A Gray-Box Impedance Reshaping Method of Grid-Connected Inverter for Resonance Damping

Oscillation phenomenon of grid-connected inverter can be caused due to time-varying grid impedance. Terminal output impedance of inverter can be reshaped by re-tuning internal parameters of inverter to mitigate the oscillation issue. However, it's not easy to obtain internal parameters of grid-connected inverter in practical operation. This paper presents a gray-box parameters identification method to identify the parameters of LCL filter, the proportional coefficient of current controller and capacitor-current-feedback coefficient from the measured terminal impedance frequency responses by using vector fitting algorithm. When instability phenomenon of multiple-paralleled inverters system occurs, the internal parameters of each inverter are first identified by using the proposed method. Then, the capacitor-current-feedback coefficients are re-tuned to stabilize the studied power system according to the impedance-based stability criterion. The re-designed parameters for the specific grid condition can be delivered to the manufacturers for further parameters modification. Simulation results are given to validate the proposed gray-box parameter identification and corresponding impedance reshaping method.


I. INTRODUCTION
Offshore wind power has been increasingly exploited due to abundant wind sources and constant wind speed [1].Voltage source converter (VSC) with LCL filter is widely adopted as an important interface to deliver electricity to utility grid [2].However, high-frequency oscillation phenomena can be triggered because of interaction between inner control loop and time-varying grid impedance [3]- [7].Therefore, stability assessment methods and damping control strategies have been developed to deal with the oscillation phenomena.
The damping methods for single VSC have been widely presented in previous work, including passive damping [8], filter-based active damping [9], [10], and multi-loop-based active damping methods [11]- [13].The passive damping methods tend to cause extra power loss and decrease system efficiency due to damping resistor [8].Also, filter-based damping methods such as notch filter should align resonant frequency of LCL filter exactly with the center frequency of the notch filter, which is not effective once the parameters of LCL filter are perturbated [9], [10].The multi-loop-based active damping methods are able to enhance stability by introducing additional control loop such as capacitor-currentfeedback active damping loop, which have been paid much attentions [11]- [13].Therefore, VSC with capacitor-currentfeedback active damping is studied in this paper.
The effect of capacitor-current-feedback active damping on output impedance of VSC is investigated in [13].However, re-tuning strategy of controller parameters is not further concerned to address stability under complicated grid conditions.In [14]- [16], controller parameters of active damping loop are tuned based on passivity-based analysis method to enhance system stability considering long transmission cable.And the design guidelines of controller parameters for these active damping methods are given.However, the accurate parameters of LCL filter and controller are important, which is difficult to be obtained from power plant operator due to industrial confidence.
Estimation and identification methods of controller and filter parameters have been proposed in [17]- [22].In [17]- [19], the equivalent inductance and resistance of VSC are estimated to support optimization design of controller parameters, so that a desirable dynamic and steady performance can be achieved.However, no specific filter or controller parameters are estimated.The parameters of LCL filter are identified in [20].In addition, the parameters of current controller and phase-locked loop (PLL) are identified with the assumption that the parameters of LCL filter are known in [21].However, both [20] and [21] cannot simultaneously identify parameters of LCL filter and current controller.A two-step parameter identification method is proposed in [22], where the parameters of voltage/current controllers and filter inductance are identified.However, the identification method is not effective for LCL-filtered VSC.
This paper presents a gray-box impedance reshaping method for VSC, where the parameters of LCL filter, current control loop and capacitor-current-feedback loop are first identified from the measured terminal impedance frequency responses by vector fitting (VF) algorithm.If instability phenomenon occurs when the VSC is connected with weak grid, parameters of current control loop and active damping loop are re-tuned Z L g L Fig. 1.Configuration of the studied power system.
to reshape the output impedance of the VSC, so that the undesired impedance interaction between the VSC and weak grid can be avoided, and the system can be brought to be stable.

II. SYSTEM DESCRIPTION AND IMPEDANCE MODELLING
In this section, the configuration of the studied power system is first described.Then, impedance model of VSC with capacitor-current-feedback active damping is established.

A. Configuration of the Studied Power System
Fig. 1 shows configuration of the studied power system with three paralleled VSCs.The control diagram of current control loop for each VSC is shown in Fig. 2, where LCL filter is used to attenuate high-frequency current ripples.PLL is used to synchronize inverter with utility grid.K vk is the capacitorcurrent-feedback coefficient.G cdqk is the transfer function of current controller in dq reference frame.G dk represents time delay (1.5 T sk ) including computation delay and PWM delay.
Stability of the studied power system as shown in Fig. 1 can be assessed by identifying the impedance ratio Z L /Z s = Z L /(Z inv1 //Z inv2 //Z inv3 ), where Z inv1 , Z inv2 and Z inv3 are output impedance of #1VSC, #2VSC and #3VSC, respectively.When instability phenomenon occurs, output impedance of each VSC can be reshaped by retuning K vk and K pk to stabilize the system.

B. Impedance Model of VSC with Capacitor-Current-Feedback Active Damping
Fig. 3 shows control diagram of grid-connected inverter with capacitor-current-feedback active damping.Output impedance of the VSC without consideration of capacitor-currentfeedback active damping is first derived by neglecting the red dashed feedback line, shown as (1) [23].The active damping loop can be equivalent to a virtual impedance Z vk paralleled to C f k [24].The equivalent virtual impedance can be derived as (2).
Therefore, output impedance of the #kVSC with consideration of capacitor-current-feedback active damping can be obtained by modifying the output impedance formula in (2), shown as (3) on the next page.
A pure resistor R vk is defined as (4).

III. THE PROPOSED GRAY-BOX IMPEDANCE RESHAPING METHOD
In this section, the gray-box parameter identification and controller re-tuning method is first developed, and the implementation procedure is then given.

A. The Proposed Parameters Identification Method
The parameters of the LCL filter L f k1 , L f k2 and C f k , the proportional coefficient of the current controller K pk and the sampling period T sk have been identified from the measured terminal impedance frequency responses using VF algorithm in [25].To implement the identification method considering capacitor-current-feedback loop, the parameter identification approach should be further developed.
Terminal impedance frequency responses can be fitted in s domain using vector fitting (VF) algorithm [26], as shown in (6).where the numerator and denominator are both in the form of M -order polynomial transfer function, and Z is non-zero only if the order of the numerator is higher than the order of the denominator.
To identify the parameters of circuit and controller, the two exponential expressions in (5) should be represented by Pade approximation using appropriate orders, so that (5) can be in the same form of ( 6), and the parameters can be identified by comparing these coefficients.Therefore, the determination of order number of Pade approximations is a critical issue.e −1.5sT sk in the nominator of ( 5) can be modelled in the form of (7) with the order of numerator n and order of denominator m [27]. where Similarly, the e −1.5sT sk in the denominator of (5) can be represented by the Pade approximation with the order of numerator q and order of denominator p, shown as (8). where (5) can be transferred to be in the form of polynomial expression by substituting ( 7) and ( 8), shown as follows, where only the several low-order components are shown in (9) for simplification.The detailed formulas are shown as (10).
On the other hand, according to the Pade approximation, (13) can be obtained.
The following equation can be obtained by combining ( 12) and ( 13), In addition, by substituting s = jω into (5), the terminal impedance can be expressed in frequency domain, as shown in (15).If both the numerator and denominator of (15) multiply the conjugate complex of the denominator, (15) can be transferred as (16).
The derivative of phase angle of Z invk (ω) can be expressed as (18).
If the derivative of the phase angle of Z invk (ω) is equal to zero at ω ck , which can be easily observed from the phase diagram of the fitted transfer function f (s) in ( 6), the following equation will be satisfied based on the calculated K pk and L f k2 .
Three unknowns (L f k1 , C f k and R vk ) exist in two equations ( 14) and (19).Therefore, the three unknown parameters can be determined using a trial-and-error method.

B. Implementation Procedure of the Proposed Gray-Box Parameters Identification and Controller Re-Tuning Method
Fig. 4 shows the implementation procedure of the proposed gray-box parameters identification and controller re-tuning method.Time-domain simulation of the studied power system is first implemented.If no harmonic instability phenomenon is observed, no further controller re-tuning operation is needed.Otherwise, the terminal impedance frequency responses for all VSCs and grid are measured.Then, the Bode diagrams of the impedance of the left part Z S and the right part Z L of PCC are plotted, and potential oscillation frequency is analytically identified from the interaction points.The proposed parameters identification method in this paper is then used to identify the parameters of all of the VSCs.With these identified parameters, the total output impedance of the paralleled VSCs Z S can be reshaped by reducing K pk or increasing K vk , so that the phase angle difference of Z S and Z L can be limited within (−180 0 , 180 0 ), which means that the unstable power system is stabilized by re-tuning controller parameters.Finally, the controller parameters re-tuning suggestion can be provided to manufacturer.

IV. SIMULATION VERIFICATION
In this section, the proposed gray-box parameters identification and controller parameters re-tuning method is validated in both single grid-connected inverter and paralleled inverters system.

Begin
Run time-domain simulation of studied power system.

Harmonic instability ?
Offline measure the terminal impedance frequency responses for all VSCs and grid.
Plot Bode diagrams of Zs and ZL, and identify the oscillation frequencies fo.
Use the proposed method to identify the internal parameters of all VSCs.
increase Kvk.  A. Case 1: Single VSC Fig. 5 shows simulation result of three-phase currents of grid-connected inverter.It can be seen that the grid-connected inverter connected with inductive grid (L g = 3 mH) is stable.However, the system becomes unstable once the grid impedance is increased (L g = 8 mH), as shown in Fig. 6.

End
System parameters of the exemplified VSC is given in Table I.The measured terminal impedance frequency response of the VSC is shown as blue solid line in Fig. 7, and frequency response of the fitted 5-order transfer function is shown as red dashed line in Fig. 7. Coefficients of the fitted polynomial transfer function are shown in Table II.According to ( 11), ( 14) and ( 19), parameters of the VSC can be obtained as (20).
By substituting T sk = 10 −4 s, ω ck = −2π1533 and ( 20) into ( 14) and (19), two constraints can be obtained according to where the detailed expressions are omitted for simplicity.
The following equation can be obtained from (21).
The solution space has one freedom.One approach is to find the accurate solution in a trial-and-error way.For example, different values of C f k can be first assumed, and the value of C f k which makes the calculated Bode diagram agree with the measured terminal impedance frequency responses is then selected, as shown in Fig. 8.
Fig. 9 shows frequency responses of the calculated terminal impedance formulas with different C f k .It can be seen that the calculated frequency response agrees with the original terminal impedance frequency responses when C f k = 3 µF and K vk = 0.2945.The identified parameters are given in Table III.
Fig. 10 shows frequency responses of VSC impedance with original controller parameters Z S bef (blue line) and grid impedance Z L with L g = 8 mH(black line).It can be seen that phase difference at the interaction point of magnitude curves is 181.36 o , which indicates that the system is unstable.The analysis results agree with the time-domain simulation result in Fig. 6.One method to stabilize the power system is to increase the capacitor-current-feedback coefficient K vk .The     the red line in Fig. 10.It can be seen that phase difference at the interaction point of magnitude curves is 178.84 o , which indicates that the system becomes stable.The simulation result in Fig. 11 shows that the system can be stabilized after K vk is re-tuned.

B. Case 2: Three Paralleled VSCs
Fig. 12 shows simulation results about grid-injected current when three VSCs are connected to grid, where the grid impedance is kept as 3 mH.Comparing with Fig. 5, it can be seen that the system becomes unstable as the number of VSCs increases.Fig. 13 shows the terminal impedance frequency responses of three VSCs with original controller parameters Z S bef (blue line) and grid impedance Z L with L g = 3 mH (black line).It can be seen that phase difference at the interaction point of magnitude curves is 182.26 o , which indicates that the system is unstable.The analysis result agrees with the time-domain simulation result in Fig. 12.The Bode diagram of VSC Z S af t impedance by increasing K vk from identified value 0.2945 to new value 2.54 is plotted as the red line in Fig. 13.It can be seen that phase difference at the interaction point of magnitude curves is 179.41 o , which indicates that the system becomes stable.The simulation result in Fig. 14 shows that the  paralleled VSC system can be stabilized once K vk is re-tuned.
V. CONCLUSION This paper presents a gray-box impedance reshaping method for VSC with LCL filter to enhance system stability.When instability phenomenon occurs, the internal controller parameters of the grid-connected inverter are identified and re-tuned to reshape the terminal impedance characteristics.Simulation results show that the parameters of the LCL filter, proportional coefficient of current controller and the capacitor-currentfeedback coefficient are identified from the measured terminal impedance frequency responses by using VF algorithm, and the re-tuned capacitor-current-feedback coefficient can stabilize the system.The proposed gray-box impedance reshaping method is able to provide parameter design and re-tuning guideline for manufacturer even if no internal parameters are known.

Fig. 6 .
Fig.6.Time-domain simulation waveforms of three-phase currents when single VSC is connected to the weak grid with Lg = 8 mH.

Fig. 7 .
Fig. 7.The measured terminal impedance frequency responses and the fitted result using VF algorithm.

Fig. 9 .
Fig. 9. Bode diagrams of the calculated impedance formulas with assumption of different values of C f k .

Fig. 10 .Fig. 11 .
Fig.10.The Bode diagrams of the terminal impedance formula before and after re-tuning capacitor-current-feedback coefficient K vk for case 1.

Fig. 12 .
Fig. 12. Simulation results of grid-connected currents when three VSCs are connected to weak grid with Lg = 3 mH.

Fig. 13 .Fig. 14 .
Fig.13.Bode diagrams of the terminal impedance formula before and after capacitor-current-feedback coefficient K vk re-tuning for case 2.
Bode diagram of VSC impedance Z S af t by increasing K vk from identified value 0.2945 to new value 2.54 is plotted as Fig. 8. Waveforms of f 3 (C f k , Kv) with different values C f k .