ECE-620 Reduced-order model of DFIG-based wind turbines

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1 ECE-620 Reduced-order model of DFIG-based wind turbines Dr. Héctor A. Pulgar Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville November 18, 2015 Dr. Héctor A. Pulgar 1 / 39

2 Motivation 1 Dr. Héctor A. Pulgar hpulgar@utk 2 / 39

3 Motivation 1 For a single wind turbine 10 state variables 8 algebraic variables For a wind farm of 400 turbines 4,000 state variables 3,200 algebraic variables Need of a wind farm simplified model Dr. Héctor A. Pulgar hpulgar@utk 2 / 39

4 Motivation 2 Need of a simplified model for analysis and design P L Droop Turbine - Generator P M P A f 1 R 1 T s1 P W 1 2HsD v w v w Wind turbine DAEs 10SV-8AV Wind turbine reduced-order model Dr. Héctor A. Pulgar hpulgar@utk 3 / 39

5 Motivation 2 Need of a simplified model for analysis and design P L Droop Turbine - Generator P M P A f 1 R 1 T s1 P W 1 2HsD v w β 2 1 s β 1 β 3 Wind turbine reduced-order model Dr. Héctor A. Pulgar hpulgar@utk 3 / 39

6 Motivation 2 Need of a simplified model for analysis and design P L Droop Turbine - Generator P M P A f 1 R H(s) 1 T s1 P W 1 2HsD v w β 2 β 3-1 s β 1 Wind turbine reduced-order model Dr. Héctor A. Pulgar hpulgar@utk 3 / 39

7 Outline 1. Selective modal analysis (SMA) 2. Examples of SMA 3. Applying SMA to a wind turbine model Dr. Héctor A. Pulgar hpulgar@utk 4 / 39

8 1. Selective modal analysis (SMA) Dr. Héctor A. Pulgar 5 / 39

9 1. Selective modal analysis (SMA) Consider a linear autonomous dynamic system of nm states as: Assume that there are: Define: h Modes of interest ẋ = Ax n State variables related to the modes of interest (relevant variables) m State variables not related to the modes of interest (irrelevant variables) r Vector of relevant state variables, r R n z Vector of irrelevant state variables, z R m Dr. Héctor A. Pulgar hpulgar@utk 6 / 39

10 1. Selective modal analysis (SMA) Thus, the system model can be rewritten as: [ ] [ ] [ ] ṙ A11 A = 12 r ż A 21 A 22 z Dr. Héctor A. Pulgar hpulgar@utk 7 / 39

11 1. Selective modal analysis (SMA) Thus, the system model can be rewritten as: [ ] [ ] [ ] ṙ A11 A = 12 r ż A 21 A 22 z Graphically, y(t) ṙ(t) r(t) A 11 A 12 z(t) ż(t) A 21 A 22 Less relevant dynamics Dr. Héctor A. Pulgar hpulgar@utk 7 / 39

12 1. Selective modal analysis (SMA) Thus, the system model can be rewritten as: [ ] [ ] [ ] ṙ A11 A = 12 r ż A 21 A 22 z Graphically, By obtaining M 0, the system can be reduced to: ṙ = (A 11 M 0 ) r y(t) ṙ(t) r(t) y(t) ṙ(t) r(t) A 11 A 11 A 12 z(t) ż(t) A 21 y(t) M 0 r(t) A 22 Approximation Less relevant dynamics Dr. Héctor A. Pulgar hpulgar@utk 7 / 39

13 1. Selective modal analysis (SMA) Assuming that r(t) = h i=1 l iv i e λit, the analytical solution for t t 0 is: t z(t) = e A22(t t0) z (t 0 ) e A22(t τ) A 21 r(τ)dτ t 0 t h = e A22(t t0) z (t 0 ) e A22(t τ) A 21 l i v i e λiτ dτ t 0 = e A22(t t0) z (t 0 ) h t l i e A22t i=1 i=1 }{{} scalar t 0 e (λii A22)τ dτa 21 v i = e A22(t t0) z (t 0 ) h ] l i e A22t (λ i I A 22 ) [e 1 (λii A22)t e (λii A22)t0 A 21 v i }{{} i=1 these terms commute = e A22(t t0) z (t 0 ) h [ l i (λ i I A 22 ) 1 Ie λit e A22(t t0) e λit] A 21 v i i=1 Dr. Héctor A. Pulgar hpulgar@utk 8 / 39 (1)

14 1. Selective modal analysis (SMA) (cont.) z(t) = e A22(t t0) [z (t 0 ) ] h l i (λ i I A 22 ) 1 A 21 v i e λit i=1 h l i (λ i I A 22 ) 1 A 21 v i e λit i=1 } {{ } forced response If A 22 0, for some t after t 0, the output of the subsystem of irrelevant dynamics can be approximated as: y(t) = A 12 z(t) h l i A 12 (λ i I A 22 ) 1 A 21 v i e λit }{{} H(λ i) i=1 Dr. Héctor A. Pulgar hpulgar@utk 9 / 39

15 1. Selective modal analysis (SMA) (cont.) Assume M 0 such that: y(t) h l i H(λ i )v i e λit M 0 i=1 h l i v i e λit = M 0 r(t) i=1 i = {1, 2,..., h}, (M 0 H(λ i ))v i = 0 M 0 [v 1, v 2,..., v h ] = [H(λ 1 )v 1, H(λ 2 )v 2,..., H(λ h )v h ] M 0 = [H(λ 1 )v 1, H(λ 2 )v 2,..., H(λ h )v h ] [v 1, v 2,..., v h ] With M 0 already determined, the reduced-order model becomes: ṙ(t) = (A 11 M 0 )r(t) Dr. Héctor A. Pulgar hpulgar@utk 10 / 39

16 2. Examples of SMA Dr. Héctor A. Pulgar 11 / 39

17 2. SMA: Example A Consider the following linear system and assume that the relevant mode is the slowest one. ẋ 1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 3 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { j j } j j R = j j [ P = x 2 v 1 = j ] Dr. Héctor A. Pulgar hpulgar@utk 12 / 39

18 2. SMA: Example A Consider the following linear system and assume that the relevant mode is the slowest one. ẋ 1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 3 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { j j } j j R = j j [ P = x 2 v 1 = j ] Dr. Héctor A. Pulgar hpulgar@utk 12 / 39

19 2. SMA: Example A Consider the following linear system and assume that the relevant mode is the slowest one. ẋ 1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 3 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { j j } j j R = j j [ P = x 2 v 1 = j x 3 ] Dr. Héctor A. Pulgar hpulgar@utk 12 / 39

20 2. SMA: Example A Consider the following linear system and assume that the relevant mode is the slowest one. ẋ 1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 3 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { j j } j j R = j j [ P = x 2 v 1 = j x 3 ] Dr. Héctor A. Pulgar hpulgar@utk 12 / 39

21 2. SMA: Example A Consider the following linear system and assume that the relevant mode is the slowest one. ẋ 1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 3 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { j j } j j R = j j [ P = x 2 v 1 = j x 3 ] Dr. Héctor A. Pulgar hpulgar@utk 12 / 39

22 2. SMA: Example A The relevant eigenvalues are λ 1 = λ b = j and λ 2 = λ b. By matrix P, the relevant state variables are x 2 and x 3. Thus, [ ] x2 r = z = x x 1 3 [ ] [ ] A 11 = A = A 1 21 = [ 1 2 ] A 22 = 5 The condition to obtain M 0 is: M 0 [v 1, v 2 ] = [H(λ 1 )v 1, H(λ 2 )v 2 ] As we have complex eigenvalues, only two equations are required. This condition is modified to: M 0 [R(v 1 ), I(v 1 )] = [R (H(λ 1 )v 1 ), I (H(λ 1 )v 1 )] Dr. Héctor A. Pulgar hpulgar@utk 13 / 39

23 2. SMA: Example A [ H(λ 1 ) = A 12 (λ 1 I A 22 ) A 21 = j j [ ] 0 H(λ 1 )v 1 = j [ ] [ ] M 0 = [ ] 0 0 = ] Finally, the reduced-order model becomes ṙ(t) = (A 11 M 0 )r(t) [ ] [ ] [ ẋ2 1 1 x2 = ẋ 3 x 3 ] The eigenvalues of the model are equal to the relevant ones Dr. Héctor A. Pulgar hpulgar@utk 14 / 39

24 2. SMA: Example A Starting from an initial point of (x 1, x 2, x 3 ) T = [ ] T 0.6 Time response using FOM x 2 x x Time [s] Dr. Héctor A. Pulgar hpulgar@utk 15 / 39

25 2. SMA: Example A Starting from an initial point of (x 1, x 2, x 3 ) T = [ ] T 0.6 Time response using ROM x 2 x Time [s] Dr. Héctor A. Pulgar hpulgar@utk 15 / 39

26 2. SMA: Example B (h < n) Consider the following linear system and assume that the relevant mode is the slowest one. ẋ1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 4 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { -2 4 j1 4 j1 } j j R = j j x 1 [ ] P = v 1 = Dr. Héctor A. Pulgar hpulgar@utk 16 / 39

27 2. SMA: Example B (h < n) Consider the following linear system and assume that the relevant mode is the slowest one. ẋ1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 4 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { -2 4 j1 4 j1 } j j R = j j x 1 [ ] P = v 1 = Dr. Héctor A. Pulgar hpulgar@utk 16 / 39

28 2. SMA: Example B (h < n) Consider the following linear system and assume that the relevant mode is the slowest one. ẋ1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 4 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { -2 4 j1 4 j1 } j j R = j j x 1 [ ] P = v 1 = x 3 Dr. Héctor A. Pulgar hpulgar@utk 16 / 39

29 2. SMA: Example B (h < n) Consider the following linear system and assume that the relevant mode is the slowest one. ẋ1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 4 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { -2 4 j1 4 j1 } j j R = j j x 1 [ ] P = v 1 = x 3 Dr. Héctor A. Pulgar hpulgar@utk 16 / 39

30 2. SMA: Example B (h < n) Consider the following linear system and assume that the relevant mode is the slowest one. ẋ1 ẋ 2 = x 1 x 2 } ẋ 3 {{ } } 1 1 {{ 4 }} x 3 {{ } ẋ A x The eigenvalues, participation factor matrix and right-eigenvectors are: Λ = diag{λ a, λ b, λ c } = { -2 4 j1 4 j1 } j j R = j j x 1 [ ] P = v 1 = x 3 Dr. Héctor A. Pulgar hpulgar@utk 16 / 39

31 2. SMA: Example B (h < n) The relevant eigenvalue is λ 1 = λ a = 2. By inspection of matrix P, the relevant state variables are x 1 and x 3. Thus, [ ] x1 r = z = x x 2 3 [ ] [ ] A 11 = A = A 1 21 = [ 0 2 ] A 22 = 3 To obtain M 0 : H(λ 1 ) = A 12 (λ 1 I A 22 ) 1 A 21 = [ ] H(λ 1 )v 1 = Thus, [ ] [ ] M 0 = = }{{}}{{} H(λ 1)v 1 v 1 [ ] [ ] Dr. Héctor A. Pulgar hpulgar@utk 17 / 39

32 2. SMA: Example B (h < n) Finally, the reduced-order model becomes ṙ(t) = (A 11 M 0 )r(t) [ ] [ ] [ ] ẋ x1 = ẋ x 3 The eigenvalues of the model are -2 and The relevant mode e 2t is preserved. The mode e t is an equivalent representation of the irrelevant dynamics. Dr. Héctor A. Pulgar hpulgar@utk 18 / 39

33 2. SMA: Example B (h < n) Starting from an initial point of (x 1, x 2, x 3 ) T = [ ] T 0.5 Time response using FOM x 1 x x Time [s] Dr. Héctor A. Pulgar hpulgar@utk 19 / 39

34 2. SMA: Example B (h < n) Starting from an initial point of (x 1, x 2, x 3 ) T = [ ] T 0.5 Time response using ROM x 1 x Time [s] Dr. Héctor A. Pulgar hpulgar@utk 19 / 39

35 3. Applying SMA to a wind turbine model Dr. Héctor A. Pulgar hpulgar@utk 20 / 39

36 3. Applying SMA to a wind turbine model Quick review of the DFIG two-axis model de qd dt = 1 T 0 (E qd (Xs X s)i ds ) ω s X m X r V dr (ω s ω r)e dd de dd dt = 1 T 0 ( E dd (X s X s)i qs ) ωs X m X r V qr dω r dt (ω s ω r)e qd = ωs [ ] T m E dd 2H I ds E qd Iqs D Manifold is associated with the following equivalent circuit: ~ jx s R s jq D R e jx V e D V D E qd -je dd I qs -ji ds I GC ~ P gen jq gen e -jq : 1 Ideal shift transformer ~ jq V e e Color definition: Blue : State variables Green : Algebraic variables Red : Inputs Machine reference Thevenin equivalent of the grid Dr. Héctor A. Pulgar hpulgar@utk 21 / 39

37 3. Applying SMA to a wind turbine model V qr, V dr, T m : P-Q controllers and pitch angle controller V wind w r Supervisory voltage controller V ref - V 1 f N K Iv K Pv s Maximum power tracking Qmin Qmax 1 T c s1 Pref Q ref - P - Q K I1 K P1 s K I3 K P3 s I qr ref I qr I dr - I dr ref Pitch-angle controller w r w ref G 2 (s) V wind w r q ref - K I2 K P2 s K I4 K P4 s V qr V dr Wind turbine T m G 3 (s) DFIG f P Q I qr I dr w r Dr. Héctor A. Pulgar hpulgar@utk 22 / 39

38 3. Applying SMA to a wind turbine model Voltage controller at the wind farm level V wind w r Supervisory voltage controller V ref - V 1 f N K Iv K Pv s Maximum power tracking Qmin Qmax Pref - P 1 T c s1 Q ref - Q K I1 K P1 s K I3 K P3 s I qr ref I qr I dr - I dr ref Pitch-angle controller w r w ref G 2 (s) V wind w r q ref - K I2 K P2 s K I4 K P4 s V qr V dr Wind turbine T m G 3 (s) DFIG f P Q I qr I dr w r Dr. Héctor A. Pulgar hpulgar@utk 23 / 39

39 3. Applying SMA to a wind turbine model Control schemes to participate in frequency regulation (P1 and P2) V wind - w r Supervisory voltage controller V ref - V 1 f N f ref - P1: Inertial response K Iv K Pv s Maximum power tracking Qmin Qmax B 1 T c s1 P2: Power reserve q 0 DP ref P ref Q ref - P - Q K pf sk df K I1 K P1 s K I3 K P3 s I qr ref I qr I dr Washout Filter - I dr ref - st w st w 1 Pitch-angle controller w r w ref G 2 (s) V wind w r q ref K I2 K P2 s K I4 K P4 s f ref - f V qr V dr Wind turbine T m G 3 (s) DFIG f P Q I qr I dr w r Dr. Héctor A. Pulgar hpulgar@utk 24 / 39

40 3. Applying SMA to a wind turbine model Test system ( I dji q ) e j (d- p 2 ) jx d R s ~ ( E dje q ) e j (d- p 2 ) jq S V S Line 1 R 1 jx 1 Transformer jx T Transformer Line 2 ~ Line 1 Load SG DFIG Simplified SG two-axis model ~ jx s R s jq D R 2 jx V 2 D V D E qd-je dd I qs-ji ds I GC ~ DFIG P genjq gen e -jq : 1 Ideal shift transformer Line 2 V L jql P LjQ L Load Dr. Héctor A. Pulgar hpulgar@utk 25 / 39

41 3. Applying SMA to a wind turbine model To reduce the model order, replace SG by a voltage source ~ V S jq S Line 1 I aji b R 1 jx 1 Transformer jx T Transformer Line 2 ~ Line 1 Load SG DFIG Constant voltage source ~ jx s R s jq D R 2 jx V 2 D V D E qd-je dd I qs-ji ds I GC ~ DFIG P genjq gen e -jq : 1 Ideal shift transformer Line 2 V L jql P LjQ L Load Dr. Héctor A. Pulgar hpulgar@utk 26 / 39

42 3. Applying SMA to a wind turbine model Further simplifications: a. Wind speed is such that pitch angle controller is inactive, i.e., θ = 0 b. Qref=0 (DFIG reactive power controller s reference) Thus, there is no voltage controller c. This single wind turbine does not participate in system frequency regulation The state and algebraic variables are: x = [ E qd, E dd, ω r, x 1, x 2, x 3, x 4 ] T y = [V qr, V dr, I qr, I dr, P gen, Q gen, I ds, I qs, I a, I b, V D, θ D, V L, θ L ] T Dr. Héctor A. Pulgar hpulgar@utk 27 / 39

43 3. Applying SMA to a wind turbine model Differential equations ( ) Ė qd 1 T E qd (Xs X s )I ds 0 Ė dd 1 T 0 ( ) ω r ω s T 2H m E D dd I ds E qd Iqs ẋ 1 [ = K I1 Cωr 3 ] P gen ẋ 2 [ ( K I2 KP 1 Cωr 3 ) ] P gen x1 I qr ẋ 3 K I3 Q gen ẋ 4 K I4 [ K P 3 Q gen x 3 I dr ] } {{ } ẋ ω s X m Xr V dr (ω s ω r)e dd ( E dd (X s X s )Iqs ) ωs X m Xr V qr (ω s ω r)e qd } {{ } f(x,y) ( 116 ) where: T m = T M = 1 ρπr 2 ω b 0.22 T b 2 S b ω r λ i = e 12.5 λ i 5 λ i }{{} C p(λ i,θ=0 ) λ λ, λ = v tip v wind = 2k p ω rr v wind v wind 3 Dr. Héctor A. Pulgar hpulgar@utk 28 / 39 [p.u.]

44 3. Applying SMA to a wind turbine model Algebraic equations [ ( V qr K P 2 KP 1 Cωr 3 ) ] P gen x1 I qr x2 [ ( ) ] V dr K P 4 KP 3 Qref Q gen x3 I dr x4 P gen E dd I ds E qd Iqs ( Rs Ids 2 I 2 ) qs (V qri qr V dr I dr ) Q gen E qd I ds E dd Iqs ( X s Ids 2 I 2 ) qs I dr E qd Xm I X m X r ds I qr E dd Xm I [0] = g(x, y) = X m X qs r V S e jθ S (R 1 j[x 1 X T ]) (I a ji b ) V D e jθ D (E qd je dd ) (Rs jx s )(Iqs ji ds) V D e jθ D V D e jθ D (R 2 jx 2 )I L V L e jθ L (P 0 V pv L jq 0 V qv L ) V L e jθ LI L I L (I d ji q)e j(δ π 2 ) I qs ji ds VqrIqrV dri dr e V jθ D D Dr. Héctor A. Pulgar hpulgar@utk 29 / 39

45 3. Applying SMA to a wind turbine model For v wind =12 m/s, P 0 =0.5 pu and Q 0 =0.1 pu, we find the equilibrium point by forcing: ẋ e = f(x e, y e ) = 0 0 = g(x e, y e ) E qd = p.u. V qr = p.u. I qs = p.u. E dd = p.u. V dr = p.u. I a = p.u. ω r = rad/s I qr = p.u. I b = p.u. x 1 = p.u. I dr = p.u. V D = p.u. x 2 = p.u. P gen = p.u. θ D = rad x 3 = p.u. Q gen = 0 p.u. V L = p.u. x 4 = p.u. I ds = p.u. θ L = rad Dr. Héctor A. Pulgar hpulgar@utk 30 / 39

46 3. Applying SMA to a wind turbine model Linearization around (x e, y e ) [ ] [ ẋ As B = s 0 C s D s ] } {{ } J [ x y [ Pgen ] = [ E1 E 2 ] [ x y ] [ K 0 Elimination of algebraic variables ( y = Ds 1 C s x): ẋ = ( A s B s Ds 1 ) C s x K v wind }{{} A sys P gen = ( E 1 E 2 Ds 1 ) C s x }{{} E ] ] v wind Dr. Héctor A. Pulgar hpulgar@utk 31 / 39

47 3. Applying SMA to a wind turbine model At the equilibrium point, the eigenvalues and matrix of participation factors of A sys are: Λ = diag{λ a, λ b, λ c, λ d, λ e, λ f, λ g} where λ a,b = ± j209.2 λ e = λ g = λ c,d = ± j λ f = P = E qd E dd ω r x 1 x 2 x 3 x 4 λ a λ b λ c λ d λ e λ f λ g Relevant mode and variable (h=n=1): λ e and ω r Dr. Héctor A. Pulgar hpulgar@utk 32 / 39

48 3. Applying SMA to a wind turbine model With r = ω r and z = [ E qd, E dd, x ] T 1, x 2, x 3, x 4, the model to reduce becomes: [ ] [ ] [ ] [ ] ωr A11 A = 12 ωr kωr v ż A 21 A 22 z 0 wind [ ] [ ] [ ] ω Pgen = r Ẽ1 Ẽ 2 z With h = n = 1 this is a very particular case: z = (λ 1 I A 22 ) 1 A 21 ω r H(λ 1 ) = A 12 (λ 1 I A 22 ) 1 A 21 M 0 [v 1 ] = [H(λ 1 ) v 1 ] M 0 = H(λ 1 ) Thus, ) ω r = (A 11 A 12 (λ 1 I A 22 ) 1 A 21 ω r k ωr v wind ) P gen = (Ẽ1 Ẽ2 (λ 1 I A 22 ) 1 A 21 ω r Dr. Héctor A. Pulgar hpulgar@utk 33 / 39

49 3. Applying SMA to a wind turbine model Take ) ω r = (A 11 A 12 (λ 1 I A 22 ) 1 A 21 ω r k v ωr wind } {{ } α ωr ) and multiply by α P = (Ẽ1 Ẽ2 (λ 1 I A 22 ) 1 A 21 to obtain: P gen = α ωr (P gen Pgen) 0 α P k }{{} (v 0 ωr wind v }{{ wind) } P gen v wind P gen = α ωr }{{} β 1 P gen α P k ωr }{{} v wind ( αωr Pgen 0 α P k ωr vwind) 0 }{{} β 2 β 3 P gen = β 1 P gen β 2 v wind β 3 P gen = P gen v wind Dr. Héctor A. Pulgar hpulgar@utk 34 / 39

50 3. Applying SMA to a wind turbine model Testing: Time-domain simulation with variable wind speed Transformer Line 2 ~ Line 1 Load SG DFIG Case SG DFIG Network Full-order model (FOM) 9 SV, 2 AV 7 SV, 8 AV 8 AV Reduced-order model (ROM) 9 SV, 2 AV 1 SV 8 AV where SV AV State variables Algebraic variables Dr. Héctor A. Pulgar hpulgar@utk 35 / 39

51 3. Applying SMA to a wind turbine model Testing: Time-domain simulation with variable wind speed 14 v wind [m/s] Time [s] 0.4 FOM ROM P gen [p.u.] Time [s] Dr. Héctor A. Pulgar hpulgar@utk 36 / 39

52 3. Applying SMA to a wind turbine model Testing: Time-domain simulation with variable wind speed T m of SG [p.u.] FOM ROM Time [s] V D [p.u.] FOM ROM Time [s] Dr. Héctor A. Pulgar hpulgar@utk 37 / 39

53 References 1. H. Pulgar-Painemal, P. Sauer, Reduced-order model of Type-C wind turbine generators, Electric Power Systems Research, Volume 81, Issue 4, April 2011, Pages H. Pulgar-Painemal, P. Sauer, Towards a wind farm reduced-order model, Electric Power Systems Research, Volume 81, Issue 8, August 2011, Pages H. Pulgar-Painemal, Wind farm model for power system stability analysis, Ph.D. Thesis, University of Illinois at Urbana-Champaign, G.C. Verghese, I.J. Perez-Arriaga, F.C. Schweppe, Selective Modal Analysis With Applications to Electric Power Systems Part II: The Dynamic Stability Problem, IEEE Transactions on Power Apparatus and Systems, vol.pas-101, no.9, pp.3126,3134, Sept Dr. Héctor A. Pulgar hpulgar@utk 38 / 39

54 Thanks for your attention Questions? Dr. Héctor A. Pulgar 39 / 39

CONTROL ASPECTS OF WIND TURBINES. Faizal Hafiz, Wind Energy Research Group, SET Center

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