Maximum power point tracking of a DFIG wind turbine system

Dissertation Maximum Power Point Tracking of a DFIG Wind Turbine System Graduate School of Natural Science & Technology Kanazawa University Division of Electrical and Computer Engineering Student ID No.: 1424042007 Name: Phan Dinh Chung Chief advisor: Prof Shigeru YAMAMOTO Date of Submission: January 6, 2017 Abstract Doubly-fed induction generator (DFIG) has been used popularly in variable speed wind turbines because the DFIG wind turbine uses a small back-to-back converter to interface to the connected grid, about 30% comparing to the wind turbine’s capacity, and provides a control ability as good as a variable speed wind turbine using a generator with a full converter The most important purpose of a variable speed wind turbine or a DFIG wind turbine, in general, is to utilize fully the kinetic energy of wind for electric generation To meet this objective, several publications provided different methods Generally, the previously proposed schemes can be listed into two groups including wind speed-based method and wind speed sensorless one With the first group, the wind turbine can give a good performance in tracking maximum power point but it requires a precise and instantaneous wind speed measurement; this requirement hardly achieve in practice With methods in the second group, an anemometer does not require but the wind turbine using these methods cannot track maximum power point efficiently under varying wind conditions In this dissertation, I proposed two methods and control laws for obtaining maximum energy output of a doubly-fed induction generator wind turbine The first method aims to improve the conventional MPPT curve method while the second one is based on an adaptive MPPT method Both methods not require any information of wind data or wind sensor Comparing to the first scheme, the second method does not require the precise parameters of the wind turbine The maximum power point tracking (MPPT) ability of these proposed methods are theoretically proven under some certain assumptions In particular, DFIG state-space models are derived and control techniques based on the Lyapunov function are adopted to i derive the control methods corresponding to the proposed maximum power point tracking schemes The quality of the proposed methods is verified by the numerical simulation of a 1.5-MW DFIG wind turbine with the different scenario of wind velocity The simulation results show that the wind turbine implemented with the proposed maximum power point tracking methods and control laws can track the optimal operation point more properly comparing to the wind turbine using the conventional MPPT-curve method The power coefficient of the wind turbine using the proposed methods can retain its maximum value promptly under a drammactical change in wind velocity while this cannot achieve in the wind turbine using the conventional MPPT-curve Furthermore, the energy output of the DFIG wind turbine using the proposed methods is higher compared to the conventional MPPT-curve method under the same conditions ii 甲（Kou） 様式 ４． （Form 4） 学 位 論 文 概 要 Dissertation Summary 学位請求論文（Dissertation） 題名（Title） Maximum Power Point Tracking of a DFIG Wind Turbine System DFIG 風力発電システムの最大電力点追従 専攻（Division）:Electrical and computer engineering 学籍番号（Student ID Number） ：1424042007 氏名（Name） ：Phan Dinh Chung 主任指導教員氏名（Chief supervisor） ：YAMAMOTO Shigeru 学位論文概要（Dissertation Summary） Doubly-fed induction generator (DFIG) has been used popularly in variable speed wind turbines because the DFIG wind turbine uses a small back-to-back converter to interface to the connected grid, about 30% comparing to the wind turbine’s capacity, and provides a control ability as good as a variable speed wind turbine using a generator with a full converter The most important purpose of a variable speed wind turbine or a DFIG wind turbine, in general, is to utilize fully wind energy for electric generation To meet this objective, several publications provided different methods Generally, the previously proposed schemes can be listed into two groups such as wind speed-based method and wind speed sensorless one With the first group, the wind turbine can give a good performance in tracking maximum power point but it requires a precise and instantaneous wind speed measurement; this requirement hardly achieve in practice With methods in the second group, an anemometer does not require but the wind turbine using these methods cannot track maximum power point efficiently under varying wind conditions In this dissertation, I proposed two methods and control laws for obtaining maximum energy output of Doubly-fed induction generator wind turbine The first method aims to improve the conventional MPPT curve method while the second one is based on an adaptive MPPT method Both methods not require any information of wind data or wind sensor Comparing to the first scheme, the second method does not require the precise parameters of the wind turbine The maximum power point tracking (MPPT) ability of these proposed methods are theoretically proven under some certain assumptions In particular, DFIG state-space models are derived and control techniques based on the Lyapunov function are adopted to derive the control methods corresponding to the proposed maximum power point tracking schemes The quality of the proposed methods is verified by the numerical simulation of a 1.5-MW DFIG wind turbine with the different scenario of wind velocity The simulation results show that the wind turbine implemented with the proposed maximum power point tracking methods and control laws can track the optimal operation point more properly comparing to the wind turbine using the conventional MPPT-curve method The power coefficient of the wind turbine using the proposed methods can retain its maximum value promptly under a drammactical change in wind velocity while this cannot achieve in the wind turbine using the conventional MPPT-curve Furthermore, the energy output of the DFIG wind turbine using the proposed methods is higher compared to the conventional MPPT-curve method under the same conditions Contents Abstract i Contents iv List of Figures v List of Tables vii Acknowledgement ix Introduction 1.1 Outline of The Dissertation DFIG-Wind Turbine 2.1 Wind turbine 2.2 DFIG 10 Controller Design and Maximum Power Strategy 15 3.1 Maximum power point tracking 15 3.2 Design RSC controller for improved MPPT scheme 19 3.3 3.4 3.2.1 RSC Controller for power adjustment 19 3.2.2 Improved MPPT scheme 20 Design RSC controller for adaptive MPPT scheme 21 3.3.1 RSC controller for rotor speed adjustment 21 3.3.2 Adaptive MPPT scheme 22 Comparison of two proposed MPPT schemes 24 iii Simulation and Discussions 27 4.1 Parameters for improved MPPT method 30 4.2 Parameters for adaptive MPPT method 30 4.3 Simulation results and disscusion 31 Conclusion 39 A DFIG Wind Turbine 41 A.1 Proof of Lemma 41 B Controller Design and Maximum Power Strategy B.1 Proof of Lemma 43 43 B.2 Proof of Lemma 43 B.3 Proof of Theorem 44 B.4 Proof of Lemma 45 B.5 Matrix inequality 46 B.6 Proof of Lemma 48 B.7 Proof of Theorem 48 Publications 55 Bibliography 56 iv List of Figures Fig 1.1 Variable speed wind turbine based on: (a) full power converter and (b) partial power converter Fig 2.1 Overall system of the doubly-fed induction generator (DFIG) wind turbine Fig 2.2 Characteristic of wind turbine: (a) C p versus λ and (b) Pm ver- sus ωr at different wind speeds as β = Fig 3.1 Wind turbine characteristic of (3.4) for β = 0: (a) C p (λ), (b) Pm (λ, Vw ) and Pmppt (ωr ), and (c) contour of wind turbine 17 Fig 3.2 Control diagram using the improved MPPT method 21 Fig 3.3 Control diagram using the adaptive MPPT method 24 Fig 4.1 Fig 4.2 δ3 (b), and ζ(ωr , Vw ) 29 δ2 Wind speed profile: (a) wind speed and (b) wind acceleration 32 Fig 4.3 Simulation results: (a) ωr (t)−ωropt (Vw (t)), (b) power coefficient ζ p (ωr , Vw ) (a), C p (λ(t)), (c) Pmax (t) − Pm (t), and (d) electrical energy output 35 Fig 4.4 Simulation results: (a) ratio kˆ opt /kopt and (b) ωropt (t) − ω ˆ ropt (t), (c) irdref (t) − ird (t), (d) xr (t) − xPQ (t) 36 Fig 4.5 Simulation results: (a) wind speed, (b) power coefficient, (c)error Pmax (t) − Pm (t), and (d) energy output 37 Fig 4.6 Simulation results: (a) wind speed, (b) power coefficient, (c) error Pmax (t) − Pm (t), and (d) energy output 38 v vi List of Tables Table 3.1 Comparision of two MPPT methods 25 Table 4.1 Parameters in simulations (DFIG[25] and wind turbine) 28 vii Proof Let us define e1 (t)     e (t) ω (t) − ω (t) r  ωrref   rref    =   e (t) i (t) − i (t)  i rref (B.21) r (3.32) can be rewritten as kd d eω (t) = −k p eωrref (t) + ei (t) dt rref (B.22) By substituting (3.31) into (2.14), we have d (irref (t) − ir (t)) = −K (irref (t) − ir (t)) dt (B.23) Then, E1 d e1 (t) = A1 e1 (t), dt (B.24) where    k  k p  d   E1 =  > 0, A1 = −    0 I  0  −1   K  (B.25) When we define a Lyapunov function as e1 (t)E1 e1 (t), V1 (B.26) its derivative is d d e1 (t) V˙ = e1 (t)E1 e1 (t) + dt dt E1 e1 (t) (B.27) ˜ we have By substituting (B.24) into (B.27), and noting that A1 + A1 = −Q, ˜ (t) ≤ −λmin (Q)e ˜ (t)e1 (t) V˙ = e1 (t) A1 + A1 e1 (t) = −e1 (t)Qe (B.28) From the Lyapunov Stability Theory, limt→∞ e1 (t) = This completes the proof B.5 Matrix inequality Let Sn denote a set of real-valued symmetrical n × n matrices and let λi (Y) denote the ith eigenvalue of Y ∈ Sn For Y ∈ Sn and x ∈ Rn , the following inequality holds [32]: λi (Y) x ≤ x Yx ≤ max λi (Y) x 46 (B.29) Definition (ex p 647 in [33]) A matrix Y = Y ∈ Rn×n is said to be positive definite if x Yx > for all nonzero vector x ∈ Rn We denote the positive definite matrix as Y > Moreover, if x Yx ≥ for all x 0, Y is said to be positive semidefinite, and we denote it as Y ≥ Lemma For Y ∈ Sn , Y > ⇔ λi (Y) > 0, (B.30) Y ≥ ⇔ λi (Y) ≥ (B.31) For Y, Z ∈ Sn , we use Y > (≥)Z to mean Y − Z > (≥)0 Lemma For any matrices Y and Z, ±Y Z ± Z Y ≤ Y Y + Z Z, (B.32) ∓Y Z ∓ Z Y ≥ −Y Y − Z Z (B.33) Proof It is trivial from ∓Y Z ∓ Z Y + Y Y + Z Z = (Y ∓ Z) (Y ∓ Z) ≥ (B.34) Lemma    Q S   −1   > ⇔ R > and Q − SR S >  S R ⇔ Q > and R − S Q−1 S > (B.35) (B.36) Proof        Q S   I SR−1  Q − SR−1 S   I          =         −1 S R I R R S I      I  Q   I Q−1 S 0       =   S Q−1 I   R − S Q−1 S 0 I  For square matrices A and B, we denote a block-diagonal matrix as   A    A ⊕ B   B 47 (B.37) (B.38) (B.39) B.6 Proof of Lemma The solutions of (3.38) and (3.39) are t ω ˆ ropt (t) = e−k3 t ω ˆ ropt (0) + k3 e−k3 (t−τ) ωr (τ)dτ, (B.40) t kˆ opt (t) = e−k4 t kˆ opt (0) + (−ωr (τ)2 ω ˆ ropt (τ) + ωr (τ)3 + k4 kopt )e−(k4 −τ) dτ (B.41) Because ωr (t) < ωrrated , we have t e−k3 (t−τ) ωr (τ)dτ ˆ ropt (0) + k3 ω ˆ ropt (t) ≤ e−k3 t ω t ≤ e−k3 t ω ˆ ropt (0) + k3 ωrrated e−k3 (t−τ) dτ ˆ ropt (0) + (1 − e ≤ ω −k3 t )ωrrated ≤ ω ˆ ropt (0) + ωrrated and kˆ opt (t) t −k4 t ≤ e kˆ opt (0) + t ωr (τ) ω ˆ ropt (τ) e −k4 (t−τ) dτ + ωr (τ)3 + k4 kopt e−k4 (t−τ) dτ t ≤ e−k4 t kˆ opt (0) + ( ω ˆ ropt (0) + ωrrated ) ωr (τ)2 e−k4 (t−τ) dτ t + ωr (τ)3 + k4 kopt e−k4 (t−τ) dτ ≤ kˆ opt (0) + (1 − e−k4 t ) k4−1 ω3rrated + kopt + ω ˆ ropt (0) + ωrrated (1 − e−k4 t )k4−1 ω2rrated ≤ kˆ opt (0) + k4−1 ω3rrated + kopt + ω ˆ ropt (0) + ωrrated k4−1 ω2rrated ≤ kˆ opt (0) + 2k4−1 ω3rrated + k4−1 ω ˆ ropt (0) ω2rrated + kopt B.7 Proof of Theorem To use     k   Jˆ  d     > E =  ⊕  I  0 I  2 48 (B.42) and   αI    X =  > 0,  I  (B.43) we define a Lyapunov function as e(t) XEe(t), V (B.44) where   eω (t)  rref     ei (t)      e(t) =  eωopt (t)      eωˆ ropt (t)   ekopt (t)     ωrref (t) − ωr (t)        irref (t) − ir (t)      ωr (t) − ωropt (Vw (t))    ωropt (Vw (t)) − ω  ˆ (t) ropt     ˆ kopt − kopt (t) (B.45) In this proof, we show that the derivative of the Lyapunov function d d V˙ = e(t) XE e(t) + E e(t) dt dt Xe(t), (B.46) satisfies V˙ < −qV + By defining the function p(t) (B.47) −qV + − V˙ > 0, we have V˙ = −qV + − p(t) Then, t V(t) = e−qt V(0) + e−q(t−τ) ( − p(τ)) dτ (B.48) e−q(t−τ) dτ (B.49) − e−qt (B.50) t < e−qt V(0) + = e−qt V(0) + q Hence, the upper bound of V(t) converges to /q This implies that V(t) will be ˆ ωopt (t)2 ≤ V(t), there exists a time to > bounded by /q as t increases Since Je such that for all t ≥ to , |eωopt (t)| = |ωr (t) − ωropt (Vw (t))| < 49 ˆ Jq (B.51) is satisfied Hereafter, we will derive (B.47) From (2.2), (3.16), and (3.36), we have Jωr (t) d ωr (t) = Pm (t) − Pe (t) dt = kopt ωr (t)3 − ζ(t)ωr (t)eωopt (t) − kˆ opt (t)ωrref (t)3 + ωr (t) k1 d ωr (t) − k2 ωr (t) − ω ˆ ropt (t) dt Then, when we use Jˆ = J − k1 and ωrref (t)3 − ωr (t)3 = ξ (ωr (t), ωrref (t)) ωr (t)eωrref (t), and ωr (t) − ω ˆ ropt (()t) = eωopt (t) + eωˆ ropt (t), we have d Jˆ ωr (t) = kopt ωr (t)2 − ζ(t)eωopt (t) − kˆ opt (t)ωr (t)2 dt − kˆ opt (t)ξ (ωr (t), ωrref (t)) eωrref (t) − k2 ωr (t) − ω ˆ ropt (t) = (kopt − kˆ opt (t))ωr (t)2 − ζ(t)eωopt (t) − kˆ opt (t)ξ (ωr (t), ωrref (t)) eωrref (t) − k2 eωopt (t) + e4 (t) = −ζ(t)eωopt (t) − k2 eωopt (t) + eωˆ ropt (t) + ωr (t)2 ekopt (t) − kˆ opt (t)ξ (ωr (t), ωrref (t)) eωrref (t) Hence, d d d Jˆ e3 (t) = Jˆ ωr (t) − Jˆ ωropt (Vw (t)) dt dt dt = − (ζ(t) + k2 ) eωopt (t) − k2 eωˆ ropt (t) + ωr (t)2 ekopt (t) d − kˆ opt (t)ξ (ωr (t), ωrref (t)) eωrref (t) − Jˆ ωropt (Vw (t)) dt (B.52) From (B.24), (3.35), (3.39), (3.38), and (B.52), we can summarize E d d e(t) = A(t)e(t) + B ωropt (t) + Cδ + Dkˆ opt (t)ξ (ωr (t), ωrref (t)) Ge(t), dt dt 50 (B.53) where                                 A           ˆ , C =   , D =   , G = A(t) =  , B = 0 0 , − J    −1  A (t)                         0 −k  0   2 − (ζ(t) + k2 )    −k Jω (t) r  k p −1       A1 = −  , A (t) = (B.54)  −k −k  3    0   K −ωr (t)2 −ωr (t)2 −k4 By substituting (B.53) into (B.46), we have V˙ = e(t) XA(t) + A(t) X e(t) + 2e(t) XB d ωropt (t) dt + 2e(t) Cδ + 2e(t) XDkˆ opt (t)ξ (ωr (t), ωrref (t)) Ge(t) (B.55) Since 2|eωopt (t)| ≤ weωopt (t)2 + w−1 and 2|eωˆ ropt (t)| ≤ weωˆ ropt (t)2 + w−1 for w > 0, we have 2e(t) XB d ˆ ωopt (t) + eωˆ ropt (t) d ωropt (Vw (t)) ωropt (t) ≤ − Je dt dt ˆ ωopt (t)| + |eωˆ ropt (t)| γ ≤ J|e ˆ ωopt (t)2 + weωˆ ropt (t)2 + w−1 Jˆ + w−1 γ ≤ w Je = e(t) Me(t) + , (B.56) where M ˆ 1, , wγdiag 0, 0, J, (B.57) w−1 γ( Jˆ + 1) (B.58) Likely, we have 2e(t) XCδ < 2k4 |ekopt (t)||δmax |kopt ≤ e(t) Ne(t) + , where N = diag(0, 0, 0, 0, vkopt k4 ), (B.59) = v−1 k4 kopt δ2max Furthermore, by applying 51 (B.32) in Lemma to set Y = αkˆ opt (t)ξ (ωr (t), ωrref (t)) G and Z = α−1 D X, we have 2e(t) XDkˆ opt (t)ξ (ωr (t), ωrref (t)) Ge(t) = e(t) G kˆ opt (t)ξ (ωr (t), ωrref (t)) D Xe(t) + e(t) XDkˆ opt (t)ξ (ωr (t), ωrref (t)) Ge(t) ≤ e(t) α2 kˆ opt (t)2 ξ (ωr (t), ωrref (t))2 G Ge(t) + α−2 e(t) (t)XDD Xe(t) (B.60) Then, by noting that α−2 XDD X = DD and defining −XA(t) − A(t) X − M − N − DD − α2 kˆ opt (t)2 ξ (ωr (t), ωrref (t))2 G G, Q(t) (B.61) + 2, (B.62) we have V˙ ≤ −e(t) Q(t)e(t) + (B.63) Q(t) − qXE > 0, (B.64) V˙ < −qe(t) XEe(t) + (B.65) Hence, if then This implies (B.47) To complete the proof, we need the next lemma Lemma 10 The five inequalities in (3.44) imply (B.64) Proof qXE and Q(t) are both block diagonal as     k   Jˆ  d   ⊕ q   , qXE = qα     I  I (B.66)   2k − αkˆ (t)2 ξ (ω (t), ω (t))2 −1  p opt r rref            Q(t) = α      K + K     −1   2ζ(t) + 2k2 − wγ Jˆ − k2 + k3       ⊕   k2 + k3 2k3 − wγ ωr (t)     ωr (t)2 (2 − vkopt )k4 52 (B.67) Hence, we show (B.64) separately as   2k − αkˆ (t)2 ξ (ω (t), ω (t))2 −1    opt r rref  p  k      d      − q   > 0,      K + K I2      −1   2ζ(t) + 2k2 − wγ Jˆ − k2 + k3       Jˆ       k2 + k3 2k3 − wγ ωr (t)2  − q   >   I   ωr (t)2 (2 − vkopt )k4 (B.68) (B.69) By noting that kˆ opt,ub ξmax ≥ kˆ opt (t)ξ (ωr (t), ωrref (t)) and applying (9) , the first two inequalities in (3.44) imply (B.68)              2 −1  ˆ   − qkd > 0,  2k − α k ξ − Ξ  p −1  opt,ub max     −1         Ξ = K + K − qI2 > To apply (B.33) in Lemma by setting   √    k2 + k3 0  , Z = 0 Y =     0 ωr (t)  √ k2 + k3 0  , ω (t) 0 (B.70) (B.71) r the off-diagonal elements of (B.69) are bounded as       k2 + k3  k + k 0         2 k2 + k3 ωr (t)  ≥ −  k2 + k3 + ωr (t)         2 ωr (t) 0 ωr (t) (B.72) By noting that ζ(t) ≥ ζmin and ωr (t) ≤ ωrrated , the last three inequalities in (3.44) imply (B.69)    ˆ − (k3 − k2 ) − > q J, ˆ  2ζmin − w Jγ         k3 − k2 − ω2rrated − wγ > q,         (2 − vkopt )k4 − ω2rrated > q (B.73) Remark We consider ξ (ωr , ωrref ) only for the region of ωr > and ωrref > 0, and the Hessian of ξ (ωr , ωrref ) is positive semidefinite as     ∂2 ξ   ωrref  ∂2 ξ ωrref      −  ∂ωr ∂ωrref   ωr  ∂ω2r ωr  ≥ =    ωrref   ∂2 ξ ∂ ξ  −   ∂ωrref ∂ωr ω2r ωr ∂ω2rref 53 (B.74) Hence, ξmax is attained at the boundary of the region [ωrmin , ωrrated ] × [ωrmin , ωrrated ] Moreover, from   ∂ξ   ω2rref      ∂ω   −   ωr  , r 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a wind turbine driving a DFIG IEEE Trans Power Electron 2008; 23(3): 1156-1169 [32] Boyd S., Ghaoui L E.l., Feron E., Balakrishnan V., Linear matrix inequalities in system and control theory Philadelphia: SIAM; 1994 [33] Boyd S., Vandenberghe L., Convex optimization New York: Cambridge University Press; 2004 60 ... capacity Therefore, the converter capacity is around 30% of DFIG capacity and this is a good point of DFIG- wind turbine For the DFIG wind turbine, thanks to the back-to-back converter, the DFIGwind... signal for the controller [8, 6] These methods are called wind- data-based methods Generally, with wind- databased methods, the MPPT ability of a wind turbine is appreciably high if accurate wind. .. to maximize the energy output of a DFIG wind turbine without any information about the wind data or an available anemometer These proposed schemes are based on the improvement of the wind turbine? ??s