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In order to make a wind power generation truly cost-effective and reliable, an advanced control techniques must be used. In this paper, we develop a new control strategy, using nonlinear generalized ...

In order to make a wind power generation truly cost-effective and reliable, an advanced control techniques must be used. In this paper, we develop a new control strategy, using nonlinear generalized predictive control (NGPC) approach, for DFIG-based wind turbine. The proposed control law is based on two points: NGPC-based torque-current control loop generating the rotor reference voltage and NGPC-based speed control loop that provides the torque reference. In order to enhance the robustness of the controller, a disturbance observer is designed to estimate the aerodynamic torque which is considered as an unknown perturbation. Finally, a real-time simulation is carried out to illustrate the performance of the proposed controller.

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- 1. Research Article Real time simulation of nonlinear generalized predictive control for wind energy conversion system with nonlinear observer Kamel Ouari a , Touﬁk Rekioua a,n , Mohand Ouhrouche b a Laboratory LTII, University of Bejaia, Algeria b Laboratory LICOME, University of Quebec at Chicoutimi, Canada a r t i c l e i n f o Article history: Received 21 February 2011 Received in revised form 16 December 2012 Accepted 8 August 2013 Available online 7 September 2013 Keywords: Nonlinear generalized predictive control (NGPC) Doubly-fed induction generator (DFIG) DFIG-based wind turbine Real-time simulation a b s t r a c t In order to make a wind power generation truly cost-effective and reliable, an advanced control techniques must be used. In this paper, we develop a new control strategy, using nonlinear generalized predictive control (NGPC) approach, for DFIG-based wind turbine. The proposed control law is based on two points: NGPC- based torque-current control loop generating the rotor reference voltage and NGPC-based speed control loop that provides the torque reference. In order to enhance the robustness of the controller, a disturbance observer is designed to estimate the aerodynamic torque which is considered as an unknown perturbation. Finally, a real-time simulation is carried out to illustrate the performance of the proposed controller. & 2013 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Recently, many new wind farms have employed wind turbines based on doubly-fed induction generator (DFIG) [1], due to their full power control capability, variable speed operation, low con- verter cost and reduced power loss [2,3]. However, DFIG constitu- tes a challenging control problem, because of its fast dynamics, and being a highly coupled and nonlinear multi-variable system. The ﬁeld-oriented vector control using cascaded PI controllers is widely used, in DFIG-based wind turbines, for reasons of simplicity and applicability [4]. However, PI-type control methods are not effective when the system to be controlled is characterized by strong nonlinearity and external disturbances. To overcome these drawbacks, various approaches have been proposed to replace PI-type controllers. Some examples are the H1 control theory, neural networks and sliding mode control [5-7]. On the other hand, model predictive control (MPC) is now regarded as one of the most robust control strategies because of its advantages, such as insensitivity to parameter variations, external disturbance rejection and fast dynamic responses [8]. Therefore, it has been applied in the industry as a promising form of advanced control [9]. Several control strategies using linear and nonlinear MPC have been proposed in the technical literature [10,15]. In [10] Chen et al. have designed a nonlinear predictive control law for multi- variable nonlinear systems based on Taylor series expansion, where the same relative degree of multi-input and multi-output system is considered. In [11,12], the principle of the MPC is combined with the well-known direct torque control (DTC) and applied to an induction motor (IM) for a fast torque response. Hedjar et al. [13] have proposed a cascaded NGPC based on Taylor series expansion for an IM. However, in these studies, the load torque is considered as a known disturbance. MPC techniques have also been proposed for DFIG-based wind turbines. In [14], the GPC has been used to control the pitch angle of windmill blades in order to reduce power ﬂuctuations. Multi- variable control strategy based on MPC techniques has been proposed for wind turbines based on DFIG in [15]. A predictive current control (PCC) strategy for doubly fed induction generators has been treated in [16]. Nevertheless, all these methods achieve a high performance only when the external disturbance (aerody- namic torque) is known. In order to provide powerful abilities in handling system disturbances and improving robustness, the nonlinear generalized predictive control must be combined with a disturbance observer [17,18]. For improvement, in this paper, the aerodynamic torque observer is integrated into the control law in order to enhance the robustness of the controller. The combination of this observer with NGPC works as a nonlinear controller. In this paper, the performance index proposed in [18] is considered to improve the performances of the DFIG-based wind turbine under unknown disturbance and parameter variations. To this end a cascaded NGPC and stator-ﬁeld-oriented control are investigated. In addition, an aerodynamic torque observer, deﬁned from predictive control, is integrated into the control law. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.08.004 n Corresponding author. Tel./fax: þ213 34205090. E-mail addresses: Ouari.Kamel@ymail.com (K. Ouari), to_reki@yahoo.fr, t_rekioua@yahoo.fr (T. Rekioua), Mohand_ouhrouche@uqac.ca (M. Ouhrouche). ISA Transactions 53 (2014) 76–84
- 2. In Section 2, the dynamic model of wind energy conversion system is exposed. In Section 3, mathematical formulation of NGPC for nonlinear system is presented. The proposed controller is applied to the DFIG in Section 4. The design of the disturbance observer and real time simulation results are given in Sections 5 and 6, respectively. Finally, Section 7 concludes the paper. 2. Mathematical model of WECS The DFIG-based wind turbine is shown in Fig. 1. It consists of a three blade wind turbine rotor coupled to a doubly-fed induction generator through a gearbox. The two pulse width modulated voltage source converters are connected back to back between the rotor terminals and the utility grid via a common DC link. 2.1. Wind turbine model The aerodynamic power extracted from the wind can be expressed as Pt ¼ 1 2 ρaπR2 Cpðλ; βÞv3 ð1Þ where v is the wind speed, ρa is the air density, R is the rotor radius and Cp is the power coefﬁcient that depends on both, pitch angle β and tip speed ratio λ. The coefﬁcient Cp is speciﬁc for each wind turbine. The tip speed ratio is deﬁned as: λ ¼ ΩtR ν ð2Þ Ωt is the turbine speed. A typical relationship between Cp and λ is shown in Fig. 2 [19]. At the point λopt, Cp¼Cp-max, the maximum power can be extracted. To this end, the turbine should always operate at λopt. Neglecting the transmission losses, the torque and speed of the wind turbine, referred to the generator side of the gearbox, are given by Tr ¼ Tt G Ωt ¼ Ωr G ð3Þ where G is the gear ratio, Ωr is the generator speed, Tr and Tt are, respectively, the generator torque and aerodynamic torque. Substituting Eq. (3) in Eqs. (1) and (2), the rotor speed and electrical grid power references are ΩrÀref ¼ λopt G R v PgridÀref ¼ 1 2ηρaπ2 CpÀ maxv3 8 < : ð4Þ where η is the system (wind turbineþDFIG) efﬁciency and ρa is the air density. Nomenclature Pt Aerodynamic power (W) ρa Air density (kg m3 ) v wind speed R Rotor turbine radius (m) Cp Power coefﬁcient Cp-max Maximal power coefﬁcient β Blades pitch angle λ Tip-speed ratio λopt Optimal tip-speed ratio Tt Aerodynamic torque (N M) Tr Generator torque (N m) Ωt Turbine speed rad/s Ωr Generator speed (rad/s) Ωr-ref Reference generator speed G Gear ratio η WECS efﬁciency Pgrid-ref Reference grid Active power (W) η System (wind turbineþDFIG) efﬁciency. Rs, Rr Per-phase stator and rotor resistances, Ls, Lr Stator, rotor self inductances M Mutual inductance P Number of pole pairs J Moment of inertia fr Coefﬁcient of friction. ωs, ωr Stator and rotor angular velocities (rd/s) s Generator slip s Dispersion ratio x State vector u Control output Tr Disturbance f(x),h(x) Functions assumed to be continuously differentiable gu(x), gT(x) continuous functions of x ℑ Finite horizon cost function Tp Predictive time ρi Relative degree of the output i Wind Blades Grid side Converter Grid DFIG Gearbox LC Filter Rotor side Converter Fig. 1. System conﬁguration. K. Ouari et al. / ISA Transactions 53 (2014) 76–84 77
- 3. 2.2. DFIG model The DFIG model can be described, according to [20], in an arbitrary (d–q) reference frame, as follows Vds ¼ RsIds þ d dt ψdsÀωsψqs Vqs ¼ RsIqs þ d dt ψqs þωsψds Vdr ¼ RrIdr þ d dt ψdrÀðωsÀωrÞψqr Vqr ¼ RrIqr þ d dt ψqr þðωsÀωrÞψdr 8 >>>>>< >>>>>: ð5Þ with: ψds ¼ LsIds þMIdr ψqs ¼ LsIqs þMIqr ψdr ¼ LrIdr þMIds ψqr ¼ LrIqr þMIqs 8 >>>>< >>>>: ð6Þ ωr ¼ P Ωr ð7Þ The notations are deﬁned as follows: Vds, Vqs, Vdr, Vqr Two-phase stator and rotor voltages, Ids, Iqs, Idr, Iqr Two-phase stator and rotor currents, Rs, Rr Per-phase stator and rotor resistances, ψds, ψqs, ψdr, ψqr Two-phase stator and rotor ﬂuxes, ωs, ωr Stator and rotor angular velocities, P Number of pole pairs, Ls, Lr Stator and rotor inductances, M Magnetizing inductance. The electrical model is completed by the mechanical equation given below J d dt Ωr ¼ TemÀTrÀf rΩr ð8Þ where J and fr are, respectively, the moment of inertia and the coefﬁcient of friction. The electromagnetic torque Tem is written as a function of stator ﬂuxes and rotor currents Tem ¼ P M Ls ðψqsIdrÀψdsIqrÞ ð9Þ If the stator ﬂux is linked to the d-axis of the frame [21,22] we have ψds ¼ ψs ψqs ¼ 0 ð10Þ Then, by using Eqs. (9) and (10), the electromagnetic torque will only depend on the q-axis rotor current Tem ¼ P MVs ωsLs Iqr ð11Þ As shown in Fig. 3, if the per phase stator resistance is negle- cted, the stator voltage vector is consequently in quadrature advance compared to the stator ﬂux vector [23]. So, the stator voltages are: Vds ¼ 0 Vqs ¼ Vs ¼ ωsψs ð12Þ By substituting Eqs. (10) and (12) in Eqs. (5) and (6), the rotor voltages, the stator active and reactive power are expressed according to the rotor currents as Vdr ¼ RrIdr þsLr d dt IdrÀsLrsωsIqr Vqr ¼ RrIqr þsLr d dt Iqr þs sωsIdr þsMVs Ls 8 < : ð13Þ Ps ¼ ÀMVs Ls Iqr Qs ¼ Vs 2 ωsLs ÀMVs Ls Idr 8 < : ð14Þ with: s ¼ 1À M LrLs ; s ¼ ωsÀωr ωs s is the generator slip 3. Mathematical formulation of NGPC The nonlinear generalized predictive control proposed by Chen et al. [17], is brieﬂy described in this section. We consider a nonlinear system of the form: _xðtÞ ¼ f ðxÞþguðxÞuðtÞþgT ðxÞTrðtÞ y ¼ hðxÞ ( ð15Þ where xAℜn is the state space vector, uAℜm is the control input and Tr Aℜ is the external disturbance. The function f(x) and h(x) are assumed to be continuously differentiable a sufﬁcient number of times. Vector function gu(x) and gT(x) are continuous functions of x. The problem consists in elaborating a control law u(t) such that the output y(tþτ) can optimally track a desired reference yr (tþτ) in presence of the disturbance Tr. The predictive control will be optimal if the ﬁnite horizon cost function ℑ is minimized ℑ ¼ 1 2 Z Tp 0 ½yðtþτÞÀyrðtþτÞT ½yðtþτÞÀyrðtþτÞdτ ð16Þ Tp is the predictive time To solve the nonlinear optimization problem (16), the predicted output y(tþτ) and the predicted reference yr(tþτ) are approximated by Taylor series expansion: yiðtþτÞ ¼ hiðxÞþ ∑ ρi k ¼ 1 τk k! Lk f hiðxÞþ τρi ρi! LguL ðρiÀ1Þ f hiðxÞuðtÞ ð17Þ where Lk f hi, denote the kth order Lie derivative of hi and ρi is the relative degree of the output yi. Fig. 2. Typical power coefﬁcient. Fig. 3. Voltage and ﬂux vectors settings. K. Ouari et al. / ISA Transactions 53 (2014) 76–8478
- 4. The Lie derivative of function hi(x) along a vector ﬁeld f(x)¼ (f1(x)…….. fn(x)) is denoted by Lf hiðxÞ ¼ ∂hiðxÞ ∂x f ðxÞ Lk f hiðxÞ ¼ Lf ðLkÀ1 f hiðxÞÞ Lgu Lf hiðxÞ ¼ ∂Lf hiðxÞ ∂x guðxÞ LgT Lf hiðxÞ ¼ ∂Lf hiðxÞ ∂x gT ðxÞ 8 >>>>>>< >>>>>>: ð18Þ The necessary condition for the optimal control is given by dℑ du ¼ 0 ð19Þ 4. Cascaded nonlinear generalized predictive controller As DFIG is characterized by two-time scales modes; i.e. elec- trical (fast) and mechanical (slow) modes, a cascaded structure is adopted for the design of the controller, see Fig. 4. The inner loop is used to regulate the d-axis rotor current and the electromag- netic torque, whereas the outer loop is employed for the speed trajectory tracking. 4.1. Inner control loop The aim of this inner loop is to regulate the d-axis rotor current and the electromagnetic torque by acting on the rotor armature voltage. The compact form of Eq. (13) can be written as follows: _xðtÞ ¼ f ðxÞþguðxÞuðtÞ y ¼ hðxÞ ( ð20Þ with: x ¼ ðIdr IqrÞT ; u ¼ ðVdr VqrÞT ; y ¼ ðTem IdrÞT The state vector x is composed of the d-axis and q-axis component of the armature rotor currents. The input vector u is made of the d-axis and q-axis components of the armature rotor voltage. The output vector y consists of the electromagnetic torque and the d-axis rotor current, while vector function f(x) and gu(x) are deﬁned as: f ðxÞ ¼ À Rr sLr Idr þsωsIqr À Rr sLr IqrÀsωsIdr þsMVs sLr Ls 0 @ 1 A; guðxÞ ¼ 1 sLr 0 0 1 sLr 0 @ 1 A The outputs to be controlled in the inner loop are deﬁned as follows: y1 ¼ h1ðxÞ ¼ Tem ¼ PMVs ωsLs Iqr y2 ¼ h2ðxÞ ¼ Idr ( ð21Þ For the outputs y1 and y2, their relative degrees ρ1 and ρ2 are equal to 1, as the ﬁrst derivative reveals the input. The resulting NGPC applied to the system (20) is given by uðtÞ ¼ ÀGuðxÞÀ1 ∑ 1 i ¼ 0 Ki½Li f hðxÞÀy½i r ðtÞ ð22Þ with: K0 ¼ 3 2Tp1 0 0 3 2Tp1 0 @ 1 A; K1 ¼ 1 0 0 1 ; GuðxÞ ¼ PMVs ωsLs 1 sLr 0 0 1 sLr 0 @ 1 A; hðxÞ ¼ Tem Idr ! ; yr ¼ TemÀref IdrÀref ! y½i r ðtÞ denotes the ith derivative of yrðtÞ and Tp1 is the prediction time of the inner loop. 4.2. Outer control loop The speed controller in the outer loop is obtained by consider- ing the mechanical dynamics of the DFIG. From Eq. (8), it follows that the mechanical equation can be expressed as: _xðtÞ ¼ f ðxÞþguðxÞuðtÞþgT ðxÞTrðtÞ y ¼ hðxÞ ( ð23Þ where x and u are, respectively, the rotor speed Ωr and the electromagnetic torque Tem. Tr is the aerodynamic torque and the output y is the rotor speed. The vector function f(x), gu(x) and gT(x) are given by: f ðxÞ ¼ Àf r J Ωr guðxÞ ¼ 1 J gT ðxÞ ¼ À1 J 8 >>>< >>>: ð24Þ The relative degree ρ of the output y is equal to 1. Then, we shall have the optimal control input as uðtÞ ¼ ÀGuðxÞÀ1 ∑ 1 i ¼ 0 kiðLi f hðxÞÀy½i r ðtÞÞþGT ðxÞTrðtÞ " # ð25Þ with k0 ¼ 3 2Tp2 ; k1 ¼ 1; hðxÞ ¼ Ωr; yr ¼ ΩrÀref GuðxÞ ¼ ∂hðxÞ ∂x guðxÞ ¼ 1 J ; GT ðxÞ ¼ ∂hðxÞ ∂x gT ðxÞ ¼ À 1 J Tp2 is the prediction time of the outer loop. 5. Nonlinear disturbance observer In this work, the aerodynamic torque is considered as an unknown disturbance, so the use of an observer is necessary to Fig. 4. Cascaded control conﬁguration. K. Ouari et al. / ISA Transactions 53 (2014) 76–84 79
- 5. estimate it. Thus, the optimal control (25) becomes: uðtÞ ¼ ÀGuðxÞÀ1 ∑ 1 i ¼ 0 kiðLi f hðxÞÀy½i r ðtÞÞþGT ðxÞ^TrðtÞ " # ð26Þ ^Tr is the estimated aerodynamic torque. The nonlinear disturbance observer is developed in the same spirit as Chen et al. [17]. An initial disturbance observer is given by _^TrðtÞ ¼ ÀψðxÞgT ðxÞ^TrðtÞþψðxÞ½_xðtÞÀf ðxÞÀguðxÞuðtÞ ð27Þ where ψ(x) is the gain to be designed to ensure the stability of the observer. It is chosen as follows: ψðxÞ ¼ ϕ0 ∂hðxÞ ∂x ð28Þ with is constant. From Eq. (23), it follows that: gT ðxÞTrðtÞ ¼ _xðtÞÀf ðxÞÀguðxÞuðtÞ ð29Þ Since there is no information about the derivative of the disturbance, we assume that _Tr ¼ 0 ð30Þ Substituting Eqs. (29) and (30) in Eq. (27), the observer error is described as _εT ðtÞþψðxÞgT ðxÞεT ðtÞ ¼ 0 ð31Þ where the observer error is: εT ¼ TrðtÞÀ^TrðtÞ From Eq. (31), it can be shown that the observer is exponen- tially stable by choosing: ψðxÞgT ðxÞ40 ð32Þ Using the Lie derivative (18) and Eq. (28), we get: ψðxÞgT ðxÞ ¼ ϕ0GT ðxÞ ψðxÞ_xðtÞ ¼ ϕ0 _yðtÞ ψðxÞf ðxÞ ¼ ϕ0Lf hðxÞ ψðxÞguðxÞ ¼ ϕ0GuðxÞ 8 >>>>< >>>>: ð33Þ Substituting Eqs. (26) and (33) in Eq. (27), we obtain _^TrðtÞ ¼ Àϕ0ðk1 _εyðtÞþk0εyðtÞÞ ð34Þ where: εy ¼ yrðtÞÀyðtÞ is the speed tracking error. Integrating Eq. (34), the estimated disturbance is calculated as follows: _^TrðtÞ ¼ Àϕ0 k1εyðtÞþk0 Z t 0 εyðτÞdτ þTrð0Þ ð35Þ with: Trð0Þ ¼ ^Trð0Þþϕ0εyð0Þ If the initial value of the estimated disturbance ^Trð0Þ is taken as ^Trð0Þ ¼ Àϕ0εyð0Þ ð36Þ Then, the estimated disturbance will be as follows: ^TrðtÞ ¼ Àϕ0 k1εyðtÞþk0 Z t 0 εyðτÞdτ ð37Þ Substituting Eq. (37) in Eq. (26), the control law becomes uðtÞ ¼ kpεyðtÞþkdðyrðtÞÀLf hðxÞÞþki Z t 0 εyðτÞdτ ð38Þ where kp, ki and kd are the proportional, integral and derivative gains, respectively. kp ¼ 3J 2Tp2 Àϕ0; kd ¼ J; ki ¼ À 3ϕ0 2Tp2 Fig. 5. Block diagram of the proposed NGPC for DFIG-based wind turbine. K. Ouari et al. / ISA Transactions 53 (2014) 76–8480
- 6. 6. Real time simulation results To evaluate the proposed controller performance, the drive system given in Fig. 5 is implemented ﬁrst in Matlab/Simulink in block diagram format. This Simulink model is then opened and compiled with RT-Lab software package. The main system parameters are listed in Tables 1 and 2. The predictive time of the inner loop Tp1 and the outer loop Tp2 are set to 0.5 ms and 8 ms, respectively. The stability of the aerodynamic torque observer is guaranteed by choosing the value of ϕ0 equal to À3. 6.1. Tracking performance under unknown aerodynamic torque In this simulation, the robustness of the proposed control law, against the aerodynamic torque variations, is tested. The d-axis rotor current reference is calculated to maintain at null value the reactive power ﬂow to the grid. The rotor speed and electrical grid power references are given in the Eq. (4). In order to limit the control effort, the reference speed signal is passed through a second order linear Table 1 Doubly fed induction machine parameters. Parameter Value Power 1.5 MW Voltage 690 V Frequency 50 Hz Rs 0.021 Ω Ls ¼Lr s 0.0137 H Jg 500 Kg m2 P 2 M 0.0135 H f 0.0071 Table 2 Wind turbine parameters. Parameter Value Turbine diameter 60 m Number of blades 3 Hub height 85 m R 36.5 m Gear box 90 0 2 4 6 8 10 12 14 16 18 9 10 11 12 13 14 15 Time (s) windspeed(m/s) Fig. 6. Wind speed proﬁle. 0 2 4 6 8 10 12 14 16 18 4 6 8 10 12 Time(s) Aerodynamictorque(kNm) Actual torque Estimed l torque Fig. 7. Actual and estimated aerodynamic torque. 0 2 4 6 8 10 12 14 16 18 -0.2 -0.1 0 0.1 0.2 Time(s) Slip Fig. 8. DFIG slip. 0 2 4 6 8 10 12 14 16 18 120 140 160 180 Time(s) Rotorspeed(rad/s) Actual speed Reference speed Fig. 9. Rotor speed trajectory tracking response. 0 2 4 6 8 10 12 14 16 18 -0.02 -0.01 0 0.01 0.02 0.03 Time (s) Speederror(rad/s) Fig. 10. Speed tracking error response. Electromagnetique torque(kNm) Actual torque Torque reference 0 2 4 6 8 10 12 14 16 18 4 6 8 10 12 Time (s) Fig. 11. Electromagnetic torque trajectory tracking. K. Ouari et al. / ISA Transactions 53 (2014) 76–84 81
- 7. ﬁlter given by FðsÞ ¼ w2 n s2 þ2ξwnsþw2 n ð39Þ Where wn¼5 ζ¼1.2 The wind speed proﬁle used in the simulation is illustrated in Fig. 6. As seen in Fig. 7, the disturbance observer gives a good estimate of the aerodynamic torque. The speed and electromag- netic torque tracking performances are satisfactory achieved as shown, in Figs. 9–11. The stator and rotor active power are plotted in Fig. 12. The sense of drainage depends on the sign of the DFIG slip (Fig. 8). Fig. 13 shows that the grid reactive power is maintained at zero value contributing to compensate the grid power factor. The rotor voltage and current at sub-synchronous, synchronous and hyper-synchronous modes are plotted in Fig. 14. 6.2. Tracking performance under DFIG's parameter variations As NGPC is a model-based approach, the accuracy of the DFIG parameters may inﬂuence the drive. The parameter variations intro- duced in the DFIG model are set to the following values: 25% in the rotor resistance at t¼4 s and 25% in the moment of inertia at t¼10 s. These variations are not taken into account in the controller. From Fig. 15, we demonstrate that the grid active power tracks perfectly its reference aiming to maximize the conversion efﬁciency. The reference tracking is not affected by the parameter variations. 6.3. Performance evaluation under predictive time change This test is performed to evaluate the performance of the controller under predictive time change. To this end, only the predictive time of the outer loop Tp2 is decreased from 20 ms to 5 ms. As shown in Figs. 16 and 17, a smaller predictive time results in a fast disturbance rejection to the detriment of the control effort Tem which is limited by the speed ﬁlter. 6.4. Performance evaluation under observer gain change For this case, only the observer gain ϕ0 is increased from À1 to À8. As shown in Figs. 18 and 19, a large gain ϕ0 results in a fast disturbance rejection at the expense of the control effort Tem which is limited by the speed ﬁlter. The observer sensitivity to the gain ϕ0 is also investigated in this test. Fig. 20 shows the observer error for ϕ0 equals to À1 and À8. These results show clearly that ^TrðtÞ converges quickly to TrðtÞ for high values of the observer gain ϕ0. 7. Conclusion A cascaded NGPC for a DFIG-based wind turbine has been presented in this paper. The control law is derived from optimization 0 2 4 6 8 10 12 14 16 18 -2 -1.5 -1 -0.5 0 0.5 Time(s) Statorandrotor activepower(Mw) Rotor active power Stator active power Fig. 12. Stator and rotor active power. 0 2 4 6 8 10 12 14 16 18 -0.2 -0.1 0 0.1 0.2 Time (s) Gridreactive power(MVAR) Actual power Reference power Fig. 13. Grid reactive power trajectory tracking. 0 2 4 6 8 10 12 14 16 18 -3 -2 -1 0 1 2 3 Time(s) Rotorvoltage(0.1kv) androtorcurrent(kA) Rotor voltage Rotor current Fig. 14. Rotor Voltage and current. 0 2 4 6 8 10 12 14 16 18 -2.5 -2 -1.5 -1 -0.5 Time(s) Gridactive power(Mw) Actual grid active power Grid active power reference Fig. 15. Grid active power trajectory tracking under DFIG's parameter variations. 0 2 4 6 8 10 12 14 16 18 -0.04 -0.02 0 0.02 0.04 Time (s) Speederror(rad/s) 0 2 4 6 8 10 12 14 16 18 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Time (s) Speederror(rad/s) Fig. 16. Speed tracking error under predictive time change: (a) for Tp2¼20 ms and (b) for Tp2 ¼5 ms. K. Ouari et al. / ISA Transactions 53 (2014) 76–8482
- 8. of an objective function that considers the control effort and the difference between the predicted outputs and the speciﬁc references. To ensure robustness against aerodynamic torque and parameter variations, a disturbance observer is designed and integrated into the control law. Furthermore, in order to limit the control effort, the reference speed signal is passed through a second order linear ﬁlter. The dynamic real time simulation shows that the grid active power tracks perfectly its reference aiming to maximize the conversion efﬁciency and the grid reactive power is maintained at zero value contributing to compensate the grid power factor. This justiﬁes the efﬁciency and the reliability of the NGPC with an aerodynamic torque observer. References [1] Lie Xu, Wang Yi. Dynamic modeling and control of DFIG-Based wind turbines under unbalanced network conditions. IEEE Transactions on Power Systems 2007;22:314–23. 0 2 4 6 8 10 12 14 16 18 6 8 10 12 Time (s) Electromagnetic torque(kNm) Actual torque Torque reference 0 2 4 6 8 10 12 14 16 18 4 6 8 10 12 Time (s) Electromagnetic torque(kNm) Fig. 17. Electromagnetic torque trajectory tracking under predictive time change: (a) for Tp2 ¼20 ms and (b) for Tp2¼5 ms. 0 2 4 6 8 10 12 14 16 18 -0.02 -0.01 0 0.01 0.02 0.03 Time (s) Speederror(rad/s) 0 2 4 6 8 10 12 14 16 18 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Time (s) Speedterror(rad/s) Fig. 18. Speed tracking error under observer gain change: (a) for ϕ0¼ À1 and (b) for ϕ0¼ À8. 0 2 4 6 8 10 12 14 16 18 4 6 8 10 12 Time (s) Electromagnetic torque(kNm) Actual torque Torque reference Electromagnetic torque(kNm) 0 2 4 6 8 10 12 14 16 18 4 6 8 10 12 Time (s) Fig. 19. Electromagnetic torque trajectory tracking under observer gain change: (a) for ϕ0 ¼ À1 and (b) for ϕ0¼ À8. 0 2 4 6 8 10 12 14 16 18 -300 -200 -100 0 100 Time (s) Observererror(Nm) 0 2 4 6 8 10 12 14 16 18 -300 -200 -100 0 100 Time (s) Observererror(Nm) Fig. 20. Observer error under observer gain change: (a) for ϕ0 ¼ À1 and (b) for ϕ0¼ À8. K. Ouari et al. / ISA Transactions 53 (2014) 76–84 83
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