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Dependable Demand Response Management in the
Smart Grid: A Stackelberg Game Approach
Sabita Maharjan, Student Member, IEEE, Quanyan Zhu, Student Member, IEEE,
Yan Zhang, Senior Member, IEEE, Stein Gjessing, Senior Member, IEEE, and Tamer Bas¸ar, Fellow, IEEE
Abstract—Demand Response Management (DRM) is a key
component in the smart grid to effectively reduce power gen-
eration costs and user bills. However, it has been an open
issue to address the DRM problem in a network of multiple
utility companies and consumers where every entity is concerned
about maximizing its own benefit. In this paper, we propose a
Stackelberg game between utility companies and end-users to
maximize the revenue of each utility company and the payoff
of each user. We derive analytical results for the Stackelberg
equilibrium of the game and prove that a unique solution exists.
We develop a distributed algorithm which converges to the
equilibrium with only local information available for both utility
companies and end-users. Though DRM helps to facilitate the
reliability of power supply, the smart grid can be succeptible
to privacy and security issues because of communication links
between the utility companies and the consumers. We study the
impact of an attacker who can manipulate the price information
from the utility companies. We also propose a scheme based on
the concept of shared reserve power to improve the grid reliability
and ensure its dependability.
Index Terms—Demand response management, dependability,
reliability, reserve power, security, smart grid, Stackelberg game.
I. INTRODUCTION
The demand of electricity consumers has been growing due
to increased use of machines and the new types of appliances
such as plug-in hybrid electric vehicles. The concern towards
the impact on environment and on the reliability of power
supply, has also been rising. However, traditional power grids
are not able to meet these demands and requirements because
of their inflexible designs and lack of prompt communications
between the supply and the demand sides. Recent blackouts [1]
have indicated the inefficiency and serious reliability issues of
the traditional grid. Therefore, it is essential to transform the
traditional power grid into a more responsive, efficient and re-
liable system. Smart grid [2] is a future power grid system that
incorporates a smart metering infrastructure capable of sensing
and measuring power consumption from consumers with the
integration of advanced information and communication tech-
nologies (ICT). Thus the power generation, distribution and
S. Maharjan, Y. Zhang and S. Gjessing are with Simula Research Labo-
ratory, Norway; and Department of Informatics, University of Oslo, Gaus-
tadalleen 23, Oslo, Norway. Email: {sabita, yanzhang, steing}@simula.no.
Q. Zhu and T. Bas¸ar are with Coordinated Science Laboratory and De-
partment of Electrical and Computer Engineering, University of Illinois at
Urbana Champaign, 1308 Main, Urbana, IL, 61801 USA. Email: {zhu31,
basar1}@illinois.edu. These authors’ research was supported in part by a grant
from the DOE and in part by NSA through the Information Trust Institute,
University of Illinois.
consumption is efficient, more economical and more reliable
in the smart grid network.
Demand Response Management (DRM), a key feature of the
smart grid, is defined as changes in electric usage by end-users
in response to changes in the price of electricity over time or
across different energy sources. The importance of DRM can
go far beyond reducing the electricity bills of consumers or
the cost of generating power. It helps to balance the demand
and supply in the power market through real-time pricing. It
can also provide short-term reliability benefits as it can offer
load relief to resolve system and/or local capacity constraints.
The recent studies on DRM can be categorized mainly into
two areas: utility company (UC) oriented and end-user ori-
ented. There has been considerable amount of work in power
systems on supply-demand balance and market clearance [3],
[4]. Such studies on power systems have focused on the
economic aspects at the planning and generation level and
have not considered user-utility as a significant component.
On the other hand, the literature on user-utility has introduced
schemes to maximize user utilities, without considering the
power generation costs or the revenue of the UCs. This has
motivated us to consider the issue of benefit maximization for
users alongside with the revenue maximization for the UCs.
Our work aims to bridge the gap between the existing two
research directions. In addition, with increasing concerns to-
wards environment, incorporating renewable energy resources
becomes important in the smart grid. This has motivated us to
include in our work renewable energy sources in addition to
traditional fossil fuel based sources.
We study the interactions among multiple UCs and multiple
consumers, who aim to maximize their own payoffs. The UCs
maximize their revenues by setting appropriate unit prices. The
consumers choose power to buy from UCs based on the unit
prices. The payoff of each consumer depends on the prices
set by all the sources. In turn, the price set by each UC
also depends on the prices of other UCs. These complicated
interactions motivate us to use a game theoretical framework
in our analysis. We develop a Stackelberg game between the
UCs and the users where the UCs play a non-cooperative game
and the consumers find their optimal response to the UCs’
strategies. The interactions between the UCs and the users are
enabled by the bidirectional communications between them.
An advanced metering infrastructure (AMI) is a commu-
nication infrastructure that enables meters and utilities to
exchange information such as power consumption, price up-
date, or outage awareness. Smart meters play the key role
of gateway between the customers0premises and the utility
network. Their functionality make them an interesting target
for attackers [5]. Therefore it is important to assess possible
consequences of attacks and develop mechanisms to maintain
the reliability and resilience of the grid in the face of unan-
ticipated events. We assess the impact of an attacker that can
manipulate the price of the UCs, and propose a scheme to
ensure the reliability of power supply in the presence of an
attacker, thus making the smart grid a dependable system.
We have three major contributions in this work.
1) We establish an analytical model for the multiple-UCs
multiple-consumers Stackelberg game and characterize
its unique Stackelberg equilibrium (SE).
2) We propose a distributed algorithm which converges to
the SE with only local information of the users and the
UCs.
3) We propose a scheme based on a common reserve to
improve the dependability of the smart grid. We also
discuss reliability of the grid when one of the sources
gets disconnected from the grid due to occurance of
some physical incidents.
The rest of the paper is organized as follows. Related work
is described in Section II. We introduce the system model and
the communication model in Section III. In Section IV, we
formulate the problem as a Stackelberg game and prove the
existence and uniqueness of the SE. We propose a distributed
algorithm for the game which converges to the SE, in Section
V. In Section VI, we study the impact of an attacker as a
possible threat to grid stability and propose a scheme based
on maintaining a shared reserve power. We provide numerical
results and discussion in Section VII. Section VIII concludes
the paper.
II. RE LATE D WOR K
There are several studies on DRM in the smart grid [6]-[10].
In [6], the authors have formulated the energy consumption
scheduling problem as a non-cooperative game among the
consumers for increasing strictly convex cost functions. In
[7], the authors have considered a distributed system where
price is modeled by its dependence on the overall system load.
Based on the price information, the users adapt their demands
to maximize their own utility. In [8], a robust optimization
problem has been formulated to maximize the utility of a
consumer, taking into account price uncertainties at each hour.
In [9], the authors have exploited the awareness of the end-
users and proposed a method to aggregate and manage end-
users’ preferences to maximize energy efficiency and user sat-
isfaction. In [10], a dynamic pricing scheme has been proposed
to incentivize costumers to achieve an aggregate load profile
suitable for utilities, and the demand response problem has
been investigated for different levels of information sharing
among the consumers in the smart grid. In [11], the unit com-
mitment scheduling problem in smart grid communications
has been studied using a partially observable Markov decision
process framework for stochastic power demand loads and
renewable energy resources. However, the analyses in [6]-[11],
are limited in the sense that there is either only one source or
a number of sources/utilities treated as one entity. Differently
in our study, we include multiple UCs and consumers whose
goal is to maximize their own payoffs, using the concept of
Stackelberg game.
We note that there is rich literature using Stackelberg games
in the context of congestion control, revenue maximization and
cooperative transmission [14]-[15]. Our approach is similar to
those in congestion control to model the behavior of end-
users, but our study involves multiple UCs, and we adopt
the non-cooperative game framework among UCs using the
Stackelberg solution concept.
DRM enhances the reliability of the grid [16] when the data
communications is perfect. However, the data communications
in the smart grid may suffer attacks such as data manipulation
or false data injection [17] from malicious nodes. In such
cases, the UC or the users may incur economic loss or physical
impact e.g., grid instability. In [18], the authors have studied
the utility-privacy tradeoffs of smart meter data and shed
light on the impact of leakage of the data on the utilities
of both the users and the suppliers. In [19], the authors
have proposed a secure routing protocol incorporating delay
due to queue building. They have investigated the tradeoffs
between efficiency, reliability and resilience in centralized and
decentralized approaches for secure routing. [20] proposes a
six-layer hierarchical security architecture for the smart grid,
identifying the security challenges present at each layer and
addressing security issues at three different layers. In [21]
the authors have developed a formal model for the C12.22
standard protocol to guarantee that no attack can violate the
security policy without being detected based on the concept of
specification-based intrusion detection. It is observed that there
is no work that addresses the impact of attacks from an out-
sider on DRM through the information exchange between the
users and the UCs. Because of the communications between
the consumers and the UCs, there are inherent vulnerabilities
that attackers can exploit to harm the utilities of either side or
to even cause physical damage on the system.
III. SYS TE M MOD EL
We consider Nend-users, which we also call customers, and
Kelectricity UCs. Fig. 1 depicts an overview of the scenario.
The utility side consists of the renewable and non-renewable
energy sources. The fossil-fuel based energy generators have
certain amount of power available all the time. The power
generated by the fossil fuel generators creates pollution to
the environment. On the other hand, the renewable energy
sources can be seen as pollution free but they do not always
have power available. When renewable energy sources are
incorporated into the system, we add uncertainties to the utility
side. There are many studies where discrete time Markov chain
models have been used to model the availability of energy
from the renewable sources (such as wind and solar energy)
[11], [23], [24]. We incorporate the renewable energy sources
too, and consider a stationary distribution for the states of the
renewable energy generators. The end-user side consists of
several consumers, which may be residential users, commer-
cial users or industries. These different types of users have
different needs for electricity. We differentiate them in terms of
2
Fig. 1. Smart grid system model with multiple energy sources and end-users
available budget which is an upper bound on their affordability
to buy power. We employ a utility function for each user
that increases with the amount of electricity the user can
consume. At the same time, we incorporate a cost constraint
for each user. The UCs and the consumers have bidirectional
communications for exchanging price and demand information
as shown in Fig. 2. The UCs can also communicate with
each other. The users receive the price information from the
utility companies and transmit their demand to them. The
data communication is carried out through the communication
channel using wireless technologies, e.g., WiFi, WiMAX, or
LTE.
In practice, the electricity generation, distribution and con-
sumption can be decomposed into three layers as described
in [22]: generators, aggregators or utility companies, and the
end-users. The acquisition of power by the utility companies
from the generators is a separate process. In this paper, we
focus on the interactions between the UCs and the end-
users. In practice, the unit price of a UC is determined
through the market by the system operator. In this paper,
the UCs play a non-cooperative game at the market level.
Different from the traditional perfect competitive market, the
UCs participate in an imperfect competition. In a perfectly
competitive market, no market participant has the ability to
influence the market price through its individual actions, i.e.,
the market price is a parameter over which firms have no
control. Consequently, each firm should increase its production
up to the point where its marginal cost equals the market price.
This is valid when the number of market participants is large
and none of the participants controls a large proportion of
the production. However, in this paper, we consider a finite
number of market participants (UCs) and each individual UC
has non-infintesimal influence in the market. This leads to
imperfect competition, where each firm determines its unit
price based on its available power.
Fig. 2. Illustration of communications between utility companies and
consumers
IV. UTI LI TY-U SE R INTERACTION: STACK EL BE RG GAME
When there are multiple UCs with different energy prices,
the cost to each user varies according to the prices set by each
UC. In addition, the price set by a UC also depends on the
prices of other UCs. Game theory provides a natural paradigm
to model the behavior of the end-users and of the UCs in this
scenario. The UCs set the price per unit power and announce
it to the users. The users respond to the price by demanding
an optimal amount of power from the UCs. Since the UCs act
first and then the users make their decision based on the prices,
the two events are sequential. Thus, we model the interactions
between the UCs and the end-users as a Stackelberg game
[25]. In our proposed game, the UCs are the leaders and the
consumers are the followers. It is a multi-leaders and multi-
followers game. The demand of the users depends on the unit
price set by the UCs as well as their own cost constraints.
In turn, the UCs optimize their unit prices according to the
response of the consumers.
A. User Side Analysis
Let xn,kbe the demand of user nfrom UC k. We define the
utility of user n,Uuser,nas
Uuser,n=αn∑
k∈K
lnβn+xn,k,∀k∈K,(1)
where αnand βnare constants. The ln function has been
widely used in economics for modeling the preference order-
ing of users and for decision making [12], [13].
The motivation behind choosing the utility function for user
nas in (1) is that it is closely related to the utility function
αn∑ln(xn,k)that leads to proportionally fair demand response
[12] [14]. If we use the utility function αn∑ln(xn,k), then a
user gets a payoff of −∞with respect to (w.r.t.) UC kwhen
xn,k=0. With βn, when xn,k=0, its benefit with w.r.t. that UC
becomes finite. A typical value of βnis 1.
3
Let ykbe the unit price set by UC kand let Cn>0 denote
the budget of user n. For a given set of prices from the UCs
{y1,y2,...,yK}, user n∈Ncalculates its optimal demand
response by solving the user optimization problem (OPuser)
max
xn:={xn,k,∀k∈K}Uuser,n(2)
s.t. ∑
k∈K
ykxn,k≤Cn,(3)
xn,k≥0; ∀k∈K.(4)
OPuser is a convex optimization problem. Hence, the stationary
solution is unique and optimal.
Let us start the analysis with Nusers and 2 UCs. We
will later generalize the results to KUCs. The optimization
problem for user nin this case is
max
xn:={xn,1,xn,2}αn
2
∑
k=1
ln(βn+xn,k)(5)
s.t. y1xn,1+y2xn,2≤Cn,(6)
xn,1,xn,2≥0.(7)
Using Lagrange’s multipliers λn,1,λn,2and λn,3for constraints
(6) and (7), we convert the constrained optimization problem
(5) - (7) to the form
Luser,n=αn
2
∑
k=1
ln β1+xn,k
−λn,1 2
∑
k=1
ykxn,k−C1!+λn,2xn,1+λn,3xn,2(8)
and the complementarity slackness conditions
λn,1 2
∑
k=1
ykxn,k−Cn!=0,(9)
λn,2xn,1=0,(10)
λn,3xn,2=0,(11)
λn,1>0,λn,2,λn,3,xn,1,xn,2≥0.(12)
The first-order optimality condition for the maximization prob-
lem is ∇Luser =0, where Luser ={Luser,n,∀n∈N}. Since the
only coupling between the users is through yk,∇Luser =0 leads
to
∂Luser,n
∂xn,k
=0,∀n∈N,k∈K,i.e., ,
(αn
βn+xn,1−λn,1y1+λn,2=0,
αn
βn+xn,2−λn,1y2+λn,3=0.(13)
The optimal demands of users can take one of the following
forms.
1) Case 1 : xn,1,xn,2>0:In this case, λn,2=λn,3=0.
Substituting λn,2and λn,3into (13) yields
xn,k=αn
λn,1yk
−βn,∀n∈N,k=1,2.(14)
Using (14) in (9) yields
1
λn,1
=
Cn+βn
2
∑
k=1
yk
2αn
.(15)
Now substituting (15) into (14) yields
xn,k=
Cn+βn
2
∑
k=1
yk
2yk
−βn,k=1,2.(16)
2) Case 2 : xn,1>0,xn,2=0:This is the case when
Cn+βn
2
∑
k=1
yk
2y2=βn. Equation (10) implies λn,2=0 and
xn,1=αn
λn,1y1
−βn.(17)
Substituting xn,1into (9), we get,
λn,1αn
λn,1
−βny1−Cn=0.
Since λn,1>0, αn
λn,1−βny1−Cn=0 which gives λn,1=αn
Cn+βny1.
Using λn,1in (17), we obtain
xn,1=Cn+βny1
y1
−βn=Cn
y1
.(18)
Equation (18) can be written as xn,1=Cn+βn(y1+y2)
2y1+
Cn−βn(y1+y2)
2y1. From
Cn+βn
2
∑
k=1
yk
2y2=βnwe get, βn(y2−
y1) = Cn. Using this, we can write xn,1=Cn+βn(y1+y2)
2y1+
βn(y2−y1)−βn(y1+y2)
2y1. After simplifying we get,
xn,1=Cn+βn(y1+y2)
2y1
−βn.(19)
3) Case 3 : xn,1=0,xn,2>0:Similar analysis can be
performed as in case 2 to obtain
xn,2=Cn
y2
=Cn+βn(y1+y2)
2y2
−βn.(20)
4) Case 4 : xn,1=0,xn,2=0:In this case, λn,1=0 and
λn,2,λn,3can be any non-negative real value. This is an extreme
case, which does not happen unless Cn=0 or yk=∞∀k∈K.
Note that in cases 1-3 discussed above, both power constraint
and the cost constraint are satisfied as equalities.
Thus using (16), (19) and (20), the demands for the general
case of Nusers and KUCs that covers cases 1−3 for a given
set of {yk}, can be formulated as
xn,k=Cn+βn∑k∈Kyk
Kyk
−βn,(21)
4
where xn,k≥0,∀k∈K,n∈N. Since xn,k≥0∀n∈N,∀k∈
K, (21) implies that
Cn+βn ∑
g∈K,g6=k
yg!≥βn(K−1)yk,∀n∈N,∀k∈K.
(22)
Conversely, user nwill demand xn,k≥0 from UC kif
yk≤"Cn+βn∑g∈K,g6=kyg
βn(K−1)#.(23)
We will derive a closed form for the necessary condition for
xn,k≥0,∀k∈K,∀n∈Nto be satisfied, in Section IV-B2.
B. Utility Side Analysis
Let P
k>0 denote the available power of UC k. Each UC
aims to sell all the available power. If it had been a single UC
case, it could have a set a very high unit price to maximize
its revenue. In this case however, there are two factors that
limit the unit price of the UCs. The first one is the budget of
the users and the second one is the competition among the
UCs. The UCs play a non-cooperative price selection game
with each other to decide the optimal unit price. We assume
that P
kis given for all k∈K. For given P
k, since the cost
of power generation is given, we define the revenue of UC k,
Ugen,kas
Ugen,k(yk,y−k) = yk∑
n∈N
xn,k,(24)
where y−kis the price of UCs other than k. Then, the
optimization problem for a UC (OPgen) is formulated as
max
y:={yk,∀k∈K}Ugen,k(yk,y−k)(25)
s.t. ∑
n∈N
xn,k≤P
k,(26)
yk>0,k∈K.(27)
Since the revenue of a UC is an increasing function in terms
of the amount of power for a fixed yk, (26) can be taken as
an equality constraint. Since we do not assume the availability
of storage with the UCs, when the available power is given,
each UC prefers to sell all its power. In order to solve OPgen,
we start by relaxing the positivity constraint (27) but will
show that the solution of (25)-(26) will lead to positive vector
{yk,∀k∈K}. Let us define Lgen,kas
Lgen,k=yk∑
n∈N
xn,k−µk ∑
n∈N
xn,k−P
k!(28)
The first order optimality condition for the UCs leads to
∂Lgen,k
∂yk=0,∀k∈K. Using (21) in ∂Lgen,k
∂yk=0 for UC k, we
obtain
(K−1)By2
k−µk"B ∑
g∈K,g6=k
yg!+C#=0,(29)
where B=∑n∈Nβnand C=∑n∈NCn. Eqn. (29) gives
Kequations. Further, ∂Lgen,k
∂ µk=0 gives Kequations, which
are actually the original constraints: (26). Solving these 2K
equations we can obtain y∗:={y∗
1,y∗
2,...,y∗
K}and µ∗:=
{µ∗
1,µ∗
2,...,µ∗
K}. Using y∗, we can compute x∗:={x∗
n,k}. Now
using (21) in the equality form of (26), we get
yk=
C+B ∑
g∈K,g6=k
yg!
KP
k+B(K−1)(30)
Substituting ykinto (29), we arrive at
µk= (K−1)B"C+B∑g∈K,g6=kyg
K(P
k+B)#= (K−1)Byk(31)
Here, B,C>0. when K=1, µk=0 and y1=C
P
k+B. This means
that there is no game when there is only one UC. Our interest
is in the case where K≥2. Then, µk>0 if yk>0. Equation
(30) can be represented in the matrix form as
Ay =F,(32)
where
A:=
P
1+D−E.... −E
−E P
2+D.... −E
.. .. .... ..
−E−E.... P
K+D
,(33)
y:={y1,y2, ....., yK}0,D:=B(K−1)
K,E:=B
Kand F:=C
K. Thus,
provided that Ais invertible, the solution of (32) is
y=A−1F.(34)
In order to obtain the closed-form of y, let us consider the
following cases:
1) Homogeneous case: When P
1=P
2=..... =P
K=P: If
all UCs have the same amount of power available, then, (32)
can be solved to obtain
yk=y:=F
−(K−1)E+D+P=C
KP >0∀k∈K.(35)
As Pincreases, ydecreases and vice versa. Now using (35)
in (22), the condition for the demand of user nfrom all UCs
to be positive is
Cn>βn(K−1)y−βn(K−1)y,(36)
i.e., Cn>0, which is always true. This indicates that when all
UCs are homogeneous in terms of the available power, they
set the same unit price and all users will buy at least some
power from them.
2) Heterogeneous case: When different UCs have different
available power: In this case, it is difficult to judge the
existence of the solution without knowing the nature of A−1
(if it exists). Interestingly, matrix Apossesses some special
properties. Let us state the following definitions and properties.
Definition 1. A real matrix A:={ai,ji,j=1,2, ....., K} ∈
RK×Kis said to be strictly diagonally dominant if it satisfies
the following condition:
|ai,i| − ∑
j6=i
|ai,j|>0i=1,2,...,K.(37)
Property 1. A strictly diagonally dominant matrix is non-
singular.
5
Property 2. If a matrix is strictly diagonally dominant by
rows and has positive diagonal entries, then, its determinant
will be positive [26].
For matrix A, since, P
k+D−(K−1)E=P
k+(K−1)B
K−
(K−1)B
K=P
k>0, Ais a strictly diagonally dominant matrix.
From Property 1, Ais invertible and (32) yields a unique
solution to y.
Theorem 1. The unique solution obtained from (33) is posi-
tive.
Proof: The solution of (33) is given by
yk=KK−1C
|A|∑
g∈K,g6=k
(B+P
g)∀k∈K.(38)
where |A|is the determinant of A. The numerator of (38)
is always positive. Property 2 implies that the denominator
is positive, and hence the solution from (38) always yields
yk>0,∀k∈K.
Theorem 2. For the case of heterogeneous generators the
necessary condition for the demands of all users from all UCs
to be non-negative, is:
Cn≥KK−1βnC
|A|" ∑
g∈K,g6=k
P
g!−(K−1)P
k#∀k,∀n.(39)
Proof: The demand of user nfrom UC kis non-
negative (xn,k≥0) if Cn+βn∑k∈Kyk≥Kβnyk,∀n∈
N,∀k∈K, i.e., if Cn≥βn(Kyk−∑k∈Kyk).
Using (38), the condition can be written as
Cn≥βnKK−1C
|A| K∑
g∈K,g6=k
(B+P
g)−∑
k∈K
∑
g∈K,g6=k
(B+P
g)!
After simplification, the required condition takes the form
given by (39).
Theorem 3. The values of price obtained from (34) maximize
the revenue and are best responses of the UCs to other UCs’
strategies.
Proof: Suppose ykis the solution obtained from (34) for
UC kand it increases its price from ykto y0
k=yk+δykwhile
the prices of other UCs remains the same. Let us assume that
yk,y0
ksatisfy (23). The demand of users from this UC will
change from xn,kto x0
n,kgiven by
x0
n,k=
Cn+βn∑g∈K,g6=kyg+y0
k
Ky0
k
−βn
.
The difference in the total demands from the users from UC
kwill be
xn,k−x0
n,k="Cn+βn∑g∈K,g6=kyg
K#(y0
k−yk)
yky0
k
.
Clearly, (xn,k−x0
n,k>0). The users will demand less power
than the power available with the UC. The difference in the
revenue of the UC because of the increase in the price will be
U0
gen,k−Ugen,k=y0
k∑
n∈N
x0
n,k−yk∑
n∈N
xn,k=−(K−1)
KBδyk.
(40)
From (40), it is clear that Ugen,k(y0
k,y−k)<Ugen,k(yk,y−k)if
δyk>0. If δyk<0, y0
k<yk, but P
kis given. Consequently, the
revenue will decrease. Therefore, the prices calculated using
(34) are the best responses to each other and are the prices
that maximize the revenue of each UC.
In addition to positivity, there are tighter limits on yksuch
that yk∈[yk,min,yk,max ]. The lower limit yk,min is due to the
associated generation costs. The UCs will not reduce their
price below yk,min. The upper bound yk,max can be fixed
according to government standards. The power available with
the UCs implicitly take into account these price limits. Thus,
the solution obtained by solving (34) should be within the
specified limits. In case if the optimal prices obtained are
outside the range, we propose the following algorithm to
calculate the unit prices.
Algorithm 1:
1) If {yk}k∈Kbe the solution obtained from (38) but yi<
yi,min or yi>yi,max for UC i, then its price will be set
as yi=yi,min or yi=yi,max.
2) The remaining UCs will use yias a known value in (30)
to obtain a square matrix of size K−Ki, where Kiis
the number of UCs for which the prices obtained are
modified. Then, the matrix of the reduced dimension is
solved to get the prices of the remaining K−KiUCs.
3) The process continues until all the prices come within
the specified range.
C. Stackelberg Equilibirum
The UCs play the non-cooperative game with each other
to set the unit price, which is at Nash equilibrium (NE)
point, and announce the prices to the users. The equilibrium
strategy for the followers in a Stackelberg game is any strategy
that constitutes an optimal response to the one adopted (and
announced) by the leader(s) [25].
Let Γgen,kand Γuser,nbe the strategy sets for UC kand
user nrespectively. Then, the strategy sets of all UCs and
all users are Γgen =Γgen,1×Γgen,2×..... ×Γgen,Kand Γuser =
Γuser,1×Γuser,2×..... ×Γuser,N, respectively. Then, y∗
k∈Γgen,k
is a Stackelberg equilibrium strategy for UC kif
Ugen,k(y∗,x(y∗)) ≥Ugen,k(yk,y∗
−k,x(yk;y∗
−k)),∀k∈K,(41)
where y∗={y∗
k},x:={x1;x2;. . . xN}is the strategy of all
users 1,2,...,Nsuch that x∈Γuser,x(y∗)is the optimal
response of all users. The optimal response of user nfor given
(y1,y2, ....., yK)∈Γgen,1×Γgen,2×..... ×Γgen,Kis
xn(y) = {ζuser,n∈Γuser,n;Uuser,n(y,ζuser,n)≥
Uuser,n(y,xn)},∀n∈N.(42)
where ζuser,n:=x∗
n∈xn(y∗). The strategy x∗
nis a corresponding
optimal strategy for user n, which is computed by using (21).
The set (y∗,x∗)is a Stackelberg equilibrium of the game
between the UCs and the users.
6
D. Existence and Uniqueness of Stackelberg Equilibrium
OPuser has a unique maximum for a given y. Therefore, the
Stackelberg game possesses a unique SE if the price setting
game among the UCs admits a unique NE.
Theorem 4. A unique Nash equilibrium exists in the price
selection game among the UCs, and thereby a unique Stack-
elberg equilibrium.
Proof: A Nash equilibrium exists for the UCs in the price
selection game if
1) yis a non-empty, convex, and compact subset of some
Euclidean space RK.
2) Ugen,k(y)is continuous in yand concave in yk,∀k∈K.
In the price selection game for the UCs, the strategy
space Γgen =Γgen,1×Γgen,2×..... ×Γgen,Kwhere yk∈Γgen,k:=
[yk,min,yk,max ],∀k∈K. Thus the strategy set is a nonempty,
convex and compact subset of the Euclidean space RK. From
(24), we see that Ugen,kis continuous in yk. Next, the second
order derivative of Ugen,kw.r.t. ykis
∂2Ugen,k
∂y2
k
=0,∀k∈K.(43)
Hence, Uk(y)is concave in yk. Therefore, NE exists in this
game.
As proven in Section IV-B, there exists only one positive
solution for the price selection game given by (38). Therefore
the NE of the UCs’ game is unique and hence the Stackelberg
game also admits a unique equilibrium.
V. DISTRIBUTED ALGORITHM
In the previous section, the consumers calculate their opti-
mal demands based on the prices provided by the UCs but the
UCs play the best response to other UCs’ strategies. In order to
calculate the unit prices, UCs need to know the power available
with other UCs too. We design a distributed algorithm that
leads to the SE of the game without each UC knowing the
parameters of the other UCs.
Each UC starts with an arbitrary price yk,1>0 and all
of them send their price information to the consumers. This
communication is enabled by the smart grid communication
infrastructure between the UCs and the consumers. Each user
decides how much to buy from each UC {xn,k,1,∀k∈K}
using (21). The UCs get this demand matrix from all users.
Then, one of the UCs will calculate the difference between
the available power and the total power demanded from it by
all users. Then, it will update its unit price using
yk,t+1=yk,t+∑n∈Nxn,k,t−P
k
σk
,(44)
where tis the iteration number and σkis the speed adjustment
parameter of UC k, which is a sufficiently large number.
Whenever a UC updates its price, it sends this information
to the users. The users again, update their demands vectors
and inform the UCs. Then, other UCs will update their prices
sequentially at alternative turns between users and UCs. This
process continues until the price values converge.
TABLE I
DISTRIBUTED ALGORITHM FOR PRICE AND DEMAND UPDATING
1: For t=1, arbitrarily choose yk,1∀k∈K
2: Repeat for t=2,3,.....
3: User n=1,2,....N
4: Do
5: Solve (2) - (4) for given ytusing (21).
6: Transmit xn,k,tto each k.
7: end
8: UC k∈K, which has not updated the price for iteration t+1
9: Calculate yk,t+1using (44).
10: If yk,t+1−yk,i=0
11: Send a no-change signal to all users.
12: Go to 8.
13: else
14: Send the new value of price to all users.
15: Go to 3.
16: end
17: If yk,t+1== yk,t∀k∈K,
18: stop.
19: else
20: Go to 2.
21: end.
We name this algorithm as Algorithm 2, which is shown
in Table I. In the table, n=1 indicates user number 1.
Theorem 5. Provided that ∀k∈K,∀n∈N,t=1,2,3,. . . ,
σk>(KP
k−βn)yk,t−βn∑g∈K,g6=kyg,t−Cn
Ky2
k,t
,(45)
Algorithm 2converges to the optimal solutions for both the
users and the UCs as long as the individual strategies are
updated sequentially.
Proof: The users’ response (21) is the optimal response
to given {yk}. The demand array of each user will converge
to a fixed set once the price set converges to a fixed point.
Consequently, it is sufficient to show the convergence of the
price vector to prove the convergence of Algorithm 2.
The algorithm will diverge only if yk,tacquires a nega-
tive value in any iteration. If ∑n∈Nxn,k,t−P
k<0, for any
k∈K,∀n∈N,t=1,2,3, . . . , then the sufficient condition
for yk,t+1not to acquire a negative value is |∑n∈Nxn,k,t−P
k
σk|<
yk,t. Since the condition should be satisfied only when
∑n∈Nxn,k,t−P
k<0, we can rewrite the condition as σk>
P
k−∑n∈Nxn,k,t
yk,t∀k∈K,∀n∈N,t=1,2,3, . . . . Using (21) and
upon simplification, the sufficient condition takes the form
(45).
Equation (44) implies that the price ykof UC kincreases
if ∑n∈Nxn,k,t−P
kis positive and the price decreases if
∑n∈Nxn,k,t−P
kis negative. Equation (44) shows that when
∑n∈Nxn,k,t−P
k=0, the price will remain unchanged. This is
the fixed point to which Algorithm 2 converges. As discussed
in section IV-B, the prices corresponding to stock clearance are
the prices that maximize the UC revenue. Therefore, if {yk,T}
are the prices at iteration Tsuch that ∑n∈Nxn,k,T−P
k=0,
then, Ugen,k(yk,T,y−k,T)≥Ugen,k(y0
k,T,y−k,T)where
Ugen,k(yk,T,y−k,T)and Ugen,k(y0
k,T,y−k,T)are the utilities
of UC kat price yk,Tand y0
k,Trespectively. Thus the fixed
point to which Algorithm 2 converges is the NE of the game
among the UCs, i.e., SE of the game between the UCs and
7
the consumers. Thereafter, the UCs will not deviate from this
point.
VI. DE PE NDA BI LI TY O F DRM
One important component of dependability of DRM is the
vulnerability due to attacks from malicious agents. While the
bidirectional communications between the UCs and the users
facilitates demand response management in a more timely and
effective manner, it also creates room for different kinds of
threats to the system from attackers.
Next, we proceed to analyze this kind of attack. Let us
assume that there is an attacker who is an outsider. The goal
of the attacker is to harm the UCs to the largest extent possible.
We constrain the attacker such that it can attack only one of
the UCs at a time by injecting the data or by manipulating the
price. This assumption is reasonable because the attacker has
physical constraints to access the UCs.
A. Attack and Its Impact
An attacker can cause two kinds of harm to the UCs and
the users: economic impact and physical impact. We consider
the case when all UCs have the same power to supply (P). We
assume that there is a range in which the price can vary. The
users have the knowledge of this range. Users get this range
during the service agreement. If the attacker manipulates the
prices in such a way that it is out of the range, the manipulation
will become obvious. So, the attacker will launch the attack
such that the resulting prices are within the specified range.
When the attacker does not manipulate any of the prices, the
price set by each UC is
yk=y=C
KP ,∀k∈K.(46)
Using (46) in (21), the demand of user nfrom UC kis
xn,k=Cn
CP.
The total demand of user nfrom all UCs is
∑
k∈K
xn,k=Cn
CKP.
Let the price of UC k,yk, be changed by the attacker through
data injection or indirect manipulation to y0
k=yk+δyand all
other prices are in-tact, where δyis a real number such that
|δy| ∈ [0,δy,max]. Positive value of δymeans that the price is
increased and negative value means the price is decreased,
δy=0 corresponds to the case when the attacker does not
act. When the price is changed by the attacker, the demand of
users will change from xn,kto x0
n,k. The attacker can change
ykbut will still keep it such that x0
n,kis non-negative. It can
be calculated by replacing ykin (21) by yk+δy, which gives
x0
n,k=CnP−δyβn(K−1)P
(C+δyKP).(47)
Equation (47) shows that the demand of each user from UC
kwill decrease if δyis positive and it will increase if δyis
negative. The total demand from UC kfrom all users will be
∑
n∈N
x0
n,k=CP −δyB(K−1)P
(C+δyKP).(48)
Now, let us see if the change in the price of UC kaffects other
UCs. For UC g6=k, the demand from user nwill be
x0
n,g=Cn
CP+δyβn
P
C.(49)
Using (49), the total demand from UC g6=ktakes the form
∑
n∈N
x0
n,g=P1+δy
B
C.(50)
Equation (48) indicates that the total demand from UC k
will decrease (increase) if ykis increased (decreased) by the
attacker compared to the case when there is no attack. If the
total demand decreases from UC k, a part of the available
power will be wasted and it will loose in terms of its revenue.
On the other hand, if the demand increases than the available
power, serious problems such as black-out may occur to the
users side. If the grid attempts to produce the shortage power
immediately, the grid might suffer physical damage. There is
an interesting observation here. Equations (49), (50) indicate
that if the price of UC kis changed, the other UCs will
also be affected. If ykincreases, the demand from the other
UCs increases. Consequently, because of the excessive demand
from the UCs other than k, problems such as grid instability
may arise at the generation side while the users will suffer
black-out. On the other hand, when ykdecreases, the demand
from the other UCs decreases. So, they will suffer economic
loss while the generators that supply the demand of the users
from UC kmay suffer physical damage. In either case (δy>0
or δy<0), the UCs and the grid will suffer both monetary
loss and physical damage.
B. Proposed Scheme with Individual Reserve Power
In a scenario where the UCs are aware that an attacker might
attempt to create monetary or physical damage, although both
kinds of loss are serious, the physical damage can make the
grid unstable and it can affect the whole infrastructure on the
UC side. Hence, we mainly focus on avoiding this problem,
for maintaining the grid dependable. We propose that each UC
should have certain reserve power in addition to the available
power to sell. Let us denote the reserve power for UC kby
P
k,res. The reserve power for each UC should be the difference
of the available power and the total possible demand from all
the users in presence of the attack. Using (48), for UC k,
whose price has been changed, the reserve power is
P
k,res =−δyP(B(K−1) + KP)
(C+δyKP).(51)
If δy>0, then P
k,res will be negative, since the demand from
UC kwill be less than the amount of power available. This
means that no reserve power is needed. Hence, we have
P+
k,res =0.(52)
For other UCs g6=k, the reserve power can be calculated as
P+
g,res =BPδy
C.(53)
Equations (52)-(53) indicate that the amount of reserve power
for each UC depends on the value of δy. Theoretically, each
8
UC can keep a large reserve so that there is always extra
power available even if the users demand more than the power
desired to be sold. In practice, this is not possible since the
UCs have to pay the cost associated to buy or produce the
reserve power. Therefore, all UCs prefer to keep as minimum
reserve power to save the associated costs, but at the same
time, it is extremely important for them to avoid the damage
to their infrastructure.
In contrast to communication problems where usually the
average performance guarantee measures are taken, we need
to consider the worst case for the power-trade scenario. The
attacker aims to maximize the harm to the UCs and/or to the
grid but without making the demands become out-of-the-range
values, e.g., negative demand from the users. The attacker
therefore, tries to manipulate ykin such a way that the demands
from all UCs is non-negative even from the user with the
lowest budget. Let us examine the following cases.
1) δy>0:If δyis positive, P+
k,res =0. The maximum harm
that the attacker can create in this case would be by letting
the user with the lowest Cnbuy zero power from UC k, which
corresponds to x0
n,k=0.
δymax =Cmin
βmin(K−1).(54)
where Cmin =min(Cn)and βmin =min(βn). If ymax is the
maximum unit price allowed, the range of δygiven by δymax
should still yield y0
k≤ymax. Thus, y0
kis given by
y0
k=yk+δyif δy≤ymax −yk,
ymax otherwise.(55)
Therefore, δymax can be calculated as
δymax =minCmin
βmin(K−1),(ymax −yk).(56)
Hence, 0 <δy≤δymax and P+
g,res can be calculated by substi-
tuting δy=δymax into (53).
2) δy<0:When δy<0, the worst impact that the attacker
can cause is by setting the prices so low that the user with the
lowest budget buys zero power from all UCs other than k. In
this case, solving x0
n,g=0∀g6=kyields
δymax =Cmin
βmin
,
where ncorresponds to the lowest budget user. However, y0
k
should not be less than yk,min. If y0
k=yk−Cmin
βmin >yk,min, then,
substituting δyin (48) and (50), and substracting the available
power of the UC P, the reserve power for UCs (using β1=
β2=..... =βN=β) are
(P−
k,res =NCn,min(K−1)β+Cn,minKP2
βC−Cn,minKP ,
P−
g,res =0.
If yk−Cmin
βmin ≤0 or if yk−Cmin
βmin >0 but y0
k<ymin, then, the
minimum y0
kthat can be chosen by the attacker is y0
k=ymin.
Using this constraint on (47), then summing over n∈Nand
subtracting the available power P, we obtain
P−
k,res =C+B(K−1)yd−KPymin
Kymin
.
where yd=C
KP −ymin . The UCs should avoid problems even in
the worst case, P
k,res =P−
k,res and P
g,res =P+
g,res. However, the
UCs may not know whether the attacker will choose δy>0
or δy<0. So, for each UC, the reserve power is
P
res =max{P+
g,res,P−
k,res}.(57)
C. Proposed Scheme with Common Reserve Power
The reserve power given by (57) is the worst case reserve
power if the UC is attacked and if its price is lowered to
the least possible value. Since, each UC needs to have this
reserve separately, while the actual number of UCs that will
be attacked is limited to 1, there is definitely a huge wastage of
the reserve power from all other UCs who are not attacked. In
particular, the UCs have to buy the reserve power from some
sources and it is obvious that each UC aims to minimize the
reserve power while ensuring that there is no power-outage.
The total reserve power needed for the individual reserve
scheme is
P
tot,res =KP
res.(58)
From the analysis in section VI-B, we can see that the total
demand from all the users from all the UCs will be
Xtot ≤KP +P
res.(59)
From (58) - (59), the total power that will not be used is
P
tot,unused ≥(K−1)P
res.(60)
The incentive for the UCs to use a common reserve is that
sharing common reserve power saves a considerable amount
of cost for each UC. Furthermore, it is also beneficial from
the overall system perspective, as there are increasing concerns
towards minimizing the waste of power.
Fig. 3. Common reserve scheme: different utility/generation companies share
a common energy reserve
Following the reasoning in Section VI-B, we can show that
the total reserve power needed in this case will be
P
tot,res =P
res.(61)
Employing this power sharing, we can see that the UCs can
maintain a reliable supply of power to the users in the presence
of the attacker with only a fraction of the reserve power
needed for the individual reserve scheme. Thus, the supply side
improves its dependability with this reserve power scheme.
9
D. Discussion
The impact on DRM when one (few) of the generator(s)
is (are) unavailable is also important. In our proposed model,
the UCs communicate with each other about their available
power before the prices are decided. If one of the generators
is not available, the prices will be calculated based on the
power available with the UCs from the remaining generators.
Our model is therefore resilient to the unavailability of one or
few generators.
There is another possible type of reliability issue with
the generators, even in the absence of an attacker. Even
when all the generators have produced the power, one of the
generators can be disconnected from the grid after the UCs
have transmitted the price information to the users. This kind
of problem arises due to physical phenomena e.g., if one of
the transmission lines suddenly gets disconnected. In this case,
the users would already be using their appliances and thus
this kind of failure is critical. The reliability can be improved
in this case also by maintaining certain reserve power. This
kind of reliability is referred to as (K−1) reliability. On
similar grounds as in Section VI, it can be inferred that a
common reserve power equal to the available power of one UC
(the maximum power in case of heterogeneous UCs) should
maintain the system stable in the face of such incidents.
VII. NUMERICAL RES ULTS
In this section, we examine how the users choose their
optimal power based on the unit prices of the UCs and how
the UCs optimize their unit prices based on their available
power and the users’ cost constraints. We also show the
convergence of the distributed algorithm. The last part of this
section shows the reserve power needed in the presence of
an attacker. We consider 3 UCs and 5 users with parame-
ters αn=1,βn=1∀n∈N. The cost limits of users are
C1=5,C2=10,C3=15,C4=20,C5=25 and the available
power of the UCs are P
1=10,P
2=15,P
3=20, respectively,
unless mentioned otherwise.
A. Stackelberg Game
Figs. 4 - 7 show how the change in the prices set by the
UCs affects the users’ utility, and in turn how the power
available and the affordability of the users affect the UCs’
equilibrium prices when the budget of user 1 varies from 2 to
42. Fig. 4 shows the total demand of each user at equilibrium.
The demand of user 1 is increasing because its budget is
increasing. The budget of other users do not change, but they
ultimately demand less power because of the increase in the
budget of user 1. Fig. 5 shows the utility of the users at
equilibrium. Fig. 6 depicts the equilibrium prices of the UCs.
The prices increase linearly as the purchasing capacity of the
users increases. UC 1 charges the highest unit price, because
it has the lowest amount of power available and the reverse
is true for UC 3. Fig. 7 shows the revenue of each UC. UC
1 has the lowest utility despite its high unit price, because
its available power is the lowest. UC 3 receives the highest
revenue although its unit price is the lowest.
Next we show the equilibrium of the Stackelberg game for
more UCs and a large number of users for a large variation in
0 10 20 30 40
0
5
10
15
20
C1
Demands
User 1 User 2 User 3 User 4, User 5
Fig. 4. User demands at equilibrium
0 10 20 30 40
1
2
3
4
5
6
C1
User Utilities
User 1 User 2 User 3 User 4, User 5
Fig. 5. User utilities at equilibrium
0 10 20 30 40
1
1.5
2
2.5
3
3.5
C1
Unit Price
y1
y2
y3
Fig. 6. Unit prices at equilibrium
0 10 20 30 40
20
25
30
35
40
C1
Revenues of the UCs
UC 1
UC 2
UC 3
Fig. 7. UC revenues at equilibrium
10
0 100 200 300 400
0
50
100
150
200
C1
Demands
Users 1
Users 2−25
Users 26−50
Users 51−75
Users 76−100
(a) User demands
0 100 200 300 400
0
5
10
15
20
C1
User Utilities
Users 1
Users 2−25
Users 26−50
Users 51−75
Users 76−100
(b) User utilities
0 100 200 300 400
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
C1
Unit Price
y1, y2
y3, y4
y5
(c) Unit price
Fig. 8. Equilibrium for a large number of users and UCs for a large variation in user budget
the budget of one of the users. Figs. 8(a)-8(c) show the user
demands, user utilities and unit prices at equilibrium for 5 UCs
and 100 electricity users. The budget of user 1 varies from 2
to 400, the budget of users 2 to 25, users 26 to 50, users 51 to
75 and users 76 to 100 are 10,15,20 and 25, respectively. The
available power of the UCs used for plots are P
1=P
2=150
units, P
3=P
4=200 units and P
5=250 units. The behaviors
of the users and the UCs are similar to Figs. 4-6.
B. Distributed Algorithm
Figs. 9 - 12 show the performance of the distributed
algorithm for σk=40,∀k∈K. The UCs reach the equilibrium
price without communicating with each other. Consequently,
the users reach their optimal demands based on the prices
from the UCs. The results show that the equilibrium price
and demand can be reached very quickly. Comparing Figs.
9 - 12 with Figs. 4 - 7, we can verify that the distributed
algorithm converges to the optimal values. Next we evaluate
0 10 20 30 40 50
10
20
30
40
50
60
70
Iterations
Demands
User 1 User 2 User 3 User 4, User 5
Fig. 9. Reaching equilibrium using distributed algorithm: User demands
the performance of the distributed algorithm for different
values of yk,1and σk. Fig. 13(a) shows the performance
when σk=10,k=1,2,3, . . . . We see from the figure that the
algorithm converges much faster (since σkis smaller) and the
unit prices converge to the same values as in Fig. 11. Figs.
13(b) and 13(c) depict that the algorithm converges to the
same values when we start from a different value for yk,1,
for σk=50 and 10, respectively ,∀k∈K. We see that the
convergence speed depends on the value of σk,∀k∈Kbut
the algorithm converges irrespective of different initial points.
0 10 20 30 40 50
5
6
7
8
9
10
Iterations
User Utilities
User 1 User 2 User 3 User 4, User 5
Fig. 10. Reaching equilibrium using distributed algorithm: User utilities
0 10 20 30 40 50
1
1.5
2
2.5
Iterations
Unit Price
y1
y2
y3
Fig. 11. Reaching equilibrium using distributed algorithm: Unit prices
0 10 20 30 40 50
22
23
24
25
26
27
28
Iterations
Revenues of the UCs
UC 1
UC 2
UC 3
Fig. 12. Reaching equilibrium using distributed algorithm: UC revenues
11
0 10 20 30 40 50
1
1.5
2
2.5
Iterations
Unit Price
y1
y2
y3
(a) yk,1=1;σk=10.
0 20 40 60 80
1
1.5
2
2.5
3
3.5
4
4.5
5
Iterations
Unit Price
y1
y2
y3
(b) yk,1=5;σk=50.
0 10 20 30 40 50
1
1.5
2
2.5
3
3.5
4
4.5
5
Iterations
Unit Price
y1
y2
y3
(c) yk,1=5;σk=10.
Fig. 13. The unit price of UCs for different initial starting points yk,1and different values of σk,k=1,2,3.
C. Dependability of DRM
We show the impact of an attacker in terms of the reserve
power needed under the two different schemes discussed in
Sections VI-B and VI-C respectively in Fig.14. We consider
both cases: when the price is increased and when the price is
decreased. We use P=10 and [ymin ,ymax] = [ 0.5C
KP ,2C
KP ]for the
plot. For the sake of illustration, we suppose that the attacker
1 1.5 2 2.5 3 3.5 4 4.5 5
0
5
10
15
20
25
30
35
40
45
y(3)
Power
Total Reserve Power − Individual Reserve
Total Reserve Power − Shared Reserve
Extra power needed
Unused power − Individual Reserve
Unused power − Common Reserve
Fig. 14. Reserve power, extra power needed and unused power
manipulates the price of the third UC. In practice, the attacker
can choose any of the UCs. This figure shows a number of
interesting facts. It verifies that the UCs save a lot in terms of
the reserve power needed if they share a common reserve. The
solid line with square markers shows the extra power needed
to cover the total demand of users. The reserve power from a
common reserve is sufficient to cover the demands of all the
users when the price is increased and also when it is decreased.
In fact, this power is extra most of the times and there is a
considerable amount of unused power in both cases (individual
reserve and common reserve). The unused power is much more
when each UC keeps its own reserve as shown by the dotted
line with ’+’ markers. The only case where the reserve power
is completely used is when the price is decreased to ymin. From
this figure, it might appear that the amount of power wasted is
still significant even with the common reserve. That is because
our analysis is based on the worst case reserve. This figure also
indicates how expensive an impact an attacker can create even
if it has access to only one UC’s price. The impact will be
similar in case of the heterogeneous UCs with the difference
in the scale of the power to be reserved.
VIII. CONCLUSION AND FUTURE WOR K
In this paper, we have proposed a Stackelberg game between
the electricity UCs for optimal price setting and the end-
users for optimal power consumption. We have derived the
SE of the game in closed form and have proved its existence
and uniqueness. We have designed a distributed algorithm for
convergence to the SE with only local information available
to the UCs. We have introduced two types of reliability issues
associated with the smart grid: reliability due to physical
disturbance and dependability in the face of an attacker. We
have investigated the impact of an attacker who manipulates
the price information from the UCs. Furthermore, we have
proposed a scheme based on the concept of shared reserve
power to ensure the reliability and dependability of the grid.
We have shown the validity of our concepts through analytical
and numerical results.
This work opens the door to some interesting extensions.
The DRM analysis incorporating the modeling of instability of
the renewable energy sources is a potential direction. Here we
have focused on a large time-scale one-period DRM scheme.
A higher resolution multi-period scheme with inter-temporal
constraints is another possible extension to this work.
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