https://doi.org/10.6113/JPE.2018.18.5.1555
ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718
A Virtual RLC Active Damping Method for LCL-Type Grid-Connected Inverters
Yiwen Geng†, Yawen Qi*, Pengfei Zheng*, Fei Guo*, and Xiang Gao*
†,*School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou, China
Abstract
Proportional capacitor–current–feedback active damping (AD) is a common damping method for the resonance of LCL-type grid-connected inverters. Proportional capacitor–current–feedback AD behaves as a virtual resistor in parallel with the capacitor. However, the existence of delay in the actual control system causes impedance in the virtual resistor. Impedance is manifested as negative resistance when the resonance frequency exceeds one-sixth of the sampling frequency (fs/6). As a result, the damping effect disappears. To extend the system damping region, this study proposes a virtual resistor–inductor–capacitor (RLC) AD method. The method is implemented by feeding the filter capacitor current passing through a band-pass filter, which functions as a virtual RLC in parallel with the filter capacitor to achieve positive resistance in a wide resonance frequency range. A combination of Nyquist theory and system close-loop pole-zero diagrams is used for damping parameter design to obtain optimal damping parameters. An experiment is performed with a 10 kW grid-connected inverter. The effectiveness of the proposed AD method and the system’s robustness against grid impedance variation are demonstrated.
Key words: LCL filter, Active damping, System delay, Virtual RLC, Grid impedance
Manuscript received Mar. 13, 2018; accepted Jun. 27, 2018
Recommended for publication by Associate Editor Jongbok Baek.
†Corresponding Author: 13913475362@139.com Tel: +86-139-1347-5362, China University of Mining and Technology
*School of Electrical and Power Engineering, China University of Mining and Technology, China
Ⅰ. INTRODUCTION
Grid-connected inverters are widely used in distributed power generation systems based on renewable energy due to the development of power semiconductor devices and power electronics [1]-[3]. When used with pulse-width modulation (PWM), grid-connected inverters acquire many advantages, such as controlled AC power factor, bidirectional power flow, and sinusoidal output currents [4]-[6]. An appropriate filter must be used between the grid and inverter to reduce the harmonics of inverter output current. LCL filters can suppress high-frequency harmonics at a smaller inductance compared with L filters. However, the existence of resonance makes system control complicated [7].
Appropriate damping methods must be applied to suppress the resonance introduced by LCL filters and ensure system stability. The passive damping method involves adding resistors in series or parallel with the filter capacitor to suppress resonance and does not require extra sensors or control loops [8]. However, this method leads to high loss and reduces the capability of the filter to attenuate high-frequency harmonics [9]. The active damping (AD) method with a notch filter was studied in [10] to address this problem. The positive resonance of an LCL filter is offset by the negative resonance of a notch filter to realize stable system operation. However, notch filters are susceptible to resonant frequency fluctuation, which can weaken damping or cause it to fail. Proportional capacitor–current–feedback AD [11] is commonly used to suppress resonance. The feedback loop is equivalent to a virtual resistor in parallel with the filter capacitor to achieve the same damping effect as that in passive damping.
Proportional capacitor–current–feedback AD is equivalent to virtual impedance due to the delay in digitally controlled systems [12]. The imaginary part of virtual impedance changes the resonance frequency, and its real part behaves as a negative resistor, which makes the system unstable [13]. The system in [14] feeds grid current back to the current control loop, and the system can operate stably by designing the parameters properly when the resonant frequency is lower than one-sixth of the sampling frequency (fs/6). Therefore, the damping region of the system is only (0, fs/6). Grid impedance cannot be ignored [15]-[17]. According to [14], resonant frequency may exceed fs/6 when grid impedance varies, and this condition results in resonance. To extend the damping region, [18] presented an AD method in which the system has a virtual RC structure in parallel with the filter capacitor. The method can ensure that the real part of virtual impedance is positive in (0, fs/3). In [19], a second-order generalized integrator was adopted to replace the proportion link of capacitor-current feedback, and the damping effect was similar to that in [18]. System stability was analyzed in detail in [20], and an improved capacitor–current–feedback AD method was proposed. The method can extend the damping region to (0, fs/4). An area equivalence method was adopted in [21] to implement time delay compensation, however the valid damping region of the system was not analyzed. In [22], capacitor voltage was fed back through a band-pass filter to form a damping loop, however the valid damping region of the system was not analyzed as well. Reference [23] used a phase-lead compensation method. However, the valid damping region is only (0, fs/4), and the method introduces high-frequency noise.
A virtual resistor–inductor–capacitor (RLC) AD method is proposed in this study to extend the damping region and maintain stable system operation when grid impedance varies considerably. The system is described in Section II, and a discrete-time domain mathematical model is presented. A stability analysis of the system with proportional capacitor–current–feedback AD is performed in Section III, and the damping coefficient range corresponding to different resonant frequency bands is studied. Section IV presents the design of the parameters of the proposed AD method and optimal damping coefficients. In Section V, experimental results are presented to verify the effectiveness of the proposed AD method and the robustness of the system against grid impedance variation.
Ⅱ. MODELING OF AN LCL-TYPE THREE-PHASE GRID-CONNECTED INVERTER
Fig. 1 shows the main circuit topology of an LCL-type three-phase grid-connected inverter. The LCL filter consists of inverter-side filter inductor L1, grid-side filter inductor L2, and filter capacitor C. Cdc and Lg are the DC-side capacitor and grid inductor, respectively. The equivalent parasitic resistances of the inductors and capacitors are small, thus disregarded in this work. udc and idc represent the DC-side voltage and current, respectively. i1k, i2k, and iCk (k=a, b, c) are the inverter-side inductor current, grid-side inductor current, and filter capacitor current, respectively. vpcck denotes the point of common coupling (PCC) voltage. vgk (k=a, b, c) denotes the grid voltage.
Fig. 1. Main circuit topology of an LCL-type three-phase grid- connected inverter.
The control strategy commonly used in grid-connected inverters is cascaded-loop control, in which the inner current loop regulates the AC-side current, and the outer voltage loop controls the DC-side voltage. This study investigates the stability of the inner current loop, and the outer voltage loop is not considered.
Fig. 2(a) shows the continuous-time domain control diagram of the inverter-side inductor current with proportional capacitor–current–feedback AD in the stationary αβ-frame. Gc(s) represents the current controller, which can reduce the static error between inverter-side inductor current i1(s) and its given i1ref(s). KPWM is the PWM gain of the inverter, and KPWM = 1 for the space vector PWM. Rad is the AD damping coefficient. 1/Ts is the transfer function of sampling, and Ts is the sampling period. Gd(s) is the delay of in a sampling period due to the fact that the computation time of the digital signal processor (DSP) microprocessor cannot be neglected, and it can be obtained as
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(a) |
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(b) |
Gh(s) is the transfer function of the zero-order holder (ZOH); it accurately describes the inherent property of PWM converters and can be expressed as
The transfer functions Gi1(s) and GiC(s) from inverter-side voltage vinv(s) to inverter-side current and filter capacitor current are expressed as Equations (3) and (4), respectively.
where ωres is the resonance angular frequency of the LCL filter and can be expressed as
Fig. 2(b) shows the discrete-time domain control diagram of the inverter-side inductor current with proportional capacitor–current–feedback AD. By applying z-transformation, GiC(s) and Gi1(s) can be rewritten as GiC(z) and Gi1(z), respectively.
A quasi-proportional resonant current controller is adopted to eliminate the steady-state error at the fundamental frequency, and it can be expressed as
where ω0 and ωi are the fundamental and cut-off angular frequencies, respectively. Kp is the proportional gain, and Kr is the resonance gain. By applying z-transformation, Gc(s) can be rewritten as Gc(z).
Ⅲ. STABILITY ANALYSIS OF CURRENT CONTROL LOOP WITH PROPORTIONAL FEEDBACK OF CAPACITANCE CURRENT
For an LCL-type inverter control system with proportional capacitor–current–feedback AD, Rad is a proportional term, as shown in Equation (10).
Gc(z) can be approximated as a proportional regulator at a high frequency [25], i.e., Gc(z)≈Kp. Therefore, the open-loop transfer function of the system can be obtained as Equation (11).
In this work, inverter-side inductor current is fed back to the current control loop, and switching frequency fsw is the same as sampling frequency fs. Therefore, the resonant frequency of the designed LCL filter fres is less than fs/2. A bode plot of open-loop transfer function of the system can be obtained from Equations (10) and (11) and Table I, as shown in Fig. 3.
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(a) |
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(b) |
Fig. 3 shows that -180° crossing may take place at LC resonant frequency fr, except for fres and fs/6. fr is shown as Equation (12).
As shown in Fig. 3, Nyquist crossing does not occur at fr [26], because the amplitude at fr is constantly negative and does not affect system stability. The open-loop transfer function of the system is expressed at fres and fs/6, which are shown in Equations (13) and (14), respectively.
By applying Jury’s criterion to Equation (11), the conditions wherein the system has no open-loop poles outside the unity circle of the z-plane can be obtained, as shown in Equations (15) and (16).
where
The case of Kad>0 can be divided into three cases according to [20].
1) 0<Kad<Kad0 and 0<fres< fs/6. The system does not have an open-loop pole outside the unit circle of the z-plane.
2) Kad>Kad0 and 0<fres< fs/6. The system has two open-loop poles outside the unit circle of the z-plane.
3) Kad>0 and fres≥fs/6. The system has two open-loop poles outside the unit circle of the z-plane.
According to Equation (13), 0° crossing occurs at fres in the three cases. Equation (14) indicates that -180° and 0° crossing occur at fs/6 in the first and second cases, respectively. Moreover, -180° and 0° crossing may take place in the third case due to the influence of the numerator in the first expression of Equation (14).
When the system adopts capacitor–current–feedback AD and the inner current loop adopts inverter-side inductor current feedback to control the grid-connected current, according to Nyquist theory, only the conditions in Case 1 can make the system stable, and it is expressed as
Similarly, the case of Kad≤0 can be divided into three cases as follows:
1) max{Kad0, Kad1}<Kad<0. The system does not have an open-loop pole outside the unit circle of the z-plane.
2) min{Kad0, Kad1}<Kad<max{Kad0, Kad1}. The system has two open-loop poles outside the unit circle of the z-plane.
3) Kad<min{Kad0, Kad1}, The system has three open-loop poles outside the unit circle of the z-plane.
In accordance with Nyquist theory, the stable condition of the system can be obtained as Equations (19) and (20).
By combining Equations (18) and (20), the conditions satisfying system stability can be achieved when 0<fres< fs/6.
From Equations (19) and (21), the range of Kad is obtained in (0, fs/2). No intersection of the range of Kad corresponding to (0, fs/6) and (fs/6, fs/2) exists. Therefore, the system with proportional capacitor-current-feedback AD can only operate stably in (0, fs/6) or (fs/6, fs/2). Generally, grid impedance variation can change the resonant frequency of an LCL filter; if this occurs, the designed damping method will lose its effect, and system stability cannot be easily guaranteed.
Ⅳ. VIRTUAL RLC AD
A. Basic Principles of Virtual RLC AD
Proportional capacitor–current–feedback AD can only guarantee that the system operates stably when resonant frequency is in (0, fs/6) because of delay. Resonant frequency might exceed fs/6 due to grid impedance variation, and the filter capacitor is in parallel with negative impedance, which results in AC-side current oscillation. Therefore, the damping region must be extended further.
Fig. 4(a) shows a control block diagram of the system with virtual RLC AD. ZA is the equivalent impedance in parallel with the filter capacitor without considering delay, and it can be expressed as
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(a) |
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(b) |
where Rd, Cd, and Ld denote the damping resistor, damping capacitance, and damping inductance, respectively, and they are positive values.
Equation (22) shows that the filter capacitor is in parallel with a series of RLC structures. In consideration of digital delay, equivalent impedance is obtained from Fig. 4(a) and shown in Equation (23). By substituting s=jω into Equation (23) and using the Euler formula, the expressions of Zeq(s) in the frequency domain can be obtained as follows:
Zeq can be represented by the structure where resistor Req is in series with reactance Xeq in consideration of the delay of the system, as shown in Fig. 4(b). Compared with proportional capacitor-current-feedback AD [17], Equation (24) adds an adjustment term consisting of damping inductance Ld and damping capacitance Cd. The adjustment term must always be positive within (0, fs/2) to improve the positive resistance characteristic of Req. The following needs to be satisfied.
As shown in Fig. 4(a), moving the capacitor–current– feedback node from the comparative link after 1/sL1 to the output of the current controller constitutes the actual capacitor–current–feedback damping loop (dotted lines), and Rad(s) can be expressed as
The capacitor current loop of the virtual RLC AD is essentially a band-pass filter. Kd, Qd, and ωd are the band-pass gain, damping factor, and center frequency of the band-pass filter, respectively, and they can be expressed as
B. Parameter Design
Equation (26) is discretized with the improved bilinear transformation method, and the expression in the z-domain can be expressed as
According to the previous analysis, the open-loop transfer function of the system in the discrete-time domain contains two cases whether there are poles outside the unit circle, and different cases have different damping coefficients. This study analyzes the situation of no pole outside the unit circle and uses Nyquist theory to design the damping coefficient and ensure the operating stability of the system in the wide damping region.
The open-loop transfer function of the system with virtual RLC AD can be obtained by substituting Equation (28) into Equation (11). Given that the order of the open-loop characteristic equation is increased from the fourth order to the sixth order, a stability analysis using Jury’s criterion can be complicated. Reference [24] showed that Req is positive at fres in the open-loop transfer function of the system without poles outside the unit circle. Therefore, to ensure system stability, the condition to make the real part of Equation (24) positive can be obtained by integrating Equation (27). The result is Equation (29).
y1=f(ω) is negative within (0, 2πfs/6) but positive within (2πfs/6, 2πfs/2). Qd is positive; thus, Equation (29) can be simplified as
Nyquist crossing does not occur in the bode diagram of the open-loop transfer function of the system to ensure the stability of closed-loop system. According to Equations (11) and (28) and Table I, the bode diagrams of the open-loop transfer function of the system are plotted in fres<fs/6 and fres>fs/6, as shown in Fig. 5.
Fig. 5. Open-loop bode diagrams of the system with virtual RLC active damping.
Fig. 5 shows that Nyquist crossing may occur at fs/6 when 0<fres≤fs/6. Nyquist crossing may also take place at fres when fs/6<fres<fs/2. To ensure system stability, the amplitude of open-loop transfer function of the system should meet Equations (32) can be derived by combining Equations (11), (28) and (31).
The value of y2 is less than 1 in (0, 2πfs/6), and because Qd<<2h, y3=f(ω) can be simplified via Equation (33).
y1 monotonically decreases from 107 to 0 in (2πfs/6, 2πfs/3) and increases from 0 to +∞ in (2πfs/3, 2πfs/2). Regardless of the value of Qd, valid damping cannot be achieved in the vicinity of fs/3. The larger the value of Qd is, the larger the invalid damping range is. y4 monotonically decreases from 90000 to 0 in (2πfs/6, 2πfs/3) and increases from 0 to +∞ in (2πfs/3, 2πfs/2). Given that Kd cannot be infinite, suppressing resonance with virtual RLC AD near fs/2 is difficult.
The parameter range in Equation (33) is strict, and to minimize the invalid damping region, Qd is within (0, 1500) and Kd is within (10000, 200000). Fig. 6 shows the zero-pole diagram of the closed-loop transfer function of the system with two different resonant frequencies (fres1 and fres3) when Qd and Kd vary, respectively.
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Fig. 6(a) shows the zero-pole diagram of the system when Kd = 50000 and Qd increases from 100 to 1500 at the 200th step size. The dominant pole is shown in a local enlarged drawing. With the increase of Qd, the position of the pair of closed-loop dominant poles becomes gradually close to the origin, and the position of the other dominant poles remains unchanged. The change trends of the closed-loop poles are consistent at different resonant frequencies. Control theory points out that the closer the dominant pole is to the origin, the more stable the system is. Hence, 1500 can be the optimal value of Qd.
Fig. 6(b) is the zero-pole diagram of the system when Qd = 1500 and Kd increases from 30000 to 150000 at the 20000th step size. The dominant pole is shown in a local enlarged drawing. With the increase of Kd, the position of the pair of closed-loop dominant poles gradually approaches the origin. Then, it gradually approaches the unit circle, and the closest to the origin is the one at Kd=90000. Similarly, the change trends of the closed-loop poles are consistent at different resonant frequencies. Therefore, 90000 can be the optimal value of Kd to achieve the best damping effect.
C. Analysis of Robustness
To verify the robustness of the designed damping coefficient against grid impedance variation, the zero-pole diagram of the closed-loop transfer function of the system when the impedance of the grid increases from 0mH to 6mH at the 1mH step size is shown in Fig. 7, where the dominant pole is shown in a local enlarged drawing.
Fig. 7. Closed-loop pole-zero map of the system with grid impedance variation.
Fig. 7 shows that Qd and Kd are set to their optimum value, i.e., 1500 and 90000, respectively. The local enlarged drawing only shows the dominant poles. In Fig. 7, each dominant pole changes its position with the increase of Lg, and all poles keep themselves in the unit circle, which implies the stability of the closed-loop system. This result reveals the good robustness of the designed damping parameters against grid impedance variation.
Ⅴ. EXPERIMENTAL RESULTS
A 10 kW grid-connected inverter prototype is built (Fig. 8) to verify the effectiveness of proposed virtual RLC AD method. The experimental circuit topology is shown in Fig. 1. The full experiment parameters are shown in Table II. In the experiments, the control scheme is implemented in a DSP+FPGA digital microcontroller, where the DSP is Texas Instruments’ TMS320F28335 and used as the main algorithm controller. The FPGA is Xilinx XC3S400 and used as the assistant controller for implementing PWM signal output, ADC control, etc.
Fig. 8. Three-phase LCL-filtered grid-connected inverter system used in the experiment.
| Parameter |
Value/Unit |
Parameter |
Value/Unit |
|||
| DC bus voltage UDC |
350V |
Phase voltage of the grid |
110V(Peak),50Hz(Frequency) |
|||
| DC-side capacitor Cdc |
2200μF |
Coefficient of AD feedback |
Kad=4 |
|||
| Inverter-side inductance L1 |
1.5mH |
Switching frequency fsw |
10kHz |
|||
| Grid-side inductance L2 |
1.2mH |
Sampling frequency fs |
10kHz |
|||
| Filter capacitor C |
9.4μF |
Proportional gain Kp |
4 |
|||
| Grid impedance Lg |
Lg1=1mH |
Lg2=2mH |
Lg3=5mH |
Resonance gain Kr |
150 |
|
| Resonance frequency fres |
1738Hz |
1624Hz |
1494Hz |
Virtual RLC coefficient |
Kd=90000 |
Qd=1500 |
Fig. 9 shows the experimental waveform of phase-A grid- side inductor current i2a, the voltage of PCC vpcca, and the total harmonic distortion (THD) of i2a, where I2an is the amplitude of the harmonics and Lg = 0. The resonant frequency exceeds fs/6, and virtual impedance behaves as a negative resistor, which results in unstable system operation and inverter resonance.
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(a) |
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Fig. 10 shows the experimental waveform of the system with virtual RLC AD. Compared with the THD of i2a in Fig. 9(b), the proposed method can effectively suppress resonance when the resonant frequency exceeds fs/6, which is consistent with the theoretical analysis.
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(a) |
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Fig. 11 shows the transient-state experimental waveforms of the system with the proportional capacitor-current-feedback and virtual RLC AD methods when the reference peak value of the AC-side current is reduced from 6A to 3A with Lg = 0. When the system uses the proportional capacitor–current–feedback AD method, output current i2a resonates. However, when the system uses the virtual RLC AD method, output current i2a can complete the dynamic change in a given switching moment, and no current oscillation occurs, indicating that the proposed method does not affect the dynamic performance of the system.
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(a) |
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(b) |
To verify the robustness of the virtual RLC AD against grid impedance variation, 5, 2, and 1mH inductance are added as the grid impedance between PCC and the grid. The experimental waveforms with the two AD methods are shown in Figs. 12 and 13.
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(b) |
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(c) |
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(d) |
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(a) |
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(b) |
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(c) |
Fig. 12 shows that when the grid impedance decreases, the proportional capacitor–current–feedback AD method cannot effectively suppress the resonance of the system. Specifically, the grid-side currents in Figs. 12(b) and 12(d) are seriously distorted due to system resonance, which can be seen from Figs. 12(c) and 12(e). However, Fig. 13 indicates that the system with virtual RLC AD is unaffected by varying the grid impedance, and the system can effectively suppress resonance. This result demonstrates that the system with virtual RLC AD has high robustness against grid impedance variation.
In summary, the experimental results illustrate that the proposed method can effectively improve system stability without affecting the dynamic performance of the system and has high robustness against grid impedance variation.
Ⅵ. CONCLUSION
This study analyzes the stability of LCL-type grid-connected inverters, and the following conclusions are obtained.
1) The damping effect of traditional proportional capacitor- current-feedback AD is affected by the existence of actual delay. Once the damping parameters are determined, the system can only operate stably in (0, fs/6) or (fs/6, fs/2).
2) To expand the damping region, this study proposes the use of a virtual RLC AD method, and designs the damping coefficients of Kd and Qd by following Nyquist theory, and obtains the range of parameters. For resonant frequency in (0, fs/2), the invalid damping region is only near fs/3 and fs/2, whose width is affected by Kd and Qd.
3) The optimal values of the damping parameters of the proposed method are obtained by using the closed-loop zero-pole diagram of system. Experiments are conducted with a 10kW grid-connected inverter prototype. The results prove that proposed AD method can effectively expand the valid damping region and exhibits high robustness against grid impedance variation.
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Yiwen Geng was born in Jiangsu Province, China, in 1977. He received his B.S., M.S. and Ph.D. degrees from the School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou, China, in 2000, 2004, and 2014, respectively. In 2006, he became a lecturer in the School of Electrical and Power Engineering, China University of Mining and Technology, where he has been working as an associate professor since 2016. His current research interests include photovoltaic inverters, harmonic mitigation, and power electronics.
Yawen Qi was born in Jiangsu Province, China, in 1992. She received her B.S. degree in electrical engineering from China University of Mining and Technology, Xuzhou, China, in 2016. She is presently working for her M.S. degree in electrical engineering in the School of Electrical and Power Engineering. Her current research interests include control of converters and harmonic mitigation.
Pengfei Zheng was born in Anhui Province, China, in 1993. He received his B.S. degree in electrical engineering and automation from Nanjing University of Posts and Telecommunications, Nanjing, China, in 2016. He is presently working for his M.S. degree in Electrical engineering in the School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests include bidirectional three-level LLC resonant converters and their control.
Fei Guo was born in Anhui Province, China, in 1994. He received his B.S. degree in electrical engineering and automation from Anhui University, Hefei, China, in 2016. He is presently working for his M.S. degree in electrical engineering in the School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests include drive systems of permanent magnet synchronous motors for electric vehicles.
Xiang Gao was born in Shanxi Province, China, in 1991. He received his B.S. and M.S. degrees in electrical engineering from China University of Mining and Technology, Xuzhou, China, in 2014 and 2017, respectively. His current research interests include control of inverters and photovoltaic generation.