Design of LLCL-filter for grid-connected converter to improve stability and robustness Min, Huang; Wang, Xiongfei; Loh, Poh Chiang; Blaabjerg, Frede

Transcription

1 Aalborg Universitet Design o LL-ilter or grid-connected converter to improve stability and robustness Min, Huang; Wang, Xiongei; Loh, Poh Chiang; Blaabjerg, Frede Published in: Proceedings o the 3th Annual IEEE Applied Power Electronics Conerence and Exposition, APEC 5 DOI (link to publication rom Publisher):.9/APEC Publication date: 5 Document Version Early version, also known as pre-print Link to publication rom Aalborg University Citation or published version (APA): Huang, M., Wang, X., Loh, P. C., & Blaabjerg, F. (5). Design o LL-ilter or grid-connected converter to improve stability and robustness. In Proceedings o the 3th Annual IEEE Applied Power Electronics Conerence and Exposition, APEC 5 (pp ). IEEE Press. I E E E Applied Power Electronics Conerence and Exposition. Conerence Proceedings, DOI:.9/APEC General rights Copyright and moral rights or the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition o accessing publications that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy o any publication rom the public portal or the purpose o private study or research.? You may not urther distribute the material or use it or any proit-making activity or commercial gain? You may reely distribute the URL identiying the publication in the public portal? Take down policy I you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to the work immediately and investigate your claim. Downloaded rom vbn.aau.dk on: august 6, 8

2 Design o LL-ilter or Grid-Connected Converter to Improve Stability and Robustness Min Huang, Xiongei Wang, Poh Chiang Loh, Frede Blaabjerg Department o Energy Technology Aalborg University Aalborg, Denmark hmi@et.aau.dk, xwa@et.aau.dk, pcl@et.aau.dk, bl@et.aau.dk Abstract The LL-ilter has recently emerged into gridconnected converters due to the improved iltering capability which ensuring a smaller physical size. An LL -based gridconnected converter has almost the same requency-response characteristic as that with the traditional L-ilter within hal o the switching requency range. The resonance requencies o the LL-ilters based grid-connected converters are sensitive to the grid impedance as well as cable capacitance, which may inluence the stability o the overall system. This paper proposes a new parameter design method or LL-ilter rom the point o stability and robustness o the overall system, when the grid-side current control is used. Based on this design method, the system can be stable without damping and also robust to the grid parameters variation. The inluence o delay and parameter variations is analyzed in details. Last, a design example or LL-ilter is given. Both simulations and experimental results are provided through a 5 kw, 38V/5 Hz grid-connected inverter model to validate the theoretical analysis in this paper. Keywords LL-ilter;robustness; stablility;delay; impedance variation;ilter design I. INTRODUCTION Most renewable energy sources and distributed generation (DG) resources are connected to the power grid through a gridconnected inverter []. However, the use o Pulse Width Modulation (PWM) scheme introduces undesirable harmonics that may disturb other sensitive loads/equipment on the grid and also result in extra power losses. Hence, a low-pass power ilter is required between the voltage source inverter (VSI) and the grid to attenuate the high-requency PWM harmonics to limit the harmonic content o the grid-injected current []. Fig. shows our dierent ilter structures. Typically, a simple series inductor L is used as the ilter interace between power converters in the renewable energy system. But it only has db/dec attenuation around the switching requency, so a high value o inductance needs to be adopted to reduce the current harmonics, which would lead to a poor dynamic response o the system and a higher power loss. In contrast to the typical L-ilter, the high order L-ilter can achieve a 6 db/dec harmonic attenuation perormance with less total inductance, signiicantly smaller size and cost, especially or applications above several kilowatts [3]. Recently, the trap ilter which is also called LL-ilter is becoming attractive or industrial applications [4]-[7], as shown in Fig. (c). Compared to the L ilter, a small inductor is inserted in the i ui L L C u i L ug u g L i Fig.. Topologies o dierent ilters or voltage source converter. branch loop o the capacitor, composing an L C series resonant circuit at the switching requency to eliminate this major harmonic component. Hence, the total inductance or capacitance o the ilter can be reduced. In order to urther reduce the size o the ilter, a multi-tuned ilter was proposed [8], but it brings the complexity to the circuit and has possible parallel resonances between the multi-tuned traps. Similarly to the L-ilter, the LL-ilter resonance is challenging the stability o the grid-connected VSI. Hence, the high order resonance should be properly damped either passively or actively [9]-[5]. The LL-ilter does not bring any extra control diiculties because an LL-based gridconnected converter has almost the same requency-response characteristic as that with the traditional L-ilter within hal o the switching requency range. In digital-controlled systems, sampling and transport delays caused by controller and the PWM modulation will aect the system stability and should be taken into account [9]-[]. The stability o the LL-ilter considering the delay eect was studied in [7]. Re. [7] comes to the conclusion that one-sixth o the sampling requency ( s /6) is regarded as a critical LL-ilter resonance requency and the system can be stable, i the resonance requency is higher than s /6 without damping method. However, long distance distribution lines will introduce inductive impedance to the grid. The resonance requency o the LL-ilter is sensitive to the grid impedance. What s more, distribution cables and interaction between multiconverters also inluence the stability o the system with the multiple resonance requencies [6]-[9]. A system is robust i ui L u i L C C L L ug C n L n L u g /5/$3. 5 IEEE 959

3 U dc i u i L L C i c u pcc Z g u g * ig Gc (s) Gd (s) ui KPWM ug i Z i c Z L ZLC L L Fig.. LL-ilter based grid-connected voltage source inverter with single-loop control. it is not very sensitive to grid and converter parameter variation. For this reason, a novel parameter design method o the LL ilter or stabilizing the system operation and improving system robustness is proposed and validated. In section II the LL-ilter based grid-connected inverter is modelled using the Norton equivalent model with grid current control. Section III analyzes the stability and robustness o the system considering the transmission cables. Then, a novel LL-ilter design method is proposed and parameters variation is analyzed in section IV. Finally, the experiments o a 5 kw grid-connected converter are carried out to veriy the theoretical analysis. II. TABLE I PARAMETERS OF THE SYSTEM Symbol Deinition Value U dc DC link voltage 65 V U g Grid phase voltage V o Grid requency 5 Hz T s Sampling period μs sw Switching requency khz MODELING OF THE LL FILTER BASED GRID- CONNECTED INVERTER A. System Description Fig. illustrates a LL-ilter based grid-connected voltage source inverter with single-loop control. The inverter output voltage and current are represented as u i and i, and the grid voltage and current are represented as u g and. The voltage u pcc is the voltage at the Point o Common Coupling (PCC). Z g is the grid impedance, which can be inductive or capacitive. U dc is the DC voltage. Table I shows the main parameters o the system which is used as an example or this study. When the grid current is controlled, the corresponding control block diagram is shown in Fig. 3. G c (s) is the implemented current controller using a Proportional Resonant (PR) controller and harmonic compensator, expressed as: G s K K s () ih c() = p + h=,5,7 s + ( ω h) where ω o = π o is the undamental angular requency, K p is the proportional gain, and K ih is the integral gain o the individual resonant requency h. G d (s) is the computational and Pulse Width Modulation (PWM) delays. Fig. 3. Block diagram o grid current control o grid-connected converter. Gi * c g s G d (s) = e λts () where T s is the sampling period o control system and λt s is delay time, K PWM is the transer unction o the inverter. Z L is the impedance o the inverter-side inductor. Z LC is the impedance o the L -C circuit. Z L is the impedance o the gridside inductor. * is the reerence grid current. B. Norton Equivalent Model Fig. 4 shows the Norton equivalent model o gridconnected converter with grid current control [9]. The dotted block is the cable capacitance C g and line impedance L g. The derivations o the terminal behavior o the grid current control are shown below. The open loop transer unctions rom to u i and to u pcc are expressed in (3) and (4), respectively. i G c g G = = u i u Z L Z L + Z L Z + Z L Z pcc= g L G = = u pcc Z L Z L+ Z u L Z + Z L Z i= Z i Z + Z T and G cl are the open-loop and closed-loop gains o the grid current control loop, which are expressed as: PWM c d (3) (4) T= K GGG (5) T Gc = (6) + T The closed-loop output admittance G c can be derived as: G u pcc G T /( ) c = = + + T G G Hence, the closed loop expression o the grid current is expressed as: C g L g * g c g c pcc ug Fig. 4. Norton equivalent model o grid-connected converter with grid current control. (7) i = G i G u (8) 96

4 The grid voltage is seen as the disturbance term in the design o the current loop controller. The resonance at the grid side will also inluence the stability o the whole system. The resonance requency o the LL-ilter r ( r=π r ) is derived as: ω = r L( L+ Lg ) + L C L+ L+ L g (9) III. STABILITY AND ROBUSTNESS ANALYSIS A. Concept o Passivity o the System Given a linear and continuous system G(s), two requirements should be met in order to obtain the passivity []: ) G(s) has no Right Hal Plane poles. ) Re { G } { G } (j ω) arg (j ω) 9, 9, ω >. For the grid-connected converter system, the cables and LL-ilters are passive i the closed-loop output admittance G c has non-negative real parts the interactions among the current control and the resonant grids will be stable [9]. But due to the presence o the delay time in the sampling and updating o PWM, a negative part could be introduced in G c. According to (3) (6), the part o G c containing a delay can be expressed as: ( L + L ) C ω G jλt s ω T = = e T KPWM Kp ( C Lω ) ( L + L) Cω = s + KPWM Kp ( C L ω ) G [ cos( λtω) jsin( λtω) ] s () It can be clearly seen rom () that a negative part is probably be presented in the closed-loop output admittance G c. Hence, the passivity o the system is dependent on the ilter parameters and the delay time. According to (), the requency boundaries o the positive or negative are expressed in (), () rc = π ( L + L ) C ( π LC ) () = / () sw s rd = (3) 4λ where sw is the switching requency and (-C L ω ) is always larger than zero beore switching requency. rc is the resonance requency o L, L and C. Hence, it can be deduced that: When < rc < s / (4λ), the system has a negative real part in the requency interval [ rc, s / (4λ)]. When s / (4λ) < rc < r, the system has negative real part in the requency interval [ s / (4λ), rc ]. Fig. 5. Bode plots o the open-loop control gain T o dierent grid inductance. Fig. 6. Bode plots o the closed-loop output admittance G c or dierent resonance requencies rc. When rc = s / (4λ), the system has no negative real part and it is the critical state. B. Stability and Robustness Analysis When λ=.5, the delay is.5t s and rd is s / 6. Fig. 5 shows the Bode plots o the open-loop gain T. The system is stable at the PCC point because the phase crosses -8º beore r due to the delay eect. Re. [7] shows that i the resonance requency is higher than s /6, the resonance peak is not required to be damped below db and i r is the resonance requency is lower than s /6, damping method is necessary to make the system stable. However, the real grid contains the grid impedance. I the grid impedance is inductive and only the grid impedance variation is considered, the resonance requency will be reduced. As shown in Fig. 5, when L g =4 mh, the resonance requency gets close to s /6. Since L is paralleled with L, r is always larger than s /6 i rc is larger than s /6 no matter how the grid inductance varies. Fig. 6 shows the Bode plots o the closed-loop output admittance G c or dierent rc. It can be seen rom Fig. 6 i rc < s /6 the phase o the output admittance is out o [-9, 9] in the requency period [ rc, s / 6]. I s /6< rc the phase o the output admittance is out o [-9, 9] in the requency period [ s / 6, rc ]. I s /6= rc the phase o the output admittance is always between [-9, 9]. This is in good agreement with the negative analysis above. As shown in Fig. 4, i the grid contains cable and the resonant grid L g -C g interacts with 96

5 TABLE II Maximum Harmonic Current Distortion in Percent o I g Individual Harmonic Order (Odd Harmonics) [%] I SC /I L < h<7 7 h<3 3 h<35 35<h THD < the LL-ilter the current control may be destabilized. So it is better to design the L, L and C to make rc to be chosen to s /6 in order to obtain the best robustness and passivity. IV. LL-FILTER PARAMETERS DESIGN PROCEDURE A. Conventioncal Design Constraints When designing a power ilter, the base impedance o the system should be known. The base values o the impedance, the inductance and the capacitance are reerred to as: where U g ω P η 7 Fig. 7. Inverter output phase voltage spectrum. U g Zb =, Cb =, Zb Lb = (4) P ω Z the line-to-line RMS voltage; the grid requency; rated active power. o The ollowing aspects o the design guideline should be satisied [3], [] and []: ) Limit the total inductance (L +L ). The upper limit or the total inductance should be less than. pu in order to limit the dc-link voltage on operation. A higher dc-link voltage will result in higher switching losses and thereby lower eiciency. ) The value o the inverter side inductor (L ). This inductor deals with high requency ripple current and it is constrained by the maximum ripple current. 3) Maximum harmonic distortion o the grid current. The lower limit o the ilter inductance is determined by the harmonic requirement o the grid-injected current according to IEEE 59-99[3], as speciied in Table II. I g is the nominal grid-side undamental current. I SC is the short circuit current o the power system. The harmonic b ω Fig. 8. The relationship between capacitance C, switching requency and delay coeicient λ. currents can be calculated by the corresponding harmonic voltage amplitudes at dierent harmonic requencies. 4) Design o the ilter capacitance. Large capacitance can provide a better high requency harmonics attenuation but it consumes more reactive power. For low voltage converter, it is considered that the maximum power actor variation at rated power is less than 5%, as it is expressed by the value o the base impedance o the system C 5%C b. 5) Resonance requency o the ilter. The resonance requency is assumed to be in a range between ten times the line requency to avoid the major low requency harmonics and one-hal o the switching requency to avoid resonance problems. B. Design Procedure According to the analysis beore, the parameter choices o the LL-ilter are very important to the system stability and robustness. The basic design guideline is given as [4] - [6]: ) Design o inverter-side inductor L. Due to the PWM, the output voltage o the inverter has high requency harmonics as shown Fig. 7. In order to smooth the inverter side current inverter-side inductor L should meet a speciic current ripple requirement. The inductance can be calculated rom equation L Udc /8 s ( αire ), I re is the rated reerence peak current, α is the inverter-side current ripple ratio, which generally is lower than 4% o the rated reerence current [4], [6]; ) Design o capacitor C. According to the analysis, C should be designed by the boundary requency rc. It can be seen rom () the switching requency L C is ixed at the switching requency. When the value o L is ixed C can be chosen according to the delay time and sampling requency. At the same time, the capacitor value should meet the reactive power requirements. Fig. 8 shows the value o the capacitance in phanges with dierent switching requencies and delays. It can be seen rom the igure that the capacitance increases with the switching requency and the delay coeicient λ increasing. The capacitance should statisied the constain that C 5%C b. 96

6 ωc Q Q L C Q 3 Q < Q < Q3 n Fig. 9. Characteristic impedance o the LC trap ilter with dierent L /C. Fig.. The relationship between capacitance, switching requency and total inductance. ωc Q L C Q Q 3 R Q < Q < Q3 Fig.. Characteristic impedance o the LC trap ilter with dierent R. 3) Design o inductor L o L -C circuit. I C is selected L can be chosen based on the switching requency to attenuate the dominant harmonics. The attenuation characteristics are inluenced by the L -C circuit quality. The L -C series resonant circuit quality actor can be taken as: L Q = R C n L (5) = (6) C where R means the sum o the equivalent series inductor resistance and the equivalent series capacitor resistance. It is a parasitic resistance and no external resistance is added to the circuit. Fig. 9 shows the characteristic impedance o the LC trap ilter with dierent L /C. It can be seen rom Fig. 9 that the trap range is wider with the Q-actor increasing. When the range o side-band harmonics around the speciic requency is relatively wide, it should be considered to obtain a larger Q-actor o the LC trap ilter branch to get better harmonic attenuation. Fig. shows that characteristic impedance o the LC trap ilter with dierent R. R will reduce the depth o the LC trap ilter. These actors tend to lower the quality actor. Normally, the Q-actor can be Q 5 [7]. 4) Selection o the grid-side inductor L. When L and C are designed, L should be designed to reduce the harmonic around the double o the switching requencies down to.3% [6], as shown in (5). 4U 3 3 o< r<.5sw Fig.. Flow chart o the parameter design procedure o LL-ilter. dc max ( J( πm), J5( πm )) G( j ωs ) π (7) I re.3% where J (πm) and J 5 (πm) are the Bessel unctions corresponding to the st and 5 th sideband harmonics at the double o the switching requency. Because the dominant harmonics around the switching requency are already attenuated by the LC trap circuit, the value o L can be selected relatively low. The resonance requency will be higher than s /6. 963

7 5) Check o the total inductance (L +L ). Fig. shows the value o the total inductance increases with the switching requency and the capacitance decreasing. It can be seen rom the Fig. that the total inductance increases with the switching requency and the capacitance is reduced. The value o the total inductance should be less than.pu in order to limit the ac voltage drop during operation and also lower the high dc-link voltage. 6) Component tolerance. The inluence o the variation o inductance and capacitance o the LL-ilter will be discussed. The parameter drit o L, L or C may result in the resonance requency rc variation. Generally, or industrial ilters the tolerances are: Capacitors: 5% and no negative tolerance; Inductors: % [8]. Considering the variation o the capacitor and inductor, rc is between [96.9% rc -3% rc ]. (a) C. Design Example A step-by-step procedure has been proposed to obtain the parameter values o the LL-ilter. The system is speciied in Table I. The rated power is 5 kw. Hence, the base impedance Z b is 9. Ω, the base capacitance C b is 9.6 μf and the base inductance is L b 9.5 mh. A design example is shown in Table TABLE III PARAMETERS OF THE FILTER Symbol Deinition Value L Inverter-side inductor. mh L Grid-side inductor.8 mh L Resonant inductor 64 μh C Capacitor 4 μf r Resonant requency.45 khz rc Frequency 67 Hz (b) Fig. 4. Transient experimental results when the grid current reerence is changed. (a) L g =. (b) L g = 4.8mH. III with ollowing steps: ) Based on the constraint o the total inductor and inverter-side current ripple, a 3% current ripple can be obtained to design the inverter inductor L. Then the inverter-side inductor is selected to be. mh. ) Considering.5T d delay and khz switching requency the capacitor value is designed as 4 μf in order to make the requency rc is s /6 and meet the constraint o 5% reactive power. 3) For the L -C resonant circuit, L can be chosen based on the chosen C and the switching requency. It is calculated to be 64 μh. 4) The grid-side inductor value o L can be calculated by the injected grid current harmonics standard. L is selected to be.8 mh. 5) Check the total inductance and the possible variation o real value o rc. (a) THD =.8% (b) Fig. 3. Experimental results o the designed LL-ilter parameters. (a) Grid current waveorm (b) Grid current spectrum. V. EXPERIMENTAL RESULTS The experimental setup consists o a 5 kw Danoss FC3 converter connected to the grid through an isolating transormer and the DC-link supplied by Delta Elektronika power sources. The control algorithm has been implemented in a dspace DS3 real time system. TABLE I shows the experimental parameters. Fig. 3(a) shows the steady state waveorms o the grid current and the L -C trap voltage or the designed parameters when the grid impedance is neglected, L g = mh. Fig. 3(b) shows the grid current spectrum. It can be seen rom the spectrum that the dominant harmonics occur around the double o the switching requency because the switching harmonics have been attenuated by the trap circuit. 964

8 the analysis. Parameters variation also inluences the system robustness and accuracy o the design. 4. A 5 kw grid-connected converter system is implemented to veriy the theoretical analysis on the LL-ilter. It can be ound that the experimental results match the theoretical analysis results well. REFERENCES (b) Fig. 5. Experimental waveorms with L g = mh, C g = 6.7 μf when (a) ollow the design method (b) design method is not ollowed (C = 8 μf). Fig. 4 shows the transient experimental results when the grid current reerence steps in the middle. Fig. 4(b) shows dynamic transition only considering the variation o grid inductance, L g =4.8 mh. The system has a good robustness in the weak grid. Fig. 5 shows the experimental results when grid impedance has the values: L g = mh, C g =6.7 μf. Fig. 5(a) shows the experimental waveorms when the design method is ollowed. rc is designed very close to s /6. Fig. 5(b) shows the experimental waveorms, when the design method is not ollowed and the capacitor C is changed to 8 μf. The LLilter interacts with the L g -C g impedance and make the unstable point to all into the period the phase o the output admittance is out o the period [-9, 9] as discussed in Fig. 6. In that case the system cannot get the passivity. VI. (a) CONUSION This paper proposed a new design method or the LLilter to improve the stability and robustness o a gridconnected system. It can be seen that:. The robustness o the LL-ilter based grid-connected system is mainly inluenced by the inverter-side inductor and the capacitor.. When L, C, L are designed to make rc to be close to the one-sixth o the sampling requency, the sensitive requency period can be disappeared or minimized. The resonance requency o the LL-ilter is always larger than one-sixth o the sampling requency no matter how the grid impedance varies. The current control also has the passivity and good robustness to the grid impedance and cable capacitance. 3. A step by step ilter design method is proposed based on [] M. Liserre, A. Dell'Aquila, and F. Blaabjerg, An overview o threephase voltage source active rectiiers interacing the utility, in Proc. Bologna Power Tech Con., 3, vol. 3, pp. 7. [] F. Blaabjerg, R. Teodorescu, M. Liserre, and A. Timbus, Overview o control and grid synchronization or distributed power generation systems, IEEE Trans. Ind. Electron., vol. 53, no. 5, pp , Oct. 6. [3] M. Liserre, F. Blaabjerg, and S. Hansen, Design and control o an Lilter-based three-phase active rectiier, IEEE Trans. Ind. Appl., vol. 4, no. 5, pp. 8-9, Sep-Oct. 5. [4] M. Huang, W. Wu, Y. Yang, and F. Blaabjerg, Step by Step Design o a High Order Power Filter or Three-Phase Three-Wire Grid-connected Inverter in Renewable Energy System in Proc. PEDG 3, pp.-8. [5] K. Dai, K. Duan, and X. Wang, Yong Kang Application o an LL Filter on Three-Phase Three-Wire Shunt Active Power Filter, in Proc. IEEE INTELEC, Sep., pp. -5. [6] W. Wu, Y. He, and F. Blaabjerg, An LL power ilter or singlephase grid-tied inverter, IEEE Trans. Power Electron., vol. 7, no., pp , Feb.. [7] M. Huang, P. C. Loh, W. Wu, and F. Blaabjerg, Stability Analysis and Active Damping or LL-ilter Based Grid-Connected Inverters, in Proc. IPEC 4, pp [8] J. M. Bloemink and T C. Green, Reducing Passive Filter Sizes with Tuned Traps or Distribution Level Power Electronics, in Proc. IEEE EPE, Aug., pp. -9. [9] S. Parker, B. McGrath, and D.G. Holmes. Regions o Active Damping Control or L Filters, IEEE Trans. Power Electron., vol. 5, no., pp.44-43, Jan. 4. [] C. Zou, B. Liu, S. Duan, and R. Li, Inluence o Delay on System Stability and Delay Optimization o Grid- Connected Inverters with L Filter, IEEE Trans. Ind. Ino., vol., no. 3, pp , Aug. 4. [] X. Wang, F. Blaabjerg, and P. C. Loh, Virtual RC damping o Liltered voltage source converters with extended selective harmonic compensation, IEEE Trans. Power Electron., early access, 4. [] X. Wang, F. Blaabjerg, and P. C. Loh, Analysis and design o gridcurrent-eedback active damping or L resonance in grid-connected voltage source converters, in Proc. IEEE ECCE 4, pp [3] M. Huang, X. Wang, P. C. Loh, and F. Blaabjerg, Resonant-inductorvoltage eedback active damping based control or grid-connected inverters with LL-ilters, in Proc. o ECCE, pp. 94-, 4. [4] W. Wu, Y. He, T. Tang, and F. Blaabjerg, A New Design Method or the Passive Damped L and LL Filter-Based Single-Phase Grid- Tied Inverter, IEEE Trans. Ind. Electron., vol. 6, no., pp , Oct. 3. [5] P. Channegowda and V. John, Filter optimization or grid interactive voltage source inverters, IEEE Trans. Ind. Electron., vol. 57, no., pp , Dec.. [6] S. Zhang, S. Jiang, X. Lu, B. Ge, and F. Z. Peng, Resonance issues and damping techniques or grid-connected inverters with long transmission cable, IEEE Trans. Power Electron., vol. 9, no., pp.-, Jan. 4. [7] O. Brune, Synthesis o a inite two-terminal network whose drivingpoint impedance is a prescribed unction o requency, MIT, Journ.Math. Phy. vol., pp. 9-36,

9 [8] X. Wang, F. Blaabjerg, and W. Wu, Modeling and analysis o harmonic stability in an AC power-electronics-based power system, IEEE Trans. Power Electron., vol. 9, no., pp , Dec. 4. [9] X. Wang, F. Blaabjerg, and P. C. Loh, Proportional derivative based stabilizing control o paralleled grid converters with cables in renewable power plants, in Proc. ECCE 4, , 4. [] A. Riccobono and E. Santi, A novel passivity-based stability criterion (PBSC) or switching converter DC distribution systems, in Proc. IEEE APEC, pp [] Robert Meyer and Axel Mertens, Design o L Filters in Consideration o Parameter Variations or Grid-Connected Converters, in Proc. IEEE ECCE, pp [] A. A. Rockhill, M. Liserre, R. Teodorescu and P. Rodriguez, Grid- Filter Design or a Multimegawatt Medium-Voltage Voltage-Source Inverter, IEEE Trans. Ind. Electron, vol. 58, no. 4,, pp [3] IEEE Recommended Practices and Requirements or Harmonic Control in Electrical Power Systems, IEEE Standard 59-99, 99. [4] J., Yang and F.C. Lee, L Filter Design and Inductor Current Ripple Analysis or 3-level NPC Grid Interace Converter, IEEE Trans. Power Electron., early access, 4. [5] Q. Liu, L. Peng, Y. Kang, S. Y. Tang, D. L. Wu, and Y. Qi, A Novel Design and Optimization Method o an L Filter or a Shunt Active Power Filter, IEEE Trans. Ind. Electron., vol. 6, no. 8, pp. 4-4, Aug. 4. [6] J. Xu, J. Yang, J. Ye, Z. Zhang, and A. Shen, An LT Filter or Three-Phase Grid-Connected Converters, IEEE Trans. Power Electron., vol. 9, no. 8, pp , Aug. 4. [7] J. K. Phipps, A transer unction approach to harmonic ilter design, IEEE Ind. Appl. Mag., vol. 3, no., pp. 68 8, Mar./Apr [8] J.C. Das, Passive Filters Potentialities and Limitations, IEEE Trans. Power Electron., vol. 4, no., pp. 3 4,