IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016
189
Demand Response Management in the Smart Grid
in a Large Population Regime
Sabita Maharjan, Member, IEEE, Quanyan Zhu, Member, IEEE, Yan Zhang, Senior Member, IEEE,
Stein Gjessing, Member, IEEE, and Tamer Başar, Life Fellow, IEEE
Abstract—In this paper, we introduce a hierarchical system
model that captures the decision making processes involved in a
network of multiple providers and a large number of consumers
in the smart grid, incorporating multiple processes from power
generation to market activities and to power consumption. We
establish a Stackelberg game between providers and end users,
where the providers behave as leaders maximizing their profit
and end users act as the followers maximizing their individual
welfare. We obtain closed-form expressions for the Stackelberg
equilibrium of the game and prove that a unique equilibrium
solution exists. In the large population regime, we show that
a higher number of providers help to improve profits for the
providers. This is inline with the goal of facilitating multiple distributed power generation units, one of the main design
considerations in the smart grid. We further prove that there
exist a unique number of providers that maximize their profits,
and develop an iterative and distributed algorithm to obtain it.
Finally, we provide numerical examples to illustrate the solutions
and to corroborate the results.
Index Terms—Consumer welfare, demand response management (DRM), large population, profit optimization, Stackelberg
game.
I. I NTRODUCTION
HE GROWING demand of electricity, the aging infrastructure, and the increasing greenhouse gas emission
are some of the challenges with the traditional power grid.
Recent blackouts [1] have further corroborated these issues,
and have fueled the need to transform the traditional power
T
Manuscript received August 10, 2014; revised September 22, 2014,
January 22, 2015, and April 26, 2015; accepted April 29, 2015. Date of publication June 1, 2015; date of current version December 19, 2015. This work
is supported in part by the projects 240079/F20, funded by the Research
Council of Norway, and in part by the European Commission FP7 Project
CROWN under Grant PIRSES-GA-2013-627490. The work of Q. Zhu was
supported in part by the Department of Energy Grant, and in part by the
National Security Agency through the Information Trust Institute, University
of Illinois. The work of T. Başar was supported by the U.S. Air Force Office
of Scientific Research Multi-Disciplinary University Research Initiative under
Grant FA9550-10-1-0573. Paper no. TSG-00800-2014.
S. Maharjan is with Simula Research Laboratory, Fornebu 1364, Norway.
Q. Zhu is with the Department of Electrical and Computer Engineering,
Polytechnic School of Engineering, New York University, Brooklyn,
NY 11201 USA (e-mail: quanyan.zhu@nyu.edu).
Y. Zhang and S. Gjessing are with Simula Research Laboratory, Fornebu,
Norway, and also with the Department of Informatics, University of
Oslo, Oslo 1325, Norway (e-mail: sabita@simula.no; yanzhang@simula.no;
steing@simula.no).
T. Başar is with the Coordinated Science Laboratory, University of
Illinois at Urbana-Champaign, Urbana, IL 61801 USA, and also with the
Department of Electrical and Computer Engineering, University of Illinois at
Urbana-Champaign, Urbana, IL 61801 USA (e-mail: basar1@illinois.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSG.2015.2431324
grid into a more responsive, efficient, and reliable system.
The smart grid [2], widely believed to be the future power
grid, offers improved efficiency, reliability, and environmental friendliness in power generation, transmission, distribution,
consumption, and management, by integration of advanced
information and communication technologies.
Demand response management (DRM) is the response system of end users to changes in electricity prices over time or to
other forms of incentives. In the smart grid, DRM plays a key
role in improving different aspects of both supply and demand
sides. For instance, DRM can result in lower bills and higher
utility efficiency for end users. DRM can also reduce the cost
of power generation or improve the revenues to retailers or utility companies (UCs). Existing studies on DRM have mostly
focused on either adjusting the demand side by load shedding
schemes when the supply is given, or on improving the profit
for the supply side when aggregate user demand is available.
Efficient DRM relies on both demand and supply sides, and an
integrated framework is needed to consider DRM in a holistic
manner. In this paper, we holistically investigate a planninglevel problem for both supply and demand sides, to cover a
large population of end users. We study regional level demand
response, incorporating the welfare of multiple providers and
a large number of consumers. We establish a DRM model to
capture the strategic behaviors of the UCs and power generation units from the supply side, and the consumers from the
demand side, in one single framework as a Stackelberg game
model. The DRM model consists of the following:
1) power generation units at the top level;
2) UCs at the middle level;
3) end users at the lowest level.
The UCs and end users interact through the unit price determined by the UCs at the middle level based on the consumer
parameters and the power generation costs. In order to capture
different reliability and efficiency requirements of the end users,
we break them into two groups: 1) residential; and 2) industrial consumers. The model allows us to develop insights into
the outcome of the strategic interactions with different types
of players in the power system. We analytically characterize
the Stackelberg equilibrium (SE) solution to the game and provide closed-form expressions for the equilibrium. Based on the
Stackelberg game model, we explore the DRM problem in a
large population regime, and introduce a new dimension of
improving providers’ profits and consumers’ welfare by adding
UCs. Since, multiple distributed power generation units form
one of the key components of the smart grid, such an alternative
c 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
1949-3053 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
190
IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016
is an important design consideration in the smart grid. Our
contributions in this paper can be summarized as follows.
1) We introduce a Stackelberg game model, incorporating
power generation units and UCs from the supply side
and end users from the demand side.
2) We derive closed-form expressions to characterize the
unique SE of the game and provide an iterative algorithm
to reach the SE in a distributed manner.
3) We propose an extended framework of the Stackelberg
game model in the large population regime. We prove
that there exist a unique number of providers that maximizes the providers’ profits, and present a distributed
algorithm to obtain the optimal number of providers.
The rest of this paper is organized as follows. Related work
is described in Section II. We introduce the system model
in Section III. In Section IV, we formulate the problem as a
Stackelberg game and prove the existence and uniqueness of
the SE. The distributed algorithm to reach the SE is presented
in Section V. In Section VI, we study the DRM problem for
a large number of end users, and develop a distributed algorithm to find the optimal number of providers for maximizing
providers’ revenues. We provide numerical results and discuss
them in Section VII. Section VIII concludes this paper.
II. R ELATED W ORK
Several studies on demand side management and DRM have
focused on either only one utility or a number of utilities
treated as one entity [3]–[8]. Mohsenian-Rad et al. [3] have
formulated an energy consumption scheduling problem as a
noncooperative game among the consumers for increasing and
strictly convex cost functions. Fan [4] has 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 [5], a robust optimization problem has been formulated to maximize the utility of a user, taking into account
price uncertainties at each hour. Wang and Groot [6] have
exploited the awareness of the users and proposed a method
to aggregate and manage end users’ preferences to maximize energy efficiency and user satisfaction. In [7], a dynamic
pricing scheme has been proposed to provide incentives for
customers to achieve an aggregate load profile suitable for
UCs, and the demand response problem has been investigated for different levels of information sharing among the
consumers. In [8], a multiresolution two-layer game is studied
using mean-field game approach to incorporate inner interactions between users in the region and outer interactions
between regions for dynamic distributed demand response in
the smart grid. References [9] and [10] have also incorporated
electric vehicles into the DRM framework.
Some recent studies have explored the DRM problem with
multiple providers (see [11]–[14]). In this paper, we study a
planning-level problem in a large population regime by considering the demand side together with the market and supply side,
bringing them into one single game framework. We note that,
there is rich literature using Stackelberg games for congestion
control, revenue maximization, cooperative transmission, and
DRM [11], [14]–[17]. Our approach is similar to those in congestion control and power control, but our game model involves
multiple providers and multiple consumers.
Maharjan et al. [11] introduced a Stackelberg game with
the supply of each UC fixed. When the number of consumers is large, the demand response problem unfolds new
and unique challenges, e.g., very high costs and poor revenues for the UCs. In fact, the SE solution may not even exist
for a given number of UCs. In this paper, we introduce a
new dimension of improving DRM performance by bringing
in additional providers to the smart grid. In addition, in our
model, we have included demands from both residential and
industrial consumers, and we have optimized power generation also. The model as well as the results are more generic
than those in [11].
Our framework and game model can be related
to [12] and [14]. Bu and Yu [14] have formulated a
four stage Stackelberg game, but with only one provider,
where as our model accommodates multiple providers.
In [12], the interaction among multiple residential users is
formulated as an evolutionary game. Each residential user has
to choose one UC to purchase power. In reality, consumers
do not choose the source of electricity or the power plant,
and hence, do not require to purchase electricity from specific
suppliers, especially for a time resolution of every hour. In
fact, in a perfect competition market, the unit price charged by
all the providers is the same, thus, not requiring consumers to
buy power from a particular UC. In our model, the consumers
play optimal response to the unit price without choosing a
particular supplier, which is more realistic and closer to the
current electricity market.
To this end, compared to related existing literature, what differentiates this paper can be summarized in three main points.
First, our focus is on a planning level problem, where the
supply demand equilibrium concerns the optimal generation
and consumption in a region, and not on the hourly or daily
scheduling of individual appliances or individual consumers
as in [3] and [13]. Second, different from the existing studies
on power systems (see [17], [18]), we derive analytical and
closed-form expressions for the unit prices, and consequently,
the supply from the providers and the demand from the consumers, which enable us to provide meaningful insights into
the behavior of and interactions between players at different
levels in the system. Third and most important, we concentrate
our analysis on a large population regime and introduce a new
dimension of improving the system welfare by increasing the
number of providers to accommodate the additional need of
the consumers.
III. S YSTEM M ODEL
Fig. 1 depicts our hierarchical system model, which consists of three levels: 1) power generation units at the top level;
2) UCs at the middle level; and 3) residential and industrial
consumers at lowest bottom level. The framework is motivated
by the hierarchy of the real power grid system. The power generation units or power plants supply power, the UCs determine
the unit price and optimal amount of power to supply, and
MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME
191
However, our formulation can be easily modified to the case
of imperfect competition. As the analysis for an imperfect
competition-based framework can be built by modifying only
certain parts of the formulation in Section IV, we provide
a sketch of the analysis for the imperfect competition scenario, mainly emphasizing what should be modified, in the
Appendix.
IV. P ROBLEM F ORMULATION : S TACKELBERG
G AME A PPROACH
Fig. 1.
Illustration of the interactions among the UCs and end users.
the bottom level represents the demand response to the price
signal from the residential consumers. The power generation
units, UCs and the consumers have bidirectional communications support to exchange price and demand information. The
data communication is carried out through the communication
channel using wireless technologies. The solid lines in the figure represent power flow, whereas the dashed lines represent
information flow.
Now, to formulate our model in precise terms, we consider N end users, which we also call consumers, and K UCs,
K := {1, 2, . . . , K}. Each UC receives power from one or
multiple sources. Let Ck (Pk ) represents the cost of supplying
Pk amount of power to the consumers, including the cost of
generating the power. We assume Ck (Pk ) is increasing and
convex ∀k, k ∈ K [3], [12]. The consumers’ side consists of
NR residential users, NR := {1, 2, . . . , NR } and NI industrial
users, NI := {1, 2, . . . , NI }, i.e., N = NR + NI .
Let xR,n denotes the demand of residential user n. The power
demand of the consumers depends on electricity price and
consumer type.
for each consumer n, the gain
Specifically,
function UR,n xR,n represents the satisfaction the consumer
gets as a function of its power demand xR,n , which is nondecreasing and concave [10], [12]. The power demand from
industrial consumers is normally on a different scale and they
have relatively stricter power requirements. Thereby, we consider the total demand from all industrial consumers m ∈ NI ,
to be given: PI ≥ 0.
The framework and the basic model for problem formulation and analysis, is that of a perfectly competitive market.
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 the
firms have no control. Consequently, each firm should increase
its production up to the point where its marginal cost equals
the market price. When each individual entity of a finite number of market participants (UCs) has noninfinitesimal influence
in the market, it leads to imperfect competition. In an imperfect competition scenario, the UCs can charge consumers with
different unit prices. In this paper, we concentrate on DRM
analysis and modeling in a large population regime where the
number of providers is usually large. In such a scenario, it
is reasonable that a provider is likely not to have the ability to influence the market price through its individual action.
Motivated by such a consideration, a perfect competition-based
model is more realistic, and hence, relevant for our scenario.
The end users are indirectly coupled through the unit price
as a result of the competitions among the UCs at the market
level. The generation units are also indirectly coupled through
the price signal. These couplings between the decisions of
the players make game theoretic approach an appealing one
for cross-level understanding of a multiplayer multilevel complex system. Since the consumers respond to prices after UCs
announce them, the hierarchical decision making process can
be modeled as a Stackelberg game [19] where the UCs behave
as leaders and the users react optimally to their strategies as
followers.
A. Demand Side Analysis
Let y be the price per unit power. For given y, user
n (n ∈ NR ) calculates its optimal demand response by solving
the user optimization problem to maximize its welfare WR,n
as follows:
(1)
max WR,n := UR,n xR,n − yxR,n
xR,n
s.t. xR,n ≥ xR,n,min
(2)
where xR,n,min is the minimum power requirement of consumer n. The above, that is, (1) and (2) characterizes a
strictly convex optimization problem for given y. Hence, the
stationary solution is unique and optimal. The first-order optimality condition for the optimizing residential user leads to
(∂WR,n /∂xR,n,k ) = 0, ∀n ∈ NR , that is
−1
UR,n
= y, ⇒ xR,n = UR,n
(3)
(y).
The condition required for constraint (2) to be satisfied can
be established by substituting (3) into (2), which requires
]
y ≤ [UR,n
xR,n =xR,n,min, ∀n ∈ NR . This can be ensured if
y ≤ ymax := UR,n
.
(4)
min x =x
n∈NR
R,n
R,n,min
For the purpose of illustration and to provide functionspecific insights, we employ two widely adopted gain
functions for residential consumers: 1) piecewise quadratic
function [12]; and 2) logarithmic function [10]. We define
the piecewise quadratic gain function of residential user
n, (n ∈ NR ), as
⎧
2
zR,n xR,n
⎪
vR,n
⎪
, if xR,n ≤
⎨ vR,n xR,n −
2
zR,n
UR,n xR,n =
zR,n
vR,n
⎪
⎪
if xR,n >
⎩
2vR,n
zR,n
(5)
192
IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016
where vR,n and zR,n are user-specific parameters ∀n ∈ NR . In
this case, (3) and (4), respectively, take the form
vR,n − y
(6)
xR,n =
zR,n
and
y ≤ ymax := min vR,n − zR,n xR,n,min .
n∈NR
(7)
The logarithmic gain function can be defined for residential
user n, (n ∈ NR ), as
UR,n xR,n = αR,n ln βR,n + xR,n , ∀k ∈ K
(8)
where αR,n and βR,n are user-specific parameters. In this case,
for given y, (3) and (4), respectively, take the form
αR,n
xR,n =
(9)
− βR,n
y
and
αR,n
.
n∈NR βR,n + xR,n,min
y ≤ ymax := min
Then, the optimization problem for each UC is as follows:
max
RUC,k
0≤y≤ymax ,Pk ∈R+
s.t. PI +
xR,n ≤
Pk
k∈K
(12)
where Pk,max is the maximum power UC k can supply. When
Pk,max is sufficiently large, PI , Pk << Pk,max . Then the second
constraint can be relaxed.
For a given y, (12) is a convex optimization problem.
Given y, the first-order optimality condition for the UCs,
(∂RUC,k /∂Pk ) = 0, gives the optimal amount of power to be
supplied by UC k as
−1
(13)
Pk = Ck
(y); ∀k ∈ K .
Normally, if the power supplies of different UCs are given,
the unit prices would be calculated based on the given power
supplies. However, we are considering here a planning level
problem where both unit price and optimal power to be supplied, are the variables. Thus, the optimal power of each UC
is calculated by backward induction, based on the optimal
unit price, a parameter obtained as a result of the profit optimization of the UCs. With the optimal demand response of
residential users (3) and UCs’ power supply (13) in response
to the price y, the objective of the UCs is to set the optimal
price y. For supply demand equilibrium, it is required that
Pk = PI +
xR,n .
(14)
k∈K
n∈NR
Substituting (3) and (13) into (14) we obtain
−1
−1
Ck
UR,n
(y) = PI +
(y).
k∈K
n∈NR
(15)
y = (G1 )−1 (PI )
(16)
−1
−1
where G1 (y) =
k∈K (Ck ) (y) −
n∈NR (UR,n ) (y).
We employ a quadratic cost function for power
generation [12], [13]. Let ak > 0 and bk , ck ≥ 0 be the
coefficients of the cost function Ck (Pk ). Then, if the total
power supplied by UC k is Pk , then the cost incurred to the
UC is
Ck (Pk ) = ak P2k + bk Pk + ck .
(17)
When the gain functions of the residential consumers are
piecewise quadratic as given by (5), (16) takes the form
bk
vR,n
+ n∈NR
2ak
zR,n
.
1
1
+
k∈K
n∈NR
2ak
zR,n
PI +
y=
If the total power supplied by UC k is Pk , the profit of
provider k is defined as
RUC,k = yPk − C(Pk ).
(11)
n∈NR
(10)
B. Supply Side Analysis
Pk ≤ Pk,max ; ∀k ∈ K
From (15), we can obtain
k∈K
(18)
Proposition 1: When the gain functions of the residential consumers are piecewise quadratic as given by (5), (18)
is the unique feasible solution to the profit maximization
problem (12) only if
PI ≤ PI,max := min vR,n − zR,n xR,n,min
n∈NR
⎛
⎞
1
1
⎠
+
×⎝
2ak
αR,n
k∈K
n∈NR
⎛
⎞
bk
vR,n
⎠.
−⎝
(19)
+
2ak
αR,n
k∈K
n∈NR
Proof: Since ak > 0, bk , ck ≥ 0, ∀k ∈ K , vR,n , zR,n > 0,
∀n ∈ NR , and PI ≥ 0, (18), implies that y > 0. For given
ak , bk , ck ∀k ∈ K , and vR,n , zR,n ∀n ∈ NR , substituting (18)
into (7), we obtain
bk
vR,n
+ n∈NR
2ak
zR,n
≤ min vR,n − zR,n xR,n,min .
1
1
n∈NR
+ n∈NR
k∈K
2ak
zR,n
(20)
PI +
k∈K
Further simplification of (20) yields (19).
Remark 1: Note that, UCs may impose their own limits on
the unit price, and usually there is a maximum limit the market
imposes, i.e., yk,min ≤ y ≤ ym,max ∀k ∈ K . Without loss of
generality, we consider ymax ≤ ym,max and y ≥ yk,min ∀k ∈ K .
Proposition 2: When the gain functions of the residential
consumers are logarithmic as given by (8), a unique feasible
solution of (12) is
−T1 + T1 2 + 8AAR
, if
(21)
y=
2A
AαR,n
AR
BA
−
PI ≤ min + BR −
α
2
n∈NR 2 βR,n + xR,n,min
min βR,n +xR,n
R,n,min
n∈NR
(22)
MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME
where
T1 = 2BR − BA − 2PI , A =
k∈K 1/ak , AR =
n∈NR αR,n , BR =
n∈NR βR,n , and BA =
k∈K bk /ak .
Proof: Substituting Ck , UR,n from (17) and (8) into (15) and
further simplification yields
αR,n
y − bk
(23)
= PI +
− βR,n .
2ak
y
k∈K
n∈NR
The solution of (23)is y = (−T1 ± T1 2 + 8AAR /2A).
Since A > 0 and T1 2 + 8AAR > T1 , the root y =
(−T1 + T1 2 + 8AAR /2A) is the only real, positive one, and
hence, feasible solution for y. Now, substituting (21) into (10)
leads to
αR,n
−T1 + T1 2 + 8AAR
. (24)
≤ min
2A
n∈NR βR,n + xR,n,min
Simplification of (24) yields
αR,n
2A min
+ T1 ≥ T1 2 + 8AAR . (25)
n∈NR βR,n + xR,n,min
Squaring both sides of (25) and upon further simplification, (25) takes the form (22).
Remark 2: If for any of the UCs, (Ck )−1 (y) > Pk,max ,
then instead of using (13), UC k supplies Pk = Pk,max . The
power supply from UC k can, therefore, be expressed as
Pk = min((Ck )−1 (y), Pk,max ).
C. Stackelberg Equilibrium: Existence and Uniqueness
Theorem 1: A unique SE exists in the hierarchical
Stackelberg game.
Proof: Given the prices y := {yk = y, ∀k ∈ K } and
power supplies P := {Pk , ∀k ∈ K }, the optimal responses
x̂ := {xR,n , ∀n ∈ NR } to y can be determined by solving the
strictly convex problem (1) in Section IV-A for residential consumers. Given these unique responses, an equilibrium exists
for the price setting game between the UCs if the following
conditions hold.
C1: y is a nonempty, convex, and compact subset of some
Euclidean space R K .
C2: RUC,k (y) is continuous in y and concave in yk , ∀k ∈ K .
Here, yk ∈ [0, ymax ] and ∀k ∈ K . Thus, the strategy set
is a nonempty, convex, and compact subset of the Euclidean
space R K . From (11), we see that RUC,k is continuous in yk .
Next, the second-order derivative of RUC,k with respect to yk is
∂ 2 RUC,k
= −2ak < 0, ∀k ∈ K .
∂y2k
(26)
Hence, RUC,k (y) is concave in yk , i.e., in y. Note that, the
supply from each UC is not dependent on the supplies of
other UCs when y and x are fixed, and it is strictly concave
in Pk , which leads to a unique solution (13). Since the optimal response of users are given by (3), using C1 and C2, we
can conclude that an SE exists for the game. As proven in
Section IV-B, there exists only one feasible solution for the
price given by (16), provided that (19) or (22) is satisfied
for the piecewise quadratic and the logarithmic gain functions, respectively. Therefore, the solutions (3), (13), and (16),
constitute the unique SE of the hierarchical DRM scheme.
193
Algorithm 1 Distributed Algorithm for Optimal DRM
1: For t = 1, control unit: arbitrarily choose y1 and announce it to
the UCs and the consumers.
2: Repeat for t=2,3, . . .
3: User n = 1, 2, . . . , NR , Do
4:
Find xR,n,t from (3) for given yt , and report the demand to
the control unit.
5: end
6: UCs k = 1, 2, . . . , K, Do
7: Update Pk,t = min((Ck )−1 (yt ), Pk,max ) for given yt , and report
the power supply to the control unit.
8: end
9: Calculate yt+1 using (27).
10: If yt+1 = yt ,
11:
Send a no-change signal to the UCs and the consumers.
12:
break.
13: else
14:
Send the new value of price to the UCs and the consumers.
15:
Go to 3.
16: end
V. D ISTRIBUTED A LGORITHM
In order to obtain the optimal demand response solution: (3), (13), and (16), a centralized solver needs to know
the exact cost functions of the UCs and the gain functions of
the consumers. Usually, UCs as well as consumers prefer not
to share their private information [20], [21]. Moreover, distributed algorithms offer scalability. In this section, we present
an algorithm in order to obtain the optimal demand response
solution in a distributed and iterative way, starting from an
arbitrary initial value [22].
For the distributed algorithm, subscript t is appended to all
variables defined in the previous sections, as the time index,
i.e., t ≥ 1 indicates the iteration number. We consider that an
intermediate entity between the supply and demand sides such
as a control unit, finds the optimal demand response solution
without knowing the cost and gain functions. The control unit
starts with an initial unit price y1 ≥ 0 and announces it to the
UCs and the consumers. Based on the unit price, the residential consumers compute their demands xR,n,t using (3), and is
sent to the control unit. The power demand from the industrial
consumers PI is known to the control unit. Each UC also computes its power supply Pk,t as Pk,t = min((Ck )−1 (yt ), Pk,max ),
and announces its power supply to the control unit. Then, the
control unit updates the unit price for the next iteration based
on the difference between the total power supply and the total
power demand as
yt+1 = yt +
n∈NR xR,n,t
+ PI −
σ
k∈K
Pk
(27)
where σ , a sufficiently large positive number, adjusts the convergence speed of the algorithm. The process repeats until
the unit price (and consequently the supplies and demands)
remains the same as in the previous iteration. The details are
presented in Algorithm 1.
194
IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016
VI. L ARGE P OPULATION R EGIME
In Section IV, we derived closed-form expressions for the
unit price y: (16), optimal demands from residential consumers
xR,n : (3), optimal power supply Pk from the UCs: (13), and
provided that certain conditions are valid: (19) and (22). In
the low to moderate demand markets, such conditions may
be readily satisfied. However, in a large population regime
the total demand from residential and industrial consumers
can be very high. This happens if either there is a large
population of residential consumers, or their minimum power
requirements are high; the number of or demand from industrial consumers increases significantly; or both are true. In
this section, we concentrate our discussions on such scenarios and extend the hierarchical DRM model to accommodate the large population needs. We introduce proportionality
between power supply units and consumers’ demands and
show that the demand response performance can be significantly improved in the large population regime by adding extra
providers.
We start with the case when the number of UCs is given,
and then move on to investigate the DRM problem with the
provision of additional providers. Since the results can be similarly derived and analyzed for quadratic and logarithmic gain
functions, we limit our discussion to the piecewise quadratic
gain function of the residential consumers. When (19) is not
valid, y = ymax . In this scenario, the supply from each UC, Pk ,
cannot be obtained directly from (13). We introduce a virtual
unit price y such that the optimal power supply Pk can be
computed for each UC k, k ∈ K as follows.
Replacing y by y in (13), and then substituting (13) into
(15), we obtain
−1 −1
Ck
y = PI +
UR,n
(28)
(ymax ).
k∈K
n∈NR
Upon simplification of (28), we get
⎛
⎞
−1
−1
y = G2 ⎝PI +
UR,n
(ymax )⎠
(29)
n∈NR
−1 where G2 (y ) =
y . For the quadratic gain
k∈K Ck
function, (29) takes the form
vR,n − ymax bk
PI + n∈NR
+ k∈K
zR,n
2ak
y =
(30)
1
k∈K
2ak
where ymax is obtained using (7). Now, Pk and xR,n can be
obtained as
−1 y − bk
y =
; ∀k ∈ K
(31)
Pk = Ck
2ak
and
−1
xR,n = UR,n
(32)
(ymax ).
Note, however, that the unit price charged to the consumers
will be ymax , not y . Thus, the UCs have to meet increased
demand requirements without increasing the unit price, consequently, reducing their profits. Since our focus is on the
planning level problem, the power generation, supply, and
demand are at a macroscale covering the population of the
electricity consumers in a region or a state. Therefore, hourly
shifting and scheduling of load is not a feasible option. Such
a condition physically indicates the need of cheaper or more
power resources to accommodate the power requirements of
the consumers. Distributed local or regional power sources are
one of the major components in the smart grid. In line with
this objective, we introduce a new dimension of optimizing
DRM by adding providers if necessary.
Let K0 denotes the set of UCs available. We introduce a
set of UCs J := {1, 2, . . . , J} that can supply additional
power to meet the demand requirements of a large number of consumers. Then, K := K0 ∪ J . Each additional
power supply unit is associated with the cost coefficients
aj > 0, bj , cj ≥ 0 ∀j ∈ J .
Suppose, with given K0 , Pk is calculated using (31) but the
unit price charged to the consumers is ymax < y . Then (18)
indicates that with every j added to the list of the providers, the
optimal y decreases for bj < 1. This means, with every additional j, the gap between y and ymax reduces. Consequently,
the amount of power supply from each UC will be closer to Pk
for given price y. Thus, with more UCs, the profit of each UC
improves, which serves as an incentive for the supply side to
add UCs. However, after a certain number of additional UCs,
the optimal unit price starts decreasing because of excessive
competition among them and the demands from the residential consumers start increasing in response to the cheaper unit
price. As a result, the profits start degrading and the welfare of
the consumers start increasing. Thus, in the large population
regime, it is necessary to scale the supply side proportionally
with the consumer population (and their demands), but for a
given population, there exists a fundamental tradeoff between
the UC profit and consumer welfare with respect to the number
of UCs.
Theorem 2: Suppose J is the number of UCs that can
be added to the supply side with the associated cost coefficients a > 0, b, c ≥ 0. For the piecewise quadratic
gain function-based
residential user-welfare
model, let E1 :=
+
2
(v
−
y
/z
)
+
2P
max R,n
n∈NR R,n
k∈
K0 bk /ak , E2 :=
I
:=
1/a
,
E
+
b
/a
+2
2P
k
3
I
k∈K0
k∈K0 k k
n∈NR (vR,n /zR,n ),
and E4 := k∈K0 1/ak + 2 n∈NR (1/zR,n ). Then, provided
bk (J + aE2 ) − aE2 )
<b<
J
bk (J + aE2 ) − aE1 )
<
if
J
E1
E2
E1
E2
or
E1
bk (J + aE2 ) − aE1 )
<b<
E2
J
bk (J + aE2 ) − aE1 )
E1
<
if
E2
J
and
bk (J + aE4 ) − aE3 )
<b<
J
bk (J + aE4 ) − aE3 )
<
if
J
E3
E4
E3
E4
(33)
MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME
Algorithm 2 Distributed Algorithm to Optimize Provider
Profits in Large Population Regime
or
E3
bk (J + aE4 ) − aE3 )
<b<
E4
J
bk (J + aE4 ) − aE3 )
E3
<
if
E4
J
195
(34)
there exists a unique number of UCs for which the profit of
each UC and the total profit are maximized.
Proof: As described earlier, the consumers will be charged
at the unit price of ymax as long as y > ymax and will be
charged at the unit price of y when y ≤ ymax . The profit of
UC k, k ∈ K when it supplies a power given by (31) is
RUC,k, = ymax y − bk
y
2a
k
y − bk 2
y − bk
+ bk
(35)
− ak
+ ck
2ak
2ak
1: For t = 1, K = K0 , j = 0 and for given NR , compute yj , ymax , yj
using (18), (7), (30), respectively. Compute Pk,j and RUC,k,j from
(13) or (31) as required, and (11), respectively. Sort J with the
cheapest source first, i.e., Jsorted = {1, 2, . . . j, . . . J}.
2: Repeat for t = 1, 2, 3, . . .
3: If (19) is true
4: break
5: else if j > J
6: break
7: else
8: for K = K0 ∪{1, 2, . . . , j} compute yj+1 , ymax,j+1 using (18), (7).
9: j ← j + 1
10: end
11: Go to 2.
where y is obtained from (29). When y ≤ ymax , the profit of
UC k, k ∈ K takes the form
RUC,k, = y y − bk
y
2ak
y − bk 2
y − bk
+ ck
+ bk
(36)
− ak
2ak
2ak
where y is given by (18). Differentiating (35) and (36) with
respect to J yields
∂|RUC,k |y
a(E1 − bE2 )(bJ + aE1 − (J + aE2 )ymax )
=
∂J
2ak (J + aE2 )3
∂|RUC,k, |y
a(E3 − bE4 )(bJ + aE3 − (J + aE4 )bk )
=−
.
∂J
2ak (J + aE4 )3
Fig. 2. Convergence of the distributed algorithm. (a) y1 = 0.2, σ = 300.
(b) y1 = 1, σ = 300.
(37)
VII. N UMERICAL R ESULTS
(38)
When y > ymax , the profit of UC k, k ∈ K , |RUC,k, |y is an
increasing function of J if (∂|RUC,k |y /∂J) > 0. From (37),
we observe that (∂|RUC,k |y /∂J) > 0 if (33) is valid. When
y ≤ ymax , the profit of UC k, k ∈ K , |RUC,k, |y is a decreasing
function of J if (∂|RUC,k, |y /∂J) < 0. Equation (38) indicates
that (∂|RUC,k, |y /∂J) < 0 if (34) is true.
Therefore, if (33) and (34) are valid, |RUC,k, |y is an increasing and |RUC,k, |y is a decreasing function of J, which implies
that a unique J exists that maximizes the profit.
We design a distributed algorithm to determine the optimal
number of UCs for maximizing their profits. For the distributed
algorithm, the additional UC index j ≥ 0 is appended as a
subscript to all variables defined in the previous sections. For
given parameters of the residential and industrial consumers,
if (19) is true, the unit price and consequently, the profits
of the UCs are optimal, i.e., no additional providers should
be added. If (19) is not valid, yj , ymax , and yj are computed
from (18), (7), and (30), respectively. The power supply of
each UC Pk,j is obtained using (31) and the profits RUC,k,j are
calculated by substituting ymax and Pk,j into (11). The cheapest
additional UC j is added, i.e., j = j + 1 and RUC,k,j are computed for all k ∈ K0 ∪ {1, 2, . . . j} again. The process will be
repeated as long as every additional UC improves the profits.
The details of the algorithm are presented in Algorithm 2.
We begin with three UCs. In practice, power generation
costs of the UCs can vary for different kinds of energy sources.
However, if the UCs supply power from similar kinds of power
plants, the costs can be similar or within a certain range. For
the purpose of illustration, we choose the cost coefficients of
these UCs as a1 = 0.1 cents, b1 = 0.2 cents, c1 = 0, and
a2 = 0.05 cents, b2 = 0.1 cents, c2 = 0, and a3 = 0.02
cents, b3 = 0.05 cents, and c3 = 0. We have selected the
quadratic and linear coefficients (ak , bk ) significantly different for each provider, to represent general scenario and to
illustrate the difference in their profits. The constant coefficient (ck ) are chosen as zero, as we are focusing on a planning
level problem, where the energy sources that do not have highstarting costs, are preferred. Note that, the same costs for each
UC or similar costs are special cases of our formulation and
analysis, and hence, are covered in our model. We consider
piecewise quadratic gain function for the residential consumers
with zR,n = 2, vR,n = 1, xR,min = 0.2 MWh, ∀n ∈ NR unless
otherwise mentioned.
A. Stackelberg Game
Fig. 2(a) and (b) shows the convergence of Algorithm 1.
Starting from a different initial values, the unit price, and consequently, the residential user-demand converge to the same
optimal values in Fig. 2(a) and (b), within 16 iterations.
196
IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016
(a)
(b)
Fig. 3. Variation with respect to the population of residential consumers (a1 = 0.1 cents, b1 = 0.2 cents, c1 = 0, a2 = 0.05 cents, b2 = 0.1 cents,
c2 = 0, a3 = 0.02 cents, b3 = 0.05 cents, c3 = 0, PI = 50 MWh, vR,n = 2, zR,n = 1, xR,min = 0.2 MWh, ∀n ∈ NR ). (a) Unit price and welfare of residential
and industrial users. (b) Total supplies and profits of the UCs.
(a)
(b)
Fig. 4. Variation with respect to the demand from industrial consumers (a1 = 0.1 cents, b1 = 0.2 cents, c1 = 0, a2 = 0.05 cents, b2 = 0.1 cents,
c2 = 0, a3 , a4 , . . . , a10 = 0.02 cents, b3 , b4 , . . . , b10 = 0.05 cents, c3 , c4 , . . . , c10 = 0, NR = 100, vR,n = 2, zR,n = 1, xR,min = 0.2 MWh, ∀n ∈ NR ).
(a) Unit price and welfare of residential and industrial users. (b) Total supplies and profits of the UCs.
Since, there exists a one-to-one mapping of the power supply from the UCs, their profits, and consumer welfare with
the unit price, it is clear that all these measures also converge.
Fig. 3(a) shows the variation of unit price, and welfare of
the residential and industrial consumers with respect to the
number of residential users for PI = 50 MWh. We observe
that the unit price increases as the number of residential
consumers increases, until NR = 100. Consequently, the residential users’ demand reduces in response to the increasing
price, and the residential user welfare also decreases. When
NR = 100, the price reaches the maximum value (ymax ). As a
result the unit price remains constant after this point. The total
cost to the industrial consumers increases with the increasing
unit price and then remains constant after y becomes ymax , as
expected. In Fig. 3(b), the increase in supply from all UCs after
NR = 100 is normal with increasing NR when y is constant.
In the region NR ≤ 100, the increase in the unit price y causes
the supplies to increase, as the increase in residential-user
population is dominant over the decrease in their demands.
The profits on the other hand attain their maximum values
when NR = 100. Until NR = 100, the increase in the unit
price y improves the profit with higher supply of power. After
NR = 100, even though the UCs need to increase their supply which consequently increases their costs, the unit price
remains the same, thus, reducing their profits. As UC 3 is the
cheapest one, it supplies the maximum power, and its profit is
the highest.
In Fig. 4(a) and (b), we illustrate the performance of our
model with respect to the power demand from the industrial
consumers. For this plot, we used NR = 100. Other parameters were the same as used for Fig. 3(a) and (b). Fig. 4(a)
shows a similar pattern to Fig. 3(a). For given NR , PI = 50
is the point until where (19) is valid. As a result, the unit
price and the demand and welfare of the residential consumers
decreases until PI = 50. After this point, the price is constant,
the demand of the residential consumers is their minimum
power requirement and their welfare is also constant. The cost
paid by the industrial users increases with increase in the total
MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME
197
B. Large Population Regime
(a)
(b)
(c)
Fig. 5.
Profits with respect to number of UCs (a1 = 0.1 cents, b1 =
0.2 cents, c1 = 0, a2 = 0.05 cents, b2 = 0.1 cents, c2 = 0, a3 , a4 , . . . , a10 =
0.02 cents, b3 , b4 , . . . , b10
=
0.05 cents, c3 , c4 , . . . , c10
=
0,
vR,n
=
2, zR,n
=
1, xR,min
=
0.2 MWh, ∀n
∈
NR ).
(a) NR = 500, PI = 50 MWh. (b) NR = 500, PI = 100 MWh.
(c) NR = 1000, PI = 50 MWh.
industrial demand, as expected. The pattern of supplies and
profits for the UCs in Fig. 4(b) is similar to that in Fig. 3(b),
The profits are maximized at PI = 50.
In this section, we extend the scope of our solutions to
cover a large population of residential consumers and a higher
demand from industrial consumers. In Section VII-A, we
observed that the residential users can only fulfill their minimum power requirement and that the profits of the UCs
decrease after certain values of NR and PI . Therefore, we start
with three UCs as specified in Section VII-A but we added 17
UCs in the list of providers, which can be deployed if they can
improve the provider profits or the consumer welfare. Without
loss of generality, for the purpose of illustration, we choose
the cost coefficients for all of the additional UCs same as that
of UC 3, i.e., a4 , a5 , . . . , a20 = a3 , b4 , b5 , . . . , b20 = b3 , and
c4 , c5 , . . . , c20 = c3 . The results can be easily extended for
any set of cost coefficients associated with the additional UCs.
Fig. 5(a)–(c) depicts the profits of the UCs and the total welfare of the residential consumers for different combinations
of NR and PI . The figures show that the profits of all UCs
increase as new UCs are added, but the profits start degrading if there are too many UCs due to excessive competition
among them, clearly indicating the optimal number of UCs
for given NR and PI . Let Kopt be the number of UCs that
maximizes the profits. Let us call the region where K < Kopt
as small-K region and the region where K > Kopt as large-K
region. For PI = 50 MWh, as NR increases from 500 to 1000,
Kopt changes from 5 to 7. When NR = 500, change in PI from
50 to 100 changes Kopt from 5 to 6.
On the other hand, the consumer welfare unfolds a different
perspective. The residential-consumers’ welfare is less for the
small-K region and it starts improving in the large-K region.
For example, in Fig. 5(a)–(c), the welfare starts drastically
increasing after K = 5, K = 6, and K = 7, respectively.
Note that, the profits of all the UCs start decreasing after
these points. Thus, the figure illustrates a fundamental tradeoff
between maximizing UC profit or optimizing consumer welfare, with respect to the number of UCs, in the large population
regime.
When the unit price is constant (ymax ) and the number of
participating UCs increases in the small-K region, the supply
from each UC decreases, thus, reducing the extra cost incurred
due to the power supply higher than the optimal value for
given y. As a result, the profits improve. The increase in the
welfare in the large-K region is governed by the cheaper unit
price. Intuitively, each UC supplies less power at a reduced
unit price, thus, lowering the profit of all UCs.
Despite this tradeoff between the provider profits and the
consumer welfare, closer observation reveals some interesting
facts. In Fig. 5(a), when K changes from 5 to 6, the total profit
decreases from 58.41 to 53.67, a reduction of about 8%. The
welfare on the other hand grows from 11.09 to 19.78, a gain of
about 78.5%. These measures for Fig. 5(b) and (c) are, respectively: when K changes from 6 to 7, 8.4% profit reduction
versus 80% improvement in welfare and when K changes from
7 to 8, 3.6% profit reduction versus 35% improvement in welfare. It can be argued that the gain in the welfare with a UC
additional to Kopt yields about ten times more gain in the residential consumer welfare compared to the loss in the total
profit for given set of chosen parameters.
198
IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016
Another interesting observation from Fig. 5(a)–(c) is that,
every additional UC in the small-K region, brings drastic
improvement in the profits of the UCs without worsening the
residential consumers’ welfare. This implies that additional
UCs essentially improve both supply and demand performances in the small-K region, a natural incentive for the
supply side to add UCs proportional to consumer population
if necessary. In the large-K region, the number of additional
UCs should be determined according to the specific requirements of the system, which prioritize the UC profits or the
consumers’ welfare.
We have proposed a hierarchical system model incorporating multiple processes from power generation to market
activities and to power consumption into one framework for
a large population of residential and industrial consumers.
We have modeled the decision processes of both supply and
demand sides as a multileader multifollower Stackelberg game,
where the UCs act as the leaders to maximize their profits, and
the residential users are the followers that optimize their individual welfare. We have developed closed form solutions for
the SE of the game and have proved that a unique SE exists.
Moreover, we have presented a distributed algorithm to reach
the optimal solution. We have introduced a new dimension
of improving the UC profits in a large population regime by
adding UCs, and developed a distributed algorithm to reach the
optimal number of UCs for profit maximization. We have also
explored the tradeoff between the provider profits and consumer welfare with respect to the number of providers. Finally,
we have provided extensive numerical results to illustrate the
performance of the proposed solutions.
A PPENDIX
S KETCH OF THE A NALYSIS FOR I MPERFECT
C OMPETITION -BASED S CENARIO
In an imperfect competition-based model, let yk be the unit
price charged by UC k, and let xR,n,k be the power consumer
n obtains from UC k. Then, the user optimization problem,
i.e., (1), takes the form
yk xR,n,k . (39)
WR,n := UR,n xR,n −
max
xR,n :={xR,n,k ,∀k∈K }
k∈K
For illustration, considering the logarithmic gain function (8)
for the consumers, for given {yk , k ∈ K }, the optimal demand
of consumer n from UC k can be expressed as
αR,n
xR,n,k =
− βR,n , ∀k ∈ K .
(40)
yk
Then, the optimization problem for UC k, k ∈ K , (12), can
be expressed as
yk Pk − C(Pk )
s.t. PI +
xR,n,k ≤
Pk
0≤yk ≤yk,max ,Pk ∈R+
n∈NR k∈K
Pk ≤ Pk,max ; ∀k ∈ K .
n∈NR
Substituting (40), (42), and Pk,I = PI /K in (43), and after
simplification, we obtain
VIII. C ONCLUSION
max
For a quadratic cost function for power generation, the optimal
power to be supplied by each UC (13), takes the form
yk − bk
; ∀k ∈ K .
(42)
Pk =
2ak
Let Pk,I be the power
that the industrial consumers get from
UC k, such that
k∈K Pk,I = PI . For simplicity, we consider P1,I = P2,I = · · · PK,I = PI /K. For supply demand
equilibrium, it is required that
xR,n,k , ∀k ∈ K .
(43)
Pk = Pk,I +
k∈K
(41)
Ky2k + K(2ak BR − bk )yk − 2ak (PI + KAR ) = 0.
(44)
The solution to (44) ∀k ∈ K is
−(2ak BR − bk ) ± (2ak BR − bk )2 + 8ak K(PI + KAR )
.
yk =
2K
(45)
From (45), it is clear that (44) possesses a unique positive
solution for the unit price of UC k, ∀k ∈ K , that is
−(2ak BR − bk ) + (2ak BR − bk )2 + 8ak K(PI + KAR )
.
yk =
2K
Note that, this formulation can be easily extended for a more
generic case such as Pi,I /Pj,I = yi /yj , i, j ∈ K , where
j∈K Pj,I = PI .
R EFERENCES
[1] U.S.—Canada Power System Outage Task Force. [Online].
Available:
http://energy.gov/oe/downloads/us-canada-power-systemoutage-task-force-final-report-implementation-task-force,
accessed
May 15, 2015.
[2] H. Farhangi, “The path of the smart grid,” IEEE Power Energy Mag.,
vol. 8, no. 1, pp. 18–28, Jan./Feb. 2010.
[3] A. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich, R. Schober, and
A. Leon-Garcia, “Autonomous demand-side management based on gametheoretic energy consumption scheduling for the future smart grid,”
IEEE Trans. Smart Grid, vol. 1, no. 3, pp. 320–331, Dec. 2010.
[4] Z. Fan, “Distributed demand response and user adaptation in
smart grids,” in Proc. IFIP/IEEE Symp. Integr. Netw. Manage. (IM),
Dublin, Ireland, May 2011, pp. 726–729.
[5] A. J. Conejo, J. M. Morales, and L. Baringo, “Real-time demand
response model,” IEEE Trans. Smart Grid, vol. 1, no. 3, pp. 236–242,
Dec. 2010.
[6] C. Wang and M. D. Groot, “Managing end-user preferences in the
smart grid,” in Proc. ACM Int. Conf. Energy-Efficient Comput. Netw.,
Passau, Germany, 2010, pp. 105–114.
[7] S. Caron and G. Kesidis, “Incentive-based energy consumption scheduling algorithms for the smart grid,” in Proc. 1st Int. Conf. Smart Grid
Commun. (SmartGridComm), Gaithersburg, MD, USA, Oct. 2010,
pp. 391–396.
[8] Q. Zhu and T. Başar, “A multi-resolution large population game framework for smart grid demand response management,” in Proc. 5th
Int. Conf. Netw. Games Control Optim. (NetGCooP), Paris, France,
Oct. 2011, pp. 1–8.
[9] S. Shao, M. Pipattanasomporn, and S. Rahman, “Grid integration of electric vehicles and demand response with customer choice,” IEEE Trans.
Smart Grid, vol. 3, no. 1, pp. 543–550, Mar. 2012.
[10] Z. Fan, “A distributed demand response algorithm and its application to
PHEV charging in smart grids,” IEEE Trans. Smart Grid, vol. 3, no. 3,
pp. 1280–1290, Sep. 2012.
[11] S. Maharjan, Q. Zhu, Y. Zhang, S. Gjessing, and T. Başar, “Dependable
demand response management in the smart grid: A Stackelberg game
approach,” IEEE Trans. Smart Grid, vol. 4, no. 1, pp. 120–132,
Mar. 2013.
MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME
[12] B. Chai, J. Chen, Z. Yang, and Y. Zhang, “Demand response management with multiple utility companies: A two-level game approach,” IEEE
Trans. Smart Grid, vol. 5, no. 2, pp. 722–731, Mar. 2014.
[13] S. Maharjan, Y. Zhang, S. Gjessing, and D. Tsang, “User-centric
demand response management in the smart grid with multiple
providers,” IEEE Trans. Emerg. Topics Comput., [Online]. Available:
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6848763
[14] S. Bu and F. R. Yu, “A game-theoretical scheme in the smart grid with
demand-side management: Towards a smart cyber-physical power infrastructure,” IEEE Trans. Emerg. Topics Comput., vol. 1, no. 1, pp. 22–32,
Jun. 2013.
[15] T. Başar and R. Srikant, “Revenue-maximizing pricing and capacity
expansion in a many-users regime,” in Proc. IEEE 21st Annu. Joint
Conf. IEEE Comput. Commun. Soc. (INFOCOM), vol. 1. New York,
NY, USA, 2002, pp. 294–301.
[16] T. Başar and R. Srikant, “A Stackelberg network game with a large number of followers,” J. Optim. Theory Appl., vol. 115, no. 3, pp. 479–490,
2002.
[17] B. F. Hobbs, C. B. Metzler, and J.-S. Pang, “Strategic gaming analysis
for electric power systems: An MPEC approach,” IEEE Trans. Power
Syst., vol. 15, no. 2, pp. 638–645, May 2000.
[18] C. J. Day, B. F. Hobbs, and J.-S. Pang, “Oligopolistic competition in
power networks: A conjectured supply function approach,” IEEE Trans.
Power Syst., vol. 17, no. 3, pp. 597–607, Aug. 2002.
[19] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory,
2nd ed. Philadelphia, PA, USA: Soc. Ind. Appl. Math., 1998.
[20] Z. Baharlouei and M. Hashemi, “Efficiency-fairness trade-off in
privacy-preserving autonomous demand side management,” IEEE Trans.
Smart Grid, vol. 5, no. 2, pp. 799–808, Mar. 2014.
[21] M. A. Rahman, L. Bai, M. Shehab, and E. Al-Shaer, “Secure distributed
solution for optimal energy consumption scheduling in smart grid,” in
Proc. IEEE 11th Int. Conf. Trust Security Privacy Comput. Commun.
(TrustCom), Liverpool, U.K., Jun. 2012, pp. 279–286.
[22] P. Samadi, A.-H. Mohsenian-Rad, R. Schober, V. W. S. Wong, and
J. Jatskevich, “Optimal real-time pricing algorithm based on utility maximization for smart grid,” in Proc. 1st IEEE Int. Conf. Smart Grid
Commun. (SmartGridComm), Gaithersburg, MD, USA, Oct. 2010,
pp. 415–420.
Sabita Maharjan
(S’09–M’13) received the
M.Eng. in wireless communication from the
Antenna and Propagation Laboratory, Tokyo
Institute of Technology, Tokyo, Japan, in 2008, and
the Ph.D. degree in network and distributed systems
from the University of Oslo, Oslo, Norway, and
the Simula Research Laboratory, Fornebu, Norway,
in 2013.
She is currently a Postdoctoral Fellow with
Simula Research Laboratory. Her current research
interests include wireless networks, network
optimization, security, game theory, smart grid communications, and
cyber-physical systems.
Quanyan Zhu (S’04–M’12) received the B.Eng.
(Hons.) degree in electrical engineering with distinction from McGill University, Montreal, QC, Canada,
in 2006; the M.A.Sc. degree in electrical engineering from the University of Toronto, ON, Canada,
in 2008; and the Ph.D. degree from the University
of Illinois at Urbana-Champaign, Urbana, IL, USA,
in 2013.
He is an Assistant Professor with the Department
of Electrical and Computer Engineering, Polytechnic
School of Engineering, New York University,
Brooklyn, NY, USA. From 2013 to 2014, he was a Postdoctoral Research
Associate with the Department of Electrical Engineering, Princeton University,
Princeton, NJ, USA. His current research interests include optimal control,
game theory, reinforcement learning, network security and privacy, resilient
control systems, and cyber-physical systems.
Dr. Zhu was a recipient of the Natural Sciences and Engineering
Research Council of Canada Graduate Scholarship, the Mavis Future Faculty
Fellowships, and the NSERC Postdoctoral Fellowship. He spearheaded the
INFOCOM workshop on Communications and Control on Smart Energy
Systems, and the Midwest Workshop on Control and Game Theory.
199
Yan Zhang (SM’10) received the Ph.D. degree in
electrical and electronic engineering from Nanyang
Technological University, Singapore, in 2005.
From 2006, he was with Simula Research
Laboratory, Fornebu, Norway. He is currently the
Head with the Department of Networks, Simula
Research Laboratory, and an Adjunct Associate
Professor with the Department of Informatics,
University of Oslo, Oslo, Norway. His current
research interests include wireless networks, cyber
physical systems, and smart grid communications.
Dr. Zhang is a Regional Editor, an Associate Editor, is on the editorial
board, and a Guest Editor of a number of international journals.
Stein Gjessing (M’90) received the Dr.Phil. degree
in computer science from the University of Oslo,
Oslo, Norway, in 1985.
He is currently a Professor of Computer Science
with the Department of Informatics, University of
Oslo, Oslo, Norway, and an Adjunct Researcher with
Simula Research Laboratory, Fornebu, Norway. His
current research interests include network and transport protocols, network resilience, cognitive radio
networks, and the smart grid. His original research
was in the field of object-oriented concurrent programming. He has researched computer interconnects such as Scalable
Coherent Interface (IEEE Standard 1596), and local area network/metropolitan
area networks such as Resilient Packet Ring (IEEE Standard 802.17).
Tamer Başar (S’71–M’73–SM’79–F’83–LF’13)
received the B.S. degree in electrical engineering
from Robert College, Istanbul, Turkey, in 1969,
and the M.S., M.Phil, and Ph.D. degrees in engineering and applied sciences from Yale University,
New Haven, CT, USA, in 1970, 1971, and 1972,
respectively.
He is with the University of Illinois at UrbanaChampaign, Urbana, IL, USA, where he is the
Swanlund Endowed Chair. He is a Professor with the
Department of Electrical and Computer Engineering,
Center for Advanced Study, Urbana. He is a Research Professor with
Coordinated Science Laboratory, Urbana, and the Information Trust Institute,
Urbana. He is the Director of the Center for Advanced Study, University of
Illinois at Urbana-Champaign. His current research interests include stochastic teams, games, and networks; security; and cyber-physical systems. He
has over 700 publications in systems, control, communications, and dynamic
games, including books on noncooperative dynamic game theory, robust control, network security, wireless and communication networks, and stochastic
networked control.
Prof. Başar was a recipient of several awards and recognitions over
the years, including the highest awards of the IEEE Control Systems
Society (CSS), the International Federation of Automatic Control (IFAC), the
American Automatic Control Council (AACC), the International Society of
Dynamic Games (ISDG), the IEEE Control Systems Award, and a number of
international honorary doctorates and professorships. He is a Member of the
U.S. National Academy of Engineering, European Academy of Sciences, and
a Fellow of the IFAC, and the Society for Industrial and Applied Mathematics.
He was the President of the IEEE CSS, the ISDG, and the AACC, and an
Editor-in-Chief of Automatica from 2004 to 2014. He is currently an Editor
of several book series.