a multichoice control strategy for a

8/20/2019 A Multichoice Control Strategy for A

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P sio

∆P LF


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E i

E j

E imi,j

E rei,j

E imio,jo E imi,j

E reio,jo E rei,j

P si,sj

i, j

P sio,sjo i, j

Qsi,sj i, j

Qsio,sjo i, j

U i,j

∆E imio,jo E imi,j

∆E reio,jo E rei,j

∆P si,sj i, j


12/97

∆Qsi,sj

i, j

γ i

γ j

θi,j

E q q

M

P e

P m

P gk k

P k k

Qgk k

Qk k

δ

ω

xd d

δ CO I

V̇ (x) V (x) λi i

V (x) ωCO I

δ̃ k k

ω̃k k


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θ̃k k

A

B

C

D

G(s)

Ri G(s) λi

T 1, T 2

T w

V L

V R

Λ

αmax

Y

ωmax αmax

φ

ξ

u

x

V HV dco

V uncontrolled


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V HV dcuncontrolled

I line

∆f


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i j

∆P 1


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P 1

MULTICHOICE

CONTROL

P 2

P 3

Q i

Q j

Controllable

Variables

∆P 2

∆P 3

∆Qi

∆Q j

∆P 1

∆P 1 ∆P 2


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•

•

•

•

•


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•

•

•

•


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Converter

reactor

Harmonic

Filter

dc Capacitor VSC Cable

Transformer CB


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1.25

1.00

0.75

0.50

0.25

0.25

0.50

0.75

1.00

1.25

0.25 0.50 0.75 1.00 1.25-1.25 -1.00 -0.75 -0.50 -0.25

P – Q DIAGRAM

Q [ pu ]

P [

p u

]

•

•

•

•

•

•

•

•


26/97

• •

• •

i d

RdCd Cd

VSC VSC

xt i xt j

P s i Qs i

Filter Filter

P s j Qs j

U i U j j xL

E i E j


27/97

j xL

j xt i j xt jE i E j

U j θ j U i θ i

I i I j

Ē i Ē j

Ē i = E ie jγ i Ē j = E j e

jγ j

E i E j γ i γ j x

ti x

tj

P si = U iE i sin(θi − γ i)

xti

Qsi = U i[U i − U iE i cos(θi − γ i)]

xti

P sj = U j E j sin(θ j − γ j )

xtj

Qsj = U j [U j − U jE j cos(θ j − γ j)]

xtj

P si Qsi γ i E i P sj Qsj γ j E j

i

j

P si = −P sj


28/97

j xLU j θ j U i θ i

P sj + j Qsj P si + j Qsi P losses

S̄ si = Ū i(Ī i)∗ = P si + jQsi

P si = btiU iE i sin(θi − γ i)= b

tiU iE i(sin θi cos γ i − cos θi sin γ i)

Qsi = bti

[U 2

i −U iE i cos(θi

−γ i)]

= bti

[U 2i − U iE i(cos θi cos γ i + sin θi sin γ i)]

bti

= 1/xti

j

P sj = btjU j E j(sin θ j cos γ j − cos θ j sin γ j)

Qsj = btj [U 2

j − U jE j(cos θ j cos γ j + sin θ j sin γ j )]

E rei = E i cos γ i E imi = E i sin γ i

E re j = E j cos γ j E im

j = E j sin γ j


29/97

S̄ si = P si + jQsi

P si = btiU i(sin θiE rei − cos θiE imi )

Qsi = bti [U 2i − U i(cos θiE rei + sin θiE imi )]

S̄ sj = P sj + jQsj

P sj = btjU j(sin θ jE re

j − cos θ jE im j )Qsj = btj [U 2 j − U j(cos θ jE re j + sin θ jE im j )]P si = −P sj

E i =

(E rei )2 + (E imi )

2 γ i = arctan

E imiE rei

E j = (E re j )

2 + (E im j )2 γ j = arctan

E im jE re j

E reio E

imio E

re jo E

im jo ∆E

rei

∆E imi ∆E re

j ∆E im

j

E rei = E reio + ∆E

rei E

imi = E

imio + ∆E

imi

E re j = E re

jo + ∆E re

j E im

j = E im

jo + ∆E im

j

P si = btiU i[sin θi(E reio + ∆E

rei ) − cos θi(E imio + ∆E imi )]

= bti

U i(sin θiE reio − cos θiE imio ) + btiU i(sin θi∆E rei − cos θi∆E imi )

= P sio + ∆P si


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Qsi P sj Qsj

P si = P sio + ∆P si

Qsi = Qsio + ∆Qsi

P sj = P sjo + ∆P sj

Qsj = Qsjo + ∆Qsj

o P Q ∆P ∆Q

P sio = btiU i(sin θiE reio − cos θiE imio )

Qsio = bti[U 2i − U i(cos θiE reio + sin θiE imio )]

P sjo = btjU j(sin θ jE re

jo − cos θ jE im jo )Qsjo = btj [U

2 j − U j(cos θ jE re jo + sin θ jE im jo )]

∆P si = btiU i(sin θi∆E rei − cos θi∆E imi )

∆Qsi =

−bti

U i(cos θi∆E rei + sin θi∆E

imi )

∆P sj = btjU j(sin θ j∆E re

j − cos θ j∆E im j )∆Qsj = −btjU j(cos θ j ∆E re j + sin θ j∆E im j )

P sjo = −P sio∆P sj = −∆P si


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δ̇ = ω

ω̇ = 1

M (P m − P e − Dω)

Ė q = 1

T do

E f − xd

xdE q +

xd − xdxd

U cos(δ − θ)

Ė d = 1

T qo −xqxq

E d −xq − xq

xqU sin(δ − θ)

P e

P e =E qU sin(δ − θ) + E dU cos(δ − θ)

xd

δ

ω

E q

E d

P m

P e

D

M

E f

xd

xd

xq


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xq

T do

T qo

U

θ U

P e

P e =E qU sin(δ − θ)

xd

xd

P e

P (U, f ) Q(U, f ) P (U ) Q(U )

P (f )

Q(f )


35/97

• P

• I

• Z

(Z ) (I ) (P )

P = P o

a1

U

U o2

+ a2U

U o+ a3

Q = Qo

a4

U

U o

2+ a4

U

U o

+ a6

U o P o Qo

a1 a6


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P = P o

U

U o

mp

Q = Qo

U

U o

mq

mp

mq

1 + af (f − f o) f

f o af

n N

P m

k


37/97

xdk E

qk

δ k

j x' dk U k kE' qk k I gk

Ȳ bus

N × N kl Ȳ buskl = j Bkl

k

k

P gk = (Ū kĪ ∗

gk )

=E qk U k sin(δ k − θk)

xdk

Qgk =

(¯U k

¯I

∗

gk )

=E qk U k cos(θk − δ k) − U 2k

xdk

k

k = 1, . . . , n

P k =

N l=1

BklU kU l sin(θk − θl) +E qk U k sin(θk

−δ k)

xdk

Qk = −N

l=1

BklU kU l cos(θk − θl) +U 2k − E qk U k cos(θk − δ k)

xdk

k = (n + 1), . . . , N


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P k =N

l=1

BklU kU l sin(θk − θl)

Qk = −N

l=1

BklU kU l cos(θk − θl)

P Lk + j QLk k

k = 1, . . . , N

0 = P k + P Lk

0 = Qk + QLk

x y

x = [δ 1 . . . δ n, ω1 . . . ωn, E

q1 . . . E

qn]

y = [θ1 . . . θN , U 1 . . . U N ]

ẋ = f (x, y)

0̇ = g(x, y)


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ẋ1 = f 1(t, x1, . . . xnx , u1 . . . ur)

ẋnx = f n(t, x1, . . . xnx , u1 . . . ur)

̇xi xi t u1 . . . ur

x =

x1

xnx

, u =

u1

u p

, f (t,x,u) =

f 1(t,x,u)

f nx(t,x,u)

nx nx

ẋ = f (t,x,u)

x u

u

ẋ = f (t, x)

u = γ (t) u = γ (x) u = γ (t, x) u = γ u

f t

ẋ = f (x)

t τ = t−a


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x = xo

xo xo

xo

f (xo) = 0

xo

xo xo t ≥ 0 xo > 0 δ = δ () > 0 x(0)

x(0) − xo < δ x(t)

x(t) − xo < , ∀ t > 0

x(t) − xo → 0 t → ∞ xo xo

dE (x)/dt E (x) x xo

E (xo) E

xo xo = 0

V : D → R D ⊂ Rnx


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V

V̇

V̇ =nx

i=1

∂ V ∂xi

f i(x)

V̇ = ∂ V ∂x1

, ∂ V ∂x2

, . . . , ∂ V ∂xnx ·

f 1(x)f 2(x)

f nx(x)

V̇ = ∂ V

∂x · f (x)

V̇ = ∇V (x) · f (x)

V

V̇

xo D ⊂ Rnx

xo V : D → R

V (xo) = 0 V (x) > 0 ∀ x ∈ D, xo V̇ (x) = ∇V (x) · f (x) ≤ 0 ∀ x ∈ D

V̇ (x) = ∇V (x) · f (x) < 0 ∀ x ∈ D xo V̇ (xo) = 0

V̇ (x) = ∇V (x) · f (x) ≤ 0, V̇ (x) x(t) D, xo.


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V (x)

V (x) V (xo) = 0

V (x) > 0

x = xo V (x) ≥ 0 x = xo

xo

∆ẋ = A∆x

A =

∂f (x)

∂x

x=xo

=

∂f 1(x)∂x1

. . . ∂f 1(x)∂xnx

∂f nx(x)∂x1

. . . ∂f n(x)∂xnx

x=xo

xo A

λi

(A − λI) = 0

I

xo


43/97

xo A

xo A

j x L

E' q U

δ̇ = ω

ω̇ = 1

M [P m − P emax sin(δ ) − Dω]

P emax =E qU

xd + xt + xL


44/97

(δ o1, ωo1) =

arcsin

P mP emax

, 0

(δ o2, ωo2) =

π − arcsin

P mP emax

, 0

(δ o3, ωo3) =

−π − arcsin

P mP emax

, 0

(δ o2, ωo2)

(δ o3, ωo3)

(δ o1, ωo1) xo1 =(δ o1, ωo1)

V = 12

M ω2 − P mδ − P emax cos(δ ) + C o

C o V = 0 xo1

˙

V =

∇V ·f (x)

V̇ = [−P m + P emax sin δ M ω] ·

ωP m−P emax sin δ−Dω

M

V̇ = −Dω2 ≤ 0

ẋ = f (x) = 0

ω̇ = 1

M [P m − P emax sin(δ ) − Dω] = 0

ω = 0 xo1

xo1 = (δ o1, ωo1)


45/97

y

x

g

g(x, y) = 0

L = (x, y) : g(x, y) = 0 L

g(x, y) = 0

L

(xo, yo) ∈ L Gy = ∂g/∂y

(Gy) = 0 (xo, yo) y = h(x)

g(x, h(x)) = 0

(xo, yo) (xo, yo)

ẋ = f (x, h(x)) = f o(x)

D = 0

mp = 0


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δ COI

= 1

M T

nk=1

M kδ k

ωCOI

= 1

M T

nk=1

M kωk

M T =n

k=1

M k

ω̇COI

= 1

M T

nk=1

M k ω̇k = 1

M T

nk=1

(P mk − P ek) = P COI M T

xo P COI = 0 δ k ωk

δ̃ k = δ k − δ COI ω̃k = ωk − ωCOI

nk=1

M kδ̃ k = 0

nk=1

M kω̃k = 0

θ̃k = θk − δ COI


47/97

˙̃δ k = δ̇ k − δ̇ COI = ωk − ωCOI = ω̃k

˙̃ωk = ω̇k − ω̇COI = 1

M k

P mk − P ek − M k

M T P COI

k = 1 . . . n)

˙̃δ k = ω̃k

˙̃ωk = 1

M k

P mk −

E qk U kxdk

sin(δ̃ k − θ̃k) − M kM T

P COI

Ė q = 1

T dok

E f k − xdk

xdkE qk +

xdk − xdkxdk

U k cos(δ̃ k − θ̃k)

0 = P k + P Lk

0 = Qk + QLk

P k Qk θ

δ

θ̃

δ̃

θ̃k − θ̃l = θk − θl δ̃ k − δ̃ l = δ k − δ l

V (ω̃, δ̃, E q, U, θ̃) = V 1 +7

k=1

V 2k + C o

V 1 =

1

2

n

k=1

M kω̃2k

V 21 = −n

k=1

P mkδ̃ k

V 22 =N

k=1

P Lkθ̃k


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V 23 =N

k=1

QLk

U kdU k

V 24 =n

k=1

1

2xdk[E

2qk + U

2k − 2E qkU k cos(δ̃ k − θ̃k)]

V 25 = −12

N k=1

N l=1

BklU kU l cos(θ̃k − θ̃l)

V 26 = −n

k=1

E f dkE

qk

xdk − xdk

V 27 =n

k=1

E 2qk2(xdk − xdk)

V 1 V 2k C o

dV dt

ω̃

∂ V ∂ ̃ω

dω̃

dt

V̇ (x)

dV 1dt

ω̃

+

dV 21

dt +

dV 24dt

δ̃

= 0

dV 22dt

+ dV 24

dt +

dV 25dt

θ̃

=

(P k + P Lk)

˙̃θk = 0

dV 23dt

+ dV 24

dt +

dV 25dt

U

=

(Qk + QLk)U̇ kU k

= 0

dV 24

dt +

dV 26dt

+ dV 27

dt

E q

= −n

k=1

T dokxdk − xdk

( Ė qk )2


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dV (x)dt

= −n

k=1

T dokxdk − xdk

( Ė qk )2 ≤ 0

u ∈ Rr

ẋ = f (x, u) , x ∈ D ⊂ Rnx D Rnx u x

xo xo ∈ D

u = u(x)

ẋ = f (x, u(x)) = f̃ (x)

xo u(x) u(x)

xo

D ⊂ Rnx xo Do ⊂ D xo V : Do → R

V (x)

V̇ (x) = ∇V (x) · f (x, u) < 0


50/97

x = xo D

V (x)

ẋ = f (x, u) = f o(x) +r

i=1

uif i(x)

x ∈ D ⊂ Rnx f o(x) f o, f

1, . . . , f r C

∞ C s s

C ∞

V (x)

∇V (x) · f o(x) ≤ 0 x ∈ D

u(x) = [u1(x), . . . , ur(x)]

ui(x) = −∇V (x) · f i(x) , i = 1, . . . , r V (x) x ∈ D

V̇ (x) = ∇V (x) · f o(x) +r

i=1

ui∇V (x) · f i(x)

≤r

i=1

ui∇V (x) · f i(x) = −r

i=1

ui(∇V (x) · f i(x))2


51/97

ẋ =

0 1−1 0

x1x2

+

01

u

= f o(x) + f 1u

u

V (x) =

3

2

x21 + x1x2 + x22

V̇ (x, u) = ∇V (x) · f (x, u)V̇ (x, u) = −x21 + x1x2 + x22 − (x1 + 2x2)u

u x1 + 2x2 = 0 u V̇ < 0 x1 + 2x2 = 0 V̇ = −5x22 u x2 x1 = −2x2 u = −x2 V̇

V̇ (x, u) = −x21 − x22 < 0 u = −x2 V̇ (x, u) x = 0

u

V̇ (x, u) < 0

∆ẋ = A∆x(t) + B∆u

∆Y = C ∆x + D∆u

∆ẋ nx × 1


52/97

∆u r × 1

∆Y

m × 1

A nx × nx

B

nx × r

C m × nx

D m × r

A

(A − λI) = 0

nx

λ = λ1, . . . , λnx A V ri

AV r

i = λiV r

i

A

λi V

ri

nx × 1 nx V R

V R = [V r1 V r2 . . . V

rnx

]nx×nx

V li

V li A = λiV l

i

A λi V li

1 × nx nx V L

V L = [(V l1)T (V l2)

T . . . (V lnx)T ]T nx×nx

nx


53/97

Λ =

λ1 0 . . . 00 λ2 . . . 0

0 0 . . . λnx

nx×nx

AV R

= V R

Λ(V R)−1AV R = (V R)−1V RΛ = Λ

A

V R

nx ξ

ξ = V L∆x = (V R)−1∆x

∆x = V Rξ

V R ξ̇ = AV Rξ + B∆u

∆Y = C V Rξ + D∆u

nx

ξ̇ = Λξ + V LB∆u

∆Y = CV Rξ + D∆u

Λ = (V R)−1AV R


54/97

Y u D = 0

ξ (s) = (sI − Λ)−1V LB∆u(s)∆Y (s) = C V R(sI − Λ)−1V LB∆u(s)

G(s) = ∆Y ∆u(s)

G(s) = C V R(sI − Λ)−1V LB

Λ

G(s) =nx

i=1

Ris − λi

Ri m × r G(s) λi

Ri = C V r

i V l

i B

H (s, k) = kH (s) k s = λi

∂λi∂k

= RiH (λi)

∆λi

∆k

= RiH (λi)

∆λi = RiH (λi, k)

∆λi = kRiH (λi)


55/97

H (s = λi) Ri

++

u(s) G(s)

H(s)POD

(s) y

G(s) H (s)

1+s T1

1+s T2

s Tw

1+s Tw

PODKp

Input

Wash-out

Filter

Lead-lag

Filter Gain

T w

T w

◦


56/97

λi

H (s = λi) φ

φ = 180◦

− arg(Ri)

Desired*

Actual*

Im

Re

arg(R i )

R i

1 + s aT

1 + s T

T 1 = aT

T 2 = T

a > 1

a

αmax

αmax = arcsin

a − 1a + 1

αmax φ


57/97

φ = arcsin

a − 1a + 1

a

a = 1 + sin(φ)

1 − sin(φ)

T

ωmax

ωmax = 1

T √

a

T λi = σi ± jωi

ωmax = ωi T

T = 1

ωi√

a

T 1

T 2


58/97


59/97

a multichoice control strategy for a

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