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IEEE TRANSACTIONS ON SMART GRID 1

Decentralized Reactive Power CompensationUsing Nash Bargaining SolutionHung Khanh Nguyen, Hamed Mohsenian-Rad, Senior Member, IEEE,

Amin Khodaei, Senior Member, IEEE, and Zhu Han, Fellow, IEEE

Abstract—We consider a distributed reactive power compen-sation problem in a distribution network in which users locallygenerate reactive power using distributed generation units to con-tribute to the local voltage control. We model and analyze theinteraction between one electric utility company and multipleusers by using the Nash bargaining theory. On one hand, usersdetermine the amount of active and reactive power generation fortheir distributed generation units. On the other hand, the elec-tric utility company offers reimbursement for each user based onthe amount of reactive power dispatched by that user. We firstquantify the benefit for the electric utility company and users inthe reactive power compensation problem. Then we derive theoptimal solution for the active and reactive power generation, aswell as reimbursement for each user under two different bar-gaining protocols, namely sequential bargaining and concurrentbargaining. Numerical results show that both the electric util-ity company and users benefit from the proposed decentralizedreactive power compensation mechanism, and the overall systemefficiency is improved.

Index Terms—Reactive power compensation, demand sidemanagement, game theory, Nash bargaining solution.

NOMENCLATURE

Indices/Sets

n Index of users/nodes.g Superscript for generation.d Superscript for demand.N Set of users/nodes.

Manuscript received April 14, 2015; revised August 5, 2015 andOctober 29, 2015; accepted November 5, 2015. The work of H. K. Nguyenand Z. Han was supported by the U.S. National Science Foundation underGrant CMMI-1434789. The work of H. Mohsenian-Rad was supported bythe U.S. National Science Foundation under Grant ECCS 1253516 and GrantECCS 1307756. The work of A. Khodaei was supported in part by theU.S. National Science Foundation, and in part by the Division of Civil,Mechanical, and Manufacturing Innovation under Grant CMMI-1434771.Paper no. TSG-00404-2015.

H. K. Nguyen and Z. Han are with the Department of Electrical andComputer Engineering, University of Houston, Houston, TX 77004 USA(e-mail: [email protected]; [email protected]).

H. Mohsenian-Rad is with the Department of Electrical Engineering,University of California at Riverside, Riverside, CA 92521 USA (e-mail:[email protected]).

A. Khodaei is with the Department of Electrical and ComputerEngineering, University of Denver, Denver, CO 80210 USA (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.2015.2500729

Parameters

pdn Active power demand of user.

qdn Reactive power demand of user.

rn + jxn Complex impedance of the link betweennode n and node n + 1.

λ Unit price of active power.π Cost parameter for compensating reactive

power.N Number of nodes or users in the network.sn Maximum apparent power that DG unit n can

support.r̂n Cumulative resistance from the substation to

node n.x̂n Cumulative reactance from the substation to

node n.pg

n,max Maximum available active power of the DGunit n.

pg0n Amount of active power that user n generates

when user n does not participate in reactivepower compensation.

C0n Cost for buying remaining active power from

the electric company when user n does notparticipate in reactive power compensation.

Cn Payment to the electric utility company if usern participates in reactive power compensation.

Vn User n’s payoff.U The electric utility company’s payoff.

Variables

pgn Active power generation of user n.

qgn Reactive power generation of user n.

zn Reimbursement that user n receives from theelectric utility company.

I. INTRODUCTION

REACTIVE power compensation is necessary in powersystems in order to assure power quality and voltage

support [1]–[3]. In traditional power systems, reactive poweris provided by synchronous generators or shunt capacitorbanks installed at specific locations of the distribution net-works [4]–[6]. However, this centralized reactive power com-pensation can be costly and it can also increase the power losson transmission and distribution lines [7], [8]. In addition, dueto the increasing number of inductive residential appliances,such as microwaves, washing machines, air conditioners, and

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2 IEEE TRANSACTIONS ON SMART GRID

refrigerators, there is a need to explore new reactive powercompensation options at distribution level [9]–[13].

With the introduction of distributed generation (DG), analternative approach to compensate reactive power is to utilizethe power electronics interfaces at DG units, see [14], [15].In [16], the optimal control schemes for reactive powerdispatch to achieve the trade-off between distribution lossreduction and voltage variation minimization using distributedphotovoltaic generators are proposed. A stochastic optimiza-tion model for real and reactive power management underthe uncertainty of solar power generation has been proposedin [17] to maximize the sum utility of users while maintainingthe voltage at every node at safe levels. In [18], a novel reac-tive power management strategy under stochastic parametersof the system has been proposed to allow system operators todetect the reactive power vulnerable part of the power grid.The work in [19] develops a convex optimization frameworkfor reactive power compensation. The authors in [20] pro-pose an online reactive power control scheme considering thestochastic nature of reactive demand and renewable generation.

Although end-user reactive power compensation via DGunits has been advocated as a viable solution to achieve highsystem efficiency [10], [11], [21], [22], users will not activelyparticipate in generating reactive power unless they have properfinancial incentives from electric utility companies or distri-bution network operators. Recent research has applied gametheory to propose incentive mechanisms to encourage users toparticipate in power management systems in the smart grid.For instance, the works in [23] and [24] investigate the demandside management problem as a noncooperative game and pro-pose a smart pricing model to encourage users to participate inenergy consumption scheduling program. In [25], the authorsformulate the reactive power compensation as a Stackelberggame and derive a pricing scheme to encourage plug-in electricvehicles in generating and consuming reactive power.

In this paper, we consider the problem of controlling reac-tive power generation from DG units in a radial network,and focus on economic incentives that a utility companyneeds to provide for users to achieve high system efficiency.Specifically, each user individually controls its DG unit todetermine the amount of active and reactive power genera-tion to partially satisfy its own demand. Based on the amountof reactive power compensation, the electric utility companyoffers reimbursement to users as financial encouragement. Themain contribution of this paper lies in the fact that we modeland analyze the interaction between the electric utility com-pany and users using the Nash bargaining theory [26]. Wequantify the benefit for users and electric utility company incollaborative reactive power compensation. The closed-formoptimal solutions for reactive and active power generation aswell as the amount of reimbursement offered to users arederived under both sequential bargaining and concurrent bar-gaining. We also investigate the connections of the optimalsolutions with the social welfare of the network.

The remainder of this paper is organized as follows.A model of decentralized reactive power compensation isformulated in Section II. We solve the reactive powercompensation problem using the Nash bargaining theory

Fig. 1. A distribution network with local reactive power compensation.

under sequential and concurrent bargaining protocols inSections III and IV, respectively. The simulation results areprovided in Section V. Section VI presents our conclusions.All analytical proofs are relegated to the Appendix.

II. SYSTEM MODEL

In this section, we describe the topology of the power dis-tribution network considered for reactive power compensationand define the payoff functions for users and utility company.

A. System Description

Without loss of generality, consider a linear distributionnetwork as in Fig. 1. The set of nodes is denoted byN = {1, . . . , N}. The reference node is connected to the sub-station, denoted by node 0. Let Pn and Qn represent active andreactive power flowing down the network from node n to noden + 1. At each node n, the complex power demand is denotedby pd

n + jqdn, where pd

n and qdn are active and reactive power

demand, respectively. The exact values of pdn and qd

n depend onthe demand condition of each node over time. However, sinceour focus in this paper is on per-time-slot analysis, we assumethat the demand of each node remains unchanged during theperiod of study. The length of each time slot is a design param-eter. In general, a shorter time slot may improve the accuracyin predicting demand; however, such improvement may alsocome at the expense of increasing computational complexitydue to the need for solving the optimal reactive power com-pensation problem more frequently. In this paper, the lengthof each time slot is assumed to be equal to the length of eachtime slot of power price so that power price does not changeduring the period of study. However, our analysis can applyto any particular choice for the length of time slots.

We further assume that each node n ∈ N has a DG unit, e.g.,a solar panel or wind turbine, which is capable of generatingpg

n active and qgn reactive power respectively. The amount of

power that a DG unit can generate must satisfy the followingconstraint:

(pg

n

)2 + (qg

n

)2 ≤ s2n, (1)

where sn is the maximum apparent power that the DG unit cansupport. Note that, our model can be applied to systems thathave only a subset of nodes have DG units by setting sn = 0


NGUYEN et al.: DECENTRALIZED REACTIVE POWER COMPENSATION USING NASH BARGAINING SOLUTION 3

for any node which is not equipped with a DG unit. For thedistribution network illustrated in Fig. 1, the power flow andvoltage for each link between nodes n and n + 1 satisfies thefollowing equations [7], [8]

Pn+1 = Pn − rnP2

n + Q2n

V2n

− pdn+1 + pg

n+1, (2)

Qn+1 = Qn − xnP2

n + Q2n

V2n

− qdn+1 + qg

n+1, (3)

V2n+1 = V2

n − 2(rnPn + xnQn) +(

r2n + x2

n

)P2n + Q2

n

V2n

, (4)

where rn + jxn is the complex impedance of the link betweennode n and node n + 1, Vn is the voltage at node n. Since thequadratic terms in (2), (3), and (4) are relatively small [7], [8],we can approximate (2), (3), and (4) as linear equations as

Pn+1 = Pn − pdn+1 + pg

n+1, (5)

Qn+1 = Qn − qdn+1 + qg

n+1, (6)

Vn+1 = Vn − (rnPn + xnQn)/V0. (7)

Since V0 is constant, we can absorb it into the voltage at eachnode and define the voltage variation �Vn between node nand node n + 1 as

�Vn = Vn+1 − Vn = −(rnPn + xnQn). (8)

Then, the total voltage deviation of the system with respect tothe reference bus can be calculated as

N−1∑

n=0

|�n| =N−1∑

n=0

(rnPn + xnQn)

=N∑

n=1

[(pd

n − pgn

)(r0 + . . . + rn−1)

+(

qdn − qg

n

)(x0 + . . . + xn−1)

]

=N∑

n=1

[(pd

n − pgn

)r̂n +

(qd

n − qgn

)x̂n

](9)

where r̂n = ∑n−1k=0 rk and x̂n = ∑n−1

k=0 xk, which are thecumulative resistance and reactance from the substation tonode n.

B. User’s Payoff Modeling

Each user has a DG unit that can generate both activeand reactive power to satisfy its demand. However, it candecide not to generate reactive power if there is no incentivefrom electric utility company for reactive power compensation.Next, we quantify the benefit that each user can receive if itdecides to participate in reactive power dispatch. We focus onthe benefit for each user from reactive power dispatch due tothe reduction of payment to the electric utility company.

We first calculate the payment of each user n ∈ N if itdecides not to participate in reactive power compensation. Inthis case, user n only generates active power from its DG unitto meet its own load demand pd

n. Let pgn,max be the maximum

available active power of the DG unit n, which depends onsolar irradiance and temperature for solar panels or wind speed

for wind turbines. Based on the amount of available capacitypg

n,max, user n will generate pg0n amount of active power to

serve its own active power demand. The amount of pg0n is

determined by user n as follow

pg0n � min

{pd

n, pgn,max

}. (10)

Note that, in (10), user n can predict its power demand pdn

and maximum available active power generation pgn,max at the

current period of study. Therefore, pg0n is a fixed parameter and

not a control variable in this problem. Moreover, equation (10)means that user n only generates active power to satisfy itsdemand and does not inject surplus active power back to grideven if there is active power available. The cost for user nto buy the remaining active power from the electric utilitycompany is obtained as

C0n = λ

(pd

n − pg0n

), (11)

where λ is the unit price of active power at the current periodof study. The unit price of active power may vary during theday. However, we assume that λ will be unchanged duringeach decision making time slot. Given the price informationfrom the electric utility company, each user n can calculate itspayment to the electric utility company since pd

n and pg0n do

not change during the period of study.To incentivize reactive power compensation, the electric

utility company offers a reimbursement zn to user n for itsamount of qg

n reactive power dispatch. Therefore, the usern’s payment to the electric utility company if participatingin reactive power compensation can be calculated as the costof purchasing remaining active power minus reimbursement:

Cn = λ(

pdn − pg

n

)− zn. (12)

In (12), the remaining amount of active power that user npurchases from the electric utility company is (pd

n −pgn). Also,

by generating qgn reactive power, user n may potentially reduce

the amount of active power pgn, constrained by (1). Moreover,

the amount of active power generation pgn cannot be greater

than the amount of active power generation in case the userdoes not participate in reactive power compensation:

pgn ≤ pg0

n . (13)

User n controls the amount of active and reactive power gen-eration {pg

n, qgn} so that it can reduce payment to the electric

utility company.Then we define the user n’s payoff as the payment reduc-

tion when compensating reactive power for the electric utilitycompany, denoted by Vn

Vn(pg

n, qgn, zn

) = C0n − Cn = zn − λ

(pg0

n − pgn

). (14)

From (14), we realize that when user n does not participatein reactive power compensation, its payoff is V0

n = 0.

C. Electric Utility Company’s Payoff Modeling

By offering financial incentive for users to locally generatereactive power, the utility company can reduce the amount of



remaining reactive power it has to provide, and thus reduce thecost for reactive power compensation. Let Qnocomp

inj = ∑Nn=1 qd

n

and Qinj = ∑Nn=1(q

dn − qg

n) be the total amount of reactivepower that node 0 has to inject into the distribution feederto satisfy all reactive power demand of the overall system,without and with reactive power compensation from users,respectively. We further define fff (Q) = πQ as the cost forthe electric utility company to compensate Q units of reac-tive power at node 0, where π is a constant parameter [20].Then the saving cost for reactive power compensation can becalculated as

�fcost � fff(

Qnocompinj

)− fff

(Qinj

) = π

N∑

n=1

qgn. (15)

Moreover, by locally compensating for reactive power, thetotal voltage deviation along the network can be reduced.Based on (9), we first determine the total voltage deviation ofthe network in case users do not participate in local reactivepower compensation as

N∑

n=1

|�on| =

N∑

n=1

[r̂n

(pd

n − pg0n

)+ x̂nqd

n

]. (16)

From (9) and (16), the reduction of voltage deviation by locallycompensating for reactive power can be computed as

�fvol �N∑

n=1

∣∣�on

∣∣−N∑

n=1

|�n| =N∑

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

].

(17)

Then the electric utility company’s payoff can be defined as thesaving cost for reactive power compensation and the reductionof voltage deviation along the distribution network

U(qqqg,pppg, zzz) =

(

�fcost −N∑

n=1

zn

)

+ α�fvol

= π

N∑

n=1

qgn −

N∑

n=1

zn

+ α

N∑

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]. (18)

where α is a positive weighted parameter to capture the trade-off between saving cost and voltage deviation.

D. Network Social Welfare Maximization

We define the social welfare as the aggregate payoff ofelectric utility company and users in the network

�(pppg,qqqg, zzz

) = U(qqqg, zzz)+

N∑

n=1

Vn(pg

n, qgn, zn

)

= π

N∑

n=1

qgn − λ

N∑

n=1

(pg0

n − pgn

)

+ α

N∑

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]

� �(

pppg,qqqg). (19)

Then, the social welfare maximization problem can be formu-lated as

max �(pppg,qqqg) (20)

s.t.{qg

n, pgn

} ∈ Xn,∀n ∈ N ,

where Xn is the set of feasible {qgn, pg

n} of user n, which isdefined as

Xn �{

qgn, pg

n

∣∣ qgn ∈

[0, qd

n

], pg

n ∈[0, pg0

n

], constraint (1)

}.

Given the complete knowledge and centralized control ofthe network, we can solve the network social welfare max-imization problem in (20) to obtain the optimal active andreactive power generation. However, this requirement is diffi-cult to fulfill in practice due to the distributed nature of thenetwork topology. Moreover, solving (20) is not able to deter-mine the amount of reimbursement for users. Therefore, in thenext section, we use the Nash bargaining theory to determinethe optimal solutions for power generation and reimbursementin a distributed fashion.

III. SEQUENTIAL BARGAINING

In this section, we first analyze the Nash bargaining solution(NBS) of the decentralized reactive power compensation undersequential bargaining protocol for a simple network consistingof one electric utility company and one user. Then we use thisresult to generalize the solution for multi-user network.

A. One-to-One Nash Bargaining Solution

In this subsection, we determine the NBS for a simple two-person bargaining, one electric utility company and one user.Let Zn be the sets of feasible zn

Zn � {zn | zn ∈ [0,+∞)}. (21)

Then the NBS is the solution of the following optimizationproblem

max[U(qg

n, pgn, zn

)− U0]

·[Vn(qg

n, pgn, zn

)− V0n

](22)

s.t.{qg

n, pgn

} ∈ Xn, zn ∈ Zn,

where U0 and V0n are the disagreement points of the electric

utility company and user n, respectively. From (18), we cancalculate the disagreement point of the electric utility com-pany, which is the electric utility company’s payoff withoutreactive power compensation U0 = U(0, 0, 0) = 0. Then wecan explicitly express the optimization problem (22) as

max[πqg

n − zn + αr̂n

(pg

n − pg0n

)+ αx̂nqg

n

]

·[zn − λ

(pg0

n − pgn

)](23)

s.t. 0 ≤ qgn ≤ qd

n,

0 ≤ pgn ≤ pg0

n ,(pg

n

)2 + (qg

n

)2 ≤ s2n,

zn ≥ 0.

By solving the optimization problem (23), we obtain theNBS for the two-person bargaining as the following theorem.



Theorem 1: The NBS (qg∗n , pg∗

n , z∗n) for the one-to-one bar-

gaining is• If (pg0

n )2 + (qdn)

2 ≤ s2n

qg∗n = qd

n, (24)

pg∗n = pg0

n , (25)

z∗n = 1

2πqd

n + α

2x̂nqd

n. (26)

• If (pg0n )2 + (qd

n)2 > s2

n

qg∗n = min

⎧⎨

⎩

(π + αx̂n

)sn

√(λ + αr̂n

)2 + (π + αx̂n

)2, qd

n

⎫⎬

⎭, (27)

pg∗n = min

⎧⎨

⎩

(λ + αr̂n

)sn

√(λ + αr̂n

)2 + (π + αx̂n

)2, pg0

n

⎫⎬

⎭, (28)

z∗n = λ

(pg0

n − pg∗n

)+ 1

2

[πqg∗

n − λ(

pg0n − pg∗

n

)

+ α[r̂n

(pg∗

n − pg0n

)+ x̂nqg∗

n

]].

(29)

Proof: See Appendix A.From the result in Theorem 1, we realize that the reimburse-

ment covers the cost incurred by reducing the active powergeneration λ(pg0

n −pg∗n ) and a half of its portion of social wel-

fare contributed to the system, i.e.,1

2[πqg∗

n − λ(pg0n − pg∗

n ) +α[r̂n(p

g∗n − pg0

n ) + x̂nqg∗n ]].

B. Generalized Sequential Bargaining for Multiple Users

In this subsection, we find the NBS for a general modelof reactive power compensation with multiple users undersequential bargaining protocol. The electric utility companywill bargain with each user n ∈ N sequentially to determine(qg

n, pgn, zn). Without loss of generality, we assume that the

electric utility company will bargain with users in the orderof 1, 2, . . . , N to obtain the NBS.

We first assume that at the current bargaining stage, theelectric utility company already finished bargaining with priorusers 1, 2, . . . , n − 1, and starts bargaining with user n. Thenthe NBS (qg∗

n , pg∗n , z∗

n) between the electric utility companyand user n is obtained via solving the following optimizationproblem

max[U[n] − U0

[n]

]·[Vn(qg

n, pgn, zn

)− V0n

](30)

s.t.{qg

n, pgn

} ∈ Xn, zn ∈ Zn.

Note that in (30), we use the subscript [n] to denote thebargaining stage index. Moreover, the disagreement point ofthe electric utility company at the current bargaining stage isU0

[n] rather than U0, which is calculated as the payoff thatelectric utility company achieved after bargaining with priorusers 1, 2, . . . , n−1. From (18), we can determine U0

[n] as thefollowing equation

U0[n] = π

n−1∑

i=1

qg∗i −

n−1∑

i=1

z∗i + α

n−1∑

i=1

[r̂i

(pg∗

i − pg0i

)+ x̂iq

g∗i

].

(31)

We further calculate the payoff of the electric utility companyat the current bargaining stage [n] as from (18)

U[n] = π

n−1∑

i=1

qg∗i −

n−1∑

i=1

z∗i + α

n−1∑

i=1

[r̂i

(pg∗

i − pg0i

)+ x̂iq

g∗i

]

+ πqgn − zn + α

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]. (32)

Therefore, from (31) and (32) the payoff gain U[n] − U0[n]

that the electric utility company receives if bargaining withuser n is

U[n] − U0[n] = πqg

n − zn + α[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]. (33)

By substituting (33) into (30), we obtain

max[πqg

n − zn + α[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]]

·[zn − λ

(pg0

n − pgn

)](34)

s.t.{qg

n, pgn

} ∈ Xn, zn ∈ Zn.

It is readily to realize that the optimization problem (34) at thebargaining stage [n] is identical in case of one-to-one bargain-ing in (23). Therefore, the optimal solution for {qg∗

n , pg∗n , z∗

n} issimilar to Theorem 1. Furthermore, we analyze the connectionbetween the bargaining result and social welfare problem asthe following theorem.

Theorem 2: The NBS {qg∗n , pg∗

n }n=1,2,...,N under the sequen-tial bargaining maximizes the social welfare problem (20).

Proof: See Appendix B.

IV. CONCURRENT BARGAINING

In this section, we find the NBS for the decentralize reactivepower compensation under the concurrent bargaining protocol,where the electric utility company bargains with users concur-rently. We also analyze the connection between the NBS andthe network social welfare problem.

The generalized NBS under concurrent bargaining is thesolution of the following optimization problem

max[U(qg

n, pgn, zn

)− U0]

·N∏

n=1

[Vn(qg

n, pgn, zn

)− V0n

]

s.t.{qg

n, pgn

} ∈ Xn, zn ∈ Zn. (35)

By solving the optimization problem (35), we obtain thefollowing result.

Theorem 3: The NBS under concurrent bargaining{qg∗

n , pg∗n }n=1,2,...,N also maximizes the social welfare prob-

lem (20) and is identical to NBS under sequential bargaining.However, the reimbursement for each user is given by

zn = λ(

pg0n − pg

n

)

+ 1

N + 1

[

π

N∑

n=1

qgn − λ

N∑

n=1

(pg0

n − pgn

)

+ α

N∑

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]]

. (36)

Proof: See Appendix C.



Remark: From the above results, we conclude that the NBS{qg∗

n , pg∗n }n=1,2,...,N in both sequential bargaining and concur-

rent bargaining will maximize the social welfare problem (20).The reimbursement for user n covers the cost incurred byreducing the active power generation λ(pg0

n − pgn) and a half

of its portion of social benefit contributed to the system,under sequential bargaining. While in concurrent bargaining,the social welfare of the system is equally divided among allusers and the electric utility company.

Based on Theorems 2 and 3, the NBS for the decentralizedreactive power compensation can be implemented by two stepsas follows. First, each user individually determines the amountof active and reactive power generation, which maximizes thesocial welfare problem. Then, in the second step, depending onthe amount of power generation from users and which bargain-ing protocol is selected, the electric utility company determinesthe amount of reimbursement offered to each user. Due to thedistributed topology of power distribution networks as well aslacking coordination among users, the sequential bargainingis more practical to be deployed in realistic applications. Forthe concurrent bargaining, it can be applied in the scenario inwhich a group of users located in the same geographical areaacts as a single entity and negotiates with the electric utilitycompany in the reactive power compensation problem.

V. SIMULATION RESULTS

In this section, we use numerical simulations to demonstratethe effectiveness of decentralized reactive power compensa-tion. We test a distribution network with N = 250 users/nodes.The voltage at the node 0 is V0 = 7.2kV . The line impedanceis (0.33+ j0.38)�/km, and the distances between neighboringnodes are drawn from a uniform distribution from 0.2km to0.3km. Each node has the active power demand uniformly gen-erated in the range [1kW, 3kW], and the corresponding reactivepower demand is generated in the range of [0kVAR, 1.8kVAR].The number of users equipped with DG units is selectedrandomly and accounts for 50% of the total users in all sim-ulations, unless otherwise stated, while the other users do nothave DG units to participate in the reactive power compen-sation. The maximum apparent capacity for all DG units issn = 2.2kVA. The amount of available active power genera-tion using renewable resources is generated randomly from auniform distribution with lower and upper limits [0.75sn, sn].The active power price is λ = ¢6.6/kWh [20] and the constantparameter π = 0.25 ∗ λ/kVARh. We set α = 1.

In Figs. 2 and 3, we plot the demand and generation pro-files for active power and reactive power, respectively, of 20randomly selected users from the set of users equipped withDG units. We compare active power generation when usersdo participate and when users do not participate in reactivepower compensation. For users who have the generation capac-ity greater than demand, they generate as much reactive powerand active power as possible to satisfy their power demand.Moreover, some users reduce the amount of active power gen-eration to increase the amount of reactive power generation.For instance, users 11 and 17 even decrease the active power

Fig. 2. The active power demand and generation profiles.

Fig. 3. The reactive power demand and generation profiles.

generation when they participate in reactive power compen-sation since at the current period, compensating for reactivepower brings higher reimbursement than generating activepower.

We further compare the reimbursement of 20 randomlyselected users under sequential bargaining and concurrentbargaining protocols in Figs. 4 and 5. Specifically, the reim-bursement that each user received covers the cost incurreddue to the reduction of active power generation and its netpayoff. For instance, users 11, 17 reduce active power genera-tion to reserve capacity for reactive compensation. Therefore,the reimbursements that they received cover these reductioncosts. Specifically, under the sequential bargaining protocol,the net payoff that each user received will be determined froma half portion of social welfare that the user contributed to thesystem, as shown in (29). Thus, each user has a different pay-off. However, under the concurrent bargaining protocol, thenet payoff each user received is equally divided from the totalsocial welfare of the system for all users and the electric utilitycompany, and hence the same for all users.



Fig. 4. The reimbursement of users in sequential bargaining.

Fig. 5. The reimbursement of users in concurrent bargaining.

We study the effect of DG unit penetration level on thesystem reliability and efficiency. Note that, for all simula-tions so far, we assume that the number of users equippedwith DG units accounted for 50% of the total number ofusers. In Fig. 6, we plot the percentage of voltage deviationof the system when the DG unit penetration level varies from10% to 80%. Furthermore, three types of weather conditionsare considered, namely, sunny, partly cloudy, and cloudy. Foreach of these weather types, pg

n,max is generated from a uni-form distribution within the ranges [0.75sn, sn] [0.5sn, 0.75sn][0, 0.25sn], corresponding to sunny, partly cloudy, and cloudy.From the figure, we realize that when the percentage of DGunit penetration level increases, the voltage variation of thesystem will decrease consequentially. This improvement hap-pens due to local compensate for reactive demand, whichrenders the voltage variation to decrease. In addition, in Fig. 7,we plot the power factor of the system, which is calculated

by PF = P0/

√P2

0 + Q20, when the DG unit penetration level

varies from 10% to 80% for three different types of weatheras well. The figure reveals that the power factor increasescorrespondingly to the increase of the penetration level.

Fig. 6. The effect of DG unit penetration level on voltage deviation.

Fig. 7. The effect of DG unit penetration level on power factor.

TABLE IPERFORMANCE COMPARISON WITH CENTRALIZED CONTROL

Finally, we compare the performance of our proposedmethod with the centralized reactive power compensation con-trol. The results are shown in Table I. To obtain the centralizedcontrol solution, we assume that the electric utility companyhas the ability to fully control the amount of reactive andactive power generation of all DG units of all users and tosolve the optimization problem in (20). Since the DG unitsare controlled by the electric utility company, the amount ofsurplus active power and reactive power can be injected backto the power grid in centralized control solution. We comparethe amount of reactive power reduction and the percentage ofvoltage deviation along the distribution network between thecentralized control and our proposed model for the case with50% users equipped with DG units. From the results in Table I,the centralized solution can help the electric utility companyreduce 71.5% amount of reactive power generation. Similarly,



a higher power quality in terms of voltage deviation is shownfor the centralized solution, where the total voltage deviationis only 6.8% while it is 8.5% in our proposed decentralizedsolution. Such performance improvement for centralized con-trol is obtained due to the fact that the surplus reactive powerand active power from users equipped with DG units can beutilized to supply demand to neighbor users who do not haveDG units.

VI. CONCLUSION

In this paper, the decentralized reactive power compensa-tion problem in a distribution network has been studied. Eachuser independently determines the amount of active and reac-tive power generation for its DG unit to locally compensatefor reactive power. Based on the amount of power dispatch,users will receive reimbursement from the electric utility com-pany. We investigate the economic interaction between usersand electric utility company using the Nash bargaining theory.Optimal solutions of power generation and reimbursement arederived under both sequential bargaining and concurrent bar-gaining protocols. Numerical results in a distribution networkwith 250 nodes/users are conducted to illustrate the effec-tiveness of the decentralized reactive power compensation inenhancing system reliability.

APPENDIX APROOF OF THEOREM 1

We can rewrite the optimization problem (23) as an equiv-alent optimization problem by taking ln of the objectivefunction

max ln[πqg

n − zn + αr̂n

(pg

n − pg0n

)+ αx̂nqg

n

)]

+ ln[zn − λ

(pg0

n − pgn

)](37)

s.t. 0 ≤ qgn ≤ qd

n,

0 ≤ pgn ≤ pg0

n ,(pg

n

)2 + (qg

n

)2 ≤ s2n,

zn ≥ 0.

The optimization problem (37) can be solved by decompos-ing into the following two steps. First, for fixed qg

n, pgn, solve

for optimal zn by setting the first derivative of the objectivefunction (37) to zero, we obtain

zn = 1

2πqg

n + λ

2

(pg0

n − pgn

)+ α

2

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

].

(38)

By substituting (38) into the problem (37), we obtain thefollowing subproblem for decision variables {qg

n, pgn}

maxqg

n,pgn

2 lnπqg

n − λ(

pg0n − pg

n

)+ α

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]

2(39)

s.t. 0 ≤ pgn ≤ pg0

n ,

0 ≤ qgn ≤ qd

n,(pg

n

)2 + (qg

n

)2 ≤ s2n.

We now solve (39) to find the optimal {qg∗n , pg∗

n }. Let considertwo cases:

• If (pg0n )2 + (qd

n)2 ≤ s2

n (total demand of active power andreactive power is less than generation capacity), the usern can generate both active and reactive power to fullysatisfy its own demand. Therefore, the optimal reactivepower generation is

qg∗n = qd

n, (40)

pg∗n = pg0

n . (41)

Then we can easily obtain the optimal value for reim-bursement

z∗n = 1

2πqd

n + α

2x̂nqd

n. (42)

• If (pg0n )2 + (qd

n)2 > s2

n, we realize that the objective func-tion (39) is an increasing function of pg

n and qgn. Therefore,

the constraint (pgn)

2 +(qdn)

2 = s2n must be hold at the opti-

mality. Then we can express pgn as a function of decision

variable qgn as

pgn =

√s2

n − (qg

n)2

. (43)

By substituting (43) to the objective function (39) andtaking the first derivative of the objective function to zero,we can find the optimal solution for qg

n and pgn as

qg∗n =

(π + αx̂n

)sn

√(λ + αr̂n

)2 + (π + αx̂n

)2, (44)

pg∗n =

(λ + αr̂n

)sn

√(λ + αr̂n

)2 + (π + αx̂n

)2. (45)

Since the reactive power and active power that user ngenerated cannot exceed its own demand, thereforewe have

qg∗n = min

⎧⎨

⎩

(π + αx̂n

)sn

√(λ + αr̂n

)2 + (π + αx̂n

)2, qd

n

⎫⎬

⎭, (46)

pg∗n = min

⎧⎨

⎩

(λ + αr̂n

)sn

√(λ + αr̂n

)2 + (π + αx̂n

)2, pg0

n

⎫⎬

⎭. (47)

Moreover, we rewrite the (38) for purpose of analysis as

zn = λ(

pg0n − pg∗

n

)

+ 1

2

[πqg∗

n + α[r̂n

(pg∗

n − pg0n

)+ x̂nqg∗

n

]

− λ(

pg0n − pg∗

n

)]. (48)

APPENDIX BPROOF OF THEOREM 2

From (39) in subproblem 2, we realize that the NBSin sequential bargaining is the optimal solution of the



following optimization problem

{qg∗

n , pg∗n

} = arg max(qg

n,pgn)∈Xn

[πqg

n + α[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]

− λ(

pg0n − pg

n

)], ∀n ∈ N . (49)

We need to show that {qg∗n , pg∗

n }n=1,2,...,N also maximize thesocial welfare optimization problem (20). First, we decou-ple the objective function of the social welfare maximizationproblem (20) into

�(pppg,qqqg) = π

N∑

n=1

qgn + α

N∑

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]

− λ

N∑

n=1

(pg0

n − pgn

)

=N∑

n=1

[πqg

n + αr̂n

(pg

n − pg0n

)

+ αx̂nqgn − λ

(pg0

n − pgn

)]

=N∑

n=1

�n(pg

n, qgn

),

where �n(pgn, qg

n) = [πqgn+αr̂n(p

gn−pg0

n )+αx̂nqgn−λ(pg0

n −pgn)].

Then we can rewrite (49) as

{qg∗

n , pg∗n

} = arg max(qg

n,pgn)∈Xn

�n(pg

n, qgn

),∀n ∈ N . (50)

From (50), for any {qg∗n , pg∗

n } �= {qgn, pg

n}, we have

N∑

n=1

�n(pg∗

n , qg∗n

)� �

(pppg∗,qqqg∗) ≥

N∑

n=1

�n(pg

n, qgn

)

� �(pppg,qqqg). (51)

Therefore, the NBS in sequential bargaining maximizes thesocial welfare.

APPENDIX CPROOF OF THEOREM 3

Since the disagreement point U0 = 0,V0n = 0,∀n, and by

taking ln of the objective function in (35), we obtain

max ln

[

π

N∑

n=1

qgn −

N∑

n=1

zn + α

N∑

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]]

+N∑

n=1

ln[zn − λ

(pg0

n − pgn

)](52)

s.t.{qg

n, pgn

} ∈ Xn, zn ∈ Zn,∀n.

We can solve the optimization problem (52) using similarmethod in Appendix A. Given the fixed qg

n, pgn, the optimal

solution zn can be obtained by setting the first derivative ofthe objective function (52) with respect to zn to zero

−1

π∑N

n=1 qgn −∑N

n=1 zn + α∑N

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]

+ 1

zn − λ(

pg0n − pg

n

) = 0,∀n. (53)

Solving the set of N equations (53), we can obtain theexpression of zn as

zn = λ(

pg0n − pg

n

)+ 1

N + 1

[

π

N∑

n=1

qgn − λ

N∑

n=1

(pg0

n − pgn

)

+ α

N∑

n=1

[r̂n

(pg

n − pg0n

)+ x̂nqg

n

]]

. (54)

By substituting (54) into the objective function of (52), weobtain

max(N + 1) ln

[1

N + 1

(

π

N∑

n=1

qgn − λ

N∑

n=1

(pg0

n − pgn

)

+N∑

n=1

α[r̂n

(pg

n − pg0n

)+ x̂nqg

n

])]

.

(55)

From (55), the optimal solution {pg∗n , qg∗

n }n=1,2,...,N

maximizes the inner term π∑N

n=1 qgn − λ

∑Nn=1(p

g0n − pg

n) +∑Nn=1 α[r̂n(p

gn − pg0

n ) + x̂nqgn], which is identical to the objec-

tive function of the social welfare problem (20). Therefore, theoptimal solutions are as in Theorem 1. And the reimbursementis given in (54).

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Hung Khanh Nguyen received the B.S. degreefrom the Department of Electrical and ElectronicEngineering, Ho Chi Minh City University ofTechnology, Vietnam, in 2010, and the M.S. degreefrom the Department of Electronics and RadioEngineering, Kyung Hee University, Korea, in 2012.He is currently pursuing the Ph.D. degree withthe University of Houston, TX, USA. His currentresearch interests are resource allocation and gametheory in smart grid.

Hamed Mohsenian-Rad (S’04–M’09–SM’14)received the Ph.D. degree in electrical and com-puter engineering from the University of BritishColumbia, Vancouver, Canada, in 2008. He iscurrently an Assistant Professor of ElectricalEngineering with the University of California atRiverside. His research interests include modeling,analysis, and optimization of power systems andsmart grids with focus on energy storage, renewablepower generation, demand response, cyber-physicalsecurity, and large-scale power data analysis. He

was a recipient of the National Science Foundation CAREER Award in2012, and the Best Paper Award from the IEEE International Conferenceon Smart Grid Communications in 2012 and the IEEE Power and EnergySociety General Meeting in 2013. He serves as an Editor for the IEEETRANSACTIONS ON SMART GRID, IEEE POWER ENGINEERING LETTERS,and IEEE COMMUNICATIONS LETTERS.

Amin Khodaei (M’11–SM’14) received the Ph.D.degree in electrical engineering from the IllinoisInstitute of Technology, Chicago, in 2010. He was aVisiting Faculty Member with the Robert W. GalvinCenter for Electricity Innovation, Illinois Instituteof Technology, from 2010 to 2012. He joined theUniversity of Denver, Denver, CO, in 2013, as anAssistant Professor. His research interests includepower system operation, planning, computationaleconomics, microgrids, and smart electricity grids.He is the Technical Co-Chair of the 2016 IEEE

PES Transmission and Distribution Conference and Exposition and the 2016North American Power Symposium, and the Chair of the IEEE MicrogridTaskforce.

Zhu Han (S’01–M’04–SM’09–F’14) received theB.S. degree in electronic engineering from TsinghuaUniversity, in 1997, and the M.S. and Ph.D. degreesin electrical engineering from the University ofMaryland, College Park, in 1999 and 2003, respec-tively.

From 2000 to 2002, he was a Research andDevelopment Engineer with JDSU, Germantown,MD. From 2003 to 2006, he was a ResearchAssociate with the University of Maryland. From2006 to 2008, he was an Assistant Professor

with Boise State University, Idaho. He is currently a Professor with theElectrical and Computer Engineering Department and the Computer ScienceDepartment, University of Houston, Texas. His research interests includewireless resource allocation and management, wireless communications andnetworking, game theory, wireless multimedia, security, and smart grid com-munication. He was a recipient of the NSF Career Award in 2010, the Fred W.Ellersick Prize of the IEEE Communication Society in 2011, the EURASIPBest Paper Award for the Journal on Advances in Signal Processing in 2015,and several best paper awards from IEEE conferences. He is currently anIEEE Communications Society Distinguished Lecturer.

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