1

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 Başar, Fellow, IEEE

Abstract—Demand Response Management (DRM) is a key
component in the smart grid to effectively reduce power generation 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 tomaximize the revenue of each utility company and the payoffof 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 theequilibrium with only local information available for both utility
companies and end-users. Though DRM helps to facilitate thereliability 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 informationfrom 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. I NTRODUCTION
The demand of electricity consumers has been growing due
to increased use of machines and the new types of appliancessuch 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 becauseof 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 thetraditional power grid into a more responsive, efficient and reliable system. Smart grid [2] is a future power grid system thatincorporates a smart metering infrastructure capable of sensing
and measuring power consumption from consumers with the
integration of advanced information and communication technologies (ICT). Thus the power generation, distribution and
S. Maharjan, Y. Zhang and S. Gjessing are with Simula Research Laboratory, Norway; and Department of Informatics, University of Oslo, Gaustadalleen 23, Oslo, Norway. Email: {sabita, yanzhang, steing}@simula.no.
Q. Zhu and T. Başar are with Coordinated Science Laboratory and Department 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 grantfrom the DOE and in part by NSA through the Information Trust Institute,
University of Illinois.

consumption is efficient, more economical and more reliablein the smart grid network.
Demand Response Management (DRM), a key feature of thesmart grid, is defined as changes in electric usage by end-users
in response to changes in the price of electricity over time oracross different energy sources. The importance of DRM cango far beyond reducing the electricity bills of consumers or
the cost of generating power. It helps to balance the demandand supply in the power market through real-time pricing. It
can also provide short-term reliability benefits as it can offerload 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 oriented. There has been considerable amount of work in powersystems on supply-demand balance and market clearance [3],[4]. Such studies on power systems have focused on theeconomic aspects at the planning and generation level andhave 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 thepower generation costs or the revenue of the UCs. This has
motivated us to consider the issue of benefit maximization forusers 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 towards environment, incorporating renewable energy resources
becomes important in the smart grid. This has motivated us toinclude 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 complicatedinteractions motivate us to use a game theoretical frameworkin our analysis. We develop a Stackelberg game between the
UCs and the users where the UCs play a non-cooperative gameand the consumers find their optimal response to the UCs’
strategies. The interactions between the UCs and the users areenabled by the bidirectional communications between them.
An advanced metering infrastructure (AMI) is a communication infrastructure that enables meters and utilities toexchange information such as power consumption, price update, or outage awareness. Smart meters play the key roleof gateway between the customers0 premises 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 maintainthe reliability and resilience of the grid in the face of unanticipated events. We assess the impact of an attacker that canmanipulate the price of the UCs, and propose a scheme toensure 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 theUCs.
3) We propose a scheme based on a common reserve toimprove 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 ofsome physical incidents.
The rest of the paper is organized as follows. Related work
is described in Section II. We introduce the system model andthe communication model in Section III. In Section IV, we
formulate the problem as a Stackelberg game and prove theexistence 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 apossible threat to grid stability and propose a scheme basedon maintaining a shared reserve power. We provide numericalresults and discussion in Section VII. Section VIII concludes
the paper.

in our study, we include multiple UCs and consumers whose
goal is to maximize their own payoffs, using the concept ofStackelberg game.
We note that there is rich literature using Stackelberg gamesin the context of congestion control, revenue maximization andcooperative transmission [14]-[15]. Our approach is similar to
those in congestion control to model the behavior of endusers, 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 communicationsin the smart grid may suffer attacks such as data manipulation
or false data injection [17] from malicious nodes. In suchcases, the UC or the users may incur economic loss or physicalimpact e.g., grid instability. In [18], the authors have studied
the utility-privacy tradeoffs of smart meter data and shedlight 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 anddecentralized approaches for secure routing. [20] proposes a
six-layer hierarchical security architecture for the smart grid,
identifying the security challenges present at each layer andaddressing 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 thesecurity 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 outsider on DRM through the information exchange between theusers 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.

II. R ELATED W ORK
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 theconsumers 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 aconsumer, taking into account price uncertainties at each hour.
In [9], the authors have exploited the awareness of the endusers and proposed a method to aggregate and manage endusers’ preferences to maximize energy efficiency and user satisfaction. 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 commitment scheduling problem in smart grid communications
has been studied using a partially observable Markov decision
process framework for stochastic power demand loads andrenewable energy resources. However, the analyses in [6]-[11],are limited in the sense that there is either only one source ora number of sources/utilities treated as one entity. Differently

III. S YSTEM M ODEL
We consider N end-users, which we also call customers, andK electricity UCs. Fig. 1 depicts an overview of the scenario.
The utility side consists of the renewable and non-renewableenergy sources. The fossil-fuel based energy generators havecertain 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 areincorporated 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 therenewable energy generators. The end-user side consists ofseveral consumers, which may be residential users, commercial users or industries. These different types of users havedifferent needs for electricity. We differentiate them in terms of2

Generators	  Distribu"on	
  Distribu"on	
  Residen"al	
  Commercial	
  U"lity	  Companies	
  Consumers	  Industrial	
  Fig. 2.
Illustration of communications between utility companies andconsumers

Fig. 1. Smart grid system model with multiple energy sources and end-users

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 theconsumers are the followers. It is a multi-leaders and multifollowers 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.

available budget which is an upper bound on their affordability
to buy power. We employ a utility function for each userthat increases with the amount of electricity the user canconsume. 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 witheach other. The users receive the price information from theutility companies and transmit their demand to them. The
data communication is carried out through the communication
channel using wireless technologies, e.g., WiFi, WiMAX, orLTE.
In practice, the electricity generation, distribution and consumption can be decomposed into three layers as describedin [22]: generators, aggregators or utility companies, and theend-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 endusers. In practice, the unit price of a UC is determinedthrough 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, theUCs participate in an imperfect competition. In a perfectlycompetitive 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 nocontrol. 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 toimperfect competition, where each firm determines its unit
price based on its available power.

A. User Side Analysis
Let xn,k be the demand of user n from UC k. We define theutility of user n, Uuser,n asUuser,n = αn ∑ ln βn + xn,k , ∀k ∈ K(1)
k∈K

where αn and βn are constants. The ln function has been
widely used in economics for modeling the preference ordering of users and for decision making [12], [13].
The motivation behind choosing the utility function for usern as 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 auser gets a payoff of −∞ with respect to (w.r.t.) UC k whenxn,k = 0. With βn , when xn,k = 0, its benefit with w.r.t. that UC
becomes finite. A typical value of βn is 1.
Let yk be the unit price set by UC k and let Cn > 0 denotethe budget of user n. For a given set of prices from the UCs{y1 , y2 , . . . , yK }, user n ∈ N calculates its optimal demand
response by solving the user optimization problem (OPuser )max

Uuser,n(2)s.t. ∑ yk xn,k ≤ Cn ,(3)

xn :={xn,k ,∀k∈K }IV. U TILITY-U SER I NTERACTION : S TACKELBERG G AME
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 thek∈K

xn,k ≥ 0; ∀k ∈ K .
3(4)2) Case 2 : xn,1 > 0, xn,2 = 0: This is the case when

OPuser is a convex optimization problem. Hence, the stationary
solution is unique and optimal.
Let us start the analysis with N users and 2 UCs. We
will later generalize the results to K UCs. The optimization
problem for user n in this case is2

Cn +βn ∑ yk
k=1
2y2

= βn . Equation (10) implies λn,2 = 0 and
xn,1 =

2max

xn :={xn,1 ,xn,2 }αn ∑ ln(βn + xn,k )

(5)
(6)

xn,1 , xn,2 ≥ 0.

(7)

αnSince λn,1 > 0, λαn −βn y1 −Cn = 0 which gives λn,1 = Cn +β
.
n y1
n,1
Using λn,1 in (17), we obtain

Using Lagrange’s multipliers λn,1 , λn,2 and λn,3 for constraints(6) and (7), we convert the constrained optimization problem(5) - (7) to the form

xn,1 =

2

Luser,n = αn ∑ ln β1 + xn,k
k=1
!



∑ yk xn,k −C1 + λn,2 xn,1 + λn,3 xn,2Cn −βn (y1 +y2 )
.
2y1

k=1

From

2y2

(9)

k=1

λn,2 xn,1 = 0,(10)λn,3 xn,2 = 0,(11)λn,1 > 0, λn,2 , λn,3 , xn,1 , xn,2 ≥ 0.

(12)

xn,1 =

xn,2 =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,2 and λn,3 into (13) yields

Cn + βn.

yg"≥ βn (K − 1)yk , ∀n ∈ N , ∀k ∈ K .

Cn + βn ∑g∈K ,g6=k yg
yk ≤βn (K − 1)

Cn + βn ∑ yk
k=1∑

Conversely, user n will demand xn,k ≥ 0 from UC k if

2

2yk

(21)

(22)

(15)

Now substituting (15) into (14) yields

xn,k =

Cn + βn ∑k∈K yk
− βn
Kyk

g∈K ,g6=k

Cn + βn ∑ yk
2αn

(20)

where xn,k ≥ 0, ∀k ∈ K , n ∈ N . Since xn,k ≥ 0 ∀n ∈ N , ∀k ∈K , (21) implies that
!

(14)

2
k=1

(19)

Cn Cn + βn (y1 + y2 )=− βn .
y22y2

xn,k =

Using (14) in (9) yields
1
=
λn,1

Cn + βn (y1 + y2 )− βn .
2y1

4) Case 4 : xn,1 = 0, xn,2 = 0: In this case, λn,1 = 0 and
λn,2 , λn,3 can 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 generalcase of N users and K UCs that covers cases 1 − 3 for a given
set of {yk }, can be formulated as,n,2αn− βn , ∀n ∈ N , k = 1, 2.
λn,1 yk

we get, βn (y2 −3) Case 3 : xn,1 = 0, xn,2 > 0: Similar analysis can be
performed as in case 2 to obtain

The first-order optimality condition for the maximization problem is ∇Luser = 0, where Luser = {Luser,n , ∀n ∈ N }. Since theonly coupling between the users is through yk , ∇Luser = 0 leadsto∂ Luser,n= 0, ∀n ∈ N , k ∈ K , i.e., ,∂ xn,k( αnβn +xn,1 − λn,1 y1 + λn,2 = 0,(13)αnβn +x − λn,1 y2 + λn,3 = 0.

xn,k == βn

βn (y2 −y1 )−βn (y1 +y2 ). After simplifying we get,
2y1

2

= 0,

Cn +βn ∑ yky1 ) = Cn . Using this, we can write xn,1 = Cn +βn2y(y11 +y2 ) +and the complementarity slackness conditions!

∑ yk xn,k −Cn

(18)

2

(8)

k=1

λn,1

Cn + βn y1
Cn− βn = .
y1
y1

Equation (18) can be written as xn,1 = Cn +βn2y(y11 +y2 ) +2− λn,1(17)

Substituting xn,1 into (9), we get,


αn
λn,1
− βn y1 −Cn = 0.
λn,1

k=1

s.t. y1 xn,1 + y2 xn,2 ≤ Cn ,αn
− βn .
λn,1 y1− βn k = 1, 2.

(16)

#
.

(23)

We will derive a closed form for the necessary condition for
xn,k ≥ 0, ∀k ∈ K , ∀n ∈ N to be satisfied, in Section IV-B2.
4

is in the case where K ≥ 2. Then, µk > 0 if yk > 0. Equation(30) can be represented in the matrix form as

B. Utility Side Analysis
Let Pk > 0 denote the available power of UC k. Each UC
aims to sell all the available power. If it had been a single UCcase, 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 ofthe users and the second one is the competition among theUCs. The UCs play a non-cooperative price selection gamewith each other to decide the optimal unit price. We assume
that Pk is given for all k ∈ K . For given Pk , since the costof power generation is given, we define the revenue of UC k,
Ugen,k as
Ugen,k (yk , y−k ) = yk ∑ xn,k ,(24)

Ay = F,
where

Ugen,k (yk , y−k )s.t. ∑ xn,k ≤ Pk ,y = A−1 F.

(25)
(26)

∑ xn,k − Pkn∈N

Cn > βn (K − 1)y − βn (K − 1)y,(28)

The first order optimality condition for the UCs leads to∂ Lgen,k∂ Lgen,k∂ yk = 0, ∀k ∈ K . Using (21) in ∂ yk = 0 for UC k, we
obtain"!
#∑yg +C = 0,

Definition 1. A real matrix A := {ai, j i, j = 1, 2, ....., K} ∈R K×K is said to be strictly diagonally dominant if it satisfies
the following condition:(29)|ai,i | − ∑ |ai, j | > 0 i = 1, 2, . . . , K.

g∈K ,g6=k

yk =

∑

(37)

j6=i

where B = ∑n∈N βn and C = ∑n∈N Cn . Eqn. (29) gives∂LK equations. Further, ∂gen,kµk = 0 gives K equations, which
are actually the original constraints: (26). Solving these 2K
equations we can obtain y∗ := {y∗1 , y∗2 , . . . , y∗K } and µ ∗ :=∗ }. Now{µ1∗ , µ2∗ , . . . , µK∗ }. Using y∗ , we can compute x∗ := {xn,kusing (21) in the equality form of (26), we get
!
C+B

(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 somepower from them.
2) Heterogeneous case: When different UCs have different
available power: In this case, it is difficult to judge theexistence of the solution without knowing the nature of A−1(if it exists). Interestingly, matrix A possesses some special
properties. Let us state the following definitions and properties.

n∈N(K − 1)By2k − µk B

(34)

As P increases, y decreases and vice versa. Now using (35)in (22), the condition for the demand of user n from all UCs
to be positive is(27)

Since the revenue of a UC is an increasing function in termsof the amount of power for a fixed yk , (26) can be taken as
an equality constraint. Since we do not assume the availabilityof 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 willshow that the solution of (25)-(26) will lead to positive vector{yk , ∀k ∈ K }. Let us define Lgen,k as!
Lgen,k = yk ∑ xn,k − µk(33)

In order to obtain the closed-form of y, let us consider thefollowing cases:
1) Homogeneous case: When P1 = P2 = ..... = PK = P: If
all UCs have the same amount of power available, then, (32)can be solved to obtainFC
yk = y :==> 0 ∀k ∈ K . (35)−(K − 1)E + D + P KPn∈N

yk > 0, k ∈ K .


....
−E
....
−E 
,
....
.. 
.... PK + D

0where y−k is the price of UCs other than k. Then, theoptimization problem for a UC (OPgen ) is formulated asmax−E
P2 + D..
−E, E := KB and F := CK . Thus,y := {y1 , y2 , ....., yK } , D := B(K−1)
K
provided that A is invertible, the solution of (32) isn∈Ny:={yk ,∀k∈K }
P1 + D
 −E
A := 
 ..
−E

(32)

Property 1. A strictly diagonally dominant matrix is nonsingular.
Property 2. If a matrix is strictly diagonally dominant byrows and has positive diagonal entries, then, its determinant
will be positive [26].
For matrix A, since, Pk + D − (K − 1)E = Pk + (K−1)B−
K(K−1)B
=
P
>
0,
A
is
a
strictly
diagonallydominant
matrix.
k
K
From Property 1, A is invertible and (32) yields a uniquesolution to y.

ygg∈K ,g6=k

KPk + B(K − 1)

Substituting yk into (29), we arrive at"
#
C + B ∑g∈K ,g6=k ygµk = (K − 1)B= (K − 1)Byk
K(Pk + B)(30)

Theorem 1. The unique solution obtained from (33) is positive.
(31)

Proof: The solution of (33) is given byHere, B,C > 0. when K = 1, µk = 0 and y1 = P C+B . This meanskthat there is no game when there is only one UC. Our interest

yk =

5

K K−1C(B + Pg ) ∀k ∈ K .
|A| g∈K∑,g6=k(38)

where |A| is the determinant of A. The numerator of (38)is always positive. Property 2 implies that the denominatoris positive, and hence the solution from (38) always yields
yk > 0, ∀k ∈ K .

specified limits. In case if the optimal prices obtained areoutside the range, we propose the following algorithm tocalculate the unit prices.
Algorithm 1:
1) If {yk }k∈K be 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 yi as a known value in (30)
to obtain a square matrix of size K − Ki , where Ki isthe number of UCs for which the prices obtained aremodified. Then, the matrix of the reduced dimension issolved to get the prices of the remaining K − Ki UCs.
3) The process continues until all the prices come within
the specified range.

Theorem 2. For the case of heterogeneous generators thenecessary condition for the demands of all users from all UCs
to be non-negative, is:"!
#K K−1 βnC
Cn ≥∑ Pg − (K − 1)Pk ∀k, ∀n. (39)|A|g∈K ,g6=k
Proof: The demand of user n from UC k is nonnegative (xn,k ≥ 0) if Cn + βn ∑k∈K yk ≥ Kβn yk , ∀n ∈N , ∀k ∈ K ,i.e.,
if
Cn ≥ βn (Kyk − ∑k∈K yk ).
Using (38), the condition can be written as!
K−1

Cn ≥ βn K |A| C

K∑(B + Pg ) − ∑g∈K ,g6=k∑

C. Stackelberg Equilibirum(B + Pg )The UCs play the non-cooperative game with each otherto 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 (andannounced) by the leader(s) [25].
Let Γgen,k and Γuser,n be the strategy sets for UC k anduser n respectively. Then, the strategy sets of all UCs andall users are Γgen = Γgen,1 × Γgen,2 × ..... × Γgen,K and Γuser =Γuser,1 × Γuser,2 × ..... × Γuser,N , respectively. Then, y∗k ∈ Γgen,kis a Stackelberg equilibrium strategy for UC k isk∈K g∈K ,g6=kAfter 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 yk is the solution obtained from (34) for0
UC k and it increases its price from yk to yk = yk + δyk while
the prices of other UCs remains the same. Let us assume that0yk , yk satisfy (23). The demand of users from this UC will0change from xn,k to xn,k given by




0
C
+
β
y
+
y
∑
n
n
g
g∈K ,g6=kk0− βn  .
xn,k = 0
Kyk

Ugen,k (y∗ , x(y∗ )) ≥Ugen,k (yk , y∗−k , x(yk ; y∗−k )), ∀k ∈ K ,where y∗ = {y∗k }, x := {x1 ; x2 ; . . . xN } is the strategy of allusers 1, 2, . . . , N such that x ∈ Γuser , x(y∗ ) is the optimal
response of all users. The optimal response of user n for given(y1 , y2 , ....., yK ) ∈ Γgen,1 × Γgen,2 × ..... × Γgen,K is

The difference in the total demands from the users from UC
k will be"
# 0
Cn + βn ∑g∈K ,g6=k yg (yk − yk )0.
xn,k − xn,k =
0
K
yk yk

xn (y) = {ζuser,n ∈ Γuser,n ;Uuser,n (y, ζuser,n ) ≥Uuser,n (y, xn )}, ∀n ∈ N .

Clearly, (xn,k − xn,k > 0). The users will demand less power
than the power available with the UC. The difference in therevenue of the UC because of the increase in the price will be
0

0

0

n∈N

n∈N

= − (K−1)
K Bδyk .

(42)

where ζuser,n := x∗n ∈ xn (y∗ ). The strategy x∗n is a corresponding
optimal strategy for user n, which is computed by using (21).
The set (y∗ , x∗ ) is a Stackelberg equilibrium of the gamebetween the UCs and the users.

0

Ugen,k −Ugen,k = yk ∑ xn,k − yk ∑ xn,k(41)

D. Existence and Uniqueness of Stackelberg equilibrium(40)

OPuser has a unique maximum for a given y. Therefore, theStackelberg game possesses a unique SE if the price setting
game among the UCs admits a unique NE.

0

From (40), it is clear that Ugen,k (yk , y−k ) < Ugen,k (yk , y−k ) if
0
δyk > 0. If δyk < 0, yk < yk , but Pk is given. Consequently, therevenue will decrease. Therefore, the prices calculated using(34) are the best responses to each other and are the pricesthat maximize the revenue of each UC.
In addition to positivity, there are tighter limits on yk suchthat yk ∈ [yk,min , yk,max ]. The lower limit yk,min is due to theassociated generation costs. The UCs will not reduce theirprice below yk,min . The upper bound yk,max can be fixedaccording 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

Theorem 4. A unique Nash equilibrium exists in the priceselection game among the UCs, and thereby a unique Stackelberg equilibrium.
Proof: A Nash equilibrium exists for the UCs in the priceselection game if
1) y is a non-empty, convex, and compact subset of some
Euclidean space R K .
2) Ugen,k (y) is continuous in y and concave in yk , ∀k ∈ K .
In the price selection game for the UCs, the strategy
space Γgen = Γgen,1 × Γgen,2 × ..... × Γgen,K where yk ∈ Γgen,k :=6

TABLE I
D ISTRIBUTED A LGORITHM FOR PRICE AND DEMAND UPDATING[yk,min , yk,max ], ∀k ∈ K . Thus the strategy set is a nonempty,
convex and compact subset of the Euclidean space R K . From(24), we see that Ugen,k is continuous in yk . Next, the second
order derivative of Ugen,k w.r.t. yk is∂ 2Ugen,k= 0, ∀k ∈ K .
∂ y2k

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 yt using (21).
6: Transmit xn,k,t to each k.
7: end8: UC k ∈ K , which has not updated the price for iteration t + 1
9: Calculate yk,t+1 using (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.

(43)

Hence, Uk (y) is concave in yk . Therefore, NE exists in thisgame.
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. D ISTRIBUTED A LGORITHM
In the previous section, the consumers calculate their optimal demands based on the prices provided by the UCs but theUCs play the best response to other UCs’ strategies. In order tocalculate 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 theparameters of the other UCs.
Each UC starts with an arbitrary price yk,1 > 0 and allof 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 usingk ∈ K , ∀n ∈ N ,t = 1, 2, 3, . . . , then the sufficient conditionx−P∑
for yk,t+1 not to acquire a negative value is | n∈N σn,k,t k | <kyk,t . Since the condition should be satisfied only when∑n∈N xn,k,t − Pk < 0, we can rewrite the condition as σk >Pk −∑n∈N xn,k,t∀k ∈ K , ∀n ∈ N ,t = 1, 2, 3, . . . . Using (21) andyk,tupon simplification, the sufficient condition takes the form
(45).
Equation (44) implies that the price yk of UC k increases
if ∑n∈N xn,k,t − Pk is positive and the price decreases if∑n∈N xn,k,t − Pk is negative. Equation (44) shows that when∑n∈N xn,k,t − Pk = 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 T such that ∑n∈N xn,k,T − Pk = 0,0
then,Ugen,k (yk,T , y−k,T ) ≥ Ugen,k (yk,T , y−k,T )where
0Ugen,k (yk,T , y−k,T ) and Ugen,k (yk,T , y−k,T ) are the utilities
0of UC k at price yk,T and yk,T respectively. Thus the fixed
point to which Algorithm 2 converges is the NE of the gameamong the UCs, i.e., SE of the game between the UCs andthe consumers. Thereafter, the UCs will not deviate from thispoint.

∑n∈N xn,k,t − Pk
,
(44)
σk
where t is the iteration number and σk is 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.
We name this algorithm as Algorithm 2, which is shownin Table I. In the table, n = 1 indicates user number 1.
yk,t+1 = yk,t +Theorem 5. Provided that ∀k ∈ K , ∀n ∈ N ,t = 1, 2, 3, . . . ,σk >(KPk − βn )yk,t − βn ∑g∈K ,g6=k yg,t −Cn,
Ky2k,tVI. D EPENDABILITY OF DRM
One important component of dependability of DRM is thevulnerability due to attacks from malicious agents. While thebidirectional communications between the UCs and the users
facilitates demand response management in a more timely andeffective manner, it also creates room for different kinds ofthreats 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 goalof 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 theprice. This assumption is reasonable because the attacker has
physical constraints to access the UCs.

(45)

Algorithm 2 converges to the optimal solutions for both theusers 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 theprice vector to prove the convergence of Algorithm 2.
The algorithm will diverge only if yk,t acquires a negative value in any iteration. If ∑n∈N xn,k,t − Pk < 0, for any
7

immediately, the grid might suffer physical damage. There isan interesting observation here. Equations (49), (50) indicatethat if the price of UC k is changed, the other UCs will
also be affected. If yk increases, the demand from the otherUCs 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 yk decreases, 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 k may suffer physical damage. In either case (δy > 0or δy < 0), the UCs and the grid will suffer both monetaryloss and physical damage.

A. Attack and Its Impact
An attacker can cause two kinds of harm to the UCs and theusers: economic impact and physical impact. We consider thecase when all UCs have the same power to supply. 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 suchthat the resulting prices are within the specified range. Whenthe attacker does not manipulate any of the prices, the priceset by each UC isC, ∀k ∈ K .
(46)
KP
Using (46) in (21), the demand of user n from UC k isyk = y =B. Proposed Scheme with Individual Reserve PowerIn a scenario where the UCs are aware that an attacker might
attempt to create monetary or physical damage, although bothkinds of loss are serious, the physical damage can make thegrid unstable and it can affect the whole infrastructure on theUC 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 k by
Pk,res . The reserve power for each UC should be the differenceof 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 isCn
xn,k = P.
C
The total demand of user n from all UCs isCn∑ xn,k = C KP.
k∈K
Let the price of UC k, yk , be changed by the attacker through0
data injection or indirect manipulation to yk = yk + δy and allother prices are in-tact, where δy is a real number such that|δy | ∈ [0, δy,max ]. Positive value of δy means that the price isincreased and negative value means the price is decreased,δy = 0 corresponds to the case when the attacker does notact. When the price is changed by the attacker, the demand of0
users will change from xn,k to xn,k . The attacker can change0
yk but will still keep it such that xn,k is non-negative. It canbe calculated by replacing yk in (21) by yk + δy , which gives
0xn,k =Cn P − δy βn (K − 1)P.
(C + δy KP)

Pk,res =0n∈N

CP − δy B(K − 1)P.
(C + δy KP)(51)

If δy > 0, then Pk,res will be negative, since the demand fromUC k will be less than the amount of power available. Thismeans that no reserve power is needed. Hence, we have

(47)+
Pk,res= 0.

Equation (47) shows that the demand of each user from UC
k will decrease if δy is positive and it will increase if δy is
negative. The total demand from UC k from all users will be∑ xn,k =−δy P (B(K − 1) + KP).
(C + δy KP)(52)

For other UCs g 6= k, the reserve power can be calculated as
BPδy
.
(53)
C
Equations (52)-(53) indicate that the amount of reserve power
for each UC depends on the value of δy . Theoretically, each
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 theUCs have to pay the cost associated to buy or produce thereserve power. Therefore, all UCs prefer to keep as minimum
reserve power to save the associated costs, but at the sametime, it is extremely important for them to avoid the damage
to their infrastructure.
In contrast to communication problems where usually theaverage performance guarantee measures are taken, we needto consider the worst case for the power-trade scenario. The
attacker aims to maximize the harm to the UCs and/or to thegrid but without making the demands become out-of-the-range
values, e.g., negative demand from the users. The attacker
therefore, tries to manipulate yk in such a way that the demands
from all UCs is non-negative even from the user with thelowest budget. Let us examine the following cases.
+
Pg,res=(48)

Now, let us see if the change in the price of UC k affects other
UCs. For UC g 6= k, the demand from user n will bePCn
P + δy βn .
(49)
CCUsing (49), the total demand from UC g 6= k takes the form


0
B
(50)
∑ xn,g = P 1 + δy C .
n∈N0xn,g =

Equation (48) indicates that the total demand from UC kwill decrease (increase) if yk is increased (decreased) by theattacker compared to the case when there is no attack. If thetotal 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 theusers side. If the grid attempts to produce the shortage power8+= 0. The maximum harm1) δy > 0: If δy is positive, Pk,resthat the attacker can create in this case would be by letting
the user with the lowest Cn buy zero power from UC k, which0corresponds to xn,k = 0.

δymax =

Cmin.
βmin (K − 1)

reserve power while ensuring that there is no power-outage.
The total reserve power needed for the individual reserve
scheme is
Ptot,res = KPres .
(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(54)

where Cmin = min(Cn ) and βmin = min(βn ). If ymax is the
maximum unit price allowed, the range of δy given by δymax
00should still yield yk ≤ ymax . Thus, yk is given by0
yk + δy if δy ≤ ymax − yk ,
yk =
(55)
ymax otherwise.
Therefore, δymax can be calculated asCmin, (ymax − yk ) .
δymax = minβmin (K − 1)

Xtot ≤ KP + Pres .

(59)

From (58) - (59), the total power that will not be used isPtot,unused ≥ (K − 1)Pres .

(60)

The incentive for the UCs to use a common reserve is thatsharing common reserve power saves a considerable amountof cost for each UC. Furthermore, it is also beneficial fromthe overall system perspective, as there are increasing concerns
towards minimizing the waste of power.

(56)+
Hence, 0 < δy ≤ δymax and Pg,res
can be calculated by substituting 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 thelowest budget buys zero power from all UCs other than k. In
0this case, solving xn,g = 0 ∀g 6= k yields

δymax =

Cmin
,
βmin
0

where n corresponds to the lowest budget user. However, yk
0should not be less than yk,min . If yk = yk − Cβmin > yk,min , then,minsubstituting δy in (48) and (50), and substracting the available
power of the UC P, the reserve power for UCs (using β1 =β2 = ..... = βN = β ) are(NC(K−1)β +Cn,min KP2−,
Pk,res= n,minβC−C KPn,min−Pg,res = 0.

Fig. 3. Common reserve scheme: different utility/generation companies share
a common energy reserve0

If yk − Cβmin ≤ 0 or if yk − Cβmin > 0 but yk < ymin , then, the
min
min
0
0
minimum yk that can be chosen by the attacker is yk = ymin .
Using this constraint on (47), then summing over n ∈ N andsubtracting the available power P, we obtain−
Pk,res=Following the reasoning in Section VI-B, we can show thatthe total reserve power needed in this case will be
Ptot,res = Pres .

Employing this power sharing, we can see that the UCs canmaintain a reliable supply of power to the users in the presenceof 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.

C + B(K − 1)yd − KPymin
.
Kymin

C
where yd = KP
−ymin . The UCs should avoid problems even in−+ . However, thethe worst case, Pk,res = Pk,res
and Pg,res = Pg,res
UCs may not know whether the attacker will choose δy > 0or δy < 0. So, for each UC, the reserve power is−+
Pres = max{Pg,res, Pk,res}.

(61)

D. Discussion(57)

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 withthe generators, even in the absence of an attacker. Even
when all the generators have produced the power, one of the

C. Proposed Scheme with Common Reserve PowerThe 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 thisreserve separately, while the actual number of UCs that will
be attacked is limited to 1, there is definitely a huge wastage ofthe reserve power from all other UCs who are not attacked. In
particular, the UCs have to buy the reserve power from somesources and it is obvious that each UC aims to minimize the9generators can be disconnected from the grid after the UCs
have transmitted the price information to the users. This kindof 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 improvedin this case also by maintaining certain reserve power. Thiskind of reliability is referred to as (K − 1) reliability. On
similar grounds as in Section VI, it can be inferred that acommon 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.

20
User 1

User 2User 3User 4,User 5

Demands

15

10

5

0
0

Fig. 4.

VII. N UMERICAL R ESULTS
In this section, we examine how the users choose theiroptimal 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 theconvergence of the distributed algorithm. The last part of thissection shows the reserve power needed in the presence ofan attacker. We consider 3 UCs and 5 users with parameters 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 P1 = 10, P2 = 15, P3 = 20, respectively,
unless mentioned otherwise.

10

20
C1

30

40

User demands at equilibrium
6

User Utilities

5

4

3

2
User 1

User 21
0

10

User 320
C

User 4,User 5

30

40

30

40

1

A. Stackelberg Game
Fig. 5.

Figs. 4 - 7 show how the change in the prices set by theUCs 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 demands of user 1 is increasing because its budget isincreasing. The budget of other users do not change, but theyultimately demand less power because of the increase in thebudget 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 theusers 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 formore UCs and a large number of users for a large variation in
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 P1 = P2 = 150
units, P3 = P4 = 200 units and P5 = 250 units. The behaviorsof the users and the UCs are similar to Figs. 4-6.

User utilities at equilibrium
3.5
y1
3

y

2

Unit Price

y3
2.5

2

1.5

1
0

Fig. 6.

10

20
C1

Unit prices at equilibrium

Revenues of the UCs

40

35

30

20
0

B. Distributed Algorithm

Fig. 7.

Figs. 9 - 12 show the performance of the distributed
algorithm for σk = 40, ∀k ∈ K . The UCs reach the equilibrium
10

UC 1
UC 2
UC 3

25

10

20
C1

UC revenues at equilibrium

30

40

200

20
Users 1
Users 2−25
Users 26−50
Users 51−75
Users 76−100

15

100

Users 1
Users 2−25
Users 26−50
Users 51−75
Users 76−100

10

50

5

0
0

0
0

Unit Price

User Utilities

Demands

150

3
2.8
2.6

y1, y22.4

y3, y4
y5

2.2
2
1.8
1.6

100

200
C1

300

400

100

(a) User demands

300

1.4
0

400

100

(b) User utilities

200
C1

300

400

(c) Unit price

Equilibrium for a large number of users and UCs for a large variation in user budget
2.5

y1
Unit Price

Unit Price

y3

1.5

1
0

10

20
30
Iterations

40

5

4.5

4.5

4

y2

2

5

y

y2

3

y3

2.5

2

(a) yk,1 = 1; σk = 10.
Fig. 13.

60

y3

2.5

1.5
40
Iterations

y2

3

1.5
20

y1

3.5

2

1
0

50

4
1

3.5

Unit Price

Fig. 8.

200
C1

1
0

80

10

(b) yk,1 = 5; σk = 50.

20
30
Iterations

4050(c) yk,1 = 5; σk = 10.

The unit price of UCs for different initial starting points yk,1 and different values of σk , k = 1, 2, 3.

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 anddemand can be reached very quickly. Comparing Figs. 9 - 12with Figs. 4 - 7, we can verify that the distributed algorithm
converges to the optimal values.

10

User 1

User 2User 3User 4,User 5User Utilities

9

8

7

70
User 1

User 2User 3User 4,User 5660
50Demands

50

Fig. 10.

40

10

20
30
Iterations

40

50

Reaching equilibrium using distributed algorithm: User utilities
2.5

30
20

y1

Fig. 9.

102030Iterations

40

50

y2

Unit Price

10
0

2

Reaching equilibrium using distributed algorithm: User demands

Next we evaluate the performance of the distributed algorithm for different values of yk,1 and σk . Fig. 13(a) shows
the performance when σk = 10, k = 1, 2, 3, . . . . We see fromthe 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

y3

1.5

1
0

Fig. 11.

1020
30
Iterations

40

50

Reaching equilibrium using distributed algorithm: Unit pricesof σk , ∀k ∈ K but the algorithm converges irrespective of11

indicates how expensive an impact an attacker can create even
if it has access to only one UC’s price. The impact will besimilar in case of the heterogeneous UCs with the differencein the scale of the power to be reserved.

28

Revenues of the UCs

27
26

VIII. C ONCLUSION AND F UTURE W ORK
In this paper, we have proposed a Stackelberg game between
the electricity UCs for optimal price setting and the endusers for optimal power consumption. We have derived theSE of the game in closed form and have proved its existence
and uniqueness. We have designed a distributed algorithm forconvergence 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 physicaldisturbance 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 haveproposed 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 analyticaland 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 wehave focused on a large time-scale one-period DRM scheme.
A higher resolution multi-period scheme with inter-temporalconstraints is another possible extension to this work.

25
UC 1
UC 2
UC 3

24
23
22
0

Fig. 12.

102030Iterations

40

50

Reaching equilibrium using distributed algorithm: UC revenues

different initial points.
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
2C
decreased. We use P = 10 and [ymin , ymax ] = [ 0.5C
KP , KP ] for theplot. For the sake of illustration, we suppose that the attacker
45
40
35

R EFERENCES

Total Reserve Power − Individual Reserve
Total Reserve Power − Shared Reserve30[1] U.S.-Canada Power System Outage Task Force, Reports, URL: https://reports.energy.gov/[2] H. Farhangi,“The Path of Smart Grid,” IEEE Power and Energy Magazine, Vol. 8, No. 1, pp. 18-28, Jan.-Feb. 2010.
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Extra power needed

Power

Unused power − Individual Reserve
25

Unused power − Common Reserve

20
15
10
5
0
1

Fig. 14.

1.5

2

2.5

3
y(3)

3.5

4

4.5

5

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 ofinteresting 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 acommon reserve is sufficient to cover the demands of all theusers when the price is increased and also when it is decreased.
In fact, this power is extra most of the times and there is aconsiderable amount of unused power in both cases (individual
reserve and common reserve). The unused power is much morewhen each UC keeps its own reserve as shown by the dottedline 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 isstill significant even with the common reserve. That is because
our analysis is based on the worst case reserve. This figure also12[14] T. Başar and R. Srikant, “Revenue-Maximizing Pricing and Capacity
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