hierarchically decentralized control of load …test a: without battery stations frequency tie-line...

第61回自動制御連合講演会

Binh-Minh Nguyen, Koji Tsumura, and Shinji Hara

The University of Tokyo, CREST

南山大学 2018/11/17

蓄電池ステーションによる電力網の負荷周波数の

階層分散制御

Hierarchically Decentralized Control of Load

Frequency for Power Networks with Battery Stations

Table of Content 1

Introduction

Theoretical results

Illustrative example

Conclusions

I. Introduction 2

Challenge of load frequency control (LFC)

B

BB

G

,1lC

G,2l

C

G

,l NC

…

Area N Area 1

Area 2

System description:

• G: traditional source

• B: battery station

• Global controller: Cg

• Local controller: Cl,i

Merit of Battery

Station: Improve both

control performance and

economy of LFC.(J. Tomic, J. Power Sources, 2007)

Challenges

How to assure the

stability of the LFC

system?

gC

…

Aggregation

…

Dis

trib

uti

on

I. Introduction 3

General idea of shared model sets

1lC

1lP

Area 1

2lC

2lP

Area 2

lNC

lNP

Area N

3lC

3lP

Area 3

⋱

⋱

nH

ξ

gC

Dis

trib

uti

on

Aggregation

gC

Dis

trib

uti

on

Aggregation

( )n N

H s I⋅

{ }( )i

diag s∆

++

Nominal model

Perturbation volume ξ

( )( )

,

,

,

,

2

, ( ) ( )

( ) (

1( )

1 1

)

) /

(

( )

gt i i

i

i gti

i

i

l

l i i

C s

CH s

F s F s

F s FB R ss

+=

+ +

I. Introduction 4

LFC control system with BS

{ }, ( )l idiag C s− { }, ( )gt idiag F s

−{ }( )

idiag F s

1N

Is

⋅

ω

δT

−

−+

LP

TLP

MP

1

i

diagR

{ }idiag B

+

+

ACE

{ },( )

bs idiag F s

−

k /t

NN1

( )gC s

BSP

battery station

governor & turbine

Dis

trib

uti

on

Aggregation

Assumption: In future, the

cost of battery system is

considerably reduced, so we

have enough battery power

to support the LFC.

{ }2,( )

idiag H s

{ }1, ( )idiag H s

1NI

s⋅T

+

+

/t

N N1

−

( )g

C s

k

−

ω

( )( ),

1,

,

,

, , ( )

( )

( ) ( )( )(

( ))

1 1 (/ )

gt i il

gt i i

i

i

i

i i

bs

l

iF s F s F sH

Cs

B R F

s

C s ss F

+=

+ +

II. Theoretical results 5

Problem setting for this presentation

{ }( )idiag H s

1NI

s⋅T

ω

k/

t

N N1

gK

s

−−

Problem setting:

Cg(s) = Kg/s

Fbs,i(s) ≈ 1

{ }2, ( )i

diag H s

{ }1, ( )i

diag H s

1N

Is

⋅T

+

+

/t

NN1

−

( )gC s

k

−

ω

( )( )

, ,

,,

1( )

1

( ) ( )

(

(

)

)

1 /( ) ( )

gt i i

gt i

i

i i

l i

l i i

C s

C

F s F s

F sH s

B Rs F s

+=

+ +

{ }( )i

diag H s

{ }( )idiag H s

1N

Is

⋅T

+

+

/t

NN1

−

gK

s

k

−

ω


Representation of LFC system by nominal model

1NI

s⋅T

k/t

N N1

gK

s

−− ( )

n NH s I⋅

{ }( )i

diag s∆

++

1NI

s⋅T−

++ ( )

n NH s I⋅

{ }( )i

diag s∆

++

1NI

s⋅G−

Hn(s): nominal model for {Hi(s)}.

Perturbation:( )

() (

)( )

nii

n

H s H ss

H s

−∆ =

Interconnection matrix

g t

N

K

N

TGQ

G = ⋅

= − −

k 1

Interconnection

Q

Proposition 1: The LFC system is stable if

(i) The nominal interconnected system is stable

where

(ii) Model matching condition is s.t.: ||Δi(s)||∞ ≤ ξ ∀i from 1 to N

(iii) Robust stability condition of nominal system is s.t.:


Stability of the overall LFC system

( ),

,

,

1 (

( )

)1

n

n

Q i

Q i

Q i

H s

sQ

H

λλ σ

ξλ∞

≤ ∀ ∈−

ɶ

ɶ

( )),(nH s Q ɶ

/( ) ( )n nH s H s s=ɶ

Remark 1: (i) is satisfied if all the eigenvalues of matrix Q are located in the stable

domain given by the nominal GFV (Hara et al, IEEE TAC 2014).( ) 1/ ( )n ns H sφ =ɶ ɶ

III. Illustrative example 8

Three-area-power-network

( )( ),

, ,

,

,

,

,

1(

(

)1

( )

)

1

1

gt i

g i t i

l i

p

l i

i

ip i

F sT s T s

LF s

Cs

T s

Ks =

=+ +

=+

Other parameters:

Tan et al, Electrical Power and Energy System, 2012.

{ }, ( )l idiag C s− { },

( )gt i

diag F s

−{ }( )idiag F s

31 Is

⋅

ω

δT

−

−+

LP

TLP

MP

1

i

diagR

{ }idiag B

+

+

ACE

{ }, ( )bs idiag F s

−

k3

/ 3t1

( )gC s

BSP

battery station

governor & turbine

12 13 12 13

21 21 23 23

31 32 31 32

where 1 / 3 ij

t t t t

T t t t t

t t t t

t i j

+ − − = − + −

− − +

= ∀ ≠


Two test cases

Test A: Without Battery Station

Test B: With Battery Station

{ }2, ( )i

diag H s

{ }1,( )

idiag H s

31 Is

⋅T

+

+

3/ 3

t1

−

gK

s

k

−

ω

{ }( )idiag H s

31 Is

⋅T

ω

k3

/ 3t1

gK

s

−−

( )

( )( )2,

1,

,

,

,

,

,

,

,

,

1 1 /

1(

( ) ( ) ( )

( ) (( ) )

( )( )

(

( )

( ) ( )

( )

))

1 1 /

l i

l i

l i

gt i i

gt i i

gt i i

gt il i i

i i

i

i i

i

C s

C

F s F s

F s F s

F s F s

B R

H sB R

s

C s

C s F s s

H

F

s

=+ +

+

= + +

( )( )

, ,

,,

1( )

1

( ) ( )

(

(

)

)

1 /( ) ( )

gt i i

gt i

i

i i

l i

l i i

C s

C

F s F s

F sH s

B Rs F s

+=

+ +

( )( )

1( )

1

( ) ( )

( ) ( )

( )

( ) 1 /

gtn n

gtn

ln

n

n n nln

H sB

F sC s

C s

F s

F s sR F

+=

+ +

( )

( )( )

1

2

( )1 1

( ) ( )( )

( )

( )

( ) ( )

( ) ( )

( ) ( )

/

1(

( ))

1 1 /

n

n n

gtn n

gtn n

gt

n

n

n n

gtn

ln

ln

ln

n n nl

H sB R

H s

C s

C

F s F s

F s F s

F s F s

F s FB s

s

C s

C s R

= + +

+

= + +

2 nominal models:

1 nominal models:


Stability of two nominal systems

The nominal system of Test B is stable if

∀ λQ,i ∈ σ(Q):,( ) 0 for all

Qn is sλφ +− ≠ ∈ɶ ℂ

3( )n

H s I⋅

31 Is

⋅T−

++

31 Is

⋅G−

Q

( ) { };/ 1; 13gGTQ Q Kσ −= − − = − −

The nominal system of Test A is stable if

∀ λT,i ∈ σ(T): 2 ,( ) 0 for all n T is sφ λ ++ ≠ ∈ɶ ℂ

∀ λG,i ∈ σ(G): 1 ,( ) 0 for all Gn is sλφ ++ ≠ ∈ɶ ℂ

( ) { }3 ; 0; 0/ 33

g t

g

KG G Kσ= =⋅k 1

2 3( )n

H s I⋅

1 3( )nH s I⋅

31 Is

⋅T

+

+

−

−

31 Is

⋅G 1(2)1(2)1(2)

1( )( )( )

n

nn

ssH sH s

φ = =ɶɶ

1( )( )( )

n

nn

ssH sH s

φ = =ɶɶ

Test A

Test B


Available range of Kg given by nominal models

Stable domain given by the nominal

GFV ( ) 1/ ( )n n

s H sφ =ɶ ɶ

Stable domain given by the nominal

GFV 1 1( ) 1/ ( )n n

s H sφ =ɶ ɶ

With Battery station, we might have more freedom to adjust

the global control gain.

Test A: Without Battery Station Test B: With Battery Station

Kg /3 ≤ 2.27 Kg /3 ≤ 7.85


Simulation results

Frequency Tie-line power

Test A: Without battery stations

Frequency Tie-line power

Test B: With Battery stations

Simulation conducted with the volume of model set ξ = 0.32

With Battery stations, the deviation of frequency and tie-line power

are quickly suppressed.

{ },( )

l idiag C s

−{ },

( )gt i

diag F s

−{ }( )idiag F s

31 Is

⋅

ω

δT

−

−+

LP

TLP

MP

1

i

diagR

{ }idiag B

+

+

ACE

k3

/ 3t1

( )g

C s

governor & turbine

{ },( )

l idiag C s

−{ },

( )gt i

diag F s

−{ }( )idiag F s

31 Is

⋅

ω

δT

−

−+

LP

TLP

MP

1

i

diagR

{ }idiag B

+

+

ACE

{ },( )

bs idiag F s

−

k3

/ 3t1

( )g

C s

BSP

battery station

governor & turbine

IV. Conclusions 13

Conclusions

Advantages of shared model set:

Any local area can be designed without understanding the others.

The global controller can be designed simply, without understanding

of the local subsystems and the details of the network structure.

Achieved result:

Apply the idea of shared model set to LFC with battery station.

Future works:

System design considering the general problem setting.

hierarchically decentralized control of load …test a: without battery stations frequency tie-line...

Documents