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ẋ = f(x) n 1 f i (i = 1, 2,..., n) x i (i = 1, 2,..., n) x(0) = x o x(t) t > 0 t < 0

x(t) x o U I xo I xo : α xo < t < β xo α xo β xo x(t) t β t α + x f(x) = 0 x x x x V 1 x x o V 1 x(t) t > 0 x o V 1 x t

x o V 1 x x ẋ = Ax x = 0

x x = x x x ẋ = A x x A = f x = f x (x ) x=x f x

ẋ = f(x, y, ϵ) ϵẏ = g(x, y, ϵ) x, y n m x s y s x f y f x = x s + x f y = y s + y f ϵ y x ϵ ẋ s = f(x s, y s ) 0 = g(x s, y s ) x = x s, y = y s n m y s = h(x s )

x s y s h ϵh x (x s, h) = g(x s, h) h x h x ϵ h(x s ) = h o (x s ) + ϵh 1 (x s ) + ϵ 2 h 2 (x s ) +... ϵ = 0 h o g(x s, h o ) = 0 ϵ 0 ϵ ϵ ẋ s = f(x s, h(x s )) x (x x s ) y f y f = y y s = y h(x s )

ϵẏ f = ϵẏ ϵẏ s = g(x s, y f + h(x s )) x s (x s, h(x s )) ϵ y f = g y y f g y ẋ = f(x, y, z, p) 0 = g o (x, y, z, p) 0 = g i (x, y, z, p), y i < 0 0 = g i+ (x, y, z, p), y i < 0 z + = h(x, y, z, p), y j = 0 ż = 0, y j 0 x, y, z, p f, g h

A z(k + 1) = A d z(k) z(k+1), z(k) k+1 k A d

E = 2π 2 fnφ

1 o 2 o 3 o

P AC = P DC + W c t = P DC + CV dc V dc t

k p1 P 1 = k p2 P 2 =... = k pn P n = f = f 0 f ref V = k q Q/(1 + st ) u Ld = 0 u Lq P (dq) ac (t) = u Lq i vq(t) Q (dq) ac (t) = u Lq i vd(t)

I lim = 1, 0I load 0, 1pu 0.86pu I βρ = V Z th V I lim

µs 10s i d i q i 2 d + i2 q > i max 1ms s

V/Z I lim

8% 12% 4200 7000E/kW

320 340C

Q = P tan ϕ R/X

Î = ˆV Z ˆV ˆ Z = 1 Y + R + jx ˆV load = ˆV Î(R + jx) P load = V load I cos ϕ Q load = V load I sin ϕ cosϕ P = P losses + P load Q = Q losses + Q load

P losses = I 2 R Q losses = I 2 X I I lim I lim I lim I lim < I max I > I lim I = I lim Z I < I lim

I < I lim 0, 9 r = 0, 01 I lim = 1, 05 V = 1, 05 R = 0, 1 X = 0, 1 V o = 1, 05 Y = 0, 01 1, 05 1, 156 Y (t) = Y o (1 + rt)

V o I lim R X r 1.05 1.05 0.1 0.1 1 S nom 105% 110% Y load = Î ˆV load

V o P loadmax V load Y 1.05 0.9 0.8588 0.9086 1.156 V max = 1.1 V min = 0.9 0.9 1.167

I < I lim I lim

P max P = P max 1 o I I lim 2 o P DG P max DG 3 o

Z = 1 Y + R + jx

ˆV th = Z th = ˆV Z Z + R + jx Z(R + jx) Z + R + jx ˆV DG = V DG 0 o ˆV th = V DG P DG V DG (R th + jx th ) V 4 DG (V 2 th + 2R th P DG )V 2 DG + P 2 DG(R 2 th + X 2 th) = 0 V 2 DG = V 2 th + 2R thp DG + (V 2 th + 2P DGR th ) 2 4P 4 DG (R2 th + X2 th ) 2 I l I Î l = V DG Z Î = Îl P DG V DG V I I lim

I = I lim I lim = V DG Z P DG V DG P DG PDG max P DG = PDG max I lim V V V th V 4 DG (I 2 lim(z 2 r + Z 2 im) + 2Z r P DG )V 2 DG + P 2 DG(Z 2 r + Z 2 im) = 0 Z r Z Z im V 2 DG = (I2 lim (Z2 r + Z 2 im) + 2Z r P DG ) + (I 2 lim (Z2 r + Z 2 im ) + 2Z rp DG ) 2 4P 2 DG (Z2 r + Z 2 im ) 2

ˆV = ˆV DG + I lim (R + jx) V o I lim P DGo P DGmax Y = Y o (1 + rt) r

1, 1026

0.6817

V DG V DG V I V o P load V load Y 1.05 0.9 1.0548 0.9 1.447 1.05 0.9 1.4039 0.8508 2.155

V o = 1 i 2 q + i 2 d < I 2 lim i d i q f V f I > I lim f

1 I lim f

f f N R + P DGo = P DG P DG = P DGmax e 10 3 P max V + V o = V

I l = V DG Z I = I l P DG V DG V DG 1 P DG I P DG I P DG I V I

P DG V o I lim P DGo P max f N K i K p

0.01 V o P DGo 2.5 1.1

P DGo P DGmax

V V DG V load

V o P load V load Y 1 0.9 0.7591 0.9 1.0418 1 0.9 1.3453 0.8027 2.32

I lim = 1.05pu P DG V DG V load 1 o 1 o 1 o 2 o 3 o 3 o

V I P V o = V ZI V o = 1 V o = V RP V o = 1.05 V o = 1 V DG V load

I lim V

V I I lim P DG PDG max

V P load V load Y 1.0742 0.9 1.2069 0.9 1.6557 1.0741 0.9 1.4372 0.8787 2.0684 V o = 1.05 V o = 1

R I I lim P DG P max DG

V P load V load Y 1.0876 0.9 1.3672 0.9 1.8756 1.0834 0.9 1.4486 0.8883 2.07

V max 1.1 1.075 1.094 V o V DG

Q DGmax = S 2 DGmax P 2 DG

Z = 1 Y + R + jx

ˆV th = Z th = ˆV Z Z + R + jx Z(R + jx) Z + R + jx ˆV th = ˆV DG P DG jq DG V DG (R th + jx th ) V 4 DG (V 2 th + 2(R th P DG + X th Q DG )V 2 DG + (P 2 DG + Q 2 DG)(R 2 th + X 2 th) = 0 V 2 DG = V 2 th + 2(R thp DG + X th Q DG ) + 2 = V 4 th + 4(P 2 DGR 2 th + 2P DG Q DG R th X th + Q 2 DGX th ) +4V 2 th(p DG R th + Q DG X th ) 4(P 2 DG + Q 2 DG)(R 2 th + X 2 th)

I l I Î l = V DG Z Î = Îl P DG V DG ˆV DG = V DG 0 o

Y o = 0.001 0.01 0.01

S DGmax = 1 Q DGmax = Q DG

Q DGmax

Q DGmax Q DGmax = Q DG I = I max V P loadmax V load Y 1 0.9 1.0119 0.9 1.3882 0.9992 0.9 1.2831 0.8241 2.1007

V V DG V load V load P load Y V o V load P loadmax Y

V o V o = V (RP + XQ)