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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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