I. INTRODUCTION

Nowadays, the number of distributed power generation systems (DPGS) connected to the grid is increasing rapidly. As a result, stricter standards for power quality have been issued [1]. The grid-connected converter plays a crucial part in injecting high quality power into the grid. Due to the LCL grid filters’ good filtering characteristic above resonance frequency, they are increasingly being used to mitigate PWM harmonics, especially in high power DPGSs whose switching frequency is limited by losses [2]. Unfortunately, the inherent resonance of the LCL filter can cause undesired oscillations or even instasbility. A simple method to damp this resonance is to add a passive resistor to the LCL filter [3]. However, this is not appropriate for high power DPGSs due to their power loss [4].

Many active damping approaches have been proposed which are more flexible and efficient than the passive damping techniques [5]-[8]. Capacitor current feedback active damping has been extensively used because it is simple to realize among the various active damping approaches [9]-[11]. However, in a digitally controlled system, there is an inevitable digital control delay between the sampling instant and the PWM updating instant which is equivalent to applying a unit time delay z-1 in the control loop [12]-[14]. Few references have thoroughly discussed the stability and damping performance of capacitor current feedback active damping when the digital control delay is taken into account, especially at low switching frequencies. Reference [15] totally neglects the influence of digital control delay. As its switching frequency as high as 10 kHz, the influence is not obvious. Digital control delay is regarded as a first-order inertia in reference [16] which is suitable for its application with a 5 kHz switching frequency. However, in high power systems, the switching frequency is limited by power losses and cannot be set very high. Furthermore, the above references design the active damping loop in the continuous time domain. However, this will lead to system instability and poor damping performance in low switching frequencies such as 2 kHz. Reference [17] demonstrates the stable region of active damping in the discrete time domain when the digital control delay is taken into account. However, it only presents the stable region in the 5 kHz switching frequency. When the switching frequency decreases, the system becomes inherently unstable no matter how the capacitor current feedback coefficient changes. Therefore, it is unsuitable to neglect the digital control delay at low switching frequencies when designing the active damping loop. This paper addresses this issue by theoretically presenting the stable regions of capacitor current feedback active damping in high, medium and low switching frequencies.

In high power applications, both the grid current and the converter current are sensed in the LCL filter system. The grid current is used in current loop control, and the converter current is used to protect IGBTs from overcurrent. The capacitor current can be calculated by the grid current minus the converter current. This means that the capacitor current feedback is equal to the grid current and converter current feedback with the same feedback coefficient which will lose one degree of control freedom. In order to increase the degree of control freedom and to expand the stable region, especially at low switching frequencies, this paper proposes a modified active damping approach with grid current and converter current feedback which can strengthen system stability and provide sufficient active damping. Moreover, the modified approach does not need any additional sensors. When the grid impedance varies, the modified approach can also work well. Simulation and experimental results based on a 300 kVA three phase prototype verify the theoretical conclusions presented in this paper.

II. MODELING OF THE LCL FILTER

A diagram of a grid connected PWM converter with an LCL filter is shown in Fig. 1, where ig, is, uc, ic and uo represent the grid current, converter current, filter capacitor voltage, filter capacitor current and converter voltage, respectively.

Fig. 1.Grid connected PWM converter with LCL filter.

The state space equation of an LCL filter in the continuous time domain yields as Equ. (1) by applying Kirchhoff laws [18], where the state vector is defined as x(t) = [ig(t) is(t) uc(t)]T .

The system matrix, input matrix and output matrix are shown as Equ. (2). The parasitic resistors of the inductors are not taken into account in the following derivation.

In a digital control system, a digital processor needs some time to make calculations [12]. Therefore, the calculated result of the present step is updated to output at the beginning of the next step which is shown in Fig. 2. As a result, there is an inevitable digital control delay between the sampling instant and the PWM updating instant which is equivalent to applying a unit time delay z-1 in the control loop. The digital control delay has a crucial influence on system stability and damping performance. It is necessary to take the digital control delay into account when designing an active damping loop, especially at low switching frequencies. In this paper, the sampling instant is located at the bottom and the summit of the carrier. Therefore, the sampling frequency is double the switching frequency.

Fig. 2.Digital control delay in digital control system.

Fig. 3 shows the grid current control structure with the digital control delay. By active damping, the current loop can be controlled with a PI controller. The acitve damping is achieved by state feedback. Choosing different feedback states corresponds to different active damping approaches. K is the feedback matrix.

Fig. 3.Control structure with digital control delay.

The discrete model is derived as Equ. (3), where the state vector is defined as x(k) = [ig(k) is(k) uc(k)]T .

The discrete system matrix and the input matrix can be calculated with:

The systme matrix and the output matrix yield as follows:

Due to the digital control delay, there is a unit time delay z-1 between the reference voltage u(k) and the output voltage uo(k) [19].

In order to exactly analyze the impact of the digital control delay on system stability and the design of the active damping loop, the state space equation taking the digital control delay into account is derived as Equ. (8), where the state vector is defined as x(k) = [ig(k) is(k) uc(k) uc(k-1)]T [20].

Whereas:

with eij and fi (i, j ∈{1,2,3}) being entries in matrices E and F.

The transfer function of the active damping loop Ig(z)/R(z) can be calculated by equation Equ. (10), where K is the feedback matrix and I is the unit matrix. The following discussions are based on this transfer function.

III. STABILITY ANALYSIS OF THE CAPACITOR CURRENT FEEDBACK ACTIVE DAMPING

Based on the discrete model in section II, the influence of the digital control delay on stability and damping performance is discussed in this section. The grid current and converter current are both sensed. The capacitor current is calculated by the grid current minus the converter current. As a result, the feedback matrix is K = [ Kic - Kic 0 0] , where Kic represents the capacitor current feedback coefficient. The capacitor voltage is not sensed so its feedback coefficient is 0. The transfer function of the active damping loop yields as Equ. (11) by substituting feedback matrix K into Equ. (10). This shows that an active damping loop with the digital control delay has four poles. Three of them move as the feedback coefficient changes. The other is located at z=1.

Fig. 4 shows the root loci of the capacitor current feedback active damping loop. This demonstrates the stable regions both while considering the digital control delay (dashed line) and while neglecting the digital control delay (solid line) in high (10 kHz), medium (5 kHz) and low (2 kHz) switching frequencies with parameters given in Table I.

Fig. 4.Root loci of capacitor current feedback when considering digital control delay and neglecting digital control delay.

TABLE IPARAMETER OF LCL FILTER

The active damping loop without the digital control delay also has four poles. Two of them move as the feedback coefficient changes. The others are located at z=1 and z=0, respectively. When the digital control delay is not taken into account, the system is always stable. However, a system with the digital control delay tends to become unstable, especially at low switching frequencies.

Fig. 4(a) demonstrates the root loci of a high (10 kHz) switching frequency. The damping performance of the dashed line (considering the delay) is nearly the same as that of the solid line (neglecting the delay). Therefore, the influence of the digital control delay is not obvious in high switching frequencies.

Fig. 4(b) demonstrates the root loci of a medium (5 kHz) switching frequency. The damping ratio of the dashed line (considering the delay) is much smaller than that of the solid line (neglecting the delay). It is easy for the system to become resonant or even unstable at a medium switching frequency when considering the digital control delay.

Fig. 4(c) demonstrates the root loci of a low (2 kHz) switching frequency. The damping performance of the dashed line (considering the delay) is different from that of the solid line (neglecting the delay). A system with the digital control delay is inherently unstable no matter how the feedback coefficient changes at a low switching frequency.

IV. MODIFIED ACTIVE DAMPING APPROACH

As discusses in section III, the capacitor current feedback is equal to the grid current and converter current feedback with the same feedback coefficient which loses one degree of control freedom. This is unsuitable at a low switching frequency. In order to increase the degree of control freedom, this paper sets different feedback coefficients for the grid current and converter current which strengthens the system stability. Moreover, the modified approach does not need additional sensors. Therefure, the feedback matrix is K = [ kig kis 0 ku] . Where, kig and kis represent the feedback coefficients of the grid current and the converter current, respectively. The capacitor voltage is not sensed so its feedback coefficient is 0. ku represents the feedback coefficient of the reference voltage u(k-1).

The transfer function of the active damping loop Ig(z)/R(z) is derived by substituting feedback matrix K and the parameters of TABLE I into Equ. (10). The characteristic equation of the active damping loop is shown as Equ. (12).

This shows that the active damping loop has four poles. Assume that the desired poles are p1,p2,p3,p4, where p3,p4 represent the pair of conjugate complex poles. Therefore, the desired characteristic equation is derived as Equ. (13).

Where:

Expressions for the feedback coefficients in terms of the desired pole locations are obtained by comparing Equ. (12) with Equ. (13). The equations are derived as Equ. (15).

In linear algebra, the solvable condition of the equation A‧x = b is that Rank(A|b)=Rank(A)=number of variable [21]. In Equ. (15), Rank(A)=number of variable=3 . Therefore, the solvable condition of Equ. (15) is Rank(A|b)=3. Use Gaussian Elimination to transform the augmented matrix A|b to an upper triangular matrix as Equ. (16).

When the fourth row of the augmented matrix A|b is equal to 0, Rank(A|b)=3 . Therefore, the solvable condition of Equ. (15) is derived as Equ. (17).

The desired poles must satisfy Equ. (17). In order to derive the root loci of the active damping loop in the discrete domain, set the dominant pole p1=0.9, and the non-dominant pole p2=0.1, and assume the pair of conjugate complex poles to be p3 = α + j‧β and p4 = α - j‧β . Substituting p1,p2,p3,p4 into Equ. (17) yields the equation of the conjugate complex poles as Equ. (18).

The root loci of the active damping loop, comparing modified approach with the conventional capacitor current feedback active damping, is shown in Fig. 5.

Fig. 5.Root loci of active damping loop comparing modified approach with conventional capacitor current feedback.

Fig. 5(a) shows the root loci of a medium (5 kHz) switching frequency. The modified approach (solid line) can provide a larger damping ratio than the conventional capacitor current feedback active damping (dashed line) in a medium switching frequency.

Fig. 5(b) shows the root loci of a low (2 kHz) switching frequency. The conventional capacitor current feedback active damping (dashed line) is unstable. However, the modified approach (solid line) can overcome this issue at a low switching frequency.

When compared with the conventional capacitor current feedback, the modified approach can strengthen stability and provide sufficient damping performance, especially in low switching frequencies. Choosing proper locations of the desired poles p1,p2,p3,p4 and substituting them into Equ. (15), the feedback matrix K can be solved out.

Fig. 6 shows a bode diagram of the active damping loop, comparing the modified approach with the conventional capacitor current feedback approach, at a 2 kHz switching frequency. It demonstrates that the conventional capacitor current feedback (marked line) causes resonance. By applying the modified approach (solid line), the resonance can be damped.

Fig. 6.Bode diagram of active damping loop comparing modified approach with conventional capacitor current feedback in 2 kHz switching frequency.

By applying the modified approach, the pair of conjugate complex poles p3,p4 can be placed away from the dominant pole p1. Therefore, the conjugate complex poles have little influence on the current loop in the low frequency band. The cut off frequency of the current loop is always set to the low frequency band so that it is suitable to neglect the conjugate complex poles when designing the current loop [22].

By neglecting the pair of conjugate complex poles, the active damping loop is simplified from fourth-order into second-order. This will simplify the design of the current loop. Fig. 6 demonstrates that dashed line (neglecting the conjugate complex poles) is nearly the same as the solid line (considering the conjugate complex poles) in the low frequency band.

Variations of the grid impedance in weak grids has an impact on active damping performance [23]. Therefore, it is necessary to do a sensitivity analysis when the grid impedance is taken into account. Fig. 7 shows the root loci of the active damping loop when the grid impedance varies from 0 μH to 225 μH which is equal to 15% pu. The conjugate complex poles move towards the circle center, and the other two poles just move a little on the real axis.

Fig. 7.Root loci of active damping loop when grid impedance varies from 0 μH to 225 μH (15% pu).

Fig. 8 shows a bode diagram of the active damping loop comparing no grid impedance with a 15% pu grid impedance. It demonstrates that the bode diagram with no grid impedance (solid line) is nearly the same the one with a 15% pu grid impedance (marked line) in the low frequency band. Moreover, the damping performance of the 15% pu grid impedance is better than that of the no grid impedance. This also demonstrates that the dashed line (neglecting the conjugate complex poles) is nearly the same as with the marked line (15% pu grid impedance). Therefore, it can be seen that the modified active damping method works well when the grid impedance varies.

Fig. 8.Bode diagram of active damping loop comparing no grid impedance with 15% pu grid impedance.

V. SIMULATION AND EXPERIMENT RESULTS

The analysis is verified with MATLAB and a 300 kVA three phase prototype. The prototype is shown in Fig. 9. The system parameters are given in Table II.

Fig. 9.Prototype of 300 kVA PWM converter.

TABLE IISYSTEM PARAMETER

The active damping performance is verified by reactive current reference steps from 0 to -500A. The conventional capacitor current feedback simulation results are unstable at a 2 kHz switching frequency. Fig. 10 shows the results of the modified approach at a 2 kHz switching frequency. Fig. 10(a) shows the simulation results, and Fig. 10(b) shows the experiment results. It can be seen that the system is stable and that the resonance is well damped. The total harmonic distortion (THD) of the grid current is 2.27% for the simulation and 2.52% for the experiment. The experimental results match the simulation results well. This demonstrates that the modified approach can strengthen the system stability and provide sufficient damping performance.

Fig. 10.Reactive current step from 0 to -500A in 2 kHz switching frequency.

Grid current waveforms are obtained by an oscilloscope whose sampling frequency is 500 kHz. In order to see the dynamic performance of Fig. 10, the grid current is transformed from the three phase stationary frame into the two phase rotary frame by MATLAB. Fig. 11 shows the step response of the reactive current component. Fig. 11(a) shows the simulation results, and Fig. 11(b) shows the experiment results. There is a one-step delay between the references and the results because of the digital control delay. The experimental results match the simulation results well. The rising time and overshoot are acceptable. This demonstrates that modified approach has good dynamic performance.

Fig. 11.Step respond of reactive current component from 0 to-500A.

The test of a DC-load step from no-load to full-load is performed with a 2Ω resistor which is parallel with a DC link capacitor. Fig. 12 shows the results of the modified approach at a 2 kHz switching frequency. Fig. 12(a) shows the simulation results, and Fig. 12(b) shows the experimental results. The DC bus is 750V and the grid current is 600A in the steady state.

Fig. 12.DC-load step from no-load to full-load in 2 kHz switching frequency.

This demonstrates that the system is stable and that the resonance is well damped. The dynamic respond is fast. The experimental results match the simulation results well.

In order to verify the modified approach when grid impedance is taken into account, a simulation of reactive current stepping from 0 to -500A is done, with the grid impedance set as 225 μH. Fig. 13 shows the simulation results. The resonance is well damped and the dynamic performance is acceptable. This shows that the modified approach still works well when grid impedance is taken into account.

Fig. 13Reactive current step from 0 to -500A when grid impedance equals 225 μH.

VI. CONCLUSION

A modified active damping approach for grid connected PWM converters with a LCL filter has been proposed and analyzed. A discrete model considering the digital control delay has been derived. The feedback coefficients of the modified approach have been yielded in terms of the desired pole locations. The stable regions of the conventional capacitor current feedback active damping and the modified active damping approach have been discussed. The root loci and bode diagram of the active damping loop have been plotted when the grid impedance varies.

This paper shows that it is acceptable for the capacitor current feedback active damping to neglect the digital control delay at a high switching frequency but that systems will become inherent unstable at a low switching frequency. Therefore, it is unsuitable to neglect the digital control delay at a low switching frequency when the designing active damping loop. The modified approach can strengthen system stability and provide sufficient damping performance. By applying the modified approach, the active damping loop can be simplified from fourth-order into second-order and the design of the current loop can be simplified. The modified approach can work well when the grid impedance varies. The theoretical results are verified by simulation and experiment results.