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IEEE TRANSACTIONS ON SMART GRID, VOL. 9, NO. 5, SEPTEMBER 2018
4181
Reactive Power Compensation Game UnderProspect-Theoretic Framing
Effects
Yunpeng Wang, Member, IEEE, Walid Saad, Senior Member, IEEE,
Arif I. Sarwat, Senior Member, IEEE,and Choong Seon Hong Senior
Member, IEEE
Abstract—Reactive power compensation is an importantchallenge in
smart power systems. However, in the context ofreactive power
compensation, most existing studies assume thatcustomers can assess
their compensation value (their Var unit)objectively. In this
paper, customers are assumed to make deci-sions that pertain to
reactive power coordination. In consequence,the way in which those
customers evaluate the compensationvalue resulting from their
individual decisions will impact theoverall grid performance. In
particular, a behavioral frame-work, based on the framing effect of
prospect theory (PT), isdeveloped to study the impact of both
objective value and sub-jective evaluation in a reactive power
compensation game. Forinstance, PT’s framing effect allows
customers to optimize asubjective value of their utility which
essentially frames the objec-tive utility with respect to a
reference point. This game enablescustomers to coordinate the use
of their electrical devices tocompensate reactive power. For the
proposed game, both theobjective case using expected utility theory
and the PT consid-eration are solved via a learning algorithm that
converges to amixed-strategy Nash equilibrium. In addition, several
key proper-ties of this game are derived analytically. Simulation
results showthat, under PT, customers are likely to make decisions
that dif-fer from those predicted by classical models. For
instance, usingan illustrative two-customer case, we show that a PT
customerwill increase the adoption of a conservative strategy
(achievinga high power factor) by 29% compared to a conventional
cus-tomer. Similar insights are also observed for a case with
threecustomers.
Index Terms—Smart grid, game theory, prospect theory,framing
effect, reactive power compensation.
Manuscript received October 17, 2015; revised January 29, 2016,
May 8,2016, August 16, 2016, and November 30, 2016; accepted
December 30, 2016.Date of publication January 16, 2017; date of
current version August 21, 2018.This work was supported by the U.S.
National Science Foundation underGrant ECCS-1549894, Grant
OAC-1541105, Grant CNS-1446621, GrantOAC-1541108, and Grant
CNS-1446570. Paper no. TSG-01346-2015.
Y. Wang is with the Electrical and Computer Engineering
Department,University of Miami, Coral Gables, FL 33146 USA, and
also with theDispatching Control Center, State Grid Beijing
Electric Power Company,Beijing 100031, China (e-mail:
[email protected]).
W. Saad is with the Wireless@VT, Bradley Department of
Electrical andComputer Engineering, Virginia Tech, Blacksburg, VA
24061 USA, and alsowith the Department of Computer Science and
Engineering, Kyung HeeUniversity, Seoul, South Korea (e-mail:
[email protected]).
A. I. Sarwat is with the Department of Electrical and
ComputerEngineering, Florida International University, Miami, FL
334174 USA(e-mail: [email protected]).
C. S. Hong is with the Department of Computer Science and
Engineering,Kyung Hee University, Seoul, South Korea (e-mail:
[email protected]).
Color versions of one or more of the figures in this paper are
availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSG.2017.2652846
I. INTRODUCTION
REACTIVE power compensation, commonly known asVar compensation,
aims to improve the efficiency ofdelivering energy in power systems
by reducing transmissionlosses. This has led to much research that
investigates howto control and manage reactive power in a smart
grid [1].Delivering energy over power lines will generate active
andreactive power, and a suitable reactive power compensationcan
decrease energy losses and increase the power factorwhich is
defined as the value of the tangent of the anglebetween active and
reactive power [2]. However, due to theaging of the devices (i.e.,
motors, switches) and the varyingenergy requirements from
end-nodes, smart grid customersmay obtain different power factors
depending on the samedevices that they are previously and currently
using. In par-ticular, in the smart grid, the power company can
requirecustomers to achieve a given power factor for efficient
deliveryof AC power [3]. Recent studies on reactive power
compen-sation have focused on analyzing coordination mechanisms,in
which some customers can support extra reactive power onbehalf of
others, as discussed in [4]–[6].
Reactive power compensation in the smart grid hasbeen
investigated in [7]–[10]. In particular, reactive powercoordination
between customers in a local area has been tech-nically introduced
at the hardware level, using voltage-source-converter technologies
that can both absorb and supply reactivepower, as discussed in
[11]–[13]. To further explore the coordi-nation between customers,
Almeida and Senna [7] proposed anactive-reactive power dispatch
procedure to minimize oppor-tunity costs via the use of marginal
pricing mechanisms tocompensate generators for power provision. The
work in [8]developed a Pareto-optimization based zonal reactive
powermarket model and a hybrid evolutionary approach was appliedin
a competitive electricity market. Xu and Chen [9] studiedthe
asynchronous generator system in a wind farm so as toefficiently
improve Var compensation between different oper-ating moments of
asynchronous generators. Soleymani [10]allowed the customer to bid
reactive power in the energy mar-ket as well as maintain the
voltage stability margin in an IEEE39 bus test system. Other
related approaches for compensatingreactive power are discussed in
[14]–[17].
The works in [7]–[10] and [14]–[17] study reactive
powercompensation using mathematical tools, such as optimiza-tion
and game theory. However, most of these existing worksassume that
customers, as the compensating nodes in thegrid, can objectively
and precisely assess their power factor
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4182 IEEE TRANSACTIONS ON SMART GRID, VOL. 9, NO. 5, SEPTEMBER
2018
compensation, i.e., Var value. However, in practice,
customersmay have subjective perceptions on how they view such
Varvalues as well as on how other customers compensate reac-tive
power. For example, operating inductive equipment (i.e.,motor,
relay, speaker, solenoid, transformer and lamp ballast,or even the
operation of switched capacitor) will change thetangent
relationship between active power and reactive powerand then change
the transmission losses. This tangent value,or power factor, will
impact the active power which, in turn,impacts a customer’s
electricity bill. To properly study suchreactive power compensation
one must therefore account fordifferent customers perceptions on
the economic gains andlosses associated with their bills, which is
directly dependenton the active power. In particular, other
considerations involvethe customers’ opinion on the usage of
electricity, the reduc-tion of transmission losses, the economic
payoffs and the effectof electricity operation and requirement.
Thus, when designingpower factor compensation and coordination
mechanisms, onemust take into account such customer-related human
factors.
The main contribution of this paper is a new
game-theoreticframework to understand how customers can coordinate
theirreactive power compensations while taking into account
theirindividual subjective perceptions on the economic gains
andlosses associated with this coordination. We formulate
thecompensation problem as a static noncooperative game, inwhich a
customer can decide whether or not to act in con-cert with others,
based on reactive power technologies (i.e.,install capacitor and
voltage support), when their inductiveloads change (such as using
speakers, cables or motors in acommunity). In this game, each
customer aims to optimizea Var utility that captures the benefits
of achieving a highpower factor and the associated costs needed to
provide reac-tive power. We allow customers to subjectively
evaluate theirobjective utility which implies that customers can
have dif-ferent ways to measure the economic benefits that they
reapfrom the power compensation game [18]–[21]. Compared torelated
works on smart grids [7]–[17], the contributions ofthis paper
include: 1) in contrast to conventional game utility,we allow
customers to subjectively evaluate their compensa-tion of Var gains
and losses and then explore the probabilityof achieving this
compensation; 2) we design a Var coordina-tion mechanism that
encourages customers to efficiently utilizethe existing
compensating devices and to reach an accept-able power factor
required by the grid; and 3) we developa distributed algorithm,
fictitious play (FP), that is proven toconverge to a mixed-strategy
Nash equilibrium of the game,thus characterizing the solution under
classical game theoryand PT. In simulations, our studies show that
insightful differ-ence between classical and PT evaluations makes
customerschange the frequency with which they participate in
reactivepower compensation, in terms of achieving power factors.
Ourresults also show that zonal compensation can be coordinatedvia
the customers’ perception of their Var gains as opposed totheir Var
losses, which can reduce the overall amount of datacollected during
reactive power compensation.
The remainder of the paper is organized as follows:Section II
presents the system model and formulates thereactive power
compensation as a noncooperative game. In
TABLE ISUMMARY OF NOTATIONS
Section III, we introduce a novel behavioral framework withPT
considerations and in Section IV we use FP to solve thegame.
Simulation results are presented in Section V whileconclusions are
drawn in Section VI.
II. REACTIVE POWER COMPENSATIONMODEL AND GAME FORMULATION
In this section, we first introduce the reactive
powercompensation model and then, formulate a noncooperativegame
between the customers. The main notations are listedin Table I.
A. Reactive Power Compensation Model
Consider a smart grid in which each customer has a
variablereactive power compensation that depends on each
customer’sowned equipment [1], [2]. Let N be the set of all N
customers.In general, for reactive power compensation, a customer
caninstall a capacitor or a voltage/current source to reduce
thepower losses and improve voltage regulation at the load
ter-minals [1]. The power company measures the active power
andgives customers their optimized power factor (PF).
However,existing Var compensation technologies, i.e., using a
capac-itor, cannot always guarantee reaching a fixed power
factor,due to the varying inductive requirement and dynamical
oper-ation, i.e., capacitor switching time. Here, we assume that
acustomer i ∈ N requires active power pi ∈ P and causesreactive
power qi ∈ Q, and thus, its apparent power si ∈ Sis s2i = p2i + q2i
and its current PF is φi ∈ �. Each customerwill compensate the
reactive power and increase its PF to apredefined PF ˜φi ∈ ˜�, as
announced by the power company.
In general, the power factor relates to a phase angle andis
defined by the ratio of active power (or real power) p andapparent
power s as shown in Fig. 1. q and q̃ are, respectively,the actual
reactive power and the required reactive power.
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WANG et al.: REACTIVE POWER COMPENSATION GAME UNDER
PROSPECT-THEORETIC FRAMING EFFECTS 4183
Fig. 1. An illustrative example of reactive power
compensation.
Then, after reactive power compensation, customer i’s
actualcompensation is qci . Here, we note that the power
companywill set a desired, compensation requirement/standard. In
thisregard, we use q̃ci to denote this required/standard
reactivepower compensation for each customer. Due to the deliveryof
AC power, there exists a capacitor between the power lineand
ground. In order to effectively deliver the active powerand
economical consideration, the power company requirescustomers to
reduce their reactive power from q to q̃ via apredefined PF. In
practice, it is hard to directly measure thePF because of the phase
angle between voltage and current.Instead, the company can collect
the energy usage of activeand apparent power, and, then, send to
the customers the tan-gent value of the angle between active and
apparent power,i.e., PF. Hence, we assume that a customer will
require aconstant active power and varying reactive power, such
thatits PF can be easily received in the process of Var
compen-sation [2]. From Fig. 1, we can compute the required
Varcompensation, i.e., customer i’s required compensation q̃ci ,as
follows:
q̃ci(
φi, ˜φi) = pi · tan θi − pi · tan˜θi,
= pi ·√
1 − φ2iφi
− pi ·√
1 − ˜φ2i˜φi
, (1)
where φi = cos θi and ˜φi = cos˜θi. In practice, it is hardto
install new equipment for compensation and the customerscan obtain
a varying PF due to their over/under compensation.Thus, there might
be a need for a Var coordination betweencustomers so as to achieve
optimal local compensation. In thestudied scenario, it is necessary
to devise a mechanism used tounderstand how the customers
compensate reactive power, andhow their usage of inductive loads
impacts the overall system,in terms of Var benefits and costs. For
example, a customercan have some inductive loads, such as speakers
in an event ormotors for pumping water and, thus, its reactive
power require-ment increases, as its PF decreases. In such a case,
it is difficultto install new capacitors; instead, compensating
reactive powerfrom other nodes will be an efficient way to maintain
the PFrequirement. For example, some customers such as electri-cal
vehicles and reactive power plants, can discharge power toincrease
PF for the total Var compensation
∑
i∈N q̃ci . Here, weassume that each customer can obtain/reach a
PF via the exist-ing compensating equipment. In this respect,
customers willhave different power requirements and can achieve a
varyingPF. Thus, the decisions made by customers will depend on
such PF as the operation of the existing devices changes, evenif
they install new compensating devices. Next, we mainlystudy the
competitive coordination between customers, usingtheir reached PFs,
which leads to a game-theoretic setting asdiscussed next.
B. Noncooperative Game Formulation
We analyze the operations of compensating reactive powerbetween
customers using noncooperative game theory [22]. Asprevious
discussed, the customers compensate reactive powerbased on their
existing equipment. Then, they must make adecision on whether to
sell (buy) reactive power to (from)the grid. For the studied model,
the compensation value isthe reactive power difference between
initial PF and the com-pensated PF reached by customers. For
example, the powercompany allows some customers to buy reactive
power fromthose supplying extra Var compensation, as total Var
compen-sation is satisfied [23]–[25]. Thus, customer i can
compensatereactive power qci and reach its PF φ
ci , while the power
company announces the standard PF ˜φi.When coordinating reactive
power compensation, customers
can interdependently determine how much reactive power mustbe
compensated (i.e., Var). We can formulate a static nonco-operative
game in strategic form � = [N , {Ai}i∈N , {ui}i∈N ],that is
characterized by three main elements: 1) the play-ers which are the
customers in the set N , 2) the strategyor action Ai: = (φi, 1],
which represents customer i’sachieved PF, and 3) the utility
function ui of any playeri ∈ N , which captures the benefit-cost
tradeoffs associ-ated with the different choices. In particular, we
hereinafterassume a discrete strategy set. The value of the utility
func-tion achieved by a customer i that chooses an action ai
isgiven by:
ui(ai, a−i) = Bi(ai, a−i) − Ci(ai, a−i), (2)where a−i = [a1, a2,
. . . , ai−1, . . . , ai+1, . . . , aN] is the vectorof actions of
all players other than i, Bi(ai, a−i) is the Varbenefits customer i
obtained if it provides surplus Var to thegrid, and Ci(ai, a−i) is
the cost in Var coordination. In practice,customer i’s action ai
corresponds to deciding on whether tosell or buy reactive power.
Then, compared to the required Varcompensation q̃c in (1), customer
i will compensate reactivepower as follows:
qci (ai) = qci (φi, ai) = pi ·√
1 − φ2iφi
− pi ·√
1 − a2iai
.
Here, before we study the benefits and costs using
reactivepower, we first design a Var coordination exchange
betweencustomers:
Ei(ai, a−i) = qci (ai) −∑
j∈N qcj(
aj)
N. (3)
In particular, Ei(·) is the Var difference between customer iand
the average compensation. In (3), customer i’s compen-sating
quantity qci depends on its action ai, i.e., q
ci (ai) as the
customers’ interactions are captured through qci . Due to
thefact that the total Var compensation
∑
i∈N qci (ai) is affected
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4184 IEEE TRANSACTIONS ON SMART GRID, VOL. 9, NO. 5, SEPTEMBER
2018
by other customers, Ei(·) can have a negative value even if
cus-tomer i’s Var compensation exceeds its standard, i.e., ai >
˜φi.Using (3), the benefit of Var exchange will be:
B(ai, a−i) ={
Ei(ai, a−i) if ai ≥ ˜φi and ∑i∈N
qci ≥∑
i∈Nq̃ci ,
0 otherwise.(4)
Moreover, the cost incurred by customer i is
C(ai, a−i) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
τi(
qci − q̃ci)+ if ai ≥ ˜φi and ∑
i∈Nqci ≥
∑
i∈Nq̃ci ,
−Ei(ai, a−i) if ai < ˜φi and ∑i∈N
qci ≥∑
i∈Nq̃ci ,
qci otherwise,(5)
where (F)+ = max{0, F} and 0 ≤ τi ≤ 1 is a penalty factorthat
weighs the losses of customer i when its Var compensationis greater
than the standard compensation q̃ci .
The utility function in (2) captures both the Var benefit aswell
as the associated costs of having a high PF. Here, when acustomer i
requires large reactive power and decreases its PF,i.e., qci <
q̃
ci or ai < ˜φi, its benefit in (4) is zero and its
utility
is Ei(ai, a−i), while total compensation satisfies power
systemrequirement. In particular, its utility depends on the
reactivepower coordination, and its Var payment would be given
tothose who provide extra reactive power compensation. On theother
hand, if a customer has a high PF, i.e., ai > ˜φi and pro-vides
extra reactive power, it might obtain a benefit due to (3).By using
a high PF, such as by over compensating the PF to0.95, one might
increase the system voltage [2] and then causevoltage oscillation
in the grid. Such an extreme high volt-age resulting from the
overcompensation will endangers theusage of equipment. Thus, a
penalty in (5) limits the extremecase, if all users pursue high
PFs. Without loss generality,we also consider another case upon
which the total reactivepower compensated by all customers cannot
meet the total Varrequirement, i.e.,
∑
i∈Nq̃ci , and assume that all customers lose
their Var values in compensation.
III. PROSPECT THEORY FOR REACTIVEPOWER COMPENSATION
In this section, we first study a conventional game
solutionusing expected utility theory to understand how the
reactivepower compensation game can reach an equilibrium.
Then,using prospect theory, we analyze the impact of
customerbehavior on this game, when customers frame their
utilityvalues with respect to a reference point.
A. Reactive Power Compensation UnderExpected Utility Theory
Owing to the varying active/reactive power requirement(i.e.,
charging/discharging, voltage support, and inductive loadusage),
the PF reached by a customer is not a fixed constant.Also, due to
the continuous operation time for the compensa-tion equipment
(i.e., switching diodes), the PFs reached aftercustomer
compensations are not discrete values but continu-ous. However, the
PF announced by the power company falls
within a discrete sample space whose distribution can be
spec-ified by a probability mass function. Here, we assume
thatcustomers can make probabilistic choices over their
discretestrategies and therefore, we are interested in studying the
gameunder mixed strategies [22] rather than under pure,
determin-istic strategies. Intuitively, a mixed strategy is a
probabilisticchoice that captures how frequently a customer will
choosea given pure strategy. Such assumption of the mixed,
prob-abilistic choices is motivated by the following factors: 1)
aprobability or frequency can represent how often a customerreaches
a power factor, and one can better understand howsuch operations
will occur over a large period of time, and2) a customer would
avoid providing individual power factorcompensation information so
as to compete with its opponents.In this respect, let σ = [σ1, σ2,
. . . , σN] be the vector of allmixed strategies. For customer i,
its σi(ai) ∈ �i is the proba-bility corresponding to its pure
strategy ai ∈ Ai, where �i isthe set of mixed strategy available to
customer i.
In traditional game theory [22], it is assumed that a
playermakes rational decisions. Such rational decisions/actions
implythat, each player will objectively choose its mixed
strategyvector so as to optimize its own utility. Indeed, under the
con-ventional expected utility theory, the utility of each
customeris simply the expected value over its mixed strategies and
thus,for any player i ∈ N , its EUT utility is given by:
UEUTi (σ ) =∑
a∈A
⎛
⎝
N∏
j=1σj
(
aj)
⎞
⎠ui(ai, a−i), (6)
where a is a vector of all chosen/played pure strategies andA =
A1 × A2 × · · · × AN .
B. Reactive Power Compensation Under Prospect Theory
Using the game-theoretic formulation in (6), a player canassess
its expected utility, where customers can objectivelyevaluate Var
payoff under EUT. However, because each cus-tomer evaluates its
economic benefits differently, such asubjective perception will
impact the overall results of thereactive power compensation game.
For example, for a 1 kWhouse usage, the compensation of 100 Var may
be consideredby a customer (i.e., require Var from grid), while
such 100 Varmight not enable a factory with 100 kW power
requirementto buy Var from grid, due to the small impact on PF.
Indeed,due to the different viewpoints on a same Var value, i.e.,
100Var, a small power customer will prefer to compensate
reactivepower, while a large power customer might ignore a
strategythat small customers choose in compensation. Thus,
customerscan make subjective evaluations that result in a deviation
fromthe utility in (2). A customer’s evaluation can consist of
bothgains and losses, when it admits a criterion. In particular,
thegain (loss) is a positive (negative) value in (2), as the
criterionis 0 for EUT. Therefore, the difference between the
subjectiveevaluation and classical, objective utility in (2)
requires one todevelop a new framework that can analyze the
compensationproblem in a smart grid.
To study the customer’s behavior, several empirical stud-ies
[18], [26]–[28] have analyzed how customer behavioraffects a
noncooperative game. In a decision-making process, a
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WANG et al.: REACTIVE POWER COMPENSATION GAME UNDER
PROSPECT-THEORETIC FRAMING EFFECTS 4185
player can evaluate its utility based on a reference, which
rep-resents how this player measures gains and losses with
respectto a certain economic reference or framing point (e.g., a
levelof “wealth”). To capture how losses loom larger than
gainsunder the perception of customers, one can map/transform
theobjective utility functions into subjective value functions
and,this transformation is the so-called framing effect. In
partic-ular, when a customer makes a decision, it will
subjectivelyevaluate its utility, i.e., based on its perception on
the Var unitsof reactive power compensation. Then,
over-compensation andunder-compensation might lead to specific
operational gains orlosses. How such gains and losses are evaluated
will be givenas a new different, customer-dependent utility, i.e.,
uPTi . Thus,taking into account a reference point and how benefits
andcosts are evaluated by each customer, the expression of
theutility will be different from that of EUT in (2).
In order to capture the effect of such evaluation, wewill use
prospect theory [18]. In particular, prospect theoryallows framing
the utilities based on the following criteria:1) Reference point: a
player can evaluate its utility using itsown individual reference
point and such evaluation representshow players act differently via
a possibly similar utility value(i.e., a same $100 can be evaluated
differently by a rich indi-vidual compared to a poor individual);
2) Gain/loss aversion:a player has different attitudes for given a
value when it cor-responds to a gain as opposed to when it
corresponds to aloss; and 3) Diminishing sensitivity: a player is
risk averse inlarge gain values and risk seeking in small losses.
Using thesethree notions, for each player i ∈ N , we can review the
utilityfunction in (2) and construct a behavioral utility function
thatcan allow the players to evaluate both gains and losses,
withthe realistic consideration of a utility reference point
[20]:
uPTi (a) ={
(
ui(a) − u0i(
a0))αi if ui(a) ≥ u0i
(
a0)
,
−ki(
u0i(
a0) − ui(a)
)βi otherwise,(7)
where u0i (a0) = ui(a0) is the utility reference point based
on
the strategy vector a0, the weighting factors αi, βi ∈ (0,
1]respectively capture the gain and loss distortions, and ki >
0is an aversion parameter to tune the impact difference
betweenlosses and gains. In this respect, the utility in (7) is a
desiredS-shape function and it is concave for gains and convexfor
losses [19]. Based on the reference point, smaller αi, βiwill cause
a greater distortion in gain and loss magnitudes.Moreover, when ki
> 1, player i evaluation will have a strongerimpact on its loss
than its gain, termed as the case “loss aver-sion” [29]. Compared
to the EUT utility function in (6), theexpected utility under PT
framing is:
UPTi (σ ) =∑
a∈A
⎛
⎝
N∏
j=1σj
(
aj)
⎞
⎠uPTi (ai, a−i), (8)
where a is the choosing action vector, as mentioned in (6),and
uPTi is the PT pure utility of the action combination.
IV. GAME SOLUTION AND PROPOSED ALGORITHM
Next, we first show the existence of a mixed NE for theproposed
game and then, we prove that using an FP-basedalgorithm customers
can reach a mixed NE in our model.
In (6) or (8), we show the expected utility using the set
ofmixed strategy over the action set Ai of each player i.
Thegame-theoretic solution for both EUT and PT can be
charac-terized by the concept of a mixed-strategy Nash
equilibrium.
Definition 1: A mixed strategy profile σ ∗ is said to be amixed
strategy Nash equilibrium if, for each player i ∈ N , wehave:
Ui(
σ ∗i , σ ∗−i) ≥ Ui
(
σ i, σ∗−i
)
, ∀σ i ∈ �i. (9)Note that the mixed-strategy Nash equilibrium
defined in (9)is applicable for both EUT and PT; the difference
would bein whether one is using (6) or (8), respectively.
Lemma 1: For the proposed reactive power compensationgame, there
exists at least one mixed strategy Nash equilibriumfor PT.
Proof: In the proposed game, a player will assess the objec-tive
utility and follow an EUT strategy using (2) and (6), whileit makes
a PT-based decision in (8) via estimating the subjec-tive tradeoffs
in (7). Under EUT, it has been shown that thereexists at least one
mixed strategy Nash equilibrium in a gamewith a finite number of
players, in which each player canchoose from finitely many pure
strategies. Under PT, both thenumber of players and the number of
their pure strategies donot change. Then, for each pure strategy,
the PT utility onlyreconstructs underlying EUT value; therefore,
there exists atleast one mixed NE in the PT game, as well as its
existencein EUT.
Corollary 1: If no customer reaches the predefined PF inthe
reactive power compensation game, i.e., ai < ˜φi,∀i ∈N , there
exists a unique, pure Nash equilibrium for bothEUT and PT.
Proof: In this case, the total reactive power compensated byall
customers does not meet the total compensation require-ment using
(2), (3), (4) and (5). In particular, ui = −qci (ai)for all
customers. For EUT, we have
∂ui∂ai
= ∂ui∂qci (ai)
· ∂qci (ai)
∂ai
= −pi · 1√1 − a2i · a2i
< 0, (10)
where φi < ai < ˜φi. Thus, player i will follow a
dominantstrategy,1 i.e., amini . Then all EUT customers will choose
theirdominant strategies as a unique, pure NE. Similarly, for
PT
∂uPTi∂ai
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−αipi · 1√1−a2i ·a2i
· (ui(a) − u00(
a0))αi−1
if ui(a) ≥ u0i(
a0)
,
−kiβipi · 1√1−a2i ·a2i
· (u00(
a0) − ui(a)
)βi−1
otherwise.
(11)
Here, both (ui(a) − u00(a0))αi−1 and (u00(a0) − ui(a))βi−1
aregreater than 0. Hence,
∂uPTi∂ai
< 0 and all PT customers willchoose the dominant strategy as
a unique, pure NE strategy. Inparticular, the unique, pure strategy
is to choose the minimumPF strategy, i.e., amini , in the strategy
set.
1A strategy is said to be a dominant strategy for a player if it
yields thebest utility (for that player) no matter what strategies
the other players choose.
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Corollary 2: If all customers exceed the predefined PF inthe
reactive power compensation game, i.e., ai > ˜φi,∀i ∈ N ,and the
penalty factor will not be equal to the ratio of allcustomers minus
one (i.e., without customer i) to the totalnumber of customers,
i.e., τi �= N−1N ,∀i, there exists a unique,pure Nash equilibrium
for both EUT and PT.
Proof: In this case, the utility of player i is:
ui = qci (ai) −∑
i∈N qci (ai)N
− τi(
qci (ai) − q̃ci(
˜φi))
=(
N − 1N
− τi)
qci (ai) −∑
l �=i,l∈N qcl (al)N
+ τiq̃ci(
˜φi)
(12)
Under both EUT and PT, the utility derivatives on player
i’sstrategy are given by:
∂ui∂ai
=(
N − 1N
− τi)
· pi · 1√1 − a2i · a2i
,
∂uPTi∂ai
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(
N−1N − τi
)
· αipi · 1√1−a2i ·a2i
· (ui(a) − u00(
a0))αi−1
if ui(a) ≥ u0i(
a0)
,(
N−1N − τi
)
· kiβipi · 1√1−a2i ·a2i
· (u00(
a0) − ui(a)
)βi−1
otherwise.
(13)
Thus, as τi �= N−1N , both EUT and PT utilities are mono-tonic
function on ai, and all players will choose their
dominantstrategies as a unique, pure NE. In particular, 1) when τi
<N−1
N , the NE is the maximum PF strategy set, 2) whenτi >
N−1N , the NE is the minimum PF strategy set, 3) when
τi = N−1N , the mixed NEs are not unique.The ratio N−1N is a
value that depends only on the num-
ber of customers. It captures how the extra compensation ofone
customer will be shared by the others. Indeed, based onvarious
conditions related to τ , Corollary 2 shows a specificcase in which
customers choose the same NE under bothEUT and PT.
Corollary 1 and Corollary 2 mainly provide the analysiswhen the
total customers’ compensation is strictly less/greaterthan the
total standard compensation. Next, we will study atwo-customer
case, when one does not satisfy its compensa-tion requirement and
exactly requires compensation from theother one.
Corollary 3: For a two-customer reactive power compen-sation
game, if both customers require different power andexceed the
predefined PF ˜φ1 = ˜φ2 = ˜φ using a pair of actions,i.e., Ai =
{v1, v2} (v1 < v2), v1+v22 = ˜φ, then, there exists aunique,
mixed Nash equilibrium for both EUT and PT.
Proof: Without loss of generality, we assume that p1 < p2
inthe following proof. When the players exceed the predefinedPF
using a pair of actions (v1, v2), the total compensation
mustsatisfy
qc1(v1) + qc2(v2) > qc1(
˜φ) + qc2
(
˜φ)
. (14)
To derive this equation, we have
p1V(v1) + p2V(v2) < p1V(
˜φ) + p2V
(
˜φ)
, (15)
where V(x) =√
1−x2x . For another action combination (v2, v1),
we need to compare qc1(v2) + qc2(v1) and qc1(˜φ) + qc2(˜φ).
Inparticular, qc1(v2)+qc2(v1)−qc1(˜φ)−qc2(˜φ) = (p1
+p2)V(˜φ)−p1V(v2) − p2V(v1). Since V(x) is decreasing and convex
in[0, 1], we have
p1V(v2)
p1 + p2 +p2V(v1)
p1 + p2 ≥ V(
p1p1 + p2 v2 +
p2p1 + p2 v1
)
= V(
˜φ + p1 − p22(p1 + p2) (v2 − v1)
)
> V(
˜φ)
. (16)
Thus, qc1(v2)+qc2(v1) < qc1(˜φ)+qc2(˜φ). This inequality
impliesthat, ui(v2, v1) = −qci (ai) for both customers and thus,
theywill lose their reactive power using a pair of actions (v2,
v1).
The above table is the utility of the proposed
noncooperativematrix game. To compare the utility values in the
matrix game,we must first define the notion of best response.
Definition 2: The best response br(a−i) of any storage uniti ∈ N
to the vector of strategies a−i is a set of strategies forseller i
such that:
br(a−i) ={
ai ∈ Ai∣
∣Ui(ai, a−i) ≥ Ui(
a′i, a−i)
, ∀a′i ∈ Ai}
.
Using the concept of best response, for any customer i ∈ N ,when
the other customers’ strategies are chosen as given bya−i, any best
response strategy in br(a−i) is at least as goodas any other
strategy in Ai. Under EUT, since u1(v1, v1) >u1(v2, v1),
customer 1 will pick the action v1 as customer 2chooses v1; for
customer 2, since u2(v1, v2) > u2(v1, v1), itwill pick the
action v2 as customer 1 chooses v1. Under PT,because the framing
utility in (7) only changes the absolutedifference between PT
(pure) utility and EUT (pure) utility, aPT customer does not change
its picking strategy as its oppo-nent holds. Thus, there exists a
unique, mixed NE under bothEUT and PT.
In particular, for EUT, as τ varies, the proposed gamecan have
three cases: 1) when τ is small, we can obtainu1(v2, v2) >
u1(v1, v2) and u2(v2, v2) > u2(v2, v1), thus, thereis a unique,
pure NE (v2, v2); 2) when τ is large, we canobtain u1(v2, v2) <
u1(v1, v2), thus, there is a unique, pureNE (v1, v2); and 3) when τ
is a median value, we can obtainu1(v2, v2) > u1(v1, v2) and
u2(v2, v2) < u2(v2, v1), thus, thereis a unique, proper mixed
NE. For PT, we will have the sameconclusion due to the framing
utility uPTi in (7).
In a practical system, we have the following scenarios:1) when τ
is small, the cost/penalty of providing reactivepower to the grid
is small and, thus, both customers willseek to compensate reactive
power. 2) When τ is large, thecost/penalty of providing reactive
power is large. However,if the total compensation cannot satisfy
the Var requirements(both customers choose a small PF strategy),
the customers’compensation action will be penalized. Thus, these
two cus-tomers will then compensate with each other so as to
avoid
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PROSPECT-THEORETIC FRAMING EFFECTS 4187
such a cost/penalty. 3) When τ is neither too large nor
toosmall, the cost/penalty might be equal to the compensation
ofchoosing the small PF strategy. Thus, customers will have amixed
strategy.
To complete such compensation between two customers,as per
Corollary 3, the grid operator can announce the PFsand based on
wireless technologies, two customers will obtainthe PF information.
Furthermore, to extend the two-by-twointeractions to a general
case, we can divide the area ofinterest into several areas where
two neighbors can have apeer-to-peer compensation. For example,
consider a scenariowith 5 customers in two areas participate in
reactive powercompensation. In particular, Area A involves Customer
A1,A2 and A3 while Area B involves Customer B1 and B2.
Inparticular, PA1 = 70, φA1 = 0.81, PA2 = 30, φA2 = 0.87,PA3 = 40,
φA3 = 0.84 and PB1 = 43, φB1 = 0.86,PB2 = 43, φB2 = 0.88. Then, the
power factors in both areascan be obtained by integrating the
customers’ active powersand factors in each area, i.e., φA = 0.83
and φB = 0.87. Then,these two areas have a pair of power factors
(actions). Thus,for the power company, the customers can be first
divided intotwo areas using a pair of power factor (even if the
number ofcustomers in each area is different).
To solve the compensation game and find an NE using asuitable
algorithm, under both EUT and PT, a fictitious play-based algorithm
is proposed in Table II. In this algorithm,the first stage involves
a simple initialization, in which eachcustomer translates its
action, i.e., the reaching PF, into theVar compensating value.
Then, we propose an iterative processbased on the fictitious play
algorithm [30] for solving the gamein the second learning stage,
under both EUT and PT. Here,the customers will observe others
strategies at time m − 1 soas to update their next strategies at
time m. In this respect, thecustomers will update their beliefs
about each other’s strategiesby monitoring their actions. We let
ai(m) be the action taken byplayer i at time m and σ aii (m), ai ∈
Ai, i ∈ N , be the empiricalfrequency, representing the frequency
that player i has chosenstrategy/action ai until time m. At any
given iteration m, thefollowing FP process is used by a player i to
update its beliefs:
σaii (m) =
m − 1m
· σ aii (m − 1) +1
m· 1{ai(m−1)=ai(m)}. (17)
The strategy chosen at time m is the one that maximizesthe
expected utility with respect to the updated empiricalfrequencies.
This expected utility would follow (6) for EUTand (8) for PT. Thus,
player i can repeatedly choose itsstrategy as:
ai(m) = arg maxai∈Ai
ui(ai, σ−i(m − 1)), (18)
where the utility here is the expected value obtained by playeri
with respect to the mixed strategy of its opponents, whenplayer i
chooses pure strategy ai. If the chosen strategy ai(m)is not a
singleton, there exists at least one strategy, in whichthe utility
of the strategy is the maximum value in a certainiteration. In
particular, if there are more than one strategy thatmaximizes the
utility in (18), we will pick the smaller purestrategy, which makes
economic sense.
TABLE IIREACTIVE POWER COMPENSATION USING PROPOSED FP
For some specific games, it is well known that FP isguaranteed
to converge to a mixed strategy NE [30], as thechoosing frequency
of players’ beliefs converge to a fixedpoint. However, to our
knowledge, such a result has not beenextended to PT, as done in the
following theorem.
Theorem 1: For the proposed reactive power compensationgame, the
proposed FP-based algorithm is guaranteed to con-verge to a mixed
NE under both EUT and PT, if the choosingfrequency of players’
beliefs converges in the FP iterativeprocess.
Proof: The convergence of FP to a mixed strategy NE forEUT under
the convergence of choosing frequency is a knownresult as discussed
in [22] and [30]. For PT, if the choos-ing frequency converges to a
fixed point, this point will be amixed strategy NE. We prove this
case using contradiction asfollows.
Suppose that {σ k} is a fictitious play process that
willconverge to a fixed point, i.e., a mixed strategy σ ∗, afterm =
n0 iterations. By contradiction, we start to assumethat the point σ
∗ = {σ ∗i , σ ∗−i} is not a mixed strategy NE.Then, 1) there must
exist a strategy σ ′i (a′i) ∈ σ ∗i , such thatσi(ai) > 0, σi(ai)
∈ σ ∗ (at least one mixed strategy of playeri is not zero) and
uPTi(
a′i, σ ∗−i)
> uPTi(
ai, σ∗−i
)
, (19)
where ui(ai, σ ∗−i) is the expected utility with respect to
themixed strategies of the opponents of player i, when playeri
chooses pure strategy ai. Here, we can choose a value �that
satisfies 2) 0 < � < 12 |uPTi (a′i, σ ∗−i) − uPTi (ai, σ
∗−i)| as σconverges to σ ∗ at iteration m = n0. Also, 3) since the
FP
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process decreases as the number of iterations n increases,
theutility distance of a pure strategy between two
consecutiveiterations must be less than � after a certain iteration
n0. Forn ≥ n0, the FP process can be written as:
uPTi(
ai, σn−i
) =∑
a∈AuPTi
(
ai, an−i)
σ n−i
≤∑
a∈AuPTi
(
ai, a∗−i)
σ ∗−i + �
<∑
a∈AuPTi
(
a′i, a∗−i)
σ ∗−i − �
≤∑
a∈AuPTi
(
a′i, an−i)
σ n−i
= uPTi(
a′i, σ n−i)
. (20)
In (20), we compute the expected utility of pure strategyai over
the probabilities of all possible cases with respect tothe
utilities. We obtained the first inequality between two
con-secutive iterations as in 3). We obtained the second
inequalityusing 1) and 2). Then, we obtained the third inequality
likethe first one, due to 3). At last we obtained the expected
utilityof pure strategy a′i.
Thus, player i would not choose ai but would ratherchoose a′i
after the nth iteration, mathematically, we willhave σi(ai) = 0.
Hence, we get σi(ai) = 0 which contra-dicts the initial assumption
that σi(ai) > 0; thus the theoremis shown.
Following the convergence to a mixed-strategy Nash equi-librium,
the last stage in the algorithm of Table II is howthe customers
compensate their reactive power in practice andexchange Var between
customers. The actual process of Stage3 is beyond the scope of this
paper and will follow economicand real-world contract
negotiations.
The algorithm in Table II shows how customers act inconcert with
each other for the purpose of reactive powercompensation. Such
process requires the power company toinvestigate customers’
perception on compensation as cap-tured by the rationality
parameters α, β and k in (11).Furthermore, the power company wants
to study the rela-tionship between EUT and PT so as to draft a
contractwith customers. Thus, we next find when the EUT utility
isexactly equal to PT result, i.e., the intersection point
betweenEUT and PT.
Theorem 2: For the proposed reactive power compensationgame, for
every customer i, there exists a threshold k0, suchthat, when ki
< k0, UPTi (σ
PT∗) > UEUTi (σEUT∗), and whenki > k0, UPTi (σ
PT∗) < UEUTi (σEUT∗).Proof: In the proposed game, the utility
derivative on k
can be obtained by (6) and (8). The partial derivative ofUi with
respect to ki depends on the expected utility, whilethe partial
derivative of ui depends on the utility of a purestrategy. In (7),
the pure PT utility is divided as two casesby the reference point;
in (8), the expected PT utility canbe also viewed as a summation of
such two cases. Usingthe PT utility uPTi of a certain pure strategy
set in (7),
we can obtain the PT expected utility, as the NE
configurationdefined in (8),
UPTi (σ ) =∑
a∈A
⎛
⎝
N∏
j=1σj(aj)
⎞
⎠uPTi (ai, a−i),
=∑
a∈A,ui>u0i
⎛
⎝
N∏
j=1σj
(
aj)
⎞
⎠uPTi (ai, a−i)
+∑
a∈A,ui=u0i
⎛
⎝
N∏
j=1σj
(
aj)
⎞
⎠uPTi (ai, a−i)
+∑
a∈A,uiu0i + UPTi (σ ) · 1uiu0i∂ki
+∂UPTi (σ
∗, ki) · 1ui k0,
UPTi (σPT∗) < UEUTi (σEUT∗). This conclusion implies that,
a
small (large) k will increase the gain (loss) evaluation,
andthen increase (decrease) the expected value under PT.
The PT framing effect is captured via three key parameters:we
have three factors, αi, βi and ki. Compared to αi and βi,
thepartial derivative of Ui with respect to ki is more linear.
Thus,it is more practical for the power company to control
localreactive power compensation via ki instead of αi and βi.
Thus,compared to other factors, the linear property of the
aversion
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PROSPECT-THEORETIC FRAMING EFFECTS 4189
Fig. 2. Customers’ mixed-strategies at the equilibrium for both
EUT and PTwith αi = 0.7, βi = 0.6, ki = 2, ∀i.
parameter k provides a useful approach for the power com-pany to
distinguish customers’ perception within the proposedcompensation
game. Theorem 2 analyzes the impact of theaversion parameter k
instead of the weighting factors α, β.This theorem investigates the
intersection between EUT andPT, and thus, it can be used to compute
when the customers’utility is more/less than the standard
compensation. In par-ticular, since the derivative of PT utility on
k is monotonicin (21), customers can have a linear outcome
regarding totheir perception on compensation gains as opposed to
that oncompensation losses.
V. SIMULATION RESULTS AND ANALYSIS
In this section, we run extensive simulations for understand-ing
how customers’ behaviors impact the Var compensationcoordination
under both EUT and PT game. For simulatingthe proposed system, we
consider a local area consisting ofa number of customers equipped
with electrical devices tocompensate reactive power, i.e., switched
capacitors, in whichcustomers’ compensation coordination depends on
their reach-ing PF in (3). To obtain the mixed Nash equilibrium
under bothEUT and PT, we use the proposed algorithm in Table
II.
A. Two-Customer Case
First, we start with the case of two customers. Here, weassume
that Customer 1 and Customer 2’s initial PF are,φ1 = 0.77, φ2 =
0.79, respectively, and their standard PFsare ˜φ1 = ˜φ2 = 0.85.
Also, we assume that the activepower p1 = 2 kW, p2 = 3 kW with a
relative penalty fac-tor τi = 0.7,∀i. In particular, both customers
choose theirstrategy from a two-strategy set Ai = {0.8, 0.9},∀i as
thecompensating PF, and their initial mixed strategy sets areσ
init1 = [0.67 0.33]T and σ init2 = [0.2 0.8]T . In general, if
weneglect the impact of the power factor, a bigger active
powerrequirement will lead to a bigger Var payment. In the
subse-quent simulations, we vary customers’ parameters, i.e., αi,
βi,ki and u0i , to gain insights on the proposed Var
compensationgame under both EUT and PT considerations.
Fig. 2 shows the resulting mixed strategies at both the EUTand
PT equilibria reached via fictitious play. In this figure, we
Fig. 3. The mixed strategy of playing each pure strategy for
both customersas the number of iterations increases.
choose αi = 0.7, βi = 0.6, ki = 2,∀i, and the reference pointu0i
(a
0) = u0i (˜φ1, ˜φ2) for two PT customers. In particular,
thereference point is chosen to coincide with the case in
whichcustomers compensate their reactive power with respect to
astandard PF ˜φ1 = ˜φ2 = 0.85 that is conveyed to customers bythe
power company, such that, u01 = −0.0256, u02 = 0.0256in (7). From
Fig. 2, we can first see that the mixed strate-gies of both
customers are different between PT and EUT.Under PT, both customers
are more likely to choose a highPF compensation action, i.e., a1 =
a2 = 0.9. When a customerchooses a low (high) PF strategy, its
reactive power compensa-tion goes below (exceeds) the standard PF
compensation of thegrid (˜φ = 0.85). Thus, under PT, customers
evaluate their pay-off based on the observation of the standard PF
compensation(i.e., u0i (a
0)), and this will make them avoid taking a low PFaction.
Indeed, since αi > βi and ki > 1,∀i, PT gains (losses)will
decrease (increase) in (7) and then, both customers tendto provide
extra Var to the grid, instead of risking prospec-tive losses if
they use a low PF value. Also, from Fig. 2, wecan see the
difference in Customer 1’s strategies chosen inEUT versus PT is
larger than that of Customer 2. In this case,Customer 2’s active
power is larger than Customer 1. Then,the framing effect on
Customer 2 is smaller than Customer 1,due to the concavity of gains
(i.e., using the large strategy) andthe convexity of losses (i.e.,
using the small strategy) in (7).Thus, Customer 2’s EUT strategy
would lightly increase viaPT considerations.
In Fig. 3, we show the values of the PT mixed strategiesof both
customers (corresponding to Fig. 2), as the numberof iterations
increases. Here, the proposed algorithm in (17)clearly converges to
a mixed NE. The mixed strategy increasesto its maximum quickly
during the first iteration in (17) andthen it decreases as the
number of iterations increases. Theconvergence criterion here is
that the difference between twoconsecutive iterations is small
enough in the proposed two-customer game. From Fig. 3, the
difference between the lasttwo iterations is less than 10−4 in 1
sec (real time, by using amachine with a 2.2GHz processor and 3GB
RAM). In practice,the company can set a suitable stop criterion for
balancing therequired communication delay and the convergence time
ofreactive power compensation.
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2018
Fig. 4. The Var expected utility under both PT and EUT as the
loss distortionparameter β varies.
Fig. 5. The Var expected utility under PT and EUT as the
reference pointu0 varies.
In Fig. 4, we study the costs of compensating reactive powerin
kVar as the loss distortion parameter β1 = β2 = β increases.In
order to singly observe the impact of the loss distortion, wehold
α1 = α2 = 1, k1 = k2 = 1 to cancel the gain distortionand aversion
effect in (7). Also, we assume u0i = 0, i = {1, 2}to neglect the
impact of reference point. Here the expectedutility for both
customers is the costs/payment of Var compen-sation, which is a
negative value in (2). In this figure, we cansee that the expected
utility increases as β increases, implyingthat large β decreases PT
costs in Var compensation. A smallβ increases the PT loss and, for
the proposed prospect model,PT customers will have much cost if
they increasingly evaluatePT losing distortion in (2). In
particular, when β = 1, the PTutility is equal to the EUT utility.
From this figure, we can alsosee that a same losing parameter β
leads to different impactson customers. For example, Customer 2’s
difference betweenPT and EUT is greater than 0.6 while that of
Customer 1is less than 0.6, when β = 0.1. This is because Customer
2requires more active power than Customer 1.
Fig. 5 shows the expected Var value under both EUT and PTas the
reference point u0 varies. For this scenario, we maintainα1 = α2 =
1, k1 = k2 = 1 to eliminate the impact of gaindistortion. Also, we
assume both customers have an equal ref-erence and a same loss
distortion, i.e., u01 = u02 and β1 = β2.
Fig. 6. Customers’ mixed-strategies at the equilibrium for both
EUT and PTin a multi-customer game.
First, we can see that the expected Var cost will increase asthe
reference increases. Because the reference point in (7)
issubtracted from the EUT utility, customer evaluation woulddepend
on a referent level. In essence, a large (active) powerrequirement
leads to more payments in practice, compared tothe EUT case. Also,
Fig. 5 shows that the distortion parame-ter β will have different
impacts on the PT utility as well asthe references u0. For example,
when the reference is a smallvalue, i.e., u0 = 0, the cost
difference between β = 1 andβ = 0.6 is around 0.2, while the
difference is less than 0.1as u0 = 0.5. This shows how the
distortion parameter impactsthe PT utility, which incorporates both
the reference and EUTutility in (7).
B. System With More Than Three Customers
In Fig. 6, we show all mixed strategies in a three-customergame.
We choose α = 0.7, β = 0.6, k = 2,∀i and Ai ={0.8, 0.82, 0.84,
0.86, 0.88, 0.9} for all customers, in which thePT reference point
is u0i (a
0) = u0i (˜φ1, ˜φ2, ˜φ3) and standardPF is 0.85. In particular,
the active power requirement vectorand the initial PF are randomly
chosen, respectively, as p =[2.4 4.1 3]T and φ = [0.77 0.78 0.77]T
. Here, we can seethat the PT mixed strategies are different from
EUT results.Accounting for the PT framing effect in (7), the
referencepoint u0 = [ − 0.1241 0.1229 0.0012]T and the
distortionparameters allow customers to evaluate more on the
losses(β, k) than the gains (α). Thus, all customers would want
toincrease their high PF strategy due to the fact that they
observea prospective losing tendency in practice. Moreover,
comparedto Fig. 2, we can see that the framing effect in (7) can
change apure strategy under PT to a more mixed strategy, even
changethe pure strategy to another pure strategy.
Using the same parameters as in Fig. 6, Fig. 7 shows thetotal
utility of all customers for the proposed multi-customergame, as
the aversion parameter k1 = k2 = k varies. In thisfigure, we can
first see that PT utility decreases as k increases.In (7), the
aversion parameter k can capture customers’ percep-tion on
evaluation, i.e., gains and losses in practice. This
resultcorresponds to Theorem 2, such that if customers
increaseevaluation on the gains compared to losses (i.e., k <
1), their
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WANG et al.: REACTIVE POWER COMPENSATION GAME UNDER
PROSPECT-THEORETIC FRAMING EFFECTS 4191
Fig. 7. Customers’ total utility under both EUT and PT, as the
framingsensitive varies.
total PT utility will be greater than that of EUT, and
viceversa. Indeed, we can see that EUT and PT utilities intersectat
a point, i.e., k = 1.04. The intersection point is not exactlyequal
to 1 due to the fact that the gain distortion parame-ter α is
greater than β which implies that customers have asmaller gaining
distortion than losing distortion. Thus, thereneeds to be a high
aversion parameter (i.e., k > 1) to bal-ance the distortion
difference between gain and loss. Second,compared the EUT and PT
results, we show the total utilityunder optimization (green line).
The optimization solution isthe total maximum utility of a pure
strategy, while EUT andPT show the expected utility over all
strategies. From this fig-ure, we can see that the performance of
EUT is always lessbelow that of the centralized optimization
approach when thepower company seeks to maximize the total utility.
Althoughthe total optimal utility is greater than the result under
EUTor PT, the optimal solution cannot capture customer behaviordue
to their independence. Thus, we can use to counter thecustomers’
behaviors and design a decentralized, customer-aware optimal
solution for reactive power compensation. Last,this figure shows
how reactive power will be compensated viazonal/local customers’
coordination. For example, comparedto the other distortion
parameters (i.e., α, β) that were evalu-ated in Fig. 4, we can see
a more linear curve as the aversionparameter k varies in Fig. 7.
For example, to collect the PTbehavior, the power company needs to
investigate how a PTcustomer may frame its objective gains via α,
and how thiscustomer may frame its objective loss via β. In this
case,the power company needs to find two types of information.For
the aversion parameter k, the power company can directlycollect the
information how a customer views the PT gainsas oppose to the PT
losses. This implies that, reactive powercompensation can be
coordinated via the customers’ percep-tion of operational gains as
opposed to losses, thus reducingthe amount of collected data.
Table III shows all mixed strategies of a system with 7customers
under both EUT and PT. In this case, we assumethat the active power
of all customer is 2.4kW with a three-strategy set, i.e., pi =
2.4,Ai = {0.86, 0.87, 0.88},∀i. Theirstandard PFs and initial mixed
strategy sets are ˜φi = 0.85,
TABLE IIIALL MIXED NE STRATEGY OF 7 CUSTOMERS
UNDER BOTH EUT AND PT, (τi = 67 , ∀i)
Fig. 8. The average utility at the equilibrium for both EUT and
PT as Nincreases.
σ initi = [0.33 0.33 0.33]T ,∀i, respectively. In line
withCorollary 2, we first set τi = 0.5. Since τi = 0.5 < 67 ,
allcustomers’ probabilities on the maximum (pure) strategy
arearound 1 and, all customers will choose 0.88 as their pure
com-pensation strategy under both EUT and PT. Similarly, whenτi =
0.9 > 67 , all customers will choose 0.86 as their com-pensation
pure strategy because all customers’ probabilities onthe minimum
(pure) strategy approach to 1 under both EUTand PT. Furthermore, to
study the difference between EUTand PT, we set τi = 67 so as to
guarantee a more “mixed”case. In Table III, the mixed strategies of
all PT customers arenot the same as the EUT strategies. The
difference betweenEUT and PT mainly pertains to their initial
strategies (i.e.,φ1 = 0.79, φ5 = 0.77) and their participating
order in the com-pensation game (i.e., we use a sequential
algorithm in (17).Last but not least, as the number of customers
increases, thecomplexity of finding an NE via FP can increase. For
exam-ple, a game with each customer having 3 strategy will have37 =
2187 combinations. Increasing the number of customerto 10 will have
69049 combinations, which can be too com-plex to solve. To solve
games with number of customers, wecan divide the system into
multiple, smaller areas and then,within each area applying the
proposed scheme to obtain an“area” NE. The “area” NE can be
considered as a player in alarge number of customer game.
In Fig. 8, we can see that the average utility decreases asthe
number of customers varies. By assuming pi = 2.4, ˜φi =0.85, σ
initi = 1/N,Ai = {0.86, 0.87},∀i, an increasing num-ber of
customers will lead to more interactions as we assumethe benefit of
Var exchange refers to all customers in (4).
-
4192 IEEE TRANSACTIONS ON SMART GRID, VOL. 9, NO. 5, SEPTEMBER
2018
Under the strategy set Ai = {0.86, 0.87}, the cost of
eachcustomer (i.e., τi(qci − q̃ci )+) in (5) is constant. For a
smallsystem (N ≤ 3), the customers might have significantly
largerbenefits to share than in the four-customer case. When N ≥
4,the benefits shared by all customers might decrease while thecost
of each customer remains constant. Then, for the proposedmodel,
thus, the shared benefits rely on both the number ofcustomers and
their initial active power and, the shared bene-fits will not
always be decreasing. Thus, due to the differenceof customers’
initial points, at N = 3, the utility curve has anon-monotonic.
VI. CONCLUSION
In this paper, we have introduced a novel game-theoreticapproach
for modeling the reactive power compensationbetween local customers
via Var coordination. We have for-mulated the Var compensation
process as a noncooperativegame between customers, in which
customers have subjec-tive perception on their economic losses and
gains. Using theframework of prospect theory, we have modeled such
per-ceptions and analyzed their impact on the system. To solvethe
proposed game, we have proposed a fictitious play-basedalgorithm
that is shown to converge to an equilibrium pointunder a PT
scenario. Simulation results have shown that theuse of
prospect-theoretic considerations can provide insightfulinformation
on the behaviors of customers engaged in reactivepower
compensation.
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Yunpeng Wang (S’12–M’15) received the bache-lor’s degree from
the North China Electric PowerUniversity, Beijing, China, in 2009,
the master’sdegree from the Stevens Institute of
Technology,Hoboken, NJ, USA, in 2012, and the Ph.D. degreefrom the
University of Miami, Coral Gables, FL,USA, in 2015, all in
electrical engineering. He iscurrently working at the State Grid
Corporation ofChina, Beijing, China. His primary research
interestsrevolve around the smart grid with a particular focuson
the design of microgrids and the control and oper-
ation of UHV power networks, with emphasis on game theory
applied topower systems.
-
WANG et al.: REACTIVE POWER COMPENSATION GAME UNDER
PROSPECT-THEORETIC FRAMING EFFECTS 4193
Walid Saad (S’07–M’10–SM’15) received thePh.D. degree from the
University of Oslo in 2010.He is currently an Associate Professor
with theDepartment of Electrical and Computer Engineering,Virginia
Tech, where he leads the Network Science,Wireless, and Security
Laboratory, Wireless@VTResearch Group. His research interests
includewireless networks, machine learning, game
theory,cybersecurity, unmanned aerial vehicles, and cyber-physical
systems. He was a recipient of the NSFCAREER Award in 2013, the
AFOSR Summer
Faculty Fellowship in 2014, the Young Investigator Award from
the Officeof Naval Research in 2015, the 2015 Fred W. Ellersick
Prize from theIEEE Communications Society and of the 2017 IEEE
ComSoc Best YoungProfessional in Academia Award, and Six Conference
Best Paper Awardsat WiOpt in 2009, ICIMP in 2010, IEEE WCNC in
2012, IEEE PIMRCin 2015, IEEE SmartGridComm in 2015, and EuCNC in
2017 for whichhe was author/co-author. From 2015 to 2017, he was
named the StephenO. Lane Junior Faculty Fellow at Virginia Tech
and, in 2017, he wasnamed College of Engineering Faculty Fellow. He
currently serves as anEditor for the IEEE TRANSACTIONS ON WIRELESS
COMMUNICATIONS, theIEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE
TRANSACTIONSON MOBILE COMPUTING, and the IEEE TRANSACTIONS ON
INFORMATIONFORENSICS AND SECURITY.
Arif I. Sarwat (M’08–SM’16) received the mas-ter’s degree from
the University of Florida in2005 and the Ph.D. degree from the
Universityof South Florida in 2010. He is currently anAssociate
Professor and the Director of FPL-FIUSolar Research Facility,
Department of Electricaland Computer Engineering, Florida
InternationalUniversity, where he leads the Energy Power
andSustainability Group. He worked with Siemens forover nine years,
winning three recognition awards.His research interests include
smart grids, plugin
hybrid and electric vehicle (PHEV & EV Systems), high
penetration renewablesystems, grid resiliency, large scale data
analysis, advance metering infrastruc-ture, smart city
infrastructure, and cyber security. He was a recipient of theNSF
CAREER Award in 2016 and multiple federal and industry
researchawards, the conference best paper awards at the resilience
week in 2017 forwhich he was author/co-author and a Journal Best
Paper Award in 2016 fromthe Journal of Modern Power Systems and
Clean Energy, the Faculty Awardfor Excellence in Research &
Creative Activities in 2016, the College ofEngineering and
Computing Worlds Ahead Performance in 2016, and theFIU TOP Scholar
Award in 2015. He has been the Chair of IEEE MiamiSection VT and
Communication since 2012.
Choong Seon Hong (S’95–M’97–SM’11) receivedthe B.S. and M.S.
degrees in electronic engineeringfrom Kyung Hee University, Seoul,
South Korea, in1983 and 1985, respectively, and the Ph.D.
degreefrom Keio University, Minato, Japan, in 1997. In1988, he
joined KT, where he worked on broad-band networks as a Member of
Technical Staff.In 1993, he joined Keio University. He workedfor
the Telecommunications Network Laboratory,KT, as a Senior Member of
Technical Staff andthe Director of the Networking Research Team
in
1999. Since 1999, he has been a Professor with the Department
ComputerScience and Engineering, Kyung Hee University. His research
interestsinclude future Internet, ad hoc networks, network
management, and net-work security. He has served as the General
Chair, a TPC Chair/Member,or an Organizing Committee Member for
international conferences suchas NOMS, IM, APNOMS, E2EMON, CCNC,
ADSN, ICPP, DIM, WISA,BcN, TINA, SAINT, and ICOIN. He is currently
an Associate Editor ofthe IEEE TRANSACTIONS ON NETWORK AND SERVICE
MANAGEMENT, theINTERNATIONAL JOURNAL OF NETWORK MANAGEMENT, and the
Journalof Communications and Networks, and an Associate Technical
Editor of theIEEE Communications Magazine. He is a member of ACM,
IEICE, IPSJ,KIISE, KICS, KIPS, and OSIA.
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