SICE202d 12 Suehiro

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Decentralized Control of Smart Grid by using Overlapping Information

Tomoharu Suehiro1 and Toru Namerikawa1

1Department of System Design Engineering Keio University Kanagawa Japan

(Tel +81-45-563-1151E-mail tomoharunlsdkeioacjp namerikawasdkeioacjp)

Abstract This paper deals with a decentralized control of smart grid by using overlapping information for load frequency

of power networks introducing distributed power generations The control objective is to minimize the cost function of

load frequency control problem We expand the state space of the system of power networks and propose a decentralized

state feedback control of subsystems for the expanded system Then we contract the decentralized feedback law to match

the original system Finally we show the effectiveness of the load frequency control by using the proposed decentralized

control method through the simulation result of decentralized large scale power network systems

Keywords Decentralized Control Distributed Generations Load Frequency Overlapping Decomposition Smart Grid

1 INTRODUCTION

The decision making problems for using different in-

formation concerning underlying uncertainties has been

studied since the 1950rsquos Some of typical problems in

these fields are called as game problem and team prob-

lem and so on In the 1970rsquos1980rsquos relations with the

decentralized control and decision making was strength-

ened and information structured systems that have mul-

tiple subsystems getting different information had been

actively studied Also decentralized control methodolo-

gies were proposed and discussed such as [1 2]

In recent years decentralized control methodologies

to apply these systems receive large attention again [3-

6] This is because new environmental and energy tech-

nologies for new electric power systems that is calledrdquosmart gridrdquo (Fig 1) where different power generators

and power storages cooperate in energetically and envi-

ronmentally optimal way are required Energy problems

and global warming have become the hottest problems

in the world Therefore distributed generations such as

wind power generations the battery systems are going

to be installed in electric power systems to save energy

resources However it is difficult to control all of these

distributed generations by oneoperation system and some

distributed generations have bad effects on system fre-

quency and fluctuation of voltage Hence it is necessary

to operate these generations in a decentralized coordi-

nated and safe way The number of studies for smart grid

is increasing (see [7]) and decentralized control method-

ologies are also studied extensively The optimal central-

ized control of an old electric power system was deeply

studied in 1970rsquos[8] Recently system frequency control

in an new power networks installing wind power genera-

tions battery energy storage system has been studied by

[9] and a distributed control methodology for such system

is proposed by [10]

In this paper we propose a decentralized control of

smart grid with information structure constraints by us-

ing overlapping information for load frequency of power

networks introducing distributed power generations Weexpand the state space of the system of power networks

propose a decentralized state feedback control of sub-

systems and contract the decentralized feedback law to

match the original system Then we simulate decentral-

ized large scale power network systems and show theeffectiveness of the load frequency control using the pro-

posed decentralized control method by the simulation re-

sult

Fig 1 smart grid

The following notations are used in this paper ℤ+ is

the nonnegative integer set ℝ shows the real space of

the n-dimension ℝ1038389times shows the real space of the mtimesn-dimension 1103925907317 shows the (1038389983084 1103925)th sub-matrix of a ma-

trix 907317 showsthe unit matrix of the ntimes n-dimension E

is an expectation operator and 1038389(983084983084983084983084983084 ) is

a set of matrices related to discrete-time algebraic Riccatiequation

2 PROBLEM FORMULATION

We consider electric power networks with two sub-

systems The overall dynamics of electric power net-

works is given by the

( + 1) = () + () + () (1)

where isin ℤ+ isin ℝ is the state isin ℝ

1038389 is the input

isin ℝ is the white noise with variance consistsof 1 isin ℝ

1 2 isin ℝ2 3 isin ℝ

3 is composed of

1 isin ℝ10383891 983084 2 isin ℝ

10383892 consists of 1 isin ℝ1 2 isin

SICE Annual Conference 2012

August 20-23 2012 Akita University Akita Japan

PR0001120000-

yen

400 copy2012 SICE

0125-125-



ℝ2 Relation of these is represented as

= [ 1 983084 2 983084 3 ] 983084 = 1 + 2 + 3983084 (2)

= [ 1 983084 2 ] 983084 = 1 + 2983084 (3)

= [ 1 983084

2 ] 983084 = 1 + 2983086 (4)

In this assumption the state of power subsystem 1 is1 the state of power subsystem 2 is 3 and the interac-

tion state between two subsystems is 2 1 is the input

for subsystem 1 and 2 is the input for subsystem 2

The matrices isin ℝtimes isin ℝ

times1038389 and isin ℝtimes

are represented as

=

⎡⎣ 11 12 13

21 22 23

31 32 33

⎤⎦ 983084

=

⎡⎣

11 12

21 22

31 32

⎤⎦ 983084 =

⎡⎣

11 12 21 22

31 32

⎤⎦ 983086 (5)

We assume the situation like Fig 2 where power subsys-

tem 1 can measure state 1 2 and power subsystem 2

can measure state 1 2 3 For this system we con-

sider a decentralized feedback control law where subsys-

tem 1 decides the input 1 from 1 2 and subsystem 2

decides the input 2 from 1 2 3

Fig 2 Electric Power System

The part of the input dealing with the state directly ()has the following structure

() = minus

⎡⎣ 1()

2()3()

⎤⎦ 983084 (6)

where is shown as Eq (7)

=983131 11 12 0

21 22 23983133

(7)

Here the following performance index is defined as

(0983084 ) = [infinsum=0

( + )]983084 (8)

where 0 is the initial state of the system isin ℝtimes is

a constant positive-semidefinite matrix and isin ℝ1038389times1038389

is a constant positive-definite matrix and are ex-

pressed by using 1 isin ℝ1times1 2 isin ℝ

2times2 3 isinℝ3times3 1 isin ℝ

10383891times10383891 2 isin ℝ10383892times10383892 as

= diag9831631

983084 2

983084 3983165983084 = diag983163

1983084

2983165983086 (9)

We consider a feedback control to minimize Eq (8) in

this situation

3 PROPOSED DECENTRALIZEDCONTROL

31 Expansion of the state space of the system

To set up a decentralized feedback control law we de-

compose the state into two overlapping components

and expand the system 983772 983772( + 1) = 983772983772() + 983772() + 983772() (10)

where 983772 isin ℝ983772 is composed of 9837721 = [ 1 983084 2 ] and 9837722 =

[ 2 983084 3 ] 9837720 is the initial state of the system 983772 983772 isinℝ983772times983772983084 983772 isin ℝ

983772times1038389 983772 isin ℝ983772times are constant matrices

and 983772 satisfies 983772 = 1 + 22 + 3

We also expand performance index as follows

983772 (9837720983084 ) = [infinsum=0

(983772 983772983772 + 983772)] (11)

where 9837720 is the initial state of the system 983772 983772 isin ℝ983772times983772 is

a constant positive-semidefinite matrix and 983772 isin ℝ1038389times1038389

is a constant positive-definite matrix 983772 and 983772 are

expressed by using 9837721 isin ℝ(1+2)times(1+2) 9837722 isin

ℝ(2+3)times(2+3) 9837721 isin ℝ


983772 = diag983163 9837721983084 9837722983165983084 983772 = diag983163 9837721983084 9837722983165983086 (12)

Here we introduce a linear transformation and

ℝ rarr ℝ983772983084 ( ) = (13)

ℝ983772 rarr ℝ983084 ( ) = (14)

These matrices satisfy = 907317 We relate (983084 ) and

( 983772983084 983772 ) by the following definition

Definition 1 The pair ( 983772983084 983772 ) includes the pair

(983084 ) if there exists a matrix such that 9837720 = 0 and

following two equations are satisfied for any input ()

( 0983084 ) = 983772( 9837720983084 )983084 for all ge 0 (15)

(0983084 ) = 983772 (0983084 ) (16)

If ( 983772983084 983772 ) includes (983084 ) then we say that ( 983772983084 983772 ) is an

expansion of (983084 ) and (983084 ) is a contraction of ( 983772983084 983772 )

We relate the matrices of ( 983772983084 983772 ) and (983084 ) by us-

ing proper dimensions matrices 983084 983084 983084 983084 as983772 = + 983084 983772 = + 983084 983772 = + 983084 983772 =

+ 983084 983772 = + With this representation

the condition for the inclusion is given by the following

theorem

Theorem 1 The pair ( 983772983084 983772 ) includes The pair(983084 ) if either

(i)

= 0983084 = 0983084 = 0983084 (17)

= 0983084 = 0 (18)

or

(ii) for 1038389 = 1983084 2983084 983084 983772 1103925 = 0983084 1103925minus1 = 0983084 1103925minus1 = 0983084 (19)

1103925minus1 = 0983084 1103925minus1 = 0983084 = 0983086 (20)

Proof The theorem can be proven based on [2]We choose matrices which satisfies (i) of this theorem as

=

⎡⎢⎣

1038389 1 0 00 1038389 2 00 1038389 2 00 0 1038389 3

⎤⎥⎦ 983084 1103925 =

⎡⎣ 1038389 1 0 0 0

0 1

21038389 2

1

21038389 2 0

0 0 0 1038389 3

⎤⎦ 983084

-126-



907317 = 0983084 907317 1038389 = 0983084 =

⎡⎢⎢⎣

0 1

212 minus

1

212 0

0 1

222 minus

1

222 0

0 minus

1

222

1

222 0

0 minus

1

232

1

232 0

⎤⎥⎥⎦ 983084

907317 1103925 = 0983084 907317 =

⎡⎢⎢⎣

0 0 0 00 1

4

2 minus

1

4

2 0

0 minus

1

42

1

42 0

0 0 0 0

⎤⎥⎥⎦ 983086 (21)

With this choice the system can be expanded to 983772 and

the matrices 983772 983772 and 983772 are described as

983772 =

983131 98377211

98377212

98377221 98377222

983133 =

⎡⎢⎣

11 12 0 13

21 22 0 23

21 0 22 23

31 0 32 33

⎤⎥⎦ 983084

983772 =

983131 98377211

98377212

98377221 98377222

983133 =

⎡⎢⎣

11 12

21 22

21 22

31 32

⎤⎥⎦

983084

983772 =

983131 983772 11 983772 12

983772 21 983772 22

983133 =

⎡⎢⎣

11 12 21 22 21 22 31 32

⎤⎥⎦ 983086 (22)

32 Decentralized Control of Two Subsystems for the

expanded system

Then we consider the new system 983772 corresponding

to the system 983772 983772 is expressed as follows

983772 983772( + 1) = 983772983772() + 983772() + 983772 () (23)

where 983772983084 983772 and 983772 are block lower triangular matri-ces of 983772983084 983772 and 983772 described as

983772 =

983131 98377211 0

98377221 98377222

983133983084 983772 =

983131 98377211 0

98377221 98377222

983133983084

983772 =

983131 983772 11 0

983772 21 983772 22

983133983086 (24)

The performance index of the system 983772 is the same as

Eq (16)

For the system 983772 which has an information structure

the decentralized feedback control to minimize Eq (8) isshown as follows

Theorem 2 [5] For the system 983772 with the infor-

mation structure where subsystem 1 can measure state 9837721

and subsystem 2 can measure state 9837721983084 9837722 the decentral-

ized feedback control to minimize Eq (8) is shown below

1() = minus( 983772 )119837721() minus ( 983772 )129837722() (25)

2() = minus( 983772 )219837721() minus 983772 9837722()

minus(( 983772 )22 minus 983772 )9837722() (26)

9837722( + 1) = ( )219837721() + ( )229837722() (27)

where = 983772 minus 983772 983772 1038389( 983772983084 983772983084 983772983084 983772983084 1983084 983772 )

and 1038389(( 983772)22983084 ( 983772)22983084 9837722983084 9837722983084 2983084 983772 ) 9837722() is theestimation of 9837722() calculated from 9837721

Proof Omitted see [5]

33 Contraction of State Feedback Law

The control law expressed above can be represented as

() = ()+() = minus 983772 983772()minus 983772 983772() where 983772()is the estimation of 983772() We use this control method into

the system 983772 and contract the method to match the sys-

tem We assume () for the system corresponding

to 983772() for the system 983772 and () = minus 983772 983772 minus 983772 983772()can contract to () = minus() minus () To find con-

dition of contractability we define contractability of the

estimation and control input

Definition 2 983772() is contractible to () if

( 0983084 ) = 983772( 9837720983084 )983084 for all ge 0983086 (28)

Definition 3 We regard () = minus 983772 983772minus 983772 983772() as

the control input of the system 983772 and () = minus()minus () as the control input of the system () =minus 983772 983772 minus 983772 983772() is contractible to () = minus() minus () if following two equations are satisfied for any

input ()( 0983084 ) = 983772 983772( 9837720983084 )983084 for all ge 0 (29)

( 0983084 ) = 983772 983772( 9837720983084 )983084 for all ge 0 (30)

We relate matrices such as 983772 = + 983084 983772 = + then the condition for the contractability is given by

the following Theorem This is a main result in this paper

Theorem 3 minus 983772 983772 minus 983772 983772() is contractible to

minus() minus () if 983772 can contract to and for 1038389 =1983084 2983084 983084 983772

1103925 = 0983084 1103925minus1 = 0983084

1103925minus1 = 0983084 = 0983086 (31)Proof ( 0983084 ) and 983772( 9837720983084 ) are expressed

as follows( 0983084 ) =

0 +sum1103925=0

983163minus1103925(1038389) + minus1103925(1038389)983165 (32)

983772( 9837720983084 ) =

983772 9837729837720 +

sum1103925=0

983163 983772 983772minus1103925 983772(1038389) + 983772 983772minus1103925 983772(1038389)983165 (33)

It can be said that Eqs (29) and (30) are equal as follows

from these Eqs and Definition 2

= 983772 983772 983084 minus1103925 = 983772 983772minus1103925 983772 (34)

minus1103925

= 983772

983772

minus1103925 983772983084 =

983772 (35)

We substitute Eqs defined previous subsections and use

Eqs in Theorem 1 for Eqs (34) and (35) By comparing

right-hand side and left-hand side of Eqs (34) and (35)

Eq (31) is derived

We choose matrices which satisfies this Theorem as

= 0983086 (36)

Then we get the decentralized control law of smart gridThe control input is shown below

983131 1()

2()

983133 = minus

983131 ( 983772 11)1 ( 983772 11)2 0

( 983772 21)1 ( 983772 21)2 + 983772 1 983772 2

983133⎡⎣ 1()

2()3()

⎤⎦

minus 983131 983772 12

983772 22minus

983772 983133 983131 2()

3()983133

(37)9831312( + 1)3( + 1)

983133 = ()21

9831311()2()

983133+ ()22

9831312()3()

983133 (38)

-127-



4 APPLICATION FOR SMART GRIDCONTROL

41 Electric Power System

We consider electric power network shown in Fig3

It focuses on Load frequency and is composed of two

power systems There are a thermal power plant and a

wind power plant in Area 1 and there are a battery sys-

tem and micro gas turbine generators in Area 2 and the

power supply is done to the electric power demand with

these power generations The detail and dynamics of the

electric power network is shown below

Battery

System

Thermal

Power Plant

Gas Turbin

System

Load

Load

ControllerSystem

Controller

System

Tie Line Power Flow(Area2 Area1)

Generator Model

Generator Model

MA1sDA1

1

MA sDA2

1

sTtie

SA1

SA2

PA1

PA2

f A1

f A2

PBcom

PGcom

PTcom

xgA1

xgA2

PB

PG

PLA2

PLA1PT

Ptie

Area1

Area2

WindPower Plant

PW

x1

xB

Fig 3 Power Network Model

411 Relationships of demand and supply

Relationships between electric demand and supply is

shown in Eq (39)

()

= 1038389() minus () (39)

1038389() is mechanical input of generator () is an

electric output of generator is a inertia constant of

generator and () is a deviation of rotating velocity

of generator System frequency changing rotation fre-

quency of rotating load power consumption shift This is

described as follows

() = () + () (40)

() is a load deviation and is a damping constant

In addition when it is assumed that all generators in onesystem are completely synchronous driving system can

be expressed as one equivalent model like Fig 4 isa inertia constant of equivalent generator

PL

MeqsD

1f

Pm1

Pm2

Pmn

Fig 4 Equivalent Generator Model

412 Tie-Line Power Flow

By DC method the tie-line power flow from Area 2 to

Area 1 1103925() is expressed like Eq (41)

1103925() = 1

( 2() minus 1()) (41)

is the reactance of the interconnected line between

Area 1 and Area 2 and 1() 2() are phase angles of

each system Using Frequency deviations of each system

1() 2() 1103925() the deviation of 1103925() is

represented as Eq (42)

1103925() = 1103925 ( 2() minus 1()) (42)

1103925 is a synchronizing coefficient If there is 50Hz area

then 1103925 = 100983087

413 Thermal Power Plant and Micro Gas Turbine Gen-

erator

In this paper thermal power plant is assumed that all

generators in the thermal power plant are completely syn-

chronous driving and can be expressed as one equivalent

model like Fig 5

RA1

1

Governor

TgA1s11

f A1

PTcom

Turbine

TTs11xgA1

PT

Fig 5 Thermal Power Plant Model

1038389() is a change command of production of elec-

tricity 1 is a regulation constant 1 is a governor

time constant and is a gas turbine constant Micro gas

turbine generator also can be expressed as one equivalent

model like Fig 5 but it is assumed that micro gas turbine

generator operates without governor free 1038389() is

a change command of production of electricity 2 is

a governor time constant and is a gas turbine con-

stant Micro gas turbine generator moves more quickly

than thermal power plant which means is smallerthan

414 Battery system

It is assumed that Battery system has a delay in re-

sponse of inverters and this is expressed with first-order

lag like Eq (43) 1038389() is a change command of

discharge and charge of electricity is an inverter time

constant The state of discharge and charge () is ex-

pressed with discharge and charge amount () and

discharge and charge efficient value like Eq (44)

() = minus 1

() + 1

1038389() (43)

983769() = minus () (44)

415 Controller

It is assumed that each controller system calculates

integration of AR 1() and 2() While the con-

troller system of Area 1 calculates by FFC method the

controller system of Area 2 calculates by TBC method

983769 1() = minus 1 1() (45)

983769 2() = minus 2 2() minus 1103925() (46)

1 2 are system constants is a standard fre-

quency and 1() 2() are deviations of frequen-

cies of each areaThe controller system of Area 2 decides a command of

production of electricity 21038389() and divides it into

-128-



1038389() and 1038389() with the ratio of each rated

capacity

1038389() =

+ 21038389() (47)

1038389() =

+ 21038389() (48)

We regard load fluctuation and wind power fluctuation as

disturbances made by reference to actual fluctuation

42 Expression of State Space on Smart Grid

By preceding section we can represent the smart grid

shown in Fig3 the state space as in Eq (49)

983769() = () + () + () (49)

() =

⎡⎣ 1()2()3()

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1() ()

1()1103925 1()

() 2() ()

2()()

()1103925 2()

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(50)

() =

983131 1()

2()

983133 =

983131 () 2()

983133 (51)

() =

983131 1()

2()

983133 =

983131 () minus 1()

minus 2()

983133 (52)

1038389 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

103838911 103838912 103838913

103838921 103838922 103838923

31 103838932 103838933

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

minus

11039251907317 1

1907317 1

0 0 2907317 11

0 0 0 0 0 0

0 minus

1

1

0 0 0 0 0 0 0 0

minus

1 11

0 minus

1 1

0 0 0 0 0 0 0 0

minus1 0 0 0 1 0 0 0 0 0 0

minus 0 0 0 0 0 0 0 0 0

0 0 0 0 minus

1907317 2

minus

11039252907317 2

1907317 2

0 0 1907317 2

0

0 0 0 0 0 0 minus

1

1

0 0 0

0 0 0 0 0 0 0 minus

1 2

0 0 0

0 0 0 0 0 0 0 0 0minus 0

0 0 0 0 0 0 0 0 0minus 1

0

0 0 0 0 minus1 minus2 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(53)

1038389 =

983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 00 01

10

0 00 00 00 0

0 2(+)

0 0

0 (+)

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(54)

1038389 = 983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1907317 1

0

0 00 00 00 0

0 1

907317 20 00 00 00 00 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(55)

To treat this system as discrete system it is converted by

the sampling time

( + 1) = () + () + () (56)

The control object is to minimize the following cost func-

tion (57) and we assume the situation that the controller

system of Area 1 decides 1038389() by 1() 2() and

the controller system of Area 2 determines 21038389()by 1() 2() 3() the parameters of the cost function

are made = diag983163100 0 0 10 10 100 0 0 10 0 10983165983084 =5907317 2

(0983084 ) = [infinsum=0

( + )] (57)

To apply the decentralized control of two subsystems

with overlapping information in preceding chapter we

expand the system (56) then we get the expanded sys-

tem (58) and the new cost function (59) by using matrices

in Eq (21)

983772( + 1) = 983772983772() + 983772() + 983772() (58)

983772 (9837720983084 ) = [infinsum=0

(983772 983772983772 + 983772)] (59)

We consider the new system 983772 corresponding to the

system 983772 derive a decentralized control of two sub-

systems and contract this decentralized control law We

verify power fluctuations and frequency deviations and

compare the proposed decentralized control a centralized

control and other decentralized controls The parameters

of power network are shown below

Table 1 Simulation Parameters

standard frequency [Hz] 50

Area1 system capacity 1[MW] 3000

Area1 inertia constant 1[s] 10

Area1 damping constant 1[pu] 1

Area1 system constant 1[ MW] 01

Area1 gas turbine constant [s] 9

Area1 governor time constant 1[s] 025

Area1 regulation constant 1[s] 005







Area2 inverter time constant [s] 1Area2 gas turbine rated capacity [MW] 900

Area2 battery rated capacity [MW] 200

Area2 discharge and charge efficient value 094

synchronizing coefficient 1103925[s] 20

sampling time [s] 1

43 Simulation Results

We simulates the effectiveness of the proposed method

by using Matlab R2011a

431 Power Fluctuation and Frequency Deviation

When applying the proposed method power fluctua-

tions and frequency deviations in each area are shown inFig 6 and Fig 7 In Fig 6 (a) the blue line is the load

fluctuation of area 1 1() the greenish yellow line is

-129-



the wind power fluctuation () the red line is fluctu-

ation of the output of the thermal power plant () and

the light blue line is fluctuation of the tie-line power flow

between area 1 and area 2 1103925() In Fig 6 (b) the blue

line is the load fluctuation of area 2 2() the green-

ish yellow line is fluctuation of the output of the micro

gas turbine generator () the red line is fluctuationof the output of the battery system () and the light

blue line is fluctuation of the tie-line power flow between

area 1 and area 2 - 1103925() From Fig 6 we can see the

outputs of controllable generators (thermal power plant

micro gas turbine generator and battery system) change

to adapt fluctuations of load and wind power That means

these generators correct imbalances between electric de-

mand and supply and stabilize power networks In addi-

tion we can also see stabilization of frequency deviations

in Fig 7 which range within plusmn09830862[Hz]

0 200 400 600 800 1000 1200-80

-60

-40

-20

0

20

40

60

80

t[s]

P o w e r F l u c t u a t i o n [ M W ]

PLA1

PW

PT

Ptie

(a)Area 1

0 200 400 600 800 1000 1200-20

-15

-10

-5

0

5

10

15

20

t[s]


PLA2

PG

PB

-Ptie

(b)Area 2Fig 6 Power Fluctuation

0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]

r e q u e n c y

e v i a t i o n

z

(a)Area 1

0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]

F r e q u e n c y D e v i a t i o n [ H z ]

(b)Area 2Fig 7 Frequency Deviation

432 Comparison of Cost Function Value

Cost function value of =800[s] is shown in Fig 8

Here in Fig 8 C is the proposed decentralized control

method Cc is the centralized control method and Co is

the decentralized control method of [2] From Fig 8 thecost value of the proposed method is second lower after

the centralized control method and lowest among the de-

centralized control methods

225

227

C o s t

C Cc Co223

Fig 8 Comparison of Cost (k=800)

5 CONCLUSION

In this paper we applied the decentralized control

of smart grid by using overlapping information to con-

trol of load frequency of power networks that installed

distributed generations We expanded the state space

of the system got a decentralized feedback law for theexpanded system and contract the decentralized feed-

back law to match the original system Then we sim-

ulated decentralized large scale power network systems

and showed the effectiveness of the load frequency con-

trol by using the proposed decentralized control method

by verifying power fluctuation and frequency deviation

and comparison between the proposed method and other

methods

REFERENCES

[1] Y C Ho and K C Chu rdquoTeam decision theory and

information structures in optimal control problemsndashPart Irdquo IEEE Transactions on Automatic Control

Vol17 No1 pp15ndash22 1972

[2] M Ikeda D D Siljak and DE White rdquoDecentral-

ized Control with Overlapping Information Setsrdquo

Journal of optimization theory and Applications


[3] M Rotkowitz and S Lall A characterization of

convex problems in decentralized control IEEE

Transactions on Automatic Control Vol51 No2

pp274ndash286 2006

[4] A Rantzer Linear Quadratic Team Theory Revis-

ited in Proceedings of the American Control Con-

ference pp1637-1641 2006[5] J Swigart and S Lall An explicit state-space solu-

tion for a decentralized two-player optimal linear-

quadratic regulator in Proceedings of the American

Control Conference pp6385ndash6390 2010

[6] T Namerikawa T Hatanaka M Fujita rdquoPredictive

Control and Estimation for Systems with Informa-

tion Structured Constraintsrdquo SICE Journal of Con-

trol Measurement and System Integration Vol4

No2 pp452ndash459 2011

[7] United States Department of Energy(DOE) rdquoSmart

Grid mdash Department of Energyrdquo

httpenergygovoetechnology-developmentsmart-grid

[8] C E Fosha and O I Elgerd rdquoThe Megawatt-

Frequency Control Problem A New Approach Via

Optimal Control Theoryrdquo IEEE Transactions on

Power Apparatus and Systems VolPAS-89 No4

pp563ndash578 1970

[9] J R Pillai B Bak-Jensen rdquoIntegration of Vehicle-

to-Grid in the Western Danish Power Systemrdquo

IEEE Transactions on Sustainable Energy Vol2


[10] T Namerikawa and T Kato rdquoDistributed Load Fre-

quency Control of Electrical Power Networks via

Iterative Gradient Methodsrdquo IEEE Conference on Decision and Control and European Control Con-

ference pp7723ndash7728 2011

-130-



ℝ2 Relation of these is represented as

= [ 1 983084 2 983084 3 ] 983084 = 1 + 2 + 3983084 (2)

= [ 1 983084 2 ] 983084 = 1 + 2983084 (3)

= [ 1 983084

2 ] 983084 = 1 + 2983086 (4)

In this assumption the state of power subsystem 1 is1 the state of power subsystem 2 is 3 and the interac-

tion state between two subsystems is 2 1 is the input

for subsystem 1 and 2 is the input for subsystem 2

The matrices isin ℝtimes isin ℝ

times1038389 and isin ℝtimes

are represented as

=

⎡⎣ 11 12 13

21 22 23

31 32 33

⎤⎦ 983084

=

⎡⎣

11 12

21 22

31 32

⎤⎦ 983084 =

⎡⎣

11 12 21 22

31 32

⎤⎦ 983086 (5)

We assume the situation like Fig 2 where power subsys-

tem 1 can measure state 1 2 and power subsystem 2

can measure state 1 2 3 For this system we con-

sider a decentralized feedback control law where subsys-

tem 1 decides the input 1 from 1 2 and subsystem 2

decides the input 2 from 1 2 3

Fig 2 Electric Power System

The part of the input dealing with the state directly ()has the following structure

() = minus

⎡⎣ 1()

2()3()

⎤⎦ 983084 (6)

where is shown as Eq (7)

=983131 11 12 0

21 22 23983133

(7)

Here the following performance index is defined as

(0983084 ) = [infinsum=0

( + )]983084 (8)

where 0 is the initial state of the system isin ℝtimes is

a constant positive-semidefinite matrix and isin ℝ1038389times1038389

is a constant positive-definite matrix and are ex-

pressed by using 1 isin ℝ1times1 2 isin ℝ

2times2 3 isinℝ3times3 1 isin ℝ


= diag9831631

983084 2

983084 3983165983084 = diag983163

1983084

2983165983086 (9)

We consider a feedback control to minimize Eq (8) in

this situation

3 PROPOSED DECENTRALIZEDCONTROL

31 Expansion of the state space of the system

To set up a decentralized feedback control law we de-

compose the state into two overlapping components

and expand the system 983772 983772( + 1) = 983772983772() + 983772() + 983772() (10)

where 983772 isin ℝ983772 is composed of 9837721 = [ 1 983084 2 ] and 9837722 =

[ 2 983084 3 ] 9837720 is the initial state of the system 983772 983772 isinℝ983772times983772983084 983772 isin ℝ

983772times1038389 983772 isin ℝ983772times are constant matrices

and 983772 satisfies 983772 = 1 + 22 + 3

We also expand performance index as follows

983772 (9837720983084 ) = [infinsum=0

(983772 983772983772 + 983772)] (11)

where 9837720 is the initial state of the system 983772 983772 isin ℝ983772times983772 is

a constant positive-semidefinite matrix and 983772 isin ℝ1038389times1038389

is a constant positive-definite matrix 983772 and 983772 are

expressed by using 9837721 isin ℝ(1+2)times(1+2) 9837722 isin

ℝ(2+3)times(2+3) 9837721 isin ℝ


983772 = diag983163 9837721983084 9837722983165983084 983772 = diag983163 9837721983084 9837722983165983086 (12)

Here we introduce a linear transformation and

ℝ rarr ℝ983772983084 ( ) = (13)

ℝ983772 rarr ℝ983084 ( ) = (14)

These matrices satisfy = 907317 We relate (983084 ) and

( 983772983084 983772 ) by the following definition

Definition 1 The pair ( 983772983084 983772 ) includes the pair

(983084 ) if there exists a matrix such that 9837720 = 0 and

following two equations are satisfied for any input ()

( 0983084 ) = 983772( 9837720983084 )983084 for all ge 0 (15)

(0983084 ) = 983772 (0983084 ) (16)

If ( 983772983084 983772 ) includes (983084 ) then we say that ( 983772983084 983772 ) is an

expansion of (983084 ) and (983084 ) is a contraction of ( 983772983084 983772 )

We relate the matrices of ( 983772983084 983772 ) and (983084 ) by us-

ing proper dimensions matrices 983084 983084 983084 983084 as983772 = + 983084 983772 = + 983084 983772 = + 983084 983772 =

+ 983084 983772 = + With this representation

the condition for the inclusion is given by the following

theorem

Theorem 1 The pair ( 983772983084 983772 ) includes The pair(983084 ) if either

(i)

= 0983084 = 0983084 = 0983084 (17)

= 0983084 = 0 (18)

or

(ii) for 1038389 = 1983084 2983084 983084 983772 1103925 = 0983084 1103925minus1 = 0983084 1103925minus1 = 0983084 (19)

1103925minus1 = 0983084 1103925minus1 = 0983084 = 0983086 (20)

Proof The theorem can be proven based on [2]We choose matrices which satisfies (i) of this theorem as

=

⎡⎢⎣

1038389 1 0 00 1038389 2 00 1038389 2 00 0 1038389 3

⎤⎥⎦ 983084 1103925 =

⎡⎣ 1038389 1 0 0 0

0 1

21038389 2

1

21038389 2 0

0 0 0 1038389 3

⎤⎦ 983084

-126-



907317 = 0983084 907317 1038389 = 0983084 =

⎡⎢⎢⎣

0 1

212 minus

1

212 0

0 1

222 minus

1

222 0

0 minus

1

222

1

222 0

0 minus

1

232

1

232 0

⎤⎥⎥⎦ 983084

907317 1103925 = 0983084 907317 =

⎡⎢⎢⎣

0 0 0 00 1

4

2 minus

1

4

2 0

0 minus

1

42

1

42 0

0 0 0 0

⎤⎥⎥⎦ 983086 (21)



983772 =

983131 98377211

98377212

98377221 98377222

983133 =

⎡⎢⎣

11 12 0 13

21 22 0 23

21 0 22 23

31 0 32 33

⎤⎥⎦ 983084

983772 =

983131 98377211

98377212

98377221 98377222

983133 =

⎡⎢⎣

11 12

21 22

21 22

31 32

⎤⎥⎦

983084

983772 =

983131 983772 11 983772 12

983772 21 983772 22

983133 =

⎡⎢⎣

11 12 21 22 21 22 31 32

⎤⎥⎦ 983086 (22)


expanded system



983772 983772( + 1) = 983772983772() + 983772() + 983772 () (23)


983772 =

983131 98377211 0

98377221 98377222

983133983084 983772 =

983131 98377211 0

98377221 98377222

983133983084

983772 =

983131 983772 11 0

983772 21 983772 22

983133983086 (24)


Eq (16)







1() = minus( 983772 )119837721() minus ( 983772 )129837722() (25)

2() = minus( 983772 )219837721() minus 983772 9837722()

minus(( 983772 )22 minus 983772 )9837722() (26)

9837722( + 1) = ( )219837721() + ( )229837722() (27)

where = 983772 minus 983772 983772 1038389( 983772983084 983772983084 983772983084 983772983084 1983084 983772 )












( 0983084 ) = 983772( 9837720983084 )983084 for all ge 0983086 (28)



input ()( 0983084 ) = 983772 983772( 9837720983084 )983084 for all ge 0 (29)

( 0983084 ) = 983772 983772( 9837720983084 )983084 for all ge 0 (30)





1103925 = 0983084 1103925minus1 = 0983084


as follows( 0983084 ) =

0 +sum1103925=0

983163minus1103925(1038389) + minus1103925(1038389)983165 (32)

983772( 9837720983084 ) =

983772 9837729837720 +

sum1103925=0

983163 983772 983772minus1103925 983772(1038389) + 983772 983772minus1103925 983772(1038389)983165 (33)



= 983772 983772 983084 minus1103925 = 983772 983772minus1103925 983772 (34)

minus1103925

= 983772

983772

minus1103925 983772983084 =

983772 (35)




Eq (31) is derived


= 0983086 (36)


983131 1()

2()

983133 = minus

983131 ( 983772 11)1 ( 983772 11)2 0

( 983772 21)1 ( 983772 21)2 + 983772 1 983772 2

983133⎡⎣ 1()

2()3()

⎤⎦

minus 983131 983772 12

983772 22minus

983772 983133 983131 2()

3()983133

(37)9831312( + 1)3( + 1)

983133 = ()21

9831311()2()

983133+ ()22

9831312()3()

983133 (38)

-127-













Battery

System

Thermal

Power Plant

Gas Turbin

System

Load

Load

ControllerSystem

Controller

System


Generator Model

Generator Model

MA1sDA1

1

MA sDA2

1

sTtie

SA1

SA2

PA1

PA2

f A1

f A2

PBcom

PGcom

PTcom

xgA1

xgA2

PB

PG

PLA2

PLA1PT

Ptie

Area1

Area2

WindPower Plant

PW

x1

xB




shown in Eq (39)

()

= 1038389() minus () (39)







() = () + () (40)




PL

MeqsD

1f

Pm1

Pm2

Pmn





1103925() = 1

( 2() minus 1()) (41)






1103925() = 1103925 ( 2() minus 1()) (42)


then 1103925 = 100983087


erator




model like Fig 5

RA1

1

Governor

TgA1s11

f A1

PTcom

Turbine

TTs11xgA1

PT












414 Battery system








() = minus 1

() + 1

1038389() (43)

983769() = minus () (44)

415 Controller





983769 1() = minus 1 1() (45)

983769 2() = minus 2 2() minus 1103925() (46)





-128-




capacity

1038389() =

+ 21038389() (47)

1038389() =

+ 21038389() (48)






983769() = () + () + () (49)

() =

⎡⎣ 1()2()3()

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1() ()

1()1103925 1()

() 2() ()

2()()

()1103925 2()

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(50)

() =

983131 1()

2()

983133 =

983131 () 2()

983133 (51)

() =

983131 1()

2()

983133 =

983131 () minus 1()

minus 2()

983133 (52)

1038389 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

103838911 103838912 103838913

103838921 103838922 103838923

31 103838932 103838933

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

minus

11039251907317 1

1907317 1

0 0 2907317 11

0 0 0 0 0 0

0 minus

1

1

0 0 0 0 0 0 0 0

minus

1 11

0 minus

1 1

0 0 0 0 0 0 0 0

minus1 0 0 0 1 0 0 0 0 0 0

minus 0 0 0 0 0 0 0 0 0

0 0 0 0 minus

1907317 2

minus

11039252907317 2

1907317 2

0 0 1907317 2

0

0 0 0 0 0 0 minus

1

1

0 0 0

0 0 0 0 0 0 0 minus

1 2

0 0 0

0 0 0 0 0 0 0 0 0minus 0

0 0 0 0 0 0 0 0 0minus 1

0

0 0 0 0 minus1 minus2 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(53)

1038389 =

983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 00 01

10

0 00 00 00 0

0 2(+)

0 0

0 (+)

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(54)

1038389 = 983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1907317 1

0

0 00 00 00 0

0 1

907317 20 00 00 00 00 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(55)


the sampling time

( + 1) = () + () + () (56)





are made = diag983163100 0 0 10 10 100 0 0 10 0 10983165983084 =5907317 2

(0983084 ) = [infinsum=0

( + )] (57)





in Eq (21)

983772( + 1) = 983772983772() + 983772() + 983772() (58)

983772 (9837720983084 ) = [infinsum=0

(983772 983772983772 + 983772)] (59)



























sampling time [s] 1








-129-



















0 200 400 600 800 1000 1200-80

-60

-40

-20

0

20

40

60

80

t[s]


PLA1

PW

PT

Ptie

(a)Area 1

0 200 400 600 800 1000 1200-20

-15

-10

-5

0

5

10

15

20

t[s]


PLA2

PG

PB

-Ptie


0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]

r e q u e n c y

e v i a t i o n

z

(a)Area 1

0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]










225

227

C o s t

C Cc Co223


5 CONCLUSION












methods

REFERENCES











pp274ndash286 2006



















pp563ndash578 1970









-130-



907317 = 0983084 907317 1038389 = 0983084 =

⎡⎢⎢⎣

0 1

212 minus

1

212 0

0 1

222 minus

1

222 0

0 minus

1

222

1

222 0

0 minus

1

232

1

232 0

⎤⎥⎥⎦ 983084

907317 1103925 = 0983084 907317 =

⎡⎢⎢⎣

0 0 0 00 1

4

2 minus

1

4

2 0

0 minus

1

42

1

42 0

0 0 0 0

⎤⎥⎥⎦ 983086 (21)



983772 =

983131 98377211

98377212

98377221 98377222

983133 =

⎡⎢⎣

11 12 0 13

21 22 0 23

21 0 22 23

31 0 32 33

⎤⎥⎦ 983084

983772 =

983131 98377211

98377212

98377221 98377222

983133 =

⎡⎢⎣

11 12

21 22

21 22

31 32

⎤⎥⎦

983084

983772 =

983131 983772 11 983772 12

983772 21 983772 22

983133 =

⎡⎢⎣

11 12 21 22 21 22 31 32

⎤⎥⎦ 983086 (22)


expanded system



983772 983772( + 1) = 983772983772() + 983772() + 983772 () (23)


983772 =

983131 98377211 0

98377221 98377222

983133983084 983772 =

983131 98377211 0

98377221 98377222

983133983084

983772 =

983131 983772 11 0

983772 21 983772 22

983133983086 (24)


Eq (16)







1() = minus( 983772 )119837721() minus ( 983772 )129837722() (25)

2() = minus( 983772 )219837721() minus 983772 9837722()

minus(( 983772 )22 minus 983772 )9837722() (26)

9837722( + 1) = ( )219837721() + ( )229837722() (27)

where = 983772 minus 983772 983772 1038389( 983772983084 983772983084 983772983084 983772983084 1983084 983772 )












( 0983084 ) = 983772( 9837720983084 )983084 for all ge 0983086 (28)



input ()( 0983084 ) = 983772 983772( 9837720983084 )983084 for all ge 0 (29)

( 0983084 ) = 983772 983772( 9837720983084 )983084 for all ge 0 (30)





1103925 = 0983084 1103925minus1 = 0983084


as follows( 0983084 ) =

0 +sum1103925=0

983163minus1103925(1038389) + minus1103925(1038389)983165 (32)

983772( 9837720983084 ) =

983772 9837729837720 +

sum1103925=0

983163 983772 983772minus1103925 983772(1038389) + 983772 983772minus1103925 983772(1038389)983165 (33)



= 983772 983772 983084 minus1103925 = 983772 983772minus1103925 983772 (34)

minus1103925

= 983772

983772

minus1103925 983772983084 =

983772 (35)




Eq (31) is derived


= 0983086 (36)


983131 1()

2()

983133 = minus

983131 ( 983772 11)1 ( 983772 11)2 0

( 983772 21)1 ( 983772 21)2 + 983772 1 983772 2

983133⎡⎣ 1()

2()3()

⎤⎦

minus 983131 983772 12

983772 22minus

983772 983133 983131 2()

3()983133

(37)9831312( + 1)3( + 1)

983133 = ()21

9831311()2()

983133+ ()22

9831312()3()

983133 (38)

-127-













Battery

System

Thermal

Power Plant

Gas Turbin

System

Load

Load

ControllerSystem

Controller

System


Generator Model

Generator Model

MA1sDA1

1

MA sDA2

1

sTtie

SA1

SA2

PA1

PA2

f A1

f A2

PBcom

PGcom

PTcom

xgA1

xgA2

PB

PG

PLA2

PLA1PT

Ptie

Area1

Area2

WindPower Plant

PW

x1

xB




shown in Eq (39)

()

= 1038389() minus () (39)







() = () + () (40)




PL

MeqsD

1f

Pm1

Pm2

Pmn





1103925() = 1

( 2() minus 1()) (41)






1103925() = 1103925 ( 2() minus 1()) (42)


then 1103925 = 100983087


erator




model like Fig 5

RA1

1

Governor

TgA1s11

f A1

PTcom

Turbine

TTs11xgA1

PT












414 Battery system








() = minus 1

() + 1

1038389() (43)

983769() = minus () (44)

415 Controller





983769 1() = minus 1 1() (45)

983769 2() = minus 2 2() minus 1103925() (46)





-128-




capacity

1038389() =

+ 21038389() (47)

1038389() =

+ 21038389() (48)






983769() = () + () + () (49)

() =

⎡⎣ 1()2()3()

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1() ()

1()1103925 1()

() 2() ()

2()()

()1103925 2()

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(50)

() =

983131 1()

2()

983133 =

983131 () 2()

983133 (51)

() =

983131 1()

2()

983133 =

983131 () minus 1()

minus 2()

983133 (52)

1038389 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

103838911 103838912 103838913

103838921 103838922 103838923

31 103838932 103838933

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

minus

11039251907317 1

1907317 1

0 0 2907317 11

0 0 0 0 0 0

0 minus

1

1

0 0 0 0 0 0 0 0

minus

1 11

0 minus

1 1

0 0 0 0 0 0 0 0

minus1 0 0 0 1 0 0 0 0 0 0

minus 0 0 0 0 0 0 0 0 0

0 0 0 0 minus

1907317 2

minus

11039252907317 2

1907317 2

0 0 1907317 2

0

0 0 0 0 0 0 minus

1

1

0 0 0

0 0 0 0 0 0 0 minus

1 2

0 0 0

0 0 0 0 0 0 0 0 0minus 0

0 0 0 0 0 0 0 0 0minus 1

0

0 0 0 0 minus1 minus2 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(53)

1038389 =

983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 00 01

10

0 00 00 00 0

0 2(+)

0 0

0 (+)

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(54)

1038389 = 983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1907317 1

0

0 00 00 00 0

0 1

907317 20 00 00 00 00 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(55)


the sampling time

( + 1) = () + () + () (56)





are made = diag983163100 0 0 10 10 100 0 0 10 0 10983165983084 =5907317 2

(0983084 ) = [infinsum=0

( + )] (57)





in Eq (21)

983772( + 1) = 983772983772() + 983772() + 983772() (58)

983772 (9837720983084 ) = [infinsum=0

(983772 983772983772 + 983772)] (59)



























sampling time [s] 1








-129-



















0 200 400 600 800 1000 1200-80

-60

-40

-20

0

20

40

60

80

t[s]


PLA1

PW

PT

Ptie

(a)Area 1

0 200 400 600 800 1000 1200-20

-15

-10

-5

0

5

10

15

20

t[s]


PLA2

PG

PB

-Ptie


0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]

r e q u e n c y

e v i a t i o n

z

(a)Area 1

0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]










225

227

C o s t

C Cc Co223


5 CONCLUSION












methods

REFERENCES











pp274ndash286 2006



















pp563ndash578 1970









-130-













Battery

System

Thermal

Power Plant

Gas Turbin

System

Load

Load

ControllerSystem

Controller

System


Generator Model

Generator Model

MA1sDA1

1

MA sDA2

1

sTtie

SA1

SA2

PA1

PA2

f A1

f A2

PBcom

PGcom

PTcom

xgA1

xgA2

PB

PG

PLA2

PLA1PT

Ptie

Area1

Area2

WindPower Plant

PW

x1

xB




shown in Eq (39)

()

= 1038389() minus () (39)







() = () + () (40)




PL

MeqsD

1f

Pm1

Pm2

Pmn





1103925() = 1

( 2() minus 1()) (41)






1103925() = 1103925 ( 2() minus 1()) (42)


then 1103925 = 100983087


erator




model like Fig 5

RA1

1

Governor

TgA1s11

f A1

PTcom

Turbine

TTs11xgA1

PT












414 Battery system








() = minus 1

() + 1

1038389() (43)

983769() = minus () (44)

415 Controller





983769 1() = minus 1 1() (45)

983769 2() = minus 2 2() minus 1103925() (46)





-128-




capacity

1038389() =

+ 21038389() (47)

1038389() =

+ 21038389() (48)






983769() = () + () + () (49)

() =

⎡⎣ 1()2()3()

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1() ()

1()1103925 1()

() 2() ()

2()()

()1103925 2()

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(50)

() =

983131 1()

2()

983133 =

983131 () 2()

983133 (51)

() =

983131 1()

2()

983133 =

983131 () minus 1()

minus 2()

983133 (52)

1038389 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

103838911 103838912 103838913

103838921 103838922 103838923

31 103838932 103838933

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

minus

11039251907317 1

1907317 1

0 0 2907317 11

0 0 0 0 0 0

0 minus

1

1

0 0 0 0 0 0 0 0

minus

1 11

0 minus

1 1

0 0 0 0 0 0 0 0

minus1 0 0 0 1 0 0 0 0 0 0

minus 0 0 0 0 0 0 0 0 0

0 0 0 0 minus

1907317 2

minus

11039252907317 2

1907317 2

0 0 1907317 2

0

0 0 0 0 0 0 minus

1

1

0 0 0

0 0 0 0 0 0 0 minus

1 2

0 0 0

0 0 0 0 0 0 0 0 0minus 0

0 0 0 0 0 0 0 0 0minus 1

0

0 0 0 0 minus1 minus2 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(53)

1038389 =

983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 00 01

10

0 00 00 00 0

0 2(+)

0 0

0 (+)

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(54)

1038389 = 983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1907317 1

0

0 00 00 00 0

0 1

907317 20 00 00 00 00 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(55)


the sampling time

( + 1) = () + () + () (56)





are made = diag983163100 0 0 10 10 100 0 0 10 0 10983165983084 =5907317 2

(0983084 ) = [infinsum=0

( + )] (57)





in Eq (21)

983772( + 1) = 983772983772() + 983772() + 983772() (58)

983772 (9837720983084 ) = [infinsum=0

(983772 983772983772 + 983772)] (59)



























sampling time [s] 1








-129-



















0 200 400 600 800 1000 1200-80

-60

-40

-20

0

20

40

60

80

t[s]


PLA1

PW

PT

Ptie

(a)Area 1

0 200 400 600 800 1000 1200-20

-15

-10

-5

0

5

10

15

20

t[s]


PLA2

PG

PB

-Ptie


0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]

r e q u e n c y

e v i a t i o n

z

(a)Area 1

0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]










225

227

C o s t

C Cc Co223


5 CONCLUSION












methods

REFERENCES











pp274ndash286 2006



















pp563ndash578 1970









-130-




capacity

1038389() =

+ 21038389() (47)

1038389() =

+ 21038389() (48)






983769() = () + () + () (49)

() =

⎡⎣ 1()2()3()

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1() ()

1()1103925 1()

() 2() ()

2()()

()1103925 2()

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(50)

() =

983131 1()

2()

983133 =

983131 () 2()

983133 (51)

() =

983131 1()

2()

983133 =

983131 () minus 1()

minus 2()

983133 (52)

1038389 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

103838911 103838912 103838913

103838921 103838922 103838923

31 103838932 103838933

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

minus

11039251907317 1

1907317 1

0 0 2907317 11

0 0 0 0 0 0

0 minus

1

1

0 0 0 0 0 0 0 0

minus

1 11

0 minus

1 1

0 0 0 0 0 0 0 0

minus1 0 0 0 1 0 0 0 0 0 0

minus 0 0 0 0 0 0 0 0 0

0 0 0 0 minus

1907317 2

minus

11039252907317 2

1907317 2

0 0 1907317 2

0

0 0 0 0 0 0 minus

1

1

0 0 0

0 0 0 0 0 0 0 minus

1 2

0 0 0

0 0 0 0 0 0 0 0 0minus 0

0 0 0 0 0 0 0 0 0minus 1

0

0 0 0 0 minus1 minus2 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(53)

1038389 =

983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 00 01

10

0 00 00 00 0

0 2(+)

0 0

0 (+)

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(54)

1038389 = 983131 103838911 103838912103838921 103838922103838931 103838932

983133 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1907317 1

0

0 00 00 00 0

0 1

907317 20 00 00 00 00 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(55)


the sampling time

( + 1) = () + () + () (56)





are made = diag983163100 0 0 10 10 100 0 0 10 0 10983165983084 =5907317 2

(0983084 ) = [infinsum=0

( + )] (57)





in Eq (21)

983772( + 1) = 983772983772() + 983772() + 983772() (58)

983772 (9837720983084 ) = [infinsum=0

(983772 983772983772 + 983772)] (59)



























sampling time [s] 1








-129-



















0 200 400 600 800 1000 1200-80

-60

-40

-20

0

20

40

60

80

t[s]


PLA1

PW

PT

Ptie

(a)Area 1

0 200 400 600 800 1000 1200-20

-15

-10

-5

0

5

10

15

20

t[s]


PLA2

PG

PB

-Ptie


0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]

r e q u e n c y

e v i a t i o n

z

(a)Area 1

0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]










225

227

C o s t

C Cc Co223


5 CONCLUSION












methods

REFERENCES











pp274ndash286 2006



















pp563ndash578 1970









-130-



















0 200 400 600 800 1000 1200-80

-60

-40

-20

0

20

40

60

80

t[s]


PLA1

PW

PT

Ptie

(a)Area 1

0 200 400 600 800 1000 1200-20

-15

-10

-5

0

5

10

15

20

t[s]


PLA2

PG

PB

-Ptie


0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]

r e q u e n c y

e v i a t i o n

z

(a)Area 1

0 200 400 600 800 1000 1200-02

-015

-01

-005

0

005

01

015

02

t[s]










225

227

C o s t

C Cc Co223


5 CONCLUSION












methods

REFERENCES











pp274ndash286 2006



















pp563ndash578 1970









-130-

SICE202d 12 Suehiro

Documents

Transcript of SICE202d 12 Suehiro