# Hosotani mechanism in the “color”-singlet plasma

Nahomi Kan<EMAIL_ADDRESS>National Institute of Technology, Gifu College,
Motosu-shi, Gifu 501-0495, Japan  Koichiro Kobayashi<EMAIL_ADDRESS>National Institute of Technology, Ohshima College, Suooshima-cho, Yamaguchi
742-2193, Japan  Kiyoshi Shiraishi<EMAIL_ADDRESS>Faculty of
Science, Yamaguchi University, Yamaguchi-shi, Yamaguchi 753–8512, Japan

###### Abstract

By projecting the partition function on the “color”-singlet state, we
investigate the Hosotani mechanism in the fermion-gauge boson plasma. The
present toy-model analysis of the one-loop effective potential at finite
temperature shows that the critical temperature of gauge symmetry breaking
increases at higher temperature in the smaller volume.

##  I Introduction

Phase transition phenomena in finite temperature systems are found in various
aspects of physics research. In the unified theory of particle physics, the
analysis of finite-temperature field theory suggests DJ  that the broken
symmetry by the Brout–Englert–Higgs (BEH) mechanism was restored in the hot
early universe. In other words, it can be said that the division of the
present fundamental interactions is the result of the phase transition in the
early universe.

The theory of the fundamental scalar field, which is indispensable for the BEH
mechanism, is unnatural in the sense of suffering huge quantum corrections.
Accordingly, the gauge-Higgs unified model has been proposed as a candidate
giving the solutions to the naturalness gh1 ; gh2 ; gh3 ; gh4 ; gh5 ; gh6 ;
gh7 . In the model, the extra-dimensional component of the gauge field plays
the role of the Higgs field. Thus, there is no potential term that corresponds
to the Higgs potential at the tree level in the theory; this is due to the
gauge symmetry, which guarantees naturalness of the theory. Therefore, in the
gauge-Higgs unified model, it is expected that the quantum effect at one (or
more) loop order will bring about a non-trivial vacuum gauge field. This
mechanism is dubbed as the Hosotani mechanism (or Wilson loop mechanism)
Hosotani . It is known that fermionic matter fields with certain suitable
representations are needed for such symmetry breaking at zero temperature. The
characteristics of the Hosotani mechanism at finite temperature have already
been studied Shiraishi ; HH1 ; MT . It has been pointed out that phase
transitions at high temperature always occur HH1  (and the transition is first
order for matter fields with the periodic boundary condition in the extra
dimension) in the Hosotani mechanism.

There is another type of phase transition in non-Abelian gauge theory. It is
the quark-hadron phase transition, which is known as hadronization from quark-
gluon plasma (QGP) Muller . Although the similar confinement is not assumed in
the gauge-Higgs scenario, non-perturbative effects are expected in the early
phase of the hot universe.111In a different context, the idea of vacuum
selection at finite temperature in supersymmetric grand unified theories,
which is attributed to the difference in degree of freedom of particles, is
discussed by the authors of the papers on “supercosmology” NT ; ELR ; NOT ;
CEHNO .  It is also well known that the AdS/CFT method illustrates the
reduction of the degrees of freedom in QGP in the non-perturbative region BCM
in a certain non-Abelian gauge theory.

Being motivated by the reduction in the number of degrees of freedom in QGP,
we revisit the general behavior of Hosotani mechanism at finite temperature
through a toy model in this paper. We adopt the color-singlet hypothesis in
the present analysis.222The projection onto the singlet sector is also
utilized in recent studies on large $N$ Yang–Mills theories at finite
temperature Sundborg ; AMMPR . This is a hypothesis that can be considered in
the hadron phase transition from quark-gluon plasma, and it is a hypothesis
that QGP whitens globally owing to the nature of the strong interaction. There
are some debates as to whether this holds true in the analysis of actual QGPs,
etc. However, since it is followed by a well-defined mathematical operation
that expresses the reduction of degrees of freedom using the group integration
technique, it is considered to be a powerful analytical method that can
approximate the aimed aspect of the effective behavior at least
“phenomenologically”.

In the following sections of this paper, we use a toy model with $SU(2)$
symmetry (instead of a large group of unified models) to find a one-loop
effective potential at finite temperature under the hypothesis of the color
singlet.

This paper is organized as follows. In Sec. II, we review the color-singlet
hypothesis and associated technique in analytical formulations. In Sec. III,
we define the effective potential for the toy model of $SU(2)$ gauge theory in
the background space $R^{D-2}\otimes S^{1}$. Numerical calculations of the
effective potential in five dimensions ($D=5$) are shown in Sec. IV and the
possible phase transition is studied. The discussion is presented in the last
section. In Appendix, an interesting and useful analysis of the effective
potential with approximations is examined.

##  II The projected partition function for the color-singlet states

In this section, we review the argument of the global color symmetry in QGP
under the color-singlet hypothesis. Using the technique shown in the present
section, we will obtain the effective potential in a toy model for the
Hosotani mechanism in the “color”-neutral plasma in the next section.

The QCD is known as a theory describing the strong force and causes the
confinement of quarks and gluons. The transition from hadronic matter to
quark-gluon plasma is considered to be a transition from local color
confinement to global color confinement at finite temperature Muller . We
should consider the restricted partition function of the color-singlet state
to realize the global color symmetry Muller ; EGR ; RT ; Turko ; EG1 ; EG2 ;
GLPZ ; GMPZ . To this end, first we define a generalized partition function
which includes the generators of the Cartan subalgebra $\hat{C}_{\alpha}$ in
the gauge group as follows:

$Z_{C}(\psi_{\alpha})={\rm
Tr\,}[e^{-\beta\hat{H}+i\psi_{\alpha}\hat{C}_{\alpha}}]\,,$ (1)

where, as usual, $\beta$ is the inverse of the temperature $T$ and $\hat{H}$
denotes the Hamiltonian.

Next, we introduce the characteristic functions. The function of the parameter
$\psi_{\alpha}$ specified by the representation $j$ of the group ($SU(3)$ for
QCD) is called as the characteristic function $\chi_{j}(\psi_{\alpha})$. The
characteristic functions satisfy

$\int
d\mu(\psi_{\alpha})\chi^{*}_{j}(\psi_{\alpha})\chi_{j^{\prime}}(\psi_{\alpha})=\delta_{jj^{\prime}}\,,$
(2)

where $d\mu(\psi_{\alpha})$ is the invariant measure of the group. For
$SU(2)$, i.e., the two-color case, it is known that

$d\mu(\psi)=\frac{\sin^{2}\frac{\psi}{2}}{2\pi}d\psi\quad(-2\pi\leq\psi<2\pi)\,,$
(3)

where $\psi$ is the single variable for the center of $SU(2)$.

We assume that the generalized partition function can be expanded by the
characteristic functions and the characteristic function for the singlet is
known to be unity. Thus, the restricted partition function $Z$ is finally
obtained by projection onto the color singlet as

$Z=\int d\mu(\psi_{\alpha})Z_{C}(\psi_{\alpha})\,.$ (4)

Here, we notice that the Jacobi’s imaginary transformation AS

$\frac{1}{\sqrt{4\pi
t}}\sum_{n=-\infty}^{\infty}e^{-\beta^{2}n^{2}/(4t)}e^{i2\pi
ny}=\frac{1}{\beta}\sum_{n=-\infty}^{\infty}e^{-[\frac{2\pi}{\beta}(n+y)]^{2}t}$
(5)

and the formula in Ref. DJ , we find that the equalities

$\displaystyle\sum_{n}\ln\left[\frac{4\pi^{2}(n+y)^{2}}{\beta^{2}}+\omega^{2}\right]$
(6) $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}\frac{dt}{t}\sum_{n}\exp\left[-\left(\frac{4\pi^{2}(n+y)^{2}}{\beta^{2}}+\omega^{2}\right)t\right]=-\frac{\beta}{\sqrt{4\pi}}\int_{0}^{\infty}\frac{dt}{t^{3/2}}\sum_{n}e^{i2\pi
ny}\exp\left[-\omega^{2}t-\frac{\beta^{2}n^{2}}{4t}\right]$ $\displaystyle=$
$\displaystyle-\frac{\beta\omega\Gamma(-\frac{1}{2})}{\sqrt{4\pi}}-\frac{2\beta\omega}{\sqrt{4\pi}}\sum_{n=1}^{\infty}2e^{i2\pi
ny}\left(\frac{2}{n\beta\omega}\right)^{1/2}K_{1/2}(n\beta\omega)=\beta\omega-\sum_{n=1}^{\infty}\frac{2}{n}e^{-n\beta\omega}e^{i2\pi
ny}$ $\displaystyle=$ $\displaystyle\beta\omega+2\ln(1-e^{-\beta\omega+i2\pi
y})$

hold, up to the terms independent of an arbitrary constant $\omega$.
Therefore, we can express $\ln Z_{C}(\psi_{\alpha})$ as

$\ln Z_{C}(\psi_{\alpha})=-\frac{1}{2}{\rm Tr\,}\sum_{n}\ln\left[\frac{(2\pi
n+\psi_{\alpha}\hat{C}_{\alpha})^{2}}{\beta^{2}}+\omega^{2}\right]\,,$ (7)

for bosonic fields and

$\ln Z_{C}(\psi_{\alpha})=\frac{1}{2}{\rm
Tr\,}\sum_{n}\ln\left[\frac{[2\pi(n+\frac{1}{2})+\psi_{\alpha}\hat{C}_{\alpha}]^{2}}{\beta^{2}}+\omega^{2}\right]\,,$
(8)

for fermionic fields, where each trace indicates the sum over possible energy
eigenvalue, $\omega$ and all degrees of freedom. Precisely speaking, the
expressions for the generalized partition function here involves the
contribution of the vacuum energy. Incidentally, the expression is convenient
for evaluation of the effective potential for the Hosotani mechanism.

##  III The $SU(2)$ toy model

Here, we consider the Hosotani mechanism at finite temperature in the $SU(2)$
gauge theory with massless fermions in the adjoint representation. We consider
$D$-dimensional spacetime and assume that the topology of space is
$R^{D-2}\otimes S^{1}$ and the circumference of the compact dimension is set
to $L$. All the fields obey the periodic boundary condition with respect to
the compact dimension.333Although various bizarre boundary conditions are
investigated by many papers including Ref. ST , we take the simplest condition
in the present paper.

Although the several conditions towards asymptotic freedom etc. may be a
necessary condition for strong non-perturbative effects implicitly assumed, we
will temporally ignore the conditions in the present toy model.

The vacuum expectation value of the extra dimensional component of the $SU(2)$
gauge field $\mathbf{A}_{y}$ is now parametrized as

$gL\langle\mathbf{A}_{y}\rangle=\frac{\theta}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\,,$ (9)

where $g$ is the $SU(2)$ gauge coupling. Note that the trivial vacuum is
associated with $\theta=0$, where the gauge invariant Wilson loop over $S^{1}$
becomes $\exp igL\langle\mathbf{A}_{y}\rangle=I$, where $I$ is the identity
matrix. The residual large gauge symmetry tells the identification
$\theta\sim\theta+4\pi$. Moreover, when $\theta=2\pi$, it gives $\exp
igL\langle\mathbf{A}_{y}\rangle=-I$; then the $SU(2)$ symmetry is unbroken.
This is due to the $Z_{2}$ symmetry in $SU(2)$ and consequently we expect that
the partition function is periodic in $\theta$ with a period $2\pi$.

The restricted partition function after projections is written as Muller ; EGR
; RT ; Turko ; EG1 ; EG2 ; GLPZ ; GMPZ

$Z(\theta)={\int_{-2\pi}^{2\pi}\frac{d\psi}{2\pi}\sin^{2}\frac{\psi}{2}}\,\,{Z_{G}(\theta,\psi)}\left[Z_{a}(\theta,\psi)\right]^{N_{a}}\,,$
(10)

where $N_{a}$ is the number of the adjoint fermion fields. In this expression,
the logarithm of the generalized partition function $Z_{G}$ for the gauge
bosons and the ghosts is formally given by

$\ln
Z_{G}(\theta,\psi)=-\frac{D-2}{2}V\sum_{A=1}^{3}\sum_{n}\sum_{k}\int\frac{d^{D-2}\mathbf{p}}{(2\pi)^{D-2}}\ln\left[\frac{(2\pi
n+C_{A}\psi)^{2}}{\beta^{2}}+\mathbf{p}^{2}+M_{A}^{2}(k)\right]\,,$ (11)

where

$C_{1}=+1\,,\quad C_{2}=-1\,,\quad C_{3}=0\,,\quad M_{A}^{2}=\left(\frac{2\pi
k+C_{A}\theta}{L}\right)^{2}\,,$ (12)

and the generalized partition function for the adjoint fermions are written by

$\displaystyle\ln Z_{a}(\theta,\psi)$
$\displaystyle=\frac{2^{[D/2]}}{4}V\sum_{A=1}^{3}\sum_{n}\sum_{k}\int\frac{d^{D-2}\mathbf{p}}{(2\pi)^{D-2}}\ln\left[\frac{[2\pi(n+\frac{1}{2})+C_{A}\psi]^{2}}{\beta^{2}}+\mathbf{p}^{2}+M_{A}^{2}(k)\right]\,,$
(13)

where $[D/2]$ is the Gauss’ symbol such that $[4/2]=[5/2]=2$.

In these expressions, the constant $V$ denotes the $(D-2)$-dimensional volume
of the hot plasma system. The definition of the volume of the system is a
crucial problem. We can consider the small objects, such as false vacuum
bubbles, or the cores of the exotic stars, or the fermion droplets in the
universe.

We should recall that the above expressions include vacuum contributions and
should be regularized by getting rid of divergences, which are irrelevant to
physical quantities. With the aid of Jacobi’s imaginary transformation AS

$\sum_{k}\exp\left[-\left(\frac{2\pi
k+\Theta}{L}\right)^{2}t\right]=\frac{L}{\sqrt{4\pi
t}}\sum_{k}\exp\left[-\frac{L^{2}k^{2}}{4t}\right]e^{-ik\Theta}\,,$ (14)

we find that the similar manipulation as in Sec. II leads to

$\ln Z_{G}(\theta,\psi)=\frac{(D-2)\Gamma\left(D/2\right)}{2\pi^{D/2}}\beta
LV{\sum_{n,k}}^{\prime}\frac{1+2\cos\left[k\theta+n\psi\right]}{\left[\beta^{2}n^{2}+L^{2}k^{2}\right]^{D/2}}\,,$
(15) $\ln
Z_{a}(\theta,\psi)=\frac{2^{[D/2]}\Gamma\left(D/2\right)}{2\pi^{D/2}}\beta
LV{\sum_{n,k}}^{\prime}\frac{(-1)^{n-1}\left\\{1+2\cos\left[k\theta+n\psi\right]\right\\}}{\left[\beta^{2}n^{2}+L^{2}k^{2}\right]^{D/2}}\,,$
(16)

where the primes on sums indicate the omission of $n=k=0$ in the summations.
By this omission, the expressions have become finite and the regularization
has been accomplished.

Each partition function can be divided into two parts, say,

$\displaystyle\ln Z_{G}(\theta,\psi)$ $\displaystyle=$ $\displaystyle\ln
Z_{G0}(\theta)+\ln Z_{GT}(\theta,\psi)\,,$ $\displaystyle\ln
Z_{a}(\theta,\psi)$ $\displaystyle=$ $\displaystyle\ln Z_{a0}(\theta)+\ln
Z_{aT}(\theta,\psi)\,\,.$ (17)

Here, contributions of vacuum fluctuations are given, with the use of the
Riemann’s zeta function $\zeta_{R}(z)$ and the $D$-th polylogarithm function
$\mbox{Li}_{D}(z)$, by

$\displaystyle\ln Z_{G0}(\theta)$ $\displaystyle=$
$\displaystyle\frac{(D-2)\Gamma\left(D/2\right)}{\pi^{D/2}L^{D}}\beta
LV{\sum_{k=1}^{\infty}}\frac{1+2\cos k\theta}{k^{D}}$ (18) $\displaystyle=$
$\displaystyle\frac{(D-2)\Gamma\left(D/2\right)}{\pi^{D/2}L^{D}}\beta
LV\left[\zeta_{R}(D)+\mbox{Li}_{D}(e^{ik\theta})+\mbox{Li}_{D}(e^{-ik\theta})\right]\,,$
$\displaystyle\ln Z_{a0}(\theta)$ $\displaystyle=$
$\displaystyle-\frac{2^{[D/2]}\Gamma\left(D/2\right)}{\pi^{D/2}L^{D}}\beta
LV{\sum_{k=1}^{\infty}}\frac{1+2\cos k\theta}{k^{D}}$ (19) $\displaystyle=$
$\displaystyle-\frac{2^{[D/2]}\Gamma\left(D/2\right)}{\pi^{D/2}L^{D}}\beta
LV\left[\zeta_{R}(D)+\mbox{Li}_{D}(e^{ik\theta})+\mbox{Li}_{D}(e^{-ik\theta})\right]\,,$

while the finite-temperature parts are written by

$\displaystyle\ln Z_{GT}(\theta,\psi)$ $\displaystyle=$
$\displaystyle(D-2)\beta LV\left[w_{b}(0,0)+2w_{b}(\theta,\psi)\right]$
$\displaystyle\ln Z_{aT}(\theta,\psi)$ $\displaystyle=$ $\displaystyle
2^{[D/2]}\beta LV\left[w_{f}(0,0)+2w_{f}(\theta,\psi)\right]\,,$ (20)

where

$w_{b}(\theta,\psi)\equiv\frac{\Gamma(D/2)}{\pi^{D/2}}\sum_{n=1}^{\infty}\sum_{k=-\infty}^{\infty}\frac{\cos
k\theta\cos n\psi}{\left[\beta^{2}n^{2}+L^{2}k^{2}\right]^{D/2}}\,,$ (21)

and

$w_{f}(\theta,\psi)\equiv\frac{\Gamma(D/2)}{\pi^{D/2}}\sum_{n=1}^{\infty}(-1)^{n-1}\sum_{k=-\infty}^{\infty}\frac{\cos
k\theta\cos n\psi}{\left[\beta^{2}n^{2}+L^{2}k^{2}\right]^{D/2}}\,.$ (22)

Incidentally, $w_{b}$ and $w_{f}$ coincide with the effective potential for
the $SU(2)$ gauge theory with torus ($T^{2}$) compactification with
appropriate boundary conditions HH2 . In our case, however, the parameter
$\psi$ is to be integrated for the “color”-singlet projection.

The partition function restricted to $SU(2)$ “color”-singlet states for the
present model is then given by

$Z(\theta)=Z_{G0}(\theta)[Z_{a0}(\theta)]^{N_{a}}\times\frac{1}{2\pi}\int_{-2\pi}^{2\pi}d\psi\,\Bigl{[}\sin^{2}{\frac{\psi}{2}}\Bigr{]}\,Z_{GT}(\theta,\psi)[Z_{aT}(\theta,\psi)]^{N_{a}}\,.$

Now, the effective potential ${\cal V}(\theta)$ is defined by

${\cal V}(\theta)\equiv-\frac{1}{\beta VL}\ln Z(\theta)\,.$ (23)

One can find that the effective potential at zero temperature becomes

${\cal V}_{0}(\theta)\equiv-\frac{1}{\beta VL}[\ln Z_{G0}(\theta)+N_{a}\ln
Z_{a0}(\theta)]\,,$ (24)

where $\ln Z_{G0}$ and $\ln Z_{a0}$ have been given by (18) and (19), which is
well-known effective potential in the Hosotani mechanism.

We should also notice that the limit of the infinite volume gives

${\cal V}(\theta)\rightarrow{\cal V}_{0}(\theta)-\frac{1}{\beta VL}[\ln
Z_{GT}(\theta,0)+N_{a}\ln Z_{aT}(\theta,0)]\,,$ (25)

because for a large volume, $\psi=0$ is a stationary point in the integrand of
(III).

If $\theta=0$ gives a global minimum of of ${\cal V}(\theta)$, the gauge
symmetry is unbroken, while if the global minimum is located at $\theta\neq
0~{}(\mbox{mod~{}}2\pi)$, the $SU(2)$ symmetry is reduced to be $U(1)$.

##  IV Numerical calculations for $D=5$

Now, we shall evaluate ${\cal V}(\theta)$ in this model. To handle the
numerical function, we use the following integral expression:

$w_{b}(\theta,\psi)=\frac{1}{2\pi^{D/2}}\int_{0}^{\infty}dt\,t^{D/2-1}\vartheta_{3}\left({\textstyle\frac{\theta}{2\pi},i\frac{L^{2}t}{\pi}}\right)\left[\vartheta_{3}\left({\textstyle\frac{\theta}{2\pi},i\frac{\beta^{2}t}{\pi}}\right)-1\right]\,,$
(26)

and

$w_{f}(\theta,\psi)=-\frac{1}{2\pi^{D/2}}\int_{0}^{\infty}dt\,t^{D/2-1}\vartheta_{3}\left({\textstyle\frac{\theta}{2\pi},i\frac{L^{2}t}{\pi}}\right)\left[\vartheta_{4}\left({\textstyle\frac{\theta}{2\pi},i\frac{\beta^{2}t}{\pi}}\right)-1\right]\,,$
(27)

where $\vartheta_{n}(v,\tau)$ is Jacobi’s theta function AS .

Hereafter, we shall concentrate ourselves on the case with $D=5$. We show
typical shapes of functions $w_{b}$ and $w_{f}$ in Fig. 1.444We used the
command FunctionInterpolation in Mathematica Mathematica . One can find that
both functions $w_{b}$ and $w_{f}$ have a period $2\pi$ for both variables
$\theta$ and $\psi$, the maximum of which is at $0$ for each variables and the
minimum of which is at $\pi$. Therefore, the extrema of the effective
potential ${\cal V}$ should be found at $\theta=0$ and $\pi$ (modulo $2\pi$).

(a) (b)

Figure 1: The functions for $\beta=1.2$ and $L=1$: (a) $w_{b}$ and (b)
$w_{f}$.

At the critical temperature, the values of ${\cal V}(0)$ and ${\cal V}(\pi)$
are equal in the present model.555This is not the case if the matter field
with a weird boundary condition exists in the model. In Fig. 2, we show the
critical lines in the parameter space spanned by $r/L$ and $\beta/L$, where,
$r$ is the radius of a ball with volume $V$ such that
$V=\frac{4\pi}{3}r^{3}$.666As is well known,
$V=\frac{\pi^{D/2-1}}{\Gamma\left(\frac{D}{2}\right)}r^{D-2}$ in the case of
$D$-dimensional spacetime.

Figure 2: Phase structure of the model. The solid line indicates the boundary
of the two phases for $N_{a}=1$, the broken line indicates that for $N_{a}=2$,
and the dotted line indicates that for $N_{a}=3$.

In the region above the line, the gauge symmetry is broken. One can find that
the $SU(2)$ symmetry is broken for any value of $r$ for a sufficiently high
temperature. For a smaller $r$, the critical temperature becomes higher; the
symmetry restoration is suppressed by the color-singletness. The suppression
is larger for a lager number of fermions, which are introduced to break the
symmetry, though it works in a much smaller volume.

The typical change in shape of the potential is exhibited in Fig. 3, which
shows the minima of the effective potential in the present model can appear at
$\theta=0$ or $\theta=\pi$ (modulo $2\pi$).

(a) (b) (c) (d)

Figure 3: The effective potential ${\cal V}$ for $N_{a}=1$, $\beta=1.2$ and
$L=1$: (a) $r=0.5$ (b) $r=0.814$ (c) $r=1$ (d) $r=\infty$.

##  V Discussion

In this paper, the effective potential at finite temperature was obtained for
the $SU(2)$ toy model based on the color-singlet hypothesis. For a smaller
volume, the critical temperature of the $SU(2)$-$U(1)$ phase transition
becomes higher.

A future task to be considered is the analysis of a more realistic general
gauge-Higgs unified model with a larger symmetry group at finite temperature.
The symmetry breaking in the $SU(3)$ gauge theory with various fermions has
been studied by lattice calculations CNHH , and it is reported that the
$SU(3)$-confined phase exists at the strong coupling regime. We should
continue to pursue the strong coupling effect and the finite size effect with
various analytical and numerical methods in order to deepen our understanding
on the Hosotani mechanism at finite temperature. We should also consider
general gauge theories in higher dimensional extra space, orbifold, and warped
spacetimes.

As a natural extension of the present analysis, we come to the idea that no
fermion number condition on the matter field in the model should be taken into
account. In addition, the Kaluza–Klein charge (originated from the momentum in
the extra dimension) is also a conserved quantity KS ; Shiraishi2 ; Shiraishi3
; McGuigan ; DLS . Thus, we can also assume the no net Kaluza–Klein charge in
the closed system. For the case of the Hosotani mechanism, however, there
appears a problematic issue that the projection onto a state of a definite
Kaluza-Klein charge breaks the residual large gauge symmetry, such as
$\theta\sim\theta+4\pi$ (or $\theta\sim\theta+2\pi$, due to $Z_{2}$) in the
present model. The treatment of these conserved charges will be challenged in
future with more elaborate investigations.

##  Appendix A An approximation scheme by finite sums

The defined functions $w_{b}$ and $w_{f}$ are expressed in the summation forms
(21) and (22). We try to approximate these by finite sums. We define the
following function:

$\displaystyle w_{app}(\theta,\psi)$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma(D/2)}{\pi^{D/2}}\sum_{n=1}^{1}\sum_{k=-2}^{2}\frac{\cos
k\theta\cos n\psi}{\left[\beta^{2}n^{2}+L^{2}k^{2}\right]^{D/2}}$ (28)
$\displaystyle=$
$\displaystyle\frac{\Gamma(D/2)}{\pi^{D/2}}\cos\psi\left[\frac{1}{\beta^{D}}+2\frac{\cos\theta}{\left(\beta^{2}+L^{2}\right)^{D/2}}+2\frac{\cos
2\theta}{\left(\beta^{2}+4L^{2}\right)^{D/2}}\right]\,.$

As an approximation, we replace both $w_{b}$ and $w_{f}$ with $w_{app}$. This
approximation is justified for a large $D$ and also for $\beta/L\ll 1$.

Figure 4: The function $w_{app}$ for $\beta=1.2$ and $L=1$.

Figure 4 shows the function $w_{app}$ for $D=5$, $\beta=1.2$, and $L=1$. The
approximation looks fine, at least for the values of parameters around this
assumption. Owing to the simplification, we can use the following formula to
evaluate the projection integral:

$\frac{1}{2\pi}\int_{-2\pi}^{2\pi}\sin^{2}\frac{\psi}{2}e^{z\cos\psi}d\psi=I_{0}(z)-I_{1}(z)\,,$
(29)

where $I_{n}(z)$ is the modified Bessel function of the first kind.

According to this approximation, the sum included in the vacuum contribution
in $Z$ is also approximated by a finite sum:

$\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{D}}\approx\cos\theta+\frac{\cos
2\theta}{2^{D}}\,.$ (30)

Figure 5 shows the reposted phase diagram (Fig. 2) and the gray lines obtained
from the above approximation in addition.

Figure 5: Phase structure of the model: The lines in Fig. 2 are reposted here
and corresponding gray lines are drawn according to the present approximation
scheme.

For a large volume, the integral over $\psi$ are dominant in the region around
the stationary points $\psi\approx 0$. Therefore the approximation of taking
$n=1$ only is good for cases with large volumes. One can see that the bending
location on the lines are well approximated.

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# Bus operators in competition: a directed location approach

Fernanda Herrera ${\dagger}$ and  Sergio I. López  School of Global Policy and
Strategy
University of California San Diego
CA 92093  Departamento de Matemáticas
Facultad de Ciencias, UNAM
C.P. 04510, Ciudad de México, México<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We present a directed variant of Salop’s (1979) model to analyze bus transport
dynamics. Players are operators competing in both cooperative and non-
cooperative games. Utility, like in most bus concession schemes in emerging
countries, is proportional to the total fare collection. Competition for
picking up passengers leads to well documented and dangerous driving practices
that cause road accidents, traffic congestion and pollution. We obtain
theoretical results that support the existence and implementation of such
practices, and give a qualitative description of how they come to occur. In
addition, our results allow to compare the base transport system with a more
cooperative one.

###### Key words and phrases:

Transport, Bus, Location games, Nash equilibrium, Mixed Strategies

###### 1991 Mathematics Subject Classification:

C710, C720, C730, R410

Fernanda Herrera gratefully acknowledges financial support from the University
of California Institute for Mexico and the United States (UC MEXUS). Sergio I.
López was partially funded by Conacyt-SNI 215989 grant.

##  1\. Introduction

In this work, we model the competition of bus operators for passengers in a
public transport concession scheme. The models -which are directed variants of
the Salop model [18], in turn a circuit adaptation of the classic Hotelling
model [13]\- are a characterization of Mexico City’s transport system.
According to a 2017 survey, 74.1% of the trips made in Mexico City by public
transport are carried out on buses with concession contracts [15].

Much like in other Latin American cities, the contracts that lay the
responsibilities, penalties and service areas, are rarely enforced by the
corresponding authorities, and in these instances, the main driver determining
the planning and operations tend to be the operator’s profit margins [12] p.
9. Leaving the task to companies or even drivers themselves, has lead to what
[10] refer to as curious old practices: driving habits adopted by bus
operators, whose salary is proportional to the fare collection, to maximize
the number of users boarding the unit. While these practices were observed and
recorded in the United Kingdom in the 1920s, they are very much present today,
particularly in cities with emerging economies and sub optimal concession
plans. The practices enlisted in [10] pertaining to driving are:

  1. (1)

Hanging back or Crowling. Operators drive slowly to pick up as many people as
possible. The idea is that long waiting times increase the number of
passengers waiting at stops. A variant is to stop altogether until the bus is
fully loaded, or the next bus catches up.

  2. (2)

Racing. When an operator deems that the number of passengers waiting at a stop
is not worth making the stop. In this case, she continues driving in the hopes
of collecting more users ahead.

  3. (3)

Overtaking, Tailing or Chasing. Attempting to pass the bus ahead, to cut in
and pick up the passengers frontwards.

  4. (4)

Turning. When an empty or nearly empty bus turns around before the end of the
route, and drives back in the opposite direction.

Many of these practices have negative consequences on the service provided to
users, and as a byproduct, on the perception of public transport. In the 2019
survey on victimization in public transport [5], carried out in Mexico City
and its metropolitan area, 50% of the interviewees deemed the quality of
concession transport to be bad, and 15% very bad. Moreover, 27% considered
that traveling in concession transport was somewhat dangerous, and 60% very
dangerous. In both of these dimensions, concession public transport did worse
than any other form of transport, including public and private types. The
matter is pressing enough that the current administration of Mexico City
stressed in its Strategic Mobility Plan of 2019 [19] p. 9: The business model
that governs this (transport) sector, (…) produces competition in the streets
for users, which results in the pick up and drop off of passengers in
unauthorized places, increased congestion and a large number of traffic
incidents each year.

A solution to these problems may involve the deregulation of public transport
to increase competition between providers, and to create incentives for
providing a differentiated product, namely better service in the form of
shorter waiting times, and safer driving practices. As an example, Margaret
Thatcher introduced the Transport Act 1985 [1], which lead to the
privatization of bus services, higher competition between companies, and a set
of norms to abide by, like keeping vehicles in good condition, avoiding
dangerous driving, and establishing routes and publishing timetables. However
successful, this type of measure seems unlikely for Mexico and other
developing Latin American countries, both for legislative reasons and
corruption in the implementation. So, with this work we aim to shed light on
the implications of a transport system where operators compete for passengers
without regulation.

To be specific, we model the situation where bus operators compete to maximize
their utility, which is proportional to the number of passengers boarding the
units. As a proxy for the number of passengers collected, we use the road
ahead up to the next bus. The strategies available to drivers are the driving
speeds. Time, like strategies themselves, is continuous. For simplicity, we do
not allow drivers to change speed any time they want, instead we assume that
they maintain a chosen speed for a given time, and let them change in the
next. While practical, the assumption also reflects the empirical observation
that bus drivers make strategic stops along the road, where they obtain
information on the game. More precisely, they pay agents that collect the
arrival times of previous buses to that particular stop, and even the identity
of the drivers themselves. This way, the operators realize whether they are
competing against known drivers, and more importantly, whether they changed
their speed. With this information, they make their decision for the next part
of the route. We obtain a simple interpretation of the results that is
consistent with the driving practices mentioned above.

To the best of our knowledge, our approach is novel, and it allows us to model
a variety of scenarios and obtain explicit descriptions of equilibria.
Furthermore, we are able to explore the time evolution of the adopted
strategies. All the results are expressed in terms of the behavior of the
operators. Given the tractability of our models, some natural theoretical
questions emerge.

Relevant literature on transport problems includes [17] modeling of the
optimal headway bus service from the point of view of a central dispatcher. In
the historical context of Transport Act 1985 [1], several scientific articles
analyzed the effect of the privatization. Under the assumption of the
existence of an economic equilibrium in the competition system, [10] classify
the driving practices into two categories: those consistent with the
equilibrium, and those who are not. They analyzed the expected timetables in
the deregulated scenario. In [9] a comparative analysis of fare and timetable
allocation in competition, monopoly and net benefit maximization (both
restricted and unrestricted to a zero profit) models is presented. Building on
from this, [16] introduces the consumer’s perspective and obtains the
equilibria prices and number of services offered by transport companies. The
possibility of predatory behavior between two enterprises competing through
fares and service level, is analyzed by [7], using the data from the city of
Inverness. In [8] the authors study the optimal policies of competing
enterprises in terms of fares, and the bus service headway, in a unique bus
stop and destination scenario. They also introduce the concept of demand
coordination which can be implemented through timetables. Assuming a spatial
directed model with a single enterprise, [6] finds the timetable that
minimizes the costs associated to service delays. The work of [4] analyzes
flight time data and finds empirical evidence to support Hotelling models.
From a non-economic perspective, [3] models competing buses in a circuit
behaving like random particles with repulsion between them (meaning they could
not pass each other). A contemporary review on transport market models using
game theory is given by [2], and a general review of control problems which
arise in buses transport systems is presented in [14].

This paper is organized as follows. In Section 2, we present the general
model, and the single and two player games. Relevant definitions, notation and
interpretations are introduced. In Section 3 we present the solutions to the
games and include in Subsection 3.3 the evolution of the strategies adopted by
the operators. That is, we look at the long-run equilibria of the games. We
also introduce a natural extension of the two player games and present the
results in Subsection 3.4. Concluding remarks are in 4, and proofs are in
Appendix A.

##  2\. The model

The assumptions of the game are the following. There are $n\leq 2$ buses, each
is driven by one of $n$ operators along a route. There is only one type of bus
and one type of driver, meaning that the buses have identical features, and
that the drivers are homogeneous in terms of skill and other relevant
characteristics.

The speed of a bus, denoted by $v$, is bounded throughout every time and place
of the road by:

(2.1) $0<v_{min}\leq v\leq v_{max},$

where the constants $v_{min}$ and $v_{max}$ are fixed, and determined by
exogenous factors like the condition of the bus, Federal and State laws and
regulations, the infrastructure of the road, etc.

Drivers can pick up passengers along any point on the route at any given time.
In other words, there are no designated bus stations, nor interval-based time
schedules in place. This scenario is an approximation to a route with a large
number of homogeneously distributed bus stops.

We allow for infinite bus capacity, so drivers can pick up any number of
passengers they come across. Alternatively, one can assume that passengers
alight from the bus almost right after boarding it, so the bus is virtually
empty and ready to pick up users at any given time. The important point to
note is that passengers that have boarded a bus will not hop on the next,
either because they never descended it in the first place, or because they
already reached their final destination if they did.

Bus users reach their pick up point at random times, so demand for transport
is proportional to the time elapsed between bus arrivals. Let $\lambda>0$
denote the mean number of passengers boarding a bus per unit of time, and let
$p\geq 0$ denote the fixed fare paid by each user. We assume that there is a
fixed driving cost $c\geq 0$ per unit of time. This cost summarizes fuel
consumption, maintenance, protection insurance for the bus and passengers,
etc.

The operators get a share of the total revenue, and consequently seek to
maximize it. Since they cannot control the number of passengers on the route,
the fare, or the driving costs, the only resource available to them is to set
the driving speed, which we assume remains constant throughout the time
interval $[0,T]$, with $T>0$. The strategy space of a bus driver is then

(2.2) $\Gamma=\\{v\geq 0:v_{min}\leq v\leq v_{max}\\},$

where $v_{min}$ and $v_{max}$ are given in (2.1). We define a mixed strategy,
$X$ or $Y$, to be a random variable taking values in the space $\Gamma$.

In what follows we define the expected utility of drivers given a set of
assumptions on the number of players and their starting positions, the fixed
variables of the models, and route characteristics. Relevant notation and
concepts are introduced when deemed necessary.

###  2.1. Single player games

We first consider a game with only one driver picking up passengers along the
road. Importantly, the fact that only one bus is covering the route implies
that commuters have no option but to wait for its arrival, the player is aware
of this.

  * •

Fixed-distance game

A single bus departs the origin of a route of length $D$. We adopt the
convention that the initial time is whenever the bus departs the origin. We
define the expected utility of driving at a given speed $v$ to be

(2.3) $u(v):=(p\lambda)T-cT,$

where $T=\frac{D}{v}$ is the time needed to travel the distance $D$ at speed
$v$.

Note that since there is no other bus picking up passengers, the expected
number of people waiting for the bus in a fixed interval of the road increases
proportionally with time. From this, one infers that the expected total number
of passengers taking the bus is proportional to the time it takes the bus to
reach its final destination.111This justifies the first summand in (2.3). The
conclusion and its implication can be expressed rigorously using a space-time
Poisson process, see for example [11] pp. 283-296.

  * •

Fixed-time game

Suppose now that the driver chooses a constant speed $v$ satisfying (2.1) in
order to drive for $T$ units of time. The bus then travels the distance
$D=Tv$, which clearly depends on $v$. We define the expected utility of
driving at a given speed $v$ to be

(2.4) $u(v):=(p\lambda)D-cT.$

The underlying assumption is that for sufficiently small $T$, there are
virtually no new arrivals of commuters to the route, so effectively, the
number of people queuing for the bus remains the same as that of the previous
instant. The requirement is that $T$ is small compared to the expected
interarrival times of commuters.

It follows that the total amount of money collected by the driver is
proportional to the total distance traveled by the bus.

###  2.2. Two player games

There are two buses picking up passengers along a route, which we assume is a
one-way traffic circuit. An advantageous feature of circuits is that buses
that return from any point on the route to the initial stop may remain in
service; this is generally not the case in other types of routes. In
particular, we assume that the circuit is a one-dimensional torus of length
$D$. For illustration purposes and without loss of generality, from now on we
require the direction of traffic to be clockwise.

We define the $D$-module of any real number $r$ as

$(r)_{mod\,D}:=\frac{r}{D}-\Big{\lfloor}\frac{r}{D}\Big{\rfloor},$

where $\lfloor z\rfloor$ is the greatest integer less than or equal to $z$.

The interpretation of $(r)_{mod\,D}$ is the following: if starting from the
origin, a bus travels the total distance $r$, then $(r)_{mod\,D}$ denotes its
relative position on the torus. Indeed, $r$ may be such that the bus loops
around the circuit many times, nonetheless $(r)_{mod\,D}$ is in $[0,D)$ for
all $r$. We refer to $r$ as the absolute position of the bus, and to
$(r)_{mod\,D}$ as the relative position (with respect to the torus). Note that
the origin and the end of the route share the same relative position, since
$(0)_{mod\,D}=0=(D)_{mod\,D}$.

Let ${\mathbf{x}}$ and ${\mathbf{y}}$ denote the two players of the game, and
let $x,y$ be their respective relative positions. The directed distance
function $d_{{\mathbf{x}}}$ is given by

(2.7) $\displaystyle
d_{{\mathbf{x}}}(x,y):=\left\\{\begin{array}[]{ll}y-x&\textrm{ if }x\leq y,\\\
D+y-x&\textrm{ if }x>y.\end{array}\right.$

Equation (2.7) has a key geometrical interpretation: it gives the distance
from $x$ to $y$ considering that traffic is one-way. The interest of this is
that the potential amount of commuters ${\mathbf{x}}$ picks up is proportional
to the distance between $x$ and $y$, namely $d_{{\mathbf{x}}}(x,y)$. See
Figure 1.

A straightforward observation is that for any real number $r$, we have

(2.8) $d_{{\mathbf{x}}}((x+r)_{mod\,D},(y+r)_{mod\,D})=d_{{\mathbf{x}}}(x,y).$

This asserts that if we shift the relative position of the two players by $r$
units (either clockwise or counterclockwise, depending on the sign of $r$),
then the directed distance $d_{{\mathbf{x}}}$ is unchanged.

One can define the directed distance $d_{{\mathbf{y}}}$ analogously,

$d_{{\mathbf{y}}}(x,y):=\left\\{\begin{array}[]{ll}x-y&\textrm{ if }y\leq
x,\\\ D+x-y&\textrm{ if }y>x.\end{array}\right.$

By definition, there is an intrinsic symmetry between $d_{{\mathbf{x}}}$ and
$d_{{\mathbf{y}}}$: we have $d_{{\mathbf{x}}}(x,y)=d_{{\mathbf{y}}}(y,x)$ and
$d_{{\mathbf{y}}}(x,y)=d_{{\mathbf{x}}}(y,x)$. Roughly speaking, this means
that if we were to swap all the labels, namely ${\mathbf{x}}$ to
${\mathbf{y}}$, $x$ to $y$,222Importantly, this switches the relative
positions of the players. and vice versa, then it suffices to plug the new
labels into the previous definitions to obtain the directed distances.

Another immediate observation is that for any pair of different positions
$(x,y)$, the sum of the two directed distances gives the total length of the
circuit,

(2.9) $d_{{\mathbf{x}}}(x,y)+d_{{\mathbf{y}}}(x,y)=D.$

This is portrayed in Figure 1.

Figure 1. Directed distances

Let us assume that players ${\mathbf{x}}$ and ${\mathbf{y}}$ have starting
positions $x_{0}$ and $y_{0}$ in $[0,D)$. The initial minimal distance is
defined to be

(2.10)
$d_{0}:=\min\\{d_{{\mathbf{x}}}(x_{0},y_{0}),d_{{\mathbf{y}}}(x_{0},y_{0})\\}.$

Now suppose that starting from $x_{0}$ and $y_{0}$, the operators drive at the
respective speeds $v_{{\mathbf{x}}}$ and $v_{{\mathbf{y}}}$, with
$v_{{\mathbf{x}}},v_{{\mathbf{y}}}$ in $\Gamma$, for $T$ units of time. Their
final relative positions are then

$x_{T}=(x_{0}+Tv_{{\mathbf{x}}})_{mod\,D}\qquad\text{and}\qquad
y_{T}=(y_{0}+Tv_{{\mathbf{y}}})_{mod\,D}.$

We orient the maximum displacement of buses by requiring $Tv_{max}$, with
$v_{max}$ given in (2.1), to be small compared to $D$. The reason for this is
to be consistent with our assumption of constant speed strategies, since they
are short-term. More precisely, we require

(2.11) $Tv_{max}<\frac{D}{2}.$

Lastly, we define the escape distance by

(2.12) $d:=T(v_{max}-v_{min}).$

This gives a threshold such that if the distance between the players is
shorter than $d$, then the buses can catch up to each other, given the
appropriate pair of speeds. If the distance is greater than $d$, this cannot
occur.

We now proceed to define the expected utility of players given the type of
game being played, namely, whether it is cooperative or non-cooperative.

  * •

Non-cooperative game

We define the utility of ${\mathbf{x}}$ given the initial positions of players
$x_{0}$ and $y_{0}$, and the strategies $v_{{\mathbf{x}}}$ and
$v_{{\mathbf{y}}}$, to be

(2.13)
$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}}):=\left\\{\begin{array}[]{cc}p\lambda\,d_{{\mathbf{x}}}(x_{T},y_{T})-cT&\textrm{
if }x_{T}\neq y_{T},\\\ p\lambda\,\frac{D}{2}-cT&\textrm{ if
}x_{T}=y_{T}.\end{array}\right.$

The definition above includes two summands: the first one gives the (gross)
expected income of ${\mathbf{x}}$, since the factor $p\lambda$ is the expected
income per unit of distance. The second term gives the total driving cost.

It is worth pointing out that for simplicity, we have assumed that the
expected income depends only on the relative final positions $x_{T}$ and
$y_{T}$. A more precise account would consider the entire trajectory of the
buses. Nevertheless, even if this could be described with mathematical
precision, the model would grow greatly in complexity without adding to its
economic interpretation.

Similarly, we define

(2.14)
$u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}}):=\left\\{\begin{array}[]{ll}p\lambda\,d_{Y}(x_{T},y_{T})-cT&\textrm{
if }x_{T}\neq y_{T},\\\ p\lambda\,\frac{D}{2}-cT&\textrm{ if
}x_{T}=y_{T}.\end{array}\right.$

By equation (2.9) and the definition of the utility functions (2.13), (2.14),
the sum $u_{{\mathbf{x}}}+u_{{\mathbf{y}}}$ is a constant that does not depend
on the driving speeds nor on the initial positions. For this reason, we
analyze the game as a zero-sum game.

  * •

Cooperative game

Players aim to maximize the collective payoff, and this amounts to solving the
global optimization of the sum $U_{{\mathbf{x}}}+U_{{\mathbf{y}}}$, which
includes the utility functions in the non-cooperative game (2.13) and (2.14).
Since the non-cooperative game is a zero-sum game, we introduce an extra term
in the utility, which gives the discomfort players derive from payoff
inequality. This assumption can be imagined in a situation where equity in
payments is desirable, specially since players have complete information.

We define the utility function to be

(2.15)
$u(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}}):=u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})+u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})-k|u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})-u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})|,$

where $k$ is a non-negative constant, and all the other elements are the same
as in the non-cooperative game.

####  2.2.1. Mixed strategies and $\varepsilon$-equilibria

For the solution of two player games, it is convenient to define the expected
utility of randomizing over the set of strategies. We also introduce the
definition of $\varepsilon$-equilibrium.

Suppose that players ${\mathbf{x}}$ and ${\mathbf{y}}$ use the mixed
strategies $X$ and $Y$.333Recall that a mixed strategy is a random variable
taking values in the set $\Gamma=\\{v\geq 0:v_{min}\leq v\leq v_{max}\\}$. We
define the utility of player ${\mathbf{x}}$ to be

$U_{{\mathbf{x}}}(x_{0},X,y_{0},Y):=\mathbb{E}[u_{{\mathbf{x}}}(x_{0},X,y_{0},Y)].$

An analogous definition can be derived for player ${\mathbf{y}}$.

Let $\varepsilon>0$. We say that a pair of pure strategies
$(v^{*}_{{\mathbf{x}}},v^{*}_{{\mathbf{y}}})$ is an $\varepsilon$-equilibrium
if for every $v_{{\mathbf{x}}}$ and $v_{{\mathbf{y}}}$ we have

$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v^{*}_{{\mathbf{y}}})\leq
u_{{\mathbf{x}}}(x_{0},v^{*}_{{\mathbf{x}}},y_{0},v^{*}_{{\mathbf{y}}})+\varepsilon,$

and

$u_{{\mathbf{y}}}(x_{0},v^{*}_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})\leq
u_{{\mathbf{y}}}(x_{0},v^{*}_{{\mathbf{x}}},y_{0},v^{*}_{{\mathbf{y}}})+\varepsilon.$

This means that any unilateral deviation from the equilibrium strategy leads
to a gain of no more than $\varepsilon$; this is why an
$\varepsilon$-equilibrium is also called near-Nash equilibrium. Note that in
particular, an $\varepsilon$-equilibrium with $\varepsilon=0$ gives the
standard definition of Nash equilibrium. However, an $\varepsilon$-equilibrium
for all $\varepsilon$ sufficiently small, need not be a Nash equilibrium,
specially if the utility function is discontinuous, which is our case.

A mixed strategies $\varepsilon$-equilibrium $(X,Y)$ is similarly defined by
replacing the utility functions with the expected utility functions in the
last definition.

##  3\. Results

In what follows, we analyze the speeds that drivers choose, both in the short
and long-run. Results on the short term are crucial to the analysis, as
implementing the optimal short-term strategies over a long period of time,
gives the long-term solution to the games.

###  3.1. Single player games

The single player games have pure strategy Nash equilibria. Although the
results are immediate, we include them in the analysis for completeness and
ease of interpretation.

######  Proposition 1.

Let $v^{*}$ in $\Gamma$ be the driving speed that maximizes the utility of the
driver. We provide an explicit description of $v^{*}$.

  1. a)

Fixed-distance game. Given the utility function defined in (2.3), we have

$\displaystyle v^{*}=\begin{cases}v_{min}&\text{if $p\lambda>c$},\\\
v_{min}\leq v\leq v_{max}&\text{if $p\lambda=c$},\\\ v_{max}&\text{if
$p\lambda<c$}.\end{cases}$

  2. b)

Fixed-time game. Given the utility function defined in (2.4), we have
$v^{*}=v_{max}$.

###### Proof.

Note that in the fixed-distance game, $p\lambda-c$ gives the driver’s expected
net income per unit of time. If this amount is positive, then the player
maximizes her utility by driving for the longest time, or equivalently, by
driving at the lowest possible speed. Conversely, a negative expected net
income leads to driving at the highest speed. Lastly, a null expected income
makes the driver indifferent between any given speed in the range.

In the fixed-time game, the total revenue is proportional to the traveled
distance, so the driver maximizes her utility by driving at the highest speed.
∎

###  3.2. Two-player games

The strategies adopted by the players strongly depend on the initial minimal
distance defined in (2.10). We cover all cases.

######  Theorem 1.

Non-cooperative game. Without loss of generality we can assume
$d_{0}=d_{{\mathbf{x}}}(x_{0},y_{0})$.

  1. a)

If $d_{0}=0$, that is, if the initial positions of the players are the same,
then the pair of strategies $(v_{max},v_{max})$ is the only Nash equilibrium.

  2. b)

If $0<d_{0}<d<d_{{\mathbf{y}}}(x_{0},y_{0})$, with $d$ the escape distance in
2.12, then for sufficiently small $\varepsilon$, the mixed strategy
$\varepsilon$-equilibria $(X,Y)$ is

$\displaystyle X=\begin{cases}v_{min}&\text{with probability
}1-\frac{d-d_{0}}{D},\\\ U&\text{with probability
}\frac{d-d_{0}}{D}\end{cases}$ and $\displaystyle
Y=\begin{cases}v_{min}&\text{with probability }q_{1},\\\ V&\text{with
probability }q_{2},\\\
v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}&\text{with probability
}1-\frac{d}{D},\end{cases}$

where $U$ is a uniform random variable on
$\Big{(}v_{min}+\frac{d_{0}}{T},v_{max}\Big{)}$, $q_{1}$ and $q_{2}$ are non-
negative numbers such that $q_{1}+q_{2}=\frac{d}{D}$ and
$q_{2}\leq\frac{d-d_{0}}{D}$, and $V$ is a uniform random variable on
$\Big{(}v_{max}-\frac{d_{0}}{T}-q_{2}\frac{D}{T},\,v_{max}-\frac{d_{0}}{T}\Big{)}$.

In other words, $X$ has an atom at $v_{min}$, and $Y$ has two atoms at
$v_{min}$ and $v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}$, and are
otherwise uniformly distributed over their respective intervals.

  3. c)

If $0<d=d_{0}<d_{{\mathbf{y}}}(x_{0},y_{0})$, then for sufficiently small
$\varepsilon$, the mixed strategy $\varepsilon$-equilibria is

$\displaystyle X=\begin{cases}v_{min}&\text{with probability
}1-\frac{2\varepsilon}{D},\\\ v_{max}&\text{with probability
}\frac{2\varepsilon}{D}\end{cases}$ and $\displaystyle
Y=\begin{cases}v_{min}&\text{with probability }\frac{2d}{D},\\\
v_{min}+\frac{\varepsilon}{T}&\text{with probability
}1-\frac{2d}{D}.\end{cases}$

  4. d)

If $d<d_{0}$, then the pair of strategies $(v_{min},v_{min})$ is the unique
Nash equilibrium.

###### Proof.

The proof is in Appendix A. ∎

By assumption (2.11), this result covers all the possible initial positions
$(x_{0},y_{0})$, so we have a complete and explicit characterization of the
equilibria. Simply put, the theorem asserts that if the players have the same
starting point, they drive at the maximum speed. If their positions differ by
at most the escape distance, then they play mixed strategies. Lastly, if the
distance between them is greater than the escape one, they drive at the
minimum speed. See Figure 2 for an illustration of the result and its cases.

Figure 2. On the rightmost side of each graph are the final positions of
players, blue for ${\mathbf{x}}$ and red for ${\mathbf{y}}$, after driving at
the optimal speed for $T$ units of time. Points represent probability mass
atoms, while continuous bars give the intervals in which the locations may be.

######  Theorem 2.

Cooperative game. Without loss of generality we assume that
$d_{0}=d_{{\mathbf{x}}}(x_{0},y_{0})$.

  1. a)

If $d_{0}=0$, then the optimal pairs of driving speeds are $(v_{min},v_{max})$
and $(v_{max},v_{min})$.

  2. b)

If $0<d_{0}$ and $d_{0}+d<\frac{D}{2}$, then the only optimal strategies are
$(v_{min},v_{max})$.

  3. c)

If $d_{0}+d>\frac{D}{2}$, then any pair $(v_{{\mathbf{x}}},v_{{\mathbf{y}}})$
such that $T(v_{{\mathbf{y}}}-v_{{\mathbf{x}}})=\frac{D}{2}$ is an optimal
strategy.

###### Proof.

The proof is direct. Since the sum
$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})+u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})$
is equal to a constant for any pair $(v_{{\mathbf{x}}},v_{{\mathbf{y}}})$, the
only quantity left to optimize is
$-k|u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})-u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})|$.
Minimization occurs when the distance between the final positions $x_{F}$ and
$y_{F}$ is the greatest possible. It is easy to check that the driving speeds
listed above do just this. ∎

An important observation is that in the case where $d_{0}=\frac{D}{2}$, which
is accounted for in c) all the optimal strategies are of the form $(v,v)$ for
a feasible speed $v$. Intuitively, this means that if the players have
diametrically opposite initial positions, then any speed is optimal, as long
as both adopt it.

###  3.3. Long-run analysis

Let us recall that the previous results are obtained for small enough $T$, the
formal requirement being stated in (2.11). It is of interest to know what
happens in longer time periods, and in particular, in the long-run. To this
end, we repeat the games infinitely many times, implementing the optimal
strategies in each stage. Of course, the strategies depend on the distance
between players, which is given by the implementation of the optimal
strategies in the previous period. It is thus convenient to define a recursive
process, and to introduce a few variables.

Consider the initial positions of ${\mathbf{x}}$ and ${\mathbf{y}}$, namely
$(x_{0},y_{0})$, with $d_{0}$ defined in (2.10). Let
$\\{(x_{n},y_{n})\\}_{n\geq 1}$ be a stochastic process with the following
property: the pair $(x_{k+1},y_{k+1})$ gives the final locations of the
players after they play their optimal strategies, taking $(x_{k},y_{k})$ as
their starting positions. It is worth noting that since equilibria in Theorem
2 involve mixed strategies, randomness is very much present in the process.

We define the distance between the buses at any (non-negative integer) time
as:

(3.1)
$d_{n}:=\min\\{d_{{\mathbf{x}}}(x_{n},y_{n}),d_{{\mathbf{y}}}(x_{n},y_{n})\\}\qquad\forall\enspace
n\geq 0.$

We also define the first time in which $d_{n}$ exceeds the escape distance $d$
(given in (2.12)), denoted by $N$, as follows

$N=\min\\{n\geq 0:d_{n}>d\\}.$

######  Theorem 3.

Non-cooperative game. If $d_{0}\neq 0,d$, we have

$\mathbb{P}(N>k)\leq\Big{(}\frac{d}{D}\Big{)}^{k}\quad\text{for all $k\geq
1$}.$

If $d_{0}=d$, then there exists a geometrically distributed random time $M$
with parameter $1-(1-\frac{2\varepsilon}{D})(\frac{2d}{D})$, taking values in
the natural numbers, with $\varepsilon$ satisfying the
$\varepsilon$-equilibrium conditions in Theorem 2, with the property that
$d_{k}=d$ for all $k<M$, and

$d_{M}=\left\\{\begin{array}[]{ll}0&\textrm{ with probability
}\qquad\frac{4\varepsilon
d}{D^{2}\Big{(}1-\Big{(}1-\frac{2\varepsilon}{D}\Big{)}\Big{(}\frac{2d}{D}\Big{)}\Big{)}},\\\
>d&\textrm{ with complementary probability.}\\\ \end{array}\right.$

###### Proof.

For the proof we refer the reader to Appendix A. ∎

Explicitly, this means that for most starting points, playing the game
repeatedly leads to a bus gap greater than the escape distance in a finite and
geometrically distributed time. From Theorem 2, we conclude that in this case,
drivers end up driving at the minimum speed. There are two exceptions to this:
if the drivers have the same starting position, or if the initial distance
between them is exactly that of escape. In the former case, the drivers choose
to go at the maximum speed forever, and in the latter, they maintain their
distance for some random time, and from then on reach the escape distance, and
drive at the minimum speed. It is with very little probability (proportional
to $\varepsilon$) that this scenario does not occur. Figure 3 shows the
evolution of the distance process $\\{d_{n}:n\geq 0\\}$ given a few initial
distances $d_{0}$.

Figure 3. Evolution of the process $\\{d_{n}:n\geq 0\\}$ for different initial
positions, non-cooperative game.

######  Theorem 4.

Cooperative game. For all $d_{0}\geq 0$ we have
$N\leq\lceil\frac{D}{2d}\rceil$, where $N=\min\Big{\\{}n\geq
0:d_{n}=\frac{D}{2}\Big{\\}}$, and $\lceil z\rceil$ is the least integer
greater than or equal to the real number $z$.

###### Proof.

First note that $N$ gives the time in which the buses reach diametrically
opposite positions in the circuit. Also, playing the optimal strategies in
Theorem 2, increases the distance between the buses by $d$. Hence, repeating
the game eventually leads to reaching the diametric distance. This means that
$N$ is at most the number of steps of size $d$ necessary to go over
$\frac{D}{2}$. Once diametrical positions are reached, the distance is
preserved forever. ∎

###  3.4. Extension

It is possible to account for perturbations like traffic lights, congestion,
or accidents, by introducing a random noise to the displacement of buses. One
could do this defining

(3.2) $x_{T}=(x_{0}+Tv_{{\mathbf{x}}}+\sigma
Z_{x})_{mod\,D}\quad\text{and}\quad y_{T}=(y_{0}+Tv_{{\mathbf{y}}}+\sigma
Z_{y})_{mod\,D},$

where $Z_{x}$ and $Z_{y}$ are independent standard normal random variables and
$\sigma\geq 0$ is a fixed parameter.

Then, the following results would be observed.

  * •

Non-cooperative game. Given that the expected value of the final positions is
unchanged, Theorem 1 remains valid. However, the repetition of this new game
leads to a new result. Since the probability of maintaining a null, or escape
distance $d$, at any positive time is zero, the long-run analysis is reduced
to two distinct cases: $0<d_{0}<d$ and $d_{0}<d$. Arguments similar to that in
the proof of Theorem 3 show that if $0<d_{0}<d$, we have $d_{N}\geq d$ in an
exponentially fast time $N$. If $d<d_{0}$, then the distance process
$\\{d_{n}\\}_{n\geq 1}$ remains above $d$ for a random time $M$, but
eventually falls below it. The expected time above is inversely proportional
to $\sigma$.

  * •

Cooperative game. The analysis collapses to the cases b) and c) of Theorem 2.
So, while the players try to reach the diametrically opposite positions, with
probability one this does not occur.

##  4\. Concluding remarks

Our theoretical results are consistent with the driving practices mentioned in
the Introduction. In particular, Theorem 1.$a$ induces (2) Racing, Theorem
1.$b$, $c$ conduce to (2) Racing and (3) Overtaking, Tailing or Chasing, and
Theorem 1.$d$ to (1) Hanging back or Crowling. It is worth noting that all of
the aforementioned are short-term strategies. As far as the time-evolution of
the game goes, Theorem 3 asserts that in the long run and with high
probability, both operators end up hanging back. Theorems 2 and 4 are intended
to contrast the drivers’ optimal strategies and ultimately the equilibria when
cooperation is desired.

In subsection 3.4, we extended the model to allow for randomness in
displacement. In this scenario no equilibrium is lasting, so the operators
alternate between racing, hanging back and chasing from time to time. We
believe this is precisely what happens in Mexico City, although proving this
would require a data driven approach analysis.

There are a few open problems worth exploring. First, one could increase the
number of players, and investigate whether equilibria still exists, and if so,
try to characterize it. Second, one may vary the distribution of the
passengers along the route, dispensing with the homogeneous assumption. Along
these lines, one may introduce traffic congestion by making the utility
function depend on space in a non-homogeneous manner. This would potentially
require strategies to depend on the player’s position. Lastly, one could
introduce decision variables like tariffs and timetables; doing so would allow
to compare the results with some that have already been addressed in the
literature.

### Declaration of interest

None.

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##  Appendix A Computations

To prove Theorem 1, it is convenient to introduce the following Lemma.

######  Lemma 1.

Let $X$ be a mixed strategy of ${\mathbf{x}}$ and $Y$ be a mixed strategy of
${\mathbf{y}}$. We define $Z$ to be a mixed random variable in the Probability
theory sense: it has both discrete and continuous components. In particular,
$Z$ is of the form

$\displaystyle Z=\left\\{\begin{array}[]{cc}z_{i}&\textrm{ with probability
}p_{i},\textrm{ for }i\in I,\\\ W&\textrm{ with probability }1-\sum_{i\in
I}p_{i},\end{array}\right.$

where $I$ is a finite or numerable set, and $W$ is a continuous random
variable with density $f_{W}(t)$ on its support, denoted by $supp(f_{W})$.
Then,

(A.2) $\displaystyle U_{\mathbf{x}}(x_{0},X,y_{0},Y)$ $\displaystyle=$
$\displaystyle\sum_{i\in
I}\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|Z=z_{i})\,p_{i}$ $\displaystyle+$
$\displaystyle\Big{(}1-\sum_{i\in
I}p_{i}\Big{)}\int_{supp(f_{W})}\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|Z=w)f_{W}(w)dw.$

If $Z=X$ and $(X,Y)$ is a mixed strategy Nash equilibrium, then

(A.3)
$\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=z_{i})=\int_{supp(f_{W})}\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=w)f_{W}(w)dw\qquad\forall
i\in I,$

and

(A.4)
$\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=w_{1})=\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=w_{2})\qquad\forall
w_{1},w_{2}\in supp(f_{W}).$

###### Proof.

Equation (A.2) is straightforwardly obtained by computing the conditional
expectancy of the random variable $u_{\mathbf{x}}(x_{0},X,y_{0},Y)$ given the
values of $Z$.

Note that if (A.3) does not occur, then there exist two different values
$z_{i}$ and $z_{j}$, such that
$\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=z_{i})\neq\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=z_{j})$.
This means that $U_{{\mathbf{x}}}$ can be increased by placing all the
probability on the value that gives the highest expectation. This leads to a
contradiction with the form of the mixed strategy $X$. Similar arguments apply
to the case where (A.3) is violated through the continuous component.

Likewise, if condition (A.4) is not fulfilled, then there are two values
$w_{1}$ and $w_{2}$ such that
$\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=w_{i})$ are different. Then,
$U_{{\mathbf{x}}}$ can be increased by restricting the support of $f_{W}$ to
the points where the maximum of the function
$g(w)=\mathbb{E}(u_{\mathbf{x}}(x_{0},X,y_{0},Y)|X=w)$ is reached. Here, the
form of the mixed strategy $X$ is violated. ∎

Proof of Theorem 1:

First, note that for optimizing the utility function (2.13), (2.14) the terms
$p\lambda$ and $c$ are irrelevant, since the $\mathop{\rm arg\,min}$ of any
function is invariant under linear transformations. Thus, there is no loss of
generality in assuming that $p\lambda=1$ and $c=0$.

By equation (2.8), we may actually assume that $0=x_{0}\leq y_{0}<D$. We then
have

$d_{0}=d_{{\mathbf{x}}}(x_{0},y_{0})=y_{0}\quad\text{and}\quad
d_{{\mathbf{y}}}(x_{0},y_{0})=D-y_{0}.$

Under the above assumption and using (2.1), (2.11) in cases a), b), c) and d),
it happens that $0<x_{T},y_{T}<D$, so we can get rid of all the $D$-modules in
the computations.

For computing the $\varepsilon$-equilibrium, we will consider the
$\varepsilon$-best reply, defined as follows. Let $\varepsilon$ be a positive
number. We say that a strategy $v^{*}_{{\mathbf{x}}}$ is ${\mathbf{x}}$’s
$\varepsilon$-best reply to ${\mathbf{y}}$’s strategy $v_{{\mathbf{y}}}$, if

$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})\leq
u_{{\mathbf{x}}}(x_{0},v^{*}_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})+\varepsilon,$

for all strategies $v_{{\mathbf{x}}}$.

To simplify notation, we write
$u_{{\mathbf{x}}}(v_{{\mathbf{x}}},v_{{\mathbf{y}}})$ and
$u_{{\mathbf{x}}}(X,Y)$ in the case of mixed strategies, instead of
$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})$ and
$u_{{\mathbf{x}}}(x_{0},X,y_{0},Y)$ if the computations do not depend on the
fixed initial positions.

  * •

Case a)
We assume that $x_{0}=y_{0}=0$. Let player ${\mathbf{y}}$ pick the strategy
$v_{{\mathbf{y}}}=v_{max}$. Then,

$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{min})=d_{{\mathbf{x}}}(Tv_{{\mathbf{x}}},Tv_{max})=Tv_{max}-Tv_{{\mathbf{x}}}\leq
Tv_{max}.$

Using (2.11), we obtain the bound

$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{min})\leq\frac{D}{2}=u_{{\mathbf{x}}}(x_{0},v_{max},y_{0},v_{max}).$

Explicitly, this means that the strategy $v_{{\mathbf{x}}}=v_{max}$ is the
best reply to $v_{{\mathbf{y}}}=v_{max}$. By symmetry, we conclude that
$(v_{max},v_{max})$ is a Nash equilibrium.

To check the uniqueness of the equilibrium, we note that ${\mathbf{y}}$’s
$\varepsilon$-best reply to a given speed $v_{{\mathbf{x}}}<v_{max}$ chosen by
${\mathbf{x}}$, is $v_{{\mathbf{y}}}=v_{{\mathbf{x}}}+\epsilon$ for
sufficiently small $\varepsilon$. On the other hand, ${\mathbf{x}}$’s
$\varepsilon$-best reply to $v_{{\mathbf{y}}}=v_{{\mathbf{x}}}+\epsilon$ is
$v_{{\mathbf{x}}}=v_{{\mathbf{y}}}+\varepsilon$. Therefore the only
equilibrium is $(v_{max},v_{max})$.

  * •

Case b)
Let us denote by $B_{{\mathbf{x}}}(v)$ ${\mathbf{x}}$’s best reply when
${\mathbf{y}}$ plays $v$. It is straightforward to show that

$\displaystyle
B_{{\mathbf{x}}}(v)=\left\\{\begin{array}[]{ll}v+\frac{d_{0}}{T}+\frac{\varepsilon}{T}&\textrm{
if }v_{min}\leq v<v_{max}-\frac{d_{0}}{T},\\\ v_{max}&\textrm{ if
}v=v_{max}-\frac{d_{0}}{T},\\\ v_{min}&\textrm{ if
}v_{max}-\frac{d_{0}}{T}<v,\end{array}\right.$

and

$\displaystyle B_{{\mathbf{y}}}(v)=\left\\{\begin{array}[]{ll}v_{min}&\textrm{
if }v<v_{min}+\frac{d_{0}}{T},\\\
v-\frac{d_{0}}{T}+\frac{\varepsilon}{T}&\textrm{ if
}v_{min}+\frac{d_{0}}{T}\leq v\leq v_{max},\end{array}\right.$

under hypothesis $b)$.

If $(X,Y)$ is a mixed strategy Nash equilibrium, then the support of the
random variable $X$ should be contained in the set of ${\mathbf{x}}$’s best
replies, the corresponding is true for variable $Y$. In this particular case,
$X$ has support on
$\\{v_{min}\\}\cup(v_{min}+\frac{d_{0}}{T},v_{max})\cup\\{v_{max}\\}$, while
$Y$ has support on
$\\{v_{min}\\}\cup(v_{min},v_{max}-\frac{d_{0}}{T})\cup\\{v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}\\}$.

Hence, a mixed strategy $X$ with the support obtained is of the form

$\displaystyle X=\left\\{\begin{array}[]{ll}v_{min}&\textrm{ with probability
}p_{1},\\\ U&\textrm{ with probability }p_{2},\\\ v_{max}&\textrm{ with
probability }1-p_{1}-p_{2},\end{array}\right.$

where $p_{1},p_{2}\in[0,1]$, and $U$ is a continuous random variable with
density $f_{U}(u)$ and support contained in
$(v_{min}+\frac{d_{0}}{T},v_{max})$. Similarly, a mixed strategy $Y$ with the
desired support is

$\displaystyle Y=\left\\{\begin{array}[]{ll}v_{min}&\textrm{ with probability
}q_{1},\\\ V&\textrm{ with probability }q_{2},\\\
v_{max}-\frac{d}{T}+\frac{\varepsilon}{T}&\textrm{ with probability
}1-q_{1}-q_{2},\end{array}\right.$

where $q_{1},q_{2}\in[0,1]$, and $V$ is a continuous random variable with
density $f_{V}(v)$ with support contained in
$(v_{min},v_{max}-\frac{d_{0}}{T})$.

To compute the density of $U$, we apply (A.4) to $Y$. Let us compute
${\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=v)$ when
$v\in(v_{min}-\frac{d_{0}}{T},v_{max}-\frac{d_{0}}{T})$:

$\displaystyle{\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=v)$ $\displaystyle=$
$\displaystyle{\mathbb{E}}(u_{{\mathbf{y}}}(X,v))$ $\displaystyle=$
$\displaystyle
p_{1}u_{{\mathbf{y}}}(v_{min},v)+p_{2}{\mathbb{E}}(u_{{\mathbf{y}}}(U,v))+(1-p_{1}-p_{2})u_{{\mathbf{y}}}(v_{max},v)$
$\displaystyle=$ $\displaystyle p_{1}(D+Tv_{min}-Tv-d_{0})$ $\displaystyle+$
$\displaystyle p_{2}\Big{[}\int_{v_{min}}^{v+\frac{d_{0}}{T}}(D+Tu-Tv-
d_{0})f_{U}(u)du+\int_{v+\frac{d_{0}}{T}}^{v_{max}}(Tu-Tv-
d_{0})f_{U}(u)du\Big{]}$ $\displaystyle+$
$\displaystyle(1-p_{1}-p_{2})(Tv_{max}-Tv-d_{0})$ $\displaystyle=$
$\displaystyle
p_{1}D+p_{2}DF_{U}\Big{(}v+\frac{d_{0}}{T}\Big{)}+p_{1}Tv_{min}+(1-p_{1}-p_{2})Tv_{max}+p_{2}T{\mathbb{E}}(U)-Tv-
d_{0},$

where $F_{U}(u)$ is the cumulative probability distribution function of the
random variable $U$.

By (A.4), we have

(A.9)
$p_{1}D+p_{2}DF_{U}\Big{(}v+\frac{d_{0}}{T}\Big{)}+p_{1}Tv_{min}+(1-p_{1}-p_{2})Tv_{max}+p_{2}T{\mathbb{E}}(U)-Tv-
d_{0}=k,$

for some constant $k$.

Since $F_{U}(v_{max})=1$, when we plug $v=v_{max}-\frac{d_{0}}{T}$, we obtain
its value

(A.10)
$k=(p_{1}+p_{2})D+p_{1}Tv_{min}-(p_{1}+p_{2})Tv_{max}+p_{2}T{\mathbb{E}}(U).$

On substituting $k$ into (A.9) we obtain

$F_{U}\Big{(}v+\frac{d_{0}}{T}\Big{)}=1-\frac{T(v_{max}-v)-d_{0}}{p_{2}D}.$

Let $u=v+\frac{d_{0}}{T}$. Then, $u\in(v_{min}+\frac{d_{0}}{T},v_{max})$ and
$F_{U}(u)=1-\frac{T(v_{max}-u)}{p_{2}D}$. From this we have
$u^{*}=v_{max}-\frac{p_{2}D}{T}$ is the value such that $F_{U}(u^{*})=0$.

The conclusion is that $U$ is uniformly distributed on the interval
$(v_{max}-\frac{p_{2}D}{T},v_{max})$, thus

(A.11) ${\mathbb{E}}(U)=v_{max}-\frac{p_{2}D}{2T}.$

In the same manner we can see that $V$ has uniform distribution on the
interval $(v_{max}-\frac{d_{0}}{T}-\frac{q_{2}}{T},v_{max}-\frac{d_{0}}{T})$,
with expectancy given by

(A.12) ${\mathbb{E}}(V)=v_{max}-\frac{d_{0}}{T}-\frac{q_{2}D}{2T}.$

To compute the values of $p_{1}$ and $p_{2}$ necessary for the
$\varepsilon$-equilibrium, we use (A.3). We first compute the conditional
expectancy of $u_{{\mathbf{y}}}(X,Y)$ given $Y$,

$\displaystyle{\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=v_{min})$ $\displaystyle=$
$\displaystyle
p_{1}u_{{\mathbf{y}}}(v_{min},v_{min})+p_{2}{\mathbb{E}}(u_{{\mathbf{y}}}(U,v_{min}))+(1-p_{1}-p_{2})u_{{\mathbf{y}}}(v_{max},v_{min})$
$\displaystyle=$ $\displaystyle
p_{1}(D-d_{0})+p_{2}\Big{[}\int_{v_{max}-\frac{p_{2}D}{T}}^{v_{max}}(Tu-
Tv_{min}-d_{0})f_{U}(u)\,du\Big{]}$ $\displaystyle+$
$\displaystyle(1-p_{1}-p_{2})(Tv_{max}-Tv_{min}-d_{0})$ $\displaystyle=$
$\displaystyle
p_{1}(D-d_{0})+p_{2}(T{\mathbb{E}}(U)-Tv_{min}-d_{0})+(1-p_{1}-p_{2})(Tv_{max}-Tv_{min}-d).$

By (A.11), we have

(A.13)
${\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=v_{min})=p_{1}D-d_{0}+(1-p_{1})(Tv_{max}-Tv_{min})-p_{2}^{2}\frac{D}{2}.$

Computing ${\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=V)$ yields

${\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=V)=\int_{v_{max}-\frac{d_{0}}{T}-\frac{q_{2}D}{T}}^{v_{max}-\frac{d_{0}}{T}}{\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=v)f_{V}(v)\,dv.$

Since we know that the integrand is constant and its value is given by
equations (A.10) and (A.11), we directly obtain

(A.14)
${\mathbb{E}}(u_{{\mathbf{y}}}(X,Y)|Y=V)=(p_{1}+p_{2})D-p_{1}(Tv_{max}-Tv_{min})-p_{2}^{2}\frac{D}{2}.$

We are left with the task of determining the expected value of
$u_{{\mathbf{y}}}(X,Y)$ conditioned on the value
$Y=v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}$,

(A.15)
$\displaystyle{\mathbb{E}}\Big{(}u_{{\mathbf{y}}}(X,Y)|Y=v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}\Big{)}$
$\displaystyle=$ $\displaystyle
p_{1}u_{{\mathbf{y}}}\Big{(}v_{min},v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}\Big{)}+p_{2}{\mathbb{E}}\Big{(}u_{{\mathbf{y}}}\Big{(}U,v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}\Big{)}\Big{)}$
$\displaystyle+$
$\displaystyle(1-p_{1}-p_{2})u_{{\mathbf{y}}}\Big{(}v_{max},v_{max}-\frac{d_{0}}{T}+\frac{\varepsilon}{T}\Big{)}$
$\displaystyle=$ $\displaystyle p_{1}(D-T(v_{max}-v_{min})-\varepsilon)$
$\displaystyle+$ $\displaystyle
p_{2}\int_{v_{max}-\frac{p_{2}D}{T}}^{v_{max}}(D-T(v_{max}-u)-\varepsilon)f_{U}(u)\,du+(1-p_{1}-p_{2})(D-\varepsilon)$
$\displaystyle=$ $\displaystyle p_{1}(D-T(v_{max}-v_{min})-\varepsilon)$
$\displaystyle+$ $\displaystyle
p_{2}(D-Tv_{max}+T{\mathbb{E}}(U)-\varepsilon)+(1-p_{1}-p_{2})(D-\varepsilon)$
$\displaystyle=$ $\displaystyle D-\varepsilon-
p_{1}T(v_{max}-v_{min})-p_{2}^{2}\frac{D}{2},$

where we used (A.11) in the last equality.

Lemma (A.3) implies that in order to have an $\varepsilon$-equilibrium, the
expressions (A.13), (A.14) and (A.15) must be equal. This system of equations
has the unique solution

$p_{1}=1-\frac{T(v_{max}-v_{min})-d_{0}}{D},\qquad
p_{2}=\frac{T(v_{max}-v_{min})-d_{0}}{D},\qquad 1-p_{1}-p_{2}=0.$

We now apply this argument again, to obtain the expectancy of the random
variable $u_{{\mathbf{x}}}(X,Y)$ conditioned on the values of $X$, as well as
the values $q_{1},q_{2}$ necessary to have an $\varepsilon$-equilibrium. In
this case, there are many solutions. Indeed, any combination $q_{1},q_{2}$
satisfying

$0\leq q_{1},q_{2},\qquad q_{1}+q_{2}=\frac{T(v_{max}-v_{min})}{D},\qquad
1-q_{1}-q_{2}=1-\frac{T(v_{max}-v_{min})}{D},$

fulfills equation (A.3).

Given that the support of $V$ is
$\Big{(}v_{max}-\frac{d_{0}}{T}-q_{2}\frac{D}{T},\,v_{max}-\frac{d_{0}}{T}\Big{)}\subseteq\Big{(}v_{min},v_{max}-\frac{d_{0}}{T}\Big{)}$,
it is necessary to impose the condition $q_{2}\leq\frac{d-d_{0}}{D}$.

  * •

Case c)

From the conditions stated in $c)$, it follows that

$\displaystyle B_{{\mathbf{x}}}(v)=\left\\{\begin{array}[]{ll}v_{max}&\textrm{
if }v=v_{min},\\\ v_{min}&\textrm{ if }v>v_{min}.\end{array}\right.$

Intuitively, under hypothesis $c)$, it always happens that $x_{T}\leq y_{T}$
for every pair of strategies $v_{{\mathbf{x}}},v_{{\mathbf{y}}}$. Equality
holds only when $v_{{\mathbf{x}}}=v_{max}$ and $v_{{\mathbf{y}}}=v_{min}$.

Similarly, one can check that

$\displaystyle B_{{\mathbf{y}}}(v)=\left\\{\begin{array}[]{ll}v_{min}&\textrm{
if }v<v_{max},\\\ v_{max}+\frac{\varepsilon}{T}&\textrm{ if
}v=v_{max},\end{array}\right.$

where last case is an $\varepsilon$-best reply.

To find the $\varepsilon$-equilibria, we define $X$ to be a random variable
such that

$\mathbb{P}(X=v_{min})=p,\qquad\mathbb{P}(X=v_{max})=1-p,\quad\text{for some
probability $p\in[0,1]$.}$

Similarly, we define a random variable $Y$ such that

$\mathbb{P}(Y=v_{min})=q,\qquad\mathbb{P}\Big{(}Y=v_{min}+\frac{\varepsilon}{T}\Big{)}=1-q,\quad\text{for
$q\in[0,1]$.}$

An $\varepsilon$-equilibrium requires
$\mathbb{E}(u_{{\mathbf{x}}}(v_{min},Y))=\mathbb{E}(u_{{\mathbf{x}}}(v_{\max},Y))$,
which is exactly the condition (A.3) when there is no continuous part for $X$.

Since

$\mathbb{E}(u_{{\mathbf{x}}}(v_{min},Y))=qu_{{\mathbf{x}}}(v_{min},v_{min})+(1-q)u_{{\mathbf{x}}}\Big{(}v_{min},v_{min}+\frac{\varepsilon}{T}\Big{)}=qd+(1-q)(d+\varepsilon),$

and

$\mathbb{E}(u_{{\mathbf{x}}}(v_{max},Y))=qu_{{\mathbf{x}}}(v_{max},v_{min})+(1-q)u_{{\mathbf{x}}}\Big{(}v_{max},v_{min}+\frac{\varepsilon}{T}\Big{)}=q\Big{(}\frac{D}{2}\Big{)}+(1-q)(\varepsilon),$

we can equalize the two equations and solve to obtain $q=\frac{2d}{D}$. Note
that (2.1) implies that $0<q<1$.

Similarly, we should have
$\mathbb{E}(u_{{\mathbf{y}}}(X,v_{min}))=\mathbb{E}\Big{(}u_{{\mathbf{y}}}\Big{(}X,v_{min}+\frac{\varepsilon}{T}\Big{)}\Big{)}$.
The explicit formulas being

$\mathbb{E}(u_{{\mathbf{y}}}(X,v_{min}))=pu_{{\mathbf{y}}}(v_{min},v_{min})+(1-p)u_{{\mathbf{y}}}(v_{max},v_{min})=p(D-d)+(1-p)\frac{D}{2},$

and

$\mathbb{E}\Big{(}u_{{\mathbf{y}}}\Big{(}X,v_{min}+\frac{\varepsilon}{T}\Big{)}\Big{)}=p\,u_{{\mathbf{y}}}\Big{(}v_{min},v_{min}+\frac{\varepsilon}{T}\Big{)}+(1-p)u_{{\mathbf{y}}}\Big{(}v_{max},v_{min}+\frac{\varepsilon}{T}\Big{)}=p(D-d-\varepsilon)+(1-p)(D-\varepsilon).$

Matching and solving the two yields $0<1-p=\frac{2\varepsilon}{D}<1$.

  * •

Case d)

Assume that player ${\mathbf{y}}$ chooses strategy $v_{{\mathbf{y}}}$
satisfying (2.1). Then

(A.18)
$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})=d_{{\mathbf{x}}}(Tv_{{\mathbf{x}}},y_{0}+Tv_{{\mathbf{y}}}).$

By assumption $d)$, we have

$T(v_{{\mathbf{x}}}-v_{{\mathbf{y}}})\leq
T(v_{max}-v_{min})<d_{{\mathbf{x}}}(x_{0},y_{0})=y_{0},$

so $y_{0}+Tv_{{\mathbf{y}}}-Tv_{{\mathbf{x}}}>0$ for every
$v_{{\mathbf{x}}},v_{{\mathbf{y}}}$. Then, (A.18) is equal to

$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})=y_{0}+T(v_{{\mathbf{y}}}-v_{{\mathbf{x}}}),$

which is bounded by

$u_{{\mathbf{x}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})\leq
y_{0}+T(v_{{\mathbf{y}}}-v_{min})=u_{{\mathbf{x}}}(x_{0},v_{min},y_{0},v_{{\mathbf{y}}}).$

We conclude that $v_{{\mathbf{x}}}=v_{min}$ is ${\mathbf{x}}$’s best reply to
any strategy $v_{{\mathbf{y}}}$ played by ${\mathbf{y}}$.

Similarly, if ${\mathbf{x}}$ chooses strategy $v_{{\mathbf{x}}}$, then

(A.19)
$u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})=d_{{\mathbf{y}}}(Tv_{{\mathbf{x}}},y_{0}+Tv_{{\mathbf{y}}}).$

We have already proven that $y_{0}+Tv_{{\mathbf{y}}}-Tv_{{\mathbf{x}}}>0$ for
every $v_{{\mathbf{x}}},v_{{\mathbf{y}}}$, so (A.19) is equal to

$u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})=D+Tv_{{\mathbf{x}}}-y_{0}-Tv_{{\mathbf{y}}}.$

We can bound the last expression by

$u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{{\mathbf{y}}})=D-y_{0}+T(v_{{\mathbf{x}}}-v_{{\mathbf{y}}})\leq
D-y_{0}+T(v_{{\mathbf{x}}}-v_{min})=u_{{\mathbf{y}}}(x_{0},v_{{\mathbf{x}}},y_{0},v_{min}).$

This implies $v_{{\mathbf{y}}}=v_{min}$ is ${\mathbf{y}}$’s best reply to any
strategy $v_{{\mathbf{x}}}$ played by ${\mathbf{x}}$. The conclusion is that
$(v_{min},v_{min})$ is the unique Nash equilibrium.

$\square$

Proof of Theorem 3:

First, note that $d_{0}>d$ implies $N\equiv 0$, and the result holds
trivially.

Assume that $0<d_{0}<d$, and suppose that $0<d_{k}<d$ for some $k\geq 0$.
Then, the strategies
$(U,v_{min}),(U,V),(U,v_{max}-\frac{d_{k}}{T}+\frac{\varepsilon}{T}),(v_{min},v_{min}),(v_{min},V)$
lead to $0<d_{k+1}<0$ with probability one.

If the strategies of ${\mathbf{x}}$ and ${\mathbf{y}}$ are instead
$(v_{min},v_{max}-\frac{d_{k}}{T}-\frac{\varepsilon}{T})$, then
$d_{k+1}=d+\varepsilon$. We can uniformly bound from below the probability
that the players adopt these strategies by

$\mathbb{P}\Big{(}(X,Y)=\Big{(}v_{min},v_{max}-\frac{d_{k}}{T}-\frac{\varepsilon}{T}\Big{)}\Big{)}=\Big{(}1-\frac{d-d_{k}}{D}\Big{)}\Big{(}1-\frac{d_{k}}{D}\Big{)}\geq\Big{(}1-\frac{d}{D}\Big{)},\quad\forall\enspace
0<d_{k}<d,$

where the inequality can be obtained by calculus (or by noting that this
probability is an inverted parabola, as a function of $d_{k}$). Therefore,

$\mathbb{P}(N>k)\leq\mathbb{P}(G>k),$

where $G$ is a geometric random variable with parameter $1-\frac{d}{D}$, and
the result follows.

Finally, assume that $d_{0}=d$. If players ${\mathbf{x}}$ and ${\mathbf{y}}$
choose $(v_{min},v_{min})$, then $d_{1}=d$. Any other strategy choice yields
$d_{1}\neq d$.

Define $M=\min\\{n\geq 1:d_{n}\neq d\\}$. By the above remark, $M$ has
geometric distribution on the natural numbers with parameter
$1-\Big{(}1-\frac{2\varepsilon}{D}\Big{)}\Big{(}\frac{2d}{D}\Big{)}$. After
$M$ trials, we are on the conditional space where ${\mathbf{x}}$ and
${\mathbf{y}}$ do not play $(v_{min},v_{min})$, instead they choose

$\displaystyle(v_{max},v_{min})$ $\displaystyle\textrm{ with probability
}\frac{\Big{(}\frac{2\varepsilon}{D}\Big{)}\Big{(}\frac{2d}{D}\Big{)}}{1-\Big{(}1-\frac{2\varepsilon}{D}\Big{)}\Big{(}\frac{2d}{D}\Big{)}},$
$\displaystyle(v_{min},v_{min}+\frac{\varepsilon}{T})$ $\displaystyle\textrm{
with probability
}\frac{\Big{(}1-\frac{2\varepsilon}{D}\Big{)}\Big{(}1-\frac{2d}{D}\Big{)}}{1-\Big{(}1-\frac{2\varepsilon}{D}\Big{)}\Big{(}\frac{2d}{D}\Big{)}},$
$\displaystyle(v_{max},v_{min}+\frac{\varepsilon}{T})$ $\displaystyle\textrm{
with probability
}\frac{\Big{(}\frac{2\varepsilon}{D}\Big{)}\Big{(}1-\frac{2d}{D}\Big{)}}{1-\Big{(}1-\frac{2\varepsilon}{D}\Big{)}\Big{(}\frac{2d}{D}\Big{)}}.$

The first election leads to $d_{M+1}=0$, while the other two give $d_{M+1}>d$.
This concludes the proof.

$\square$


0Front MatterrootNode

I hereby certify that the submitted work is my own work, was completed while registered as a candidate for the degree stated on the Title Page, and I have not obtained a degree elsewhere on the basis of the research presented in this submitted work.

Georgios Tzounas, 2020.
All Rights Reserved.

CHAPTER: LIST OF ACRONYMS AND ABBREVIATIONS

agc[AGC]Automatic Generation Control
aiits[AIITS]All-Island Irish Transmission System
avr[AVR]Automatic Voltage Regulator
bbd[BBD]Bordered-Block Diagonal
cih[CIH]Contour Integration with Hankel matrices
cirr[CIRR]Contour Integration with Rayleigh-Ritz
coi[CoI]Center of Inertia
csr[CSR]Compressed Sparse Row
dae[DAE]Differential-Algebraic Equation
ddae[DDAE]Delay Differential-Algebraic Equation
dde[DDE]Delay Differential Equation
emt[EMT]Electro-Magnetic Transient
entsoe[ENTSO-E]European Network of Transmission System Operators for Electricity
erd[ERD]Explicitly Restarted and
ess[ESS]Energy Storage System
facts[FACTS]Flexible AC Transmission System

FO[FO]Fractional Order
foc[FOC]Fractional-Order Controller
foi[FOI]Fractional-Order Integral
fopi[FOPI]Fractional-Order Proportional Integral
FOPID[FOPID]Fractional-Order Proportional Integral Derivative
fopss[FOPSS]Fractional-Order Power System Stabilizer

gd[GD]Generalized Davidson
gep[GEP]Generalized Eigenvalue Problem
grpr[GRPR]Generator Reactive Power Regulator
hvdc[HVDC]High Voltage Direct Current
IO[IO]Integer Order
itm[ITM]Implicit Trapezoidal Method
lep[LEP]Linear Eigenvalue Problem
LM[LM]Largest Magnitude
LRP[LRP]Largest Real Part
lti[LTI]Linear Time-Invariant
Passing Interface
nnz[NNZ]Number of Non-Zero
ode[ODE]Ordinary Differential Equation
omib[OMIB]One-Machine Infinite-Bus
ora[ORA]Oustaloup's Recursive Approximation
pde[PDE]Partial Differential Equation
pf[PF]Participation Factor
PI[PI]Proportional Integral
pid[PID]Proportional Integral Derivative
pod[POD]Power Oscillation Damper
pr[PR]Proportional Retarded
pss[PSS]Power System Stabilizer
Communication Interface
rocof[RoCoF]Rate of Change of Frequency
SM[SM]Smallest Magnitude
sssa[SSSA]Small-Signal Stability Analysis
statcom[STATCOM]STATic synchronous COMpensator
svc[SVC]Static Var Compensator
svr[SVR]Secondary Voltage Regulation
tdi[TDI]Time Domain Integration
tg[TG]Turbine Governor
vsc[VSC]Voltage Source Converter
wadc[WADC]Wide-Area Damping Controller
wams[WAMS]Wide-Area Measurement System
wscc[WSCC]Western Systems Coordinating Council

CHAPTER: NOTATION

This section states the notation adopted throughout the thesis.

§.§.§ Vectors and Matrices

$a$, $A$ scalar

$\bfg a$, $\bfb a$ vector

$\bfg A$, $\bfb A$ matrix

$\bfg A\T$ matrix transpose

$\bfg A\CT$ matrix conjugate transpose (Hermitian)

$\bfg I_\nd$ identity matrix of dimensions $\nd \times \nd$

$\bfg 0_{\nd,m}$ zero matrix of dimensions $\nd \times m$

§.§.§ Sets and Units

$\mathcal{C}^n$ $n$ time continuously differentiable functions

$\mathbb{C}$ complex numbers

$\jj$ unit imaginary number

$\mathbb{I}$ imaginary numbers

$\mathbb{N}$ natural numbers

$\mathbb{R}$ real numbers

$\mathbb{Z}$ integer numbers

§.§.§ Time and Frequency Domain

$a(t)$ time domain quantity

$\Dt a(t)$ first order derivative

$\ddot a(t)$ second order derivative

$a^{(n)}(t)$ $n$-th order derivative (of fractional order or of integer order $\geq 3$)

$a(s)$ frequency domain quantity

$\mathcal{L}$ Laplace transform

$s$ complex Laplace variable

$z$ spectral transform

§.§.§ Eigenvalues and Eigenvectors

${\rm f}_{\rm n}$ natural frequency of eigenvalue

$\reigvel$ element of right eigenvector

$\reigv$ right eigenvector

$\reigvmat$ right modal matrix

$\leigvel$ element of left eigenvector

$\leigv$ left eigenvector

$\leigvmat$ left modal matrix

$\zeta$ damping ratio of eigenvalue

$\lambda$ eigenvalue

${\rm Im}(\lambda)$ imaginary part of eigenvalue $\lambda$

${\rm Re}(\lambda)$ real part of eigenvalue

$\ninf$ multiplicity of infinite eigenvalue

$\nf$ number of finite eigenvalues

$\PF$ participation factor

$\PFmat$ participation matrix

§.§.§ Parameters

$B$ susceptance

$c$ constant

$D$ damping coefficient

$G$ conductance

$h$ time step size

$K$ control gain

$M$ mechanical starting time

$R$ resistance

$\mathcal{R}$ droop constant of primary frequency control

$T$ time constant

$X$ reactance

$\tau$ time delay

§.§.§ Variables and Functions

$e$ electromotive force

$P$ active power

$Q$ reactive power

$t$ time

$u$ input signal

$v$ voltage magnitude

$\mathcal{V}$ Lyapunov function

$w$ output signal

$x$ state variable

$y$ algebraic variable

$\delta$ angular position

$\theta$ voltage phase angle

$\uptau$ torque

$\omega$ angular speed

§.§.§ Superscripts and Subscripts

$\rm d$ direct axis of the $\rm dqo$ transform

$d$ delayed quantity

$\rm m$ mechanical

$\rm min$ minimum

$\rm max$ maximum

$o$ initial condition

$\rm q$ quadrature axis of the $\rm dqo$ transform

$\rm r$ rotor

$\rm ref$ reference

$\rm s$ stator

CHAPTER: CONCLUSIONS AND FUTURE WORK

This thesis proposes novel
sssa-based techniques with application to modal analysis, robust control, and numerical integration of power systems.
The objective of this chapter is to summarize the main conclusions of the thesis and support directions for future work.

* Modal analysis: The study on modal analysis shows that the classical assumptions made when computing pf, namely that the system is modeled as a set of ode and that all eigenvalues are distinct, are not binding.
In fact, considering a singular system of differential equations with eigenvalue multiplicities, allows extracting a generalized expression of pf in the form of eigen-sensitivities, from which
pf in the classical sense arise as a special case.
Moreover, prior to this work, algebraic variables of a power system model were either eliminated or treated as states with infinitely fast dynamics and, as such, their pf to finite modes were
considered to be null.
The proposed formulation shows that it is possible to quantify the pf of algebraic variables of a power system, and
in general of any function of the system variables, through the definition of appropriate input/output vectors of the system's state-space model.

Future work will focus on studying the
effect of network constraints on the design of control schemes based on the
proposed modal analysis.
A control signal/actuator selection
that is based on the pf of line power flows and takes into account information on the
current/power capacity limits of transmission lines is a relevant example. Such information is readily available to system operators and thus it can be effectively included to the analysis.

Moreover, the proposed formulation of pf
can be extended to include the analysis
of systems of
fractional differential equations. Such analysis
will allow measuring the coupling between the variables of a power system models and its fractional order dynamics, such as the ones introduced by foc discussed in Chapter <ref>.
Moreover, extension of the proposed modal participation analysis is also relevant for systems of ddae.
This will allow an efficient assessment of control signals impacted by time delays, including wide-area controllers and controllers with intentional delays, such as the ones described in
Chapter <ref>. In this case, the
calculation of pf will be a challenging task, since the existence of an infinite dimensional spectrum makes it difficult to determine the coupling between eigenvectors and system variables in an accurate and efficient way.

* Fractional Control: The contributions of the thesis are a systematic study of controllers based on fractional calculus and a technique to carry out eigenvalue analysis
and assess the small-signal stability of power systems with inclusion of exact fractional dynamics of Caputo type.
Furthermore, the properties of ora are investigated and, through time domain simulations, it is shown that foc
perform better than their conventional IO versions for synchronous machine agc,
ess frequency control, and
statcom voltage control,
while they require only a small additional tuning effort.

A relevant extension of the work on foc is the study of practical
aspects, such as potential modeling and stability issues introduced with the inclusion of control saturation limits.

With this regard, a preliminary study on foc control limits for power system applications can be found in <cit.>.

Furthermore, time domain simulations in Chapter <ref> are based on the widely employed ora, but there exist also other methods that approximate fractional order dynamics, see e.g. <cit.>. To the best of our knowledge, a systematic study that provides an eigenvalue-based comparison of approximation methods for fractional order dynamics with application to power systems is still missing and it is worth consideration.

* Delay-based Control: The study on time-delayed control
focuses on the stability boundary of power systems with
delay-based pss, as well as on the
relationship between the
existence of delay-independent stability and connected stability domains in the delay-control gain space.
Analytical results based on the omib system, as well as a numerical analysis
of the IEEE 14-bus system,
illustrate that in contrast to their bad reputation, large delays
may achieve best damping characteristics, provided that the pss is properly configured.
It is shown that one such possible configuration is
to employ a two-channel pss, which permits a fully connected stability region in the delay-gain space. Then, non-linear dynamics can be tuned
to achieve high damping,
by fully exploring the parameter space and without introducing instabilities.

A possible future work direction is the design of an adaptive control scheme so that the delay and gain are automatically tuned,
following a stable path and through consecutive quasi-steady state
shifts of the system equilibrium. This is particularly interesting in case that part of the delay is inherent, i.e. represents measurement and/or
communication latency. In this case, high damping can be achieved by
adding an artificial controlled delay on top of the inherent delay.
This is a novel approach, whose effectiveness can be assessed in comparison with other techniques used for mitigating the
destabilizing effect of communication delays, such as delay compensation methods <cit.>.

* Numerical Integration:
Finally, this work proposes an “one-step delay" approximation technique for the numerical integration of the dae utilized to study power system transient stability, and provides a
first evaluation of its accuracy, convergence and computational burden.
Chapter <ref> shows that the proposed approach
allows exploiting delays equal to the time step of the numerical integration, in order to reduce
the coupling of the equations of conventional power system dae models for transient stability analysis. sssa-based techniques are discussed for the
selection of the variables of a power system dae model that
can be delayed, as well as for the estimation of the maximum admissible time delay, so that the accuracy of trajectories is not compromised.

The next step is to embed the proposed one-step delay approximation technique in algorithms that apply state-of-the art parallelization techniques.
This will typically require to exploit a current injection-based power system model formulation, in order
to take advantage of the bbd structure of
the corresponding Jacobian matrix, see
e.g. <cit.>.

Delays are known to have a destabilizing effect. For example, the latency induced by communication
networks affects wide area measurement systems and controllers <cit.>.
Despite their bad reputation, delays, if properly added, form novel controllers that are surprisingly
beneficial on damping power system electromechanical oscillations.
In practice, an undesired delay may stem from a communication network.
Conventional solutions
attempt to compensate the destabilizing delay-effect <cit.>. This work indicates that an alternative may be
to properly introduce an extra delay.