# A Coalitional Game for Demand-Side Management in a Low-Voltage Resistive
Micro-Grid with Multiple Electricity Retailers ††thanks: *Sponsored by
Mexico’s CONACyT, scholarship number: 440742. ††thanks: This work was
supported by EPSRC under Grants No EP/S001107/1 and EP/S031863/1. ††thanks: F.
Genis Mendoza, and P. R. Baldivieso-Monasterios are with the Department of
Automatic Control and Systems Engineering, The University of Sheffield, Mappin
Street, S1 3JD Sheffield, United Kingdom. {fgenismendoza1,
<EMAIL_ADDRESS>G Konstantopoulos is with the
Department of Electrical and Computer Engineering, University of Patras, Rion,
26500, Greece<EMAIL_ADDRESS>††thanks: D. Bauso is with the
Jan C. Willems Center for Systems and Control, ENTEG, Fac. Science and
Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The
Netherlands and Dipartimento di Ingegneria, Università di Palermo, viale delle
Scienze, Palermo, Italy<EMAIL_ADDRESS>
Fernando Genis Mendoza, Pablo R. Baldivieso-Monasterios*, Dario Bauso and
George Konstantopoulos

###### Abstract

An existing challenge in power systems is the implementation of optimal demand
management through dynamic pricing. This paper encompasses the design,
analysis and implementation of a novel on-line pricing scheme based on
coalitional game theory. The setting consists of a network with multiple
energy retailers competing to attract consumers by announcing a price in a
hierarchical leader-follower structure. The process of coalition formation
under such a pricing scheme can be viewed as a game for which we show that a
Stackelberg equilibrium exists, _i.e._ , given a price, consumers will respond
by conforming to a reciprocal power consumption quantity. We propose a
coalition formation algorithm and perform a game-theoretic stability analysis
on the resulting coalitions. We integrate the pricing setting with a resistive
micro-grid dynamic model. In this context we analyse the behaviour of the
integrated system, bridging the gap between market and physical layers of the
problem. Simulations provide a comparison of profits generated by the proposed
scheme against a more traditional single retailer scheme, while simultaneously
showing convergence towards a steady-state equilibrium. Additionally, we shed
light into the system’s physical response when subject to our proposed pricing
scheme.

###### Index Terms:

Micro-grid, coalitional games, Stackelberg equilibrium, on-line pricing,
resistive network, stability.

##  I Introduction

With the advent of the smart grid paradigm, more ways to distribute power have
emerged, bringing changes to the electricity market. Here, governments and
general consumers now seek and switch to better providers and sources of power
that enable them to fulfill certain requirements while still being profitable.
Such paradigm has also enabled a higher degree of communication between
consumers and energy retailers, providing new set-ups where these interact and
cooperate to optimise their outputs. Dynamic pricing schemes in electrical
systems establish a viable way to optimise and shift consumption during peak
times without compromising generation and distribution systems [1]. These
schemes allow retailers to make tariff changes based on demand whilst helping
users minimise costs of their consumption [2]. An underlying assumption is
that both suppliers and consumers are rational, namely they aim to optimise
their behaviour, and as consequence, a price change entails a change in
consumption [1].

The landscape of dynamic pricing algorithms in the literature is vast. In this
brief literature review, we focus on describing pricing mechanisms in game-
theoretic terms. The framework of cooperative games, as mentioned in [2],
provides a natural method for describing and modeling pricing in electric
networks. The use and formulation of cooperative coalitional games in micro-
grids was first introduced in [3], where the proposed algorithm focuses on
reducing power losses and costs by forming coalitions of neighboring micro-
grids. A game where local micro-grids cooperate without participation of a
main grid is presented in [4] by assuming a known non-flexible demand. In [5]
the introduction of auction theory is used to define the pairing of micro-
grids that buy and sell. In [6], the authors propose forming coalitions in a
macro station, and using the Shapley value to distribute profits. A similar
approach is used in [7], where a case study is performed with data of an
existing micro-grid and a utility grid with fictitious interconnections. In
[8] and [9], we can see, from a numerical point of view, the relation between
coalition formation and changes in energy prices. A study of the case where
greedy prosumers do not align with the micro-grid’s decision is presented in
[10], here a balanced game is proposed without the need of calculating the
imputation set. A notion of fairness is introduced by [11] using particle
swarm optimisation and using the nucleolus as a game solution concept. A
bidding system for cooperating prosumers is presented in [12], however
constraints on power capacity and losses are ignored. The use of evolutionary
game theory in conjunction with coalitions is proposed in [13], where the
price is a quadratic function of the consumption. In summary, coalitional game
theory represents a powerful modelling tool to understand and enhance pricing
behaviours within an electric network.

We have, however, identified two fronts on which the existing body of
literature has had limited contributions. Existing approaches often neglect
the effect of pricing on the physical network. Additionally, with a few
exceptions [4, 14, 15], the coalitional games proposed in the literature do
not consider end consumers as players. In this paper, we aim to tackle both of
these challenges. The pricing scheme is modelled as a Stackelberg-based game
with incentive strategies where both _retailers_ and _consumers_ are players.
Retailers adjust their prices periodically, which in consequence will cause
them to lose or gain consumers, and at the same time, adjust the demand and
supply of energy in the network. We build upon our previous results in [16] to
analyse the game equilibria and coalition stability properties. We employ the
conductance matrix of the network as a tool in a cost-saving coalitional game;
this, to the best of the author’s knowledge, has not been done before. From
the physical perspective, we analyse the interactions between dynamic pricing
and a modified droop based control laws governing the voltage dynamics. We
leverage on the bounded integral control theory from [17]. Our approach aims
to provide a rigorous framework for modern energy trading platforms such as
[18] and [19].

The contributions and structure of this paper are:

  * •

We propose a coalitional game framework in Section III, where there are
multiple competing retailers in a micro-grid. To the best of the authors’
knowledge, this is a novel problem setup in micro-grid literature. We analyse
the stability, in a game-theoretic sense, of the coalitions generated by our
algorithm. Additionally, we introduce the basis for a novel method where a
cost-saving game is linked and formulated directly from the network
conductance matrix.

  * •

In Section IV, we show how consumers can improve their welfare by sharing
risks. The analysis further stresses the benefits of the coalitional approach
in a pricing setting.

  * •

We derive in Section V stability conditions, from a dynamical systems
perspective, for the physical system subject to our pricing algorithm. To the
best of the author’s knowledge, this is the first instance of such combined
analysis.

_Notation and definitions_ : A _game_
$(\mathcal{N},\\{\mathcal{H}_{i}\\}_{i\in\mathcal{N}},\\{u_{i}\\}_{i\in\mathcal{N}})$
is a triplet composed by a set of players $\mathcal{N}$, a collection action
sets $\mathcal{H}_{i}$, and a collection of payoff functions
$u_{i}\colon\mathcal{H}_{i}\to\mathbb{R}$. An _action profile_ for a set of
$|\mathcal{N}|$ players is an ordered tuple
$(h_{1},\ldots,h_{|\mathcal{N}|})$. The _best response set_ of player
$i\in\mathcal{N}$ is defined as
$\mathcal{Q}_{i}(h_{-i}):=\arg\max\\{u_{i}(h_{i},h_{-i})\colon
h_{i}\in\mathcal{H}_{i}\\}$ where
$h_{-i}=(h_{j})_{j\in\mathcal{N}\setminus\\{i\\}}$. For
$\mathcal{N}=\\{1,2\\}$, an _action profile_ $(h_{1}^{S},h_{2}^{S})$ is a
_Stackelberg Equilibrium_ for player 1 if
$h_{2}^{S}\in\mathcal{Q}_{2}(h_{1}^{S})$ and $u_{1}(h_{1}^{S},h_{2}^{S})\geq
u_{1}(h_{1},h_{2})$, $\forall
h_{1}\in\mathcal{H}_{1},h_{2}\in\mathcal{Q}_{2}(h_{1})$. A coalitional game is
defined by the pair $(\mathcal{N},\nu)$ with $\mathcal{N}$ a set of players
and $\nu\colon 2^{\mathcal{N}}\to\mathbb{R}$ its characteristic function. A
pay-off vector is an element $a\in\mathbb{R}^{|\mathcal{N}|}$ such that
$a_{i}\in\mathbb{R}$ represents the gains of player $i\in\mathcal{N}$. The
_core_ , $\mathcal{C}\subset\mathbb{R}^{|\mathcal{N}|}$, of a coalitional game
$(\mathcal{N},\nu)$ satisfies
$\mathcal{C}=\\{a\in\mathbb{R}^{|\mathcal{N}|}\colon\sum_{i\in
S}a_{i}\geq\nu(S),~{}\forall S\subset\mathcal{N}\\}$. A cooperative game
$(\mathcal{N},\nu)$ with $\nu(\emptyset)=0$ is: a _convex game_ if $\nu(S\cup
T)+\nu(S\cap T)\leq\nu(S)+\nu(T)$ $\forall S,T\subseteq\mathcal{N}$; a
_permutationally convex_ (PC) game if there exists a permutation $\pi$ of
$\mathcal{N}$ such that $\forall S\subseteq\mathcal{N}\setminus[k]$ and $k>j$,
$\nu([k]\cup S)-\nu([k])\leq\nu([j]\cup S)-\nu([j])$ where
$[k]=\\{k\\}\cup\\{i\in\mathcal{N}\colon\pi(i)<\pi(k)\\}$.

A power network is defined by a connected, undirected, and weighted graph
$\mathcal{G}=\mathcal{(N,E)}$, where $\mathcal{N}$ is the set of nodes, and
$\mathcal{E}\subseteq\mathcal{N}\times\mathcal{N}$ the set of edges defining
the interconnection topology. A $k-$path from node $i$ to $j$ is a sequence of
$k$ edges $\phi(i,j)=\\{e_{1},\ldots,e_{k}\\}\subset\mathcal{E}$. The mapping
$\omega\colon\mathcal{E}\to\mathbb{R}$ defines the weight of each edge such
that $\omega(i,j)=\omega_{ij}\in\mathbb{R}$. For undirected graphs, the
mapping $\omega(\cdot,\cdot)$ is symmetric and can be characterised by a
symmetric adjacency matrix
$A\in\mathbb{R}^{|\mathcal{N}|\times|\mathcal{N}|}$. The matrix
$B\in\mathbb{R}^{|\mathcal{N}|\times|\mathcal{N}|}$, such that $B_{ij}=1$ if
$A_{ij}>0$ and $B_{ij}=0$ otherwise, and $A$ determine the connectivity
properties of $\mathcal{G}$. From [20], $B^{k}$ and $A^{k}$ with
$k\in\mathbb{N}$ determine the number of $k-$paths and the sum of products of
weights of all $k-$paths respectively from nodes $i$ to $j$. The total weight
of $\mathcal{G}$ is
$\omega(\mathcal{E})=\sum_{e\in\mathcal{E}}\omega_{e}=\frac{1}{2}\sum_{i}\sum_{j}A_{ij}$.
The out degree and Laplacian matrices of $\mathcal{G}$ are
$D\coloneqq{\textnormal{diag}}(\sum_{j=1}^{N}A_{ij})$, and $\mathcal{L}=D-A$
respectively. The neighbours of node $i$ is
$\mathcal{N}_{i}=\\{j\in\mathcal{N}\colon\exists(i,j)\in\mathcal{E}\\}$; the
out-degree of node $i$ is $\delta_{i}=|\mathcal{N}_{i}|$. A tree is an
undirected graph in which any two vertices are connected by a sequence of
edges. The _Minimum Spanning Tree_ (MST) of $\mathcal{G(N,E)}$ is a tree
$T=(\mathcal{N,E^{*}})$ such that
${\mathcal{E}^{*}}={\arg\min}\\{\omega(\mathcal{E}^{\prime})\colon\mathcal{E}^{\prime}\subset\mathcal{E},\forall
i\in\mathcal{N},\exists j\in\mathcal{N},(i,j)\in\mathcal{E}^{\prime}\\}$.

##  II System Model, Definition and Preliminaries

In this section, we make precise the notions of retailers and consumers. In
our demand-side management problem, electricity prices regulate user
consumption. The consumer response to announced prices involves a hierarchical
structure fitting within a Stackelberg game framework [21]. In this setting,
_leaders_ , (retailers) play first by selecting a price; and a subset of
_followers_ (consumers) select their power consumption at a given price. When
consumers and retailers cannot obtain more benefit by modifying consumption or
price, then the game is at an equilibrium.

###  II-A Coalitional setting

To study the coalitional behaviour of our scheme, we employ a game-theoretic
framework. The set of players in the micro-grid is
$\mathcal{N}=\\{1,\ldots,N\\}$ and involves $N$ players. This is partitioned
into two non-overlapping sets: the set of retailers
$\mathcal{R}\subset\mathcal{N}$ and the set of consumers
$\mathcal{B}\subset\mathcal{N}$ with $\mathcal{R}\cup\mathcal{B}=\mathcal{N}$
and $\mathcal{R}\cap\mathcal{B}=\emptyset$. Since $\mathcal{R}$ and
$\mathcal{B}$ form a partition of $\mathcal{N}$, then
$N=|\mathcal{B}|+|\mathcal{R}|$ where $|\mathcal{B}|$ is the number of
consumers, and $|\mathcal{R}|$ is the number of retailers. Besides this basic
partition, we seek a pairing between a retailer and a subset of consumers.

######  Definition 1 (Retailer’s coalition).

For $r\in\mathcal{R}$, the _retailer coalition_ is
$S_{r}\coloneqq\\{r\\}\cup\mathcal{B}_{r}$ where
$\mathcal{B}_{r}=\\{b_{1},\ldots,b_{k}\\}\subset\mathcal{B}$.

The above definition assigns to each element in the retailer set, a
corresponding subset of consumers; this definition considers the case where a
retailer $r\in\mathcal{R}$ does not succeed in attracting any consumers,
_i.e._ , $\mathcal{B}_{r}=\emptyset$ and $S_{r}=\\{r\\}$. We invoke the
following assumptions:

######  Assumption 1.

The collection of sets $\\{S_{r}\\}_{r\in\mathcal{R}}$ forms a covering of
$\mathcal{N}$, _i.e._ , $\bigcup_{r}S_{r}=\mathcal{N}$.

A consequence of Assumption 1 is that each consumer $b\in\mathcal{B}$ needs to
be assigned to one retailer. Coalitions that have more than one retailer or
that share consumers are considered nonviable, this is formalised as follows.

######  Assumption 2.

[Viable coalitions]

  1. 1.

For $r\in\mathcal{R}$, its associated retailer coalition
$S_{r}\subset\mathcal{N}$ satisfies $\forall s\in\mathcal{R}\setminus\\{r\\}$,
$S_{r}\cap\mathcal{R}\setminus\\{s\\}=\emptyset$.

  2. 2.

For any $r,s\in\mathcal{R}$, $S_{r}\cap S_{s}=\emptyset$.

Coalitions satisfying Assumptions 1 and 2 are considered viable coalitions and
exclude all those containing more than one or no retailer.

###  II-B Consumer and Retailer Profit Functions

In our problem setting, both consumers and retailers are considered to be
price-taking rational agents, _i.e._ , both aim to maximise their profit for
producing or consuming energy. The profit function for a retailer
$r\in\mathcal{R}$ is
$\Pi_{r}(S_{r},\lambda_{r},P)={\lambda_{r}}P-C_{r}((1+\alpha_{r}^{loss})P,S_{r}),$
where $C_{r}(\cdot,\cdot)$ is a function corresponding to the cost of
producing $(1+\alpha_{r}^{loss})P$ units of power distributed among consumers
in $S_{r}$ at a price $\lambda_{r}$ and a network dependent loss coefficient
$\alpha_{r}^{loss}\in[0,1]$. Similarly, every consumer $b$ that has opted to
consume from $r$ calculates its profit as
$\Pi_{b}(\zeta,r)=U_{b}(\zeta,r)-{\lambda_{r}}\zeta$ where
$U_{b}(\cdot,\cdot)$ is the utility from consuming $\zeta$ amount of power.
Following [1], we impose restrictions on the choice of functions:

######  Assumption 3.

For each $(r,b)\in\mathcal{R}$ and $b\in\mathcal{B}$, the function
$C_{r}(\cdot,S_{r})\colon\mathbb{R}\to\mathbb{R}$
($U_{b}(\cdot,r)\colon\mathbb{R}\to\mathbb{R}$) is continuous, monotonically
increasing, and convex (concave).

Each retailer in our setting seeks to maximise its profits by selling power to
a subset of consumers $\mathcal{B}_{r}$ at a price ${\lambda_{r}}$. Similarly,
each consumer seeks to maximise its benefit by consuming ${P^{d}_{{b}}}$ power
from a retailer $r\in\mathcal{R}$. This process is captured by the following
coupled optimisation problems:

$\displaystyle\mathbb{P}_{r}(P^{d}_{\mathcal{S}_{r}})\colon$
$\displaystyle\mathop{\max}\limits_{{B}\subset\mathcal{B},~{}\lambda\in[\underline{\lambda},\bar{\lambda}]}~{}~{}\Pi_{r}(\\{r\\}\cup
B,\lambda,\sum_{b\in{B}}P_{{b}}^{d}),$ (1a)
$\displaystyle\mathbb{P}_{b}(r)\colon$
$\displaystyle\mathop{\max}\limits_{\zeta\in[\underline{\zeta},\bar{\zeta}]}~{}\Pi_{b}(\zeta,r),$
(1b)

where $P^{d}_{\mathcal{S}_{r}}=\\{P_{b}^{d}\\}_{b\in\mathcal{S}_{r}}$ is the
collection of power demands in the coalition of $r$ and
$[\underline{\lambda},\bar{\lambda}]\subset\mathbb{R}$ and
$[\underline{\zeta},\bar{\zeta}]\subset\mathbb{R}$ are the price and power
consumption limits respectively. Each retailer $r\in\mathcal{R}$ cannot
provide power to the consumers beyond its own generation capabilities

$(1+\alpha_{r}^{loss})\biggr{|}\sum_{b\in S_{r}}P^{d}_{{b}}\biggl{|}\leq
P_{r}^{\textnormal{max}}.$ (2)

The optimisation problems defined in (1) are of mixed-integer type which makes
their use in an online setting problematic. This problem, however, is of
interest conceptually because its solution yields both the optimal price for
each retailer $\lambda_{r}^{*}$, the optimal consumption for each consumer
${P_{b}^{d}}^{*}$, and the optimal partition
$\\{\mathcal{B}_{r}^{*}\cup\\{r\\}\\}_{r\in\mathcal{R}}$ of $\mathcal{N}$.

##  III Coalitional Game with Multiple Retailers

In this section, we propose an algorithmic way of handling the coupled
optimisation problems $\mathbb{P}_{r}(\cdot)$ and $\mathbb{P}_{b}(\cdot)$ for
each $r,b\in\mathcal{N}$. In fact, Algorithm 1 defines the coalition formation
procedure that employs game-theoretic concepts to achieve a Stackelberg
equilibrium of a cooperative game. We show that the outcome of this coalition
formation procedure is in fact a solution to (1) and is a stable solution of
the game.

###  III-A Cost Definition and Minimum Spanning Tree Problem

For a retailer $r\in\mathcal{R}$, its associated coalition
$S_{r}\subset\mathcal{N}$ induces a connection sub-graph
$\mathcal{G}_{r}=(S_{r},\mathcal{E}_{r})$ denoted here as _retailer’s cost
network_. The set of edges $\mathcal{E}_{r}$ comprises all links of the form
$(r,b)$ with $b\in\mathcal{B}_{r}$; the associated weights $\omega(r,b)$
determine the _direct connection_ costs and may represent different factors
such as physical position with respect to the supplier, power losses, or fees
imposed. The links of the type $(b,d)$ with $b,d\in\mathcal{B}_{r}$ and their
associated weights represent _aggregate connection_ and may represent benefits
of joining a coalition. The total value $c\colon 2^{\mathcal{N}}\to\mathbb{R}$
of coalition $(S,\mathcal{E}_{S})$ is given by the MST, _i.e._ ,
$c(S)=\omega(\mathcal{E}^{*}_{S})$.

$r_{j}$$(r_{j},b_{1})$$(r_{j},b_{2})$$(r_{j},b_{3})$$(b_{2},b_{3})$$(b_{1},b_{2})$$b_{3}$$b_{1}$$r_{1}$10030804090$b_{2}$
Figure 1: Example cost network from retailer $r_{1}$, and its MST (dashed).
Other retailers $r_{j}$ also assign similar costs (red).

######  Example 1 (MST for a coalition ${S}_{r}$).

Consider a coalition $S_{r_{1}}=\\{r_{1},b_{1},b_{2},b_{3}\\}$, with a cost
network given in Fig. 1. This network consists of five different edges
$\mathcal{E}_{r_{1}}=\\{(r_{1},b_{1}),(r_{1},b_{2}),(r_{1},b_{3}),(b_{1},b_{2}),(b_{1},b_{3})\\}$.
From the graph, there are seven possible trees that contain all nodes. The MST
is found to be the one containing the edges
$\\{(r_{1},b_{2}),(b_{1},b_{2}),(b_{1},b_{3})\\}$, yielding the cost of the
coalition as $c(S_{r_{1}})=150$.

Now we can define the characteristic function of the cooperative game
$(\mathcal{N},\nu)$ defined for all viable coalitions $S_{r}$:

$\nu({S_{r}})=\sum\limits_{b\in{\mathcal{B}_{r}}}\omega({r},{b})-c(S_{r}).$
(3)

The value for nonviable subsets, see Assumption 2, $X\subset\mathcal{N}$ is
$\nu(X)=0$. Our goal is to design a cost-saving game which creates an
incentive for consumers to join a larger coalition to increase their profits
by means of savings. This requirement can be formalised by requiring that
savings are larger in larger coalitions.

######  Assumption 4 (Coalition savings).

For a retailer $r\in\mathcal{R}$, if its associated coalition $S_{r}$ consists
of two or more consumers, then for all $b\in S_{r}$,
$\nu(\\{r,b\\})\leq\nu(S_{r})$.

In coalitional game theory, the concept of Shapley value is used to define
fair imputations [21]. Consider that consumers enter a coalition $S_{r}$ in a
given order $\sigma=\\{r,b_{i_{1}}\ldots b_{i_{k}}\\}$ where the ordering
number of buyer $b_{i_{k}}$ is denoted as $\sigma^{-1}(b_{i_{k}})$. For
example, if $S_{r}=\\{r,b_{3},b_{4},b_{6}\\}$ with the ordering
$\sigma=(r,b_{4},b_{3},b_{6})$, then $\sigma^{-1}(b_{4})=2$. The set of
predecessors of $b$ is $\rho_{b}^{\sigma}\coloneqq\\{s\in
S_{r}\colon\sigma^{-1}(s)<\sigma^{-1}(b)\\}$. As a result, the marginal value
for $b$ given a sequence $\sigma$ is
$m_{{b}}^{\sigma}=\nu(\rho_{{b}}^{\sigma}\cup\\{{b}\\})-\nu(\rho_{{b}}^{\sigma})$
and in vector form
$m^{\sigma}=(m^{\sigma}_{b_{1}},\ldots,m^{\sigma}_{b_{k}})$. The _Shapley
value_ is then calculated as the average of the marginal vector over all
permutations of sequences, $\Phi(\nu)=\frac{1}{{k!}}\sum_{\sigma}{m^{\sigma}}$
where $k$ is the total number of consumers in the coalition. The resulting
vector outputs the corresponding portion of savings imputed to each consumer
$b\in S_{r}$.

###  III-B Retailer’s Cost Network Derivation

A remaining issue is determining the weights of the retailer network
$(S_{r},\mathcal{E}_{r})$ for $r\in\mathcal{R}$. For a given
$r\in\mathcal{R}$, we propose the following method to compute the cost
network:

  1. 1.

_Obtain the adjacency matrices_ $A$ and $B$ from the network Laplacian
$\mathcal{L}\in\mathbb{R}^{N\times N}$,

  2. 2.

The weight $\omega(r,b)$ is the geometric average of all $n-$paths from $r$ to
$b$ within the micro-grid $\mathcal{G}$. Given $\gamma,\xi>0$ and setting
$n=1$,

$\omega(r,b)=\gamma\biggr{(}\frac{[A^{n}]_{rb}}{[B^{n}]_{rb}}\biggl{)}^{\frac{1}{n}}+n\xi,$

if $[A^{n}]_{rb}=0$ then set $n=n+1$. Otherwise terminate.

  3. 3.

For a pair of consumers $(b_{i},b_{j})$, the weight is
$\omega(b_{i},b_{j})=\gamma A_{{b_{i}}{b_{j}}}+\beta\xi$ if
$A_{{b_{i}}{b_{j}}}>0$ where $\beta>0$ and $\beta\neq\xi$, and
$\omega(b_{i},b_{j})=0$ otherwise.

The scalars $\gamma$, $\xi$, and $\beta$ represent prices related to edge
conductance, direct connection fees, and aggregate connection fees
respectively. We note that the cost increases linearly with the number of
walks needed to connect two nodes.

###  III-C Price, Consumption, and Coalition Formation

In this section, we define both cost $C_{r}(\cdot,\cdot)$ and utility
$U_{b}(\cdot,\cdot)$ function for retailers and consumers respectively. We
also establish a link between these functions and those arising from the
supplier cost network and how the combination of both leads to a solution of
the coalition formation problem. We propose for each $r\in\mathcal{R}$ and
$b\in\mathcal{B}$ the following cost and utility functions

$\displaystyle C(P,S_{r})$
$\displaystyle={\alpha_{{r}}}{(\lambda{P})^{2}}+\nu(S_{r}),$ (4a)
$\displaystyle U(P^{d}_{b},r)$
$\displaystyle={\alpha_{{b}}}{(P^{d}_{{b}})^{\frac{1}{6}}}+\kappa_{{r}}\delta_{{b}}^{{r}}-\omega(r,b)$
(4b)

where $\alpha_{r}>0$ and $\alpha_{b}>0$. For the former, the first term is
related to the generation costs for each retailer and the second is the
savings from the consumers connected to $r\in\mathcal{R}$ which will be
rewarded back via imputation. The utility function employs a concave function
similar to that of [22, 23]; a base payment $-\omega(r,b)$ for joining
$S_{r}$; and a _subsidy_ ${\kappa_{{r}}\delta_{{b}}^{{r}}}\geq 0$ to consumer
$b$ from retailer $r$. The subsidy term is equivalent to the Bahnzaf power
index employed in cooperative games dictating how pivotal is a player within a
coalition, see [24]. A form of the subsidy term is also used in [1] as an
incentive tool. The potential subsidy is announced by the retailer to the
consumer as an incentive for the latter to join its coalition, since in MST
games the players with more connections in the cost network can potentially
hold more value [21], enabling consumers to consequently increase their
profits. The subsidy is calculated after the coalition is formed; its value is
equal to each consumer’s Shapley value $\Phi(\nu)$ imputation. The following
lemma provides some properties of both functions.

######  Lemma 1.

For any $r\in\mathcal{R}$, the functions $C(\cdot,S_{r})$ and $U(\cdot,r)$
defined in (4) satisfy Assumption 3.

Our approach aims to solve the optimisations (1) in an iterative way. A
consumer $b\in\mathcal{B}$ evaluates prices $\lambda_{r}$, base payments
$\omega(r,b)$, and potential subsidies $\kappa_{r}\delta_{{b}}^{{r}}$ for all
$r\in\mathcal{R}$. Consumer $b$ chooses $r^{*}$ and then optimises its profit
$\Pi_{b}(\cdot,r^{*})$ to yield ${P_{b}^{d}}^{*}$ and reports to $r^{*}$. Each
retailer evaluates submissions from consumers according to its generation
constraints (2): if the constraint is violated, then retailer $r$ rejects
consumers with the least profit in its coalition. When a consumer $b$ is
rejected by $r$, then it chooses a different coalition from a retailer
contained in the set $\mathcal{R}\setminus\\{r\\}$. After all consumers have
chosen their retailers, savings are distributed via the Shapley value. This is
repeated _ad infinitum_.

Data: $\forall r\in\mathcal{R}$: $\tilde{S}_{r}$, $P_{r}^{g}$, and
$\Lambda_{r}$; $\forall b\in\mathcal{B}$, $P_{b}^{d}$

1 repeat

2 $\forall r\in\mathcal{R}$, _measure_ $\sum_{{b}\in S_{r}}P_{b}^{d}$ and
_evaluate_ (1a) to obtain $\lambda_{r}$;

3 $\forall r\in\mathcal{R}$, Broadcast $\lambda_{r}$ and
$\kappa_{r}\delta_{{b}}^{{r}}$ to all $b\in\mathcal{B}$;

4 $\forall b\in\mathcal{B}$, solve
$\max\\{\Pi_{b}(\zeta,r)\colon\zeta\in[\underline{\zeta},\bar{\zeta}],~{}r\in\mathcal{R}\\}$;

5 $\forall b\in\mathcal{B}$, announce ${P_{b}^{b}}^{*}$ to
$r^{*}=\arg\max\Pi_{b}(r,\zeta)$;

6 repeat

7 if  _for $r\in\mathcal{R}$, (2) does not hold_ then

8 repeat

9 $r$ rejects $\hat{b}\in S_{r}$ with the lowest $P_{b}^{d}$;

10

11 until  _( 2) holds_;

12 Each rejected $\hat{b}\in\mathcal{B}$ by $r\in\mathcal{R}$ solves
$\max\\{\Pi_{\hat{b}}(\zeta,s)\colon\zeta\in[\underline{\zeta},P_{s}^{g}+\sum_{b\in
S_{s}}P_{b}^{d}],~{}s\in\mathcal{R}\setminus\\{r\\}\\}$;

13 Each $\hat{b}\in\mathcal{B}$ announces its new coalition $\hat{r}^{*}$ and
demand ${P_{\hat{b}}^{d}}^{*}$;

14

15 end if

16

17 until  _no consumer $b\in\mathcal{B}$ is rejected_;

18until  _$\\{S_{r}\\}_{r\in\mathcal{R}}$ is unchanging_;

19$\forall r\in\mathcal{R}$, calculate $\nu(S_{r})$ and $\Phi(\nu)$;

20 $\forall r\in\mathcal{R}$, distribute $\Phi(\nu)$ among $b\in
S_{r}\cap\mathcal{B}$;

21 $\forall b\in B$, consume agreed demand ${P_{b}^{d}}^{*}$;

Algorithm 1 Coalition Formation

###  III-D Properties and Stability of the Coalitional Game

In this section, we show that coalitions formed using Algorithm 1 are stable
in a game-theoretic sense. Our proposed algorithm leads to a Stackelberg
equilibrium between each retailer and their respective consumers. For
$r\in\mathcal{R}$ and $S_{r}\subset\mathcal{M}$, the MST induces a partial
order in the elements of $S_{r}$ with respect to $r$, _i.e._ , $b\succ_{r}d$
if $d$ lies in the path connecting $b$ to $r$. We use this order relation to
define a PC game. However, we will first investigate existence of a solution
for the _minimum cost spanning tree_ (MCST) game, namely the existence of a
non-empty core given a retailer’s coalition $S_{r}$. Non-emptiness of the core
is a known property of convex games; moreover, [25] has shown that MCST and
convex games are PC games and all PC games possess a non-empty core. We are
now ready to present the following results:

######  Theorem 1.

Suppose Assumptions 1 and 2 hold. For $r\in\mathcal{R}$, the MCST $(S_{r},c)$
is a PC game with $c\colon 2^{S_{r}}\to\mathbb{R}$ an MST cost.

###### Proof.

For a consumer $b\in\mathcal{B}\cap S_{r}$, and $[b]\subset S_{r}$ an MST
ordered set with $b\succ_{r}b_{j}$ where $b_{j}\in[b]$ satisfies
$c([b])>c([b_{j}])$ and $[b_{j}]\subset[b]$. Now, given a finite $T\subset
S_{r}\setminus[b]$, there exist a path $\phi(v_{0},v_{f})$ in
$(S_{r},\mathcal{E}_{r})$ such that its end points $v_{0},v_{f}\in S_{r}$
satisfy $v_{0}\in T$ and $v_{f}\in[b]$. The minimum spanning tree for
$T\cup[b]$ contains all the edges from the path $\phi(r,b)$ and all those
edges $\mathcal{E}_{b}^{T}=\bigcup_{t\in T}\bigcup_{b\in[b]}\phi(b,t)$, _i.e._
, $c(T\cup[b])=\omega(\mathcal{E}_{b}^{T})+c([b])$. There are two possible
cases: the set $T$ contains elements $d\succ_{r}b$ or $d\not\succ_{r}b$. For
the former, the increment $c(T\cup[b])-c([b])=\omega(\mathcal{E}_{b}^{T})$
which implies

$\begin{split}c(T\cup[b])-c([b])&=\omega(\mathcal{E}_{b}^{T})+c([b_{k}])-c([b_{k}])\\\
c(T\cup[b])-c([b])&=c(T\cup[b_{k}])-c([b_{k}])\end{split}$

for all $b_{k}\succ_{r}b$. For the second case, the set $T\subset
S_{r}\setminus[b]$ contains at least a player that dominates
$b\in\mathcal{B}\cap S_{r}$. The set of edges $\mathcal{E}_{b}^{T}$ contains
two components: the path $\phi(d,b)$ and the remaining edges
$\mathcal{E}_{b}^{T}\setminus\phi(d,b)$. Clearly, the increment

$c(T\cup[b])-c([b])=\omega(\phi(d,b))+\omega(\mathcal{E}_{b}^{T}\setminus\phi(d,b)).$

On the other hand, for $b\succ_{r}b_{k}$,

$c(T\cup[b_{k}])-c([b_{k}])=\omega(\phi(d,b_{k}))+\omega(\mathcal{E}_{b_{k}}^{T}\setminus\phi(d,b_{k})).$

Since $b_{k}\succ_{r}d$, then $\omega(\phi(d,b_{k}))>\omega(\phi(d,b))$ and
$\omega(\mathcal{E}_{b_{k}}^{T}\setminus\phi(d,b_{k}))=\omega(\mathcal{E}_{b}^{T}\setminus\phi(d,b))$.
As a result,

$c(T\cup[b])-c([b])<c(T\cup[b_{k}])-c([b_{k}])$

for all $T\subset S_{r}\setminus[b]$. The MCST $(S_{r},c)$ is a PC game. ∎

######  Theorem 2.

Suppose Assumptions 1 and 2 hold. For $r\in\mathcal{R}$, The MCST $(S_{r},c)$
has a non-empty core.

###### Proof.

The proof is an adaptation of [25, Theorem 1] to our setting. ∎

With the non-emptiness of the core for cost networks secured, we turn our
attention to the game involving multiple retailers. Our objective is to show
that competition among retailers results in greater profits. To this aim, the
next result establishes subadditivity of the game.

######  Theorem 3.

Suppose Assumption 2 holds. The coalitional game with multiple energy
retailers $(\mathcal{N},\nu)$ with value function $\nu(\cdot)$ defined in (3)
is subadditive, _i.e._ , for $r,s\in\mathcal{R}$

$v(S_{r}\cup S_{s})\leq v(S_{s})+v(S_{r}).$ (5)

###### Proof.

For $r,s\in\mathcal{R}$ and from Assumption 2, $\nu(S_{r}\cup S_{s})=0$. On
the other hand, $\nu(S_{r})\geq 0$ since (3) depends on the sum of direct
connection costs $\omega(r,b)$ with $b\in\mathcal{B}\cap S_{r}$ which is at
worst equal to the MST cost $c(S_{r})$. Therefore (5) holds for any
$r,s\in\mathcal{R}$. ∎

As mentioned in [12], subadditivity is not enough to show the stability of the
game or the satisfaction of the members of a given coalition, _i.e._ , there
is no incentive to disband coalitions. To do this, we utilize the following
result on concavity.

######  Theorem 4.

Suppose Assumption 2 holds. The game with multiple energy retailers
$(\mathcal{N},\nu)$ satisfies for any $r,s\in\mathcal{R}$,

$\nu(S_{r}\cup S_{s})+\nu(S_{r}\cap S_{s})\leq v(S_{r})+v(S_{s}).$ (6)

###### Proof.

As consequence of (1) and (2) in Assumption 2, $v(S_{i}\cup S_{j})=0$ and
$v(S_{i}\cap S_{j})=v(\emptyset)=0$. From Theorem 5, $v(S_{r})\geq 0$ holds
for all $r\in\mathcal{R}$. Therefore (6) holds. ∎

A property of all $N$-player concave games is that they are balanced [26],
meaning that for any concave game, there exists an equilibrium point. We can
state the following result.

######  Theorem 5.

The coalitional game with multiple energy retailers $(\mathcal{N},v)$ is
balanced.

###### Proof.

The proof is adapted from [26, Theorem 1]. ∎

######  Remark.

In both Theorem 1 and 4 similar inequalities are used to denote either
convexity and concavity respectively. As mentioned in [25], the difference
lies in the nature of the characteristic function used in the game: for the
former $c(\cdot)$ is a cost function and measures how expensive is for players
to form a coalition, whereas for the latter $\nu(\cdot)$ represents savings
made by players joining a coalition.

A consequence of Theorem 5 and Assumption 2 is that $\nu(\mathcal{N})=0$
since, by definition, the grand coalition contains all retailers. This fact
renders conventional cooperative game methods, see [21, 12], unusable. To
overcome this difficulty, we recur to notion of $\mathbb{D}_{hp}$-stability
introduced by [27].

######  Definition 2 ($\mathbb{D}_{hp}$ stability).

For $r\in\mathcal{R}$, the coalition $S_{r}\subset\mathcal{N}$ is
$\mathbb{D}_{hp}$-stable if the following conditions are satisfied:

  1. 1.

given a collection $\\{P_{r_{1}},\ldots,P_{r_{L}}\\}$ resulting from an
arbitrary partition of $S_{r}$, such that $\cup_{j=1}^{L}P_{r_{j}}=S_{r}$:

$\nu(S_{r})\geq\sum_{j=1}^{L}\nu(P_{r_{j}}),~{}\forall i\in\mathcal{R},$ (7)

  2. 2.

given a set $\mathcal{T}\subset\mathcal{R}$ with
$|\mathcal{T}|\leq|\mathcal{R}|$:

$\sum_{r\in\mathcal{T}}\nu(S_{r})\geq\nu(\bigcup\limits_{r\in\mathcal{T}}S_{r}).$
(8)

Our next result establishes the $\mathbb{D}_{hp}$ stability of the coalitions
formed in the game $(\mathcal{N},\nu)$.

######  Theorem 6.

Suppose Assumption 1 and 2 hold. The retailer coalitions
$S_{r}\subset\mathcal{N}$ are $\mathbb{D}_{hp}$ stable for the game with
multiple retailers $(\mathcal{N},\nu)$.

###### Proof.

For a fixed $r\in\mathcal{R}$ and associated coalition
$S_{r}\subset\mathcal{N}$, consider a collection
$\\{P_{r_{1}},\ldots,P_{r_{L}}\\}$. It follows that $r\in P_{r_{j}}$ for some
$j\in\\{1,\ldots,L\\}$ which implies $\nu(P_{r_{j}})\geq 0$, and
$\nu(P_{i_{k}})=0$ for the rest. The cost associated with $P_{r_{j}}$
satisfies $c(S_{r})\geq c(P_{r_{j}})$ since the MST of $P_{r_{j}}$ is
contained in the one corresponding to $S_{r}$. Condition (7) follows directly
from

$\nu(S_{r})-\nu(P_{r_{j}})\geq c(S_{r})-c(P_{r_{j}})$

since $c(\\{r,b_{k}\\})>0$ for all $b_{k}\in S_{r}\setminus P_{r_{j}}$.

For the second condition, (1) and (2) imply that
$\nu(\cup_{r\in\mathcal{T}}S_{r})=0$. From the value formulation for a
coalition (3) and Assumption 2, $\nu(S_{i})\geq 0$. From the above, condition
(8) holds. ∎

The coalitional game as formulated, following Algorithm 1, leads to a
partition of the network in coalitions. The prices and demands from each
retailer and consumer in such partition operate at a Stakelberg equilibrium.
We refer the reader to our previous result in [16, Theorem 2], where a similar
market setup is presented together with the ways in which such an equilibrium
is derived and characterised. We illustrate the existence and convergence
towards a Stackelberg equilibrium for our game with the numerical results in
Section V.

##  IV Risk Sharing and Reduction of Statistical Dispersion

In this section we provide a brief insight into the statistical implications
for consumers when considering their demand to be a random variable. Motivated
by [28], we consider the statistical properties of consumer demand in order to
show that coalition formation entails lower risks. In the proposed game, the
situation where a retailer coalition $S_{r}$ contains less consumers than
expected constitutes a risk for consumers since subsidies are lower. Consumers
can lower this risk by acting together in joining a retailer coalition which
allows them to increase collective profit and sharing the risk.

For each $b\in\mathcal{B}$, consider $\alpha_{b}$ to be a random variable
$\alpha_{{b}}(t)\sim N(\mu_{b},\sigma^{2})$. A direct consequence is that the
power demand $P_{b}^{d}(t)\in[\underline{P}_{b},\bar{P}_{b}]$ is a stochastic
process; the vector demand is $P^{d}=(P_{1},\ldots
P_{|\mathcal{B}|})\in\mathcal{P}=\prod_{b\in\mathcal{B}}[\underline{P}_{b},\bar{P}_{b}]$.
The cumulative distribution function (CDF), compactly supported on
$\mathcal{P}\subset\mathbb{R}^{|\mathcal{B}|}$, at each time $t$ is given by

$\Phi(Q^{d};t)=\mathbb{P}\\{P^{d}(t)\leq Q^{d}\\},$ (9)

and denotes the probability of the random process $P^{d}(t)$ taking a value
less than or equal to $Q^{d}$. The power consumption of a coalition
$S_{r}\subseteq\mathcal{N}$ for $r\in\mathcal{R}$ would be then be represented
by a sum of stochastic processes $P^{d}_{S_{r}}(t)=\sum_{b\in
S_{r}\cap\mathcal{B}}{P^{d}_{b}(t)}$. The corresponding random process is
denoted by $\mathbf{P}^{d}_{S_{r}}=\\{{P}^{d}_{S_{r}}(t)<P^{d}\\}$ with
support $\mathcal{P}_{S_{r}}=[\underline{P}_{S_{r}},\bar{P}_{S_{r}}]$ with
$\underline{P}_{S_{r}}=\sum_{b\in S_{r}\cap\mathcal{B}}\underline{P}_{b}$ and
similarly for $\bar{P}_{S_{r}}$. From this, the time averaged CDF is

$F_{S_{r}}(P^{d})=\frac{1}{T}\int_{t_{0}}^{t_{f}}\Phi_{S_{r}}(P^{d},t)dt.$
(10)

The associated quantile function $F^{-1}_{S_{r}}:[0,1]\to\mathcal{P}_{S_{r}}$
is $F^{-1}_{S_{r}}(p)=\inf\\{x\in[0,1]|F_{S_{r}}(x)\geq p\\}$ for any
$p\in[0,1]$. On the other hand, the total consumer profit for a given price
$\Lambda_{r}$ and subsidies $\kappa_{r}\delta_{b}^{r}$ is
$\Pi_{S_{r}}(P^{d},r)=\sum_{b\in S_{r}\cap\mathcal{B}}{\Pi_{b}(P^{d}_{b},r)}$.
The expected consumer profit is also a stochastic process and is given by:

$J_{S_{r}}(r)=\mathbb{E}~{}\Pi_{S_{r}}(\mathbf{P}^{d}_{S_{r}},r).$ (11)

The individual profit of a consumer $b\in\mathcal{B}$ such that
$S_{r}=\\{r,b\\}$ is denoted as $\Pi_{\\{r,b\\}}$. The following result shows
that risk sharing through coalitions leads to an increase in profit _almost
surely_.111A property $P$ holds _almost surely_ if the points where its
complement does not hold has measure zero..

######  Proposition 1.

Suppose Assumptions 1 and 2 hold. For $r\in\mathcal{R}$, the following holds
_almost surely_ ,

$\Pi_{S_{r}}(\mathbf{P}^{d}_{S_{r}},r)\geq\sum_{b\in
S_{r}\cap\mathcal{B}}{\Pi_{\\{r,b\\}}(\mathbf{P}^{d}_{b_{j}},r)}.$ (12)

###### Proof.

To prove the result, we employ the properties of the sum of random variables
[29], homogeneity and superadditivity of functions of stochastic processes
[28]. Condition (12) is fulfilled as a direct consequence of Assumption 4,
where the value of a coalition is always greater or equal than the value of a
single consumer with a retailer. ∎

The above result establishes that coalitions always bring larger collective
profits for consumers. These benefits can be directly attributed to the
attenuation of statistical dispersion from aggregation; this phenomena has
been also explored in [30] where the optimal expected profit directly depends
on the deviation of the coalitional value-at-risk (CVaR) or _coalitional
shortfall deviation_ [31]. For any $q\in(0,1)$, the CVaR deviation of
$\mathbf{P}^{d}_{S_{r}}\sim F_{S_{r}}$ is defined as
$\mathcal{D}_{q}(\mathbf{P}^{d}_{S_{r}}):=\mathbb{E}[\mathbf{P}^{d}_{S_{r}}]-\mathbb{E}[\mathbf{P}^{d}_{S_{r}}|\mathbf{P}^{d}_{S_{r}}\leq
F_{S_{r}}^{-1}(q)]$. This deviation measures the difference between the
expected value and the probability of $q$ being near the bounds of the
probability distribution. Then, the reduction of dispersion that is induced by
consumers joining a coalition $S_{r}$ and aggregating their demand is
$\mathbf{\Delta}_{S_{r}}:=\sum_{b\in
S_{r}}\mathcal{D}_{q}(\mathbf{P}^{d}_{b})-\mathcal{D}_{q}(\mathbf{P}^{d}_{S_{r}})$.
The non-negativity of $\mathbf{\Delta}_{S_{i}}$ for all
$S_{r}\subset\mathcal{N}$ follows immediately. This establishes that CVaR is a
measure of the reduction in statistical dispersion. Moreover, there is an
inversely proportional relation between expected profits and statistical
dispersion of aggregate consumption.

##  V Implementation with Physical Dynamics

In this section, we analyse the effect of the proposed pricing scheme on the
voltages of a low voltage and resistive micro-grid, see Fig. 2. An element of
the adjacency matrix $A_{ij}$ corresponds to a conductance $1/R_{ij}$ between
nodes $i$ and $j$ and the network conductance matrix $G=\mathcal{L}+R^{-1}$
where $R={\textnormal{diag}}(\\{R_{ii}\\}_{i\in\mathcal{N}})$ is the impedance
of each node [32].

###  V-A Micro-Grid and Demand Dynamics

b5b4b2b1b3r1r2r3

Figure 2: Resistive micro-grid in a network representation, comprised by
loads/consumers  and generators/retailers  , resistive distribution lines, and
shunt conductances.

We consider islanded micro-grids since these allow to reframe the application
within the proposed multiple retail scenario222The grid connected case can be
addressed by adding an extra retailer node with additional properties such as
wider generation capabilities.. We consider a network where both retailers and
consumers provide grid support by operation under $P\sim V$ droop control. The
droop controller used in our setting is similar in functionality to the
traditional one [33], albeit this version uses bounded integrators [34] which
are capable of voltage constraint satisfaction. The dynamics of each node
$i\in\mathcal{N}$ are

$\begin{split}\tau_{v,i}\dot{V}_{i}&=\big{(}{V^{*}}-V_{i}+\\\
&-{k_{i}}\sum_{j\in\mathcal{N}}G_{ij}V_{i}V_{j}+k_{i}{P_{i}^{\textnormal{set}}}\big{)}\bigg{(}1-\frac{(V_{i}-V^{*})^{2}}{\Delta
V_{i}^{2}}\bigg{)},\end{split}$ (13)

where $\tau_{v,i},k_{i}>0$ are the time constant and droop coefficient,
$V^{*}\in\mathbb{R}$ is the rated voltage, and $\Delta V_{i}>0$ define desired
voltage bounds. The following result shows that voltages remain within
prespecified bounds.

######  Lemma 2.

If $V_{i}(0)\in(V^{*}-\Delta V,V^{*}+\Delta V)$, $\forall i\in\mathcal{N}$,
then $\forall t\geq 0$ and $\forall i\in\mathcal{N}$,
$V_{i}(t)\in[V^{*}-\Delta V,V^{*}+\Delta V]$.

###### Proof.

The proof follows from contradiction. Suppose there exists a voltage violating
the constraints. By continuity of solutions, there exist $i\in\mathcal{N}$ and
$\hat{t}>0$ such that $|V_{i}(\hat{t})-V^{*}|=\Delta V$. This point would be
an equilibrium of (13), _i.e._ , $\dot{V}_{i}=0$. This implies that the
voltage trajectory $V_{i}(t)$ cannot leave the set $(V^{*}-\Delta
V,V^{*}+\Delta V)$. This proves our result. ∎

The vector $P^{set}$ contains the power reference (demanded or consumed) for
each node. For consumers, the $P_{i}^{\textnormal{set}}$ contains the first
order demand response [22, 35] for all $b\in\mathcal{B}$

$\tau_{i}\dot{P}_{b}^{\textnormal{set}}=P_{b}^{d}-P_{b}^{\textnormal{set}},$
(14)

where $\tau_{i}>0$ is the associated time constant and the input $P_{b}^{d}$
is the output of (1b). The overall state vector is
$x=\\{V_{b},V_{r},P_{b}^{\textnormal{set}}\\}_{b\in\mathcal{B},r\in\mathcal{R}}$.

###  V-B Physical Stability Analysis

In this section, we analyse the stability of the micro-grid given by (13) and
(14). To ensure that our problem is well posed, we require the following
assumption:

######  Assumption 5.

For constant inputs $P_{r}^{\textnormal{set}}\in\mathbb{R}$ for all
$r\in\mathcal{R}$ and $P_{b}^{d}$ for all $b\in\mathcal{B}$, there exists an
equilibrium point
$\\{\bar{V}_{{b}},\bar{V}_{{r}},\bar{P}_{{b}}^{\textnormal{set}}\\}$ $\forall
b\in\mathcal{B},r\in\mathcal{R}$ for the coupled system (13)-(14) such that
$|V_{i}-V^{*}|\leq\Delta V$ for all $i\in\mathcal{N}$.

If Assumption 5 does not hold, the equilibrium voltage would lie, following
Lemma 2, on the boundary of the constraint sets.

######  Proposition 2.

Suppose Assumption 5 holds. The solutions of (13)-(14) satisfy
$\lim_{t\to\infty}|x-\bar{x}|=0$ with
$\bar{x}=\\{\bar{V}_{{b}},\bar{V}_{{r}},\bar{P}_{{b}}^{\textnormal{set}}\\}_{b\in\mathcal{B},r\in\mathcal{R}}$
and $x(0)\in\mathcal{A}$ a neighbourhood of $\bar{x}$ if for all
$i\in\mathcal{N}$

$k_{i}\Delta
V\sum_{j\in\mathcal{V}}\frac{1}{R_{ij}}<\frac{1}{2}+k_{i}\frac{V^{*}}{R_{ii}}.$
(15)

###### Proof.

The Jacobian of (13)-(14) at
$\bar{x}\in\mathbb{R}^{|\mathcal{N}|+|\mathcal{B}|}$ is:

$J(\bar{x})=\begin{bmatrix}\Upsilon&-\kappa\\\
\mathbf{0}&-\tau^{-1}\end{bmatrix},$ (16)

where $\Upsilon=\frac{\partial f_{v}}{\partial
V}\bigl{|}_{\bar{x}}\in\mathbb{R}^{|\mathcal{N}|\times|\mathcal{N}|}$,
$\kappa=\frac{\partial f_{v}}{\partial
P^{\textnormal{set}}}\bigl{|}_{\bar{x}}\in\mathbb{R}^{|\mathcal{N}|\times|\mathcal{B}|}$,
and $\tau\in\mathbb{R}^{|\mathcal{B}|\times|\mathcal{B}|}$ is a diagonal
matrix of time constants and $f_{v}(\cdot,\cdot)$ is the voltage vector field
of (13). The required stability condition follows from the eigenvalues of
$J(\bar{x})$; negative eigenvalues guarantee a stable system at $\bar{x}$. We
exploit the structure of $J(\bar{x})$ to compute its
eigenvalues333$\mathbb{I}_{n}$ denotes the identity in $\mathbb{R}^{n}$.,

$\det(\eta\mathbb{I}_{|\mathcal{N}|+|\mathcal{B}|}-J(\bar{x}))=\det(\eta\mathbb{I}_{|\mathcal{N}|}-\Upsilon)\det(\eta\mathbb{I}_{|\mathcal{B}|}+\tau^{-1}).$

Clearly, $-\tau^{-1}$ contributes only negative eigenvalues, we, therefore,
only analyse the behaviour of the first factor. We note that the quantity
$\xi=(1-(V^{*}-V_{i})^{2}/\Delta V_{i}^{2})$ is positive by Assumption 5 for
all $i\in\mathcal{N}$, and $\Upsilon$ can be factored as
$\Upsilon=\xi\tilde{\Upsilon}$ where
$\tilde{\Upsilon}_{ii}=c_{i}\tau_{v,i}^{-1}(-1-k_{i}(\sum_{j\neq
i}G_{ij}V_{j}+2G_{ii}V_{i}))$ and
$\tilde{\Upsilon}_{ij}=-c_{i}\tau_{v,i}^{-1}k_{i}G_{ij}V_{i}$. As a result,
the determinant we seek is:
$\det(\eta\mathbb{I}_{|\mathcal{N}|}-\tilde{\Upsilon})$. We use Gershgorin
discs $\bm{\Delta}_{i}(C_{i},R_{i})$ with centre at
$C_{i}=\tilde{\Upsilon}_{ii}$ and radius $\varepsilon_{i}=\sum_{j\neq
i}|{\tilde{\Upsilon}_{ij}}|$ to analyse the eigenvalue location. If condition
(15) holds, the eigenvalues of
${\textnormal{diag}}(\\{\tilde{\Upsilon}_{ii}\\}_{i\in\mathcal{N}})$ lie on
the left side of the complex plane. The equilibrium $\bar{x}$ is
asymptotically stable in a neighbourhood
$\mathcal{A}\subset\mathbb{R}^{|\mathcal{N}|+|\mathcal{B}|}$ where the
linearisation holds. ∎

The above results allow us to check the stability of the micro-grid based on
(15) without evaluating directly the equilibrium point which may be difficult
to compute; a computation of the equilibrium point requires a solution of the
power flow equations. Furthermore, using a continuity argument, our result
holds for equilibrium points on the boundary of $(V^{*}-\Delta
V_{i},V^{*}+\Delta V_{i})$.

##  VI Simulations

###  VI-A Coalition Formation and Profit Calculation

To illustrate our scheme, we have formulated two scenarios. The first one
consists of a micro-grid with a single retailer, whereas the second one
considers two additional retailers; both scenarios have five consumers
$\mathcal{B}=\\{b_{1},\ldots,b_{5}\\}$. The parameters for both scenarios are
listed in Table I and their cost networks of different retailers are shown in
Fig. 3. The games are played periodically, we use a period of $10$ seconds for
both scenarios.

TABLE I: Parameters for Retailers and Consumers.

Retailer | $\alpha_{{r}_{i}}$ | $\kappa_{{r}_{i}}$ | $\underline{\lambda}_{i}$ | $\overline{\lambda}_{i}$ | $P^{g}_{i}$
---|---|---|---|---|---
$r_{1}$ | 1e-4 $\$^{\frac{1}{2}}$ | 65 $ | 0.01 $/W | 4 $/W | 30 kW
$r_{2}$ | 7e-5 $\$^{\frac{1}{2}}$ | 64 $ | 0.01 $/W | 2 $/W | 30 kW
$r_{3}$ | 5e-5 $\$^{\frac{1}{2}}$ | 63 $ | 0.01 $/W | 3.5 $/W | 30 kW

Consumer | $\alpha_{{b}_{j}}$ | $P^{d_{rated}}_{{b}_{j}}$ | $\underline{\zeta}_{{b}_{j}}$ | $\overline{\zeta}_{{b}_{j}}$
---|---|---|---|---
$b_{1}$ | 1800 W6 | 3 kW | 0 kW | 6 kW
$b_{2}$ | 150 W6 | 3.5 kW | 0 kW | 7 kW
$b_{3}$ | 140 W6 | 2.8 kW | 0 kW | 5.6 kW
$b_{4}$ | 100 W6 | 4 kW | 0 kW | 8 kW
$b_{5}$ | 1600 W6 | 1.5 kW | 0 kW | 3 kW

$b_{3}$$b_{4}$$b_{5}$$b_{1}$$b_{2}$$r_{1}$550475535300280245540455$b_{3}$$b_{4}$$b_{5}$$b_{1}$$b_{2}$$r_{3}$460495525230400340540250500$b_{3}$$b_{4}$$b_{5}$$b_{1}$$b_{2}$$r_{2}$505520515425225350365510525
Figure 3: Cost networks defined by each retailer for the same set of
consumers.

Figure 4 shows the rationality of players, _i.e._ , consumers tend to consume
more (less) given a lower (higher) price, and that retailers tend to lower
(raise) their price when consumption is low (high). This is more evident in
the single retailer scenario where both prices and demands settle. In
addition, Fig. 4 shows that some coalitions choose not to consume, this
behaviour is the result of consumers choosing a coalition maximising their
profit and leaving some retailers without consumers.

Figure 4: Price and coalition consumption responses over time.

Figure 5 shows individual power demand for all $b\in\mathcal{B}$; each
consumer is able to consume more power in the multiple retailer scenario. Each
consumer power demand settles above their rated consumption even for those who
do not prioritise consumption, _i.e._ , $\alpha_{b}$ is small. The profit
obtained by each consumer, see Fig. 6, shows that greater profits can be
obtained in the latter scenario. The case of consumer $b_{4}$ is of particular
interest: this buyer is forced to pay a connection fee in the single retailer
scenario; whereas it has no longer to pay such fee when there are more
retailers.

Figure 5: Consumers’ individual power demand over time. Figure 6: Consumers’
immediate profits for every time period.

The profits of retailers, see Fig. 7, also change from scenario to scenario;
as expected the profits of $r_{1}$ decrease in case of competition. When
retailers fail to attract consumers, their profits can become negative as seen
for $r_{2}$. The benefits of a multiple retailer scenario are clear from Figs.
6 and 7, consumers gain more from retailer competition. Lastly, Fig. 8 shows
the evolution of the choice of coalition for each consumer.

Figure 7: Retailers’ immediate profits for every time period. Figure 8:
Coalition selection over time

###  VI-B Game and Physical Dynamics Integration

In this section, we show the response of the physical system to the changing
set-points arising from the proposed pricing scheme. The parameters are the
following: droop control time constant $\tau_{vi}=0.1$s, load response time
constant $\tau_{i}=3$s, nominal voltage $V^{*}=220$V, voltage deviation
$\Delta V=11$V, droop coefficients $k_{i}=0.05V_{i}^{*}/P_{i}^{rated}$ and
$c_{i}=(\pi\Delta V)/(0.1k_{i}P_{i}^{rated})$, where the $P_{i}^{rated}$
vector consists of the $P_{{r}}^{g}$ and $P_{{b}}^{Lrated}$ values for the
corresponding retailer and consumers respectively. The coalition selection
game is played every $T_{S}=60$s.

Figure 9: Resistive network node powers when subject to the proposed
coalitional game. Figure 10: Resistive network node voltages when subject to
the proposed coalitional game.

The response of the physical system to changes in power set-points, see Fig. 9
and 10, for both retailers and consumers is seen through the voltage changes
(voltage deviations allow for current to flow between nodes). We show that all
voltages are bounded around the nominal voltage, and the power losses are
small comparing Fig. 9 with 5.

##  VII Conclusions and Future Directions

In this paper we have proposed an on-line pricing scheme that encompasses the
concepts of a hierarchical structure with the Stackelberg game and coalitional
games with competing players. The definitions and the algorithm for the game
have been established and the stability of the game has been shown. We have
described a potential implementation of the present scheme integrated with the
physical micro-grid dynamics, together with its corresponding stability
analysis. A comparison between single and multiple retailer scenarios has been
shown numerically, demonstrating the advantages of the latter from an economic
point of view. Future directions of this work dwell in the incorporation of
disconnection penalties, quality of the service, among other factors that can
be included into the algorithm. Additionally, several approaches/variations
for deriving the proposed cost networks can be investigated. Finally, with the
same market setup and dynamics as basis, a plethora of optimisation methods
can also be implemented for the design of novel dynamic pricing algorithms.

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One of the most active research areas in the world is the search for effective
materials for use in the fields of optoelectronics and photovoltaics. The
potential of materials like ATiO3 (A=Ti,Sr) is yet largely untapped. Ab initio
studies based on density functional theory (DFT) have been used to
comprehensively explore the structural, electronic, elastic, and optical
properties of Ti2O3 and SrTiO3. In this study, the ground state properties
were computed with spin-orbit coupling (SOC), without spin-orbit coupling, and
with the inclusion of Hubbard U parameter. Ti2O3 has been found to have
electronic bandgaps of 0.059 eV without SOC, 0.131 eV with SOC, and 1.665 eV
with Hubbard U. For SrTiO3, electronic bandgaps of 1.612 eV, 1.761 eV, and
2.769 eV have been obtained, respectively, without SOC, with SOC, and with
Hubbard U. Ti-4d orbitals have been observed to dominate near the top of the
valence band in each and every instance. SOC did not significantly affect the
bandgaps and calculated lattice parameters for ATiO3 (A=Ti,Sr). ATiO3
(A=Ti,Sr) is mechanically stable at absolute zero pressure, according to the
mechanical stability test. The optical band gap has been seen to increase when
Hubbard U is taken into account. In general, the Hubbard U parameter enhances
bandgap and optical property predictions. Ti2O3 and SrTiO3 are good UV-Vis
absorbers and appropriate for photovoltaic applications owing to the optical
absorption coefficient curves being found to cover the ultraviolet to visible
(UV-Vis) region.

Key words: Optical properties, Electronic and Elastic properties, DFT + U,
Spin orbit coupling, ATiO3 (A=Ti,Sr).

# An insight into the properties of ATiO3 (A=Ti,Sr) materials for photovoltaic
applications

Lynet Allan  Corresponding author<EMAIL_ADDRESS>Department
of Physics, School of Physical Sciences, University of Nairobi, P.O.Box
30197-00100 Nairobi Kenya.  Julius M. Mwabora  Department of Physics, School
of Physical Sciences, University of Nairobi, P.O.Box 30197-00100 Nairobi
Kenya.  Musembi J. Robinson  Department of Physics, School of Physical
Sciences, University of Nairobi, P.O.Box 30197-00100 Nairobi Kenya.

##  I Introduction

The search for sustainable energy resources has become a key field of study
due to rising global energy demands and the negative environmental pollution
generated by the burning of fossil fuels Eddiouane  _et al._ (2018); Samat
_et al._ (2016); Umebayashi  _et al._ (2002). Titanium based oxides have been
under investigation for the past few decades owing to their wide range of
applications in environmental remediation and solar energy conversion,Samat
_et al._ (2019); Umebayashi  _et al._ (2002); Giustino (2014). Titanium is one
of the earth-abundant elements, and its oxides including titanium dioxide
(TiO2 ) and ATiO3 (A= Ba,Ca,Fe) have gained significant attention for
applications in technologies of electronics, energy conversion, catalysis,
sensing, and so onZhang  _et al._ (2010); Anderson (1963); Inturi  _et al._
(2014); Li  _et al._ (2018). Despite its large band gap, TiO2 has been
researched for its potential modification for solar harvestingFahmi  _et al._
(1993); Zhang and Kilin (2012). However, there hasn’t been much focus on
reduced TiO2 compounds like Ti2O3. Previously,Li (2016), Yang Yang et al
observed a cubic perovskite Ti2O3 unit cell made at room temperature at the
Ti/SrTiO3 interface, although they did not investigate the material’s
properties. We thoroughly investigate the structural, electrical, elastic, and
optical properties of Ti2O3 and SrTiO3 materials for photovoltaic applications
as a result of this. Ti2O3 is Corundum structured and crystallizes in the
trigonal R3c space group with Ti3+ atoms bonded to six equivalent O-2 atoms to
form a mixture of edge, corner, and face-sharing TiO6 octahedra as reported
elsewhere Guo  _et al._ (2012). Similarly, SrTiO3 is (Cubic) Perovskite
structured and crystallizes in the tetragonal I4/mcm space group with Sr2+
bonded to twelve O-2 atoms to form SrO12 cuboctahedra that share corners with
twelve equivalent SrO12 cuboctahedra. In this study, the stability and
properties of these two compounds will be explored their suitability for use
in solar harvesting applications. Most recentlyAllan  _et al._ (2022a), the
properties of titanium oxynitride compounds TinN2O2n-1 with n=3, obtained from
pristine Ti2O3, were investigated. Ti2O3 displayed metallic properties, yet it
had been reported to have semi-conducting properties experimentallyAoki  _et
al._ (2019). The failure of first-principles DFT to properly describe the
electronic band gaps of strongly correlated materials, in which the electron-
electron interaction has a significant impact, is a well-known problemSamat
_et al._ (2016). Many approaches to resolving this discrepancy have been
proposed. The inclusion of the Hubbard model (DFT+U) in electronic structure
calculations is one of them Samat  _et al._ (2016). The DFT+U method has been
used for improved prediction of the gaps as well as to exhaustively determine
the elastic and mechanical properties of the compounds. The study of the
elastic properties such as elastic constants provides fundamental information
on how these materials behave under external strainBurke  _et al._ (2007).
Besides this, the effect of spin orbit coupling (SOC) has also been analyzed.
To the best of our knowledge, there are a few experimental reports in the
literature on the properties of Ti2O3 and SrTiO3. Hence, this work aimed at
investigating the properties of the two titanites containing Ti3+ ions with a
specific focus on their suitability for use in photovoltaic applications.
Specifically, we explore the structural, electronic, elastic, mechanical, and
optical properties of Ti2O3 and SrTiO3 compounds using the DFT method with SOC
and Hurbbard U effects. This study opens new doors towards some new titanite-
based technologies. This paper is organized as follows: Sec.II accounts for
the technicalities that may be needed for future reproducibility in our
computations. We offer detailed analysis of results and discussions in
Sec.III. The structural properties of both Ti2O3 and SrTiO3 with and without
SOC effects are discussed in Sec.III.1 while the eEffect of SOC and Hubbard U
on electronic Properties in Sec.III.2. The elastic constants and mechanical
properties of the compounds under study are outlined in Sec.III.3. In
Sec.III.4, we discuss the optical properties of Ti2O3 and SrTiO3. Sec.IV
outlines the conclusions of this work.

##  II Computational details

The structural, electronic, elastic, mechanical, and optical properties of
Ti2O3 and SrTiO3 compounds were computed within the DFTBurke  _et al._ (2007);
Kohn and Sham (1965) as implemented in the quantum espresso (QE) code
Giannozzi  _et al._ (2009). The Ti2O3 and SrTiO3 crystal structure input files
were downloaded from the materials project database Jain  _et al._ (2013) and
the materials cloud Talirz  _et al._ (2020) input generator implemented in QE
was used to generate the PWscf input files for DFT calculations. The electron-
ion interaction was denoted by using the projector augmented-wave function
(PAW) methodKresse and Joubert (1999). The Generalized Gradient Approximation
(GGA) with the Perdew-Burke-Ernzerhof (PBE) Perdew  _et al._ (1996) was chosen
to define the exchange-correlation effect of the electrons. The optimized
cutoff energy of 50Ry and 6 × 6 × 4 Monkhorst-Pack grid for Brillouin zone
integration were used for the calculations. Geometry optimization was
performed by computing the total energy per unit cell at several lattice
constant values to obtain the ground state structural properties. Based on the
optimized lattice constants, the elastic, electronic, and optical properties
were calculated. The effects of spin orbit coupling (SOC) was included in the
structural and electronic properties, while the well-known DFT underestimation
problem was countered but the inclusion of Hubbard U correction on the
electronic structure calculations. The choice of the effective Hubbard U
values were peged on earlier works done on same materials while assesing the
impact of the inclusion of U on the lattice parameter(a0) of the compounds.

##  III Results and Discussions

###  III.1 Structural Properties

Ti2O3 and SrTiO3 compounds adopt trigonal and tetragonal crystal system as
reported elsewhere Talirz  _et al._ (2020). In order to achieve the
equilibrium structure, one has to calculate the lattice parameter that
minimizes the DFT total energy, the optimized crystal structures are shown in
Figure 1. Table 1 shows bond strength for Ti2O3 and SrTiO3.

Figure 1: Optimized Crystal structure for (a) SrTiO3 and (b) Ti2O3 compounds Material | Bond | Bond length
---|---|---
Ti2O3 | Ti-O | 2.04
Ti-Ti | 2.37
SrTiO3 | Sr-Ti | 3.41
Sr-O | 2.78
Sr-Sr | 3.95
Table 1: Bond lengths for Ti2O3 and SrTiO3 compounds

From Table 1, the Ti-O bond is stronger than the Ti-Ti bond in Ti2O3, while
the Ti-O bond is stronger than both Ti-Sr and Sr-O bonds in SrTiO3 . The Sr-Sr
bond is the weakest with long bond lengths; this difference is attributed to
the fact that the volume per atom tends to increase with the increase in
atomic radius and therefore affects crystal lattice basisMooser and Pearson
(1956). The total energy at various lattice constant values is computed with
SOC and without SOC effects are presented in figure 2 and figure 3 for Ti2O3
and SrTiO3 compounds, respectively. Table 2 gives a summary of the calculated
lattice parameters and other structural properties.

Figure 2: Total energy versus lattice constants computed with and without SOC effects for Ti2O3 compound. Figure 3: Total energy versus lattice constants computed with and without SOC effects for SrTiO3 compound. Structural Properties | Ti2O3 | SrTiO3
---|---|---
| Without soc | With soc | Without soc | With soc
Lattice Parameters a0(a.u) | 9.933 | 10.404 | 10.535 | 10.849
Lattice parameter c/a | 0.520 | 0.571 | 1.353 | 1.403
Bulk modulus Bo (GPa) | 166.4 | 137.0 | 217.1 | 218.2
| Ground state energy
---
Eo (ry)
-673.7 | -291.1 | -1253.07 | -1585.16
| Equilibrium volume
---
Vo (a.u)3
980.14 | 1126.09 | 1167.17 | 1277.00
Table 2: Effect of SOC on Structural properties of Ti2O3 and SrTiO3 compounds

From figure 2 and figure 3, SOC did not have significant effects on the
lattice parameters. However, when the data was fitted to Murnaghan’s equation
of stateMorgan and Watson (2010), minimum changes were noted on the
equilibrium volume and ground state energies of the two compounds as indicated
in table 2

###  III.2 Effect of SOC and Hubbard U on Electronic Properties

The electronic band structures and projected density of states (PDOSs) of the
Ti2O3 and SrTiO3 compounds were computed by using the optimized crystal
structures and presented on figure 4 and figure 5. The band structures were
calculated in three stages; i) without SOC effects, ii) with SOC effect, and
iii) with Hubbard U corrections. The choice of U values were pegged on the
earlier works on the same materials by refAllan  _et al._ (2022a) and
refAdewale and Chik (2019) for Ti2O3 and SrTiO3 , respectively. For Ti2O3 and
SrTiO3 , Hubbard U=5 eV and U=7 eV, respectively, proved to have no
significant effects on the lattice constants and were therefore used for the
band structure calculations.

Figure 4: The calculated (a) band structures of Ti2O3 compounds without SOC
effects(blue), with SOC(red) and with DFT+U (U=5 eV) (black), (b)The PDOS for
Ti2O3 compound without SOC and without DFT+U effects. Figure 5: The calculated
(a) band structures of SrTiO3 compounds without SOC effects (blue), with
SOC(red) and with DFT+U (U=5 eV) (black), (b)The PDOS for SrTiO3 compound
without SOC and without DFT+U effects.

The Ti2O3 and SrTiO3 compounds have narrow bandgaps of Ti2O3 has been found to
have electronic bandgaps of 0.059 eV and 1.612 eV, respectively, without SOC,
0.131 eV and 1.761 eV, respectively, with SOC, and 1.665 eV and 2.769 eV,
respectively, with Hubbard U as seen in figures 4 and 5(a). The summary of the
calculated bandgaps is tabulated in table 3.

Materials | Calculated Band Gaps |  |  | Experimental gaps |
---|---|---|---|---|---
| Without SOC | With SOC | With Hubbard U |  |
Ti2O3 | 0.0598 | 0.1311 | 1.667 (U=5eV) | 1.67Li (2016) |
SrTiO3 | 1.612 | 1.7612 | 2.769(U=7eV) | 2.85Van Benthem  _et al._ (2001) |
Table 3: The calculated and experimental bandgaps

From the band structure calculations, SOC did not have significant effect on
the bandgaps, however the DFT+ U improved the prediction of the band gaps with
respect to the experimental values. The maxima of the valence bands and the
minima of the conduction bands occur at different symmetry points (Z-$\Gamma$)
for Ti2O3 and SrTiO3 and ($\Gamma$–M) for SrTiO3 in the Brillouin zone,
implying that the two compounds are indirect bandgap semiconductors. The
projected density of states, which describes the available states for
electrons to occupy when projected on atomic orbitals were calculated.
However, since the effects of SOC and without Hubbard U did not have
significant impact on the types of dominant states in the PDOS, only the
results of the PDOS without the effects of SOC and Hubbard U are presented in
figures 4(b) and 5(b). The orbital contributions to the formation of valence
bands and conduction bands are described by the PDOS in the energy region -4.5
eV to 4 eV. As illustrated in figure 4(b), the entire band structure is
dominated with O-2p and Ti-4d orbitals, an indication that Ti2O3 exhibits semi
metallic properties without Hubbard U effects, this is attributed to the well-
known poor description of the electronic structure by the DFT method. The
bands in the region -4.5 to -2.5 eV are majorly formed by O-2p while Ti-4d
states forms the bands in the energy region of -1.0 eV to 4 eV in the Ti2O3
band structure shown in figure 5. In the case of SrTiO3 compound (Figure
5(b)), the valence band is majorly formed by the O-2p states in the energy
range of -4.5eV to Fermi level, while the conduction band is majorly formed by
the Ti-4d orbitals.

###  III.3 Mechanical and Elastic properties

The Ti2O3 and SrTiO3 compounds adopt trigonal and tetragonal crystal structure
featuring six independent elastic constants. The independent elastic constants
for trigonal Ti2O3 are C11 C12 C13 C14 C33 C44 Chung and Buessem (1967). The
necessary and sufficient conditions for trigonal elastic Ti2O3 stability is
given by equation(1);

$\displaystyle\begin{split}C_{11}>\left|C_{12}\right|;C_{44}>0;\\\
C_{13}^{2}<\frac{1}{2}C_{33}\left(C_{11}+C_{12}\right);\\\
C_{14}^{2}<\frac{1}{2}C_{44}\left(C_{11}-C_{12}\right)=C_{44}C_{66};\\\
C_{66}=\frac{1}{2}(C_{11}-C_{12})\end{split}$ (1)

For Tetragonal SrTiO3 , the six independent elastic constants are C11 C12 C13
C14 C33 C44 and C66 Chung and Buessem (1967). The necessary and sufficient
conditions for elastic stability of the tetragonal crystal system are ;

$\displaystyle\begin{split}C_{11}>\left|C_{12}\right|;C_{44}>0;\\\
2C_{13}^{2}<C_{33}\left(C_{11}+C_{12}\right);C_{66}>0\end{split}$ (2)

Table 4 shows the calculated elastic constants for both the compounds, from
the results, both the compounds meet the stability criteria, at DFT and DFT+U
levels, and are therefore mechanically stable. The bulk modulus B, Young’s
modulus E, shear modulus G , Pugh’s ratio B/G, and Poisson’s ratio n are
presented in table 5 .

Table 4: Computed elastic constants Cij (GPa) of Ti2O3 and SrTiO3 compounds Compound |  | C11 | C12 | C13 | C14 | C33 | C44 | C66
---|---|---|---|---|---|---|---|---
Ti2O3 | DFT | 350.3 | 138.9 | 175.7 | 83.04 | 276.0 | 114.7 | 105.6
DFT+U | 285.3 | 102.1 | 113.4 | 3.073 | 228.4 | 121.5 | 91.6
SrTiO3 | DFT | 338.3 | 110.6 | 105.7 | - | 351.6 | 116.5 | 122.3
DFT+U | 372.5 | 116.7 | 115.2 | - | 383.3 | 130.3 | 133.0
Compound |  | B | E | G | B/G | n
---|---|---|---|---|---|---
Ti2O3 | DFT | 217.0 | 250.2 | 95.7 | 2.26 | 0.31
| DFT+U | 161.3 | 239.4 | 95.5 | 1.69 | 0.25
SrTiO3 | DFT | 185.8 | 292.4 | 118.1 | 1.57 | 0.24
| DFT+U | 202.5 | 322.9 | 130.8 | 1.54 | 0.23
Table 5: Mechanical properties of Ti2O3 and SrTiO3 compounds

Bulk modulus measures the resistance against volume change resulting from
applied external pressure Allan  _et al._ (2022b). Large B value predicts hard
materials; thus, from the computed values of the bulk modulus, both Ti2O3 and
SrTiO3 are not hard materials. It is clear that for Ti2O3, DFT+U gives a more
accurate prediction of the mechanical properties with reference to the values
obtained (in table 2) when lattice constants were fitted to Murnaghan’s
equation of stateMurnaghan (1944). Additionally, the bond lengths of the
crystal structures are correlated to the size of the B. The shorter the bond
lengths, the larger the B value Manyali (2012). From the structural
properties, the obtained bond lengths in Ti2O3 are shorter than those in the
SrTiO3 crystal structure thus the higher value of B in Ti2O3 compound by DFT
calculations. The ductile (ionic) and brittle (covalent) nature of materials
is determined by Pugh’s ratio B/G and Poisson’s ratio, n Allan  _et al._
(2022b). The restriction for brittleness is B/ G $<$ 1.75; otherwise, the
material is said to be ductile. In addition, the Cauchy pressure (C12 -C44 )
also confirms the ductility and brittleness of materials. A positive value of
the Cauchy pressure indicates ductility and brittleness otherwise. The
critical value for the Poisson’s ratio is 0.26, with materials having n as
0.26 exhibiting a ductile nature and brittleness otherwise. From the
calculated values, all the three methods are consistent in classifying the
materials as brittle. Poisson’s ratio n 0.1, 0.25, and 0.33 for pure covalent,
ionic, and metallic bonds, respectively, Manyali (2012). Thus, we can conclude
that Ti2O3 and SrTiO3 compounds are ductile and strongly dominated by ionic
character. The stiffness of a material is determined by applying Young’s
modulus value Manyali (2012). The higher the value of E, the stiffer the
material, therefore, SrTiO3 compound is stiffer than Ti2O3.

###  III.4 Optical Properties

Optical properties of solids, which dependent on their band gap, are very
interesting to know how the solid interacts with light. To highlight optical
properties such as: reflection transmission and absorption, we need to
calculate the dielectric function $\varepsilon\left(\omega\right)$, which
reflects the response of electrons in solid to electromagnetic irradiation.
The dielectric function $\varepsilon\left(\omega\right)$ is given by :

$\varepsilon\left(\omega\right)=\varepsilon_{1}\left(\omega\right)+i\varepsilon_{2}\left(\omega\right)$
(3)

$\varepsilon_{1}\left(\omega\right)$ and $\varepsilon_{2}\left(\omega\right)$
are the real and the imaginary parts of the dielectric function. The real part
is associated to the electronic polarizability of the material while the
imaginary part is correlated to the electronic absorption of materialAslam
_et al._ (2021). The real part and the imaginary parts describe, respectively,
the dispersion and the absorption of the electromagnetic radiation by the
medium which it crosses. The spectrum is calculated by summing the electric
dipole operator matrix elements between the occupied and unoccupied wave
functions over the Brillouin zone while respecting the selection rules. This
is mainly connected with the electronic structures and characterizes the
linear response of the material to electromagnetic radiation. It therefore
governs the propagation behavior of radiation in a medium. The imaginary part
of the dielectric function represents the electron transition between the
valence and the conduction bands. The other optical properties like as: the
reflectivity $R\left(\omega\right)$ , the absorption
$\alpha\left(\omega\right)$ and the refractivity $n\left(\omega\right)$, can
be derived from $\varepsilon_{1}\left(\omega\right)$ and
$\varepsilon_{2}\left(\omega\right)$. These are presented in Figure 6, 7 and 8
are computed by using the equations presented as follows Samat  _et al._
(2019); Aslam  _et al._ (2021); Arbab (2019):

$\alpha\left(\omega\right)=\sqrt{2}\omega\left(\sqrt{\varepsilon_{1}^{2}\left(\omega\right)+\varepsilon_{2}^{2}\left(\omega\right)}-\varepsilon_{1}\left(\omega\right)\right)^{1/2}$
(4)

${n\left(\omega\right)=\sqrt{2}\omega\left(\frac{\sqrt{\varepsilon_{1}^{2}\left(\omega\right)+\varepsilon_{2}^{2}\left(\omega\right)}-\varepsilon_{1}\left(\omega\right)}{2}\right)^{1/2}}$
(5)

$K\left(\omega\right)=\sqrt{2}\omega\left(\frac{\sqrt{\varepsilon_{1}^{2}\left(\omega\right)+\varepsilon_{2}^{2}\left(\omega\right)}-\varepsilon_{1}\left(\omega\right)}{2}\right)^{1/2}$
(6)

$L\left(\omega\right)=\frac{\varepsilon_{2}(\omega)}{\varepsilon_{1}^{2}\left(\omega\right)+\varepsilon_{2}^{2}(\omega)}$
(7)

$R\left(\omega\right)=\frac{{(n-1)}^{2}+K^{2}}{{(n+1)}^{2}+K^{2}}$ (8)

Figure 6: Optical Properties for Ti2O3 Figure 7: Optical Properties for SrTiO3
Figure 8: Energy loss $L\left(\omega\right)$ for ATiO3

The imaginary part ($\varepsilon_{2}\left(\omega\right)$) of the dielectric
wave function describes the photon absorption in crystalline materialsGuo  _et
al._ (2012). The peaks in ($\varepsilon_{2}\left(\omega\right)$)curves result
from the electronic transitions from the valence to the conduction bands. The
absorption onsets in $\alpha\left(\omega\right)$ curves refer to the materials
bandgaps which lie within the visible region, $<$3.1 eV for for Ti2O3 and
SrTiO3 . compounds, and are consistent with the bandgaps from the band
structure, an implication of strong inter-band transitions. This makes for
Ti2O3 and SrTiO3 compounds promising candidates for photovoltaic applications.
Additionally, narrow bandgaps facilitate faster electron transitions as
opposed to wide bandgaps Samat  _et al._ (2019). The key feature of
the($\varepsilon_{1}\left(\omega\right)$) curve
is($\varepsilon_{1}\left(\omega\right)$) Energy=0), also referred to as the
static valueArbab (2019). This static value is correlated to the material’s
refractive index as n ($\omega$) . Starting from Energy 0, the
($\varepsilon_{1}\left(\omega\right)$) plot attained major peaks at low energy
regions, $<$1.5eV and $<$2.7 eV for Ti2O3 and SrTiO3, respectively. The photon
transmission persisted until the Re($\omega$) values became negative at energy
regions of 8eV - 12eV. At this energy region, the incident photon radiations
are assumed to be fully attenuatedMurtaza and Ahmad (2011) and the compounds
assert a metallic behaviourMurtaza  _et al._ (2011). From n ($\omega$) curves,
the calculated refractive indices obtained at zero energies are 2.3 (DFT$+$U)
and 2.5 (without U) for SrTiO3 and 2.5 (with Hubbard U) for SrTiO3 . The major
refractive index peaks for SrTiO3 and Ti2O3 (with Hubbard U) reside within the
visible region. Additionally, the Ti2O3 and SrTiO3 were found to have high
optical absorption in the energy range of 8eV-10eV and 10eV-12eV,
respectively. The optical absorption coefficients of Ti2O3 and SrTiO3
compounds cover the UV-Vis regions. The absorption coefficients calculated
covered a wide range of the electromagnetic spectrum in the energy regions
2.0–13.5 eV; this demonstrates that these compounds can be utilized for
photovoltaic applications. The materials’ surface behaviour and energy loss by
fast electrons entering a medium are determined by reflectivity and energy
loss function, respectively Arbab (2019). The main peaks of the reflectivity
curves are observed in the regions 6 eV -11 eV and 2 eV-6 eV for Ti2O3 and
SrTiO3, respectively, the reflectivity decreased beyond this region. There was
no significant absorption in the visible regions, as depicted in the loss
spectrum for both compounds in figures 8(a) and 8(b). The major absorption
peak occurred at higher energy regions 9.97–12.71 eV. The major absorption
peak occurred at higher energy regions $>$12 eV and $>$10 eV for SrTiO3 and
Ti2O3, respectively. The optical properties results obtained in this work are
in agreement with the results obtained previously on the related
materialsSamat  _et al._ (2019); Aslam  _et al._ (2021), .

##  IV Conclusion

We have studied the structural, electronic, elastic, mechanical, and optical
properties of Ti2O3 and SrTiO3 .compounds using the DFT (with and without SOC
effects), and DFT+U methods as implemented in the QE package. Interestingly,
SOC did not have significant effects on the structural and electronic
properties of the compounds. Equilibrium lattice constants of 9.93 a.u without
SOC and 10.40 a.u with SOC were obtained for Ti2O3 compound, whereas for
SrTiO3 the calculated equilibrium lattice constants were 10.53 a.u and 10.84
a.u without SOC and with SOC, respectively . Ti2O3 was found to have
electronic bandgaps of 0.059 eV without SOC and 0.131 eV with SOC while SrTiO3
electronic bandgaps were 1.612 eV, and 1.761 eV, respectively, without SOC and
with SOC. When the Hubbard U effects were in cooperated in the electronic
structure calculations, the bandgaps for Ti2O3 and SrTiO3, were predicted as
1.665 eV and 2.769 eV, respectively. These were in fairly good agreement with
reported experimental results. The formation of the conduction band was
primarily by Ti-4d, while the valence bands were dominated by O-2p orbitals in
both compounds. Both Ti2O3 and SrTiO3 were found to be mechanically stable at
zero pressure, ductile, and ionic, thus their potentiality for resilient
materials application, these properties were confirmed by DFT+U studies.
Additionally, the optical properties of Ti2O3 and SrTiO3 were improved by
DFT+U studies. The calculated bandgaps, high refractive indices, high
absorption coefficients, and wide energy coverage of the absorption
coefficients spectra was mainly in the UV-Vis regions of the electromagnetic
spectrum. This work suggests that Ti2O3 and SrTiO3 compounds are suitable for
photovoltaic applications.

##  V Acknowledgments

This work was supported by the Partnership for Skills in Applied Sciences,
Engineering and Technology (PASET)— Regional Scholarship Innovation Fund
(RSIF). The authors acknowledge the Centre for High-Performance Computing,
CHPC, Cape for HPC resources.

##  VI Data availability

The source files and data will be assessed upon request from the authors.

##  VII conflicts of interest

The authors declare that they have no conflicts of interest.

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