# Stability Assessment of Current Controller with Harmonic Compensator for LCL-Filtered Grid-Connected Inverter under Distorted Weak Grid

^{*}

*Applied Sciences*: Invited Papers in Electrical, Electronics and Communications Engineering Section)

## Abstract

**:**

## 1. Introduction

- (1)
- The two typical current controllers with harmonic compensators for the LCL-filtered grid-connected inverter are implemented to analytically investigate their performances under distorted weak grid by means of the stability assessment tools and comprehensive evaluation results.
- (2)
- By the movement of closed-loop poles and disturbance rejection responses, the stability margin of each controller is well investigated. It is clearly addressed that the stability is weakened under the grid impedance variation by the addition of harmonic resonant controllers. The theoretical results are validated by simulation and experiments.
- (3)
- The full-state feedback current control method with augmented harmonic resonant compensators has well proved its robustness for a wide range of grid impedance variations (up to 14 times of grid-side inductors in the high region) by theoretical analysis and evaluation results.
- (4)
- In order to validate the presented theoretical analyses, comprehensive simulation and experimental results based on 2 kVA grid-connected inverter are presented under the grid environment including both uncertain grid impedance and distorted harmonics.

## 2. System Description and Current Controller

#### 2.1. System Model of Grid-Connected Inverter

_{DC}denotes the DC-link voltage; R

_{1}, R

_{2}, L

_{1}, and L

_{2}are the filter resistances and filter inductances, respectively; C

_{f}is the filter capacitance, and L

_{g}is the grid inductance due to weak grid. When the grid impedance does not exist in Figure 1, inverter system can be expressed mathematically in the synchronous reference frame (SRF) as [15,16]:

_{1}is the inverter-side current, i

_{2}is the grid-side current, v

_{c}is the capacitor voltage, v

_{i}is the inverter output voltage, and ω is the grid angular frequency. The system matrices

**A**,

**B**,

**C**, and

**D**are expressed as:

_{g}. To follow the term in the conventional works, the term “the grid impedance” is used in this study. On the other hand, to simply represent the quantity, the grid impedance variation is denoted by L

_{g}variation.

_{s}of 10 kHz as follows:

#### 2.2. Direct Current Control Based on Capacitor Current Damping

_{g}. In this scheme, the active damping is realized through the virtual resistance based on capacitance current to achieve stable grid current control loop. In Figure 2, the function of the proportional gain K

_{c}is the same as the virtual resistance in capacitor branch for the purpose of restraining the resonance of the LCL filter.

#### 2.3. Integral-Resonant State Feedback Control

_{g}. To ensure asymptotic reference tracking as well as disturbance rejection for the harmonics in the orders of 6th and 12th in the SRF, the integral and resonant control terms are augmented in the state feedback control. In the discrete-time state-space, the integral and resonant terms are expressed as [16,31]:

**A**

_{ci},

**B**

_{ci},

**A**

_{ch}, and

**B**

_{ch}are expressed as:

**R**is a positive definite matrix.

**Q**and

**R**are selected by an iterative selection process and verified by both the simulation and experimental results. Furthermore, to implement full-state feedback current controller, a full-state observer is employed in the stationary frame to estimate the system states without installing an extra sensing device [16].

## 3. Stability Analysis under Weak Grid

_{R}= ω

_{R}/2π) and critical frequency defined as 1/6 of the switching frequency. Particularly, three LCL filter designs can be considered: the LCL filter has the resonance frequency higher than the critical frequency, around the critical frequency, and lower than the critical frequency. The system stability is analyzed by investigating the closed-loop eigenvalues of the inverter system with two presented controllers under the grid impedance variation in distorted weak grid. Finally, the design guidelines for the presented current controllers are given to achieve both system dynamic performance and strong robustness under weak grid conditions. It is worth noting that other negative effects of weak grid such as unbalanced grid voltages or grid frequency variation are beyond the scope of this paper.

#### 3.1. Frequency Response of LCL Filter under Grid Impedance Change

_{2}and L

_{g}is combined as L

_{2g}= L

_{2}+ L

_{g}, the transfer function from the inverter voltage to the grid-side current is obtained from Figure 2 as [5,30]:

_{g}under weak grid condition. In fact, as the grid inductance is increased, the resonance peak is shifted toward low frequency region. Thus, since the variation of L

_{g}is often uncertain and unpredictable, the damping method designed at a fixed frequency is not effective under weak grid condition.

_{g}= 0 mH) to very weak grid (L

_{g}= 21 mH).

_{g}is varied in the LCL filter parameter of Case 1. As the grid inductance is increased, the resonance frequency of the LCL filter is reduced and shifted toward the critical frequency. It is worthwhile to note that the LCL filter in Case 1 has the resonance frequency quite far from the critical frequency (high region). Even under severe weak grid condition, this resonance frequency is not shifted to the low region. On the contrary, the LCL filter in Case 2 represents that f

_{R}possibly moves from the high region to the low region depending on the value of L

_{g}. Finally, the LCL filter parameter set in Case 3 is also employed to validate the controllers under the low region.

#### 3.2. Closed-Loop Stability of Direct Grid Current Control with Active Damping under Grid Impedance Change

_{p}

_{1}, and five resonant gains at the fundamental and selected harmonic frequencies K

_{r}

_{1}, K

_{r}

_{5}, K

_{r}

_{7}, K

_{r}

_{11}, and K

_{r}

_{13}. The controller gains are selected in the stiff grid condition. Under weak grid, the closed-loop pole locations designed in the stiff grid move from the designed locations as L

_{g}is increased.

_{g}is increased from stiff grid to weak grid. The chosen LCL filter parameters produce the resonance frequencies higher than the critical frequency. To suppress the current harmonic distortion caused by distorted grid, the PR controllers are incorporated in the orders of 5th, 7th, 11th, and 13th in the stationary frame. In spite of large variation of L

_{g}, most of the closed-loop system poles remain in the stable region as is shown in Figure 6. On the other hand, the poles of the PR controllers in 11th and 13th orders move toward the stability boundary of the unit circle as L

_{g}is increased. Eventually, when L

_{g}is increased larger than 7 mH, the system operation becomes unstable. This fact indicates that this control method is very poor to mitigate the current harmonics in the presence of distorted grid and grid impedance variation.

_{f}values of 10 μF, and 30 μF, respectively, which produce the resonance frequencies around the critical frequency, and lower than the critical frequency. Similar to Figure 6, the resonant controllers are incorporated in the orders of 5th, 7th, 11th, and 13th in the stationary frame, and L

_{g}is varied from 0 to 10 mH. As the resonance frequency is selected in low frequency region, the system is more vulnerable to the uncertainty in L

_{g}as shown in these two figures. In Figure 7, one of the resonant controller poles in 11th and 13th leaves the stability boundary before L

_{g}reaches 2 mH. In low resonance frequency region of the LCL filter in Figure 8, the resonant controller poles in 11th and 13th always remain outside of the stability region, which indicates that this control scheme fails to stabilize the system in the presence of the uncertainty of grid impedance under distorted grid.

#### 3.3. Closed-Loop Stability of Integral-Resonant State Feedback LQR Control under Grid Impedance Change

_{g}is gradually increased from stiff grid to weak grid. While most of the closed-loop system poles remain in the stable region in spite of large variation of L

_{g}, the poles of the 6th resonant controller move toward the stability boundary of the unit circle as L

_{g}is increased. However, as compared with Figure 6 which shows unstable poles at 10 mH of L

_{g}, Figure 9 shows that the system is only unstable when L

_{g}is increased beyond 14 mH.

_{f}values of 10 μF and 30 μF, respectively, when L

_{g}is varied from stiff grid to weak grid. These filter capacitors produce the resonance frequencies around the critical frequency, and lower than the critical frequency. Similarly, as the grid impedance increases, the poles of the 6th resonant controller move outside the unit circle. However, in the integral-resonant state feedback control, the stability limit is much extended since the instability occurs with larger value of L

_{g}than the direct grid current controller. As a result, the integral-resonant state feedback control can be regarded as more robust to uncertainty in the grid impedance change. In addition, it is confirmed that the current control is more likely to be unstable for small variation of L

_{g}as the resonance frequency of the LCL filter gets smaller.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF without the grid impedance under three different grid frequencies of 60, 55, and 50 Hz. As shown in this figure, the closed-loop poles are almost overlapped in each frequency value, and are maintained in the stable region regardless of the frequency change.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF from stiff grid to very weak grid is shown in Figure 13a. The disturbance rejection response also exposes the reason of the system instability when the grid impedance increases. As seen from Figure 13a, the bandwidth of the harmonic compensators is significantly reduced when the grid is weaker. Moreover, the high peak exceeds 0 dB when L

_{g}reaches 21 mH, which causes the instability in system. The interpretation from the frequency response shows the maximum stability margin of L

_{g}is 14 mH. Similar conclusions are inferred from Figure 13b,c, in which the maximum stability margins in the critical and low regions are 7 mH and 4 mH, respectively. The theoretical analysis will be validated by the simulation and experiment in the next section.

_{g}< 4 mH in Case 1, and in the region of L

_{g}< 2 mH in Case 2. In Case 3, the direct grid current controller cannot stabilize the inverter system. On the other hand, the integral-resonant state feedback current controller in Figure 14b greatly extends the stability limit for three LCL filter cases. The inverter system is stable in the region of L

_{g}< 14 mH in Case 1, in the region of L

_{g}< 7 mH in Case 2, and in the region of L

_{g}< 4 mH in Case 3. As a result, a relative stability is enhanced in the LCL filter design having high resonance frequency. Moreover, integral-resonant state feedback control has an improved relative stability than direct grid current control in view of the harmonic compensation and effective resonance damping.

## 4. Performance Assessment under Distorted Weak Grid Condition

#### 4.1. System Configuration

#### 4.2. Simulation Results

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages without grid impedance in Case 1. Figure 17a shows three-phase distorted grid voltages which contain the 5th, 7th, 11th, and 13th harmonics with the magnitude of 5% of the fundamental component. Figure 17b,c show grid-side three-phase current waveforms and the fast fourier transform (FFT) result for a-phase current with the harmonic limits specified by the grid interconnection regulation IEEE Std. 1547 [36]. As is clearly shown, the current harmonic distortion caused by distorted grid is well suppressed and the total harmonic distortion (THD) value of current is 3.59%, which represents that this controller effectively deals with the resonance of the LCL filter and low-order harmonic disturbance in grid voltages.

_{g}is suddenly increased to 7 mH in Case 1 at 0.5 s under the same distorted grid condition. With this value of L

_{g}, the poles of the PR controllers in 11th and 13th orders for harmonic suppression are located outside of the unit circle. As a result, grid phase-currents are gradually oscillating, and eventually, the entire system becomes unstable.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages without the grid impedance in Case 2. Decrease of the resonance frequency improves the harmonic suppression of the LCL filter. Thus, the grid current quality in Figure 19 is much improved as compared with Figure 17, producing only the THD of 2.54%. However, reducing the resonance frequency makes the inverter resonant controller to be more vulnerable to uncertain grid impedance.

_{g}has a step change from 0 to 2 mH in Case 2 at 0.5 s with the same conditions and control parameters as Figure 19. Currents become unstable as soon as uncertain grid impedance is applied. Figure 19 and Figure 20 show a strong agreement with the stability analysis in Figure 7. Obviously, with the LCL filter designed at lower frequency band, the direct current controller including the PR compensators is more vulnerable to uncertain grid impedance.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under ideal grid voltages without the grid impedance in Case 3. Under the ideal grid voltages, this scheme provides reasonable current waveforms without the harmonic resonant controllers incorporated in the orders of 5th, 7th, 11th, and 13th. However, as shown in Figure 21, as soon as the resonant controllers start at 0.5 s, the system instantly becomes unstable, which accords closely with the stability analysis given in Figure 8.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF without the grid impedance in Case 1. Test grid voltages are the same as Figure 17a. As can be clearly observed in Figure 22, the grid currents are quite sinusoidal without negative impact from the distorted grid voltages, resulting in the THD value of 3.96%.

_{g}= 0. It is worth noting that the measured grid voltages in Figure 23a are different from Figure 17a since L

_{g}produces additional inductive voltage drop by the grid current as is shown in Figure 15. Clearly, the instability of the direct current controller is observed in Figure 18 with the same level of weak grid. On the contrary, the output currents of the full-state feedback controller in Figure 23a shows stable sinusoidal waveforms with the THD value reduced to 2.16%. The full-state feedback control is tested further under more severe weak grid conditions with L

_{g}increased to 14 mH in Figure 24. Though the voltages at the PCC contain more distorted harmonics due to the weak grid, the current controller can still produce stable high-quality injected currents with the THD of 2.09%. These results well match up the stability analysis given in Figure 9. In terms of current quality, the THD value of current is smaller as L

_{g}is increased. This complies well with the frequency responses in Figure 5, in which the resonance frequency of the LCL filter is smaller as L

_{g}is increased.

_{g}is increased to 21 mH in Case 1 under distorted grid. In this Case, the system loses the stability. From the simulation tests in Figure 18 and Figure 25, it is confirmed that the use of the state feedback controller provides a more flexible option to design a current control of an LCL-filtered inverter system in the environment of uncertain grid impedance under distorted weak grid condition because the stability region is extended.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF. Figure 26 shows the simulation results under distorted grid voltages without the effect of the grid impedance in Case 2. The grid current waveforms are satisfactory with the THD of 3.86%.

_{g}of 7 mH in Case 2 under the same voltage conditions of Figure 17a. As shown in Figure 10, the poles of the closed-loop current control still remain inside the stable region when L

_{g}varies from 0 to 7 mH. As a result, high-quality sinusoidal grid-injected currents can be obtained, which well demonstrates the validity of the stability analysis. The THD value is much smaller than that of Figure 26 due to additional inductance.

_{g}increases to 14 mH in Case 2. According to the analysis in Figure 10, the integral-resonant state feedback controller designed for the LCL parameter set of Case 2 shows unstable currents for this value of the grid impedance.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF are given. The state feedback current controller is designed for the given LCL filter set to produce a good grid-side current at the stiff grid condition as shown in Figure 29. However, as presented in Figure 11, low region resonance frequency of the LCL filter is extremely sensitive to uncertainty caused by the weak grid condition, causing the instability by only small grid impedance change.

_{g}is suddenly increased from 3 mH to 7 mH in Case 3 at 0.5 s. While stable currents are observed with 3 mH of L

_{g}, the change to 7 mH of L

_{g}produces unstable grid currents.

#### 4.3. Experimental Results

_{g}= 7 mH and L

_{g}= 14 mH are applied with the same conditions of Figure 31a. An external inductor is used to emulate the grid impedance as in Figure 15. Similar to the simulation in Figure 23, it is observed that the measured grid voltages are altered due to the inductive voltage drop by the grid current and L

_{g}. In addition, the harmonic rejection of the inverter is greatly enhanced with additional grid impedance.

_{g}reaches 21 mH as shown in Figure 34. This also accords closely with the stability analysis in Figure 9 and the simulation in Figure 25. Figure 35, Figure 36 and Figure 37 represent the experimental results of grid-side three-phase currents by the integral-resonant state feedback controller under the weak grid condition (L

_{g}= 0 to 14 mH) with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF as the LCL parameters. The current controller is also designed by using the given LCL filter set at stiff grid condition with distorted grid voltage of the 5th, 7th, 11th, and 13th harmonics. Until L

_{g}increases to 14 mH, the inverter system maintains the stability, giving desirable grid currents. However, as L

_{g}becomes larger than 14 mH, the grid currents become unstable, which is well matched to the simulation results.

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF as the LCL parameters. This LCL filter has the resonance frequency smaller than the critical frequency. It is shown that only 7 mH of L

_{g}makes the inverter system unstable.

_{g}.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Block diagram of the LQR (linear quadratic regulator) -based integral-resonant state feedback current controller.

**Figure 4.**Frequency responses of LCL filter parameters without grid impedance for Cases 1, 2, and 3.

**Figure 5.**Frequency responses of LCL filter with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under grid impedance variation (L

_{g}= 0 to 21 mH).

**Figure 6.**Location of the closed-loop poles for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid and L

_{g}variation.

**Figure 7.**Location of the closed-loop poles for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid and L

_{g}variation.

**Figure 8.**Location of the closed-loop poles for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under distorted grid and L

_{g}variation.

**Figure 9.**Location of the closed-loop poles for augmented integral-resonant state feedback LQR controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid and L

_{g}variation.

**Figure 10.**Location of the closed-loop poles for augmented integral-resonant state feedback LQR controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid and L

_{g}variation.

**Figure 11.**Location of the closed-loop poles for augmented integral-resonant state feedback LQR controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under distorted grid and L

_{g}variation.

**Figure 12.**Location of the closed-loop poles for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under different grid frequencies of 60, 55, and 50 Hz.

**Figure 13.**Frequency responses for the augmented integral-resonant state feedback LQR controller under grid impedance variation: (

**a**) Case 1; (

**b**) Case 2; (

**c**) Case 3.

**Figure 14.**Comparison of the stability limit under the grid impedance variation for different LCL parameter sets: (

**a**) Direct grid current controller; (

**b**) Integral-resonant state feedback LQR controller.

**Figure 17.**Simulation results for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages without grid impedance in Case 1: (

**a**) Distorted grid voltages; (

**b**) Grid-side three-phase currents; (

**c**) FFT result for a-phase grid-side current.

**Figure 18.**Simulation result for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages and L

_{g}= 7 mH in Case 1.

**Figure 19.**Simulation result for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages without grid impedance in Case 2.

**Figure 20.**Simulation result for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages and L

_{g}= 2 mH in Case 2.

**Figure 21.**Simulation result for direct grid current controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under ideal grid voltages without grid impedance in Case 3.

**Figure 22.**Simulation results for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages without grid impedance in Case 1: (

**a**) Grid-side three-phase currents; (

**b**) FFT result for a-phase grid-side current.

**Figure 23.**Simulation results for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages and L

_{g}= 7 mH in Case 1: (

**a**) Distorted grid voltages; (

**b**) Grid-side three-phase currents.

**Figure 24.**Simulation result for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages and L

_{g}= 14 mH in Case 1.

**Figure 25.**Simulation result for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages and L

_{g}= 21 mH in Case 1.

**Figure 26.**Simulation result for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages without grid impedance in Case 2.

**Figure 27.**Simulation result for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages and L

_{g}= 7 mH in Case 2.

**Figure 28.**Simulation result of grid-side three-phase currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages and L

_{g}= 14 mH in Case 2.

**Figure 29.**Simulation result for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under distorted grid voltages without grid impedance in Case 3.

**Figure 30.**Simulation result for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under distorted grid voltages when L

_{g}is changed from 3 mH to 7 mH in Case 3.

**Figure 31.**Experimental results for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages without grid impedance in Case 1: (

**a**) Distorted grid voltages; (

**b**) Grid-side three-phase currents.

**Figure 32.**Experimental results for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages and L

_{g}= 7 mH in Case 1: (

**a**) Distorted grid voltages; (

**b**) Grid-side three-phase currents.

**Figure 33.**Experimental result of grid-side currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages and L

_{g}= 14 mH in Case 1.

**Figure 34.**Experimental result of grid-side currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 4.5 μF under distorted grid voltages and L

_{g}= 21 mH in Case 1.

**Figure 35.**Experimental result of grid-side currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages without grid impedance in Case 2.

**Figure 36.**Experimental result of grid-side currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages and L

_{g}= 7 mH in Case 2.

**Figure 37.**Experimental result of grid-side currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 10 μF under distorted grid voltages and L

_{g}= 14 mH in Case 2.

**Figure 38.**Experimental result of grid-side currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under distorted grid voltages without grid impedance in Case 3.

**Figure 39.**Experimental result of grid-side currents for integral-resonant state feedback controller with L

_{1}= 1.7 mH, L

_{2}= 1.0 mH, and C

_{f}= 30 μF under distorted grid voltages and L

_{g}= 7 mH in Case 3.

**Table 1.**LCL (inductive-capacitive-inductive) filter parameters with grid impedance under weak grid.

Cases | LCL Filter Parameters | f_{R} |
---|---|---|

Case 1 | L_{1} = 1.7 mH, L_{2} = 1.0 mH, C_{f} = 4.5 µF | 2991 Hz |

Case 2 | L_{1} = 1.7 mH, L_{2} = 1.0 mH, C_{f} = 10 µF | 2006 Hz |

Case 3 | L_{1} = 1.7 mH, L_{2} = 1.0 mH, C_{f} = 30 µF | 1158 Hz |

Parameters | Symbol | Value | Units |
---|---|---|---|

DC-link voltage | V_{DC} | 400 | V |

Filter resistance | R_{1}, R_{2} | 0.5 | Ω |

Nominal filter capacitance | C_{f} | 4.5 | μF |

10.0 | μF | ||

30.0 | μF | ||

Filter capacitor resistance | R_{cf} | 16 | mΩ |

Nominal inverter-side filter inductance | L_{1} | 1.7 | mH |

Nominal grid-side filter inductance | L_{2} | 1.0 | mH |

Grid voltage (line-to-line rms) | e | 220 | V |

Nominal grid frequency | f_{g} | 60 | Hz |

LCL Filter | Grid Inductance | THD in Simulation | Harmonic Magnitude in Experiment Integral-Resonant State Feedback Control | ||||
---|---|---|---|---|---|---|---|

Integral-Resonant State Feedback Control | Direct Grid Current Control | 2 h | 5 h | 7 h | 11 h | ||

Case 1 | L_{g} = 0 mH | 3.96% | 3.59% | <0.8% | <0.4% | <0.2% | <0.1% |

L_{g} = 7 mH | 2.16% | - | <1.5% | <0.2% | <0.1% | <0.1% | |

L_{g} = 14 mH | 2.09% | - | <0.8% | <0.6% | <0.9% | <0.1% | |

L_{g} = 21 mH | - | - | - | - | - | - | |

Case 2 | L_{g} = 0 mH | 3.86% | 2.54% | <0.7% | <0.1% | <0.2% | <0.1% |

L_{g} = 7 mH | 1.12% | - | <2.2% | <0.1% | <0.2% | <0.1% | |

L_{g} = 14 mH | - | - | |||||

Case 3 | L_{g} = 0 mH | 3.04% | - | <0.7% | <0.2% | <0.2% | <0.1% |

L_{g} = 7 mH | - | - |

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## Share and Cite

**MDPI and ACS Style**

Yoon, S.-J.; Tran, T.V.; Kim, K.-H.
Stability Assessment of Current Controller with Harmonic Compensator for LCL-Filtered Grid-Connected Inverter under Distorted Weak Grid. *Appl. Sci.* **2021**, *11*, 212.
https://doi.org/10.3390/app11010212

**AMA Style**

Yoon S-J, Tran TV, Kim K-H.
Stability Assessment of Current Controller with Harmonic Compensator for LCL-Filtered Grid-Connected Inverter under Distorted Weak Grid. *Applied Sciences*. 2021; 11(1):212.
https://doi.org/10.3390/app11010212

**Chicago/Turabian Style**

Yoon, Seung-Jin, Thuy Vi Tran, and Kyeong-Hwa Kim.
2021. "Stability Assessment of Current Controller with Harmonic Compensator for LCL-Filtered Grid-Connected Inverter under Distorted Weak Grid" *Applied Sciences* 11, no. 1: 212.
https://doi.org/10.3390/app11010212