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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-
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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
10383891times10383891 9837722 isin ℝ10383892times10383892 as
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
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8122019 SICE202d 12 Suehiro
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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)
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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
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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 system capacity 2[MW] 900
Area2 inertia constant 2[s] 10
Area2 damping constant 2[pu] 2
Area2 system constant 2[ MW] 01
Area2 gas turbine constant [s] 1
Area2 governor time constant 2[s] 1
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
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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]
P o w e r F l u c t u a t i o n [ M W ]
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
Vol34 No2 pp280ndash310 1981
[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
No1 pp12ndash19 2011
[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
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8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 26
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
10383891times10383891 9837722 isin ℝ10383892times10383892 as
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
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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)
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8122019 SICE202d 12 Suehiro
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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
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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 system capacity 2[MW] 900
Area2 inertia constant 2[s] 10
Area2 damping constant 2[pu] 2
Area2 system constant 2[ MW] 01
Area2 gas turbine constant [s] 1
Area2 governor time constant 2[s] 1
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
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8122019 SICE202d 12 Suehiro
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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]
P o w e r F l u c t u a t i o n [ M W ]
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
Vol34 No2 pp280ndash310 1981
[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
No1 pp12ndash19 2011
[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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 36
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)
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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
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8122019 SICE202d 12 Suehiro
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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 system capacity 2[MW] 900
Area2 inertia constant 2[s] 10
Area2 damping constant 2[pu] 2
Area2 system constant 2[ MW] 01
Area2 gas turbine constant [s] 1
Area2 governor time constant 2[s] 1
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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 66
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]
P o w e r F l u c t u a t i o n [ M W ]
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
Vol34 No2 pp280ndash310 1981
[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
No1 pp12ndash19 2011
[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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 46
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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 56
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 system capacity 2[MW] 900
Area2 inertia constant 2[s] 10
Area2 damping constant 2[pu] 2
Area2 system constant 2[ MW] 01
Area2 gas turbine constant [s] 1
Area2 governor time constant 2[s] 1
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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 66
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]
P o w e r F l u c t u a t i o n [ M W ]
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
Vol34 No2 pp280ndash310 1981
[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
No1 pp12ndash19 2011
[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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 56
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 system capacity 2[MW] 900
Area2 inertia constant 2[s] 10
Area2 damping constant 2[pu] 2
Area2 system constant 2[ MW] 01
Area2 gas turbine constant [s] 1
Area2 governor time constant 2[s] 1
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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 66
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]
P o w e r F l u c t u a t i o n [ M W ]
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
Vol34 No2 pp280ndash310 1981
[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
No1 pp12ndash19 2011
[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-
8122019 SICE202d 12 Suehiro
httpslidepdfcomreaderfullsice202d-12-suehiro 66
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]
P o w e r F l u c t u a t i o n [ M W ]
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
Vol34 No2 pp280ndash310 1981
[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
No1 pp12ndash19 2011
[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-