ISSN Article. Maximum Energy Output of a DFIG Wind Turbine Using an Improved MPPT-Curve Method

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1 Energies 2015, 8, ; oi: /en OPEN ACCESS energies ISSN Article Maximum Energy Output of a DFIG Win Turbine Using an Improve MPPT-Curve Metho Dinh-Chung Phan 1,2 an Shigeru Yamamoto 3 * 1 Faculty of Electrical Engineering, Danang University of Science an Technology, 54-Nguyen Luong Bang, Danang , Vietnam; chungpy996@yahoo.com 2 Division of Electrical Engineering an Computer Science, Grauate School of Natural Science an Technology, Kanazawa University, Kakuma, Kanazawa, Ishikawa , Japan 3 Faculty of Electrical an Computer Engineering, Institute of Science an Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa , Japan * Author to whom corresponence shoul be aresse; shigeru@se.kanazawa-u.ac.jp; Tel.: Acaemic Eitor: Free Blaabjerg Receive: 1 August 2015 / Accepte: 13 October 2015 / Publishe: 19 October 2015 Abstract: A new metho is propose for obtaining the maximum power output of a oubly-fe inuction generator (DFIG) win turbine to control the rotor- an gri-sie converters. The efficiency of maximum power point tracking that is obtaine by the propose metho is theoretically guarantee uner assumptions that represent physical conitions. Several control parameters may be ajuste to ensure the quality of control performance. In particular, a DFIG state-space moel an a control technique base on the Lyapunov function are aopte to erive the control metho. The effectiveness of the propose metho is verifie via numerical simulations of a 1.5-MW DFIG win turbine using MATLAB/Simulink. The simulation results show that when the propose metho is use, the win turbine is capable of properly tracking the optimal operation point; furthermore, the generator s available energy output is higher when the propose metho is use than it is when the conventional metho is use instea. Keywors: oubly-fe inuction generator; Lyapunov function-base control; maximum power point tracking; maximum energy

2 Energies 2015, Introuction Generally, generator win turbines are ivie into two groups: fixe-spee win turbines (FSWT) an variable-spee win turbines (VSWT). For FSWTs, a squirrel-cage inuction generator (SCIG) [1] is often employe to operate at a fixe rotor spee an is often irectly connecte to the gri. In contrast, since VSWTs operate at variable rotor spees, the generator is often connecte to the gri via a converter system for synchronization [2,3]. For VSWTs base on synchronous generators (SG), permanent magnetic synchronous generators (PMSG) or SCIGs, a full converter must be installe [2]; however, for VSWTs that use a oubly-fe inuction generator (DFIG), a partial converter must be installe at the rotor sie [3]. The essential objective for VSWTs is to obtain maximum power output to maximize energy conversion efficiency. To acquire maximum power output, many methos have been propose [2,4 9]. Generally, these methos are base on win spee measurement [5,6] or win spee sensor-less [7,8] approaches. The maximum power point tracking (MPPT) ability of the win turbine is achievable when a precise win spee measurement is available. Unfortunately, win spee measurement is unreliable because of the win s rapi fluctuation [10]. Other methos that o not use an anemometer, such as the MPPT-curve metho [8,9] an perturbation an observation (P&O) [4], cannot track the maximum power point either exactly or quickly, because they operate basically on the generator s output. Such methos are mainly applie either to photovoltaic power systems where the inertia of the generator is zero or to PMSG win turbines with a DC/DC converter [2,4]. The MPPT-curve metho inexes the current rotor spee (or power output), as well as the win turbine s MPPT curve to etermine a reference power output (or rotor spee) [8,9]. It oes not require any perturbation signal an is robust; however, this metho cannot track the maximum point quickly because of the high inertia of the generator win turbine system. To control the win turbine, traitional proportional-integral (PI) control is popularly use for many purposes, such as rotor spee control, current control, power control, an so on [8,9,11]. However, stability with PI control is not theoretically ensure [12,13]; thus, sliing moe control [14 16] has been recently evelope. Unfortunately, sliing moe control is only applicable to the power in the DFIG s stator sie [14] or the rotor spee [15] via the rotor-sie converter (RSC) an current components in the gri sie-converter (GSC) [16]. Hence, there is not currently a control metho irectly applicable to the total power output of DFIG an DC voltage. This paper proposes a new scheme to maximize the energy output of a DFIG win turbine. Aitionally, new controllers for controlling the total power output will be applie to the DFIG s converters an the DC voltage, on the DC link will be esigne base on Lyapunov function control. The propose MPPT scheme will be analyze an verifie via the simulation of a 1.5-MW DFIG win turbine using MATLAB/Simulink. Simulation results will be evaluate an compare to the results of a win turbine using the conventional MPPT-curve metho with PI controllers.

3 Energies 2015, Moel of a DFIG-Win Turbine Generally, a DFIG win turbine [17] appears as shown in Figure 1. Figure 1. Overall system of the oubly-fe inuction generator (DFIG) win turbine Win Turbine In the generator-win turbine system [14], the ynamic equation is written as: Jω r (t) t ω r(t) = P m (t) P e (t) (1) where ω r an P m are the rotor spee an mechanical power of the win turbine, respectively; P e is the electrical power of the generator an J is the inertia of the generator-win turbine system. The mechanical power is expresse by: P m (t) = 1 2 ρπr2 C p (λ(t), β(t))v 3 w(t) (2) where R is the blae size; ρ is the air ensity; V w is the win spee; C p is the power coefficient; β is the pitch angle of the blae system an λ is the tip spee ratio efine by: λ(t) = Rω r(t) V w (t) (3) The maximum mechanical power P max is efine as: P max (t) = 1 2 ρπr2 C pmax (β)v 3 w(t) (4) An example of C p (λ, β) [18] an P m when β = 0 is shown in Figure 2.

4 Energies 2015, (a) (b) Figure 2. Characteristics of the win turbine when β = 0: (a) C p versus λ; (b) P max versus ω r. Figure 2a inicates that when β = 0, there exists a maximum point C pmax := max λ C p (λ, 0) = 0.48 attaine by λ opt = As can be seen in Figure 2b, the win turbine works in the optimal power control region when the rotor spee is between the minimum spee ω rmin an the rate spee ω rrate or when the win spee is between the minimum spee V wmin an the rate spee V wrate. In this region, it is esirable for the win turbine to operate on the locus of the maximum power point P max. This locus, name the MPPT-curve, is escribe by: P mppt (t) = k opt ω 3 r(t) (5) where: k opt = 1 C 2 ρπr5 pmax (6) λ 3 opt The objective of this paper is to propose an MPPT-scheme an a controller, such that the win turbine operates on the MPPT curve as ω rmin ω r (t) < ω rrate. In fact, max λ C p (λ, β) is ecrease when β increases [19], an max λ C p (λ, β) only reaches to max λ,β C p (λ, β) at β = β min, where β min is the minimum value of the pitch angle in its operation range an β min is normally zero. In other wors, to obtain max λ,β C p (λ, β), the pitch angle β is normally set at β min. Hence, without loss of generality, the following assumption is use in this research. Assumption 1. The pitch angle β is kept at β min = 0 as ω rmin ω r (t) < ω rrate DFIG In the qframe, the DFIG can be escribe as [19,20]: { where v s = i s = v s (t) = R s i s (t) + L s t i s(t) + L m t i r(t) + ω s Θ(L s i s (t) + L m i r (t)) v r (t) = R r i r (t) + L r i t r(t) + L m i (7) t s(t) + ω s s(t)θ(l m i s (t) + L r i r (t)) [ ] T [ ] T v s v sq is the stator-sie voltage; vr = v r v rq is the rotor-sie voltage; [ ] [ ] T 0 1 i r i rq is the rotor-sie current an Θ =. 1 0 [ i s i sq ] T is the stator-sie current; ir =

5 Energies 2015, ω, R, L an s represent rotational spee, resistance, inuctance an rotor slip, respectively; subscripts r, s an m stan for rotor-sie, stator-sie an magnetization. Note that ω s is normally constant. The rotor slip of the DFIG is efine by: The power output of the generator is escribe by: s(t) = 1 ω r(t) ω s (8) P e (t) = P s (t) + P r (t) = (1 s(t))p s (t) = ω r(t) ω s P s (t) (9) where P s an P r are the stator-sie an rotor-sie power, respectively. Assumption 2. The stator flux is constant, an the -axis of the q-frame is oriente with the stator flux vector. Hence, [ ] [ ] Ψ s (t) Ψ s Ψ s (t) = = L s i s (t) + L m i r (t) (10) Ψ sq (t) 0 Then, L s t i s(t) + L m t i r(t) = 0 (11) Moreover, the resistance of the stator wining can be ignore, i.e., R s = 0. By substituting Equations (10) an (11) an R s = 0 into Equation (7), we have: v s (t) = because V s = v s (t). From Equations (10) an (12), we have: [0 ω s Ψ s ] T = [ 0 V s ] T (12) L s i s (t) + L m i r (t) = 1 ω s [V s 0] T (13) By substituting Equation (13) into the secon part of Equation (7), we have: where: t i r(t) = A 1 (t)i r (t) + 1 σ v r(t) 1 (t) (14) σ = L r L2 m, L [ s A 1 (t) = 1 σ R r σω s s(t) ] σω s s(t) R r Ṽ s = L m V s, L s [ ] (15) 0 1 (16), 1 (t) = Ṽs σ s(t) Lemma 1. A state-space representation of the active power P s an the reactive power Q s on the stator sie is given by: t x 1(t) = A 1 (t)x 1 (t) Ṽs σ v r(t) + c 1 (t) (17)

6 Energies 2015, where: [ ] Q s (t) x 1 (t) =, c 1 (t) = A 1(t) P s (t) L s ω s [ V 2 s 0 ] + Ṽs 1 (t) = V R s 2 r ω s (18) σl s L r s(t) In aition, uner Assumption 2, a state-space representation of the DFIG (from Equation (7)) is escribe by: where: x(t) = [ ] Q s (t) P e (t) t x(t) = A(t)x(t) + B(t)v r(t) + (t) (19) A(t) = C(t) 1 ( A 1 (t)c(t) Ċ(t) ) = B(t) = Ṽs σ C 1 (t) C(t) = (t) = C(t) 1 c 1 (t) = V s 2 σl s ω s [ ω 2 s ω r (t) s(t) (20) 1 R r σ ω R r (21) t r(t) ω s ω r (t)s(t) σ ω r (t) 1 ω 0 s (22) 0 ω r (t) ] R r L r ω r (t)s(t) (23) Proof. The power in the stator sie are given as [20]: [ ] [ Q s (t) i sq (t) x 1 (t) = = P s (t) i s (t) ] [ ] i s (t) v s (t) i sq (t) v sq (t) (24) By substituting Equations (12) an (13) into Equation (24), we have: [ ] x 1 (t) = V s i s (t) = Ṽsi r (t) + 1 Vs 2 L s ω s 0 (25) Then, from Equations (25) an (14), (17) changes as follows: t x 1(t) = Ṽs t i r(t) = A 1 (t)ṽsi r (t) Ṽs σ v r(t) + Ṽs 1 (t) [ ] = A 1 (t)x 1 (t) Ṽs σ v r(t) A 1(t) Vs 2 + L s ω s 0 Ṽs 1 (t) By applying a transformation: x 1 (t) = C(t)x(t) (26) to Equation (17), we have: t x 1(t) = Ċ(t)x(t) + C(t) t x(t) = A 1(t)x 1 (t) Ṽs σ v r(t) + c 1 (t) (27)

7 Energies 2015, Hence, Equation (19) changes to be: ( ) t x(t) = C(t) 1 A 1 (t)c(t) Ċ(t) x(t) Ṽs σ C(t) 1 v r (t) + C(t) 1 c 1 (t) (28) Here, it is easy to check Equations (23) an (21) by using: Ċ(t) = ω s ω r (t) ω (29) 2 t r(t) 2.3. Converter The overall converter system is shown in Figure 1, where the RSC an GSC are linke by a DC circuit [3,20]. This DC circuit, calle the DC link, consists of a capacitor. Normally, the GSC is connecte to the gri via a filter (represente by a resistor R f, an inuctor L f in series an a power factor correction pfin parallel) [20]. This R f L f circuit can be escribe in the qframe as: v g (t) = v s (t) + R f i g (t) + L f t i g(t) + L f ω s Θi g (t) (30) [ ] T [ ] T where v g = v g v gq is the voltage output of the GSC; ig = i g i gq is the current output of the GSC an R f an L f are the resistance an inuctance, respectively, of the R f L f circuit. Assumption 3. Power loss in converters an the R f L f circuit is neglecte [19,21]. Uner Assumption 3, the DC link is escribe as [21]: t V c(t) = 1 CV c (t) (P r(t) P g (t)) (31) where C is the capacitance of the capacitor; V c is the DC voltage on the DC link an P g is the active power output of the GSC [20]. Assumption 4. The q frame in which the axis is oriente with the voltage vector v s, so v s (t) = V s an v sq (t) = 0. Uner Assumption 4, the active power output of the GSC connecte to the gri is escribe by: P g (t) = v s (t)i g (t) + v sq (t)i gq (t) = V s i g (t) (32) Lemma 2. Uner Assumption 4, a state-space representation of the GSC connecte to the gri is escribe as: [ ] t i g(t) = A 2 i g (t) + 1 v g (t) 1 V s (33) L f L f 0 where I is the ientity matrix. A 2 = R f L f I ω s Θ (34)

8 Energies 2015, Proof. From Equation (30), it is easy to obtain: ( t i Rf g(t) = I + ω s Θ L f From Assumption 4, we have Equation (33). This completes the proof. 3. Controller Design an Maximum Power Strategy ) i g (t) + v g(t) v s (t) L f (35) Assumption 5. We can measure i r, i s, i g, v s, V c an ω r. In aition, we can manipulate v r, v rq, v g an v gq an know parameters R s, R r, L s, L r an L m. Assumption 6. The q/abc transformation block, the PWM an the IGBT-valves in converters operate properly Rotor-Sie Control The objective of RSC is to maintain, at the esire references, the reactive power in the stator sie Q s an the total active power output P e of the DFIG. [ ] T x T r = Q sref P eref (36) From Equations (9), (14) an (25), to ajust Q s an P e corresponing to i r an i rq, v r an v rq must be regulate, respectively. To perform this task, previous stuies employe PI controllers [9], [22]. In this stuy, a new control law is propose. Lemma 3. (RSC control) Uner Assumptions 2 an 5, when we can measure ω t r(t) for any esire reference x r (from Equation (36)), if we use any positive efinite matrix P, v r (t) = B(t) (A(t)x(t) 1 + P e x (t) ) t x r(t) + (t) (37) e x (t) = x r (t) x(t) (38) for the DFIG (from Equation (19)), then it is ensure that: Proof. By applying Equations (37) to (19), we have: which is equivalent to: lim e x(t) = 0 (39) t t x(t) = P e x(t) + t x r(t) (40) t e x(t) = P e x (t) (41) It is obvious that when we efine a Lyapunov function V (e x ) = e T x e x, its time erivative becomes: t V (e x) = 2e x (t) T t e x(t) = 2e x (t) T P e x (t) < 0 (42) for any e x (t) 0. Hence, e x (t) converges to zero.

9 Energies 2015, From Lemma 3, if the rotor voltage v r is esigne as Equation (37), the power output of DFIG will converge to its reference value (given Equation (36)) Maximum Output Power Control The main objective of this section is to propose a new MPPT scheme that improves the conventional MPPT-curve metho [22] so that P m approaches the neighbor of P max as ω rmin ω r (t) < ω rrate, as shown in Figure 2b. From Equation (4) an Figure 2a, P m (t) only reaches the neighbor of P max as λ approaches the neighbor of λ opt. Therefore, in this stuy, a new strategy is propose, such that λ approaches the neighbor of λ opt. Assumption 7. The win turbine operates in the optimal power control region, as shown in Figure 2b, an λ min λ(t) λ max, where λ min = Rω rmin. In aition, there exists a constant γ, such that V wrate V t w(t) γ. For the win turbine (from Equations (1) (4)) an C p (λ) (shown in Section 4), we have Remark 1, as follows. Remark 1. From (1) an (3) an from the efinition of k opt, we have: ( P m (t) k opt ωr(t) 3 = ρπr5 ωr(t) 3 Cp (λ(t)) C ) pmax = ζ(t) (λ(t) λ 2 λ 3 (t) λ 3 opt ) (43) opt where: ζ(t) = ρπr5 2λ 3 opt ω 3 r(t) λ(t) λ opt ( ) C pmax C p (λ(t)) λ3 opt λ 3 (t) (44) is positive, continuous an boune in λ min λ(t) λ max. Theorem 1. Uner Assumption 7, suppose that we use a positive constant α < J, k opt (given in Equation (6)), an P eref (given in Equation (36)) for the RSC control (given in (37)) as: P eref (t) = k opt ω 3 r(t) αω r (t) t ω r(t) (45) If there exists a positive constant 0 < χ < 2ζ(t), such that: P := 2P λ(t) > 0 (46) 0 χ ηωr(t) 2 for the efinite matrix P > 0 in Equation (37) an all t, then there exists a time t 0 > 0, such that for e λ (t) = λ(t) λ opt, e λ (t) 2(J α)γ max ω 2 r(t) (2ζ(t) χ)v w (t) (47) for all t t 0.

10 Energies 2015, Proof. First, let e x (t) = [ e Q e P ] T = xr (t) x(t). We efine a Lyapunov caniate as: The time erivative of the Lyapunov function is: Since λ(t) = Rω r (t)/v w (t), we have: V (e x, e λ ) = e x (t) T e x (t) + e 2 λ(t) (48) V (e x, e λ ) = 2e x (t) T t e x(t) + 2e λ (t) λ(t) (49) t λ(t) λ(t) = t ω r (t) From Equation (1) an e P (t) = P eref (t) P e (t), we have: t ω r(t) λ(t) V w (t) t V w(t) (50) t ω r(t) = 1 Jω r (t) (P m(t) + e P (t) P eref (t)) (51) By substituting P eref from Equations (45) into (51), we get: t ω r(t) = 1 ηω r (t) (P m(t) k opt ω 3 r(t) + e P (t)) (52) where η = J α > 0. By substituting Equations (43) into (52), we have: t ω r(t) = 1 ηω r (t) (e P (t) ζ(t)e λ (t)) (53) From Equations (53) an (50), we have: where δ 1 (t) := we have: 2e λ (t) t λ(t) = δ 1(t) ( 2e P (t)e λ (t) 2ζ(t)e 2 λ(t) ) 2λ(t) V w (t) e λ(t) t V w(t) (54) λ(t) ηω 2 r(t) > 0. Since: 2e P (t)e λ (t) χ 1 e 2 P (t) + χe 2 λ(t) (55) t V w(t) γ (56) 2λ(t)e λ(t) V w (t) t V w(t) 2λ(t) e λ(t) γ (57) V w (t) 2e λ (t) t λ(t) χ 1 δ 1 (t)e 2 P (t) δ 2 (t)e 2 λ(t) + δ 3 (t) e λ (t) (58) where: an: δ 2 (t) := (2ζ(t) χ)δ 1 (t) > 0 δ 3 (t) := 2λ(t)γ V w (t) 0

11 Energies 2015, From Equations (42) an (58), (49) becomes: V (e x, e λ ) 2e T x (t)p e x (t) + χ 1 δ 1 (t)e 2 P (t) δ 2 (t)e 2 λ(t) + δ 3 (t) e λ (t) (59) Obviously, 2e T x (t)p e x (t) + χ 1 δ 1 (t)e 2 P (t) = e T x (t) P e x (t) < 0 Hence, ( V (e x, e λ ) δ 2 (t) e λ (t) e λ (t) δ ) 3(t) δ 2 (t) e λ (t) ( e λ (t) 3 ) (60) δ 2 (t) where: 3 = max δ 3(t) δ 2 (t) = 2ηγ max ω 2 r(t) (2ζ(t) χ)v w (t) (61) Then, V (e x, e λ ) ecreases when: e λ (t) > 3 = 2ηγ max ω 2 r(t) (2ζ(t) χ)v w (t) (62) This completes the proof of Theorem 1. From Theorem Equation (1), if the reference power P eref in Equation (36) is calculate as Equation (45), the tip-spee ratio λ of the win turbine will converge to the neighbor of λ opt an, hence, P m will approach its maximum value Gri-Sie Control From Equation (3), if V c on the DC link is kept at a constant value, all power output generating in the rotor sie will be elivere to the connecte gri. Moreover, if the i gq component is maintaine at zero, the power loss in the R f L f circuit will be minimize. Consequently, the conversion energy efficiency of the DFIG-win turbine will be maximize. Without loss of generality, in this section, the objective of GSC is to maintain V c (t) an i gq (t) at esire references V cref (t) an i gqref (t), respectively. From Equations (31) (33), we nee to ajust v g. Theorem 2. For any esire reference V cref (t) an i gqref (t), for GSC (given in Equation (33)), we use: ( ) [ ] v g (t) = L f t i V s gr(t) + Q(t)e i (t) A 2 i g (t) + (63) 0 1 ( Pr (t) CV i gr (t) = c (t) V V t cref(t) ) ke v (t) s (64) i gqref (t) e v (t) = V cref (t) V c (t) (65) e i (t) = i gr (t) i g (t) (66)

12 Energies 2015, an if there exists a constant k > 1/2 an a positive efinite matrix Q(t), such that: [ ] Q(t) > 0 (67) V c (t) 0 0 it is ensure that: lim e v(t) = 0 an lim (i gqref (t) i gq (t)) = 0 t t Proof. By substituting Equations (63) into (33), we have: By efining e i (t) = t i g(t) = t i gr(t) + Q(t) (i gr (t) i g (t)) (68) [ T e i1 (t) e i2 (t)] = igr (t) i g (t), Equation (68) means: t e i(t) = Q(t)e i (t) (69) Furthermore, by efining e v (t) = V cref (t) V c (t), from Equation (31), we have: t e v(t) = t V 1 cref(t) CV c (t) (P r(t) V s i g (t)) (70) where we use P g (t) = V s i g (t). To use Equation (70), we can show: e i1 (t) = CV c(t) V s t e v(t) ke v (t) (71) When we introuce a Lyapunov function: an its time erivative is expresse as: V (e v, e i ) = C V s e 2 v(t) et i (t)e i (t) (72) V (e v, e i ) = 2C V s e v (t) t e v(t) + e T (t) t e i(t) = 2 V c (t) e v(t) (e 5 (t) + ke v (t)) e T i (t)q(t)e i (t) = 2k 1 V c (t) e2 v(t) 1 V c (t) (e v(t) + e i1 (t)) V c (t) e2 i1(t) e T i (t)q(t)e i (t) Hence, if Equation (67) hols, then V (e v, e i ) < 0 for all nonzero e v an e i. This completes the proof of Theorem 2. From Theorem 2, if v g is esigne as Equation (63), where V cref an i gqref are set at a constant an zero, respectively, the power loss in the converter an R f L f will be minimize. Control iagrams for the RSC an GSC are inicate in Figure 3. (a) (b) Figure 3. Control of DFIG: (a) rotor-sie converter (RSC) controller; (b) gri sie-converter (GSC) controller.

13 Energies 2015, Performance Valiation This section compares the simulation results of the 1.5-MW DFIG win turbine using the propose MPPT scheme with that of the same win turbine using the conventional MPPT-curve metho using PI control [9,22]. MATLAB/Simulink was use for the simulation. The parameters use for the win turbine in the simulation are given in Table 1 [23]. Table 1. Parameters of the win turbine. Name Symbol Value Unit The length of blae R m Rate rotor spee ω rrate 22 rpm Minimum rotor spee ω rmin 11 rpm Rate win spee V wrate 12 m/s The power coefficient of the win turbine [18] is: where: ( ) C p (λ, β) = β 5 e λ i λ λ i 1 λ i = 1 λ β β with air ensity ρ = kg/m 3 [20] an system inertia J kg m 2 [17]. A win spee profile (shown in Figure 4) was use, in which t V w(t) γ = 0.44 m/s 2. To test the propose MPPT strategy an controllers, the win velocity was always below the rate win spee, 12 m/s. (a) (b) Figure 4. Win spee profile: (a) V w (t) an (b) t V w(t). The rate rotor spee an the minimum one are ω rrate = 2.3 ra/s an ω rmin = 1.15 ra/s, respectively. From the C p (λ, β) equation an β = 0, we can obtain λ opt = an λ max = The minimum value of λ is efine by: λ min = Rω rmin V wrate = = 3.4, an V wmin = Rω rmin λ opt = = 5m/s

14 Energies 2015, The reference values settings for the RSC an GSC control (from (37) an Equations (63)), with Q sref (t) = 0, V cref = 1150 V an i gqref (t) = 0, are: [ ] 2 0 P =, Q(t) = V c (t) an k = We use P eref for the RSC control (from Equation (45)) with α = 0.3 J an k opt = kg m 2. We have: max δ 1 (t) = λ max ηω 2 rmin = (73) where η = J α = 0.7 J = kg m 2. With 5 m/s V w (t) 12 m/s an 1.15 ra/s ω r (t) 2.3 ra/s, ζ versus ω r an V w are shown in Figure 5a. (a) (b) Figure 5. (a) ζ an (b) δ 3 δ 2 versus ω r an V w. From Figure 5a, we can obtain min ζ(t) = ; we choose χ = 1. With χ = 1, δ 3 δ 2 versus ω r an V w are shown in Figure 5b. From that figure, we obtain 3 = Obviously, the conition in Equation (46) is satisfie, an Equation (47) gives the boun of the tip-spee ratio λ as: λ(t) λ opt 3 = Hence, λ(t) We compare the efficiency of the maximum power output in Figure 6. In Figure 6a, the power coefficient C p is always maintaine at its maximum value C pmax = 0.48 when the win spee varies insignificantly. However, eterioration of C p uring a perio of rapi change in win conitions still occurs. Due to the large inertia of the system when the win velocity increases or ecreases quickly, the rotor spee of the turbine cannot respon instantaneously. The tip-spee ratio λ cannot keep its optimum value λ opt ; an hence, the ecrease in C p is unavoiable. Compare to the conventional MPPT-curve metho, in the propose metho, C p retains C pmax promptly, mainly because its inertia seems to be reuce from J to J α; thus, λ retains λ opt. When the MPPT-curve metho is use, the minimum value of C p uring the interval of rapi ecrease in win spee is 0.45; when the propose metho is use, that value is

15 Energies 2015, Figure 6b epicts that, with the propose scheme, λ only varies in a narrow range, < λ(t) < 8.989, which is theoretically ensure by Theorem 1. By contrast, with the conventional scheme, λ vacillates in a larger range. Concerning mechanical power, Figure 6c epicts the error between P max an P m. Obviously, uring perios of stable win conitions, there is no ifference between the conventional an propose methos. However, when the win velocity ramatically varies, the error in the propose metho is significantly smaller than that in the other, mainly because the power coefficient C p, with the conventional metho, is reuce significantly uring suen variations in win conitions. (a) (b) (c) () Figure 6. Simulation results using the propose maximum mechanical power metho: (a) power coefficient; (b) operation range of λ; (c) error between P max an P m ; () electrical energy output. Figure 6 inicates that uring the perio of increasing win velocity [20 s,40 s], a higher mechanical energy part is store in its mechanical system to accelerate the rotor spee by the propose metho than that by the conventional MPPT-curve metho. Hence, the electrical energy output in this perio is smaller. However, this mechanical energy part is release uring eceleration of the rotor spee, as seen in [60 s,75 s]. Obviously, by the propose metho, the total electrical energy output of the generator is higher than that by the conventional one, as shown in Figure 6, mainly because P m, in the propose metho, has a higher value. This affirms the improve efficiency of the propose metho.

16 Energies 2015, MPPT tracking ability is inicate in Figure 7 where the win spee profile shown in Figure 4a is use. As that figure shows, with the propose metho, the curve P m versus V w is approximate to the ieal curve, P max versus V w, while, with the conventional scheme, this is impossible (see Figure 7b). In other wors, the win turbine with the propose metho can track the MPPT better than that with the conventional one. (a) (b) Figure 7. P m (t) versus V w (t): (a) propose scheme an (b) conventional MPPT-curve metho. Concerning the propose control laws an scheme, Figure 8a argues that both e Q (t) = Q sref (t) Q s (t) an e P (t) = P eref (t) P e (t) are approximately zero, which means that Q s an P e always converge to their reference values, Q sref an P eref, respectively; in other wors, Lemma 3 is ensure. Likely, Theorem 2 is also guarantee from Figure 8b because e v (t) = V cref (t) V c (t) an e i2 (t) = i gqref (t) i gq (t) are very small. In other wors, the controllers suggeste for the RSC an GSC have goo performance. (a) (b) Figure 8. (b) GSC. Error between reference signal an actual output in the controller: (a) RSC

17 Energies 2015, When a rapi win profile as shown in Figure 9a is use, the simulation results are emonstrate in Figures 9b 9. Obviously, the win turbine by the propose metho has better performance in the power coefficient C p, the mechanical power an the electrical energy output than by the conventional one. (a) (b) (c) () Figure 9. Simulation results with the rapi win profile: (a) win spee profile; (b) power coefficient; (c) error between P max an P m ; () electrical energy output. 5. Conclusions This paper proposes an MPPT metho for DFIG win turbines. The propose MPPT metho ensures that the win turbine can track the maximum power operation point better than can a win turbine with the conventional MPPT-curve metho, as verifie via the simulation of a 1.5-MW DFIG win turbine. These simulation results inicate that C p reache C pmax, λ was varie aroun λ opt an the electrical energy output of the generator was higher than that achieve with the conventional metho. Furthermore, with the esigne controllers, the error between esire values an actual ones converge to zero. Thus, the propose control metho is proven to achieve stable operations. Author Contributions The propose metho an simulation results were carefully iscusse by both authors. The secon author was the avisor of the first one.

18 Energies 2015, Conflicts of Interest The authors eclare no conflict of interest. References 1. Leon, A.E.; Farias, M.F.; Battaiotto, P.E.; Solsona, J.A.; Valla, M.I. Control strategy of a DVR to improve stability in win farms using squirrel-cage inuction generators. IEEE Trans. Power Syst. 2011, 26, Barakati, S.M.; Kazerani, M.; Aplevich, J.D. Maximum power tracking control for a win turbine system incluing a matrix converter. IEEE Trans. Energy Convers. 2009, 24, Qiu, Y.; Zhang, W.; Cao, M.; Feng, Y.; Infiel, D. An electro-thermal analysis of a variable-spee oubly-fe inuction generator in a win turbine. Energies 2015, 8, Abullah, M.; Yatim, A.H.M.; Tan, C.; Saiur, R. A review of maximum power point tracking algorithms for win energy systems. Renew. Sustain. Energy Rev. 2012, 16, Ganjefar, S.; Ghassemi, A.; Ahmai, M. Improving efficiency of two-type maximum power point tracking methos of tip-spee ratio an optimum torque in win turbine system using a quantum neural network. Energy 2014, 67, Lei, Y.; Mullane, A.; Lightboy, G.; Yacamini, R. Moeling of the win turbine with a oubly fe inuction generator for gri integration stuies. IEEE Trans. Energy Convrers. 2006, 21, Jeong, H.G.; Seung, R.H.; Lee, K.B. An improve maximum power point tracking metho for win power systems. Energies 2012, 5, Tapia, A.; Tapia, G.; Ostolaza, J.; Saenz, J. Moeling an control of a win turbine riven oubly fe inuction generator. IEEE Trans. Energy Convers. 2003, 18, Fernanez, L.; Garcia, C.; Jurao, F. Comparative stuy on the performance of control systems for oubly fe inuction generator (DFIG) win turbines operating with power regulation. Energy 2008, 33, Wagner, H.; Mathur, J. Introuction to Win Energy Systems: Basics, Technology an Operation; Springer-Verlag: Berlin/Heielberg, Germany, 2013; pp Kim, H.W.; Kim, S.S.; Ko, H.S. Moeling an control of PMSG-base variable-spee win turbine. Electr. Power Syst. Res. 2010, 80, Yang, L.; Xu, Z.; Stergaar, J.; Dong, Z.Y.; Wong, K.P.; Ma, X. Oscillatory stability an eigenvalue sensitivity analysis of a DFIG win turbine system. IEEE Trans. Energy Convers. 2011, 26, Mishra, Y.; Mishra, S.; Li, F.; Dong, Z.Y.; Bansal, R.C. Small signal stability analysis of a DFIG-base win power system uner ifferent moes of operation. IEEE Trans. Energy Convers. 2009, 24, Hu, J.; Nian, H.; Hu, B.; He, Y.; Zhu, Z.Q. Direct active an reactive power regulation of DFIG using sliing-moe control approach. IEEE Trans. Energy Convers. 2010, 25, Barambones, O.; Cortajarena, J.A.; Alkorta, P.; Durana, J.M.G. A real-time sliing moe control for a win energy system base on a oubly fe inuction generator. Energies 2014, 7,

19 Energies 2015, Khemiri, N.; Kheher, A.; Mimouni, M.F. Win energy conversion system using DFIG controlle by backstepping an sliing moe strategies. Int. J. Renew. Energy Res. 2012, 2, Beltran, B.; Ahme-Ali, T.; Benbouzi, M. High-orer sliing-moe control of variable-spee win turbines. IEEE Trans. In. Electron. 2009, 56, Soliman, M.; Malik, O.P.; Westwick, D.T. Multiple moel preictive control for win turbines with oubly fe inuction generators. IEEE Trans. Sustain. Energy 2011, 2, Wu, B.; Lang, Y.; Zargari, N.; Kouro, S. Power Conversion an Control of Win Energy System; Wiley: Hoboken, NJ, USA, Aba, G.; López, J.; Roríguez, M.A.; Marroyo, L.; Iwanski, G. Doubly Fe Inuction Machine: Moelling an Control for Win Energy Generation; Hanzo, L., E.; John Wiley & Sons: Hoboken, NJ, USA, Kim, Y.; Chung, I.; Moon, S. Tuning of the PI controller parameters of a PMSG win turbine to mmprove control performance uner various win spees. Energies 2015, 8, Hughes, F. M.; Anaya-Lara, O.; Jenkins, N.; Strbac, G. A power system stabilizer for DFIG-base win generation. IEEE Trans. Power Syst. 2006, 21, Singh M.; Santoso, S. Dynamic moels for win turbines an win power plants. Available online: (accesse on 22 May 2015). c 2015 by the authors; licensee MDPI, Basel, Switzerlan. This article is an open access article istribute uner the terms an conitions of the Creative Commons Attribution license (