# Generalized Stochastic Dynamic Aggregative Game for Demand-Side Management
in Microgrids with Shared Battery

Shahram Yadollahi1 Hamed Kebriaei1 Senior Member, IEEE  and Sadegh Soudjani2
Member, IEEE 1 Shahram Yadollahi and Hamed Kebriaei are with the School of
ECE, College of Engineering, University of Tehran, Tehran, Iran. Emails:
<EMAIL_ADDRESS>kebriaei@ut.ac.ir)2 Sadegh Soudjani is with the
School of Computing, Newcastle University, United Kingdom. Email:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we focus on modeling and analysis of demand-side management in
a microgrid where agents utilize grid energy and a shared battery charged by
renewable energy sources. We model the problem as a generalized stochastic
dynamic aggregative game with chance constraints that capture the effects of
uncertainties in the renewable generation and agents’ demands. Computing the
solution of the game is a complex task due to probabilistic and coupling
constraints among the agents through the state of charge of the shared
battery. We investigate the Nash equilibrium of this game under uncertainty
considering both the uniqueness of the solution and the effect of uncertainty
on the solution. Simulation results demonstrate that the presented stochastic
method is superior to deterministic methods.

Stochastic dynamic game, chance constraints, microgrids, shared battery,
renewable energy sources.

##  1  Introduction

Multi-agent coordination for energy systems has emerged as a highly effective
approach to achieve energy savings and maintain stability in microgrid
systems. To accomplish this objective, concepts and techniques from game
theory has been utilized due to their ability to incorporate user behavior and
optimization perspectives [1].

In this paper, we study demand-side management (DSM) in microgrids as a
generalized stochastic dynamic aggregative game with uncertainties in
renewable energy and demand, using a shared battery and selfish residential
consumers. Additionally, operational constraints were taken into account in
modeling the problem. To avoid suffering from peak loads, a term related to
the cumulative exchange of power between the agents and the grid appears in
the cost function of the agents as part of electricity tariff. This leads to
the aggregative form of the proposed game model. Unlike previous studies, a
more comprehensive form of stochastic constraints is considered in the form of
chance constraints. Due to the incorporation of a shared battery and
accounting for the impact of uncertain sources, the state of charge (SoC) of
the battery has stochastic dynamics shared between the agents. In addition, as
we impose a constraint on the SoC of the shared battery, we have both dynamic
and static stochastic coupling constraints among the agents, which shape the
proposed game in a generalized, stochastic, and dynamic form. Then, we propose
a series of reformulations and guaranteed under-approximations over the cost
functions, stochastic dynamics, and chance constraints such that the game is
converted into a static generalized aggregative form. Finally, we verify the
conditions under which a Nash-seeking method can obtain the game’s equilibrium
point.

Related Works. Game theory analyzes strategic interactions in demand-side
management, modeling consumer behavior and optimizing incentives for efficient
resource use. The main types of games studied in this context are dynamic
games that focus on the dynamic relationships among the agents (e.g., shared
battery resources, dynamic pricing, etc.), and generalized aggregative games,
which account for the coupling constraints among the agents. Comprehensive
reviews on the analysis and decomposition of dynamic games can be found in [2,
3]. Various theoretical approaches have been employed to solve dynamic games
utilizing optimal control theory to study open-loop and closed-loop Nash
equilibrium [3, 4, 5]. For deterministic finite horizon discrete-time dynamic
games, the state dynamic equation can be treated as a finite number of
constraints which transforms the problem into a generalized aggregative game
[6], however, increasing the number of constraints can impose high
computational cost. Stochastic dynamic games have also been studied for both
continuous-time [7, 8] and discrete-time systems [9, 10]. However, these works
do not study chance constraints on the system’s state or coupling constraint
on the control inputs of agents. In contrast to previous studies that use
almost-sure satisfaction of constraints [11, 12], we employ chance constraints
due to their flexibility and potential for better outcomes with a predefined
confidence. Imposing deterministic constraints on estimated random variables
have also been considered in [13, 14, 15], but this approach overlooks the
inherent stochastic nature of the uncertainties, increasing the risk of
practical implementation failures. From an application perspective, DSM is
addressed through dynamic games with [16, 17] or without [18, 19] the presence
of shared battery. DSM has also been studied through generalized aggregative
games[20, 21, 13]. In [21, 13], the impact of shared dynamics was not taken
into account. These works studied the game in deterministic setups without
considering uncertainty sources in microgrid systems. In this paper, we are
dealing with a DSM game with stochastic shared dynamics of battery and
coupling chance constraint on the state, referred to as a generalized
stochastic dynamic aggregative game.

Contributions. The contributions of this paper are:

  * •

Introducing a novel framework for demand-side management as a generalized
stochastic dynamic aggregative game, featuring constraints in the form of
stochastic dynamics and chance constraints.

  * •

Providing a guaranteed under-approximation for the game in the form of a
generalized static aggregative game with deterministic constraints, in which,
the feasible set of the resulted game is a subset of the feasible set of the
original game.

  * •

Analyzing the existence and uniqueness of the generalized Nash equilibrium
(GNE) for the game, and proposing a semi-decentralized algorithm for Nash
seeking.

##  2 review on some mathematical concepts and tools

###  2.1 Chernoff-Hoeffding inequality

Consider a sequence of random variables, $Z_{i}$, where $i$ ranges from $1$ to
$n$, each with bounded support between $a_{i}$ and $b_{i}$. Let each of these
variables possess an expected value, $\mathbf{E}\\{Z_{i}\\}$, belonging to the
moment interval, $\mathbb{M}_{i}$. Now, let’s introduce a new random variable,
$Z$, defined as the sum of all $Z_{i}$ from $i=1$ to $n$. By employing the
Chernoff-Hoeffding inequality, we have the following inequalities:

$\displaystyle\mathbf{Pr}\left\\{Z-\mathbf{E}\left\\{Z\right\\}\leq-\zeta\right\\}\leq\mathrm{exp}\left(\frac{-\zeta^{2}}{\nu\sum_{i=1}^{n}\left(b_{i}-a_{i}\right)^{2}}\right)\,\
\forall\zeta\geq 0$
$\displaystyle\mathbf{Pr}\left\\{Z-\mathbf{E}\left\\{Z\right\\}\geq\zeta\right\\}\leq\mathrm{exp}\left(\frac{-\zeta^{2}}{\nu\sum_{i=1}^{n}\left(b_{i}-a_{i}\right)^{2}}\right)\
\ \forall\zeta\geq 0\ .$

In this context, $\nu=\chi(\hat{G})/2$ is a positive constant as referred in
[22]. Here, $\hat{G}$ signifies the undirected dependency graph of the random
variables $Z_{1},...,Z_{n}$, and $\chi(\hat{G})$ represents the chromatic
number of this graph. The chromatic number of $\hat{G}$, is defined as the
minimum number of colors required to color the vertices of $\hat{G}$ such that
no adjacent vertices share the same color. For instances when the variables
are independent, the chromatic number of the graph $\hat{G}$ is one.

###  2.2 Operator theoretic definitions

  * •

For a closed set $S\subseteq\mathbb{R}^{n}$, the mapping
$\mathrm{proj}_{S}:\mathbb{R}^{n}\longrightarrow S$ denotes the projection
onto $S$, i.e., $\mathrm{proj}_{S}(x)=\mathrm{argmin}_{y\in S}\|y-x\|$.

  * •

A set-valued mapping
$\mathcal{F}:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ is $l-$Lipschiz
continuous, with $l>0$, if $\|u-v\|\leq l\|x-y\|$ for all
$x,y\in\mathbb{R}^{n}$,$u\in\mathcal{F}(x),v\in\mathcal{F}(y)$.

  * •

A set-valued mapping
$\mathcal{F}:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ is monotone if
$(u-v)^{T}(x-y)\geq 0$ for all $x\neq
y\in\mathbb{R}^{n},u\in\mathcal{F}(x),v\in\mathcal{F}(y)$.

  * •

A set-valued mapping
$\mathcal{F}:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ is strictly
monotone if $(u-v)^{T}(x-y)>0$ for all $x\neq
y\in\mathbb{R}^{n},u\in\mathcal{F}(x),v\in\mathcal{F}(y)$.

  * •

A set-valued mapping
$\mathcal{F}:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ is $\eta-$strongly
monotone with $\eta>0$, if $(u-v)^{T}(x-y)\geq\eta\|x-y\|^{2}$ for all
$x,y\in\mathbb{R}^{n},u\in\mathcal{F}(x),v\in\mathcal{F}(y)$.

###  2.3 Generalized Variational Inequality

Consider a closed convex set $S\subset\mathbb{R}^{n}$, a set-valued mapping
$\Psi:S\rightrightarrows\mathbb{R}^{n}$, and a single-valued mapping
$\psi:S\rightarrow\mathbb{R}^{n}$. The generalized variational inequality
problem $\mathrm{GVI}(S,\Psi)$, seeks to find $\mathbf{x}^{*}\in S$ and
$\mathbf{g}^{*}\in\Psi(\mathbf{x}^{*})$ the following condition holds for all
$\mathbf{x}\in S$:

$(\mathbf{x}-\mathbf{x}^{*})^{T}\mathbf{g}^{*}\geq 0.$

If $\Psi(\mathbf{x})={\psi(\mathbf{x})}$ for all $\mathbf{x}\in S$, then the
generalized variational inequality problem $\mathrm{GVI}(S,\Psi)$ reduces to
the standard variational inequality problem $\mathrm{VI}(S,\psi)$.[23]

###  2.4 Gershgorin circle theorem

The Gershgorin circle theorem may be used to bound the spectrum of a square
matrix.
Let $A$ be a complex $n\times n$, with entries $a_{ij}$. For $i=1,\dots,n$ let
$R_{i}$ be the sum of absolute values of non-diagonal entries in $i-$th row:

$\displaystyle R_{i}=\sum_{j=1,j\neq i}^{n}|a_{ij}|$

Let $D(a_{ii},R_{i})\subset\mathbb{C}$ , be a closed disc centered at $a_{ii}$
with radius $R_{i}$. Such a disc is called Gershgorin disc.

###### Lemma 1

Every eigenvalue of $A$ lies within at least one of the Gershgorin discs
$D(a_{ii},R_{i})$.

Notations. $\mathbb{R}$ denotes the set of real numbers.
$\mathbf{0}_{\tau}(\mathbf{1}_{\tau})$ denotes a vector with dimention
$\tau\times 1$ that all elements equal to $0(1)$.
$\mathbf{1}_{\tau\times\tau}$ denotes a matrix with dimention $\tau\times\tau$
that all elements equal to $1$. $\mathbf{I}_{\tau}$ denotes a $\tau\times\tau$
identity matrix. $A\otimes B$ denotes the Kronecker product between matrices
$A$ and $B$. Suppose that we have $\mathit{N}$ vectors,
$\mathbf{x}_{1},...,\mathbf{x}_{N}\in\mathbb{R}^{n}$, then we define
$\mathbf{x}\triangleq\left[\mathbf{x}_{1}^{T},...,\mathbf{x}_{N}^{T}\right]^{T}$.
$\mathbf{M}_{\tau}$ is a $\tau\times\tau$ lower triangular matrix such that
$M_{\tau}(i,j)=1$ if $i\leq j$, and zero otherwise.

##  3 System Model

We investigate a grid-connected community microgrid as shown in Figure 1,
which comprises $N$ selfish residential households serving as agents in the
system. Each household has the capacity to meet its own energy requirements
via the power grid and a shared battery, which is recharged using renewable
resources. Furthermore, we incorporate in our model the uncertainty in both
the renewable energy sources and the energy demand. The microgrid operates
under a tariff scheme with a retailer communicating the tariff information to
the households. Each household uses this information to decide its battery
discharging profile for minimizing the total cost of electricity consumption,
which includes the cost of purchasing electricity from the retailer and the
cost of using the battery to store and discharge energy. We consider the
system where agents interact solely with a coordinator. All the decisions are
taken over the day-ahead horizon.

###  3.1 Shared Battery Model

Battery plays an important role in microgrid systems by providing energy
storage solutions that can balance energy supply and demand, and contribute in
peak load shaving by proving a grid-free source of energy in peak hours. We
consider that the battery is charged only through renewable energy and
discharged by consumption of the agents from the battery. The state of charge
(SoC) of the shared battery $x^{t}$ is considered to have the following
dynamics

$x^{t+1}=x^{t}+\eta\Delta t\left[r^{t}-\sum_{j=1}^{N}u_{j}^{t}\right].$ (1)

Figure 1: System model. Red, blue and purple solid lines show respectively the
energy supplied by the Distributed Network Provider, shared battery and
renewable energy. Green dashed line shows Essential Information Exchange (e.g.
demands, common tariff).

At time $t$, $u_{j}^{t}$ is the discharging decision of the battery by the
$j^{\text{th}}$ agent and $r^{t}$ is the power generated by renewable energy
sources, which is unknown and stochastic. The parameter $\eta$ is the
charging/discharging efficiency of the shared battery, and $\Delta t$ is the
sampling time.

Due to the stochastic input in (1), we consider the following chance
constraints on the SoC and its final value:

$\displaystyle\mathbf{Pr}\\{\underline{x}\leq x^{t}\leq\bar{x}\\}\geq
1-\delta_{x}^{t}\hskip 14.22636pt\forall t\in\left\\{0,\dots,\tau\right\\},$
(2) $\displaystyle\mathbf{Pr}\\{|x^{\tau}-r|\leq\epsilon\\}\geq
1-\delta_{x}^{\tau,\mathrm{final}},$ (3)

where $\underline{x},\bar{x},\delta_{x}^{t}$, and
$\delta_{x}^{\tau,\mathrm{final}}$ are positive constants in $[0,1]$ such that
$\underline{x}<\bar{x}$, $r\in\left[\underline{x},\bar{x}\right]$ is a
positive constant, and
$\epsilon\in\left(0,\textrm{min}\left\\{r-\underline{x},\bar{x}-r\right\\}\right)$.
The chance constraint in (3) guarantees that the SoC at the final time step
lies within a specific range around $r$ with a certain confidence. We also
have the following constraint:

$0\leq u_{i}^{t}\leq\bar{u}_{i}\hskip 14.22636pt\forall
t\in\left\\{0,\dots,\tau-1\right\\}.$ (4)

###  3.2 Power exchange model

The load balance equation for household $i$ at time $t\in\\{0,...,\tau-1\\}$
can be written as $g_{i}^{t}=d_{i}^{t}-u_{i}^{t}$, where $g_{i}^{t}$ is power
exchange of the $i^{\text{th}}$ agent with the grid and $d_{i}^{t}$ is its
stochastic power demand. Let $g^{t}\triangleq\sum_{i=1}^{N}g_{i}^{t}$. The
retailer imposes the following constraints on the allowable power exchange
with the community of the microgrid:

$\mathbf{Pr}\left\\{0\leq g^{t}\leq\bar{g}\right\\}\geq
1-\delta_{g}^{t}\quad\forall t\in\\{0,\dots,\tau-1\\},$ (5)

where $\bar{g}$ is the maximum power supply of the retailer, and
$\delta_{g}^{t}\in\left[0,1\right]$.

###  3.3 Electricity tarrifs

All the agents in the neighbourhood are billed using common electricity
tariffs, modelled as

$\pi(g^{t})=K_{ToU}^{t}+\sum_{j=1}^{N}g_{j}^{t}k_{c}^{N}\;,$ (6)

where $\pi(g^{t})$ is the common electricity tariff for agents at time $t$ and
$K_{ToU}^{t}$ is the conventional time-of-use pricing tariff that could depend
on hour of the day. The positive constant $k_{c}^{N}$ influences the cost in
proportion to the peak power consumption creates a balance between the
microgrid’s objectives and ensuring fairness, peak shaving, and stability in
the tariff system. As mentioned in [11, 24], $k_{c}^{N}$ is selected to be
inversely proportional to the number of agents (this dependency is denoted by
the superscript $N$), which performs normalization in aggregative term of grid
electricity use of agents. The tariff function (6) is based on the hypothesis
that increasing the price of electricity at times of peak demand will motivate
a rational household to schedule the shared battery such that the community
peak can be shaved [25, 26].

###  3.4 Cost Function of Each Agent in Model

The cost function of each agent (household) is

$\displaystyle J_{i}=\mathbf{E}\Bigg{\\{}\sum_{t=0}^{\tau-1}\Bigg{[}$
$\displaystyle\pi\left(g^{t}\right)g_{i}^{t}+\sum_{j=1}^{N}\left(\alpha^{dch}\left(u_{j}^{t}\right)^{2}+\beta^{dch}u_{j}^{t}\right)\Bigg{]}\Bigg{\\}},$
(7)

where $\alpha^{dch},\beta^{dch}$ are positive constants. The cost function has
a conventional form used frequently in the literature (see e.g., [13, 11]). It
consists of two parts: one related to the cost of electricity and the other to
battery degradation, which serves as a proxy for the shared battery lifespan.

Let $\mathbf{u}_{i}=[u_{i}^{0},u_{i}^{1},\dots,u_{i}^{\tau-1}]^{T}$. We are
now equipped to delineate the game-theoretic setup for our demand-side
management model, as

$\displaystyle\mathcal{G}=\begin{cases}\textbf{Players:}\ \textrm{A set of
residential agents}\ \mathcal{N}=\\{1,2,\dots,N\\}\\\ \textbf{Strategies of
Agents:}\ \mathbf{u}_{i}\hskip 28.45274pti\in\mathcal{N}\\\ \textbf{Cost
Functions:}\ J_{i}\left(\mathbf{u}_{i},\mathbf{u}_{-i}\right)\hskip
21.33955pti\in\mathcal{N}\\\ \textbf{Stochastic Dynamic:}\ \eqref{SoC dynamics
of the battery}\\\ \textbf{Constraints:}\begin{cases}\textbf{Local:}\ \eqref{u
constaint1}\\\ \textbf{Coupling}\ \textbf{(Chance Constraints):}\ \eqref{Soc
constraint},\eqref{reformulation last time SoC},\eqref{retailer
constraint}\end{cases}\end{cases}$

As demonstrated above, $\mathcal{G}$ evidently corresponds to a generalized
stochastic dynamic aggregative game.

##  4 Approximation with Generalized Static Aggregative Game

We raise the following assumption for constructing an under-approximation of
the feasible set of the game.

###### Assumption 1

The initial state $x^{0}$ is known and $\underline{x}\leq x^{0}\leq\bar{x}$.
Additionally, $r^{t}$ and $d_{i}^{t}$ are random variables with bounded
support $[a^{t},b^{t}]$ and $[c_{i}^{t},f_{i}^{t}]$, and their mean values are
available to each agent. Random variables $r^{t}$ and $d_{i}^{t}$ could in
general be dependent with known dependency graph.

The above assumption is based on the fact that the power generated by
renewable energy sources and power demand of agents are bounded. Let us denote
the input constraints in (4) in vector format as
$\mathbf{0}_{\tau}\leq\mathbf{u}_{i}\leq\bar{u}_{i}\mathbf{1}_{\tau}$.

###### Proposition 1

The feasible domain of the chance constraint (2) can be under-approximated
with the deterministic inequalities

$\displaystyle\left(\underline{x}-x^{0}\right)\mathbf{1}_{\tau}-\rho\mathbf{M}_{\tau}\mu_{r}+\mathbf{q}_{x,1}+\sum_{j=1}^{N}\rho\mathbf{M}_{\tau}\mathbf{u}_{j}\leq
0,$ (8)
$\displaystyle\left(x^{0}-\bar{x}\right)\mathbf{1}_{\tau}+\rho\mathbf{M}_{\tau}\mu_{r}+\mathbf{q}_{x,2}-\sum_{j=1}^{N}\rho\mathbf{M}_{\tau}\mathbf{u}_{j}\leq
0,$ (9)

where
$\mu_{r}=\left[\mathbf{E}\left\\{r^{0}\right\\},\mathbf{E}\left\\{r^{1}\right\\},\dots,\mathbf{E}\left\\{r^{\tau-1}\right\\}\right]^{T}$,
$\mathbf{q}_{x,1}=\left[q_{x,1}^{1},q_{x,1}^{2},\dots,q_{x,1}^{\tau}\right]^{T}$,
$\mathbf{q}_{x,2}=\left[q_{x,2}^{1},q_{x,2}^{2},\dots,q_{x,2}^{\tau}\right]^{T}$,
with
$q_{x,1}^{t}\triangleq\sqrt{-\rho^{2}\nu_{r}^{t}\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}\mathrm{Ln}(\tilde{\delta}_{x}^{t})}$,
$q_{x,2}^{t}\triangleq\sqrt{-\rho^{2}\nu_{r}^{t}\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}\mathrm{Ln}(\tilde{\delta}_{x}^{t}-\delta_{x}^{t})}$
, $0\leq\tilde{\delta}_{x}^{t}\leq 1$,
$0\leq\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\leq 1$, and
$\rho\triangleq\eta\Delta t$. Moreover, $\nu_{r}^{t}=\chi(\hat{G}_{r}^{t})/2$
with $\hat{G}_{r}^{t}$ being the undirected dependency graph of the random
variables $r^{0},\ldots,r^{t-1}$, and $\chi(\hat{G}_{r}^{t})$ represents the
chromatic number of this graph (see [22]).

######  Proof 4.1.

The proof is based on deriving from (1) the explicit form of $x^{t}$ as

$x^{t}=x^{0}+\sum_{k=0}^{t-1}\left\\{-\rho\left(\sum_{j=1}^{N}u_{j}^{k}\right)+\rho
r^{k}\right\\}$ (10)

and using Chernoff-Hoeffding inequality [22].
It is obvious that:

$\displaystyle\mathbf{Pr}\\{\underline{x}\leq x^{t}\leq\bar{x}\\}\geq
1-\delta_{x}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\left(\underline{x}\leq
x^{t}\right)\bigcap\left(x^{t}\leq\bar{x}\right)\right\\}\geq
1-\delta_{x}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\left(x^{t}\leq\underline{x}\right)\bigcup\left(\bar{x}\leq
x^{t}\right)\right\\}\leq\delta_{x}^{t}\Longleftarrow$
$\displaystyle\begin{cases}\mathbf{Pr}\left\\{x^{t}\leq\underline{x}\right\\}\leq\tilde{\delta}_{x}^{t}\\\
\mathbf{Pr}\left\\{\bar{x}\leq
x^{t}\right\\}\leq\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\end{cases}.$

where $\tilde{\delta}_{x}^{t}$ is real value in $[0,1]$ such that
$0\leq\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\leq 1$. Consequently, we establish
certain sufficient conditions for $\mathbf{Pr}\\{\underline{x}\leq
x^{t}\leq\bar{x}\\}\geq 1-\delta_{x}^{t}$. Initially, we commence with the
condition
$\mathbf{Pr}\left\\{x^{t}\leq\underline{x}\right\\}\leq\tilde{\delta}_{x}^{t}$.

$\displaystyle\mathbf{Pr}\left\\{x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho
r^{k}\leq\underline{x}\right\\}\leq\tilde{\delta}_{x}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho
r^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\leq\underline{x}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right\\}\leq\tilde{\delta}_{x}^{t}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\sum_{k=0}^{t-1}\rho
r^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\leq\underline{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right\\}\leq\tilde{\delta}_{x}^{t}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\sum_{k=0}^{t-1}r^{k}-\sum_{k=0}^{t-1}\mu_{r}^{k}\leq\frac{1}{\rho}\left(\underline{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\right\\}$
$\displaystyle\leq\tilde{\delta}_{x}^{t}$

In order to employ the Chernoff-Hoeffding inequality, it is essential that the
following condition be satisfied:

$\displaystyle\left(\underline{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\leq
0$

If the aforementioned condition is met, we can, based on the Chernoff-
Hoeffding inequality, state the following:

$\displaystyle\mathbf{Pr}\left\\{\sum_{k=0}^{t-1}r^{k}-\sum_{k=0}^{t-1}\mu_{r}^{k}\leq-\frac{1}{\rho}\left(-\underline{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\right\\}$
$\displaystyle\leq\mathrm{exp}\left(\frac{-\left[\frac{1}{\rho}\left(-\underline{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\right]^{2}}{\nu^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}}\right)$

So it is necessary to have:

$\displaystyle\mathrm{exp}\left(\frac{-\left[\frac{1}{\rho}\left(-\underline{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\right]^{2}}{\nu^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}}\right)\leq\tilde{\delta}_{x}^{t}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\left(-\underline{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)^{2}\geq-\rho^{2}\nu^{t}\mathrm{Ln}\tilde{\delta}_{x}^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}$

Clearly, the term
$-\rho^{2}\nu^{t}\mathrm{Ln}\tilde{\delta}_{x}^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}$
is non-negative, thus we can state:

$\displaystyle-\underline{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}$
$\displaystyle\geq\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\tilde{\delta}_{x}^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}}$
$\displaystyle\mathrm{Or}$
$\displaystyle-\underline{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}$
$\displaystyle\leq-\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\tilde{\delta}_{x}^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}}$

But
$-\underline{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\geq\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\tilde{\delta}_{x}^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}}$
is only valid, base on
$\left(\underline{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\leq
0$. Hence, we arrive at the following conclusion:

$\displaystyle\underline{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}$
$\displaystyle+\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\tilde{\delta}_{x}^{t}\sum_{k=0}^{t-1}(b^{k}-a^{k})^{2}}\leq
0$

Now, we commence with the condition $\mathbf{Pr}\left\\{\bar{x}\leq
x^{t}\right\\}\leq\delta_{x}^{t}-~{}\tilde{\delta}_{x}^{t}$.

$\displaystyle\mathbf{Pr}\left\\{\bar{x}\leq
x^{t}\right\\}\leq\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\bar{x}\leq
x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}+\sum_{k=0}^{t-1}\rho
r^{k}\right\\}\leq\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\leq\sum_{k=0}^{t-1}\rho
r^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right\\}$
$\displaystyle\leq\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\sum_{k=0}^{t-1}r^{k}-\sum_{k=0}^{t-1}\mu_{r}^{k}\geq\frac{1}{\rho}\left(\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\right\\}$
$\displaystyle\leq\delta_{x}^{t}-\tilde{\delta}_{x}^{t}$

In order to employ the Chernoff-Hoeffding inequality, it is essential that the
following condition be satisfied:

$\displaystyle\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\geq
0$

If the aforementioned condition is met, we can, based on the Chernoff-
Hoeffding inequality, state the following:

$\displaystyle\mathbf{Pr}\left\\{\sum_{k=0}^{t-1}r^{k}-\sum_{k=0}^{t-1}\mu_{r}^{k}\geq\frac{1}{\rho}\left(\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\right\\}$
$\displaystyle\leq\mathrm{exp}\left(\frac{-\left[\frac{1}{\rho}\left(\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)\right]^{2}}{\nu^{t}\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}}\right)$

Like the previous part, it is necessary to have:

$\displaystyle\left(\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\right)^{2}$
$\displaystyle\geq-\rho^{2}\nu^{t}\mathrm{Ln}\left(\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\right)\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}$

It is obvious that
$-\rho^{2}\nu^{t}\mathrm{Ln}\left(1-\delta_{t}-\tilde{\delta}_{t}\right)\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}$
is non-negative, so:

$\displaystyle\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}$
$\displaystyle\geq\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\left(\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\right)\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}}$
$\displaystyle\mathrm{Or}$
$\displaystyle\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}$
$\displaystyle\leq-\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\left(\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\right)\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}}$

But
$\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\geq\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\left(\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\right)\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}}$
is just valid, base on
$\bar{x}-x^{0}-\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}-\sum_{k=0}^{t-1}\rho\mu_{r}^{k}\geq
0$. Hence, we arrive at the following conclusion:

$\displaystyle-\bar{x}+x^{0}+\sum_{k=0}^{t-1}\sum_{j=1}^{N}(-\rho)u_{j}^{k}$
$\displaystyle+\sum_{k=0}^{t-1}\rho\mu_{r}^{k}+\sqrt{-\rho^{2}\nu^{t}\mathrm{Ln}\left(\delta_{x}^{t}-\tilde{\delta}_{x}^{t}\right)\sum_{k=0}^{t-1}\left(b^{k}-a^{k}\right)^{2}}\leq
0$

######  Proposition 4.2.

The feasible domain of the chance constraint (3) can be under-approximated
with the deterministic inequalities

$\displaystyle
r\\!-\\!\epsilon\\!-\\!x^{0}\\!-\\!\rho\mathbf{1}_{\tau}^{T}\mu_{r}\\!+\\!q_{x,\mathrm{final},1}\\!+\\!\sum_{j=1}^{N}\rho\mathbf{1}_{\tau}^{T}\mathbf{u}_{j}\leq
0,$ (11) $\displaystyle
x^{0}\\!-\\!r\\!-\\!\epsilon\\!+\\!\rho\mathbf{1}_{\tau}^{T}\mu_{r}\\!+\\!q_{x,\mathrm{final},2}\\!+\\!\sum_{j=1}^{N}\left(-\rho\mathbf{1}_{\tau}^{T}\right)\mathbf{u}_{j}\leq
0,$ (12)

where
$q_{x,\mathrm{final},1}=\sqrt{-\rho^{2}\nu_{r}^{\tau}\sum_{k=0}^{\tau-1}\left(b^{k}-a^{k}\right)^{2}\mathrm{Ln}(\tilde{\delta}_{x}^{\tau,\mathrm{final}})}$,
$q_{x,\mathrm{final},2}=\sqrt{-\rho^{2}\nu_{r}^{\tau}\sum_{k=0}^{\tau-1}\left(b^{k}-a^{k}\right)^{2}\mathrm{Ln}(\delta_{x}^{\tau,\mathrm{final}}-\tilde{\delta}_{x}^{\tau,\mathrm{final}})}$,
with $0\leq\tilde{\delta}_{x}^{\tau,\mathrm{final}}\leq 1$, and
$0\leq\delta_{x}^{\tau,\mathrm{final}}-\tilde{\delta}_{x}^{\tau,\mathrm{final}}\leq
1$.

######  Proof 4.3.

The proof is like Proposition 1.

######  Proposition 4.4.

The feasible domain of the chance constraint (5) can be under-approximated
with the deterministic inequalities

$\displaystyle-\sum_{j=1}^{N}\mu_{d_{j}}+\mathbf{q}_{g,1}+\sum_{j=1}^{N}\mathbf{u}_{j}\leq
0,$ (13)
$\displaystyle-\bar{g}\mathbf{1}_{\tau}+\sum_{j=1}^{N}\mu_{d_{j}}+\mathbf{q}_{g,2}-\sum_{j=1}^{N}\mathbf{u}_{j}\leq
0,$ (14)

where
$\mu_{d_{j}}=\left[\mathbf{E}\left\\{d_{j}^{0}\right\\},\mathbf{E}\left\\{d_{j}^{1}\right\\},\dots,\mathbf{E}\left\\{d_{j}^{\tau-1}\right\\}\right]^{T}$,
$\mathbf{q}_{g,1}=\left[q_{g,1}^{0},q_{g,1}^{1},\dots,q_{g,1}^{\tau-1}\right]^{T}$,
$\mathbf{q}_{g,2}=\left[q_{g,2}^{0},q_{g,2}^{1},\dots,q_{g,2}^{\tau-1}\right]^{T}$,
with
$q_{g,1}^{t}=\sqrt{-\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}\mathrm{Ln}(\tilde{\delta}_{g}^{t})}$,
$q_{g,2}^{t}=\sqrt{-\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}\mathrm{Ln}(\delta_{g}^{t}-\tilde{\delta}_{g}^{t})}$,
$0\leq\tilde{\delta}_{g}^{t}\leq 1$, and
$0\leq\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\leq 1$. Moreover,
$\nu_{d}^{t}=\chi(\hat{G}_{d}^{t})/2$ with $\hat{G}_{d}^{t}$ being the
undirected dependency graph of the random variables
$d_{1}^{t},\ldots,d_{N}^{t}$.

######  Proof 4.5.

We know, that $g_{i}^{t}=d_{i}^{t}-u_{i}^{t}$ and
$g^{t}=\sum_{j=1}^{N}g_{j}^{t}=\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}u_{j}^{t}$.
We also considered $\mathbf{Pr}\left\\{0\leq g^{t}\leq\bar{g}\right\\}\geq
1-\delta_{g}^{t}$ as retailer constraint. Now like what we have done in
Proposition 1, we have:

$\displaystyle\mathbf{Pr}\left\\{0\leq g^{t}\leq\bar{g}\right\\}\geq
1-\delta_{g}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\left(0\leq
g^{t}\right)\bigcap\left(g^{t}\leq\bar{g}\right)\right\\}\geq
1-\delta_{g}^{t}\Longleftrightarrow$ $\displaystyle
1-\mathbf{Pr}\left\\{\left(g^{t}\leq
0\right)\bigcup\left(g^{t}\geq\bar{g}\right)\right\\}\geq
1-\delta_{g}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{\left(g^{t}\leq
0\right)\bigcup\left(g^{t}\geq\bar{g}\right)\right\\}\leq\delta_{g}^{t}\Longleftrightarrow$
$\displaystyle\mathbf{Pr}\left\\{g^{t}\leq
0\right\\}+\mathbf{Pr}\left\\{g^{t}\geq\bar{g}\right\\}\leq\delta_{g}^{t}\Longleftarrow$
$\displaystyle\begin{cases}\mathbf{Pr}\left\\{g^{t}\leq
0\right\\}\leq\tilde{\delta}_{g}^{t}\\\
\mathbf{Pr}\left\\{g^{t}\geq\bar{g}\right\\}\leq\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\end{cases}$

We know:

$\displaystyle\mathbf{Pr}\left\\{g^{t}\leq
0\right\\}=\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}u_{j}^{t}\leq
0\right\\}=$
$\displaystyle\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}u_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\leq-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\right\\}=$
$\displaystyle\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\leq\sum_{j=1}^{N}u_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\right\\}$

For identifying a sufficient condition that ensures
$\mathbf{Pr}\left\\{g^{t}\leq 0\right\\}\leq\tilde{\delta}_{g}^{t}$ is
satisfied, we use Chernoff-Hoeffding inequality. For using Chernoff-Hoeffding
inequality, it is necessary the following condition satisfied:

$\displaystyle\sum_{j=1}^{N}u_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\leq 0$

If the aforementioned condition is met, we can, based on the Chernoff-
Hoeffding inequality, state the following:

$\displaystyle\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\leq-\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)\right\\}$
$\displaystyle\leq\mathrm{exp}\left(\frac{-\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)^{2}}{\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\right)$

So it is necessary to have:

$\displaystyle\mathrm{exp}\left(\frac{-\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)^{2}}{\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\right)\leq\tilde{\delta}_{g}^{t}\Longleftrightarrow$
$\displaystyle\frac{-\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)^{2}}{\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\leq\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\Longleftrightarrow$
$\displaystyle-\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)^{2}\leq\nu_{d}^{t}\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}\Longleftrightarrow$
$\displaystyle\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)^{2}\geq-\nu_{d}^{t}\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}$

It is obvious that
$-\nu_{d}^{t}\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}$
is non-negative, so:

$\displaystyle\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)\geq\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}$
$\displaystyle\mathrm{Or}$
$\displaystyle\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)\leq-\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}$

But
$\left(\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}\right)\geq\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}$
is just valid. So, we consider:

$\displaystyle\sum_{j=1}^{N}u_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left(\tilde{\delta}_{g}^{t}\right)\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\leq
0$

Now we discuss similarly about
$\mathbf{Pr}\left\\{g^{t}\geq\bar{g}\right\\}\leq\delta_{g}^{t}-\tilde{\delta}_{g}^{t}$.
We know:

$\displaystyle\mathbf{Pr}\left\\{g^{t}\geq\bar{g}\right\\}=\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}u_{j}^{t}\geq\bar{g}\right\\}=$
$\displaystyle\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}u_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\geq\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\right\\}=$
$\displaystyle\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\geq\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\right\\}$

For identifying a sufficient condition that ensures
$\mathbf{Pr}\left\\{g^{t}\geq\bar{g}\right\\}\leq\delta_{g}^{t}-\tilde{\delta}_{g}^{t}$
is satisfied, we use Chernoff-Hoeffding inequality. For using Chernoff-
Hoeffding inequality, it is necessary the following condition satisfied:

$\displaystyle\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\geq
0$

If the aforementioned condition is met, we can, based on the Chernoff-
Hoeffding inequality, state the following:

$\displaystyle\mathbf{Pr}\left\\{\sum_{j=1}^{N}d_{j}^{t}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}\geq\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\right\\}$
$\displaystyle\leq\mathrm{exp}\left(\frac{-\left(\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\right)^{2}}{\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\right)$

So it is necessary to have:

$\displaystyle\mathrm{exp}\left(\frac{-\left(\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\right)^{2}}{\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\right)\leq\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\Longleftrightarrow$
$\displaystyle\frac{-\left(\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\right)^{2}}{\nu_{d}^{t}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\leq\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\Longleftrightarrow$
$\displaystyle-\left(\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\right)^{2}\leq\nu_{d}^{t}\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\left(\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\right)^{2}\geq-\nu_{d}^{t}\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}$

It is obvious that
$-\nu_{d}^{t}\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}$
is non-negative, so:

$\displaystyle\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\geq\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}$
$\displaystyle\mathrm{Or}$
$\displaystyle\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\leq-\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}$

But
$\bar{g}-\sum_{j=1}^{N}\mu_{d_{j}}^{t}+\sum_{j=1}^{N}u_{j}^{t}\geq\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}$
is just valid. So, we consider:

$\displaystyle-\bar{g}+\sum_{j=1}^{N}\mu_{d_{j}}^{t}-\sum_{j=1}^{N}u_{j}^{t}+\sqrt{-\nu_{d}^{t}\mathrm{Ln}\left\\{\delta_{g}^{t}-\tilde{\delta}_{g}^{t}\right\\}\sum_{j=1}^{N}\left(f_{j}^{t}-c_{j}^{t}\right)^{2}}\leq
0$

All the inequalities in (8)–(9) and (11)–(14) can be written as

$\sum_{j=1}^{N}A\mathbf{u}_{j}\leq b,$ (15)

with appropriately defined matrices $A$ and $b$, like,

$\displaystyle A=\begin{bmatrix}\rho\mathbf{M}_{\tau}\\\ \dots\dots\\\
-\rho\mathbf{M}_{\tau}\\\ \dots\dots\\\ \rho\mathbf{1}_{\tau}^{T}\\\
\dots\dots\\\ -\rho\mathbf{1}_{\tau}^{T}\\\ \dots\dots\\\ \mathbf{I}_{\tau}\\\
\dots\dots\\\ -\mathbf{I}_{\tau}\\\
\end{bmatrix},b=-\begin{bmatrix}\left(\underline{x}-x^{0}\right)\mathbf{1}_{\tau}-\rho\mathbf{M}_{\tau}\mu_{r}+\mathbf{q}_{x,1}\\\
\dots\dots\\\
\left(x^{0}-\bar{x}\right)\mathbf{1}_{\tau}+\rho\mathbf{M}_{\tau}\mu_{r}+\mathbf{q}_{x,2}\\\
\dots\dots\\\
r-\epsilon-x^{0}-\rho\mathbf{1}_{\tau}^{T}\mu_{r}+q_{x,\mathrm{final},1}\\\
\dots\dots\\\
x^{0}-r-\epsilon+\rho\mathbf{1}_{\tau}^{T}\mu_{r}++q_{x,\mathrm{final},2}\\\
\dots\dots\\\ -\sum_{j=1}^{N}\mu_{d_{j}}+\mathbf{q}_{g,1}\\\ \dots\dots\\\
-\bar{g}\mathbf{1}_{\tau}+\sum_{j=1}^{N}\mu_{d_{j}}+\mathbf{q}_{g,2}\end{bmatrix}\\!,$

and with explicit local constraint or local decision set

$\Omega_{i}=\left\\{\mathbf{u}_{i}\in\mathbb{R}^{T}|\mathbf{0}_{\tau}\leq\mathbf{u}_{i}\leq\bar{u}_{i}\mathbf{1}_{\tau}\right\\}.$
(16)

Reformulation of the cost function. We can rewrite the cost function (7) as

$J_{i}=\mathbf{u}_{i}^{T}G\mathbf{u}_{i}+T_{i}\mathbf{u}_{i}+\left[\frac{1}{N}\sum_{j=1}^{N}\mathbf{u}_{j}^{T}\right]H\mathbf{u}_{i}+c_{i}\
,$ (17)

where

$\displaystyle G$ $\displaystyle=\alpha^{dch}\mathbf{I}_{\tau},\hskip
2.84544ptH=Nk_{c}^{N}\mathbf{I}_{\tau},\hskip 2.84544pt$ $\displaystyle T_{i}$
$\displaystyle=\left(-K_{ToU}-k_{c}^{N}\mu_{d_{i}}-k_{c}^{N}\sum_{j=1}^{N}\mu_{d_{j}}+\beta^{dch}\mathbf{1}_{\tau}\right)^{T}$
$\displaystyle c_{i}$
$\displaystyle=\mu_{d_{i}}^{T}K_{ToU}+\sum_{t=0}^{\tau-1}\Biggl{[}k_{c}^{N}\mathbf{E}\left\\{\sum_{j=1}^{N}d_{i}^{t}d_{j}^{t}\right\\}$
$\displaystyle+\alpha^{dch}\sum_{j=1,j\neq
i}^{N}\left(u_{j}^{t}\right)^{2}+\left(-k_{c}^{N}\mu_{d_{i}^{t}}+\beta^{dch}\right)\sum_{j=1,j\neq
i}^{N}u_{j}^{t}\Biggr{]}.$ $\displaystyle K_{ToU}$
$\displaystyle=\left[K_{ToU}^{0},K_{ToU}^{1},\dots,K_{ToU}^{\tau-1}\right]^{T},$
$\displaystyle\mu_{{r}^{t}}$
$\displaystyle=\mathbf{E}\left\\{r^{t}\right\\},\mu_{d_{i}^{t}}=\mathbf{E}\left\\{d_{i}^{t}\right\\}.$

The above under-approximations and reformulation of the cost function gives
the following game $\mathcal{G^{\prime}}$:

$\displaystyle\mathcal{G^{\prime}}=\begin{cases}\textbf{Players:}\ \textrm{A
set of residential agents}\ \mathcal{N}=\\{1,2,\dots,N\\}\\\
\textbf{Strategies of Agents:}\ \mathbf{u}_{i}\hskip
28.45274pti\in\mathcal{N}\\\ \textbf{Cost Functions:}\
J_{i}\left(\mathbf{u}_{i},\mathbf{u}_{-i}\right)\hskip
21.33955pti\in\mathcal{N}\\\
\textbf{Constraints:}\begin{cases}\textbf{Local:}\ \eqref{local constraint}\\\
\textbf{Coupling\ (deterministic and static):}\ \eqref{coupling
constraint}\end{cases}\end{cases}$

where
$\mathbf{u}_{-i}=\mathrm{col}\left(\left\\{\mathbf{u}_{j}\right\\}_{j\neq
i}\right)$. Each agent seeks to minimize its cost function,
$J_{i}(\mathbf{u}_{i},\mathbf{u}_{-i})$, while taking into account both the
coupling constraint (15) and the local constraint (16).

##  5 Game Theoretical Analysis

Utilizing the coupling constraint and local decision set, we define the
collective feasible set $\mathcal{U}$ as

$\mathcal{U}=\Omega\cap\left\\{(\mathbf{y}_{1},\dots,\mathbf{y}_{N})\in\mathbb{R}^{\tau
N}|\sum_{i=1}^{N}A\mathbf{y}_{i}-b\leq 0\right\\},$ (18)

where $\Omega=\Omega_{1}\times\Omega_{2}\times\dots\times\Omega_{N}$.
Additionally, we define the feasible decision set of agent
$i\in\left\\{1,\dots,N\right\\}$ as

$\mathcal{U}_{i}(\mathbf{u}_{-i})=\left\\{\mathbf{y}_{i}\in\Omega_{i}|A\mathbf{y}_{i}\leq
b-\sum_{j=1,j\neq i}^{N}A\mathbf{u}_{-i}\right\\}.$

Our local optimization problem can be formalized as a game theory setup:

$\begin{cases}\mathrm{min}_{\mathbf{u}_{i}\in\Omega_{i}}J(\mathbf{u}_{i},\mathbf{u}_{-i})\\\
\mathrm{s.t.}\quad A\mathbf{u}_{i}\leq b-\sum_{j=1,j\neq
i}^{N}A\mathbf{u}_{j}\end{cases}\,\textrm{for all}\ i\in\\{1,\dots,N\\}$ (19)

In this context, we use the notion of Generalized Nash instead of Nash, since
the coupling among the agents is not only in their cost functions but also in
their constraints.

######  Definition 5.1 (Generalized Nash Equlibrium).

The collective strategy $\mathbf{u}^{*}$ is a generalized Nash equilibrium
(GNE) of the game in (19) if $\mathbf{u}^{*}\in\mathcal{U}$ and for all
$i\in\\{1,\dots,N\\}$

$J_{i}\left(\mathbf{u}_{i}^{*},\mathbf{u}_{-i}^{*}\right)\leq\inf\left\\{J_{i}(\mathbf{y},\mathbf{u}_{-i}^{*})|\mathbf{y}\in\mathcal{U}_{i}(\mathbf{u}_{-i}^{*})\right\\}.$

######  Remark 5.2.

In our study, we can easily demonstrate that for each $i\in\\{1,\dots,N\\}$
and $\mathbf{y}\in\mathcal{U}_{-i}$, the local cost function
$J_{i}(\cdot,\mathbf{y})$ is strictly convex (since $G$ is positive definite)
and continuously differentiable. Additionally, for each agent, its local
decision set is non-empty, compact (i.e., closed and bounded), and convex
(since it is a hypercube, see (16)).

We also consider the following relax assumption.

###### Assumption 2

The collective feasible set satisfies Slater’s constraint qualification.

######  Proposition 5.3.

Under Assumption 2, the GNE exists in our proposed game setup.

######  Proof 5.4.

Based on Remark 5.2, Assumption 2 and by utilizing Brouwer’s fixed point
theorem [27, Proposition 12.7], it is evident that a GNE exists.

Our study aims to identify a subset of the GNE that exhibits favorable
properties such as economic fairness and enhanced social stability. Typically,
this subset can be determined by solving an appropriate variational
inequality, known as a generalized variational inequality (GVI). However,
since the local cost functions are continuously differentiable, we utilize a
specific type of GVI from [23]. Define the pseduo-gradient
$F:\mathcal{U}\rightarrow\mathbb{R}^{\tau N}$ as

$F(\mathbf{u})=\mathrm{col}\left(\left\\{\partial_{\mathbf{u}_{i}}J(\mathbf{u}_{i},\mathbf{u}_{-i})\right\\}_{i\in\\{1,\dots,N\\}}\right),$

which is single-valued with $F(\mathbf{u})=\Gamma\mathbf{u}+\Lambda$, where

$\displaystyle\Gamma$ $\displaystyle=\mathbf{I}_{N\times
N}\otimes\left[2\left(G+\frac{H^{T}}{N}\right)\right]+\left(\mathbf{1}_{N\times
N}-\mathbf{I}_{N\times N}\right)\otimes\left(\frac{H^{T}}{N}\right)$
$\displaystyle=\left(2\alpha^{dch}+k_{c}^{N}\right)\left(\mathbf{I}_{N\times
N}\otimes\mathbf{I}_{\tau}\right)+k_{c}^{N}\left(\mathbf{1}_{N\times
N}\otimes\mathbf{I}_{\tau}\right)$ $\displaystyle\Lambda$
$\displaystyle=\begin{bmatrix}T_{1}&T_{2}&\dots&T_{N}\end{bmatrix}^{T}.$

Based on Remark 5.2 and [27, Proposition 12.4], any solution of standard
variational inequality problem $\mathrm{VI}(\mathcal{U},F)$ is a generalized
Nash equilibrium of (19), However, the converse is not necessarily true.

######  Proposition 5.5.

The variational GNE exists and is unique.

######  Proof 5.6.

It has been demonstrated in [27, Proposition 12.11] that under all the system
model properties stated in Remark 5.2 and Assumption 2, a sufficient condition
for the existence (uniqueness) of a variational GNE in our game setup (19) is
$F$ being monotone (strictly monotone). The definition of monotonicity (strict
monotonicity) implies that if $\Gamma$ is positive semidefinite (positive
definite), then $F$ is monotone (strictly monotone). Since
$\left(2\alpha^{dch}+k_{c}^{N}\right)\left(\mathbf{I}_{N\times
N}\otimes\mathbf{I}_{\tau}\right)$ and $k_{c}^{N}\left(\mathbf{1}_{N\times
N}\otimes\mathbf{I}_{\tau}\right)$ are two commuting square matrices, then
based on [28, Theorem 1.3.12], the eigenvalues of $\Gamma$ are either
$(2\alpha^{dch}+k_{c}^{N}+NK_{c}^{N})$ or $(2\alpha^{dch}+k_{c}^{N})$, both
strictly positive, so $\Gamma$ is positive definite and $F$ is also strictly
monotone.

######  Remark 5.7.

$F$ is $\zeta$-strongly monotone with
$\zeta\in\left(0,2\alpha^{dch}+k_{c}^{N}\right)$. This can be demonstrated by
utilizing the definition of strong monotonicity, in a manner analogous to
Proposition 5.5.

######  Remark 5.8.

Based on Lipschitz definition and Gershgorin Theorem, $F$ is
$\mathit{l_{F}}$-Lipschitz with
$\mathit{l_{F}}>2\alpha^{dch}+\left(N+1\right)k_{c}^{N}$.

As mentioned in [23], projected-gradient algorithms for generalized
equilibrium seeking in aggregative games are preconditioned forward-backward
methods. Based on this, we propose a semi-decentralized preconditioned
forward-backward algorithm to solve variational inequality problem
VI($\mathcal{U},F$), as presented in Algorithm 1. In the proposed algorithm,
we define $\alpha_{i}$ as real values that belong to the interval
$\left(0,\frac{2\zeta}{l_{F}^{2}}\right)$ for all $i\in\\{1,\dots,N\\}$ and
$\gamma\in\left(0,\gamma_{max}\right)$. We set
$\gamma_{max}\triangleq\frac{1}{\|\mathbf{1}_{N}^{T}\otimes
A\|^{2}}\left[\frac{1}{\alpha_{i,max}}-\frac{1}{2\frac{\zeta}{l_{F}^{2}}}\right]$.

Initialization: $k\leftarrow 1$,
$\mathbf{u}_{i}^{1}\leftarrow\mathbf{u}_{i}^{\mathrm{init}}$,
$\lambda^{1}\leftarrow\lambda^{\mathrm{init}}$

Repeat $\displaystyle\hskip 4.26773pt{\footnotesize{\textbf{Agents:}}}\hskip
2.84544pt\forall i\in\\{1,\dots,N\\}$ $\displaystyle\hskip
8.5359pt\mathbf{u}_{i}^{k+1}\leftarrow\mathrm{proj}_{\Omega_{i}}\left[\left(\mathbf{I}_{\tau\times\tau}-2\alpha_{i}\left(G+\frac{H^{T}}{N}\right)\right)\mathbf{u}_{i}^{k}-\alpha_{i}A^{T}\lambda^{k}-\alpha_{i}T_{i}^{T}\right]$
Coordinator: $\displaystyle\hskip
8.5359pt\lambda^{k+1}\leftarrow\mathrm{proj}_{\mathbb{R}_{\geq
0}^{m}}\left[\lambda^{k}+\gamma\left(2\
A\sum_{j=1}^{N}\mathbf{u}_{j}^{k+1}-A\sum_{j=1}^{N}\mathbf{u}_{j}^{k}-b\right)\right]$
$\displaystyle\hskip 4.26773ptk\leftarrow k+1$ Until for all agents
$\|\mathbf{u}_{i}^{k+1}-\mathbf{u}_{i}^{k}\|\leq\epsilon_{u}^{\mathrm{stop}}$
, $\|\lambda^{k+1}-\lambda^{k}\|\leq\epsilon_{\lambda}^{\mathrm{stop}}$

Algorithm 1 Preconditioned Forward Backward

Moreover, we denote $\alpha_{i,min}$ and $\alpha_{i,max}$ as the minimum and
maximum values of $\alpha_{i}$ across all $i\in\\{1,\dots,N\\}$ respectively,
i.e., $\alpha_{i,min}\triangleq\min_{i\in\\{1,\dots,N\\}}\alpha_{i}$ and
$\alpha_{i,max}\triangleq\max_{i\in\\{1,\dots,N\\}}\alpha_{i}$. In this
context, the parameters $l_{F}$ and $\zeta$ are determined according to Remark
5.7 and Remark 5.8, respectively.

The sequence
$\left(\mathrm{col}(\mathbf{u}^{k},\lambda^{k})\right)_{k=0}^{\infty}$defined
by Algorithm 1, with step sizes
$\alpha_{i}\in\left(0,\frac{2\zeta}{l_{F}^{2}}\right)$, for all
$i\in\\{1,\dots,N\\}$, and $\gamma\in\left(0,\gamma_{max}\right)$, with
$\gamma_{max}\triangleq\frac{1}{\|\mathbf{1}_{N}^{T}\otimes
A\|^{2}}\left[\frac{1}{\alpha_{i,max}}-\frac{1}{2\frac{\zeta}{l_{F}^{2}}}\right]$
, globally converges to variational GNE based on [23, Theorem 1]. More details
about the Algorithm are expressed in Appendix.

##  6 Simulation

In this section, we consider a microgrid system consisting of $N=20$ identical
residential users. The parameters of the cost functions in (7) are assigned as
$\alpha^{dch}=8$, $\beta^{dch}=10$, and $k_{c}^{N}=0.015$. Additionally, the
conventional time-of-use pricing tariff is provided in Table 1. The initial
SoC is $x_{0}=0.5$ with minimum and maximum SoC of $\underline{x}=0.1$ and
$\bar{x}=0.9$. The battery has a total energy capacity of $E=20000$ units with
efficiency $\eta=1/E$. The upper bound on the discharging power for each user
$i$ is $\bar{u}_{i}=E/\left(N\Delta t\right)$. In the chance constraints (2),
(3), and (5), we set $\delta_{x}^{t}=0.8$,
$\delta_{x}^{\tau,\mathrm{final}}=0.9$, $r=0.6$, $\epsilon=0.05$,
$\bar{g}=600$ and $\delta_{g}^{t}=0.8$. The time horizon is $\tau=24$ hours
with time step $\Delta t=1$ hour. We consider $d_{i}^{t}$ and $r^{t}$ to have
a bounded support with 25% deviation around their mean value. In our
configuration, we set
$\tilde{\delta}_{x}^{t}=\tilde{\delta}_{x}^{\tau,\mathrm{final}}=\tilde{\delta}_{g}^{t}=0.05$
and $\nu_{r}^{t}=\nu_{d}^{t}=1$. For Algorithm 1, $l_{F}=16.31$,
$\gamma=5.33\times 10^{-4}$, and $\zeta=4.24$. Additionally,
$\alpha=10^{-4}\mathbf{diag}\\{p\\}\otimes(\mathbf{I}_{\footnotesize{\tau}\times\footnotesize{\tau}})$,
where $p=[251,311,317,172,131,70,28,152,129,74,32,121,80,223,\\\
35,71,292,33,251,183]$.

Table 1: conventional time-of-use pricing tariff. Time(t) | $0-4$ | $5-14$ | $15-16$ | $17-21$ | $22-24$
---|---|---|---|---|---
Tariff($K_{ToU}^{t}$) | 29.45 | 30 | 29.5 | 30.5 | 29.5
Figure 2: Profile of $u_{i}^{t}$ in stochastic and deterministic approaches.
Figure 3: Profile of power exchange of all the agents with the grid in
stochastic and deterministic approaches

According to Proposition 5.5, the variational generalized Nash equilibrium is
unique in our stochastic approach. We apply our stochastic method for DSM and
compare its results with two deterministic methods. In these deterministic
methods, we consider two worst-case scenarios for agent demand (lower bound
and upper bound of demands) and use the mean value of renewable energy at each
time as its deterministic value.

The profile of $u_{i}^{t}$ in the stochastic and deterministic approaches can
be viewed in Figure 2. This figure illustrates that in periods when prices and
demand are relatively low, the stochastic approach utilizes less battery
energy compared to the deterministic methods. Conversely, during intervals
with higher demand and electricity prices, the battery is employed to a
greater extent under the stochastic method. Figure 3 illustrates the means of
power exchange profile of the agents with the grid. The results in Figure 3
indicate that the stochastic approach of this paper performs peak shaving more
effectively than the deterministic methods. Simulation results also reveal
that, while the stochastic approach achieves more effective peak shaving than
the deterministic methods, the power exchange profile of the agents with the
grid increases more frequently compared to the deterministic approaches. We
also compare, numerically, the cost function of our stochastic approach with
the deterministic methods for 1,000 random demand values and display the
histogram in Figure 4. As evident from the figure, the expected cost for our
stochastic approach is lower, showing that it incurs lower costs for agents.

Figure 4: Histogram comparing the random costs of stochastic and deterministic
approaches.

##  7 CONCLUSIONS

This paper studied a microgrid system where residential users use a shared
battery charged through renewable sources and the grid. The model considered
uncertain variables including user demand and renewable energy, as well as
dynamic and stochastic coupling constraints. The paper used game theory to
analyze the Nash equilibrium for demand-side management, examining existence
and uniqueness. A semi-decentralized algorithm for Nash seeking was also
proposed. The simulation results demonstrated the advantages of the proposed
setup.

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## APPENDIX

## Some Explanation about Proof of Game

In the section dedicated to game analysis, it is necessary to demonstrate that
function $F$ possesses certain characteristics (monotone, strictly monotone,
$\eta$-strongly monotone and $l_{f}-\mathrm{Lipschitz}$). Subsequent sections
will establish the equivalency of these properties with other concepts in
linear algebra (Positive definitive or Positive semi-definite), and the
corresponding results.

###  7.1 The Monotonicity of $F$

We start with definition of monotonticity.

$\displaystyle\left(F(u)-F(v)\right)^{T}(u-v)\geq
0\Longleftrightarrow\left(\Gamma u-\Gamma v\right)^{T}(u-v)\geq 0$
$\displaystyle\Longleftrightarrow\left(\Gamma\left(u-v\right)\right)^{T}(u-v)\geq
0\Longleftrightarrow$
$\displaystyle\left(u-v\right)^{T}\left(\Gamma^{T}\right)\left(u-v\right)\geq
0\Longleftrightarrow\left(u-v\right)^{T}\left(\Gamma\right)\left(u-v\right)\geq
0$

We have ascertained that for $F$ to be monotone, the necessary and sufficient
condition is that $F$ must be positive semi-definite. Further, we are also
endeavoring to pinpoint conditions leading to $\Gamma$ being positive semi-
definite.

###  7.2 strictly Monotonicity of $F$

We start with definition of monotonticity.

$\displaystyle\left(F(u)-F(v)\right)^{T}(u-v)>0\Longleftrightarrow\left(\Gamma
u-\Gamma v\right)^{T}(u-v)>0$
$\displaystyle\Longleftrightarrow\left(\Gamma\left(u-v\right)\right)^{T}(u-v)>0\Longleftrightarrow$
$\displaystyle\left(u-v\right)^{T}\left(\Gamma^{T}\right)\left(u-v\right)>0\Longleftrightarrow\left(u-v\right)^{T}\left(\Gamma\right)\left(u-v\right)>0$

We have ascertained that for $F$ to be strictly monotone, the necessary and
sufficient condition is that $F$ must be positive definite. Further, we are
also endeavoring to pinpoint conditions leading to $\Gamma$ being positive
definite.

###  7.3 $\zeta$-strongly monotonicity of $F$

We start with definition of $\zeta$-strongly monotone of $F$.

$\displaystyle\left(F(u)-F(v)\right)^{T}(u-v)\geq\zeta\|u-v\|^{2}\Longleftrightarrow$
$\displaystyle\left(\Gamma u-\Gamma
v\right)^{T}(u-v)\geq\zeta(u-v)^{T}(u-v)\Longleftrightarrow$
$\displaystyle\left(\Gamma\left(u-v\right)\right)^{T}(u-v)\geq\zeta(u-v)^{T}(u-v)\Longleftrightarrow$
$\displaystyle\left(u-v\right)^{T}\left(\Gamma^{T}-\zeta
I\right)\left(u-v\right)\geq 0\Longleftrightarrow$
$\displaystyle\left(u-v\right)^{T}\left(\Gamma-\zeta
I\right)\left(u-v\right)\geq 0$

Now we try to find $\zeta$ such that $\Gamma-\zeta I$ becomes positive semi-
definite.

###  7.4 $l_{f}$-Lipschitz Continuity of $F$

We start with definition of $l_{f}$-Lipschitz of $F$.

$\displaystyle\|F(u)-F(v)\|\leq l_{f}\|u-v\|\Longleftrightarrow$
$\displaystyle\|\Gamma u-\Gamma v\|\leq
l_{f}\|u-v\|\Longleftrightarrow\|\Gamma u-\Gamma v\|^{2}\leq
l_{f}^{2}\|u-v\|^{2}\Longleftrightarrow$ $\displaystyle\|\Gamma(u-v)\|^{2}\leq
l_{f}^{2}\|u-v\|^{2}\Longleftrightarrow$
$\displaystyle\left(u-v\right)^{T}\Gamma^{T}\Gamma\left(u-v\right)\leq
l_{f}^{2}\left(u-v\right)^{T}\left(u-v\right)\Longleftrightarrow$
$\displaystyle\left(u-v\right)^{T}\left(l_{f}^{2}I-\Gamma^{2}\right)\left(u-v\right)\geq
0$

Now we try to find $l_{f}$ such that $l_{f}^{2}I-\Gamma^{2}$ becomes positive
semi-definite. First, we find $\Gamma^{2}$:

$\displaystyle\Gamma^{2}=$ $\displaystyle\left\\{I_{N\times
N}\otimes\left[2\left(G+\frac{H^{T}}{N}\right)\right]+\left(\mathbf{1}_{N\times
N}-I_{N\times N}\right)\otimes\frac{H^{T}}{N}\right\\}^{2}$ $\displaystyle=$
$\displaystyle I_{N\times
N}\otimes\left[4\left(\left(\alpha^{dch}+k_{c}^{N}\right)^{2}I_{\tau}\right)\right]$
$\displaystyle+$ $\displaystyle\left(\mathbf{1}_{N\times N}-I_{N\times
N}\right)\otimes\left[2\left(\alpha^{dch}+k_{c}^{N}\right)k_{c}^{N}I_{\tau}\right]$
$\displaystyle+$ $\displaystyle\left(\mathbf{1}_{N\times N}-I_{N\times
N}\right)\otimes\left[2k_{c}^{N}I_{\tau}\left(\left(k_{c}^{N}+\alpha^{dch}\right)I_{\tau}\right)\right]$
$\displaystyle+$ $\displaystyle\left[\left(N-2\right)\mathbf{1}_{N\times
N}+I_{N\times N}\right]\otimes\left(\left(k_{c}^{N}\right)^{2}I_{\tau}\right)$

The diagonal elements of $\Gamma^{2}$ are defined by the expression
$4\left(\alpha^{dch}+k_{c}^{N}\right)^{2}+\left(N-1\right)\left(k_{c}^{N}\right)^{2}$.
Moreover, the summation of the absolute values of off-diagonal elements within
a row is given by
$4k_{c}^{N}(\alpha^{dch}+k_{c}^{N})(N-1)+(N-2)(N-1)\left(k_{c}^{N}\right)^{2}$.
Therefore, according to the Gershgorin Circle Theorem, the following
conditions are necessary.

$\displaystyle
l_{F}^{2}-4\left(\alpha^{dch}+k_{c}^{N}\right)^{2}-(N-1)\left(k_{c}^{N}\right)^{2}>0$

So

$\displaystyle\mathit{l_{F}}>$
$\displaystyle\Biggl{\\{}4\left(\alpha^{dch}+k_{c}^{N}\right)^{2}$
$\displaystyle+\left(N-1\right)\left[\left(N-1\right)\left(k_{c}^{N}\right)^{2}+4k_{c}^{N}\left(\alpha^{dch}+k_{c}^{N}\right)\right]\Biggr{\\}}^{\frac{1}{2}}$
$\displaystyle=2\alpha^{dch}+(N+1)k_{c}^{N}$

## Details about algorithm

In this paper we exploited preconditioned forward-backward splitting method to
compute the generalized Nash equilibria of the game [23]. Since the
convergence characterization of the forward-backward splitting method is well
established, the advantage of the proposed design is that global convergence
follows provided that some mild monotonicity assumptions on the problem data
are satisfied.
In the classical Nash Equilibrium concept, each player $i$ chooses their
strategy $u_{i}$ to minimize their individual cost function
$J_{i}(u_{i},u_{-i})$, where $u_{-i}$ denotes the strategies of all other
players. In the GNE, there’s an added complexity: the players’ strategy
choices are also subject to some shared/coupled constraints. By writing the
KKT condition for finding GNE, this setup leads to a monotone inclusion
problem of the form:

$0\in\mathcal{P}(w)+\mathcal{Q}(w)$

where $w=\begin{bmatrix}\mathbf{u}\\\ \lambda\end{bmatrix}$, that $\lambda$ is
Lagrangian multiplier in the KKT conditions. Moreover,

$\displaystyle\mathcal{P}:\begin{bmatrix}\mathbf{u}\\\
\lambda\end{bmatrix}\longrightarrow\begin{bmatrix}F(\mathbf{u})\\\
b\end{bmatrix},$ $\displaystyle\mathcal{Q}:\begin{bmatrix}\mathbf{u}\\\
\lambda\end{bmatrix}\longrightarrow\begin{bmatrix}\mathbf{N}_{\Omega}(\mathbf{u})\\\
\mathbf{N}_{\mathbb{R}^{m}_{\geq
0}}(\mu)\end{bmatrix}+\begin{bmatrix}0&(\mathbf{1}_{N}^{T}\otimes A)^{T}\\\
-(\mathbf{1}_{N}^{T}\otimes A)&0\end{bmatrix}\begin{bmatrix}\mathbf{u}\\\
\lambda\end{bmatrix}.$

that $\mathbf{N}_{S}$ denotes normal cone operator for set $S$. $\mathcal{P}$
and $\mathcal{Q}$ are monotone operators contains the game’s cost functions
and constraints. In Remark 4 and Remark 5, we establish sufficient conditions
under which $F$ is $\zeta-\mathrm{strongly\ monotone}$ and
$l_{f}-\mathrm{Lipschitz}$ . If $F$ satisfied these properties, we can also
show, similar to Lemma 1 in [23], that $\mathcal{Q}$ is maximally monotone and
$\mathcal{P}$ is $(\zeta/l_{f}^{2})$-cocoercive.
This monotone inclusion problem represents the fixed-point problem associated
with finding the GNE. Let
$\mathrm{zer}(\mathcal{P}+\mathcal{Q})=\\{w\in\mathbb{R}^{Nn+m}\mid
0\in(\mathcal{P}+\mathcal{Q})(w)\\}$. Similarly, define
$\mathrm{fix}(V_{\phi}\circ U_{\phi})=\\{w\in\mathbb{R}^{Nn+m}\mid
w=(V_{\phi}\circ U_{\phi})(w)\\}$. Furthermore, let
$V_{\phi}=(\mathrm{Id}-\phi^{-1}\mathcal{P})$ and
$U_{\phi}=(\mathrm{Id}+\phi^{-1}\mathcal{Q})^{-1}$, where $\mathrm{Id(.)}$
denotes the identity operator and, positive definite matrix $\phi$ is
preconditioning matrix. In Lemma 2 of [23], it has been shown that:

$w\in\mathrm{zer}(\mathcal{P}+\mathcal{Q})\Longleftrightarrow
w\in\mathrm{fix}(V_{\phi}\circ U_{\phi}),$ (20)

The forward backward algorithm is the Banach–Picard iteration [29] applied to
the mappings $V_{\phi}\circ U_{\phi}$ in $w\in\mathrm{fix}(V_{\phi}\circ
U_{\phi})$ , i.e.,

$\displaystyle
w^{k+1}=(\mathrm{Id}+\phi^{-1}\mathcal{Q})^{-1}\circ(\mathrm{Id}-\phi^{-1}\mathcal{P})(w^{k})$
(21)

In numerical analysis, $U_{\phi}$ represents a forward step with size and
direction defined by $\phi$, while $V_{\phi}$ represents a backward step.
Directly from the iteration in (20), we have that

$\displaystyle(\mathrm{Id}-\phi^{-1}\mathcal{P})(w^{k})\in(Id+\phi^{-1}\mathcal{Q})(w^{k+1})$
$\displaystyle\Longleftrightarrow-\mathcal{P}(w^{k})\in\mathcal{Q}(w^{k+1})+\phi^{-1}(w^{k+1}-w^{k})$
(22)

The choice of the preconditioning matrix $\phi$ in (Details about algorithm)
plays a key role in the algorithm design (it is important in convergence of
our GNE seeking algorithm), and is set base on Theorem 1 in [23]. So, We
consider $\phi$ like what considered in [23], as

$\displaystyle\phi=\begin{bmatrix}\alpha^{-1}&-(\mathbf{1}_{N}^{T}\otimes
A)^{T}\\\ -(\mathbf{1}_{N}^{T}\otimes A)&\gamma^{-1}\mathbf{I}\end{bmatrix}$

where
$\alpha=\mathrm{diag}\left(\alpha_{1},\alpha_{2},\dots,\alpha_{N}\right)\otimes\mathbf{I}_{\tau}$
and coefficients $\\{\alpha_{i}\\}_{i=1}^{N}$ and $\gamma$ are chosen such
that $\phi$ is positive definite (based on design guidelines and Theorem 1 in
[23]). By substituting the the above $\phi$ in (Details about algorithm) and
doing some manipulation we can easily reach to

$\displaystyle-\begin{bmatrix}F\left(\mathbf{u}^{k}\right)\\\
b\end{bmatrix}\in\begin{bmatrix}\mathbf{N}_{\Omega}\left(\mathbf{u}^{k+1}\right)\\\
\mathbf{N}_{\mathbb{R}^{m}\geq 0}\left(\lambda^{k+1}\right)\end{bmatrix}+$
$\displaystyle\begin{bmatrix}\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\lambda^{k+1}+\alpha^{-1}\left(\mathbf{u}^{k+1}-\mathbf{u}^{k}\right)-\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\left(\lambda^{k+1}-\lambda{k}\right)\\\
-\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k+1}-\left(\mathbf{1}_{N}^{T}\otimes
A\right)\left(\mathbf{u}^{k+1}-\mathbf{u}^{k}\right)+\gamma^{-1}\left(\lambda^{k+1}-\lambda^{k}\right)\end{bmatrix}$

First we consider the following equations

$\displaystyle-b=$ $\displaystyle-\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k+1}-\left(\mathbf{1}_{N}^{T}\otimes
A\right)\left(\mathbf{u}^{k+1}-\mathbf{u}^{k}\right)$
$\displaystyle+\gamma^{-1}\left(\lambda^{k+1}-\lambda^{k}\right)$ (23)
$\displaystyle-F\left(\mathbf{u}^{k}\right)=$
$\displaystyle\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\lambda^{k+1}+\alpha^{-1}\left(\mathbf{u}^{k+1}-\mathbf{u}^{k}\right)$
$\displaystyle-\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\left(\lambda^{k+1}-\lambda^{k}\right)$ (24)

Based on (Details about algorithm), we can find

$\displaystyle\lambda^{k+1}=\lambda^{k}+\gamma\left(2\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k+1}-\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k}-b\right)$ (25)

By substituting (25) in (Details about algorithm), we can reach to

$\displaystyle\mathbf{u}^{k+1}=\mathbf{u}^{k}-\alpha\left(F\left(\mathbf{u}^{k}\right)+\left(\mathbf{1}_{N}^{T}\otimes
A\right)\lambda^{k}\right)$ (26)

As it has been mentioned in main manuscript,
$F\left(\mathbf{u}^{k}\right)=\Gamma\mathbf{u}^{k}+\Lambda$, so (26) can
express as

$\mathbf{u}^{k+1}\longleftarrow\left(\mathbf{I}-\alpha\Gamma\right)\mathbf{u}^{k}-\alpha\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\lambda^{k}-\alpha\Lambda.$

Now by applying normal cone operator to $\mathbf{u}^{k+1}$ we have

$\mathbf{u}^{k+1}\longleftarrow\mathrm{proj}_{\Omega}\left[\left(\mathbf{I}-\alpha\Gamma\right)\mathbf{u}^{k}-\alpha\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\lambda^{k}-\alpha\Lambda\right].$

Also by applying normal cone operator to $\lambda^{k+1}$ we have

$\displaystyle\lambda^{k+1}$
$\displaystyle\leftarrow\mathrm{proj}_{\mathbb{R}_{\geq
0}^{m}}\left[\lambda^{k}+\gamma\left(2\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k+1}-\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k}-b\right)\right]$ (27)

So, Algorithm 2 can be used as a centralized algorithm for GNE seeking.
The centralized algorithm presented was modified to a semi-decentralized one
that employs a coordinator to become GNE seeking computationally scalable for
large scale networks. To arrive at the semi-decentralized algorithm, we
undertook these steps. Let’s delve deeper into the following update term in
Algorithm 2

$\mathbf{u}^{k+1}\longleftarrow\mathrm{proj}_{\Omega}\left[\left(\mathbf{I}-\alpha\Gamma\right)\mathbf{u}^{k}-\alpha\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\lambda^{k}-\alpha\Lambda\right].$

where
$\mathbf{u}=[\mathbf{u}_{1}^{T},\mathbf{u}_{2}^{T},\dots,\mathbf{u}_{N}^{T}]^{T}$.
By expanding the right hand side, we found that $\mathbf{u}_{i}$ is updated as

$\mathbf{u}_{i}^{k+1}\longleftarrow\mathrm{proj}_{\Omega_{i}}\left[\left(\mathbf{I}_{\tau\times\tau}-2\alpha_{i}(G+\frac{H^{T}}{N})\right)\mathbf{u}_{i}^{k}-\alpha_{i}A^{T}\lambda^{k}-\alpha_{i}T_{i}^{T}\right].$

Moreover, we can easily find that $\lambda$ is updated as

$\lambda^{k+1}\longleftarrow\mathrm{proj}_{\mathbb{R}_{\geq
0}^{m}}\left[\lambda^{k}+\gamma\left(2\
A\sum_{j=1}^{N}\mathbf{u}_{j}^{k+1}-A\sum_{j=1}^{N}\mathbf{u}_{j}^{k}-b\right)\right]$

Fig.1 in the paper shows the information flow between the agents and
coordinator. As we can see, the coordinator does not need to know the local
objectives of the agents and only receives the data of $\mathbf{u}_{j}^{k}$
and $\mathbf{u}_{j}^{k+1}$ from the agents and then updates the value of
$\lambda$ accordingly. Such a semi-decentralized scheme has been also used in
some other studies like [30, 31]. So we propose a semi-decentralized
algorithm, as presented in Algorithm 1.

Initialization: $k\leftarrow 1$,
$\mathbf{u}_{i}^{1}\leftarrow\mathbf{u}^{\mathrm{init}}$,
$\lambda^{1}\leftarrow\lambda^{\mathrm{init}}$

Repeat $\displaystyle\mathbf{u}^{k+1}$
$\displaystyle\leftarrow\mathrm{proj}_{\Omega}\left[\left(\mathbf{I}-\alpha\Gamma\right)\mathbf{u}^{k}-\alpha\left(\mathbf{1}_{N}^{T}\otimes
A\right)^{T}\lambda^{k}-\alpha\Lambda\right]$ $\displaystyle\lambda^{k+1}$
$\displaystyle\leftarrow\mathrm{proj}_{\mathbb{R}_{\geq
0}^{m}}\left[\lambda^{k}+\gamma\left(2\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k+1}-\left(\mathbf{1}_{N}^{T}\otimes
A\right)\mathbf{u}^{k}-b\right)\right]$ $\displaystyle k$
$\displaystyle\leftarrow k+1$ Until
$\|\mathbf{u}^{k+1}-\mathbf{u}^{k}\|\leq\epsilon_{u}^{\mathrm{stop}}$,
$\|\lambda^{k+1}-\lambda^{k}\|\leq\epsilon_{\lambda}^{\mathrm{stop}}$

Algorithm 2 Preconditioned Forward Backward (Centralized Algorithm)