Adaptive Vectorial Filter for Grid Synchronization of Power Converters Under Unbalanced and/or Distorted Grid Conditions

1 Adaptive Vectorial Filter for Grid Synchronization of Power Converters Under Unbalanced and/or Distorted Grid Conditions Sergio Vazquez, Member,IEEE, Juan A. Sanchez, Manuel R. Reyes, J. I. Leon, Member,IEEE and J. M. Carrasco, Member,IEEE Abstract— This paper presents a new synchronization scheme for detecting multiple positive-/negative-sequence frequency harmonics in three-phase systems for grid-connected power converters. The proposed technique is called MAVF-FLL because it is based on the use of multiple adaptive vectorial filters (AVFs) working together inside a harmonic decoupling network (HDN), resting on a frequency-locked loop (FLL) which makes the system frequency adaptive. The method uses the vectorial properties of the three-phase input signal in the αβ reference frame in order to obtain the different harmonic components. The MAVF-FLL is fully designed and analyzed addressing the tuning procedure in order to obtain the desired and predefined performance. The proposed algorithm is evaluated by both simulation and experimental results, demonstrating its ability to perform as required for detecting different harmonic components under a highly unbalanced and distorted input grid voltage. I. INTRODUCTION THE DISTRIBUTED power generation networks and smartgrids are characterized by a high penetration of renewable energies, like wind or PV, and the inclusion of energy storage systems. The integration of these energy resources has been achieved by means of using power electronic devices as grid interface [1]–[3]. On the other hand, system operators have introduced new and high demanding grid codes oriented to ensure a proper performance of these devices in order to achieve a reliable, efficient and safe operation of the grid [4], [5]. As a consequence, companies and academia have paid a great effort to develop new control strategies for grid connected power converters. The new control schemes allow to achieve an optimal and efficient operation of power electronics converters [6]–[12]. The new grid codes impose to power converters to stay connected and supporting the grid service even if the voltage at its point of common coupling (PCC) is distorted and unbalanced. In this way, power converters should be properly synchronized with the grid. Therefore, the synchronization Manuscript received September 06, 2012. Accepted for publication March 25, 2013. This work was supported by the Spanish Economy and Competitiveness under project ENE2012-36897. Copyright c 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs- [email protected]. S. Vazquez, J.A. Sanchez, J.I. Leon and J.M. Carrasco are with the Electronic Engineering Department, University of Seville (Spain), (e-mail: [email protected]). M.R. Reyes is with the Power Electronics area at Win Inertia (Spain), (e-mail: [email protected]). algorithm accuracy plays a fundamental role in the performance achieved by the control strategies. The main task of this mechanism is to provide synchronization with the voltage at the PCC under any circumstance. Several strategies can be used to synchronize the power converter with the grid voltage. The most conventional is the phase-locked loop (PLL) technology [13]–[15]. This technique has been extensively used in both single and three-phase systems, where it is based on synchronous reference frame (SRF-PLL) [16]. The main drawback of SRF-PLLs is that their performance under unbalanced and/or distorted voltage are reduced. This makes this technique unsuitable for the power converters under the new grid codes requirements [17]. Several proposals have been developed to cope with this issue such as the use of two SRFs and a decoupling network to separate the effects of the positive- and negative-sequence voltage components [18], or the use of three single-phase PLLs combined with a positive-sequence calculator without using any SRF [19]. Another approach consists of using the frequency-locking concept for synchronizing. The main advantage of the frequency-locked loop (FLL) is that it estimates the frequency of the input grid voltage, which is a signal that is not affected by sudden changes [20]. Therefore it is more convenient than the phase-angle based method used by the PLL algorithms. A system consisting of a synchronization scheme based on multiple second-order generalized integrator resting on an FLL (MSOGI-FLL) was presented in [21]. Its main advantage is that it allows to perform an analysis of the input signal in real-time, providing an accurate estimation of the fundamental and harmonic components of the grid voltage. On the other hand, solutions based on the application of parallel adaptive notch filters (ANFs) for estimating the harmonic content in real-time can also be found in the literature. A single-phase scheme was presented in [22] and later was extended to three-phase systems by applying three single-phase synchronization schemes based on a direct frequency-adaptive estimation algorithm [23], [24]. Other synchronization methods are also presented in the literature, like a delayed signal cancellation scheme shown in [25] or the use of vectorial filters as a synchronization scheme proposed in [26]–[28], where, complex coefficient band-pass filters are used to extract the positive-/negative-sequence frequency harmonics. This paper presents a new synchronization scheme for 2 detecting multiple positive-/negative-sequence frequency harmonics in three-phase systems. The proposed technique has been named MAVF-FLL because it is based on the use of multiple adaptive vectorial filters (AVFs). These AVFs have been used previously for controlling active filters with excellent results [29]. In the proposed synchronization approach, the AVFs work together inside a harmonic decoupling network (HDN), resting on an FLL. Comparing the proposed AVF with MSOGI-FLL, the main difference is the way used to arrive to the AVF expression. The new method uses a vectorial approach where the β-axis measurement is influencing the filtered α-axis signal and vice versa. On the other hand, the MSOGI-FLL is rather a particular band-pass filter that can be applied to single-phase, as well as three-phase signals represented in any reference frame (natural abc frame or the αβ frame). In the three-phase case, the vectorial properties of the signals are then not considered until the filtered signals are used for sequence separation. The AVF presented in this paper is however inherently connected to the vectorial properties of three-phase signals in the αβ reference frame, and is therefore specifically adapted to three-phase systems. Although it is possible to use it also for single-phase systems by considering these cases as a highly unbalanced three-phase system [26]. When brought into comparison with the cascaded delay signal cancellation PLL method [25], the proposed AVF strategy uses complex coefficient filters for grid synchronization. While in [25] it is applied the symmetric component theory to extract the fundamental-frequency positive sequence by employing finite impulse response (FIR) filters to attenuate the rest. On the other hand, compared with previous complex coefficient filters used for positive-/negative-sequence harmonic detection, [26]–[28], the approach followed to obtain the AVF coefficients uses the vectorial properties of the threephase input signal in the αβ reference frame in order to obtain the final complex coefficient filter. In this way, the proposed scheme lies on using a vectorial approach which is different from previous methods, because the expression for the synchronization algorithm is derived from the vector properties of the input signals providing useful characteristics for the tuning of the design constants. The main advantage of the method is that these constants can be chosen to provide a desired and predefined performance. Besides, the presented scheme can be easily implemented in low-cost DSP hardware with low computational cost permitting to estimate in real time the fundamental and harmonic components of the input signal. The paper is divided as follows. In section II the AVF is derived. Then, in section III its functional block is analyzed and the criteria for tuning the design constants are established. After that, the frequency adaptation loop is proposed and analyzed in sections IV and V. The performance under unbalanced and distorted input voltage is studied in sections VI and VII. Finally section VIII shows the experimental results of the proposed algorithm in order to evaluate the effectiveness of the proposal when it is implemented in a real system based on a DSP platform. II. THE ADAPTIVE VECTORIAL FILTER In this section, the adaptive vectorial filter (AVF) is designed. In order to develop the AVF lets suppose an input vector aαβ defined in the stationary reference frame αβ by aαβ =Ap{cos(ωt) sin(ωt)}.(1) Lets assume parameter ωis a constant or slowly variant value. This assumption can be considered true for reconstructing the grid voltages, because the grid frequency can not change suddenly. Thus the following analysis is a valid approximation for this application. In that case, it is possible to calculate the derivative over the time of aαβ as daαβ dt =ωJaαβ,(2) where matrix Jis J={0−1 1 0 }.(3) Lets bαβ be the output vector of the AVF. Then the scalar and vectorial products of aαβ and bαβ are sp =bT αβaαβ (4) vp =bT αβJaαβ.(5) From (4) and (5) it can be obtained the derivative over the time of sp and vp dsp dt =(dbT αβ dt +ωbT αβJ)aαβ (6) dvp dt =(dbT αβ dt +ωbT αβJ)Jaαβ.(7) As the objective is to reconstruct aαβ from bαβ then it must be satisfied that sp →sp∗=aT αβaαβ (8) vp →vp∗= 0.(9) Taking into account (8) and (9), equations (6) and (7) can be manipulated to define the expression of bαβ in such a way that it achieves these objectives. If ˜sp is defined as ˜sp ,sp −sp∗,(10) then (6) can be transformed in d˜sp dt +ksp ˜sp +dsp∗ dt −ksp ˜sp =(dbT αβ dt +ωbT αβJ)aαβ.(11) From (11) it can be noticed that choosing dsp∗ dt −ksp ˜sp ,(dbT αβ dt +ωbT αβJ)aαβ,(12) then 3 d˜sp dt +ksp ˜sp = 0.(13) It can be observed that (13) represents a first order linear system and that ˜sp tends exponentially towards zero when ksp >0. Thus, condition represented in (8) is fulfilled. Equation (12) can be simplified regarding (4) and that sp∗ is a constant. In that case, (12) is reduced to kspaT αβaαβ =(dbT αβ dt +ωbT αβJ+kspbT αβ)aαβ,(14) therefore, the dynamics representing bαβ must satisfy dbT αβ dt +ωbT αβJ+kspbT αβ =kspaT αβ +k1aT αβJ. (15) Similar analysis can be done starting from (7). In that case ˜vp is defined as ˜vp ,vp −vp∗,(16) and therefore (7) can be manipulated to be d˜vp dt +kvp ˜vp +dvp∗ dt −kvp ˜vp =(dbT αβ dt +ωbT αβJ)Jaαβ.(17) Now if it is chosen dvp∗ dt −kvp ˜vp ,(dbT αβ dt +ωbT αβJ)Jaαβ,(18) then d˜vp dt +kvp ˜vp = 0.(19) Again, it can be observed that (19) corresponds to a first order linear system with ˜vp tending exponentially towards zero when kvp >0. Therefore condition in (9) is also achieved. Now (18) can be simplified considering (5) and that vp∗is a constant equal to 0. Taking into account these factors, (18) can be rewritten as 0 = (dbT αβ dt +ωbT αβJ+kvpbT αβ)Jaαβ ,(20) and as a consequence the dynamics of bαβ should be defined as dbT αβ dt +ωbT αβJ+kvpbT αβ =k2aT αβ.(21) Notice that conditions (8) and (9) should be satisfied simultaneously. In order to achieve this, design constants ksp,k1, kvp and k2should be chosen suitably. In this case, the choice for these constants is Fig. 1. Block diagram of the AVF structure (scalar notation). AVF Fig. 2. Block diagram of the AVF structure (vectorial notation). ksp =kvp =k2=ωkf(22) k1= 0.(23) Under this situation (15) and (21) lead to the same equation dbαβ dt =ωJbαβ +ωkf(aαβ −bαβ),(24) that finally represents the dynamic of bαβ. It should be noticed that (24) is used to reconstruct an input signal of frequency ω. In this way, if ωis considered as an unknown parameter that can be calculated in real time, then (24) can be made adaptive as will be shown in section IV. The value of the output signal can be calculated using the block diagram presented in Fig. 1 that has been expressed in its scalar form. As can be observed, the coupling between the αand βaxes is presented because the β-axis signals are influencing the filtered α-signal and vice versa. This block diagram can be drawn in a compact way by using a vectorial notation as shown in Fig. 2. III. ANALYSIS AND TUNING OF THE AVF To tune the design constant kfof the AVF, it is necessary to consider the final closed loop expressions for (13) and (19) when ksp and kvp are chosen equal to ωkf. Assuming ωto 4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 −100 −50 0 50 100 Time (s) vabc (V) va vb vc va * Fig. 3. Response of the AVF for a 50 Hz input signal. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 −100 −50 0 50 100 Time (s) vabc (V) va vb vc va * Fig. 4. Response of the AVF for a 250 Hz input signal. be constant or slowly variant, it is possible to solve these equations. ˜sp = ˜spoe−ωkft(25) ˜vp = ˜vpoe−ωkft(26) The value kfthen can be chosen for a certain predefined settling time Ts. For instance, if in Tsit is desired to have only a 1 % of the initial error value then kf≈4.6 ωTs .(27) Besides, Tscan be selected proportional to the number of periods of the input signal that should be used to reconstruct the signal. For instance if parameter kis chosen as the number of periods with k > 0then Ts=kT0,(28) finally, considering that ω=2π T0 ,(29) then the design constant can be calculated as kf≈0.73 k.(30) Fig. 3 and Fig. 4 show the response of the AVF when the input signal frequency is 50 Hz and 250 Hz respectively with an amplitude of 100 V in both cases. The simulations have been performed using a discrete implementation of (24). The sampling frequency is 10 kHz and the number of periods to reconstruct the input signal is chosen to be k= 1 thus the value of kfis finally 0.73. It can be noticed that in both cases −50 −40 −30 −20 −10 0 Magnitude (dB) 10 100 1.000 10.000w −90 −45 0 45 90 Phase (deg) Bode Diagram Frequency (rad/s) k=4 k=2 k=1 k=0.5 Fig. 5. Bode diagram of the AVF for ωequal to 50 Hz and different values of k. the input signal is reconstructed properly and in one period as expected. However, the effect of the sampling frequency is clearly visible in Fig. 4. Thus, the sampling frequency should be chosen carefully when it is necessary to reconstruct high frequency signals. The analysis of the behavior of the AVF can be done through its transfer function. To obtain such a transfer function, (24) should be expressed in a more suitable form. {bα bβ}=ωkf (s+ωkf)2+ω2{(s+ωkf)−ω ω(s+ωkf) }{aα aβ},(31) now, using complex notation and considering Aαβ =aα+jaβ(32) Bαβ =bα+jbβ(33) the transfer function can be obtained from (31) like H(s) = Bαβ Aαβ =ωkf(s+ωkf+jω) (s+ωkf)2+ω2.(34) Fig. 5 shows the bode diagram of (34). In this diagram, ω has been chosen to be 50 Hz, because this value is commonly adopted as the grid frequency in Europe. Other value could be adopted and the same conclusions would be addressed. Several plots have been represented for different values of parameter k, which finally defines the AVF design constant kfthrough (30). For any value of k, the gain and phase of the AVF for the frequency ωare zero. For any other frequency the input signal is attenuated and shifted. This behavior ensures the extraction of the desired component. Besides, it can be noticed that increasing kthe filter is more selective. However krepresents the number of cycles needed to reconstruct the desired frequency component. Thus a tradeoff between selectivity and speed should be considered when the input signal contents more than one harmonic component. 5 IV. FREQUENCY ADAPTATION LOOP The frequency parameter ωof the AVF is critical to achieve the correct reconstruction of the input signal. As it can be observed in Fig. 5, if the input signal has a different frequency than the AVF parameter then this signal will be attenuated and therefore the output signal will never achieve the desired value. For this reason, it is necessary to develop a frequency adaptation loop allowing to calculate the correct parameter ω. This task can be performed by a FLL which allow to calculate the actual value of ω[20], [21], [30]. Lets suppose that the frequency parameter of the AVF is bω. Under this assumption (24) is transformed in dbαβ dt =bωJbαβ +bωkf(aαβ −bαβ ).(35) When (35) is used to obtain the output signal, by introducing it in (17), the equation of the dynamic for ˜vp changes to d˜vp dt +kfbω˜vp =speω, (36) where eωis the error between the input signal frequency ωand bω, defined as eω,bω−ω. (37) To estimate bωa Lyapunov function approach is used [31]. Lets define the positive semidefinite function W W=1 2˜vp2+1 2γeω2,(38) where γ > 0is the design constant used in the proposed adaptive law. The time derivative of (38) along the trajectories of ˜vp is ˙ W= ˜vp (speω−kfbω˜vp) + 1 γeωdeω dt ,(39) which is made negative semidefinite by proposing the adaptive law deω dt =−γsp ˜vp. (40) This yields ˙ W=−kfbω˜vp2.(41) The stability and convergence of the proposed adaptive law can be done using Lasalle’s arguments. As W≥0and ˙ W≤0 then should be true that lim t→∞ ˙ W= 0,(42) taking into account expression (41) and that kf= 0 and bω= 0, the only possibility is lim t→∞ ˜vp = 0.(43) Finally, regarding (36) and considering that sp = 0, then lim t→∞ eω= 0.(44) Thus (44) ensures that the error between ωand bωtends to 0 and therefore that AVF can be properly tuned to the input signal frequency. To calculate bωit is necessary to consider that ωis a constant or slowly variant signal. Under this assumption deω dt ≈dbω dt ,(45) So the final expression to calculate bωis dbω dt =−γsp ˜vp. (46) It should be noticed that if the input signal has a very high content of negative-sequence or any other harmonic component, the angular speed of the input vector may oscillate and the amplitude of these oscillations may be high. Therefore under this situation the last assumption will not be valid and these variations should be considered to calculate ω. This situation only appears for very severe grid faults and in this case the power converter should be disconnected from the grid for a safe operation [32]. In spite of this, for the grid synchronization application it should be noticed that ωis a signal that does not suffer sudden variations during normal grid operation or under most of grid faults. Thus the proposed analysis can be considered as valid. V. ANALYSIS AND TUNING OF THE FREQUENCY ADAPTATION LOOP Tuning the design constant γis not immediate due to the fact that the proposed adaptive law has a highly non linear behavior. As a first approximation considering definitions (4), (5) and (16) and that vp∗= 0 then sp =|aαβ||bαβ|cosθ (47) ˜vp =|aαβ||bαβ|sinθ (48) where θis the angle between the input and output vector. Lets rewrite the dynamic of eω, (40), as deω dt =−ξ |aαβ|2|bαβ|2sp ˜vp, (49) where ξis the new design constant. The main advantage of this formulation is that the behavior of (49) will not depend on the module values of the input and output vectors. To calculate the value of ξit can be considered that when the system is reaching its steady state it can be assumed that sp ≈ |aαβ||bαβ|(50) vp ≈|aαβ ||bαβ | kfbω∞eω, (51) where bω∞is the value achieved by the estimator in steady state and (51) can be derived from (36). Replacing (50) and (51) in (49) allows to approximate the behavior of the dynamic of eωas deω dt =−ξ kfbω∞eω. (52) This expression corresponds to a first order linear system that can be easily integrated as eω=eωoe−ξ kf bω∞t(53) 6 Frequency adaptation loop Fig. 6. Block diagram of the frequency adaptation loop. AVF Frequency Adaptation Loop Fig. 7. Basic synchronization system scheme. From (53) it is possible to calculate ξto achieve a desired settling time Tω. For instance, if for Tωit is desired to have only a 1 % of the initial error value then ξ≈4.6kfbω∞ Tω .(54) Now, considering (45) and replacing bω∞by bωin (54). The expressions used to estimate the AVF frequency parameter are ξ,4.6kfbω Tω (55) dbω dt =−ξ |aαβ|2|bαβ|2sp ˜vp. (56) Comparing (46) and (56) it is clear the gain γof the frequency adaptation loop is given by γ=ξ |aαβ|2|bαβ|2,(57) where, ξis calculated by means of (55). It should be noticed that when the grid voltage is approaching to zero design constant γhas to be limited to a maximum value [33]. Fig. 6 presents the block diagram of the frequency adaptation loop. Besides, Fig. 7 shows the scheme of the proposed synchronization system including the AVF and the frequency adaptation loop. Fig. 8 and Fig. 9 show the performance achieved by the proposed frequency adaptation loop. In Fig. 8 the frequency of the input signal is changed suddenly from 50 Hz to 55 Hz and after 0.15 s it is restored to its previous value. A similar situation is presented in Fig. 9 but here the frequency step is from 50 Hz to 45 Hz and backward. The response of the proposed adaptive law is shown for several values of settling 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 48 50 52 54 56 58 Time(s) ω (Hz) ref T ω =0.02 T ω =0.05 T ω =0.1 T ω =0.15 ^ 1 2 3 4 1 2 3 4 1 2 3 4 Fig. 8. Frequency adaptation loop response for an input signal frequency step from 50 Hz to 55 Hz and backward with different values of Tω. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 42 44 46 48 50 52 Time(s) ω (Hz) ref T ω =0.02 T ω =0.05 T ω =0.1 T ω =0.15 ^ 1 2 3 4 1 2 3 4 1 2 3 4 Fig. 9. Frequency adaptation loop response for an input signal frequency step from 50 Hz to 45 Hz and backward with different values of Tω. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 35 40 45 50 55 Time(s) ω (Hz) ref T ω =0.02 T ω =0.05 T ω =0.10 T ω =0.15 ^ 1 2 3 4 1 2 3 4 1 2 3 4 Fig. 10. Frequency adaptation loop response for an input signal frequency step from 50 Hz to 45 Hz and backward with different values of Tω. 0.09 0.1 0.11 0.12 0.13 0.14 0.15 − 100 − 50 0 50 100 Time(s) Va(V) ref T ω =0.02 T ω =0.05 T ω =0.10 T ω =0.15 1 2 3 4 1 2 3 4 Fig. 11. AVF response for an input signal frequency step from 50 Hz to 45 Hz and backward with different values of Tω. time. As it is shown, the proposed frequency adaptation loop performs as expected and the frequency estimation is achieved in the defined time interval. It can be noticed in Fig. 8 and Fig. 9 that for Tωin the range as the settling time of the AVF both dynamics interact and an overshoot appears. It should be considered that (53) is an approximation for the steady state behavior but the actual dynamic of eωduring transient is defined by (49) which has a non linear behavior, therefore there is a trade-off between settling time and overshoot. This must be considered when Tω 7 0.08 0.1 0.12 0.14 0.16 0.18 0.2 42 44 46 48 50 52 54 Time(s) ω (Hz) ref ω ^ ^ (a) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 100 − 50 0 50 100 Time(s) Vabc p(V) ref Va p Va p Vb p Vc p (b) Fig. 12. System response: (a) Estimated frequency, (b) Positive output voltages. is chosen, especially if not only a frequency step is imposed in the input signal but also amplitude and phase changes are also included. Under this situation the non linear behavior of the adaptive law clearly appears during the transient response and higher overshoots can be produced. This effect is shown in Fig. 10 where an input reference step from 100∠0oto 50∠− 30ois done including a frequency step from 50 Hz to 45 Hz and backward. The results show that the proposed frequency adaptation loop works properly also under these conditions. Besides, the defined settling time is also achieved. It should be noticed that the transient response will vary for different amplitude or phase changes. This is because in (56) the values of sp and ˜vp depend on the actual module and position of the input and output vectors. For the sake of simplicity, Fig. 11 only shows the output of the AVF for one phase. It can be observed that for increasing values of Tωthe AVF needs more time to reconstruct the input signal. Thus the trade-off between speed and overshoot should be considered in order to achieve a desired performance. VI. AVF PERFORMANCE UNDER UNBALANCED INPUT CONDITIONS One of the most important tasks of a grid synchronization system is to obtain the positive- and negative-sequence of the grid voltage under unbalanced conditions. For this reason the scheme presented in Fig. 7 is tested when a fault occurs in the input voltage. The pre-fault voltage is 100∠0owith 50 Hz and the post-fault voltage is 50∠−30oand 25∠110ofor the positive- and the negative-sequence respectively. Besides, it has been also considered a sudden change in the grid frequency from 50 Hz to 45 Hz. The design parameters are kf= 0.73 and Tω= 0.1s. Fig. 12 shows the obtained results. It can be noticed that the performance is not good. In Fig. 12a it can be observed that the estimated frequency does not achieve AVF AVF Frequency Adaptation Loop Fig. 13. Synchronization system scheme under unbalanced conditions. 0.08 0.1 0.12 0.14 0.16 0.18 0.2 42 44 46 48 50 52 54 Time(s) ω (Hz) ref ω ^ ^ (a) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 100 − 50 0 50 100 Time(s) Vabc p(V) Va p Vb p Vc p ref Va p (b) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 100 − 50 0 50 100 Time(s) Vabc n(V) Va n Vb n Vc n ref Va n (c) Fig. 14. System response: (a) Estimated frequency, (b) Positive output voltages, (c) Negative output voltages. the actual input frequency value and moreover it has a non negligible ripple. On the other hand, Fig. 12b shows that the input positive-sequence is not reconstructed properly. Finally, 8 this scheme does not allow to calculate the negative-sequence. For all these reasons, it is clear that it is necessary to modify the proposed synchronization scheme. It was shown in section III that the AVF has unity gain and zero phase only for the tuned frequency. For any other frequency it attenuates the input. From Fig. 5 it can be observed that attenuation for other components depends on the value of design constant kf. Therefore, if the input signal has several components, then one option will consist on decreasing the value of kfproviding the desired attenuation. As a counterpart, lower values of kfimply higher settling times. As a consequence, this choice reduces the performance of the system and it should be avoided if possible. Besides, this solution does not solve the problem of reconstructing the negative-sequence of the input signal. The proposed solution is to use two independent AVFs tuned for the positive- and negative-sequence respectively. Also, a cross-feedback network is also included. This network mainly consists on subtracting the undesired component at the input of each AVF. It should be noted that only one frequency adaptation loop is necessary because the frequency for the negative-sequence AVF can be calculated as −bω. Besides, the inputs for the frequency adaptation loop are those corresponding to the positive-sequence signals. The scheme of the proposed solution is shown in Fig. 13. It should be noticed that structural differences between previous synchronization schemes and the proposed solution are present. For instance, the MSOGI-FLL requires a positive- and negative-sequence network calculation that is applied to the output signals from the SOGI-based filter [21]. The same approach is followed when ANFs are used [23], [24]. In the proposed AVF, it is needed the decoupling network before filtering the input signals due to the vectorial approach. Compared with other complex coefficient filter solutions, in [26] the decoupling network is not used, but a cascaded structure is proposed to extract the positive- and negative-sequence. On the other hand, the proposed cross-feedback network also appears in [28], in order to achieve the desired behavior. Fig. 14 plots the obtained results under the previous fault conditions when the proposed modified synchronization scheme is used. Now it can be observed that the system performs as expected. In Fig. 14a is presented the estimated input frequency, as it can be noticed the steady state achieves the actual frequency value in the defined settling time. Regarding the positive- and negative-sequence, Fig. 14b and Fig. 14c display that both signals can be reconstructed properly maintaining the desired performance. VII. SYNCHRONIZATION UNDER UNBALANCED AND DISTORTED INPUT CONDITIONS Unbalanced and distorted input voltages condition is a common situation after a grid fault. Following the same reasoning proposed in section VI, it is clear that it will be necessary to use one AVF for each harmonic and sequence that should be detected. Besides, the cross-feedback network is also included, thus the final synchronization scheme is shown in Fig. 15. It should be noticed that this kind of cross-feedback network AVF AVF AVF AVF Frequency Adaptation Loop Fig. 15. Synchronization system scheme under unbalanced and distorted conditions. TABLE I SYSTEM VARIABLES. Parameter Description 1st harmonic positive-sequence 50∠−30o 1st harmonic negative-sequence 25∠110o 5th harmonic negative-sequence 20∠0o 7th harmonic positive-sequence 20∠0o 11th harmonic negative-sequence 20∠0o has been previously used in other synchronization structures like in [28] or the MSOGI-FLL [21] where it was called harmonic detection network (HDN). This solution presented good results and for the proposed AVF scheme allows to remove the undesired harmonics components before the input signals are filtered. The proposed scheme has been tested during a grid fault. Initially, the pre-fault grid voltage is 100∠0owith 50 Hz. The post-fault voltage is composed by several harmonics with different sequences. Besides, it has been also considered a sudden change in the grid frequency from 50 Hz to 45 Hz. The values for each harmonic are summarized in Table I and the resulting input voltages are plotted in Fig. 16. 9 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 150 − 100 − 50 0 50 100 150 Time(s) Vabc(V) Va Vb Vc Fig. 16. Unbalanced and distorted input voltages for testing the proposed synchronization scheme. To obtain the harmonic components of the input voltages five AVF have been used. Each one tuned to the appropriate harmonic and sequence, that is, positive 1st harmonic, negative 1st harmonic, negative 5th harmonic, positive 7th harmonic and negative 11th harmonic. As the input voltages have high order frequency components, the sampling frequency has been set to 20 kHz in order to obtain a more accurate reconstruction of the 11th harmonic. It should be noticed that the number of samples in the reconstructed signal decrease when the harmonic order increases. The gains for the AVF tuned at the fundamental frequency have been set to kp f,1= 0.73 and kn f,1= 0.73. The gain for the rest of components are kn f,5=kp f,7=kn f,11 = 0.073. This makes the AVFs for the high order harmonics more selective but does not diminish the performance of the system because the overall settling time is defined by the lower frequency component. Finally, the settling time for the frequency adaptation loop is set to Tω= 0.1s. Fig. 17 shows the response of the proposed synchronization system under the aforementioned unbalanced and highly distorted grid conditions. The plot in Fig. 17a confirms that the frequency adaptation loop works properly and exhibits the expected behavior. Fig. 17b and Fig. 17c present the detection of the fundamental positive- and negative-sequences. It can be noticed that the signals are reconstructed almost perfectly. Finally, Fig. 17d, Fig. 17e and Fig. 17f display the behavior of the system for the reconstruction of the high order harmonic components. It can be observed that the proposed scheme provides and accurate detection of these components even when a step in the input frequency has been considered. Finally, it should be noticed that the behavior of the MAVFFLL was tested without considering the effects of dc or interharmonics disturbances. However, it must be pointed out that the AVF attenuates the dc signal as can be observed from its bode diagram in Fig. 5. Besides, the MAVF-FLL is able to estimate inter-harmonics components as well, just by adding AVF blocks tuned at the proper frequency. VIII. EXPERIMENTAL RESULTS The behavior of the proposed synchronization algorithm has been tested in a real system. For this purpose a power converter prototype connected to a programmable three-phase ac source has been used. The experimental setup consists on a three-phase two-cell H-Bridge cascaded converter working as a STATCOM. The scheme of the laboratory setup is depicted in Fig. 18 and the main system parameters are summarized in Table II. 0.08 0.1 0.12 0.14 0.16 0.18 0.2 44 45 46 47 48 49 50 51 Time(s) ω (Hz) ref ω ^ ^ (a) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 100 − 50 0 50 100 Time(s) Vabc,1 p(V) ref Va,1 p Va,1 p Vb,1 p Vc,1 p (b) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 30 − 20 − 10 0 10 20 30 Time(s) Vabc,1 n(V) ref Va,1 n Va,1 n Vb,1 n Vc,1 n (c) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 30 − 20 − 10 0 10 20 30 Time(s) Vabc,5 n(V) ref Va,5 n Va,5 n Vb,5 n Vc,5 n (d) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 30 − 20 − 10 0 10 20 30 Time(s) Vabc,7 p(V) ref Va,7 p Va,7 p Vb,7 p Vc,7 p (e) 0.08 0.1 0.12 0.14 0.16 0.18 0.2 − 30 − 20 − 10 0 10 20 30 Time(s) Vabc,11 n(V) ref Va,11 n Va,11 n Vb,11 n Vc,11 n (f) Fig. 17. Simulation system response: (a) Estimated frequency, (b) Positive 1st harmonic output voltages, (c) Negative 1st harmonic output voltages, (d) Negative 5th harmonic output voltages, (e) Positive 7th harmonic output voltages, (f) Negative 11th harmonic output voltages.