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Sensorless MRAS control of emerging doubly‐fed reluctance wind generators

2021; Institution of Engineering and Technology; Volume: 15; Issue: 9 Linguagem: Inglês

10.1049/rpg2.12123

1752-1424

M. R. Agha Kashkooli, Milutin Jovanović,

Wind Energy Research and Development

IET Renewable Power GenerationVolume 15, Issue 9 p. 2007-2021 ORIGINAL RESEARCH PAPEROpen Access Sensorless MRAS control of emerging doubly-fed reluctance wind generators Mohammad-Reza Agha-Kashkooli, Corresponding Author Mohammad-Reza Agha-Kashkooli mohammad.kashkooli@northumbria.ac.uk orcid.org/0000-0001-7537-5497 Faculty of Engineering and Environment, Northumbria University Newcastle, Newcastle upon Tyne, UK Correspondence Mohammad-Reza Agha-Kashkooli, Faculty of Engineering and Environment, Northumbria University Newcastle, Newcastle upon Tyne, UK. Email: mohammad.kashkooli@northumbria.ac.ukSearch for more papers by this authorMilutin Jovanović, Milutin Jovanović Faculty of Engineering and Environment, Northumbria University Newcastle, Newcastle upon Tyne, UKSearch for more papers by this author Mohammad-Reza Agha-Kashkooli, Corresponding Author Mohammad-Reza Agha-Kashkooli mohammad.kashkooli@northumbria.ac.uk orcid.org/0000-0001-7537-5497 Faculty of Engineering and Environment, Northumbria University Newcastle, Newcastle upon Tyne, UK Correspondence Mohammad-Reza Agha-Kashkooli, Faculty of Engineering and Environment, Northumbria University Newcastle, Newcastle upon Tyne, UK. Email: mohammad.kashkooli@northumbria.ac.ukSearch for more papers by this authorMilutin Jovanović, Milutin Jovanović Faculty of Engineering and Environment, Northumbria University Newcastle, Newcastle upon Tyne, UKSearch for more papers by this author First published: 23 February 2021 https://doi.org/10.1049/rpg2.12123AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract A new model reference adaptive system based estimation technique for vector control of real and reactive power of a brushless doubly fed reluctance generator without a shaft position sensor is proposed. The rotor speed is being precisely observed in a closed-loop manner by eliminating the error between the measured and estimated inverter-fed (secondary) winding current angles in a stationary frame. Contrary to the existing model reference adaptive system observer designs reported in the brushless doubly fed reluctance generator literature, the reference model is entirely parameter-free and only utilises direct measurements of the secondary currents. Furthermore, the current estimates coming from the adaptive model are obtained using the measured voltages and currents of the grid-connected (primary) winding, which has provided prospects for much higher accuracy and superior overall performance. The realistic simulations, preliminary experimental results, and the accompanying parameter sensitivity studies have shown the great controller potential for typical operating conditions of variable speed wind turbines with maximum power point tracking. 1 INTRODUCTION A conventional slip-ring doubly fed induction generator (DFIG), and its brushless alternative (BDFG), allow the use of a partially-rated power converter in wind power applications, which is not only low cost but also more reliable than a fully-rated counterpart of electrically-excited or permanent magnet synchronous generators [1, 2]. The BDFG is an attractive maintenance-free solution and it overcomes the main DFIG limitations by the absence of brush gear. It has two standard sinusoidally distributed stator windings of different pole numbers and applied frequencies as shown in Figure 1. The primary is connected to a fixed voltage and frequency grid, whereas the secondary is back-to-back converter fed at controllable voltage and frequency. The bi-directional power flow on the secondary side enables the BDFG operation in super- or sub-synchronous speed modes. The indirect magnetic coupling between the windings for the torque production is provided by the rotor with the pole number equal to the sum of the stator pole-pairs [3]. This makes the BDFG 'natural' synchronous speed half that of the equivalent DFIG. Hence, a more efficient, compact medium speed two-stage gearbox can be used for BDFG turbines [4, 5] bringing significant reliability and economic advantages over DFIG drive trains using a complicated three-stage one [6]. Furthermore, the BDFG has been proven to have superior low voltage fault ride through capabilities to DFIG due to the relatively higher leakage inductances [7, 8], as well as the potential for competitive frequency support provision [9]. FIGURE 1Open in figure viewerPowerPoint A BDFRG-based wind energy conversion system schematic A modern cage-less reluctance rotor version of the BDFG, the brushless doubly-fed reluctance generator (BDFRG), presented in Figure 1 is particularly promising [10, 11]. It can offer better efficiency, facilitated less parameter dependent dynamic modelling and much simpler control compared to the cage induction rotor counterpart with nested-loop structures [12-14]. The latest state-of-the-art hybrid reluctance rotor designs with assistive conductive bars have augmented further the torque density and efficiency [15]. Two distinct control concepts have been considered for the BDFRG [16]: hysteresis control (HC) [17, 18], and vector control (VC) [19, 20]. Variable switching rates and higher total harmonic distortion are the known drawbacks of direct power HC strategies. VC can resolve these issues offering superior power quality and has been widely adopted for wind energy conversion systems (WECS). However, the utilisation of shaft encoders for voltage or flux oriented VC undermines the mechanical robustness and reliability. The development of rotor position and/or speed estimation techniques for sensorless operation of both DFIG [21-24] and BDFRG [25-28] has been getting increasingly popular for this reason. A Luenberger observer has been effectively used to estimate the rotor speed and position for field oriented control (FOC) of the BDFRG [25, 26]. However, this method is highly parameter dependent, and sensitivity analyses have not been done. The alternative MRAS speed observers with the same dedicated controller as in ref. [25] are presented in refs. [27-29]. The secondary-flux estimation is prone to the well-known back-emf integration errors caused by the more pronounced resistive effects at low fundamental frequency converter voltages and currents in much the same manner as with the rotor winding of DFIGs [21]. For this reason, it has been unable to guarantee the controller stability over the entire speed range [27]. Using the secondary reactive [28] or real power [29] as the reference model basis has improved the MRAS observer performance. However, the adaptive model in ref. [28] relies on all self and mutual inductances of the machine, being particularly sensitive to the secondary one, which is difficult to determine accurately by off-line testing due to the large leakage of the BDFRG secondary winding [30]. In contrast, the real power based MRAS presented in ref. [29] requires only the secondary resistance (Rs) and mutual inductance knowledge. As such, it is less parameter dependent than that in ref. [28]. However, the parameter sensitivity analysis is only performed to Rs variations, but not to the mutual inductance uncertainties, the latter being crucial for the observer performance. The MRAS observer designs in refs. [31, 32] bring additional advantages by introducing the measurable secondary currents as the reference model outputs, while the corresponding estimates are entirely based on the primary quantities at fixed line frequency. However, the requirement for the flux identification and resistance (Rp) knowledge of the primary winding is the main drawback of the adaptive models in refs.[31, 32]. Furthermore, the associated results are produced for a small BDFRG prototype with relatively limited industrial interest for wind power applications. This paper presents a novel rotor angular velocity and position MRAS observer for encoder-less vector control of the grid-connected BDFRG. A simpler implementation and inferior parameter sensitivity are achieved in comparison with the state observer in refs.[25, 26]. The absence of troublesome secondary flux estimation in the low frequency region makes it clearly superior to that in ref. [27]. The measured secondary current stationary-frame components used as the reference model outputs can provide higher reliability and fewer parameter dependence than the MRAS observers in refs.[28, 29]. Moreover, the proposed primary power based adaptive model design overcomes the limitations of the counterparts in refs. [31, 32] by avoiding the primary voltage integration issues and the need for Rp in flux calculations, hence offering improved accuracy and easier implementation. The small-signal analysis of the parameter mismatches and the excellent controller response are supported by the realistic simulation and experimental studies of a large-scale wind turbine. 2 BDFRG OPERATION The key rotor angular velocity and position relationships for the BDFRG with the ωp supply frequency of the primary and ωs for the secondary winding, can be written as [19, 20, 25]: ω rm = ω p + ω s p r = ω p p r 1 + ω s ω p = ω syn 1 + ω s ω p , (1) θ rm = θ p + θ s p p + p s = θ p + θ s p r , (2)where pp and ps denote the windings pole pairs, and p r = p p + p s is the rotor poles number. Figure 2 illustrates the meanings of θp and θs reference frames angles in Equation (2). When the secondary is DC (i.e ω s = 0 ), the BDFRG operates in synchronous mode at ω syn , which is half that of a pr-pole DFIG given Equation (1). Thus, the BDFRG can be classified as a medium-speed machine requiring a two-stage gearbox unlike the vulnerable three-stage one with DFIG wind turbines [33]. FIGURE 2Open in figure viewerPowerPoint The BDFRG sensorless primary-voltage oriented controller: (A) Block scheme; (B) Phasor diagram The mechanical power input of the BDFRG for the maximum wind energy extraction in steady-state can be expressed using Equation (1) as follows: P m = T e · ω rm = T e · ω p p r ︸ P p + T e · ω s p r ︸ P s = P s · 1 + ω p ω s , (3)where the electro-magnetic torque ( T e < 0 ) and the primary power ( P p < 0 ) with the assumed motoring (BDFRM) convention. The bi-directional flow of the secondary power (Ps) allows the machine operation above and below the synchronous speed, i.e. in super-synchronous mode ( ω s > 0 ) when P s < 0 , and at sub-synchronous speeds ( ω s < 0 ) for P s > 0 as shown in Figure 3. FIGURE 3Open in figure viewerPowerPoint Reference power flow and field distribution in the simulated 8/4-pole BDFRG with a 6-pole multi-barrier rotor design [9] If a variable speed range of the BDFRG wind turbine is symmetric around ω syn , i.e. [ ω min = ω syn − Δ ω r , ω max = ω syn + Δ ω r ], it can be defined by the following ratio: r = ω max ω min = ω p + ω s ω p − ω s ⟹ ω s ω p = r − 1 r + 1 . (4)Therefore, for a typical r = 2 , the secondary frequency is limited to ω s = ω p / 3 and P s ≈ 0.25 P m from Equation (3). This implies that a fractional converter can be used by analogy to DFIG. 3 DYNAMIC MODELLING AND CONTROL The BDFRG(M) d − q model in rotating reference frames (Figure 2(B)) can be represented by the following set of equations using standard space vector notation [19, 20]: v ̲ p = R p i ̲ p + d λ ̲ p d t + j ω p λ ̲ p , (5) v ̲ s = R s i ̲ s + d λ ̲ s d t + j ω s λ ̲ s , (6) λ ̲ p = L p i ̲ p + L m i ̲ s m ∗ = L p ( i p d + j i p q ) + L m ( i m d − j i m q ) , (7) λ ̲ s = L s i ̲ s + L m i ̲ p m ∗ = σ L s i ̲ s + L m L p λ ̲ p ∗ = σ L s i ̲ s + λ ̲ m , (8) T e = 3 2 p r λ p d i p q -- λ p q i p d = 3 2 p r λ m d i s q -- λ m q i s d , (9)where Lm is the magnetising inductance, Lp and Ls are the primary and secondary self-inductances, σ = 1 − L m 2 / ( L p L s ) is the leakage coefficient, λm is the mutual flux linkage, and the superscript '*' denotes complex conjugate [34]. It is important to emphasise that ωp rotating i ̲ sm is the frequency (but not amplitude) modulated secondary current vector ( i ̲ s ) running at ωs as shown in Figure 2(B). Furthermore, the relative angular displacements and magnitudes of i ̲ s m and i ̲ s from the respective flux vectors λ ̲ p and λ ̲ m are the same so the following relationship applies in the corresponding reference frames under FOC conditions (i.e. with the dp-axis aligned with λ ̲ p , and the ds-axis with λ ̲ m ) [19, 20, 25]: i ̲ s m = i m d + j i m q = i s m e j γ ︸ d p − q p frame ⇔ i ̲ s = i s d + j i s q = i s e j γ ︸ d s − q s frame . (10) The fundamental power expressions, P p = 1.5 · Re { v ̲ p i ̲ p ∗ } and Q p = 1.5 · Im { v ̲ p i ̲ p ∗ } , can be simplified by aligning the qp-axis with the primary voltage vector (i.e v p q = v p , v p d = 0 ) as depicted in Figure 2. Hence, P p = 1.5 v p i p q and Q p = 1.5 v p i p d . However, as i p d and i p q are the d p − q p frame components and cannot be varied directly, Pp and Qp should be expressed in terms of the corresponding controllable secondary current counterparts, i s d and i s q in the d s − q s frame. Neglecting the winding resistance (Rp), which is a valid approximation for large generators, the key FOC form relations can be developed by using Equations (5) and (7) with λ p d ≈ λ p = v p / ω p , λ p q ≈ 0 , and i m d = i s d and i m q = i s q from Equation (10): P p = 3 2 v p i p q = 3 2 λ p q + L m i m q L p v p ≈ 3 2 L m L p v p i s q , (11) Q p = 3 2 v p i p d = 3 2 λ p d − L m i m d L p v p ≈ 3 v p 2 L p v p ω p − L m i s d , (12) Note that ignoring Rp not only allows straightforward, yet reasonably accurate, calculations of the primary flux magnitude ( λ p ≈ v p / ω p ), but significant simplifications, and a faster execution, of the observer-control algorithm by avoiding to deal with the voltage integration issues (e.g. dc offset) to identify λ p q and λ p d in Equations (11) and (12). Furthermore, since vp and ωp are both constant, Pp and Qp can be controlled in an inherently decoupled manner through i s q and i s d respectively as illustrated in Figure 2(A). The inductance knowledge is not required for this purpose as any uncertainties can be effectively taken care of by the properly tuned PI regulators as will be shown by the results presented in Section 5. 4 MRAS OBSERVER DESIGN In a MRAS observer, the rotor speed information is retrieved by comparing the actual state variables and their rotor position dependent estimates generated by the reference and adaptive models, respectively. A suitable adaptation mechanism is devised to drive the differential between the models outputs to zero, which is a prerequisite for the algorithm to converge. Hence, an accurate speed estimation is possible in a stable, closed-loop fashion [35]. 4.1 Reference and adaptive models The proposed MRAS observer layout is presented in Figure 4(A). The secondary current components in a stationary α − β frame are used as the reference model outputs. The obvious advantages of such a selection are the complete parameter independence and high current sensor-alike accuracy afforded by the immediately accessible i s α and i s β from measurements. For a star-connected secondary winding with a positive phase sequence and an isolated neutral point, the latter can be calculated as: i ̲ s = i s e j δ = i s α + j i s β = i s a + j i s a + 2 i s b 3 . (13) FIGURE 4Open in figure viewerPowerPoint The proposed MRAS observer: (A) Block scheme; (B) Phasor diagram Under the FOC conditions according to Equation (10), the magnetically coupled secondary currents in the ωp frame, i ̂ m q and i ̂ m d , are mirror images of the actual ones in the ωs frame, i ̂ s q and i ̂ s d , and they can be estimated from the measured Pp and Qp using Equations (11) and (12): i ̂ m q = i ̂ s q ≈ L ̂ p v p L ̂ m ( v p α i p α + v p β i p β ︸ ) 2 3 P p , (14) i ̂ m d = i ̂ s d ≈ v p ω p L ̂ m − L ̂ p v p L ̂ m ( v p β i p α − v p α i p β ︸ ) 2 3 Q p , (15)where L ̂ m and L ̂ p are the inductance estimates and (Figure 2(B)): v ̲ p = v p e j ( θ p + π / 2 ) = v p α + j v p β = v a b + v a c 3 + j v b c 3 (16) A common d s − q s to α − β frame conversion transformation, i ̲ α β = i ̲ d q e j θ s , can now be applied to derive: i ̂ s α = i ̂ s d cos ( θ ̂ r − θ p ) − i ̂ s q sin ( θ ̂ r − θ p ) , (17) i ̂ s β = i ̂ s d sin ( θ ̂ r − θ p ) + i ̂ s q cos ( θ ̂ r − θ p ) , (18)where θ ̂ s = θ ̂ r − θ p (Figure 4(B)) and θ ̂ r = p r θ ̂ rm are the estimated secondary frame and rotor 'electrical' positions. The error between the reference and adaptive model outputs is defined as the cross product of the true and estimated secondary-current vectors of the form: ε = i ̲ ̂ s × i ̲ s i s 2 = i ̂ s · i s i s 2 sin δ err = i ̂ s α i s β − i ̂ s β i s α i s α 2 + i s β 2 , (19)where δ err = δ − δ ̂ is their angular phase displacement (Figure 4(B)). By continuously updating the adaptive model with the improved position estimates as shown in Figure 4(B), δ err should eventually converge to zero when the estimated ( i ̂ s ) and actual (is) current vectors overlap both rotating at ωs. An adaptive scheme is developed on the basis of a small signal analysis of ε with the following approximations in steady-state, δ err ≈ 0 , cos δ err ≈ 1 and i ̂ s ≈ i s : d ε d t = ε ̇ = i ̂ s · i s i s 2 cos δ err d δ d t − d δ ̂ d t ≈ d δ d t − d δ ̂ d t , ≈ ω s − ω ̂ s = ( ω r − ω p ) − ( ω ̂ r − ω p ) = Δ ω r , (20) ε ≈ ∫ ( ω r − ω ̂ r ) d t = ω r − ω ̂ r s = Δ ω r s . (21) A prerequisite for the small-signal stability of the closed-loop MRAS observer (i.e ε ̇ ≈ 0 ) would be satisfied if Δ ω r ≈ 0 in Equation (20). The required damping and zero steady-state error for the estimation response are achieved by using a PI as the adaptation mechanisms for MRAS [35]. This PI controller is designed based on Equation (21) to drive ε to zero. Dividing ε by i s 2 provides a straightforward on-line adaptive PI tuning under variable loading conditions of the BDFRG. The raw angular velocity estimates ( ω ̂ r ) generated by the observer are further processed using a low-pass filter to improve the quality of the corresponding mechanical output ( ω ̂ rm ), which is necessary for accurate MPPT as illustrated in Figures 4(A) and 5(A). FIGURE 5Open in figure viewerPowerPoint (A) Aerodynamic and mechanical WECS models; (B) Turbine output power and Maximum Power Point Tracking (MPPT) curve The θ ̂ r is estimated by integrating the ω ̂ r signal by analogy to a conventional technique used for incremental encoders. A conventional PLL, shown in Figure 4(A), is used to obtain the angular frequency (ωp) and phase (θp) of sinusoidal three-phase primary voltages in a rotating d − q frame as detailed in ref. [36]. This PLL is designed for normal stiff grid conditions considered in this paper, and it can effectively eradicate noise and dc-offset in voltage measurements. The impact of weak grids on the PLL performance is studied in refs. [37, 38] and the corresponding methods are presented in ref. [39]. 4.2 Parameter knowledge uncertainties The machine parameter dependence of the adaptive model is evident from Equations (14)–(18) used to estimate the secondary α β currents. Therefore, any mismatch between the applied ( L ̂ m,p ) and actual L m,p values, caused by off-line testing inaccuracies, magnetic saturation or otherwise, can influence the observer performance in different ways. For instance, the i ̂ s q and i ̂ s d estimates using Equations (14) and (15) are prone to errors in L ̂ p / L ̂ m , the latter also being affected by L ̂ m . The situation is even worse for i ̂ s α and i ̂ s β coming from Equations (17) and (18) where further estimation inaccuracies can be introduced by erroneous θ ̂ r . In order to illustrate the effects of all these variations on the position estimation quality, Equations (17) and (18) can be manipulated into a more insightful single equation of the form: tan ( θ ̂ r − θ p ) = tan θ ̂ s = i ̂ s d i ̂ s q tan δ ̂ − 1 i ̂ s d i ̂ s q + tan δ ̂ , (22)where using Equations (14) and (15): i ̂ s d i ̂ s q = 3 v p 2 2 ω p L ̂ p P p − Q p P p = V p 2 X ̂ p P p − Q p P p , (23)Substituting now for Equation (23), Equation (22) can be further developed as follows: tan ( θ r ̂ − θ p ︸ θ s ̂ ) = V p 2 tan δ ̂ − X ̂ p · ( Q p tan δ ̂ + P p ) V p 2 + X ̂ p · ( P p tan δ ̂ − Q p ) . (24) The MRAS satisfies Equation (24) by maintaining ε ≈ 0 in steady-state regardless of the inductance mismatch. Thus, the estimated ( i ̲ ̂ s ) and actual ( i ̲ s ) secondary current vectors are always nearly in phase (i.e. δ err ≈ 0 ) rotating at the same velocity ω ̂ s ≈ ω s , albeit with different magnitudes ( i ̂ s ≠ i s ) as shown in Figure 4(B) so that δ ̂ ≈ δ or: tan δ ̂ = i ̂ s β i ̂ s α ≈ tan δ = i s β i s α ⇔ i ̂ s β i s β ≈ i ̂ s α i s α ≈ i ̂ s i s ≠ 1 . (25) The wrong L ̂ p causes the rotor position estimation ( θ ̂ r ) errors, which can be understood from Equations (22)–(24). Namely, with the tan δ ̂ estimates being fairly accurate by the MRAS strategy as mentioned above, the θ ̂ r inaccuracies coming from Equations (22) and (24) are essentially induced by the incorrect i ̂ s d / i ̂ s q due to the erroneous X ̂ p = ω p L ̂ p values in Equation (23) considering that both the line-to-line rms voltage (Vp) and angular frequency (ωp) are fixed by the primary winding grid connection. It may be easily concluded from Equations (23) and (24) that for a given operating point determined by Pp and Qp, the larger the X ̂ p , the lower i ̂ s d / i ̂ s q and θ ̂ r become. Therefore, in case of deviations from the exact Xp, a smaller θ ̂ r error ( Δ θ r ) would be produced if X ̂ p > X p than for X ̂ p < X p . Also, note from Equation (22) that since θ ̂ r − θ p = θ ̂ s , Δ θ r : Equation (1) propagates to Equations (17) and (18) bringing additional errors in i ̂ s α and i ̂ s β making i ̂ s α ≠ i s α , i ̂ s β ≠ i s β , hence i ̂ s ≠ i s within the constraints defined by Equation (25); Equation (2) manifests itself as a misalignment between the estimated ( d ̂ s − q ̂ s ) and actual ( d s − q s ) secondary frames (Figure 4(B)) as formulated below: Δ θ r = θ r − θ ̂ r = ( θ s + θ p ) − ( θ ̂ s + θ p ) = θ s − θ ̂ s = Δ θ s (26) 5 SIMULATION RESULTS The MRAS observer (Figure 4(A)) is applied to achieve the sensorless control (Figure 2(A)) of the BDFRG-based WECS (Figure 5(A)) simulated in Simulink®. The following practical effects were incorporated to make the simulations as realistic as possible: discrete-time implementation, high frequency white noise and transducer dc offset in the measured signals, and detailed IGBT power-electronics converter models. A family of typical GE®turbine characteristics for various wind and generator speeds, including the MPPT trajectory, used for the simulation studies are displayed in Figure 5(B) [40, 41]. The BDFRG wind turbine specifications can be found in Table 1 and its optimised 'ducted' rotor design details (Figure 3) in ref. [9]. TABLE 1. The BDFRG wind turbine: Parameters and Ratings Parameter Value Power (MW) 1.5 Primary voltage (V rms) V p = 690 Secondary voltage (V rms) V s = 230 Primary current (kA rms) I p = 1.1 Secondary current (kA rms) I s = 1.2 Stator resistance (mΩ) R p = 7 , R s = 14.2 Stator self-inductance (mH) L p = 4.7 , L s = 5.7 Mutual inductance (mH) L m = 4.5 Stator pole-pairs p p = 4 , p s = 2 Winding connections Y/Y Generator speed (rev/min) n rm = 30 ω rm / π = 600 Turbine speed (rev/min) n t = 30 ω t / π = 20 Gearbox ratio g = 30 5.1 Variable speed wind energy conversion Figure 6 illustrates the precise rotor speed ( n ̂ rm ) and position ( θ ̂ r ) estimation and effective power control in a desired base speed range of the WECS. The wind profile with a fast slew rate of ±1 m/s is chosen to examine the MRAS observer performance during deceleration from 600 rev/min through the synchronous 500 rev/min and down to 350 rev/min followed by an acceleration back to the rated 600 rev/min. The absolute speed error ( Δ n rm ) does not exceed 2.5 rev/min with about 1 rev/min mean throughout. This small dc offset has been introduced by the first-order low-pass filter used to handle the noisy ω ̂ r / p r estimates coming from the observer (Figure 4(A)) as shown in the zoomed snapshot on the speed sub-plot. Such a filtering has been necessary for reliable MPPT as per Figure 5(A). The θ ̂ r values for the observed ω ̂ r are accurate to Δ θ r ≈ 0 . 6 ∘ electrical and pr times less, i.e. 0.1° mechanical (not shown), in average. For this test, the BDFRG is operated at unity power factor ( Q p ref = 0 in Figure 2(A)) resulting in the virtually constant i s d ≈ 0.4 kA and zero i p d given Equation (12) with the machine magnetisation being entirely provided from the secondary side. The power producing i s q and i p q change proportionally to the Pp variations considering Equation (11). Note that the secondary power (Ps) is indeed bi-directional in nature as indicated in Figure 3 to balance out the difference between the required MPPT reference input ( P m ref ) and respective primary winding share ( P p ref ) according to Equation (3) and Figure 5(A). The Ps waveform shows that the BDFRG converter injects power (i.e. P s < 0 ) to the grid in the super-synchronous region (i.e. above 500 rev/min) and consumes (i.e. P s > 0 ) in the sub-synchronous mode (i.e. below 500 rev/min). The adaptive model can obviously produce the majority of quality i ̂ s α and i ̂ s β outputs judging by | i s − i ̂ s | ≈ 0.01 kA and | δ err | ≤ 1 ∘ (Figure 4(B)) in average sense. Such a high estimation accuracy, coupled with very little error in θ ̂ r , hence the secondary frame angles ( θ ̂ s ) bearing in mind Equation (26), imply the smooth, intrinsically decoupled power (current) control as expected from Equations (11) and (12). This can be clearly seen from the Pp and Qp curves, and the closely related i s q ≈ i p q and i s d ≈ i p d counterparts, on the pertaining sub-plots. Another important observation from an operational point of view is certainly the secondary winding phase sequence reversal from positive to negative (and vice-versa while riding through the synchronous speed around 110 s this transient being difficult to see clearly with the scaling adopted) during the speed mode transition around 35 s time instant corresponding to ω s < 0 in Equations (1) and (3) and Figure 3. FIGURE 6Open in figure viewerPowerPoint Sensorless operation of the grid-connected BDFRG wind turbine with MRAS observer based MPPT and reactive power control 5.2 Disturbance rejection evaluation Since the adaptive model relies on the current–power expressions (14) and (15) to estimate the secondary current stationary frame components given by Equations (17) and (18), the MRAS observer is subjected to Pp and Qp step changes to test its robustness at a constant wind speed. Figure 7 shows that sudden ±0.3 MVAr/MW variations in the respective power references (Figure 5(A)) do not affect the marginal estimation errors similar to those in Figure 6. The observer