Design of LCL and LLCL filters for single‐phase grid connected converters
2016; Institution of Engineering and Technology; Volume: 9; Issue: 9 Linguagem: Inglês
10.1049/iet-pel.2015.0922
ISSN1755-4543
AutoresMajid Sanatkar‐Chayjani, Mohammad Monfared,
Tópico(s)Islanding Detection in Power Systems
ResumoIET Power ElectronicsVolume 9, Issue 9 p. 1971-1978 Research ArticlesFree Access Design of LCL and LLCL filters for single-phase grid connected converters Majid Sanatkar-Chayjani, Majid Sanatkar-Chayjani orcid.org/0000-0002-1333-5673 Department of Electrical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, IranSearch for more papers by this authorMohammad Monfared, Corresponding Author Mohammad Monfared m.monfared@um.ac.ir Department of Electrical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, IranSearch for more papers by this author Majid Sanatkar-Chayjani, Majid Sanatkar-Chayjani orcid.org/0000-0002-1333-5673 Department of Electrical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, IranSearch for more papers by this authorMohammad Monfared, Corresponding Author Mohammad Monfared m.monfared@um.ac.ir Department of Electrical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, IranSearch for more papers by this author First published: 01 July 2016 https://doi.org/10.1049/iet-pel.2015.0922Citations: 29AboutSectionsPDF 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 Compared with the traditional L filters, the high order LCL and LLCL filters for grid integration of voltage source converters offer better switching harmonic attenuation even with smaller passive elements. However, the inherent resonance of the high order filters complicates the control of the converter and also the design of filter parameters, especially at presence of digital delays and grid impedance variations. This study proposes a simple design procedure for the LCL and the LLCL filters of a delay-based stabilised converter that satisfies the predefined constraints on the converter current ripple, grid current harmonics and the reactive power of the capacitor and at the same time, ensures adequate stability of the simple single loop current control against the resonance. Moreover, the effect of grid impedance on the filter performance and controller stability is taken into account, thus the proposed method is robust against a wide range of grid impedance variations. Experimental results for a 3 kW test rig under steady state and transient conditions confirm the effectiveness of the proposed filter design algorithm. 1 Introduction The rapidly increasing energy demand and growing environmental concerns cause further attention to renewable energy sources (RESs). A current controlled voltage source converter (VSC) is an industrially accepted solution to interface the RES to power grid [1]. The quality of the current injected by VSCs to the utility grid is a main concern today. In order to prevent the harmonics and switching noises of the VSCs from penetrating the utility grid, some kind of passive low-pass filters, such as L, LCL, LLCL or other higher order filters are used in the output stage of the converter. The harmonic attenuation capability of LCL and LLCL filters are significantly higher than the traditional L filters, even with smaller passive elements, which offers lower volume, cost and losses [2]. Although, in spite of the prominent advantages of high order filters, the inherent resonance nature of them challenges the stability of the current control loop that may even lead to instability under certain circumstances. To deal with the resonance problem and stabilise the closed loop current control system, several methods are presented in recent studies [3-17]. Passive damping, using an additional resistor in the filter structure, is the simplest and most common method, which suffers from major drawbacks, such as power losses, reduced attenuation at switching frequencies and deteriorated performance when connected to a weak grid [3-5]. To ensure stability against the resonance and at the same time avoid additional losses, the active damping methods are proposed. In these methods, the resonance damping is achieved by modifying the control algorithm, which requires the addition of either some kind of digital filters to the control loop or extra sensors to the circuit [6-17]. Filter based methods include two distinct controllers in the current control loop. While the main controller regulates the low-frequency dynamics, the added one, which can be seen as a filter, attenuates the high-frequency resonances [6-9]. The grid impedance variations can weaken the damping effect and turn the stabilised system to instability. Active damping with extra control variables, such as the capacitor voltage [10, 11] or the capacitor current [12-17] are proposed. These techniques offer adequate damping in a selective way, at the cost of added circuit cost and control complexity, especially with considering digital implementation delays [16, 17]. An interesting technique to yield stability over a certain range of resonance to sampling frequency ratios, in a simple single-loop current control system is the delay-based stabilisation [15, 18-21]. In contrast to active damping solutions proposed yet, no digital filter is inserted in the current control loop and no extra variable, except the controlled current is sensed or controlled. In these approaches, different factors that influence the stability, considering the effect of the digital control delay are identified first. Then, the sampling and/or the resonance frequency are carefully selected such that adequate stability margins are obtained for the single-loop control with a certain delay presented in the loop. For instance, considering the 1.5Ts delay, the grid current can be controlled directly without any passive or active damping method if the ratio of the resonance frequency to the sampling frequency lies between 1/6 and 1/2 (i.e. 1/6 < fres/fs < 1/2). Other works are also available that change the total loop delay as another tuning factor, which lets a wider range for above ratio between the two frequencies. The nature of this damping characteristic caused by the delay is recently discussed in details by many authors [18-20]. Despite the advantages of low cost and simplicity of the control algorithm, the stability of these methods is highly affected by grid impedance variations. The change of grid impedance can push the resonance frequency outside the stable region. Regardless of the adopted stabilisation technique, designing the parameters of the high-order filters (LCL and LLCL) is also an important as well as complicated issue. In most of the previous works, the parameter design procedure is a trial and error method with the goal of size and cost reduction [22-27]. In this way, the LCL filter parameters are determined so that the converter-side inductor limits the converter current ripple, the capacitor value is simply chosen based on the reactive power limits and the grid-side inductor is decided to meet the grid current harmonic attenuation in accordance to IEEE-1547.2-2008 standard [28]. A similar approach is also proposed for higher order filters, such as the LLCL [24], the LTCL [25] and the LCL-LC filters [26]. In combination with the delay-based stabilisation methods, it is evidently difficult to find the best compromise between various, sometimes conflicting, requirements of output current ripple, harmonic attenuation, reactive power limits, size of components and the stability margins. Consequently, this paper proposes a simple design procedure for the LCL and the LLCL filters of a delay-based stabilised converter that ensures stability of the closed loop system, and at the same time meets the reactive power, current ripple and harmonic limits requirements. The effect of grid impedance variations on the system stability is also considered, which ensures robust operation even with a weak grid. The rest of the paper is organised as follows: Section 2 formulates the switching harmonics of the single phase converter with the unipolar asymmetrical modulation. In Section 3, the characteristics of the LCL and the LLCL filters in both low and high frequency ranges are investigated. In Section 4, the single loop current control of the converter system, with delay-based stabilisation is explained. The step by step design procedure for both LCL and LLCL filters is presented in Section 5. The experimental results are given in Section 6 and Section 7 concludes the paper. 2 Harmonics in the output of the single phase VSC The power stage of the grid-connected single phase VSC is shown in Fig. 1. The switching harmonics of the VSC output voltage v appear in the grid current. As mentioned before, to adequately suppress the switching harmonics in the current, the VSC is connected to the grid through a high-order low-pass LCL or LLCL filter. Fig. 1Open in figure viewerPowerPoint Grid-connected single phase VSC The most common modulation method for the single phase VSC is the unipolar asymmetrical (double-update) regular sampled pulse-width modulation (PWM), for which the output voltage of the VSC is expressed as [29] (1)where q = 2 m + (2n − 1)/Mf and Vdc is the dc link voltage, Ma and Mf are the amplitude and the frequency modulation indices, respectively, ωc and ω0 are the carrier and the fundamental frequencies, respectively, (Mf = ωc/ω0) and Jn(x) is the integral of Bessel function, which is expressed as . In the asymmetrical unipolar PWM, the sampling frequency is ωs = 2ωc and obviously, the first sideband is located at 2ωc. From (1), the amplitude of the high frequency switching components (i.e. sideband harmonics) can be derived as (2). The two first sidebands are plotted in Fig. 2. (2) Fig. 2Open in figure viewerPowerPoint Normalised harmonics amplitudes of the VSC output voltage a First sideband b Second sideband The lower the frequency of the sideband, the greater the harmonic amplitude; that is, the most dominant harmonics, as demonstrated in Fig. 2, are located around the first and the second sidebands. 3 LCL and LLCL filters characteristics Both LCL and LLCL filters are shown in Fig. 1. The transfer functions of the converter voltage (v) to the converter and grid currents (i1 and i2) for the LLCL filter are formulated as [24] (3)where L'2 = L2 + Lg and Lg represents the equivalent inductance of the grid. Apparent from (3), the resonance frequency of the LLCL filter is ωres = [(L1 + L'2)/(Cf (L1L'2 + L1Lf + L'2Lf))]1/2. The transfer functions for the LCL case can be readily derived by setting Lf equal to zero. The transfer functions Y11 and Y21 determine the filter attenuation for i1 and i2 currents, respectively. Fig. 3 shows the Bode plots of Y21 for the LCL and the LLCL filters. Fig. 3Open in figure viewerPowerPoint Bode plots of Y21 for the LCL and the LLCL filters As it can be seen in Fig. 3 and presented in (4), the LCL filter offers a high attenuation at high frequencies, where the gain rolls-off at −60 dB/decade, while this rate is limited to −20 dB/decade for the LLCL case. However, the LLCL filter has a pair of complex-conjugate zeros in its transfer function located at the first sideband frequency (as the dominant switching harmonic). Therefore, the LLCL filter can significantly attenuate the first sideband harmonics. Although the high frequency roll-off rate of LLCL is the same as the L filter (with L = L1 + L′2), but the attenuation is higher with a factor of 1/(1 + L1||L′2/Lf). (4)In the low frequency range, as calculated in (5) and shown in Fig. 3, the characteristics of both LCL and LLCL filters are identical and the same as a simple L filter with L = L1 + L′2. Therefore, the design of the current controller, regardless of the type of the filter, follows a similar procedure. (5)Furthermore, the filters also present similar characteristics at high frequencies in terms of the converter current ripple attenuation, as presented in (6). As a result, regardless of the type of the filter, the converter current ripple is imposed only by the L1 value. (6)From the stability point of view, Bode plots of Fig. 3 show a sharp phase transition through −π along with a high magnitude peak at the resonance frequency that readily leads to instability. 4 Single-loop control of grid current with delay-based stabilisation method The proportional-resonant (PR) controller is commonly used to eliminate the steady-state error in the stationary reference frame. The open loop transfer function of the system, in presence of a delay Td, can be derived as (7)where GPR is the transfer function of the PR controller. Open loop Bode plots for both LCL and LLCL filters are shown in Fig. 4. Fig. 4Open in figure viewerPowerPoint Open loop Bode plot of grid current loop for LCL and LLCL filters 4.1 Stabilisation of the single-loop controller A digitally implemented control system has several delays, mainly originated from the sampling, analogue-to-digital conversion and computations. All these can be modelled with one sampling period (Ts) delay. Taking into account the 0.5Ts PWM transport delay of the converter system, a total delay of Td = 1.5Ts exists in the control loop [30]. The delay introduces a linear phase lag and changes the phase characteristic of open loop system. In delay-based stabilisation method, the resonance frequency (compared with the sampling one) is selected low enough to take benefit of this phase lag. This way, the phase lag originated from the loop delay causes the sharp phase change occurs below −π. If this sudden phase fall ends above −3π, then the single loop system remains stable. Based on this analysis, the stable range of resonance frequency for Td = 1.5Ts is already obtained as fs/6 < fres < fs/2 [18, 20]. Hence, Fig. 4 is plotted with the assumption that the resonance frequency is located in the stable rage of fs/6 < fres < fs/2; therefore, the sharp phase change lies between −π and −3π. In this situation, three phase margins can be recognised. At the low frequency range, the phase margin PM1 characterises the current control loop performance and is determined in the controller design stage, which is presented in the next subsection. Indeed, the dynamic performance of the closed loop system is imposed by the stability margin PM1, while stability margins PM2 and PM3 have negligible effects on the dynamic performance and only must be positive to ensure the stability. In practice, PM2 and PM3 should be selected large enough to remain positive in presence of parameter uncertainties of the filter circuit and ensure stability against the resonance under all circumstances. The latter issue is fully attended in Section 5. 4.2 Tuning the controller parameters As mentioned before, the characteristics of both LCL and LLCL filters at the low frequency range are identical and the same as a simple L filter with L = L1 + L2. Therefore, the PR controller for the current loop can be designed in the same way as for a system with the L filter. The tuning of PR constants is performed according to directions of [15]. Considering the negligible effect of the resonant terms of the PR controller on the gain crossover frequency, ωgc1 (for PM1), then ωgc1 can be calculated as (8)where PM1d is the desired value for the PM1. To achieve the unity loop gain at the crossover frequency, the proportional gain is set to (9)The resonant gain is selected so that a negligible phase contribution at ωgc1 is achieved, i.e. (10) 5 LCL and LLCL filters design Previous works have already proposed some criteria for selecting the parameters of high-order filters based on the limits on the current ripple of the inductors and the reactive power of the capacitor while the stability against the resonance is achieved with a passive or active damping method [22-26]. In this paper, the tuning of filter parameters is conducted in conjunction with the delay-based stabilisation technique. Indeed, the proposed step-by-step filter design algorithm satisfies the predefined constraints on the converter current ripple, grid current harmonics and the reactive power of the capacitor and at the same time, ensures adequate stability of the simple single loop current controller. Moreover, the effect of grid impedance on the filter performance and controller stability is taken into account, thus the proposed method is robust against a wide range of grid impedance variations. In the following, the limits on the converter current ripple, the grid current harmonics and the reactive power are first presented, then the stability constraints of the closed-loop system are derived and finally a simple and systematic procedure to design the filter parameters is presented to simultaneously meet all the constraints. 5.1 Limits on the converter current ripple, the grid current harmonics and the reactive power of the filter capacitor To design the parameters of a LCL or LLCL filter, some limits on current ripples and harmonics and reactive power should be considered, which are already discussed in [22, 24, 25]: i. At the fundamental frequency, both filters act as a shunt capacitor that supplies reactive power. The reactive power by capacitor must be limited to a per cent, we call it x1, of the rated power (generally x1 = 2.5 to 5%). ii. The converter side current ripple mainly determines the current rating of converter switches and is decided to be a per cent, we call it x2, of the peak rated current of the converter IP,rated (generally x2 = 15 to 40% [24]). In both cases, the maximum current ripple can be derived as [24] (11) iii. The IEEE-1547.2-2008 standard recommend that all harmonics higher than 35th in the grid current must be lower than a per cent, x3 = 0.3%, of the rated current. To address the above requirement, as already discussed in Section 3, in the case of an LCL filter, the first sideband and in the case of an LLCL filter, the second sideband must be limited as (12)where the Vsb1,max and Vsb2,max are the maximum values of all sideband amplitudes in the expected range of variations for Ma, which can be replaced from (2) or readily from Fig. 2. 5.2 Constraints for stability against the resonance However by setting the resonance frequency in accordance to fs/6 < fres < fs/2 (sharp phase change occurs between −π and −3π) the stability of single-loop current regulator is achieved, but sufficient damping at resonance frequency, in terms of the phase margins PM2 and PM3, must be provided to ensure positive margins under the expected range of filter parameter uncertainties. In a weak grid, the grid impedance variations lead to a wide uncertain range of the resonance frequency that must be also explored. Therefore, according to the grid impedance condition, the expected range of variation of resonance frequency should be first determined, then the filter parameters must be chosen such that the resonance frequency range always remains inside the stable range of fs/6 < fres < fs/2 with appropriate margins. In the case of a PR controller, where the phase change due to the resonance network just affects the low frequencies (obvious in Fig. 4), then the open-loop system phase in vicinity of the resonance frequency can be assumed as (13)As it can be seen in Fig. 4, the gain crossover frequencies for PM2 and PM3 are slightly lower and higher than the resonance frequency, respectively. With a good approximation, both gain crossover frequencies can be replaced with the resonance frequency. Therefore, the criterion that the sharp phase change lies in the stable range of −π to −3π, with some margins as we already called PM2 and PM3, respectively, imposes that (14)Hence, the appropriate resonance frequency range that meets the desired phase margins PM2d and PM3d can be concluded as (15)It is worth mentioning that if one replace PM2d = PM3d = 0 and Td = 1.5Ts, then (15) simplifies to fs/6 < fres < fs/2. On the other hand, the possible variation range for the resonance frequency with variation of Lg between zero and infinity can be derived as (16)The resonance frequency decreases with the increase of Lg and vice versa. Therefore, based on (14) the minimum values for PM2 and PM3 occur for infinite and zero value of Lg, respectively. Consequently, in order to ensure that the resonance frequency range of (16) always lies within the stable range of (15), two stability constraints must be met (17) (18)In the case of the LCL filter, one must replace Lf = 0. Satisfaction of these two constraints guarantees the stability of the closed-loop system with sufficient phase margins while the grid impedance changes widely. 5.3 Choosing the proper phase margins As mentioned before, the values of the PM2d and PM3d must be selected properly to avoid instability due to uncertainties and drifts of filter parameters. A change in the filter parameters, may lead to a decrease of the minimum or an increase of the maximum value of the resonance frequency, already predicted by (16), which may push it outside the stable range. Considering (16), the worst case for the ωres,min occurs when all filter parameters have positive uncertainty (i.e. the actual values be higher than the designed ones), and for the ωres,max it occurs when all filter parameters have negative uncertainty (i.e. the actual values be smaller than the designed ones). Assuming that the actual values of filter parameters, with a subscript ‘a’, are related to the designed values as La = kLL and Ca = kCC, then, the actual resonance frequency can be derived as (19)For the given values of kL and kC, with the goal of achieving the zero phase margin for the worst case of parameter uncertainties, the values of the PM2d and PM3d can be derived using (15), (17) and (18) as (20) 5.4 Proposed design procedure Two sets of limits and constraints are already presented in Sections 5.1 and 5.2. Satisfying them simultaneously results in a filter with good harmonic attenuation and sufficient stability margins, regardless of the grid impedance conditions. After choosing the limits x1, x2 and x3 and the range of parameters uncertainty (kL and kC), according to the application requirements, the LCL and LLCL filter parameters can be calculated according to the following straightforward step-by-step procedure: i. For both filters, the capacitor value is initially determined from the reactive power limit as (21)where Prated is the rated power and Vg is the rms value of the grid voltage. The second limit for the capacitor in the case of the LCL filter is obtained by removing L1 from (11) and (17) (22)and in the case of the LLCL filter and with the extra assumption that the filter zero is exactly located at the first sideband, i.e. 1/(Lf Cf)1/2 = ωs, we have (23)Hence, the minimum of (21) and (22) determines the Cf value for the LCL and the minimum of (21) and (23) determines the Cf value for the LLCL filter. ii. In the LLCL filter, the first sideband is extremely attenuated by selecting the Lf value as (24) iii. With the calculated values for Cf and Lf, the value of L1 can be determined using (17) as: (25)It should be noted that the ripple limit on the converter side current is considered in selection of Cf, therefore with this value for L1 the ripple limit is also satisfied. iv. The value of L2 should be determined to achieve a high attenuation of grid current harmonics based on the standards and at the same time obtain a guaranteed stability. Therefore, (4) and (12) are combined to provide a limit for the grid current attenuation for the LCL and the LLCL filters, presented in (26) and (27), respectively. Since the worst case of harmonic attenuation occurs in a stiff grid with Lg = 0, the following results are calculated with the assumption that Lg = 0. (26) (27) The values of Vsb1,max and Vsb2,max can be determined from (2) or Fig. 2 for the expected range of Ma. From the stability point of view, the limit on the value of L2 is achieved using (18) as (28)Again for the LCL filter, one must replace Lf = 0. Hence, the maximum of (26) and (28) determines the L2 value for the LCL and the maximum of (27) and (28) determines the L2 value for the LLCL filter. Fig. 5a summarises the four step design algorithm for the LCL and the LLCL filters. Fig. 5Open in figure viewerPowerPoint Four step design algorithm for the LCL and the LLCL filters a Design flow for the LCL and the LLCL filters b Design example 5.5 Design example The proposed step-by-step design procedure for a single phase converter with Vg = 220 Vrms/50 Hz, Vdc = 388 V (Ma = 0.8), Prated = 3 kW and fc = 8 kHz (fs = 2fc) is performed in the following and the results are summarised in Fig. 5b. Choosing the value of parameter uncertainties depends on the type of used capacitors and inductors. Here, the deviation of ±30% and ±20% is considered for inductances and capacitors, respectively (i.e. 0.7 < kL < 1.3 and 0.8 < kC < 1.2). Therefore, the proper stability margins can be calculated from (20) as PM2d = 23° and PM3d = 68°. i. With the reactive power limit of x1 = 5%, the capacitor should be <9.86 µF and with the current ripple limit of x2 = 30%, the maximum allowed value is 4.3 µF for the LCL and 4.1 µF for the LLCL filter. In each case, the lower value considering the available capacitors should be chosen. Therefore, the selected values are 4 µF for both LCL and the LLCL filter. ii. With the chosen value for the capacitor of the LLCL filter, the value of Lf is calculated from (24) as 25 µH. iii. The value of L1 is calculated from (25) as 570 and 540 µH for the LCL and the LLCL filter, respectively. iv. The limit on the value of L2 is calculated from (26) and (27). With a dc link voltage of 388 V, which means Ma ≥ 0.8 (considering the grid impedance volt-drop), the maximum value of the converter voltage for the first and the second sidebands is determined from Fig. 2 as 0.32Vdc and 0.12Vdc, respectively. Thus the value of L2 is calculated as 940 and 150 µH, for the LCL and the LLCL filter, respectively. Furthermore, the value of L2 is calculated from (28) as 260 and 210 µH, for the LCL and the LLCL filter, respectively. Hence, the maximum value for L2 is chosen as 940 and 210 µH, for the LCL and the LLCL filter, respectively. The designed filter parameters and the resultant limits and constraints are summarised in Table 1. As expected, the achieved total inductance value for the LCL filter is higher than the LLCL filter. For both filters, the Cf value is determined by the stability constraint PM2d. The L2 value is determined by the stability constraint PM3d for the LLCL filter and by the harmonics attenuation limit (x3) for the LCL filter. Table 1. Designed filter example Filter type Filter parameters Design limits and constraints L1, μH L2, μH Cf, μF Lf, μH PM2, deg PM3, deg x1, % x2, % x3, % LCL 570 940 4 – 23 127 2.03 27.9 0.3 LLCL 540 210 4 25 23 68 2.03 29.2 0.23 6 Experimental verification To confirm the theoretical achievements, a 3 kW single phase VSC is constructed with the filter parameters already calculated in the previous section. The control algorithm is implemented on a TMS320F28335 floating point digital signal controller from Texas Instruments. For both LCL and LLCL filters, the current controller parameters are calculated from (8)–(10), with a desired phase margin of PM1d = π/3 as: ωgc1 = 5.58 krad/s, kp = 8.4 (LCL)/4.2 (LLCL) and kr = 940 (LCL)/470 (LLCL). Besides the fundamental component, the set of harmonics to be regulated are selected as n = 3, 5, 7 and 9, based on the grid dominant harmonics. The PR controller is discretised by the impulse invariant method [31]. 6.1 Steady state performance Fig. 6 shows the steady state performance of the converter with both designed filters and Lg ≃ 0. Although the grid voltage is highly polluted with a THDv = 3.8%, the injected current is highly sinusoidal with a THDi of about 1.2% (LCL) and 2.1% (LLCL). These results indicate that the filter parameters, as well as the current controller are designed correctly. The spectra of grid current for both filters confirm adequate switching harmonic attenuation according to design constraints. The amplitude of the dominant switching harmonic for the LCL filter (at the first sideband) and the LLCL filter (at the second sideband) is 0.28 and 0.23%, respectively, which both meet the IEEE–1547 standard requirements. Fig. 6Open in figure viewerPowerPoint Steady state waveforms and grid current harmonic spectrum with Lg ≃ 0 a LCL filter b LLCL filter The low frequency spectra (zoomed view of Fig. 6 for n < 50), compared with the IEEE 1547 limits are shown in Fig. 7. As it can be seen, both filters successfully meet the IEEE 1547 standard limits. The low frequency harmonics appeared in the grid current are mainly due to the grid voltage background harmonics and the non-idealities of the real system, such as the switching dead times and the inductor core non-linearity and saturation. These harmonics are attenuated primarily by the inductors in the filter structure. The harmonic compensation network in the current controller can also improve the results. It should be noted that, the higher magnitude of the low frequency harmonics and consequently higher value for THDi of the LLCL filter compared with the LCL one is mainly due to the lower total inductance (L = L1 + L2) of the LLCL filter, which makes the converter performance, in terms of the quality of the currents, more susceptible to non-idealities of the real system. Obviously, this effect decreases in presence of a higher grid impedance value and as it can be seen in Fig. 8, in such a situation, Lg = 3.7 mH, almost similar THDis are obtained for both LCL and LLCL filters. It can be concluded that, for the filters with lower total inductance, which have the advantages of lower cost and size, the low frequency harmonic suppression level is lower and the THDi value is higher. Moreover, Fig. 8 confirms that both converter systems remain stable and keep their excellent performance, even in presence of a very weak grid, where a large range of impedance values and background harmonic distortions are possible. Fig. 7Open in figure viewerPowerPoint Low frequency harmonics of grid current compared with IEEE 1547 limits Fig. 8Open in figure viewerPowerPoint Steady state waveforms with Lg = 3.7 mH a LCL filter b LLCL filter The ripple content of the converter side current is almost equal to the capacitor current iC. As it can be seen in Fig. 6, the maximum peak current ripple is about 6 A, which translates to 30% of the peak rated current IP,rated. 6.2 Dynamic performance The dynamic performance of the VSC is investigated in response to step changes in the reference active power from 2 to 3 kW and vice versa, which is shown in Fig. 9. In the case of Lg ≃ 0, results confirm the excellent transient response of both LCL and LLCL filtered converters, where the converter current changes almost instantaneously with a negligible oscillation. The transient performance with Lg = 3.7 mH shows that the converter remains stable, however the transient response is slowed down by the effect of the grid impedance. This appreciable performance can be attributed to the proper tuning of the stability margins of the converter system, PM1d to PM3d. It should be noted that, in all transient waveforms any oscillation corresponding to the filter resonance does not appear. Fig. 9Open in figure viewerPowerPoint Dynamic performance in response to step changes in the reference active power from 2 to 3 kW and vice versa with Lg ≃ 0 and Lg = 3.7 mH a LCL filter b LLCL filter 7 Conclusion In this paper, a simple and straightforward design algorithm for the LCL and the LLCL filtered VSCs with a delay-based stabilised single-loop current controller is proposed that takes into account both filtering requirements and stability issues simultaneously. In this approach the filter parameters are designed directly and without any iteration or trial and error procedure. The constraints on the filter parameters due to the stability requirements are carefully extracted for the delay-based stabilisation method, considering the effect of grid impedance variations. 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