})\left(\frac{\partial S}{\partial
r}\right)^{2}\left|{}_{r\rightarrow
r_{H}}\right.-\left[(r_{H}^{2}+a_{kn}^{2})\frac{\partial S}{\partial
t}+a_{kn}\frac{\partial S}{\partial\varphi}+qQr_{H}\right]^{2}=0.$ (26)

Because the quantum tunneling radiation of a black hole is a property of the
radial direction of the black hole, we care the radial component of the
Hamilton principal function. From Eq.(26),

$\frac{\partial S}{\partial r}\left|{}_{r\rightarrow
r_{H}}\right.=\pm\frac{(r_{H}^{2}+a_{kn}^{2})\sqrt{1+2a}}{\Delta\left|{}_{r\rightarrow
r_{H}}\right.\sqrt{1+2a+2cmk^{2}_{r}}}\left(\frac{\partial S}{\partial
t}+\frac{a_{kn}\frac{\partial
S}{\partial\varphi}+qQr_{H}}{r_{H}^{2}+a_{kn}^{2}}\right).$ (27)

According to Eq.(15), we set

$S=-\omega t+R(r)+\Theta(\theta)+j\varphi,$ (28)

where $\omega$ is particle energy, $j$ is a constant which describes the
$\varphi$ component of general momentum. Eq.(27) can be reducible to

$\frac{\partial S}{\partial r}\left|{}_{r\rightarrow
r_{H}}\right.=\frac{dR}{dr}\left|{}_{r\rightarrow
r_{H}}\right.=\pm\frac{(r_{H}^{2}+a_{kn}^{2})\sqrt{1+2a}}{\Delta\left|{}_{r\rightarrow
r_{H}}\right.\sqrt{1+2a+2cmk^{2}_{r}}}(\omega-\omega_{0}),$ (29)

where

$\omega_{0}=\frac{qQr_{H}+a_{kn}j}{r_{H}^{2}+a_{kn}^{2}},$ (30)

where $\omega$ is the particle energy, $\omega_{0}$ is the chemical potential,
which means the minimal energy of emission particles. Integrating the above
equation from the inner side to the outer side of $r_{H}$ with the residue
theorem, we obtain

$\displaystyle S_{\pm}=R_{\pm}$ $\displaystyle=\pm\int
dr\frac{(r_{H}^{2}+a_{kn}^{2})\sqrt{1+2a}}{\Delta\left|{}_{r\rightarrow
r_{H}}\right.\sqrt{1+2a+2cmk^{2}_{r}}}(\omega-\omega_{0}),$ (31)
$\displaystyle=\pm\frac{i\pi}{2}\frac{(r_{H}^{2}+a_{kn}^{2})\sqrt{1+2a}}{(r_{H}-M)\sqrt{1+2a+2cmk^{2}_{r}}}(\omega-\omega_{0}),$

where subscript + and - denote outgoing and incoming module, respectively.
According to the tunneling theory of black hole, the tunneling rate is

$\displaystyle\Gamma$ $\displaystyle=\exp[-2({\rm Im}S_{+}-{\rm Im}S_{-})]$
(32) $\displaystyle=\exp(-\frac{\omega-\omega_{0}}{T^{{}^{\prime}}_{H}}),$

where $T^{{}^{\prime}}_{H}$ is Hawking temperature after modification.

$\displaystyle T^{{}^{\prime}}_{H}$
$\displaystyle=\frac{(r_{H}-M)(1+2a+2cmk^{2}_{r})^{1/2}}{2\pi(r^{2}_{H}+a_{kn}^{2})\sqrt{1+2a}}$
(33) $\displaystyle=T_{KNh}(1+2a+2cmk^{2}_{r})^{1/2}/\sqrt{1+2a},$

where $T_{KNh}$ is the Hawking temperature without LIV modification,

$T_{KNh}=\frac{(r_{H}-M)}{2\pi(r^{2}_{H}+a_{kn}^{2})}.$ (34)

##  4 Summery and Discussions

In this paper, we have modified the dynamical equation of Dirac particles in
the curved spacetime, considering LIV theory. Comparing Eq.(1) and Eq.(14), we
simplify the complicate derivation process. By solving Eq.(14), the new
Hamilton principal function $S$ of Dirac particle is obtained. Since the
quantum tunneling rate and Hawking temperature at the horizon of black hole
depend on the imaginary part of $S$, so the new quantities related to Hawking
radiation are obtained naturally. These results are valuable for further study
of LIV theory and quantum gravitational theory. Because LIV theory modifies
the Hawking temperature at the event horizon of Kerr-Newman black hole, it
will lead to the correction of black hole entropy. According to the first law
of thermaldynamics of black hole,

${\rm d}M=T{\rm d}S+V{\rm d}J+U{\rm d}Q,$ (35)

where $V$ and $U$ are the rotation potential and electric potential of black
hole respectively. From Eq.(33) and Eq.(34), the entropy with LIV theory
correction in the Kerr-Newman black hole is

$\displaystyle S_{H}=\int dS_{H}$ $\displaystyle=\int\frac{dM-VdJ-UdQ}{T_{H}}$
(36) $\displaystyle=\int\frac{\sqrt{1+2a}}{\sqrt{1+2a+2cmk_{r}^{2}}}dS_{KNh},$

where $S_{KNh}$ denotes the entropy without modification,

$dS_{KNh}=\frac{dM-VdJ-UdQ}{T_{KNh}}.$ (37)

It can be seen from Eq.(33) and Eq.(36) that the increase or decrease of black
hole’s Hawking temperature and entropy are related to the value of constants
$a$ and $c$ in Eq.(1). As $c$ is positive, the new Hawking temperature is
about $\frac{\sqrt{1+2a+2cmk_{r}^{2}}}{\sqrt{1+2a}}$ times higher than that
without modification, but the entropy will decrease. Above results are from
semi-classical approximation and based on spin $3/2$ fermions. Beyond semi-
classical approximation can refer our recent paper zha20c . For spin $3/2$
fermions, the wave function in Eq.(1) should be substituted by

$\Psi=\left(\begin{array}[]{c}A_{V}\\\ B_{V}\end{array}\right){\rm
e}^{\frac{{\rm i}}{\hbar}S},$ (38)

where $A_{V}=[A\ \ B]^{T}$, $B_{V}=[A\ \ B]$yan10 . For arbitrary spin
fermions, $\Psi$ in Eq.(1) should be substituted by
$\Psi_{\alpha_{1}\ldots\alpha_{k}}$, where the value of $\alpha_{k}$
corresponds to different spin. The larger $\alpha_{k}$, the higher the spin
is. Furthermore, the above conclusions are also applicable to the other
spherically symmetric and axisymmetric charged black holes. We will do further
research at this aspect in the future.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(grant U2031121), the Science Foundation of Sichuan Science and Technology
Department(grant 2018JY0502) and the Natural Science Foundation of Shandong
Province (grant ZR2020MA063, ZR2019MA059).

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# Topology-based Approximations for $\mathcal{N}-1$ Contingency Constraints in
Power Transmission Networks

Amin Shokri Gazafroudi<EMAIL_ADDRESS>Fabian Neumann<EMAIL_ADDRESS>Tom Brown<EMAIL_ADDRESS>Institute for Automation and Applied
Informatics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
Institute of Energy Technology, Technical University of Berlin, Einsteinufer
25, 10587 Berlin, Germany

###### Abstract

It is crucial for maintaining the security of supply that transmission
networks continue to operate even if a single line fails. Modeling
$\mathcal{N}-1$ security in power system capacity expansion problems
introduces many extra constraints if all possible outages are accounted for,
which leads to a high computational burden. Typical approaches to avoid this
burden consider only a subset of possible outages relevant to a given dispatch
situation. However, this relies on knowing the dispatch situation beforehand,
and it is not suitable for investment optimization problems where the
generation fleet is not known in advance. In this paper, we introduce a
heuristic approach to model the fully secured $\mathcal{N}-1$ feasible space
using a smaller number of constraints in a way that only depends on the
topology of transmission networks. In our proposed approach, the network’s
security is modelled by comparing the polytope of the feasible space of nodal
net power obtained from the security-constrained linearized AC optimal power
flow problem. To approximate this polytope, a buffer capacity factor is
defined for transmission lines in the $\mathcal{N}-0$ secure case, thereby
avoiding the introduction of many additional constraints. In this way, three
approaches are introduced for obtaining a buffer capacity factor consisting of
approximate, robust and line-specific approaches. Finally, the performance of
our proposed approaches is assessed in different scales of transmission
networks for determining the proposed buffer capacity factors, contingency
analysis and economic evaluation. Moreover, we find that our proposed
heuristics provide excellent approximations of the fully secured
$\mathcal{N}-1$ solutions with a much lower computational burden.

###### keywords:

Linear optimal power flow , polytope , power system , reliability , security ,
transmission network.

††journal: International Journal of Electrical Power & Energy Systems

##  1 Introduction

###  1.1 Motivation

Security and reliability are defined as two of the most important goals of
power systems to supply power demand continuously [1]. Thus, $\mathcal{N}-1$
criteria have been introduced to retain the power system in a secure state
when a single outage of power system components occurs [2]. Conventionally,
$\mathcal{N}-1$ security is modelled by many extra constraints to account for
all outages, which leads to a high computational burden, even when the flow
physics is linearized (a common approach for operational and capacity
expansion problems). Typical approaches to avoid this burden are to consider
only a subset of possible outages which are relevant for a given dispatch
situation, iteratively adding the outage constraints until the network can be
demonstrated by experimentation to be $\mathcal{N}-1$ secure [3]. This
approach depends on the dispatch situation and is unsuitable for investment
optimization, where many dispatch situations are considered simultaneously and
the generation is not known in advance. In the following section we summarise
different approaches in the literature for modelling security constraints, so-
called security-constrained linear optimal power flow (SCLOPF), to guarantee
the power grid’s reliability and prevent line overloading and cascading
failures in the system.

###  1.2 Related Works

Authors in [2, 4] presented how the current injection method retains the power
system’s security in the optimal power flow problem by compensating power flow
of the failed line based on injecting the virtual currents to prevent
overloading in rest of the lines in the transmission network. Authors in [5]
presented a mixed-integer program for the linearized AC optimal dispatch
problem with transmission switching considering $\mathcal{N}-1$ constraints.
In [5], it has been evaluated how transmission switching can keep the security
of the power system based on $\mathcal{N}-1$ criteria. The transmission
switching problem was also considered by authors in [6, 7]. A joint unit
commitment and transmission switching problem considering $\mathcal{N}-1$
constraints has been presented in [6]. A bi-level stochastic optimal switching
model has been introduced for determining line switching strategy considering
$\mathcal{N}-1$ limitations in [7]. A preventive SCOPF problem has been
proposed in [8] where authors assessed the performance of their proposed model
in planning and operation problems of the power system. In the preventative
SCOPF problems, the proposed problem consists of decision-making variables
which are feasible for both pre-contingency (before single line outages) and
post-contingency (after single line outages), simultaneously [9]. Ref. [10]
proposed a risk-based decomposed model for contingencies in the power system.
In [11], contingencies relaxation has been proposed. Authors in [12] presented
a security-constrained multi-objective optimal power flow method considering
$\mathcal{N}-1$ security constraints. Ref. [13] proposed a mixed-integer
programming model for linear optimal power flow considering $\mathcal{N}-1$
conditions for preventing cascading failures in the power network. In [14], an
algorithm has been proposed for reducing the constraints introduced by
contingency scenarios of the SCOPF problem and finding the feasible space for
nodal power injected based on a geometric algorithm. In [15], authors
introduced an optimal topology control model considering $\mathcal{N}-1$
constraints in the power systems. In [16], an umbrella constraint discovery
problem has been introduced where the SCOPF problem is approximated by the
linearised AC power flow. Authors in [17] proposed a framework for modeling
zonal electricity markets and developed cutting-plane algorithms for
considering $\mathcal{N}-1$ security conditions.

###  1.3 Contributions

Although various models have been introduced in the literature to model
$\mathcal{N}-1$ security conditions in different applications of power
systems, most of them are dependent on properties of components of the system,
e.g. generators and loads, which increases the computational burden beyond
what is solvable in reasonable time in the context of multi-period capacity
expansion models. Moreover, there exist numerous alternative approaches with
regards to contingency conditions in light of new technologies such as large-
scale energy storage systems, e.g. batteries, that can be placed at the
connecting nodes of transmission lines outage which can compensate as virtual
lines for some time. In this paper, we present a topology-based
$\mathcal{N}-1$ security method for transmission line outages in a preventive
SCOPF problem that can be applied in power system expansion problems. In other
words, the power system’s security is solely studied based on the topology of
the transmission network without relying on information about generators,
loads and energy storage systems. Thus, this paper proposes a heuristic model
for $\mathcal{N}-1$ security constraints of the transmission grid by adding
buffer capacity factor for transmission lines. In this way, we propose three
heuristic approaches, consisting of robust, approximate and line-specific,
without adding constraints for every single line outage. Our proposed models
for $\mathcal{N}-1$ security constraints are based on a $\mathcal{N}-0$
secured linear optimal power flow formulation by adding a buffer capacity
factor for each transmission line. For instance, the loading of a line may be
limited to 70% of its real thermal capacity as an industry-standard [18, 19].
This way, the remaining 30% of the thermal capacity is kept as a reserve to
guarantee security and reliability if another line fails [20]. However, the
approximation has never been justified or shown to be robust, nor have line-
specific factors been developed. Using a buffer capacity factor on the line
capacities instead of the full set of $\mathcal{N}-1$ constraints saves many
heavy constraints and non-zero values in the constraint matrix. However, a
price is paid for the lower computational burden as the heuristic approaches
presented herein may be too optimistic or too conservative. The main
contributions of this paper can be summarized as follows:

  * 1.

We propose a heuristic approach for $\mathcal{N}-1$ security constraints of
the transmission power grid based on adding a buffer capacity factor for
transmission lines that dramatically reduces number of necessary constraints.

  * 2.

We investigate three approaches to find the buffer capacity factor for all or
individual transmission lines.

  * 3.

The performance of the fully secured $\mathcal{N}-1$ network is compared with
our proposed heuristic approaches.

  * 4.

A sensitivity analysis of the proposed approaches is performed in different
network topologies and German transmission networks.

The rest of this paper is organized as follows. Section 2 presents the problem
formulation for the linearized AC optimal power flow problem under
$\mathcal{N}-1$ security constraints. In Section 3, our heuristic approaches
to model $\mathcal{N}-1$ secure network are introduced. Then, the simulation
results are discussed in Section 4. Finally, our findings are summarised in
Section 5.

##  2 Problem Formulation

###  2.1 Linearized AC Optimal Power Flow Problem

We consider the linearized AC optimal power flow (LOPF) problem that seeks to
minimize the operational cost of generator dispatch under network constraints
with fixed capacities. Nonlinear and non-convex formulations of power flow
have not been considered in this paper because they cannot be solved within
reasonable time in the context of large capacity planning problems. The
problem is classified as a linear problem (LP) and no capacity expansion of
generation, storage or transmission infrastructure is assumed. The objective
function for the LOPF problem is given by (1).

$\min_{g_{s,t},f_{\ell,t}}\sum_{s,t}o_{s}g_{s,t},$ (1)

where $o_{s}g_{s,t}$ represents operation cost for generator $s$ at time $t$.
This way, $g_{s,t}$ represents the dispatched power of generator $s$ at time
$t$. The LOPF problem is constrained by physical equations and constraints
which are presented in the following. Eq. (2) presents the balancing
constraint based on Kirchhoff’s Current Law (KCL) requiring matching
generation and demand at bus $i$ and time $t$

$\sum_{s}M_{i,s}g_{s,t}-d_{i,t}=\sum_{\ell}K_{i,\ell}f_{\ell,t},\quad\forall
i,\forall t$ (2)

where $M_{i,s}$ maps generator $s$ at bus $i$. In other words, $M_{i,s}=1$ if
generator $s$ is located at bus $i$, otherwise $M_{i,s}=0$. Besides, $d_{i,t}$
represents electricity demand at bus $i$ and time $t$, and $f_{\ell,t}$ states
power flow in line $\ell$ and time $t$. Moreover, $K_{i,\ell}$ represents the
incidence matrix between buses and lines as represented by (3)

$K_{i,\ell}=\begin{cases}1,\quad\hskip 4.26773pt\text{if bus $i$ is started
bus of line $\ell$,}\\\ -1\quad\text{if bus $i$ is ended bus of line
$\ell$,}\\\ 0\quad\hskip 7.11317pt\text{if bus $i$ is not connected to line
$\ell$.}\end{cases}$ (3)

Moreover, maximum and minimum limitations of dispatched power of generators
are expressed by (4)

$0\leq g_{s,t}\leq\overline{G}_{s,t},\quad\forall s,\forall t$ (4)

where $\overline{G}_{s,t}$ represents the time-dependent potential of power
generation for generator $s$ at time $t$. Also, Eq. (5) states the $CO_{2}$
constraint for avoiding $CO_{2}$ emission exceeds a desired target level.

$\sum_{s,t}\frac{e_{s}}{\eta_{s}}g_{s,t}\leq\mathcal{E}_{CO_{2}},$ (5)

where $e_{s}$ is the specific emission of the fuel, $\eta_{s}$ is the
generator efficiency, and $\mathcal{E}_{CO_{2}}$ is a fraction of the $1990$
emissions in electricity [21].

###  2.2 Power Flow Constraints

In this section, nodal and line-specific power flow equations and constraints
of the LOPF problem are represented. Power nodal balancing is one of the
fundamental equations in the power system. In this paper, power loss is
ignored for simplicity. Then, we obtain

$\sum_{i=1}^{\mathcal{B}}p_{i,t}=0,$ (6)

where $p_{i,t}=\sum_{s}M_{i,s}g_{s,t}-d_{i,t}$, and it represents net power
injection at bus $i$ and time $t$. Thus, positive (negative) value of
$p_{i,t}$ presents network generation (load). Here, bus $i=1$ is chosen as a
slack bus in the grid. In other words, $p_{2,t},\dots,p_{\mathcal{B},t}$ are
independent values and $p_{1,t}$ is the dependent value at each time snapshot.
Authors in [22, 23, 24, 25] present the relation between power flow and nodal
net power is given by (7)

$\displaystyle
f_{\ell,t}=\sum_{i=2}^{\mathcal{B}}PTDF_{\ell,i}p_{i,t},\quad\forall\ell,\forall
t.$ (7)

where $PTDF_{\ell,i}$ represents power transfer distribution factor (PTDF)
between line $\ell$ and bus $i$ which is given by $PTDF=BK^{T}\Lambda^{*}$.
Moreover, $B$ is a diagonal matrix of branch susceptance as it arrays are
given by (8)

$B_{\ell,k}=\begin{cases}\frac{1}{x_{\ell}},\quad\text{if }\ell=k\\\
0\quad\quad\text{otherwise}\end{cases}$ (8)

where $x_{\ell}$ is a reactance for line $\ell$. Moreover, $\Lambda^{*}$ is a
pseudo-inverse matrix of $\Lambda$, as nodal susceptance matrix, which is
obtained by $\Lambda=KBK^{T}$. Thus, as $i=1$ is assumed as a slack bus, then
$PTDF_{\ell,1}=0$. Maximum and minimum constraints of power flow are
represented in (9)

$\displaystyle-F_{\ell}\leq f_{\ell,t}\leq F_{\ell},\quad\forall\ell,\forall
t$ (9)

where $F_{\ell}$ is the capacity of line $\ell$. To model the $\mathcal{N}-1$
secure network, it is necessary to define the $\mathcal{N}-0$ secure network.
The $\mathcal{N}-0$ secure network is only secure as long as there is no line
outage. According to $\mathcal{N}-0$ security conditions, Eq. (10) is obtained
based on (7) and (9)

$\displaystyle-F_{\ell}\leq\sum_{i=2}^{\mathcal{B}}PTDF_{\ell,i}p_{i,t}\leq
F_{\ell},\quad\forall\ell,\forall t.$ (10)

In this way, $\mathcal{B}-1$ independent nodal net powers
($p_{2,t},\dots,p_{\mathcal{B},t}$) make the $\mathcal{B}-1$ dimensional space
at each time snapshot based on (10). In other words, at time $t$, Eq. (10)
constrains the $(\mathcal{B}-1)$–dimensional polytope with at most
$2\mathcal{N}$ faces where $\mathcal{N}$ is number of transmission lines in
the network. We define this as the $\mathcal{N}-0$ secure polytope
$Q^{\mathcal{N}-0}$.

###  2.3 Security Constraints

In this section, security constraints are presented that guarantee no
overloading occurs in transmission lines if there is a single line outage
($\mathcal{N}-1$ criteria). In this way, if line $k$ fails, the flow in line
$\ell$ at time $t$ after the outage of line $k$ ($f_{\ell,t}^{k}$) is related
to the flows before the line outage in lines $\ell$ ($f_{\ell,t}$) and $k$
($f_{k,t}$) based on the line outage distribution factor (LODF) represented in
Eq. (11) [26]. Based on Eqs. (7) and (11), Eq. (12) is obtained [26]

$\displaystyle
f_{\ell,t}^{k}=f_{\ell,t}+LODF_{\ell,k}f_{k,t},\quad\forall\ell,\forall t$
(11) $\displaystyle
f_{\ell,t}^{k}=\sum_{i=2}^{\mathcal{B}}[PTDF_{\ell,i}+LODF_{\ell,k}PTDF_{k,i}]p_{i,t},\forall\ell,\forall
t$ (12)

where the $PTDF$ and $LODF$ matrices are model coefficients and calculated
before outage of line $k$. Moreover $LODF$ is calculated according to Refs.
[26, 27, 28] by Eq. (13)

$\displaystyle LODF_{\ell,k}=\frac{[PTDF\cdot K]_{\ell,k}}{1-[PTDF\cdot
K]_{\ell,\ell}},\quad\forall\ell,$ (13)

where $LODF_{\ell,k}=-1$ in case of $\ell=k$. If there is an outage in one
line (e.g. line $k$ fails), the power flow in lines after a single line outage
is limited their corresponding thermal capacity which is given by (14)

$\displaystyle-F_{\ell}\leq f_{\ell,t}^{k}\leq
F_{\ell},\quad\forall\ell,\forall t.$ (14)

Additionally, according to (13) and (14), Eq. (2.3) is obtained:

$\displaystyle-
F_{\ell}\leq\sum_{i=2}^{\mathcal{B}}[PTDF_{\ell,i}+LODF_{\ell,k}PTDF_{k,i}]p_{i,t}\leq
F_{\ell},$ $\displaystyle\quad\forall\ell,\forall t.$ (15)

Thus, for a fully secured $\mathcal{N}-1$ network, both Eqs. (10) and (2.3)
must hold. This adds $2\mathcal{N}(\mathcal{N}-1)$ constraints from Eq. (2.3)
to the $2\mathcal{N}$ constraints from Eq. (10) ($\mathcal{N}-0$ secure
network) at each time snapshot. In this way, at time $t$, the fully secured
$\mathcal{N}-1$ network can be represented by an $(\mathcal{B}-1)$–dimensional
polytope $Q^{\mathcal{N}-1}$ consisting of $2\mathcal{N}^{2}$ constraints
which is often computationally infeasible for large networks that consider a
wide range of operating conditions. Thus, an approximation approach is needed
to model $\mathcal{N}-1$ security conditions for large networks. Our proposed
heuristic methods are described in the following section.

##  3 Methodology

In our proposed methodology, we approximate the shape of the $\mathcal{N}-1$
secure polytope $Q^{\mathcal{N}-1}$ by taking the $\mathcal{N}-0$ secure
polytope $Q^{\mathcal{N}-0}$ and introducing buffer capacity factors for each
line to constrain it further. In this approach, we modify Eq. (9) to:

$\displaystyle-c_{\ell}F_{\ell}\leq f_{\ell,t}\leq
c_{\ell}F_{\ell},\quad\forall\ell,\forall t$ (16)

where $c_{\ell}$ represents buffer capacity factor for line $\ell$ which is
between $0$ and $1$, and it expresses the share of line $\ell$ nominal
capacity that can be used. Thus, we define a new $(\mathcal{B}-1)$–dimensional
polytope $Q^{\textrm{heuristic}}$ with at most $2\mathcal{N}$ faces in the
space of possible nodal net powers. By definition, it is a subset of the
$\mathcal{N}-0$ secure polytope $Q^{\textrm{heuristic}}\subset
Q^{\mathcal{N}-0}$ as long as $c_{\ell}\leq 1$ for all lines.

###  3.1 Proposed heuristic approaches

We present three different approaches – consisting of robust, approximate, and
line-specific – to choose the buffer capacity factors to get the best match
between the polytope $Q^{\textrm{heuristic}}$ and the $\mathcal{N}-1$ polytope
($Q^{\mathcal{N}-1}$). While the same buffer capacity factor is allocated for
all lines in robust and approximate approaches, the specific amount of buffer
capacity factors are determined for each line in the line-specific approach.
Additionally, while $Q^{\textrm{heuristic}}$ is entirely inside the
$Q^{\mathcal{N}-1}$ in the robust and line-specific approaches,
$Q^{\textrm{heuristic}}$ is not entirely inside the $Q^{\mathcal{N}-1}$ in the
approximate approach and the volume of $Q^{\textrm{heuristic}}$ is equal to
the volume of the $Q^{\mathcal{N}-1}$ in the approximate one. For contracting
$Q^{\mathcal{N}-1}$, firstly, we must shape $Q^{\mathcal{N}-0}$ and polytope
of each single line failure scenario of the network. In this way,
$Q^{\mathcal{N}-\ell_{1}}$ and $Q^{\mathcal{N}-\ell_{n}}$ present the
polytopes if only lines $\ell_{1}$ and $\ell_{n}$ are failed, respectively.
Hence, $Q^{\mathcal{N}-1}$ represents the common space between
$Q^{\mathcal{N}-0}$ and polytope of all single line failure scenario

$\displaystyle
Q^{\mathcal{N}-1}=Q^{\mathcal{N}-0}\cap_{i}Q^{\mathcal{N}-\ell_{i}}$ (17)

####  3.1.1 Robust approach

The robust approach is the most conservative one. In this approach, the buffer
capacity factor is the same for all lines $c_{\ell}=c^{R}$, and $c^{R}$ is
chosen so that the resulting polytope $Q^{R}$ is entirely inside the polytope
of the fully secured $\mathcal{N}-1$ network, $Q^{R}\subseteq
Q^{\mathcal{N}-1}$, as shown in Fig. 1. This guarantees that any nodal
imbalance inside $Q^{R}$ will always be $\mathcal{N}-1$ secure. However, it
may be over-constrained such that there are more cost-effective dispatch
solutions outside $Q^{R}$ that are still $\mathcal{N}-1$ secure, i.e. there
may be better solutions in the set $Q^{\mathcal{N}-1}-Q^{R}$.

####  3.1.2 Approximate approach

This approach also has the same buffer capacity factor for every line,
$c_{\ell}=c^{A}$, but now $c^{A}$ is chosen such that the resulting polytope
$Q^{A}$ has the same volume $V^{A}$ as the fully secured $\mathcal{N}-1$
polytope $Q^{\mathcal{N}-1}$, i.e. $V^{A}=V^{\mathcal{N}-1}$ which is less
constraining than the robust approach where $V^{R}\leq V^{\mathcal{N}-1}$.
This means the feasible space is the same size as the $\mathcal{N}-1$
polytope, in the hope that the economic dispatch is better approximated by
having a larger feasible space than the conservative robust approach. Now,
like the robust case, we may still have $\mathcal{N}-1$ secure solutions that
are outside $Q^{A}$, i.e. the set $Q^{\mathcal{N}-1}-Q^{A}$ is non-trivial,
but we may also have solutions in $Q^{A}$ that are now no longer
$\mathcal{N}-1$ secure, i.e. the set $Q^{A}-Q^{\mathcal{N}-1}$ is non-trivial.
Since $V^{A}=V^{\mathcal{N}-1}$, we have that the volume of these two non-
trivial spaces is equal as represented by Eq. 18. The step-by-step
implementation diagram of our proposed robust and approximate approaches are
illustrated in Fig. 1.

$\displaystyle V(Q^{\mathcal{N}-1}-Q^{A})=V(Q^{A}-Q^{\mathcal{N}-1})$ (18)

Figure 1: Step-by-step diagram of our proposed robust and approximate
approaches to find buffer capacity factor for transmission lines.

####  3.1.3 Line-specific approach

This approach builds on the robust approach by using the freedom to set the
individual buffer capacity factor for each line, $c_{\ell}$, in a way that
keeps the overall polytope be inside $Q^{\mathcal{N}-1}$ while increasing the
volume of the polytope. Thus, the maximum power transfer capacity for each
line is determined based on the redundant capacity factor, $t_{\ell}$,
according to the Edmond-Karp algorithm [29, 30]. This way, if $t_{\ell}>1$, it
represents that power flow above the nominal capacity of line $\ell$ does not
cause overloading in the rest of the lines in the network in the case of a
single line outage of line $\ell$. The redundant capacity factor $t_{\ell}$
for line $\ell$ is calculated based on the LOPF problem in the network after
(i) removing line $\ell$ with start node $i$ and end node $j$, (ii) adding
power generation at node $i$ and load at node $j$ until one line becomes
overloaded in the transmission network, and vice versa. Based on $t_{\ell}$
and $Q^{\mathcal{N}-1}$, $c_{\ell}$ is determined iteratively as presented in
Algorithm 1. The algorithm starts with the robust polytope, then increases
$c_{\ell}$ for each line based on its corresponding $t_{\ell}$, then
incrementally reduces $c_{\ell}$ until the resulting polytope is robust, i.e.
within $Q^{S}\subseteq Q^{\mathcal{N}-1}$.

1: Determine robust buffer capacity factor ($c^{R}$)

2: Determine redundant capacity factor for each line ($t_{\ell}$)

3: Update line capacity ($F_{\ell}^{{}^{\prime}}\leftarrow t_{\ell}F_{\ell}$)

4: Determine $c^{R^{\prime}}$ based on $F_{\ell}^{{}^{\prime}}$

5: Set initial value for line-specific buffer capacity factor
($c_{\ell}\leftarrow c^{R^{\prime}}t_{\ell}$)

6: Update value for line-specific buffer capacity factor

($c_{\ell}\leftarrow\max(c^{R},\min(c_{\ell},1))$)

7: for $l\leftarrow 1$ to $\mathcal{N}$ do

8: while $Q^{\textrm{S}}\not\subset Q^{\mathcal{N}-1}$ and $c_{\ell}>c^{R}$ do

9: $c_{\ell}\leftarrow c_{\ell}-\delta,\quad\forall\delta\in[0,1]$ (e.g.
$\delta=0.01$)

10: end while

11: end for

Algorithm 1 Proposed algorithm to determine line-specific buffer capacity
factor ($c_{\ell}$).

###  3.2 Finding the volume of the polytope

As highlighted in Section 3.1.2, $c^{A}$ is found based on comparison between
$V^{A}$ and $V^{\mathcal{N}-1}$. To determine the volume of the polytopes
generated from our proposed approximate approach, a set $X$ of random samples
is used, where $X$ is a $\mathcal{B}$–dimensional vector and represents
$p_{i}$ for a single time. In this way, we use a quasi-random low-discrepancy
series, e.g. the Halton sequence [31] to improve the convergence over pseudo-
random sampling. Thus, the space in the bounding box of Eqs. (19) and (20) is
covered more efficiently based on the $\mathcal{N}-1$ polytope defined by Eq.
(2.3).

$\displaystyle\overline{X}$ $\displaystyle=\max\\{X\>|\>AX-b=0\\},$ (19)
$\displaystyle\underline{X}$ $\displaystyle=\min\\{X\>|\>AX-b=0\\},$ (20)

where $A$ and $b$ are the left-hand side and the right-hand side of Eq. (2.3),
respectively. In this way, $b$ is a $\mathcal{N}$– dimensional vector and $A$
is a $\mathcal{N}\times\mathcal{B}$– dimensional matrix. Besides,
$\underline{X}$ and $\overline{X}$ present lower and upper bounds of $X$,
respectively. Finally, samples constrained to (21) are selected for
determining the volume of the polytope.

$\displaystyle AX\leq|b|,$ (21)

where $x_{hit}$ is number of selected samples from $X$ and $x_{miss}$ is total
number of generated samples from $X$ in the bounding box and are not selected.
This way, the volume of the polytope is calculated based on a hit-miss
integration by Eq. (22):

$\displaystyle
V=\frac{(\overline{X}-\underline{X})x_{hit}}{x_{hit}+x_{miss}},$ (22)

where $V$ represents volume of the polytope.

###  3.3 Methodology for large numbers of buses

Our proposed heuristic approaches to find buffer capacity factors are based on
estimating volumes of polytopes. It is noteworthy that estimating volumes of
polytopes does not perform well on large-dimensional spaces because of the
curse of dimensionality. To circumvent this problem, we apply the algorithms
proposed in Section 3.1 on many smaller subsets of the buses separately.
Moreover, a large number of subsets of buses is necessary for accurate
results, which is a drawback for this method. For instance, for a network with
50 buses, we can take a group of $\mathcal{B}_{z}$ buses, e.g. $5$, buses and
explore the $\mathcal{B}_{z}$–dimensional polytope defined by the space of
possible dispatches of these $\mathcal{B}_{z}$ buses, with all other buses
having zero power dispatch. Thus, all line constraints are still active, and
this is equivalent to only examining $\mathcal{B}_{z}$ columns of the matrix
$A$ in equation (21), but keeping all the rows that represent all the line
constraints in the model as represented in (23).

$\begin{array}[]{r@{\,}l}&\begin{matrix}\mspace{95.0mu}\overbrace{\rule{28.45274pt}{0.0pt}}^{\mathcal{Z}}\mspace{35.0mu}\overbrace{\rule{28.45274pt}{0.0pt}}^{\mathcal{Z}}\end{matrix}\\\
A=&\begin{pmatrix}a_{1,1}&\cdots&a_{1,i}&\cdots&a_{1,i^{\prime}}&\cdots&a_{1,\mathcal{B}}\\\
\vdots&&\ddots&&\vdots&&\vdots\\\
a_{\ell,1}&\cdots&a_{\ell,i}&\cdots&a_{\ell,i^{\prime}}&\cdots&a_{\ell,\mathcal{B}}\\\
\vdots&&\vdots&&\ddots&&\vdots\\\
a_{\mathcal{N},1}&\cdots&a_{\mathcal{N},i}&\cdots&a_{\mathcal{N},i^{\prime}}&\cdots&a_{\mathcal{N},\mathcal{B}}\end{pmatrix}\end{array}$
(23)

In this way, the corresponding matrix $A$ for subset $\mathcal{Z}$ is given in
(24).

$A^{S}_{\mathcal{Z}}=\begin{pmatrix}a_{1,i}&\cdots&a_{1,i^{\prime}}\\\
\vdots&\ddots&\vdots\\\
a_{\mathcal{N},i}&\cdots&a_{\mathcal{N},i^{\prime}}\end{pmatrix}$ (24)

Moreover, matrix $A$ is represented as consisting of $A^{S}$ for all subsets.

$A=\begin{pmatrix}A^{S}_{1}&A^{S}_{2}&\cdots&A^{S}_{\mathcal{Z}}\end{pmatrix},\quad\forall
z.$ (25)

Geometrically this is equivalent to taking cross-sections of the polytope
along particular subsets of dimensions while setting the coordinates of other
dimensions to zero. By sampling many such subsets of nodes, corresponding to
different cross-sections of the polytope, we can build information on the
whole polytope. In this way, subsets can be selected by random selections or
by clustering buses that are geographically close to each other using k-means
algorithm [32]. In this way, $c^{R}_{z}$ and $c^{A}_{z}$, which are
represented as robust and approximate buffer capacity factors for cluster $z$,
respectively, are determined according to robust and approximate approaches
described in Section 3.1. According to the definition of $c^{R}$, we have

$\displaystyle c^{R}=\min\\{c_{\ell}\\},\quad\forall\ell.$ (26)

As presented in Eq. (26), the $c^{R}$ for the system must be the minimum of
the line-specific buffer capacity factors of all lines. In this way, if we
split the network to $\mathcal{Z}$ subsets, subset $z$ contains its
corresponding lines and robust buffer capacity factor, $c_{z}^{R}$, which is
the minimum of line-specific buffer capacity factor for lines belonged to
subset $z$. Thus, the $c^{R}$ for the network split to several clusters can be
defined as the minimum of robust buffer capacity of all subsets as represented
in (27).

$\displaystyle c^{R}=\min\\{c_{z}^{R}\\},\quad\forall z.$ (27)

However, for determining $c^{A}$, we need to figure out the relation between
the buffer capacity factor and the volume of $Q^{A}_{z}$ as compare to the
volume of $Q^{\mathcal{N}-1}_{z}$. In this paper, we define a criteria so-
called volume ratio ($\Upsilon_{z}$) for subset $z$, which is given in (28).
Regarding to the definition of the approximate approach in Section 3.1, one of
the conditions for finding $c^{A}$ for the whole polytope is $\Upsilon=1$. In
this way, we propose a criterion which is called the common volume ratio,
$\Psi$, as represented in (29). Thus, $c^{A}$ is chosen to minimise the
differences of the individual volume ratios from unity, see (30).

$\displaystyle\Upsilon_{z}=\frac{V(Q^{A}_{z})}{V(Q^{\mathcal{N}-1}_{z})},\quad\forall
z.$ (28) $\displaystyle\Psi(c)=\sum_{z}(\Upsilon_{z}(c)-1)^{2},\quad\forall
c.$ (29) $\displaystyle c^{A}=c,\hskip 8.5359pt\min_{c}\Psi(c).$ (30)

An alternative method for finding line-specific capacity factors ($c_{\ell}$)
is proposed for large-scale networks based on the method for finding the
robust subset factor ($c^{R}_{z}$). In this way, the network is split to
$\mathcal{N}$ subsets in which buses $i$ and $j$ belong in the same subset if
they connect directly with a transmission line. This way, matrix
$A^{S}_{\ell}$ for the subset of line $\ell$ where is ended to buses $i,j$ as
represented in (31)

$A^{S}_{\ell}=\begin{pmatrix}a_{1,i}&a_{1,j}\\\ \vdots&\vdots\\\
a_{\mathcal{N},i}&a_{\mathcal{N},j}\end{pmatrix},\quad\forall\hskip
2.84544pt|K_{i,\ell}|\cdot|K_{j,\ell}|=1.$ (31)

In other words, each subset consists of only two buses that are connected with
a transmission line and its line-specific buffer capacity factor, $c_{\ell}$,
equals the robust buffer capacity factor for the corresponding subset,
$c^{R}_{z=l}$, as seen in (32)

$\displaystyle c_{\ell}=c^{R}_{z}\hskip 8.5359pt,\quad\forall\hskip
2.84544pt|K_{i,\ell}|\cdot|K_{j,\ell}|=1.$ (32)

Figure 2: 3-bus triangular network (a) $\mathcal{N}-0$ scenario (b)
$\mathcal{N}-1$ scenarios.

(a) $\mathcal{N}-0$ secure network scenario

(b) Line 1 failure scenario

(c) Line 2 failure scenario

(d) Line 3 failure scenario

Figure 3: Polygon of feasible region between $p_{1}$ and $p_{2}$ in a 3-bus
triangular network.

(a)

(b)

(c)

Figure 4: Polygon of feasible region between $p_{1}$ and $p_{2}$: for fully
secured $\mathcal{N}-1$ network (a), based on the robust approach and fully
secured $\mathcal{N}-1$ network (b), based on the approximate approach and
fully secured $\mathcal{N}-1$ network (c) in a 3-bus triangular network.

##  4 Simulation results

In this section, the performance of our proposed approaches to determine
buffer capacity factors are studied. The LOPF problem is simulated in PyPSA
[21]. The simulation results of our proposed approaches are evaluated in
different test grids, including a 3-bus triangular network, some small-scale
network topologies and a 50-bus Germany network as a large-scale network
derived from PyPSA-Eur, an open-source model of the European electricity
transmission system [33].

###  4.1 3-bus network

In this section, the performance of our proposed methodologies is studied in a
3-bus triangular network. Fig. 2a displays the $\mathcal{N}-0$ scenario when
there exist no line outage in the network. The $\mathcal{N}-1$ scenarios are
also illustrated in Fig. 2b where only one line outage has occurred in each of
the scenarios. Assuming bus $1$ is a slack bus, there are two nodal net power
injections ($p_{1}$ and $p_{2}$) in the system. Thus, there exists a
2-dimensional polytope (so-called polygon) [34]. Fig. 3 demonstrates the
feasible region between $p_{1}$ and $p_{2}$ in $\mathcal{N}-0$ and
$\mathcal{N}-1$ scenarios, respectively. According to Fig. 3, feasible region
between $p_{1}$ and $p_{2}$ for the fully secured $\mathcal{N}-1$ network is
illustrated in Fig. 4a. As mentioned in Section 3.1, we propose three
approaches consisting of robust, approximate and line-specific approaches in
this paper. Fig. 4c shows the polygons of feasible regions between $p_{1}$ and
$p_{2}$ based on the approximate approach and fully secured $\mathcal{N}-1$
network in a 3-bus triangular grid. The total volumes of both polygons, as
well as the volumes of mismatched regions, are equal, as shown in Fig. 4c. On
the other hand, the polygons of feasible regions between $p_{1}$ and $p_{2}$
based on the robust approach and fully secured $\mathcal{N}-1$ network are
illustrated in Fig. 4b. The polygon that came from the robust approach is
completely covered by the polygon of fully secured $\mathcal{N}-1$ network,
and this is a reason why this approach is called robust. In other words, the
feasible region of the robust approach is a subset of the feasible of the
fully secured $\mathcal{N}-1$ network, so $c^{R}$ (which is not a line-
dependant factor) guarantees the reliability of the system in case of a
single-line outage in the transmission network.

###  4.2 Small-scale networks

Figure 5: Topology of small-scale networks. Figure 6: Volume-in ($\Gamma$) of
the proposed heuristic approaches in small-scale European countries.

In this section, the performance of the proposed approaches for small-scale
network topologies is evaluated European networks, which is derived from
highly aggregated European transmission network, without cross-border
connections with their neighborhoods as shown in Fig. 5. buffer capacity
factor of each line obtained by the line-specific for small-scale European
networks. Table 1 presents the values of $c^{A}$ and $c^{R}$ for small-scale
networks. As seen in Table 1, $c^{R}$ is less than $c^{A}$ in all networks.
Moreover, $c^{A}$ is around 0.7 for all small-scale networks. However, $c^{R}$
is not around 0.7 for all networks which is because of their specific
topology. Table 1, $c_{\ell}$ for some of transmission lines are higher than
$c^{R}$. Additionally, the relation between the redundant capacity and buffer
capacity factor is represented in Fig. LABEL:fig:cl_tl. Thus, increment of
$t_{\ell}$ is in line with the increment of $c_{\ell}$. In other words, the
lines with higher redundant capacity could have higher buffer capacity factor
as seen in Fig. LABEL:fig:cl_tl. Besides, the volume-in criteria ($\Gamma$),
which is defined in Eq. (33), obtained by approximate, robust and line-
specific approaches, is displayed in Fig. 6.

$\displaystyle\Gamma=\frac{V(Q^{heuristic}\cap
Q^{\mathcal{N}-1})}{V(Q^{\mathcal{N}-1})}$ (33)

According to Fig. 6, $\Gamma$ is maximum in the approximate approach.
Moreover, $\Gamma$ of the line-specific approach is higher than the robust one
presenting the effectiveness of the line-specific approach. However, note that
even if the volume-in is high for the approximate approach, there will be
parts of $Q^{A}$ outside $Q^{\mathcal{N}-1}$, making some solutions infeasible
for $\mathcal{N}-1$ .

Table 1:  Robust and approximate buffer capacity factors for small-scale networks. Network | $c^{A}$ | $c^{R}$ | Network | $c^{A}$ | $c^{R}$
---|---|---|---|---|---
(a) | 0.66 | 0.59 | (d) | 0.71 | 0.62
(b) | 0.84 | 0.74 | (e) | 0.8 | 0.78
(c) | 0.72 | 0.67 | (f) | 0.7 | 0.55

###  4.3 Large-scale network: German transmission system

(a)

(b)

Figure 7: German transmission network case study (a) map of German
transmission network with 50 buses in which green and red lines show HVDC and
HVAC transmission lines, respectively, and (b) yearly energy generation by
energy carriers in German transmission network.

In this section, the buffer capacity factor for the Germany network as a
large-scale case study is found by our proposed methodology for a large number
of buses as presented in Section 3.3. Fig. 7(a) displays 50-bus German
transmission. Besides, Fig. 7(b) shows the total annual energy generation of
energy carriers in the German transmission network. Additionally, a $90\%$
carbon emission reduction scenario is considered in this paper so that
$\mathcal{E}_{CO_{2}}$ is just 10% of the electricity sector emissions in
Germany in 1990. Table 2 presents buffer capacity factors for different
subsets of the 50-bus Germany network based on the proposed clustering
algorithm. The subsets represented in Table 2 are obtained by the k-means
clustering algorithm and based on geographic proximity. As seen in Table 2, we
assume 13 subsets for the 50-bus Germany network and determine $c^{R}$ and
$c^{A}$ for each subset.

As seen in Table 2, $c^{R}$ for the 50-bus Germany network is $0.59$ which is
obtained based on Eq. (26) which is the minimum of robust buffer capacity
factor for each subset. On the other hand, Fig. 8(a) shows the impact of
buffer capacity factor on $\Upsilon$ in all subsets of the Germany network. As
in k-means clustering, each bus belongs only to one cluster, we miss lines
that cross clusters. In other words, the results came from k-means clustering
do not guarantee $\mathcal{N}-1$ security of our proposed robust approach.
Thus, the robust buffer capacity factor is found by 100 random samples to
cover all transmission lines in the transmission network, and $c^{R}$ is found
$0.59$ which confirms outcomes presented by Table 2.

(a)

(b)

Figure 8: Impact of buffer capacity factors based on k-means clustering on (a) volume ratio in Germany network (b) common volume ratio criteria in Germany network. Table 2:  Robust and approximate buffer capacity factors based on k-means clustering for clusters of the 50-bus Germany network. Cluster | $c^{R}$ | $c^{A}$ | Cluster | $c^{R}$ | $c^{A}$
---|---|---|---|---|---
0 | 0.62 | 0.77 | 7 | 0.66 | 0.75
1 | 0.67 | 0.76 | 8 | 0.66 | 0.77
2 | 0.73 | 0.81 | 9 | 0.65 | 0.73
3 | 0.8 | 0.81 | 10 | 0.71 | 0.83
4 | 0.71 | 0.75 | 11 | 0.78 | 0.85
5 | 0.59 | 0.68 | 12 | 0.61 | 0.73
6 | 0.77 | 0.83 | Total | 0.59 | 0.74

As seen in Fig. 8(a), increment of the buffer capacity factor increases
$\Upsilon$ for all subsets. However, their corresponding increment patterns
are not the same. Moreover, $c^{A}$ for each subset is different as presented
in Table 2. As shown in Fig. 8(b) and according to Eq. (30), there exists a
minimum in the common volume criteria curve with buffer capacity equals $0.74$
as presented in Table 2. Fig. 9 illustrates the map of line-specific buffer
capacity factors in Germany network which is calculated based on Eq. (32).
According to Fig. 9, $c_{\ell}$ is in range of $0.59$ and $1$ for each line of
the transmission network where lines with higher $c_{\ell}$ are more secure.
Additionally, Fig. 10 shows the impact of buffer capacity factor on total
number of secure transmission lines. As seen in Fig. 10, for $c\leq 0.59$ all
transmission lines are secured which proves $c^{R}$ equals $0.59$ for the
German transmission network.

Figure 9: Line-specific buffer capacity factor in Germany network. Figure 10:
Total number of secure transmission lines with different buffer capacity
factors in Germany network.

Additionally, the economic impact of the different buffer capacity factor
approaches is studied in the German transmission network. In this analysis,
operations are optimized for a full year of weather and load data for the 50
buses of the Germany network, assuming a generation fleet that yields a 90%
CO2 reduction compared to 1990. This leads to a $78\%$ penetration of wind and
solar power generation based on Fig. 7(b). Fig. 11(a) illustrates the impact
of the buffer capacity factor on the yearly operational cost in Eq. (1), in
the 50-bus Germany network. According to Fig. 11(a), increasing the amount of
the buffer capacity factor decreases the operational cost of the system.
Moreover, the operational cost of the system in the fully secured
$\mathcal{N}-1$ network is equal to the operational cost of the system with
buffer capacity in the range of $[0.75,0.8]$ as seen in Fig. 11(a). As the
approximate buffer capacity factor, $c^{A}$, is found to be $0.74$ in Section
4.3, it can be concluded that the operational costs of the system in the fully
secured $\mathcal{N}-1$ network is close to the operational cost of the system
with $c^{A}$ which confirms the intuitions from the industry that a buffer
capacity factor of $0.7$ is suitable [20]. It is also noticeable that the
close operational system costs do not necessarily imply that the operational
results are the same. Moreover, Fig. 11(a) illustrates that the operational
cost based on our proposed heuristic approach with the line-specific approach
is higher than the operational cost in the fully secured $\mathcal{N}-1$ case
and less than the proposed robust approach. Furthermore, it has been found
that operational costs of the system for a fully secured $\mathcal{N}-1$,
approximate $\mathcal{N}-1$ with line-specific buffer capacity factors, and
approximate $\mathcal{N}-1$ with approximate buffer capacity factor are very
close. On the other hand, it is noteworthy that the approximate buffer
capacity does not guarantee security of all transmission lines, whereas all
lines are secured based on line-specific buffer capacity factor. Hence, the
line-specific approach is the best approach among our proposed approaches to
approximate the fully secured $\mathcal{N}-1$ conditions in the network, while
using uniform approximate buffer capacity factor can only provide
approximation of operational costs of the fully secured $\mathcal{N}-1$.

(a)

(b)

Figure 11: Impact of buffer capacity factor on (a) system annual operational cost in Germany network (b) total snapshots with overloading and overloaded lines in Germany network. Table 3:  Computational time and memory usage of $\mathcal{N}-0$ problem with uniform/line-specific buffer capacity factors and $\mathcal{N}-1$ fully secured problem. | Computational | Memory
---|---|---
| time [Sec] | usage [MB]
$\mathcal{N}-0$ with uniform |  $\approx$ 270 |  $\approx$ 860
buffer capacity factor |  |
$\mathcal{N}-0$ with line-specific | 326 | 860
buffer capacity factor |  |
$\mathcal{N}-1$ fully secured | 3786 | 5582

According to Fig. 11(a), the robust case that guarantees no overloading is
$4\%$ more expensive than the fully secured $\mathcal{N}-1$ case since its
feasible space is smaller. Moreover, the operational cost of the line-specific
approach is $2\%$ lower than the robust one. Thus, while line-specific and
robust approaches guarantee no overloading caused by a single-line outage in
the network, the operational cost of the line-specific case is less than the
robust one showing the benefit of the line-specific approach. Table 3 presents
the computational time and memory usage of our proposed heuristic approaches
with the uniform (approximate and robust) and line-specific buffer capacity
factors in comparison with the fully secured $\mathcal{N}-1$ problem. As seen
in Table 3, the computational time and memory usage in our proposed method is
significantly less than the fully secured case, which is because of fewer
constraints in our proposed approximate $\mathcal{N}-1$ case. Moreover, Fig.
11(b) shows total cases and snapshots where the transmission network is
overloaded in at least one line for different buffer capacity factors. As
shown in Fig. 11(b), there exists no overloading in the system for the buffer
capacity factor less than $0.59$ (as $c^{R}$ has been obtained $0.59$
according to Fig. 10 and Table 2). In other words, Fig. 11(b) proves that the
transmission network with a robust buffer capacity factor is guaranteed to be
fully secured, i.e. no transmission line is ever overloaded.

##  5 Conclusion

In this paper, we have introduced a novel heuristic method for modeling the
fully secured network considering $\mathcal{N}-1$ contingencies based on the
topology of the transmission network in the security-constrained linearized AC
optimal power flow problem. To this end, the network’s secure feasible space
has been determined by comparing the polytopes of the feasible region among
nodal net power. Additionally, secure feasible regions have then been
approximated using far fewer constraints by using the $\mathcal{N}-0$
constraints with buffer capacity factors that have been obtained by line-
specific, robust and approximate approaches to constrain the maximum loading
of lines. The algorithms have been applied on smaller subsets and clusters of
the buses to find line-specific, robust and approximate buffer capacity
factors for networks with many buses. According to the simulations results of
the test networks, it has been found that:

  * 1.

The industry heuristic of $0.7$ buffer capacity is roughly suitable for the
Germany network to provide fast results (since there are fewer constraints)
while not producing significant overloading, and giving a system cost similar
to the actual $\mathcal{N}-1$ case.

  * 2.

For absolute robustness while having fast results, you need to shrink the
feasible space smaller than the true $\mathcal{N}-1$ polytope, which leads to
overestimation of the system cost. This can be mitigated by a novel line-
specific buffer capacity, which better estimates the shape of the
$\mathcal{N}-1$ polytope while remaining robust.

  * 3.

These speed-ups are particularly important when we include investment
optimisation, which tends to increase the computational burden. Including
fully secured $\mathcal{N}-1$ constraints while optimising over a full year to
dimension generation, storage, and transmission assets would be impossible, so
these heuristics provide a good alternative that is still accurate.

In the future, maintaining $\mathcal{N}-1$ security preventatively can be
superseded with methods that use distributed flexibility to deal with
contingencies like line failures reactively as they arise, thus allowing
network assets to be fully used in regular operation.

## Acknowledgement

This paper was conducted as part of the CoNDyNet2 project, which was supported
by the German Federal Ministry of Education and Research under grant number
03EK3055E.

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