1 y(t)m b k u(t) ẋ = [ 0 1 k m b m x + [ 0 1 m u, x = [ ẏ y (1) y b k m u

( ) LPV( )

1 y(t)m b k u(t) ẋ = [ 0 1 k m b m x + [ 0 1 m u, x = [ ẏ y (1) y b k m u

m 1 m m 2, b 1 b b 2, k 1 k k 2 (2) [m b k ( ) k 0 b m ( )

2 ẋ = Ax, x(0) 0 (3) (x(t) 0) ( ) V (x) V (x) = x T P x > 0 x 0, V (0) = 0 (4) P T = P P > 0 (5)

x(t) V (x) < 0 x(t) 0 (6) V (x(t)) t V (x) 0 lim V (x(t)) = t x( )T P x( ) = 0 x( ) = 0 (7)

3 V (x) = ẋ T P x + x T P ẋ, ẋ = Ax = (Ax) T P x + x T P (Ax) = x T (A T P + P A)x < 0 x 0 (8) A T P + P A < 0 (9) P > 0

4 A T P + P A + 2σP < 0 (10) P > 0 V (x) = x T (A T P + P A)x < x T ( 2σP )x = 2σV (x) V (x(t)) < e 2σt V (x(0)) (11) x T (x)p x(t) < e 2σt x T (0)P x(0) x(t) < ce σt x(0) (12) x(t) σ

5 ẋ = A(θ)x, θ : (13) (u = 0) [ 0 1 ẋ = k m b m x = A(m, b, k)x, θ = [m b k V = x T P x V (x) = x T P x > 0 x 0, V (x, θ) < 0 x 0, θ (14)

6 V (x, θ) = x T (A T (θ)p + P A(θ))x P > 0 A T (θ)p + P A(θ) < 0 θ (15) ẋ = (2 + θ)x, 1 θ 1 A T (θ)p + P A(θ) = (2 + θ)p P (2 + θ) = 2(2 + θ)p P = 1 A T (θ)p + P A(θ) = 2(2 + θ) < 0 θ

A T (θ)p + P A(θ) < 0 θ P > 0? θ A(θ) θ

7 m [m 1, m 2 m 1, m 2 m = λm 1 + (1 λ)m 2 = m 2 λ(m 2 m 1 ), λ [0, 1 (16) m = α 1 m 1 + α 2 m 2, α 1 = λ, α 2 = 1 λ α 1 + α 2 = 1, α i 0(17) m [m 1, m 2 b [b 1, b 2 m = α 1 m 1 + α 2 m 2, α 1 + α 2 = 1, α i 0 (18) b = β 1 b 1 + β 2 b 2, β 1 + β 2 = 1, β i 0 (19) [m b

b θ 2 θ 4 θ 1 θ 3 m θ 1 = [ m1 b 1, θ 2 = [ m1 b 2, θ 3 = [ m2 b 1, θ 4 = [ m2 b 2 (20) θ = [ m b = = α 1 β 1 [ m1 b 1 [ (β1 + β 2 )(α 1 m 1 + α 2 m 2 ) (α 1 + α 2 )(β 1 b 1 + β 2 b 2 ) [ [ m1 m2 + α 1 β 2 + α b 2 β 1 2 b 1 (21) [ m2 + α 2 β 2 (22) b 2

λ 1 = α 1 β 1, λ 2 = α 1 β 2, λ 3 = α 2 β 1, λ 4 = α 2 β 2 λ i 0 λ 3 + λ 3 + λ 3 + λ 4 = α 1 (β 1 + β 2 ) + α 2 (β 1 + β 2 ) = α 1 + α 2 = 1 θ = λ 1 θ 1 + λ 2 θ 2 + λ 3 θ 3 + λ 4 θ 4 ( ) k 0 b m

m b = λ 1 m 1 b 1 k +λ 5 k 1 m 2 b 1 k 1 + λ 2 m 1 b 1 k 2 + λ 6 m 2 b 1 k 2 + λ 3 m 1 b 2 k 1 + λ 7 m 2 b 2 k 1 + λ 4 m 1 b 2 k 2 + λ 8 m 2 b 2 k 2 (23) λ i 0, 8 λ i = 1 i=1

8 [ ẋ = 0 1 k m b m x + [ 0 1 m u, x = [ ẏ y (24) m 1 m = α 1 1 1 + α 2 m 1 m 2 (α 1 + α 2 = 1 ) [ 0 1 k m b m = α 1 [ 0 1 k m 1 b m 1 + α 2 [ 0 1 k m 2 b m 2 [ 0 1 m = α 1 [ 0 1 m 1 + α 2 [ 0 1 m 2

m, b b/m? b m = (α 1 1 1 + α 2 )(β 1 b 1 + β 2 b 2 ) m 1 m 2 = α 1 β 1 b 1 m 1 + α 1 β 2 b 2 m 1 + α 2 β 1 b 1 m 2 + α 2 β 2 b 2 m 2 = λ 1 b 1 m 1 + λ 2 b 2 m 1 + λ 3 b 1 m 2 + λ 4 b 2 m 2 A(m, b) = λ 1 A(b 1, m 1 ) + λ 2 A(b 2, m 1 ) + λ 3 A(b 1, m 2 ) + λ 4 A(b 2, m 2 ) B = λ 1 B(m 1 ) + λ 2 B(m 1 ) + λ 3 B(m 2 ) + λ 4 B(m 2 )!( )

9 ẋ = ( N λ i A i )x, x(0) 0, λ i 0, N λ i = 1 (25) i=1 i=1 N N ( λ i A i ) T P + P ( λ i A i ) < 0 λ i (26) i=1 i=1 N λ i (A T i P + P A i ) < 0 λ i (27) i=1 P

λ i = 1 λ j = 0(j i) (27) A T i P + P A i < 0 (28) i( ) (28) i λ i N λ i (A T i P + P A i ) < 0 λ i i=1 A T i P + P A i < 0 i (29) (29)

ẋ = 0 k/j M 0 1 0 1 x + 1/J M 0 0 k/j L 0 0 y = [1 0 0x u J M, J L, k J 1 J L J 2, k 1 k k 2

10 ẋ = ( N λ i A i )x + ( N λ i B i )u, λ i 0, N λ i = 1 (30) i=1 i=1 i=1 u = F x (31) N N ẋ = ( λ i A i + λ i B i F )x = i=1 λ i 0, i=1 N λ i = 1 i=1 N i=1 λ i (A i + B i F )x (32)

F, P > 0 s.t. (A i + B i F ) T P + P (A i + B i F ) < 0 i (33) A T i P + P A i + (B 1 F ) T P + P B i F < 0 i (34) (B 1 F ) T P, P B i F LMI BMI( ) BMI LMI Q = P 1, X = F Q F = XQ 1 (35) Q(A i + B i F ) T + (A i + B i F )Q < 0 i (36) QA T i + A i Q + X T Bi T + B i X < 0 i (37) Q, X LMI

11 LPV LPV( ) ẋ = A(p(t))x + B(p(t))u (38) y = C(p(t))x (39) p(t) p(t) p(t) = [p 1 (t) p 2 (t) A(p(t)) = A 0 + p 1 (t)a 1 + p 2 (t)a 2, B(p(t)) = B 0 + p 1 (t)b 1 + p 2 (t)b 2 C(p(t)) = C 0 + p 1 (t)c 1 + p 2 (t)c 2 LPV LPV

12 δ = ω ω 0 ω = ω 0 M P M ω 0 M P e D M (ω ω 0) E q = 1 T d E q + x d x d T d0 x V s cos δ + 1 V f dσ T d0 (40) P e = E qv s sin δ x dσ (41) Vt Vs Infinite bus Generator LT Transformer HT Transmission line I

(δ 0, ω 0, E q0, V f0 ) x 1 = δ δ 0, x 2 = ω ω 0, x 3 = E q E q0, u = V f V f0 x ẋ 1 = x 2 ẋ 2 = d 1 sin δx 3 + d 1 E q0(sin δ sin δ 0 ) + d 2 x 2 ẋ 3 = d 3 x 3 + d 4 (cos δ cos δ 0 ) + d 5 u (42)

(sin δ sin δ 0 )/(δ δ 0 ) (cos δ cos δ 0 )/(δ δ 0 ) sin δ sin δ 0 = sin δ sin δ 0 δ δ 0 x 1 cos δ cos δ 0 = cos δ cos δ 0 δ δ 0 x 1 (43) δ w 1 (δ) = sin δ sin δ 0 δ δ 0 w 2 (δ) = sin δ w 3 (δ) = cos δ cos δ 0 δ δ 0 (44) δ(t) w(t)

ẋ 1 = x 2 ẋ 2 = d 1 w 2 (δ)x 3 + d 1 E q0w 1 (δ)x 1 + d 2 x 2 ẋ 3 = d 3 x 3 + d 4 w 3 (δ)x 1 + d 5 u LPV ẋ = A(w)x + bu (45) 0 1 0 A(w) = d 1 E q0w 1 (δ) d 2 d 1 w 2 (δ), b = 0 T 0 d 4 w 3 (δ) 0 d 3 d 5 A(w) = A 0 + w 1 A 1 + w 2 A 2 + w 3 A 3 (46)

13 LPV ẋ = A(p(t))x + B(p(t))u (47) y = C(p(t))x (48) p(t) p(t) ẋ K = A K (p(t))x K + B K (p(t))y (49) u = C K (p(t))x K + D K (p(t))y (50) p(t)

14 u = F (w)x (51) F (w) = F 0 + w 1 F 1 + w 2 F 2 + w 3 F 3 (52) ẋ = A cl (w)x (53) A cl (w) = (A 0 + bf 0 ) + w 1 (A 1 + bf 1 ) + w 2 (A 2 + bf 2 ) + w 3 (A 3 + bf 3 ) σ A cl (w) T P + P A cl (w) + 2σP < 0, P > 0 (54)

V f ( u) F (w) F (w)x 1 2 F (w)x 1 2 2 < γ, X = P 1 (55) γ G(w)X 1 2 2 < γ G(w)X 1 G T (w) < γ ( ) γi G(w) G T > 0 (56) (w) X : G(w) = G 0 + w 1 G 1 + w 2 G 2 + w 3 G 3 (57) w LMI

15 δ 0 = 0.1962, w 0 = 1, E qo = 1.23, V fo = 2.37 V s = 1.0 D = 0.15 M = 12.922 T do = 6.55 x d = 0.8258 x s = 0.0558 x q = 0.535 x d = 0.1045 [p.u [p.u [sec. [sec. d [p.u [p.u q [p.u d [p.u

σ = 1.44 δ [1, 90 0.05

0.4 0.35 0.3 0.25 δ [rad 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 Time [s 1

3 2.5 2 1.5 ω [p.u 1 0.5 0 0.5 1 0 1 2 3 4 5 6 7 8 9 10 Time [s 2

5 4.5 4 3.5 3 Vf [p.u 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Time [s 3 V f

3 2.5 2 Pe [p.u 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Time [s 4 P e