# An Improved LCL Filter Design in Order to Ensure Stability without Damping and Despite Large Grid Impedance Variations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. LCL Filter Mathematical Model

_{g}

_{(a,b,c)}(respectively i

_{2(a,b,c)}) refer to the grid voltage vector components (respectively the grid current vector components) in the stationary reference frame, while V

_{i}

_{(a,b,c)}(respectively i

_{i}

_{(a,b,c)}) refer to the components of the output power converter voltage vector (respectively the components of the output power converter current vector) in the stationary reference frame. L

_{i}(respectively L

_{2}) refer to the converter side inductor of the LCL filter (respectively the grid side inductor of the LCL filter), while R

_{i}(respectively R

_{2}) refer to the internal resistance of the converter side inductor, (respectively the internal resistance of the grid side inductor). C

_{f}refer to the LCL filter capacitor, while V

_{c}

_{(a,b,c)}(respectively i

_{c}

_{(a,b,c)}) refer to the voltage across the filter capacitor vector components (respectively the capacitor current vector components) in the stationary reference frame. L

_{g}(respectively R

_{g}) refer to the inductive part of the grid impedance (respectively the resistive part of the grid impedance), while ${L}_{2}^{g}$ (respectively ${R}_{2}^{g}$) refer to the filter grid side inductor in series with the grid inductor (respectively the internal resistance of the filter grid side inductor in series with the internal resistance of the grid inductor).

_{2}and the input voltages V

_{i}and V

_{g}. It is computed based on the superposition principle. The first transfer function is obtained by setting the V

_{g}input equal to zero. According to Equations (1a)–(1c), it is given by (2a). While the second transfer function is obtained by setting the V

_{i}input equal to zero and is given by Equation (2b):

_{1}and F

_{2}and is given by the following equation:

_{n}and i

_{n}are respectively the voltage and current n-harmonic components.

_{2n}and the converter voltage V

_{in}is expressed by Equation (4a). According to this equation, the LCL filter resonance frequency (that corresponds to zero impedance) is given by Equation (4b):

_{in}and the converter voltage V

_{in}can be approximated as in (5a) [35]. According to Equations (4a) and (5a), the transfer function between the grid current i

_{2n}and the converter current i

_{in}for high frequencies, is given by Equation (5b). At the switching frequency, the previous equation becomes equal to (5c):

## 3. LCL Filter Design Methodology

- -
- The line-to-line RMS grid voltage U
_{g}; - -
- The rated active power of the system P;
- -
- The rated frequency of grid voltage f
_{g}; - -
- The switching frequency of the converter f
_{sω}; - -
- The saturation current of the LCL filter inductors I
_{sat}.

#### 3.1. Resonance Frequency Condition

_{res}depends on:

- -
- The filter inductors L
_{i}and L_{2}; - -
- The grid inductor L
_{g}; - -
- The filter capacitor C
_{f}.

_{i}and L

_{2}can be considered constant since their corresponding saturation current is not exceeded. However, the grid inductor L

_{g}can have a large set of values. Based on [17,18,19,20], the ratio R

_{g}/X

_{g}varies according to the grid configuration (low, medium or high voltage lines, wires length…) and conditions (weak or stiff grid). It includes also the leakage inductance of the transformer. The capacitor value has a small error that depends on capacitor accuracy. Low cost capacitors have an accuracy that typically varies between ±5%. The range of resonance frequency variation is given by Equation (6) since the resonance frequency is a decreasing function for both L

_{g}and C

_{f}variables:

_{res}must be higher than 10 times the grid frequency f

_{g}and less than half of the switching frequency f

_{sω}[9]. So, in order to avoid resonance problems due to large grid impedance variations and capacitor values errors, Equation (7) must be verified:

_{c}

_{min}and f

_{c}

_{max}can be defined. The first one (f

_{c}

_{min}) is equal to f

_{sω}/6, while the second one (f

_{c}

_{max}) is equal to f

_{sω}/2. In [31], it is concluded that the PI-based current control can be achieved without active or passive damping if the resonance frequency f

_{res}is inside the interval [f

_{c}

_{min}, f

_{c}

_{max}] as shown in Equation (8):

_{res}should be placed in a stable region as shown in Equation (9), which is deduced from (7) and (8):

#### 3.2. Maximum Value of the Total Inductor

_{TBase}is the base value of the total inductor value and Z

_{Base}is the base impedance [9]. Consequently, the maximum value of the total inductor is expressed by Equation (10d):

#### 3.3. Minimum DC-Link Voltage

_{T}equal to the sum of the two inductor values L

_{i}and L

_{2}. This is mainly due to the fact that the LCL filter is designed so that the capacitor has great impedance value for fundamental signals. Based on Figure 1b and neglecting the influence of different resistors, the relationship between the converter and grid voltages can be expressed in complex form as follows:

_{imax}is given by Equation (12):

_{dcmin}is computed according to Equation (13):

#### 3.4. Maximum LCL Filter Capacitor Value

_{c}denotes the reactive power consumed by the filter capacitor and λ is a positive factor chosen generally equal to or lower than 5% [36,37,38]. According to Equations (14a) and (14b), the maximum value of the filter capacitor can be expressed as in Equation (14c):

#### 3.5. Tuning of the Converter Side Inductor

_{dc}/3 to V

_{dc}/3 [35]. Figure 4 shows a waveform example of the converter current i

_{i}with regard to the applied converter voltage V

_{i}. In this figure, t

_{on}and t

_{off}refer to the time taken by the control signal at high and low logical level, respectively. T

_{sω}is the switching period.

_{i}must verify Equation (16) in order to avoid inductor saturation problems. According to Equations (15b) and (16), the minimum converter side inductor value can be deduced based on Equation (17):

#### 3.6. Tuning of the Grid Side Inductor

_{2}by its expression given by Equation (18a) and supposing the grid inductor L

_{g}equal to zero, Equation (5c) becomes equal to Equation (19a), where δ is the harmonic attenuation rate. It represents the relation between the converter current and the grid current at the switching frequency. The positive solution of Equation (19a) is given by Equation (19b):

_{1}= L

_{i}C

_{f}ω

_{sω}

^{2}− 1. Based on (18a) and (19b), the grid side inductor can be expressed as follows:

_{2}by its expression given by Equation (20), f

_{res}

_{min}(expressed by Equation (6)) becomes equal to (21a). Also, f

_{res}

_{max}(expressed by Equation (6)) becomes equal to (21b):

_{2}= L

_{i}+ a

_{1}L

_{g}

_{max}+ a

_{1}L

_{i}, a

_{3}= (L

_{i}+ a

_{1}L

_{g}

_{max})L

_{i}C

_{f}

_{max}, b

_{2}= L

_{i}+ a

_{1}L

_{g}

_{min}+ a

_{1}L

_{i}, b

_{3}= (L

_{i}+ a

_{1}L

_{g}

_{min})L

_{i}C

_{f}

_{min}. According to Equations (9), (21a) and (21b), f

_{res}

_{min}and f

_{res}

_{max}must verify the conditions expressed by Equations (22) and (23), respectively. These equations provide a condition on the value of the harmonic attenuation rate δ which ensures a resonance frequency variation included in the stable region. On the other hand, the desired harmonic attenuation rate must be greater than a minimum harmonic attenuation rate δ

_{min}that corresponds to a

_{max}. This condition is given by Equation (24):

_{2}is deduced based Equation (18a).

#### 3.7. LCL Filter Parameter Verification

_{Cf}and Z

_{L}

_{2}are the capacitor impedance and the filter grid side inductor impedance, respectively.

#### Algorithm of the LCL filter design methodology

## 4. Case Study for LCL Filter Design

#### 4.1. Application Design Methodology

_{g}equal to 400 V, a rated power P equal to 4 kW, a rated frequency of grid voltage f

_{g}equal to 50 Hz and a switching frequency f

_{sω}equal to 10 kHz. Based on the design methodology presented and detailed in Section 3, the LCL filter parameters are computed as follows:

- -
- Maximum value of the total inductor L
_{T}_{max}

_{T}

_{max}is equal to 12.7 mH.

- -
- Minimum DC-link voltage V
_{dc}_{min}

_{2max}and voltage V

_{g}

_{max}are equal to 10 A and 325 V, respectively. According to Equation (12), the maximum converter voltage V

_{i}

_{max}is equal to 328 V. Consequently, based on Equation (13), the required minimum DC-link voltage V

_{dc}

_{min}is equal to 567 V. We choose V

_{dc}equal to 600 V.

- -
- Maximum LCL filter capacitor value C
_{f}_{max}

_{f}

_{max}is equal to 4 µF. A value of 2 µF is chosen for the LCL filter capacitor.

- -
- Tuning of the converter side inductor L
_{i}

_{sat}is equal to 12 A and the maximum converter current I

_{i}

_{max}is equal to 10 A. So, according to Equation (15a), the maximum converter current ripple Δ

_{i}

_{max}is equal to 4 A. Based on Equation (17), the minimum converter side inductor value is equal to 2.5 mH. A value of 5 mH is chosen for the converter side inductor which presents 40% of the total LCL filter inductor value.

- -
- Tuning of the grid side inductor L
_{2}

_{res}

_{min}is obtained for weak grid conditions (L

_{g}= L

_{g}

_{max}= 13 mH) and maximum capacitor value (C

_{f}= C

_{f}

_{max}), while the maximal value f

_{res}

_{max}is obtained for stiff grid conditions (L

_{g}= 0) and minimum capacitor value (C

_{f}= C

_{f}

_{min}). Based on Equations (22) and (23), the desired harmonic attenuation rate must obey to conditions given by the following equations:

- -
- Resonance frequency

_{i}= 5 mH and L

_{2}= 2 mH). As it is shown in this figure, the range of resonance frequency variation is limited between f

_{c}

_{min}and f

_{c}

_{max}even for the worst case of C

_{f}and L

_{g}. Consequently, for the chosen LCL filter parameters and for the worst case of L

_{g}and C

_{f}, the resonance frequency is placed in a stable region where no damping is required.

_{2(a,b,c)}. It is based on the voltage oriented PI control, which is designed in the dq synchronous reference frame. In this figure, i

_{dq}and V

_{dq}denote respectively the dq-axis current and voltage of dq transformation, while i* and V* denote respectively the reference current and voltage. Since the simplified block diagram of an LCL filter in the dq frame can be considered the same as in the abc frame (by neglecting the decoupling terms on d and q axis) [9], the open and closed loop transfer functions of the whole controlled system are given by Equations (31a) and (31b), respectively:

_{p}and K

_{i}) (used for grid-side current regulation) were tuned according to the optimum criterion method [9].

_{C}transfer function (Equation (31b)) when L

_{g}varies from 0 to 13 mH (with a step of 1 mH). As shown in this figure, the system stability is ensured without damping even for large grid impedance variation. Moreover, since the converter side inductor is realized using the iron-powder core with distributed air gap (Figure 13), its inductance value can change over the time. The robustness of the system against converter side inductor variations was investigated for the obtained filter parameters. To this purpose, Figure 9 shows the Bode diagram of the F

_{O}transfer function (Equation (31a)) when L

_{i}varies from 3.5 mH to 6.5 mH (5 mH ± 30%). According to this figure, for all cases of the L

_{i}variations, the gain margin G

_{m}and phase margin P

_{m}are larger than 19 dB and 45.9 degree, respectively. So, the system stability is ensured even for large converter side inductor variations.

#### 4.2. Simulation Results

_{2a}with regard to the grid voltage V

_{ga}during steady state operation. It is commented that the grid current and the grid voltage are in phase. Hence, the power factor is close to the unity, which corresponds to grid code requirements. In order to test the robustness of the designed LCL filter, additional inductors of 13 mH are inserted in series with the filter grid side inductor. Figure 12b shows the simulation results of the grid current in case of weak grid conditions (L

_{g}= 13 mH). It should be noted that the system remains stable without damping despite of a large variation of the grid inductor value. The obtained simulation results show that, even without damping, the system stability is guaranteed under stiff as well as weak grid conditions. Finally, it should be noted that the obtained LCL filter parameters are relatively small (Table 1) which can not only save money, but also enhance the dynamic response of the system. Simulation results indicate the effectiveness and the robustness of the designed LCL filter and therefore the efficiency of the proposed design methodology.

## 5. Experimental Results

- -
- A 20 kVA three phase high voltage power converter.
- -
- An auto transformer that varies the voltage peak magnitude (in the AC side).
- -
- An LCL filter (composed of three inductors (5 mH/10 A) with an internal resistor of 0.1 Ω, three capacitors (2 µF/400 V) and three inductors (2 mH/10 A) with an internal resistor of 0.1 Ω).
- -
- A capacitor for the dc-link (1100 μF/800 V).
- -
- Measurement board that provides current and voltage measurements.
- -
- Three inductors (4.5 mH/10 A) used in order to emulate the large grid impedance variation.
- -
- The STM32F4-Discovery digital solution, which is used for the implementation of the control algorithm.

_{ia}and the grid current i

_{2a}for L

_{g}= ${L}_{g}^{1}$. The converter current THD is equal to 17%, while the grid current THD is equal to 3.5%. As depicted in Figure 14, the current harmonics and the THD value are reduced using the designed LCL filter. Moreover, the obtained grid current THD is below 5% which meets grid code requirements. Also, the stable operation of the system is ensured without the use of any damping method. Figure 15a,b present the high frequency spectra of the measured converter and grid currents. Based on these figures the largest near switching frequency current harmonic component is equal to 63% on the converter side and 6% on the grid side. Thus, the harmonic attenuation rate δ is equal to 10%.

_{ga}with regard to the grid current i

_{2a}during steady state operation. The power factor is close to the unity since the grid current and voltage are in phase. Figure 16b shows the waveforms of the converter current i

_{ia}and the grid current i

_{2a}for L

_{g}= ${L}_{g}^{1}+{L}_{g}^{2}$. It should be noted, based on this figure, that the stable operation of the system is ensued without damping despite of the large grid impedance variation. Moreover, it can be noted from experimental results that some low-frequency harmonics appear in the measured grid current. However, these odd harmonics could be neglected since they come from sensors noise and the external control loop of the dc-link voltage V

_{dc}[40]. Finally, it is worth noting that the designed LCL filter provides high filtering performances with minimized size, weight, losses and cost. Also the obtained experimental results are quite closely similar to those obtained in simulation. Finally, it should be noted that experimental results indicate the high filtering performances and reliability of the designed LCL filter and therefore the efficiency of the proposed design methodology.

## 6. Conclusions

## Acknowledgments

## Author Contributions

^{®}and experimental results using a prototyping platform. The others co-authors contributed by supervising the research work and by providing facilities.

## Conflicts of Interest

## References

- Liserre, M.; Sauter, T.; Hung, J.Y. Future energy systems: Integrating renewable energy sources into the smart power grid through industrial electronics. IEEE Trans. Ind. Electron. Mag.
**2010**, 4, 18–37. [Google Scholar] [CrossRef] - Roy, N.K.; Pota, H.R. Current status and issues of concern for the integration of distributed generation into electricity networks. IEEE Syst. J.
**2015**, 9, 933–944. [Google Scholar] [CrossRef] - Tsili, M.; Papathanassiou, S. A review of grid code technical requirements for wind farms. IET Renew. Power Gener.
**2009**, 3, 308–332. [Google Scholar] [CrossRef] - Cracium, B.I.; Kerekes, T.; Sera, D.; Teodorescu, R. Overview of recent grid codes for PV power integration. In Proceedings of the 13th International Conference on Optimization of Electrical and Electronic Equipment (OPTIM), Brasov, Romania, 24–26 May 2012; pp. 959–965.
- Lettl, J.; Bauer, J.; Linhart, L. Comparison of different filter types for grid connected inverter. In Proceedings of the 29th Progress in Electromagnetics Research Symposium (PIERS 2011), Marrakesh, Morocco, 20–23 March 2011; pp. 1426–1429.
- Elsaharty, M.A.; Ashour, H.A. Passive L and LCL filter design method for grid-connected inverters. In Proceedings of the 2014 IEEE Conference in Innovative Smart Grid Technologies, Kuala Lumpur, Malaysia, 20–23 May 2014; pp. 13–18.
- Fu, X.; Li, S. A Novel Neural Network Vector Control for Single-Phase Grid-Connected Converters with L, LC and LCL Filters. Energies
**2016**, 9, 328. [Google Scholar] [CrossRef] - Cao, W.; Liu, K.; Ji, Y.; Wang, Y.; Zhao, J. Design of a Four-Branch LCL-Type Grid-Connecting Interface for a Three-Phase, Four-Leg Active Power Filter. Energies
**2015**, 8, 1606–1627. [Google Scholar] [CrossRef] - Liserre, M.; Blaabjerg, F.; Hansen, S. Design and control of an LCL-filter-based three-phase active rectifier. IEEE Trans. Ind. Appl.
**2005**, 41, 1281–1291. [Google Scholar] [CrossRef] - Popescu, M.; Bitoleanu, A.; Preda, A. A new design method of an LCL filter in active dc-traction substations. In Proceedings of the 2016 IEEE International Power Electronics and Motion Control Conference, Varna, Bulgaria, 25–28 September 2016; pp. 876–881.
- Park, M.; Chi, M.; Park, J.; Kim, H.; Chun, T.; Nho, E. LCL-filter design for grid connected PCS using total harmonic distortion and ripple attenuation factor. In Proceedings of the 2010 IEEE International Conference on Power Electronics Conference, Sapporo, Japan, 21–24 June 2010; pp. 1688–1694.
- Wu, Z.; Aldeen, M.; Saha, S. A novel optimization method for the design of LCL filters for three-phase grid-tied inverters. In Proceedings of the 2016 IEEE Innovative Smart Grid Technologies, Melbourne, Australia, 28 November–1 December 2016; pp. 214–220.
- Tang, Y.; Yao, W.; Loh, P.C.; Blaabjerg, F. Design of LCL filters with LCL resonance frequencies beyond the nyquist frequency for grid-connected converters. IEEE J. Emerg. Sel. Top. Power Electron.
**2016**, 4, 3–14. [Google Scholar] [CrossRef] - Yanga, J.; Lee, F.C. LCL filter design and inductor current ripple analysis for a three-level NPC grid interface converter. IEEE Trans. Power. Electron.
**2015**, 30, 4659–4668. [Google Scholar] - Jayalath, S.; Hanif, M. Generalized LCL-Filer Design Algorithm for Grid-connected Voltage Source Inverter. IEEE Trans. Ind. Electron.
**2017**, 64, 1905–1915. [Google Scholar] [CrossRef] - Park, K.B.; Kieferndorf, F.; Drofenik, U.; Pettersson, S.; Canales, F. Weight Minimization of LCL Filters for High Power Converters. IEEE Trans. Ind. Appl.
**2017**. [Google Scholar] [CrossRef] - Pan, D.; Ruan, X.; Bao, C.; Li, W.; Wang, X. Optimized controller design for LCL-type grid-connected inverter to achieve high robustness against grid-impedance variation. IEEE Trans. Ind. Electron.
**2014**, 62, 1537–1547. [Google Scholar] [CrossRef] - Klaus, H.; Klaus-Dieter, D. Elektrische Engergieversorgung, 3rd ed.; Vieweg: Braunschweig, Germany, 2010. [Google Scholar]
- Xu, J.; Xie, S.; Tang, T. Improved control strategy with grid-voltage feedforward for LCL-filter-based inverter connected to weak grid. IET Power Electron.
**2014**, 7, 2660–2671. [Google Scholar] [CrossRef] - He, J.; Wei Li, Y.; Bosnjak, D.; Harris, B. Investigation and active damping of multiple resonances in a parallel-inverter-based microgrid. IEEE Trans. Power Electron.
**2012**, 28, 234–246. [Google Scholar] [CrossRef] - Cehn, C.; Xiong, J.; Lei, J.; Zhang, K. Time delay compensation method based on area equivalence for active damping of LCL-type converter. IEEE Trans. Power Electron.
**2016**, 32, 762–772. [Google Scholar] - Wang, X.; Blaabjerg, F.; Chiang Loh, P. Grid-current-feedback active damping for LCL resonance in grid-connected voltage-source converters. IEEE Trans. Power Electron.
**2016**, 31, 213–223. [Google Scholar] [CrossRef] - Yao, W.; Yang, Y.; Xiaobin, Z.; Blaabjerg, F.; Loh, P.C. Design and Analysis of Robust Active Damping for LCL Filters Using Digital Notch Filters. IEEE Trans. Power Electron.
**2017**, 32, 2360–2375. [Google Scholar] [CrossRef] - Ben Said-Romdhane, M.; Naouar, M.W.; Slama-Blkhodja, I.; Monmasson, E. Robust Active Damping Methods for LCL Filter Based Grid Connected Converters. IEEE Trans. Power Electron.
**2016**. [Google Scholar] [CrossRef] - Lorzadeh, I.; AskarianAbyaneh, H.; Savaghebi, M.; Bakhshai, A.; Guerrero, J.M. Capacitor Current Feedback-Based Active Resonance Damping Strategies for Digitally-Controlled Inductive-Capacitive-Inductive-Filtered Grid-Connected Inverters. Energies
**2017**, 9, 642. [Google Scholar] [CrossRef] - Xin, Z.; Ching, P.; Wang, X.; Blaabjerg, F.; Tang, Y. Highly accurate derivatives for LCL-filtered grid converter with capacitor voltage active damping. IEEE Trans. Power Electron.
**2016**, 31, 3612–3625. [Google Scholar] [CrossRef] - Hyo-Min, A.; Chang-Yeol, O.; Won-Yong, S.; Jung-Hoon, A.; Byoung-Kuk, L. Analysis and design of LCL filter with passive damping circuits for three-phase grid-connected inverters. In Proceedings of the 9th International Conference on Power Electronics and ECCE Asia (ICPE-ECCE Asia), Seoul, Korea, 1–5 June 2015; pp. 652–658.
- Xiongfei, W.; Beres, R.; Blaabjerg, F.; Poh, C. Passivity-based design of passive damping for LCL-filtered voltage source converters. In Proceedings of the 2015 IEEE Energy Conversion Congress and Exposition, Montreal, QC, Canada, 20–24 September 2015; pp. 3718–3725.
- Beres, R.N.; Wang, X.; Blaabjerg, F.; Liserre, M.; Bak, C.L. Optimal Design of High-Order Passive-Damped Filters for Grid-Connected Applications. IEEE Trans. Power Electron.
**2016**, 31, 2083–2098. [Google Scholar] [CrossRef] - Beres, R.N.; Wang, X.; Blaaberj, F.; Liserre, M.; Bak, C.L. Optimal design of High-order passive-damped filters for grid-connected applications. IEEE Trans. Power Electron.
**2016**, 31, 2083–2098. [Google Scholar] [CrossRef] - Parker, S.G.; McGrath, B.P.; Holmes, D.G. Regions of active damping control for LCL filters. IEEE Trans. Ind. Appl.
**2014**, 50, 424–432. [Google Scholar] [CrossRef] - Yi, T.; Changwoo, Y.; Rongwu, Z.; Blaabjerg, F. Generalized stability regions of current control for LCL-filtered grid-connected converters without passive or active damping. In Proceedings of the 2015 IEEE Energy Conversion Congress and Exposition, Montreal, QC, Canada, 20–24 September 2015; pp. 2040–2047.
- Jianguo, W.; Jiu, D.Y.; Lin, J.; Jiyan, Z. Delay-dependent stability of single-loop controlled grid-connected inverters with LCL filters. IEEE Trans. Power Electron.
**2016**, 31, 743–757. [Google Scholar] - Gohil, G.; Bede, L.; Teodorescu, R.; Kerekes, T.; Blaabjerg, F. Line Filter Design of Parallel Inverleaved VSCs for High-Power Wind Energy Conversion Systems. IEEE Trans. Power Electron.
**2015**, 30, 6775–6790. [Google Scholar] [CrossRef] - Rockhill, A.A.; Liserre, M.; Teodorescu, R.; Rodriguez, P. Grid-Filter Design for a Multimegawatt Medium-Voltage Voltage-Source Inverter. IEEE Trans. Power Electron.
**2011**, 58, 1205–1216. [Google Scholar] [CrossRef] [Green Version] - Ben Said-Romdhane, M.; Naouar, M.W.; Slama.Belkhodja, I.; Monmasson, E. Simple and systematic LCL filter design for three-phase grid-connected power converters. Math. Comput. Simul.
**2016**, 130, 181–193. [Google Scholar] [CrossRef] - Ren, B.; Sun, X.; An, S.; Cao, X.; Zhang, Q. Analysis and Design of an LCL Filter for the Three-level Grid-connected Inverter. In Proceedings of the 7th International Conference on Power Electronics and Motion Control, Harbin, China, 2–5 June 2012; pp. 2023–2027.
- Sanatkar-Chayjani, M.; Monfared, M. Design of LCL and LLCL filters for single-phase grid connected converters. IET Power Electron.
**2016**, 9, 1971–1978. [Google Scholar] [CrossRef] - 519-1992-IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems; The Institute of Electrical and Electronics Engineers Inc.: New York, NY, USA, 1993.
- Karimi-Ghartemani, M.; Khajehoddin, S.A.; Jain, P.; Bakhshai, A. A systematic approach to DC-Bus control design in single-phase grid-connected renewable converters. IEEE Trans. Power Electron.
**2013**, 28, 3158–3166. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Power circuit of the three phase grid connected power converter with LCL filter; (

**b**) single phase equivalent circuit; (

**c**) block diagram of the LCL filter.

**Figure 9.**Bode diagram of the open loop system when L

_{i}varies from 3.5 mH to 6.5 mH (5 mH ± 30%).

**Figure 10.**Simulation results during steady state operation (

**a**) power converter current i

_{ia}response; (

**b**) grid current i

_{2a}response.

**Figure 12.**Simulation results during steady state operation (

**a**) grid voltage V

_{ga}and current i

_{2a}waveforms (

**b**) grid current i

_{2a}response for L

_{g}= 13 mH.

**Figure 14.**Measured power converter current i

_{ia}(1 A/100 mV) and grid current i

_{2a}(1 A/100 mV) for L

_{g}= ${L}_{g}^{1}$.

**Figure 16.**(

**a**) Grid voltage V

_{ga}(50 V/100 mV) and current i

_{2a}(1 A/100 mV) waveforms at steady state operation; (

**b**) measured power converter current i

_{ia}(1 A/100 mV) and grid current i

_{2a}(1 A/100 mV) for L

_{g}= ${L}_{g}^{1}+{L}_{g}^{2}$.

Parameter | Value | |
---|---|---|

System | U_{g} | 400 V |

P | 4 kW | |

f_{sω} | 10 kHz | |

f_{g} | 50 Hz | |

V_{dc} | 600 V | |

LCL filter | C_{f} | 2 µF |

L_{i} | 5 mH | |

L_{2} | 2 mH | |

PI controller | K_{p} | 2.4 |

K_{i} | 592 | |

Grid inductance | L_{g} | L_{g}_{min} = 0 mH and L_{g}_{max} = 13 mH |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Said-Romdhane, M.B.; Naouar, M.W.; Belkhodja, I.S.; Monmasson, E.
An Improved LCL Filter Design in Order to Ensure Stability without Damping and Despite Large Grid Impedance Variations. *Energies* **2017**, *10*, 336.
https://doi.org/10.3390/en10030336

**AMA Style**

Said-Romdhane MB, Naouar MW, Belkhodja IS, Monmasson E.
An Improved LCL Filter Design in Order to Ensure Stability without Damping and Despite Large Grid Impedance Variations. *Energies*. 2017; 10(3):336.
https://doi.org/10.3390/en10030336

**Chicago/Turabian Style**

Said-Romdhane, Marwa Ben, Mohamed Wissem Naouar, Ilhem Slama Belkhodja, and Eric Monmasson.
2017. "An Improved LCL Filter Design in Order to Ensure Stability without Damping and Despite Large Grid Impedance Variations" *Energies* 10, no. 3: 336.
https://doi.org/10.3390/en10030336