reactive power compensation game under prospect-theoretic...

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.

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


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


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.


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


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


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


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.


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


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).


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.


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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