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energies

Article

Game-Theory Modeling for Social Welfare

Maximization in Smart Grids

Yu Min Hwang 1ID , Issac Sim 1, Young Ghyu Sun 1ID , Heung-Jae Lee 2and Jin Young Kim 1 ,*

1Department of Wireless Communications Engineering, Kwangwoon University, Seoul 01897, Korea;

yumin@kw.ac.kr (Y.M.H.); dltkr34@kw.ac.kr (I.S.); yakrkr@kw.ac.kr (Y.G.S.)

2Department of Electrical Engineering, Kwangwoon University, Seoul 01897, Korea; hjlee@kw.ac.kr

*Correspondence: jinyoung@kw.ac.kr; Tel.: +82-02-940-5567

Received: 30 June 2018; Accepted: 14 August 2018; Published: 3 September 2018

Abstract:

In this paper, we study the Stackelberg game-based evolutionary game with two players,

generators and energy users (EUs), for monetary proﬁt maximization in real-time price (RTP)

demand response (DR) systems. We propose two energy strategies, generator’s best-pricing and

power-generation strategy and demand’s best electricity-usage strategy, which maximize the proﬁt

of generators and EUs, respectively, rather than maximizing the conventional uniﬁed proﬁt of the

generator and EUs. As a win–win strategy to reach the social-welfare maximization, the generators

acquire the optimal power consumption calculated by the EUs, and the EUs obtain the optimal

electricity price calculated by the generators to update their own energy parameters to achieve

proﬁt maximization over time, whenever the generators and the EUs execute their energy strategy

in the proposed Stackelberg game structure. In the problem formulation, we newly formulate

a generator proﬁt function containing the additional parameter of the electricity usage of EUs

to reﬂect the inﬂuence by the parameter. The simulation results show that the proposed energy

strategies can effectively improve the proﬁt of the generators to 45% compared to the beseline scheme,

and reduce the electricity charge of the EUs by 15.6% on average. Furthermore, we conﬁrmed the

proposed algorithm can contribute to stabilization of power generation and peak-to-average ratio

(PAR) reduction, which is one of the goals of DR.

Keywords:

smart grid; game-theoretic modeling; social welfare maximization; Stackelberg game;

pricing and power-generation strategy

1. Introduction

With advanced communication networks and intelligent controllable electrical devices or energy

users (EUs), the smart grid makes it possible to achieve the distributed control and the distributed

energy management (DEM) [

1

]. DEM plays a key role in the distributed monitoring, controlling,

scheduling, and optimization of the proﬁt of both generators and demands for the implementation of

demand response (DR) programs [

2

]. DR, deﬁned as the energy-usage changes of users in response to

varying electricity prices or to incentive payments [

3

], induces EUs to consume less energy during

periods of high wholesale market prices or at the peak power consumption. This price-based DR

program can be optimally implemented through a continuous interaction between the users and

the service provider. The user needs to adjust the electricity usage in consideration of the varying

electricity price over time, while the service provider also needs to properly adjust the electricity price

with the amount of power generation to motivate users to evenly use electricity over time [

4

]. The most

efﬁcient use of the smart grid, for example, is the increasing of the energy efﬁciency through measures

such as the decrease of the peak-to-average ratio (PAR) of the energy demand; however, in reality,

this will determine the extent of the monetary proﬁts in the generator and demand sides.

Energies 2018,11, 2315; doi:10.3390/en11092315 www.mdpi.com/journal/energies

Energies 2018,11, 2315 2 of 23

To maximize the monetary proﬁt (or social welfare), various studies on smart-home scheduling

have been completed, as follows: the formulation of a linear programming problem for smart-home

scheduling in consideration of the uncertainty of energy consumption [

5

], a Markov-chain model of the

scheduling problem and the development of the backtrack algorithm based on a decision threshold [

6

],

a dynamic-programming algorithm to schedule home appliances in consideration of multiple power

levels [

7

], the consideration of the distributed-load-management problem as a congestion game

with a dynamic pricing strategy to discourage the energy consumption at peak hours [

8

], and the

deployment of pricing strategies by local aggregators to control the energy load [

9

,

10

]. In [

11

],

an optimal management system of battery energy storage to enhance the resilience of the microgrid

is proposed while maintaining its operational cost at a minimum level. In [

12

], a two stage energy

management strategy for the contribution of plug-in electrical vehicles (PEVs) in demand response

programs of commercial building microgrids is addressed for energy management optimization.

In [

13

], by forming coalitions for gaining competitiveness in the energy market, a smart transactive

energy (TE) framework in which home microgrids (H-MGs) can collaborate with each other in a

multiple H-MG system is presented and analyzed. In [

14

], an optimization-based algorithm in which

an objective function premised on economic strategies, distribution limitations and the overall demand

in the market structure is proposed with emphasizing optimum use of electrical/thermal energy

distribution resources, while maximizing proﬁt for the owners of the H-MGs.

Over the past few years, related studies that are based on game-theory modeling for the energy

management in a real-time price (RTP)-based DR have also been conducted. In [

15

], a cake-cutting game

(CCG) for the selection of discriminate prices for different users was investigated. In [

4

], an RTP-based

DR algorithm for the achievement of an optimal load control regarding the devices in a facility that is

obtained through the forming of a virtual electricity-trading process is proposed. In [

16

], an aggregate

game is adopted for the modeling and analysis of the energy-consumption control in the smart grid

and Nash seeking strategies are developed. In [

17

,

18

], the Stackelberg game was leveraged to model

the interaction between the demand-response aggregators and generators. In [

19

], the authors propose

a light-weight DR scheme for managing energy consumption based on a non-iterative Stackelberg

model and historical real-time pricing without iterations for the massive smart manufacturing systems.

In [

20

], a multiagent-based energy market for multi-microgrid systems using game-theoretic and

hierarchical optimization approaches is proposed to achieve the optimal operation of smart microgrids

in distribution systems. In [

21

], an advanced retail electricity market based on game theory for the

optimal operation of H-MGs and their interoperability within active distribution networks is proposed

and the optimal solution is achieved using the Nikaido-Isoda Relaxation Algorithm (NIRA) in a

non-cooperative gaming structure. In [

22

] and [

23

], a consensus-based distributed-energy-management

algorithm for both sides of an indirectly connected network is proposed. However, the following

shortcomings relative to the game structure and formulas for proﬁt maximization can be identiﬁed in

the previous research, which have been addressed in this paper:

1.

It is necessary to study an optimization methodology that maximizes the monetary proﬁt of the

generators and the demands, respectively, rather than the optimization methodology based on

the uniﬁed proﬁt of the generators and the demands [24,25].

2. As an revolutionary game based on the Stackelberg game structure, it is necessary to study that

the players (demands and generators) obtain the each other’s energy parameters, such as the

time-varying electricity price and an amount of electricity consumption with spying on each

other’s energy strategy to effectively and adaptively maximize their proﬁt.

In view of these needs for research, in this paper, the Stackelberg game model where the generators

and the demands that are the players of the stackelberg game alternatively maximize their respective

proﬁts using their own energy strategy over time while watching each other’s energy strategies is

studied. The contributions of this paper are as follows:

Energies 2018,11, 2315 3 of 23

1.

We propose two energy strategies for proﬁt maximization for both the generators and

EUs (Generator’s Best-Pricing and Power-Generation Strategy in Section 3.1, Demand’s Best

Electricity-Usage Strategy in Section 3.2) based on the Stackelberg game as an evolutionary game

where the players (demands and generators) alternately perform their energy strategy with

spying on each other’s energy strategy to update their own energy parameters to achieve proﬁt

maximization over time in smart grid demand response. Whenever the generators and the EUs

execute their energy strategy in the proposed Stackelberg game structure, the generators acquire

the optimal power consumption calculated by the EU, and the EUs obtain the optimal electricity

price calculated by the generator. To the best knowledge of the authors, this game structure based

on the aforementioned parameter exchange (optimal power consumption and optimal electricity

price) for proﬁt maximization in the smart grid demand response is ﬁrst studied.

2.

We newly formulate a generator proﬁt function including the additional parameter,

the electricity-consumption of EUs, compared with that of the conventional proﬁt function [

26

]

since the proﬁt of generators can be inﬂuenced by the electricity-consumption of EUs.

3.

We greatly improve the monetary proﬁt of the generators and EUs using the proposed two energy

strategies by optimizing the amount of the power generation and the electricity price in the

generator side, and electricity consumption in the EU side.

4.

As one of the simple and powerful electricity-usage control strategy of the demand that

is applicable to the time-varying electricity market, we newly propose a market-adaptive

electricity-usage scheduling algorithm which maximizes the demand’s proﬁt by calculating

the amount of power that should be consumed according to the time-varying electricity price.

5.

The proposed proﬁt maximization algorithm can solve the existing PAR reduction problem

because the energy usage immediately increases as the electricity price goes down, and the

energy usage goes down as the price goes up according to the proposed game structure. It is

also possible to alleviate the problem that the price greatly ﬂuctuates because if the price changes

greatly, the power plant will lose its beneﬁts and will not be able to withstand it. The problem of

making power generation stable is consistent with reducing PAR, which is one of DR’s ultimate

goals. Stable power generation can reduce useless power generation, which makes our energy

resources the most efﬁcient to use, and can contribute to addressing issues such as global warming

in the energy industry.

The Stackelberg game model is used to realistically model the proﬁt maximization scenarios

wherein two players, the generators and the demands, repeatedly modify their energy strategies

by checking on each other’s energy strategy in the order of time. It is conﬁrmed here that the

monetary proﬁt of the both is optimized when the two players compete strategically over time. Further,

a modeling of the demand proﬁt and the optimization of the EU energy usage is performed based on a

time-varying electricity price. Since the electricity price at different times can be signiﬁcantly different

even in a single day, it is necessary to optimally schedule the electricity usage for all of the demands,

while the electricity-price changes are monitored. It is assumed here that the comprehensive judgment

and control of the EU electricity usage are optimally performed by the smart-home scheduler. In this

paper, we provide the proﬁt maximization methodology based on the proﬁt of the generators and EUs,

respectively, not the uniﬁed proﬁt of the generators and EUs.

The rest of this paper is organized as follows: in Section 2, we formulate the new proﬁt function of

the generators and the proﬁt maximization problem for the generators and EUs, respectively, based on

the Stackelberg game. In Section 3, we propose two proﬁt maximization algorithms as a solution of the

problem, and present the schematic overview and application of the proposed algorithms. In Section 4,

simulation results are presented and the practical beneﬁts and advantages of the proposed algorithm

are described. Finally, we conclude the paper in Section 5.

Energies 2018,11, 2315 4 of 23

2. System Model and Problem Formulation

For the problem formulation, a smart grid with a network of

K

-distributed EUs and

I

generators

was considered. It was assumed that each EU

k∈ K ={1, . . . , K}

and each generator

i∈ I ={1, . . . , I}

are directly connected as the communication topology of the network (i.e., any two nodes are connected

by a directed path). In this paper, it should be noted that the assumption of the strongly connected

communication network renders the distributed-energy-management problem to be more general

and relaxes the undirected-connection assumption [

22

]. A block-processing model was adopted to

schedule the load demands according to periodic time blocks. We divide the total scheduling time

into ttime periods

t∈ T ={1, . . . , T}

with a constant length

tl

. We consider that scheduling horizon

for an EU is 24 h (a single day), that is

tl

and

T

are set to 1 h and 24, respectively, in this paper.

All parameters and functions used in this paper are listed in Table 1. Furthermore, the systemic model

of the electricity buying and selling is shown in Figure 1. The notion used is listed in Table 1. The users

are equipped with an advanced metering infrastructure (AMI) and an energy-management controller

(EMC) [

22

]. Load information for each EU is exchanged between these two modules. The AMI is

used to schedule, control and optimize the electricity usage for each EU and enables bidirectional

communication between the EUs and the DEM which is connected to the generators. DEM plays a role

in optimizing the beneﬁts of both the EU network (demand) and the generator network (energy supply)

based on the DR programs and real-time pricing to improve the energy-usage efﬁciency. Further,

the following three key optimization variables (the electricity price, the generation power, and the

demand power) are employed to realize the coordination of the generators and the EUs.

In the proposed evolutionary game, we assume that the players have perfect rationality.

They always act in a way that maximizes their proﬁt, and are capable of obtaining the energy

information to calculate the best response to other players’ energy strategies. The tractability of the

perfect rationality game can be realized by using the infrastructure, such as smart meter, EMC, DEM

and wired/wireless communications to obtain maximum proﬁt of the cooperative and win-win players.

Energies 2017, 10, x FOR PEER REVIEW 4 of 22

nodes are connected by a directed path). In this paper, it should be noted that the assumption of the

strongly connected communication network renders the distributed-energy-management problem to

be more general and relaxes the undirected-connection assumption [22]. A block-processing model

was adopted to schedule the load demands according to periodic time blocks. We divide the total

scheduling time into t time periods ∈={1, … , } with a constant length . We consider that

scheduling horizon for an EU is 24 h (a single day), that is and are set to 1 h and 24, respectively,

in this paper. All parameters and functions used in this paper are listed in Table 1. Furthermore, the

systemic model of the electricity buying and selling is shown in Figure 1. The notion used is listed in

Table 1. The users are equipped with an advanced metering infrastructure (AMI) and an energy-

management controller (EMC) [22]. Load information for each EU is exchanged between these two

modules. The AMI is used to schedule, control and optimize the electricity usage for each EU and

enables bidirectional communication between the EUs and the DEM which is connected to the

generators. DEM plays a role in optimizing the benefits of both the EU network (demand) and the

generator network (energy supply) based on the DR programs and real-time pricing to improve the

energy-usage efficiency. Further, the following three key optimization variables (the electricity price,

the generation power, and the demand power) are employed to realize the coordination of the

generators and the EUs.

In the proposed evolutionary game, we assume that the players have perfect rationality. They

always act in a way that maximizes their profit, and are capable of obtaining the energy information

to calculate the best response to other players’ energy strategies. The tractability of the perfect

rationality game can be realized by using the infrastructure, such as smart meter, EMC, DEM and

wired/wireless communications to obtain maximum profit of the cooperative and win-win players.

Figure 1. Systemic model of social-welfare maximization in a smart grid with demand response (DR)

systems.

Figure 1.

Systemic model of social-welfare maximization in a smart grid with demand response

(DR) systems.

Energies 2018,11, 2315 5 of 23

Table 1. Notation list.

Notations Meanings of Notations

K={1, . . . , K},k∈ K The set of energy users (EUs)

I={1, . . . , I},i∈ I The set of generators

T={1, . . . , T},t∈ T Total scheduling time set

tlThe constant time length

pt(cents/kWh) The electricity price per unit energy

Pi,t(kWh) The amount of the actual providable power

PL

i,t(kWh) The transmission losses

PT

i,t(kWh) The total amount of generated power

Ci,tPT

i,t(cents) The cost to generate power for generator i

aicents/kWh2,bi(cents/kWh), and ci(cents)The ﬁtting parameters of the cost function Ci,tPT

i,t

diThe coefﬁcient for power loss

Pm

iand PM

iThe minimum and maximum bound of PT

i,t

UG

i,tThe proﬁt function for generator i

UG

tThe total proﬁt for all generators

RtReal proﬁt of generators

Mi,tMaximum achievable proﬁt of the generator i

Pk,tThe amount of the electricity usage of UE k

ˆ

Pk,tThe amount of electricity to be used in the future

divided by the remaining time

∆Pk,tThe additional or abandoned electricity usage

UEU

kThe electricity charge for UE k

UEU The total electricity charge for all UEs

G=(N,A,U)

The Stackelberg game where Nis a player set that is

composed of generators and EUs, Ais the constraint

set, and Uis the proﬁt set

pmand pMThe minimum and maximum prices of the electricity

Pm

kand PM

kThe minimum and maximum electricity-usage

ϕkThe minimum required electricity consumption

η?The optimum solution of P1

P?

i,t,p?

tand P?

k,tThe optimized value of Pi,t,ptand Pk,t

FA set of feasible solutions

LGThe Lagrange dual function of P3

βi,t,γi,t,δt,εt, and θtThe Lagrange multipliers

NG

Mand NG

S

The number of iterations for the master-loop and

slave-loop in Algorithm 1

ω1,ω2,ω3,ω4, and ω5The iteration steps for optimization

ptThe base price as the historical average of pt

twThe window size of the time slots

UEU The total electricity charge accumulated for T

2.1. Generator Proﬁt

As the player of the Stackelberg game for each time t, each generator iaims to maximize its own

proﬁt by adjusting and optimizing the electricity price per unit energy

pt

(cents/kWh) and the amount

of the actual providable power

Pi,t

(kWh), excluding the transmission losses

PL

i,t

(kWh), in the total

amount of generated power

PT

i,t=Pi,t+PL

i,t

(kWh). To design a practical electric-power-transmission

system, the parameters of the modeling of the transmission-loss factors were considered since these

are inevitable in a power grid [

22

]. According to the micro-incremental transmission losses of each

generator [

24

], the transmission-loss amount

PL

i,t

that is induced by the ican be represented using the

following simple quadratic function:

PL

i,t=diPT

i,t2(1)

Energies 2018,11, 2315 6 of 23

where

di

is the loss coefﬁcient. The total amount of generated power

PT

i,t

is derived by jointly solving

the Equation (1) and PT

i,t=Pi,t+PL

i,tas follows:

PT

i,t=Pi,t+PL

i,t=Pi,t+1−2Pi,tdi−p1−4Pi,tdi

2di

(2)

In reference to [

25

–

27

], the cost

Ci,tPT

i,t

can be represented by the quadratic function of

PT

i,t

,

as follows:

Ci,tPT

i,t=aiPT

i,t2+biPT

i,t+ci(3)

where

aicents/kWh2

,

bi(cents/kWh)

, and

ci(cents)

are the ﬁtting parameters of the cost function,

and the minimum (maximum) bound of

PT

i,t

is denoted by

Pm

iPM

i

. Therefore, a new proﬁt

function

UG

i,t(Pi,t,ptPk,t)

and the real-proﬁt-to-maximum-achievable-proﬁt ratio (RMR), with the

latter composed of the two proﬁt functions for the ifor each time t, was deﬁned, as follows:

UG

i,t(Pi,t,ptPk,t)=Real proﬁt

Maximum achievable proﬁt =Rt(pt|Pk,t)

Mi,t(Pi,t,pt)

=pt∑K

k=1Pk,t

ptPi,t−Ci,t(Pi,t+PL

i,t)

(4)

where

Pk,t

is the amount of the electricity usage of each EU kfor each t. It becomes evident that

the newly deﬁned proﬁt function of (4) is more reasonable and practical compared with that of [

26

],

since the additional parameter

Pk,t

is considered in (4) to reﬂect the relationship between the real proﬁt

and the maximum achievable proﬁt. The generator total proﬁt for each tis represented as the sum of

the proﬁts of the i, as follows:

UG

t(Pi,t,ptPk,t)=Rt(ptPk,t)

Mt(Pi,t,pt)=Rt(ptPk,t)

∑I

i=1Mi,t(Pi,t,pt)(5)

2.2. EU Proﬁt

It was supposed that EUs schedule their energy consumption in consideration of the time-varying

electricity price to minimize their electricity charge without reducing the total amount of electricity

that should be used. The electricity charge accumulated until the

t

for the

Pk,t

of the kis written as

UEU

k(Pk,tpt), as follows:

UEU

k(Pk,tpt)=

t

∑

t=1

UEU

k,t(Pk,tpt)=

t

∑

t=1

ptPk,t=

t

∑

t=1

ptˆ

Pk,t+∆Pk,t(6)

where

ˆ

Pk,t

is the amount of electricity to be used in the future divided by the remaining time, and

∆Pk,t

is the additional or abandoned electricity usage for the current tin consideration of the current

electricity price pt.

Remark 1.

Reducing the total electricity usage itself will obviously reduce the electricity charge, but its usage

is not intended. In this paper,

∆Pk,t

was modeled to reduce the electricity charge through the adjusting of the

∆Pk,t

while the total electricity consumption was retained. For example, the

∆Pk,t

increased at the low

pt

and

decreased at the high

pt

in consideration of the setting of different prices for different periods by the generators.

Then, the total electricity charge accumulated until the

t

, and then

UEU

was introduced as the sum of each k

electricity charge, as follows:

UEU (Pk,tpt)=

K

∑

k=1

UEU

k(Pk,tpt)(7)

Energies 2018,11, 2315 7 of 23

2.3. Optimization of the Problem Formulation Based on the Stackelberg Game Model

In the game model of the present study, the generator acts as a follower to observe the electricity

usage of the EUs for each tand the maximization of its proﬁt; then, the EU acts as a leader to observe

the generator electricity price of each tand the minimization of its electricity charge. Based on the

game model, the Stackelberg game was formulated as

G=(N,A,U)

, where

N

is a player set that

is composed of generators and EUs,

A

is the constraint set, and

U

is the proﬁt set. Regarding P1,

the generator proﬁt,

UG

t

is deﬁned as the extent to which the generator can increase the proﬁt given the

EU electricity usage. In this paper, the constraints of the power generation unit were not considered.

In other words, we assume that the power generators are able to ﬂexibly adjust the amount of the

power generation to maximize their proﬁt by the proposed dynamic algorithm in Section 3. To solve

P1, it was assumed that the generator can observe the EU electricity usage

Pk

. In P2,

UEU

t

is the EU

electricity charge that is for the maximization of its proﬁt in consideration of the current electricity

price. The optimization problem for the two players is formulated as follows:

(P1)max

{Pi,t,pt∈AG}η=UG

t(Pi,t,ptPk,t)(8)

subject to:

AG=

(Pi,t,pt|Pk)

Pm

i≤Pi,t≤PM

i

pm≤pt≤pM

K

∑

k=1

Pk,t≤I

∑

i=1

Pi,t

(9)

(P2)min

{Pk,t∈AEU }UEU(Pk,tpt)(10)

subject to:

AEU =

(Pk,tpt)

Pm

k≤Pk,t≤PM

k

T

∑

t=1

Pk,t≥ϕk

(11)

Here,

pm

and

pM

are the minimum and maximum prices of the electricity, and

Pm

k

and

PM

k

are the

minimum and maximum electricity-usage values of each k, respectively. For each responsive demand,

a certain adjustable range of the electricity usage maximizes its own proﬁt through the adjusting of the

Pk,t

for the t. Furthermore, the minimum required electricity consumption

ϕk

of each EU is available

for the entire time and an electricity amount should be efﬁciently used. This enables the EUs to shift

heavy consumption loads from the peak-price time slots to the nonpeak-price time slots [28].

3. Proﬁt Maximization

In this section, the two proﬁt-optimization algorithms which can maximize the social welfare,

which are for the generators and the EUs, are proposed based on the time-hierarchy structure of the

Stackelberg game.

3.1. Generator’s Best-Pricing and Power-Generation Strategy

In this subsection, to maximally increase the generator proﬁt with the knowledge of the UE

energy consumption, the generator proﬁt is maximized as a part of the Stackelberg game. Firstly,

to successfully acquire the P1 maximum proﬁt, the nonconvex function of P1 was transformed into

the convex function using nonlinear fractional programming [

29

], since the solving of the nonconvex

function of P1 is extremely complex, and the optimum values can only be found using a brute-force

approach. Then, the Lagrangian dual decomposition was applied as a greedy-type iterative solution to

the transformed convex function to estimate the optimum argument set

{Pi,t,pt}

, where the constraint

Energies 2018,11, 2315 8 of 23

set

AG

is guaranteed. By exploiting the properties of nonlinear fractional programming [

29

], P1 is

equivalent to P3, as follows:

(P3)max

{Pi,t,pt∈AG}R(ptPk,t)−ηM(Pi,t,pt)(12)

subject to

AG

, where

η?

is the P1 optimum solution when the

{Pi,t,pt}

is equal to the optimal argument

set

nP?

i,t,p?

to

. To mathematically prove that P3 is the convex and equivalent function, the theorem for

the transformation was given with the deﬁning

F

as a set of feasible solutions and the maximum proﬁt

as η?in the maximization problem P1, as follows [29]:

Theorem 1. η?=max

{Pi,t,pt∈AG}UG

t(Pi,t,ptPk,t)∀{Pi,t,pt}∈F=R(p?

t|Pk,t)

M(P?

i,t,p?

t)

if and only if,

max

{Pi,t,pt∈AG}R(ptPk,t)−η?M(Pi,t,pt)=R(p?

tPk,t)−η?MP?

i,t,p?

t=

0, for

R(ptPk,t)≥

0and

M(Pi,t,pt)>0.

Theorem 1 represents P1 in the fractional form that can equivalently be transformed into the

subtractive form of P3. To prove Theorem 1, the transformed function F(η)was deﬁned as follows:

F(η)=max

{Pi,t,pt∈AG}R(ptPk,t)−ηM(Pi,t,pt)(13)

By following the approaches of [29], it was possible to prove Theorem 1.

Proof of Theorem 1. Convexity and equivalence.

Lemma 1. F(η)is exactly monotonically decreasing for η(i.e., F(η0)>F(η00 )if η0<η00 .

Proof

. Let

η00

maximize

F(η00 )

, then

F(η00 ) = max

{P

i,t,pt∈AG}R(ptP

k,t)−η00 M(P

i,t,pt)=R(p00

tP

k,t)−

η00 M(P00

i,t,p00

t)<R(p00

tP

k,t)−η0M(P00

i,t,p00

t)≤R(p0

tP

k,t)−η0M(P0

i,t,p0

t) = max

{P

i,t,pt∈AG}R(ptP

k,t)−η0M(P

i,t,pt)

=F(η0)

, where

R(p00

tPk,t)−η0M(P00

i,t,p00

t)≤Ri,t(p0

tPk,t)−η0M(P0

i,t,p0

t)

is reasonable,

because

R(p0

tPk,t)−η0M(P0

i,t,p0

t)

is the maximized value when the input value

η0

is given in

the function F(·).

Lemma 2. Let any set be {P0

i,t,p0

t}and the set satisﬁes η0=R(p0

t|Pk,t)

M(P0

i,t,p0

t),then F(η0)≥0.

Proof.F(η0) = max

{Pi,t,pt∈AG}R(ptPk,t)−η0M(Pi,t,pt)≥R(p0

tPk,t)−η0M(P0

i,t,p0

t) = 0.

As shown in Lemma 1 and Lemma 2, it is natural that the transformed function

F(η)

is convex,

since

F(η)

is monotonically decreasing and converges to zero. Further, the convergence of

F(η)

to zero

is representative of the generator proﬁt,

η

, reaching the maximum value. Thus, it became possible to

see P3 as equivalent to P1, and P3 is used as an equivalent objective function of P1 in the rest of this

paper. Furthermore, to illustrate the solving of P1 through P3 as pseudocode, we construct the iterative

algorithm, Algorithm 1: Generator’s Proﬁt Maximization, in page 10. In Algorithm 1, the lines 3–8

are performed by the generators with the operation of the master-loop algorithm that is based on the

slave-loop algorithm.

Remark 2.

When the number of master-loop iterations,

NG

M

, continues, the

η

increases and converges to

η?

if

F(η)<τ

, as shown in the lines 3–8 in Algorithm 1. Note that, for the convergence of the function

F(η)

,

the threshold parameter

τ

is set to approximately positive zero, as represented by

τ≈+

0. Furthermore,

Energies 2018,11, 2315 9 of 23

we have introduced nonlinear fractional programming [

29

] to ensure that our proposed solution has stability

in reaching equilibrium. From Proof of Theorem 1, we conﬁrmed that the P3 is not only equivalent to the P1,

but also a function which is monotonically decreasing and converges to zero (Lemma 1 and Lemma 2). The P3

is monotonically decreasing converges to zero when the iteration number

NG

M

of Master-loop algorithm in

Algorithm 1 increases, which means the equilibrium of P1 can be obtained with stability.

The slave-loop algorithm of Algorithm 1 line 5, can be considered as the solution of P3. By using

the Lagrange dual method [

30

], it was then possible to solve the convex optimization problem of P3 to

propose the slave-loop algorithm. To estimate the optimal arguments of

P?

i,t

and

p?

t

, the Lagrange dual

function of P3 was derived as follows:

LG(Pi,t,pt,βi,t,γi,t,δt,εt,θt)=Rt(ptPk,t)−ηMt(Pi,t,pt)−βi,t∑I

i=1Pm

i−Pi,t−

γi,t∑I

i=1Pi,t−PM

i−δt(pm−pt)−εtpt−pM−θt∑K

k=1Pk,t−∑I

i=1Pi,t(14)

where

βi,t

,

γi,t

,

δt

,

εt

, and

θt

are the Lagrange multipliers. The Lagrange dual problem of P3 can be

formulated as follows:

min

{βi,t,γi,t,δt,εt,θt}max

{Pi,t,pt∈AG}LG(Pi,t,pt,βi,t,γi,t,δt,εt,θt)(15)

By solving the following formulas of (16) and (17), which were derived using the

Karush–Kuhn–Tucker (KKT) conditions [

30

], it is possible to simply derive the optimal values,

P?

i,t

and

p?

t, as follows:

∂LG

∂Pi,t

=η(ai+bidi)

dip1−4diPi,t

−η+βi,t−γi,t+θt−ηpt(=0, Pi,t>0.

<0, otherwise. (16)

∂LG

∂pt

=−ηPi,t+δt−εt+∑K

k=1Pk,t

I(=0, pt>0.

<0, otherwise. (17)

P?

i,t="(A−ηbi)∗(2ηai+ηbidi+diA)

(2ηai+2diA)2#+

(18)

p?

t="ai+aiB−1

2+bidiB−1

2

di

+βi,t−γi,t+θt

η#+

(19)

where

A=ηpt−βi,t+γi,t−θ

, and

B=

1

−

4

di(δi,t−εi,t+K

∑

k=1

Pk,t)/Iη

,

[X]+=max{X, 0}

. Moreover,

the Lagrange multiplier

βi,t

,

γi,t

,

δt

,

εt

and

θt

can be updated by using gradient methods in a distributed

manner, as follows:

βi,tNG

S+1="βi,tNG

S−ω1

I

∑

i=1Pm

i−P?

i,t#+

(20)

γi,tNG

S+1="γi,tNG

S−ω2

I

∑

i=1P?

i,t−PM

i#+

(21)

δtNG

S+1="δtNG

S−ω3

I

∑

i=1

(pm−p?

t)#+

(22)

εtNG

S+1="εtNG

S−ω4

I

∑

i=1p?

t−pM#+

(23)

Energies 2018,11, 2315 10 of 23

θtNG

S+1="θtNG

S−ω5 K

∑

k=1

Pk,t−

I

∑

i=1

P?

i,t!#+

(24)

where the iteration steps

ω1

,

ω2

,

ω3

,

ω4

, and

ω5

are positive values, which are like a learning rate,

for a more rapid convergence of the algorithm, and the parameter

NG

S

is the number of iterations

for the slave loop in the line 5 of Algorithm 1: Generator ’s Proﬁt Maximization. The pseudocode of

the iterative slave-loop algorithm is proposed in Algorithm 3. This is the operating structure of the

master- and slave-loop algorithms, where the result of the slave-loop algorithm is the input of the

master-loop algorithm.

In this paper, we assumed that there are multiple power generators and multiple EUs as game

players in the proposed game algorithm. Each proﬁt of the power generators and EUs, respectively,

was deﬁned in Sections 2.1 and 2.2. However, we maximize the sum of the proﬁts of the power

generators and the EUs, so there exist two proﬁt sums for the power generators and the EUs, given by

Equations (5) and (7), respectively. If the proﬁt sums are successfully maximized by the proposed

algorithm, the proﬁt of each player can be distributed according to the pre-deﬁned proﬁt function

given by Equations (4) and (6) in Section 2.

The detailed description of the operation of Algorithm of Social-Welfare Maximization is as

follows. The Algorithm of Social-Welfare Maximization is the hierarchical bi-level iterative algorithm

with the Stackelberg-loop iteration number

t∈ T ={1, . . . , T}

, and is composed of the two

pseudocode tables, Algorithms 1 and 2. We deﬁne the generators as the “leader”, and the energy users

as the “follower” as a game player in the proposed game structure. Originally

t

denotes time set, but it

is recognized as an iteration number. If the

T

is set to be 24 and the time interval is 1 h, the Algorithm of

Social-Welfare Maximization repeats 24 times until the Stackelberg-loop iteration number

t

becomes 24.

Whenever the iteration for

t

is performed, the proposed Algorithms 1 and 2 are performed in succession.

They play their algorithm in the Stackelberg game and do not play the game at the same time but

play alternately over time to “interact” with each other and maximize their proﬁt. Algorithm 1 is for

generator’s proﬁt maximization, and iterates with the master-loop iteration number

NG

M

until the line

6,

R(p?

tPk,t)−η?M(P?

i,t,p?

t)<τ

, is satisﬁed. Note that the threshold parameter

τ

for the convergence

of the function

R(p?

tPk,t)−η?M(P?

i,t,p?

t)

should be set to around positive zero,

τ≈+

0, by the proof

of Lemmas 1 and 2. Furthermore, before the line 6 is performed, the optimal value P?

i,tand p?

tshould

be calculated in the line 5 as an outcome of the Slave-loop Algorithm of Algorithm 1, Algorithm 3,

with the slave-loop iteration number

NG

S

. The Slave-loop Algorithm is performed to solve the Lagrange

dual problem P3 based on Equations (18)–(24). When the line 6 is satisﬁed, the maximized proﬁt for

the generators is obtained with the optimal electricity price

p?

t

and the optimal power generation

P?

i,t

.

To interact with each other and effectively maximize the proﬁt, if Algorithm 1 ends, the optimal price

p?

t

calculated from Algorithm 1 is passed to the input of Algorithm 2, and Algorithm 2 is performed.

Algorithm 2 is for energy user’s proﬁt maximization, and the speciﬁc methodology is described in

next Section 3.2. If Algorithm 2 is successfully performed and the maximum EU’s proﬁt is calculated

with the optimal electricity usage of the EUs

P?

k,t

, the optimal electricity usage

P?

k,t

is also passed to the

input of Algorithm 1 (which means the “interact”) and

t

is incremented by one. We have described this

passing of values as “spy on” in Introduction. By running the algorithm repeatedly and alternatively

over time and spying on each other’s energy parameters, they can effectively maximize their own

proﬁt. The Algorithm of Social-Welfare Maximization (combination of Algorithms 1 and 2, lines 1–18)

is as follows:

Energies 2018,11, 2315 11 of 23

Algorithm 1.

Generator’s Proﬁt Maximization (lines 3–8, performed by the generator, master-loop algorithm).

1: Input: time set t∈ T ={1, . . . , T}.

2: While t≤T,do

3:

Input

: active generator set

i∈ I ={1, . . . , I}

,

Pi,t

,

pt

,

Pk,t=ˆ

Pk,t

,

η=

0,

τ≈+

0 (positive zero),

NG

M=

1.

4: Initialize η=UG

tPi,t,ptPk,t=R(pt|Pk,t)

M(Pi,t,pt)=0.

5: Update P?

i,tand p?

tfrom Slave-loop algorithm (“Algorithm 3”).

6: If Rp?

tPk,t−η?M(P?

i,t,p?

t)<τ

7: Return optimal parameters {P?

i,t,p?

t}and optimal proﬁt of generators, η?=R(p?

t|Pk,t)

M(P?

i,t,p?

t).

8: else Update η=R(p?

t|Pk,t)

M(P?

i,t,p?

t)and NG

M=NG

M+1, then Go to line 5.

Algorithm 2. Energy-User’s Proﬁt Maximization (Market-adaptive Electricity-Usage Scheduling Algorithm,

lines 9–17, performed by the EU).

9: Input:p?

t, active EU set k∈ K ={1, . . . , K},Pk,t,ϕk,t,T,tw,Pk,t.

10: For k∈ K do

11: Calculate ptwith p?

taccording to (26).

12: Calculate ∆Pk,twith ptaccording to (25).

13: Calculate P?

k,twith ∆Pk,taccording to (27).

14: Update ˆ

Pk,t+1with ˆ

Pk,tand ∆Pk,taccording to (28).

15: end for

16: Calculate UEU

k,twith (7).

17: Let t=t+1.

18: end while

Algorithm 3. Slave-loop Algorithm of the Generator ’s Proﬁt Maximization.

1: Input: active generator set i∈{1, . . . , I},Pi,t,pt,Pk,t,ai,bi,ci,di,Pm

i,PM

i,pm,pM,βi,t,γi,t,. δt,εt,θt,. ω1,

ω2,ω3,ω4,ω5,NG

S=1.

2: While βi,t,γi,t,δt,εtand θtare not converged do

3: for i∈ I do

4: Update P?

i,t=P?

i,tNG

S+1with (18), p?

t=p?

tNG

S+1with (19), βi,tNG

S+1with (20),

γi,tNG

S+1with (21), δtNG

S+1with (22), εtNG

S+1with (23), and θtNG

S+1with (24).

5: end for

6: Let NG

S=NG

S+1.

7: end while

3.2. Demand’s Best Electricity-Usage Strategy

While the total electricity consumption was retained, the electricity charge of the EU was

minimized with the knowledge of the current electricity price

pt

, and this is another part of the

Stackelberg game along with Algorithm 2: Energy User’s Proﬁt Maximization (Market-adaptive

electricity-usage scheduling algorithm) which is newly proposed in this subsection to solve P2.

Algorithm 2 is one of the electricity-usage controlling strategies of the demand that is applicable

to the time-varying electricity market. Firstly, it was assumed that the

Pm

k

and

PM

k

are the minimum

and maximum electricity-usage values for each tof each EU, respectively, and the total sum of the

electricity usage of the entire time for each EU should be greater than or equal to the minimum requisite

electricity consumption

ϕk

. Furthermore, the entire time of the scheduling of the electricity usage of

the EU was set to T∈ T ={1, . . . , T}.

To initialize the electricity usage of each scheduling time before the applying of the proposed

algorithm, it was assumed that each EU is supposed to consume

ϕk/T

during every scheduling time,

and the electricity of

ϕk

is used for the entire time

T

. Further,

ˆ

Pk,t

was set as the amount of electricity

that the kis expected to use during the current t, and the initial value of

Pk,t

is allocated to

ϕk/T

. Then,

Energies 2018,11, 2315 12 of 23

in consideration of the electricity price of every scheduling time, the

ˆ

Pk,t

can be increased or decreased

by the ﬂuctuation of the electricity price. The amount of change in the electricity usage that is due to

changes in the electricity price, ∆Pk,t, is as follows:

∆Pk,t=ˆ

Pk,t∗1−pt

pt(25)

pt=1−1

tw∗pt−1+1

tw

∗pt(26)

P?

k,t=ˆ

Pk,t+∆Pk,t(27)

where

pt

is the base price as the historical average of

pt

based on the electricity prices of the previous

time slots,

tw

is the window size of the time slots, and

P?

k,t

is the adjusted electricity usage according to

the proposed algorithm. In Equation (25), a calculation of the amount of the change in the electricity

usage is performed using the ratio of the base price to the current price, thereby acknowledging that

the price that breaks past the average is a substantially increased price and the electricity consumption

is reduced by the rate of increase, and vice versa. Then, in Equation (27), the electricity usage is

updated for the current scheduling time.

To obtain insight regarding

pt

according to

tw

in (26), it was assumed that the

tw

values are 1,

3, and 5. When

tw

is 1,

∆Pk,t

is not generated in its structure, thereby meaning that the electricity

consumption is calculated regardless of the electricity-price ﬂuctuation, and this can be used for

the control of the proposed algorithm result. Alternatively, the increasing

of tw

to 3 or 5 means the

determination of how sensitively

∆Pk,t

is able to react to the current market price, since it was assumed

that

pt

is the average of the prices in the previous

tw

occasions. This provides an opportunity to

adaptively reﬂect the market characteristics to the rapidly ﬂuctuating or the gentle market. To retain the

total electricity consumption in the proposed algorithm, the

∆Pk,t

is uniformly collected or distributed

for each remaining time block. If the

∆Pk,t

is negative, the

∆Pk,t

x amount is divided by the total

remaining time

T−t

, and it is then distributed among each of the scheduling time blocks, and

ˆ

Pk,t+1

is

updated as follows:

ˆ

Pk,t+1=ˆ

Pk,t−∆Pk,t/(T−t)(28)

By performing (28), it is possible to constantly retain the electricity that is for consumption,

ϕk

.

We propose Algorithm 2: Energy User’s Proﬁt Maximization reﬂecting the whole description and

formulas in Section 3.2 as a proﬁt maximization algorithm performed by the EU.

Remark 3.

Algorithm 2 was proposed to solve the problem P2. The equilibrium of P2 is involved in the iteration

t in Algorithm of Social-Welfare Maximization, which gradually reaches the equilibrium point whenever t

increases. According to the maximization structure of the proposed algorithm, if the pre-deﬁned total energy

consumption ϕkare all distributed according to the algorithm, the equilibrium can be stably reached.

3.3. Complexity Analysis

The computational complexity of the Algorithm of Social-Welfare Maximization based on the

optimization technique used in this paper can be evaluated as follows. First, the complexity of

the gradient method updating dual variables to obtain the optimal price

p?

t

and the optimal power

generation

P?

i,t

in Slave-loop Algorithm (Algorithm 3) of Algorithm 1 linearly increases with the

number of generators Iand the number of iterations

NG

S

, i.e.,

ONG

SI

where

O{·}

is Big O notation.

Second, as provided in Section 3.1, the dual function P3 is always convex by proof of Theorem 1,

and the gradient method was employed to update

{βi,t,γi,t,δt,εt,θt}

toward the optimal solution with

guaranteed convergence [

29

]. Thus, in Master-loop Algorithm of Algorithm 1, the complexity of the

Dinkelbach method [

29

] to update

η

is independent of I and linearly increases with the number of

Energies 2018,11, 2315 13 of 23

iterations

T2

, i.e.,

ONG

MNG

SI

. Third, when we consider the number of EUs

K

in Algorithm 2 and the

number of iteration

t

, the complexity becomes

OtNG

MNG

SIK

. Therefore, the total complexity of the

proposed algorithm is

OtNG

MNG

SIK

. For comparisons, the complexity of the exhaustive search [

31

]

is roughly O{(K+I)(K+I)}, where Kand Iare the number of generators and EUs, respectively.

3.4. Schematic Overview and Application of Proposed Algorithms

In this paper, the two game players (generators and EUs) are supposed to participate in the

Stackelberg game, and play the game in order to maximize their monetary proﬁt in the smart grid

demand response. We proposed two algorithms (Algorithm 1: Generator’s Proﬁt Maximization and

Algorithm 2: Energy User’s Proﬁt Maximization) in Sections 3.1 and 3.2, respectively, as an energy

strategy to achieve proﬁt maximization, and the generators and the EUs play Algorithms 1 and 2

alternately in time, respectively, as shown in Figure 2. The proposed game structure is “dynamic game”

from the following two reasons:

1.

The game players, generators and EUs, interact with observing the each other’s energy strategy

for proﬁt maximization. The generators observe the electricity consumption of the EUs, and the

EUs observe the electricity price of the generators in the proposed game operation.

2.

The proposed proﬁt maximization game was constructed as an iterative algorithm where the

energy strategies (Algorithms 1 and 2) are repeatedly performed up to the speciﬁed number of

times, for example 24 times as 24 h a day.

They play their algorithm in the Stackelberg game and do not play the game at the same time

but play alternately over time since the EU needs to know how much the generator has set the

current electricity price in order to optimally control his energy consumption

P?

k,t

, and the generator

needs to know how much power the EU is currently consuming in order to determine the electricity

price

p?

t

and the amount of power generation in each game play (the game play means the algorithm

operation). By running the algorithm repeatedly and alternatively over time and spying on each other’s

energy parameters, they can effectively maximize their own proﬁt. The Algorithm for Social-Welfare

Maximization is detailed in Algorithms 1 and 2 as the Stackelberg-loop algorithm with the iteration

number

T

. The total architecture of the proposed social welfare maximization including Algorithms 1

and 2 is described in Figure 2as an overview of the algorithms proposed in this paper.

Energies 2017, 10, x FOR PEER REVIEW 13 of 22

2. The proposed profit maximization game was constructed as an iterative algorithm where the

energy strategies (Algorithms 1 and 2) are repeatedly performed up to the specified number of

times, for example 24 times as 24 h a day.

They play their algorithm in the Stackelberg game and do not play the game at the same time

but play alternately over time since the EU needs to know how much the generator has set the current

electricity price in order to optimally control his energy consumption ,

⋆, and the generator needs

to know how much power the EU is currently consuming in order to determine the electricity price

⋆ and the amount of power generation in each game play (the game play means the algorithm

operation). By running the algorithm repeatedly and alternatively over time and spying on each

other’s energy parameters, they can effectively maximize their own profit. The Algorithm for Social-

Welfare Maximization is detailed in Algorithms 1 and 2 as the Stackelberg-loop algorithm with the

iteration number . The total architecture of the proposed social welfare maximization including

Algorithms 1 and 2 is described in Figure 2 as an overview of the algorithms proposed in this paper.

Figure 2. The diagram of the proposed iterative algorithm for social welfare maximization in smart

grid.

The proposed methodology to achieve the social-welfare maximization can provide the

advantage to maximize the monetary profit of the generators and EUs, but we also reveal the

following considering points expected in real-world implementation; 1. Algorithm 1 proposed to

maximize the profit of the generator in this paper derives a sub-optimal solution that can reduce the

computational complexity compared to the brute-force approach to be suitable for real-time

optimization in smart grid demand response. The performance of the algorithm can vary slightly

depending on the initial point and the values of the variables that make up the Algorithm; 2.

Algorithms 1 and 2 proposed in this manuscript can be implemented in a smart meter or the EMC as

a form of software to automatically control the power consumption of energy user’s appliances and

facilities. It is a system that can operate only in limited areas equipped with the AMI; 3. Generators

are required to have the ability to flexibly control their power generation to participate in the

proposed algorithm. We need to consider practical implementations to enable real-time operation of

the proposed cooperative and simultaneous usage of coalitional game theory methods.

1. In the EU side, the implementation method of the proposed profit maximization algorithm is as

follows: The proposed profit maximization algorithm and formulas is applicable to general

demand response applications between generators and EUs such as, residential households,

electrical appliances, new smart appliances and internet of things (IoT) devices as a real-world

Figure 2.

The diagram of the proposed iterative algorithm for social welfare maximization in smart grid.

Energies 2018,11, 2315 14 of 23

The proposed methodology to achieve the social-welfare maximization can provide the advantage

to maximize the monetary proﬁt of the generators and EUs, but we also reveal the following considering

points expected in real-world implementation; 1. Algorithm 1 proposed to maximize the proﬁt of the

generator in this paper derives a sub-optimal solution that can reduce the computational complexity

compared to the brute-force approach to be suitable for real-time optimization in smart grid demand

response. The performance of the algorithm can vary slightly depending on the initial point and the

values of the variables that make up the Algorithm; 2. Algorithms 1 and 2 proposed in this manuscript

can be implemented in a smart meter or the EMC as a form of software to automatically control the

power consumption of energy user’s appliances and facilities. It is a system that can operate only

in limited areas equipped with the AMI; 3. Generators are required to have the ability to ﬂexibly

control their power generation to participate in the proposed algorithm. We need to consider practical

implementations to enable real-time operation of the proposed cooperative and simultaneous usage of

coalitional game theory methods.

1.

In the EU side, the implementation method of the proposed proﬁt maximization algorithm is

as follows: The proposed proﬁt maximization algorithm and formulas is applicable to general

demand response applications between generators and EUs such as, residential households,

electrical appliances, new smart appliances and internet of things (IoT) devices as a real-world

scenario. In the EU side which can be used at high priced hours, we can effectively reduce

electricity charges by adjusting the energy usage according to the electricity price with

the proposed Market-Adaptive Electricity-Usage Scheduling Algorithm (Algorithm 2 in the

manuscript). To implement the Market-Adaptive Electricity-Usage Scheduling Algorithm,

automatic electricity usage controller can be needed to be implemented and connected to EU

applications (the residential households, IoT devices and etc.). The automatic electricity usage

controller can be developed by porting a functional software which quickly and dynamically

performs the automatic electricity usage control to an AMI or an EMC. In the case of AMI and

EMC, it is possible to transmit and receive electricity price information in real time through

power line communications in smart grid. Based on this, the proposed algorithm can be fully

implemented and operate to achieve energy usage optimization and electricity charge savings.

To summarize, EUs should choose their appliances or facilities to automatically control their

energy usage to maximize monetary proﬁt, and if they are connected to a smart meter or EMC

equipped with our proposed algorithms, the proposed proﬁt maximization system will simply

be able to operate. We think that the proposed system can be implemented in the direction of

utilizing existing infrastructure such as the smart meter and EMC.

2.

On the generator side, the implementation method of the proposed proﬁt maximization algorithm

is as follows: In order to realize the optimal power generation and optimal pricing based

on the proposed algorithm in the generators side, the generators should be able to integrate

and manage the total power generation and the electricity price by forming a coalitional for

proﬁt maximization themselves. Or as a top authority for power generators that are already

integrated and managed by the government can implement the proposed coalitional game theory

methods. Or a third party such as a power retailer that runs various demand response programs

can implement the proposed game theory. If the proposed algorithm is implemented and

operated, it should be able to interact with another game player, energy user, with through power

line communication or wireless local area network (WLAN) on the smart grid for information

communication, such as real-time electricity price and electricity consumption exchange required

by this algorithm.

In view of appropriate time intervals for this real-time operation and implementation, we consider

the time interval of one hour is reasonable, and we can think about a smaller or larger interval based

on this one hour. For example, a 10-min interval that is smaller than 1 h is expected to cause confusion

because too much dynamic power generation and power consumption changes for both the generators

Energies 2018,11, 2315 15 of 23

and the energy users can be caused. On the other hand, if we set a time interval greater than one hour

to a time interval, the proposed proﬁt maximization system can be somewhat inefﬁcient if we run the

system once a day at a speciﬁc time because we have a fairly wide variation in power consumption

trends during a day. Therefore, it is reasonable to set the time interval appropriately between minimum

1 h and maximum 24 h in consideration of country, region and environment in which the proposed

proﬁt maximization system operates.

4. Simulation Results

In this section, numerical results are provided to demonstrate the effectiveness of the proposed

algorithm. The system setup is as follows: six generators and 12 EUs are considered based on the

IEEE 39-BUS system [

32

]. The graph of the communication network of these generators and EUs

shows that they are strongly connected. The parameters of the EUs and the generators are given in

Tables 2and 3[

32

]. The constant time length

tl

was set to 1 h, and the total scheduling time was

set to

T=

24 (one day) with the minimum requisite electricity consumption

ϕk=T∗Pm

k+PM

k/

2.

Further, the initial value of the

ˆ

Pk,t

was set to

ˆ

Pk,t=ϕk/T

, and it was assumed that the lower bound

of the electricity price per unit of energy is nonzero, whereas the upper bound is 50 (cents/kWh).

The window size

tw

was set to be from 1 to 10. In the following results, the maximum number of

slave-loop iterations was set to 6.

Table 2. Parameters of generators for IEEE 39-BUS [32].

Generator Parameters

Node aibicidiPm

iPM

i

1 0.0024 5.56 30 0.00021 60 339.69

2 0.0056 4.32 25 0.00031 25 479.10

3 0.0072 6.60 25 0.00011 28 290.4

4 0.0047 3.14 16 0.00022 40 306.34

5 0.0091 7.54 6 0.00041 35 593.80

6 0.0046 4.76 12

0.000121

30 443.41

Table 3. Parameters of EUs for IEEE 39-BUS [32].

Demand Parameters

Node Pm

kPM

kNode Pm

kPM

k

1 50 100.34 7 80 137.93

2 100 159.13 8 50 84.19

3 40 80.56 9 50 104.06

4 30 123.98 10 78 119.36

5 80 109.55 11 103 176.19

6 40 76.34 12 67 147.26

Figure 3shows the evolution of the generation power (kWh)

Pi,t

and the electricity price

(cents/kWh)

pt

of each generator optimized by the proposed iterative Algorithm 1. “Iterations” in

Figure 3is the number of master-loop iterations

NG

M

of Algorithm 1, and the result was averaged over

1000 independent simulations, each of which involved different scheduling time slots. From Figure 3,

it is evident that the convergence of the generation power and the price can be achieved within seven

iterations on average to maximize the

UG

t

. All of the values in each iteration change with satisfying the

AG

in (9). To maximize the generator proﬁt, the generation power of each generator converges to a

different optimal value, which is affected by the constant parameters such as

ai

,

bi

,

ci

,

di

,

Pm

i

, and

PM

i

,

and the price converges to its optimal value in conjunction with the generation power.

Energies 2018,11, 2315 16 of 23

Energies 2017, 10, x FOR PEER REVIEW 15 of 22

Table 2. Parameters of generators for IEEE 39-BUS [32].

Generator Parameters

Node

1

0.0024

5.56

30

0.00021

60

339.69

2

0.0056

4.32

25

0.00031

25

479.10

3

0.0072

6.60

25

0.00011

28

290.4

4

0.0047

3.14

16

0.00022

40

306.34

5

0.0091

7.54

6

0.00041

35

593.80

6

0.0046

4.76

12

0.000121

30

443.41

Table 3. Parameters of EUs for IEEE 39-BUS [32].

Demand Parameters

Node

Node

1

50

100.34

7

80

137.93

2

100

159.13

8

50

84.19

3

40

80.56

9

50

104.06

4

30

123.98

10

78

119.36

5

80

109.55

11

103

176.19

6

40

76.34

12

67

147.26

Figure 3 shows the evolution of the generation power (kWh) , and the electricity price

(cents/kWh) of each generator optimized by the proposed iterative Algorithm 1. “Iterations” in

Figure 3 is the number of master-loop iterations

of Algorithm 1, and the result was averaged

over 1000 independent simulations, each of which involved different scheduling time slots. From

Figure 3, it is evident that the convergence of the generation power and the price can be achieved

within seven iterations on average to maximize the . All of the values in each iteration change

with satisfying the in (9). To maximize the generator profit, the generation power of each

generator converges to a different optimal value, which is affected by the constant parameters such

as , , , , , and , and the price converges to its optimal value in conjunction with the

generation power.

Figure 3. Generation power , and electricity price versus iteration with = 6 generators.

Figure 4 demonstrates the evolution of the generator profit calculated with the employment of

the optimal values of the generation power and the electricity price from Figure 3. The optimization

Figure 3. Generation power Pi,tand electricity price ptversus iteration with I=6 generators.

Figure 4demonstrates the evolution of the generator proﬁt calculated with the employment of the

optimal values of the generation power and the electricity price from Figure 3. The optimization process

is performed using the proposed master- and slave-loop algorithms in Algorithm 1. Even though

the generator power curves in Figure 3show ﬂuctuations with the iterations, it is evident that all

of the curves continually increase as the iterations continue. Furthermore, a baseline scheme was

set as a proﬁt maximization strategy which optimizes only one of the two optimization parameters

(Pi,t,pt)

compared to the proposed Algorithm 1, where the other optimization parameter was ﬁxed.

The “ﬁxing” means that the ﬁxed parameter was not optimized but its value is constant, and the ﬁxed

value of

Pi,t

was set as just one of the possible values to

Pm

i+PM

i/

2. In Figure 4, it is conﬁrmed that

the proﬁt of the proposed algorithm is greater than that of the baseline scheme, thereby conﬁrming the

proper operability of the algorithm. Furthermore, the proposed optimization problem in this paper

is a non-convex optimization problem (P1 in Section 2) and the global optimum for the non-convex

optimization problem is usually only achieved by using a brute-force approach (or exhaustive search).

In this paper, to solve P1, Lagrange dual method and non-linear fractional programming were applied

to ﬁnd a sub-optimal point close to the global optimum. From Figure 4, the global optimum point was

further suggested, and it was conﬁrmed that the difference from the sub-optimal point is within 1%

when the iteration is converged to 7. Since the sub-optimal point we found may differ depending on

the initial point in the proposed proﬁt maximization algorithm. So we propose a method to calculate

multiple sub-optimal points with multiple initial points to determine the more maximized point among

the multiple sub-optimal points. In the implementation phase, this algorithm should be devised to

generate multiple initial points and calculate the sub-optimal point corresponding to the multiple

initial points.

Energies 2018,11, 2315 17 of 23

1

Figure 4. Generator proﬁt versus iteration.

Remark 4.

It is not possible to run the baseline scheme by ﬁxing the

pt

. When the

pt

is ﬁxed, it is obvious that

the Pi,twill be set to the minimum value to maximize the proﬁt according to the structure of Equation (4).

In Figures 5–7, the electricity price, the electricity usage of one of the EUs according to the price,

and the accumulated electricity charge are depicted over time. All of the results were simulated for

T=

24 h (one day) with 1-h intervals, and the window size was set to be

tw=

3. Figures 5–7show the

changes in the electricity price as Case #1: side-crawl trend, Case #2: rising-tide trend, and Case #3:

falling-tide trend, respectively, and these are case studies where the possible trends were analyzed in

the real market to determine the effectiveness of the proposed Algorithm 2. Please note that, regarding

the totals of Algorithms 1 and 2, Algorithm of Social-Welfare Maximization, the electricity prices of

all of these cases were calculated using Algorithm 1, while the electricity usage and the electricity

charge were calculated using Algorithm 2. Also, these three cases are three of the results that were

obtained by the performance of the entire Algorithm of Social-Welfare Maximization over 1000 times.

From Figures 5–7, the changes of the electricity usage of EU 1 show that the proposed Algorithm 2

operates in a market-adaptive manner (EU 1 is merely a representative of the EUs, and the rest of the

EUs show the same tendency), as the expectation of this is described in Section 3.2. Based on this

market-adaptive manner, the electricity usage is reduced when the electricity is expensive, whereas

the electricity usage increases when the electricity is cheap, thereby meaning that the total electricity

consumption is the same, but the electricity charge can be considerably reduced. Please note that

this means that the proposed algorithm can alleviate the existing PAR reduction problem because

the electricity usage immediately responds to the price according to the results of Figures 5–7. In the

real world implementation, it is also possible to alleviate the problem that the price greatly ﬂuctuates

because if the price changes greatly, the generators will lose its proﬁts and will not be able to withstand

it. We can contribute to stabilization of power generation and PAR reduction, which is one of the ultimate

goals of DR through our proposed algorithm. Figures 5–7show that the average electricity usage is

the same in any trend according to the

AEU

, and the electricity charge can be greatly reduced from

13–18%. The comparator, which consumes the same power at all times, is “unadjusted” in the legend.

Furthermore, from Figures 5–7, it can be seen that the greater the variability in the market, the greater

the possibility of the adaptation to the market that facilitates the attainment of a greater benefit.

Energies 2018,11, 2315 18 of 23

Energies 2017, 10, x FOR PEER REVIEW 17 of 22

Algorithm 2 operates in a market-adaptive manner (EU 1 is merely a representative of the EUs, and

the rest of the EUs show the same tendency), as the expectation of this is described in Section 3.2.

Based on this market-adaptive manner, the electricity usage is reduced when the electricity is

expensive, whereas the electricity usage increases when the electricity is cheap, thereby meaning that

the total electricity consumption is the same, but the electricity charge can be considerably reduced.

Please note that this means that the proposed algorithm can alleviate the existing PAR reduction

problem because the electricity usage immediately responds to the price according to the results of

Figures 5–7. In the real world implementation, it is also possible to alleviate the problem that the price

greatly fluctuates because if the price changes greatly, the generators will lose its profits and will not

be able to withstand it. We can contribute to stabilization of power generation and PAR reduction,

which is one of the ultimate goals of DR through our proposed algorithm. Figures 5–7 show that the

average electricity usage is the same in any trend according to the , and the electricity charge can

be greatly reduced from 13–18%. The comparator, which consumes the same power at all times, is

“unadjusted” in the legend. Furthermore, from Figures 5–7, it can be seen that the greater the

variability in the market, the greater the possibility of the adaptation to the market that facilitates the

attainment of a greater benefit.

Figure 5. Electricity price with time (Case #1: side-crawl trend) and the electricity usage adjusted

accordingly, and the confirmation of the electricity-charge saving.

Figure 5.

Electricity price with time (Case #1: side-crawl trend) and the electricity usage adjusted

accordingly, and the conﬁrmation of the electricity-charge saving.

Energies 2017, 10, x FOR PEER REVIEW 18 of 22

Figure 6. Electricity price with time (Case #2: rising-tide trend) and the electricity usage adjusted

accordingly, and the confirmation of the electricity-charge saving.

Figure 7. Electricity price with time (case #3: falling tide trend) and the electricity usage adjusted

accordingly and confirmation of electricity charge savings.

Figure 6.

Electricity price with time (Case #2: rising-tide trend) and the electricity usage adjusted

accordingly, and the conﬁrmation of the electricity-charge saving.

Energies 2018,11, 2315 19 of 23

Energies 2017, 10, x FOR PEER REVIEW 18 of 22

Figure 6. Electricity price with time (Case #2: rising-tide trend) and the electricity usage adjusted

accordingly, and the confirmation of the electricity-charge saving.

Figure 7. Electricity price with time (case #3: falling tide trend) and the electricity usage adjusted

accordingly and confirmation of electricity charge savings.

Figure 7.

Electricity price with time (case #3: falling tide trend) and the electricity usage adjusted

accordingly and conﬁrmation of electricity charge savings.

We can think of the proposed game methodology as a type of DR because the demand is responsive

to price as the reviewer commented, but it is not exactly a DR program. In general DR should include

the ability to convert excess power consumption to optimal power consumption (“load reduction”),

as well as to provide price elasticity to eliminate inefﬁciencies due to ﬁxed prices. In the proposed

game methodology, we do not reduce the power consumption of the energy user, but optimize how

efﬁciently the speciﬁed power consumption will be consumed in a given period of time. In Figures 5–7,

to maximize the proﬁt of the EU, we estimated the adjusted electricity usage which should be consumed

for each hour when we know an amount of energy we should use during the day. When the electricity

price changes as shown in the top ﬁgure of Figures 5–7, respectively, the electricity charge can be

reduced by 13%, 18% and 16% when calculating the electricity usage as shown in the middle ﬁgure of

Figures 5–7based on the proposed algorithm. We studied in this paper how to use the load efﬁciently

within the reduced load when the energy user receives an instruction to reduce the load on the DR.

Remark 5.

In the proposed algorithm, the proﬁt function of each generator and EU is deﬁned mathematically in

Section 2so that the proﬁt generation and distribution can be fair. Each generator and EU does not need special

skills for fairness because it is an independent entity and can take its own proﬁts by the deﬁned proﬁt function.

Figure 8represents the way the market volatility (degree of change of the electricity price) provides

the EU with beneﬁts when the window size changes from 1 to 10. The beneﬁt function is ﬁrst given in

Figure 8, as follows:

Beneﬁt =1−UEU

UEU ∗100 (29)

Energies 2018,11, 2315 20 of 23

where

UEU

is the total electricity charge that is accumulated for the

T

when the electricity usage is

unadjusted and allocated in a totally ﬂat manner. Note that the reduction percentages in Figures 5–7

were calculated using Equation (29). Also note that, in Figure 8, the peak-to-average ratio (PAR) is an

approximate indicator of the electricity usage as it is known, and it was assumed that the change in

the PAR reﬂects the volatility of the electricity price, because the electricity usage ﬂuctuates when the

electricity price ﬂuctuates according to Algorithm 2. It is evident that Algorithm 2 responds sensitively

to the changes in the electrical price; that is, the smaller that

tw

is, the greater the beneﬁt that is derived,

with the exception of the case where

tw

is 1. Here,

tw

is 1, and this means that the electricity usage

has not been adaptively adjusted to the market using Equation (26), while it is also accurate that the

PAR is 1. In the meantime, it is possible to observe the trends of the increasing beneﬁt as the PAR

is increased, and this means that, as the market volatility is increased, Algorithm 2 can increase the

beneﬁt. The purpose of this simulation, however, is not the raising of the PAR to increase the beneﬁt.

The purpose is the demonstration of the beneﬁt that is attained from the proposed algorithm when the

EU has some PAR.

Energies 2017, 10, x FOR PEER REVIEW 20 of 22

Figure 8. Benefits versus window sizes with varying PAR.

Summary of Simulation Results and Insights

We showed the results of maximizing the profit of the generators through Algorithm 1 in Figures

3 and 4, and we demonstrated the results of maximizing the profit of the EUs based on Algorithm 2

in Figures 5–7. In Figures 3 and 4, we confirmed that the profit of the generators can be improved to

about 45% compared to existing (baseline scheme) scheme, and the electricity charge of the EUs can

be reduced by 15.6% on average compared to that of when algorithm was not applied. Please note

that the amount of power consumption of the EUs is same when the algorithm was applied and not

applied. From Figures 3–7, we confirmed that the proposed profit maximization algorithms

effectively improves the monetary profit of generators and EUs. We summarize the gain of monetary

profit from the proposed algorithms in Table 4.

Table 4. The gain of monetary profit from the proposed algorithm.

Energy Strategy

Gain of Profit

Algorithm 1: Generator Profit

Maximization

About 45% (compared to existing (beseline scheme) scheme)

Algorithm 2: Energy User

Profit Maximization

15.6% on average (compared to that of when algorithm was not applied)

To provide and investigate the influence of the changes of the PAR on the monetary profit

generated from the proposed algorithms, Figure 8 showed the change of the profit of the EUs. From

Figure 8, we confirmed that the EUs gain more profit as the PAR increases. What this means that

more EUs will participate in this game and algorithms when the PAR increases, and it can lead to

PAR reduction because the monetary gain that can be obtained for the current PAR has no choice but

to be limited, and it can be predicted that as many people share the profit, the PAR would decrease.

This also suggest the proposed algorithm not only can contribute to maximization of the profit of the

generators and the EUs, but also the desired goal of demand response, PAR reduction, at the same

time.

5. Conclusions

In this paper, to maximize the monetary profit in real-time price DR systems, we formulated the

Stackelberg game-based non-convex optimization problem, and proposed two energy strategies,

Algorithm 1: Generator’s Profit Maximization and Algorithm 2: Energy User’s Profit Maximization,

as an optimal solution. In the problem formulation, we newly formulated the generator profit

Figure 8. Beneﬁts versus window sizes with varying PAR.

Summary of Simulation Results and Insights

We showed the results of maximizing the proﬁt of the generators through Algorithm 1 in Figures 3

and 4, and we demonstrated the results of maximizing the proﬁt of the EUs based on Algorithm 2 in

Figures 5–7. In Figures 3and 4, we conﬁrmed that the proﬁt of the generators can be improved to

about 45% compared to existing (baseline scheme) scheme, and the electricity charge of the EUs can

be reduced by 15.6% on average compared to that of when algorithm was not applied. Please note

that the amount of power consumption of the EUs is same when the algorithm was applied and not

applied. From Figures 3–7, we conﬁrmed that the proposed proﬁt maximization algorithms effectively

improves the monetary proﬁt of generators and EUs. We summarize the gain of monetary proﬁt from

the proposed algorithms in Table 4.

Table 4. The gain of monetary proﬁt from the proposed algorithm.

Energy Strategy Gain of Proﬁt

Algorithm 1: Generator Proﬁt Maximization About 45% (compared to existing (beseline scheme) scheme)

Algorithm 2: Energy User Proﬁt Maximization 15.6% on average (compared to that of when algorithm was not applied)

Energies 2018,11, 2315 21 of 23

To provide and investigate the inﬂuence of the changes of the PAR on the monetary proﬁt

generated from the proposed algorithms, Figure 8showed the change of the proﬁt of the EUs.

From Figure 8, we conﬁrmed that the EUs gain more proﬁt as the PAR increases. What this means that

more EUs will participate in this game and algorithms when the PAR increases, and it can lead to PAR

reduction because the monetary gain that can be obtained for the current PAR has no choice but to be

limited, and it can be predicted that as many people share the proﬁt, the PAR would decrease. This also

suggest the proposed algorithm not only can contribute to maximization of the proﬁt of the generators

and the EUs, but also the desired goal of demand response, PAR reduction, at the same time.

5. Conclusions

In this paper, to maximize the monetary proﬁt in real-time price DR systems, we formulated

the Stackelberg game-based non-convex optimization problem, and proposed two energy strategies,

Algorithm 1: Generator ’s Proﬁt Maximization and Algorithm 2: Energy User’s Proﬁt Maximization,

as an optimal solution. In the problem formulation, we newly formulated the generator proﬁt function

to reﬂect the inﬂuence of the electricity usage of EUs. To solve the non-convex optimization problem,

nonlinear fractional programming and the Lagrange-multiplier method were adopted in proposing

the energy strategy for the generators. Also, we newly proposed the energy strategy for the EUs

based on the time-window-based market-adaptive manner. We greatly improve the monetary proﬁt of

the generators and EUs using the proposed two energy strategies by optimizing the amount of the

power generation and the electricity price in the generator side, and electricity consumption in the EU

side. In Figures 3and 4, we conﬁrmed that the proﬁt of the generators can be improved to about 45%

compared to the existing (baseline) scheme, and the electricity charge of the EUs can be reduced by

15.6% on average compared to that of when algorithm was not applied. Furthermore, we conﬁrmed

that the simulation result from Figure 8suggests the proposed algorithm can contribute to not only

maximization of the proﬁt of the generators and the EUs, but also the desired goal of demand response,

PAR reduction, at the same time.

Author Contributions:

Y.M.H. has contributed to the theoretical approaches, simulation and preparing the paper;

I.S. has contributed to the theoretical approaches and literature review; Y.G.S. has contributed to the theoretical

approaches and preparing the paper; H.-J.L. has contributed to the theoretical approaches, literature review and

paper writing; J.Y.K. has designed and supervised the paper.

Funding:

This work was, in part, supported by Basic Science Research Program through the National Research

Foundation of Korea funded by the Ministry of Education (NRF-2016R1D1A1B03933872), and in part supported

by “Human Resources Program in Energy Technology (No. 20174010201620)” of the Korea Institute of Energy

Technology Evaluation and Planning (KETEP), granted ﬁnancial resource from the Ministry of Trade, Industry &

Energy, Republic of Korea, and in part by Kwangwoon University in 2018.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access

article distributed under the terms and conditions of the Creative Commons Attribution

(CC BY) license (http://creativecommons.org/licenses/by/4.0/).