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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
II. Theoretical results 6
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.:
II. Theoretical results 7
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
+ − − = − + −
− − +
= ∀ ≠
III. Illustrative example 9
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:
III. Illustrative example 10
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
III. Illustrative example 11
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
III. Illustrative example 12
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.