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Capacitor-current-feedback active damping is an effective method to suppress the $LCL$ -filter resonance in grid-connected inverters. However, due to the variation of grid impedance, the $LCL$ -filter resonance frequency will vary in a wide range, which challenges the design of the capacitor-current-feedback coefficient. Moreover, if the resonance frequency is equal to one-sixth of the sampling frequency $(f_{s}/6)$ , the digitally controlled $LCL$ -type grid-connected inverter can be hardly stable no matter how much the capacitor-current-feedback coefficient is. In this paper, the optimal design of the capacitor-current-feedback coefficient is presented to deal with the wide-range variation of grid impedance. First, the gain margin requirements for system stability are derived under various resonance frequencies. By evaluating the effect of grid impedance on gain margins, an optimal capacitor-current-feedback coefficient is obtained. With this feedback coefficient, stable operations will be retained for all resonance frequencies except $f_{s} /6$ . Second, in order to improve system stability for a resonance frequency of $f_{s} /6$ , a phase-lag compensation for the loop gain is proposed. Finally, a 6-kW prototype is tested to verify the proposed design procedure.

Due to the effect of the computation and modulation delays on the capacitor-current-feedback active damping, the digitally controlled LCL-filtered inverter tends to be unstable as the LCL-filter resonance frequency approaching to one-sixth of the sampling frequency. Therefore, to guarantee sufficient stability margins, the guideline for choosing the LCL-filter resonance frequency is proposed in this paper. After the resonance frequency is selected, a systematic design method is proposed to facilitate the selection of the proper controller parameters. With this design method, a satisfactory region of the controller parameters for meeting the system specifications is obtained, from which the proper controller parameters can be easily determined. Moreover, it is convenient and explicit to optimize the system performance according to the satisfactory region. A 6-kW prototype is built and tested. The simulation and experimental results validate the theoretical analysis.

For the LCL-type grid-connected inverter, the capacitor-current-feedback active-damping is equivalent to a resistor in parallel with the filter capacitor to damp the LCL filter resonance. This active-damping method has no power loss and has been widely used. Based on the capacitor-current-feedback active-damping and the proportional-integral (PI) regulator as the grid current regulator, this chapter proposes a step-by-step controller design method for the LCL-type grid-connected inverter. By carefully examining the steady-state error, phase margin, and gain margin, a satisfactory region of the capacitor-current-feedback coefficient and PI regulator parameters for meeting the system specifications is obtained. With this satisfactory region, it is very convenient to choose the controller parameters and optimize the system performance. Besides, the proposed design method is extended to the situations where PI regulator with grid voltage feedforward scheme or proportional-resonant (PR) regulator is adopted. Finally, design examples of capacitor-current-feedback coefficient and current regulator parameters are presented for a single-phase LCL-type grid-connected inverter, and experiments are performed to verify the proposed design method.

The capacitor-current-feedback active-damping is an effective approach for damping the resonance peak of the LCL filter. When the LCL-type grid-connected inverter is digitally controlled, the control delay will be generated. This will result in different behavior of the capacitor-current-feedback active-damping from that with analog control. In this chapter, the mechanism of the control delay in the digital control system is introduced first. Then, a series of equivalent transformations of the control block diagram considering the control delay are performed, and it reveals that the capacitor-current-feedback active-damping is no longer equivalent to a virtual resistor in parallel with the filter capacitor, but a virtual frequency-dependent impedance. A forbidden region for choosing the LCL filter resonance frequency is presented in order to guarantee the system stability. Then, the controller design for digitally controlled LCL-type grid-connected inverter with capacitor-current-feedback active-damping is studied. Since the control delay leads to a phase lag and consequently changes the location of −180°-crossing in the phase curve of the loop gain, the system stability might be guaranteed even without damping the resonance of LCL filter. For this case, the necessary condition for system stability is studied, and the controller design method is presented. Finally, the controller parameters design examples for the grid current regulator with and without the capacitor-current-feedback active-damping are given, and the effectiveness of the theoretical analysis is verified by the experimental results.

The control challenges of LCL-type grid-connected inverter arise from the resonance problem. At the resonance frequency, the LCL filter resonance causes a sharp phase step down of −180° with a high resonance peak. This resonance peak would easily lead to system instability and should be damped. In this chapter, the resonance hazard resulted by the LCL filter is reviewed first, and then, the existing passive- and active-damping solutions are described systematically to reveal the relationship among them. Among the six basic passive-damping solutions, adding a resistor in parallel with capacitor shows the best damping performance, but it results in a high power loss. In order to avoid the power loss in the damping resistor, the active-damping solutions equivalent to a resistor in parallel with capacitor are derived, and the capacitor-current-feedback active damping is superior for its simple implementation and effectiveness. This chapter provides the basis for the study of the control techniques of LCL-type grid-connected inverter in the following chapters.

An LCL filter has two individual inductors. In order to reduce the volume of magnetic components, magnetic integration of these two inductors is introduced in this chapter. First, the integration method of the two inductors of an LCL filter is proposed, and the magnetic circuit model of integrated inductors is built. Then, based on this model, the coupling caused by the nonzero reluctance of the common core is analyzed, and the coupling effect on the ability of attenuating high-frequency harmonics of LCL filter is evaluated. According to the harmonic limits of the grid current, the maximum allowable coupling coefficient is derived, which provides the guidelines for selecting cross-sectional area and magnetic material of the common core. Finally, with the help of Ansoft Maxwell software, design examples of integrated magnetics for both single-phase and three-phase LCL filters are presented, and experiments are performed to verify the proposed method.

The full-feedforward scheme of grid voltages for three-phase LCL-type grid-connected inverter has been introduced in Chap. 7, and the injected grid current harmonics and imbalance caused by grid voltages can be effectively suppressed. However, during the derivation of the full-feedforward scheme, the grid impedance is assumed to be zero. For weak grid condition, as the grid impedance becomes large, the full-feedforward scheme might cause the instability of the grid-connected inverter. In this chapter, the stationary α-β frame-controlled three-phase LCL-type grid-connected inverter with the full-feedforward scheme is taken as the example to analyze the system stability under weak grid condition, and the weighted-feedforward scheme is proposed to achieve a trade-off between the stability and the harmonic suppression. With the weighted-feedforward scheme, the extracted and weighted grid voltages are fed forward, and the injected grid current harmonics can be effectively suppressed while the stability of grid-connected inverter under weak grid condition is also guaranteed. Finally, the experimental results verify the analysis in this chapter.

Due to the significance of extracting the grid voltage information, the grid synchronization system plays an important role in the control of grid-connected power converters, and various grid voltage synchronization schemes have been proposed. This chapter adopts the complex-vector-filter method (CVFM) to analyze the grid synchronization systems. With this method, the pairs of scalar signals, for example, the α- and β-axis components in the stationary α-β frame, are combined into one complex vector. As a consequence, the grid synchronization systems can be described with the complex transfer functions, which is very convenient to evaluate the steady-state performance, for example, the fundamental and harmonic sequences decoupling/cancellation, and dynamic performance of these systems. Besides, the CVFM also provides a more generalized perspective to understand and develop the grid synchronization systems. Therefore, some of the representative systems are reanalyzed with the CVFM in this chapter. A generalized second-order complex-vector filter and a third-order complex-vector filter are proposed with the CVFM to achieve better dynamic performance or higher harmonic attenuation. Moreover, a brief comparison of the complex-vector filters analyzed in this chapter is presented. The effectiveness of the CVFM and the proposed two complex-vector filters are verified by the simulation and experimental results.

As illustrated in Chap. 8, in the digitally controlled LCL-type grid-connected inverters, proportional feedback of the capacitor current is equivalent to a frequency-dependent virtual impedance connected in parallel with the filter capacitor due to the control delay including the computation and pulse-width modulation (PWM) delays. This virtual impedance leads to the change of the LCL filter resonance frequency. At the frequencies higher than one-sixth of the sampling frequency (f s /6), the virtual impedance contains a negative resistor component. So, if the actual resonance frequency is higher than f s /6, a pair of open-loop right-half-plane (RHP) poles are generated. As a result, the LCL-type grid-connected inverter is easier to be unstable if the resonance frequency is moved closer to f s /6 due to the variation of grid impedance. Meanwhile, the computation and PWM delays also reduce the control bandwidth greatly and thus impose a severe limitation on the low-frequency gains. Therefore, it is desirable to reduce the control delay so as to improve the stability and the control performance of the grid-connected inverter. In this chapter, the influence of the control delay on the LCL-type grid-connected inverter is firstly analyzed. Then, the real-time sampling method [1] and real-time computation method with dual sampling modes [2] are proposed to reduce or even remove the computation delay. Finally, the experimental results from a 6-kW prototype verify the effectiveness of the proposed methods.

The grid-connected inverter plays an important role in injecting high-quality power into the power grid. The injected grid current is affected by the grid voltage at the point of common coupling (PCC). This chapter studies the feedforward scheme of the grid voltage for single-phase LCL-type grid-connected inverter. First, the mathematical model for the LCL-type grid-connected inverter with capacitor-current-feedback active-damping is presented, and then it is simplified through a series of equivalent transformations. After that, a full-feedforward of the grid voltage is proposed to eliminate the effect of the grid voltage on the steady-state error and harmonics in the injected grid current. The feedforward function consists of three parts, namely proportional, derivative, and second-derivative components. A comprehensive investigation shows that if the grid voltage contains only the third harmonic, the proportional feedforward component is adequate to suppress the harmonic distortion in the grid current caused by the grid voltage; when the grid voltage contains harmonic distortion up to the thirteenth harmonic, the proportional and derivative components are required; and when the grid voltage contains harmonic distortion higher than the thirteenth harmonic, the second-derivative component must be incorporated, i.e., the full-feedforward scheme is necessary.

As the penetration of distributed power generation systems goes high, the power grid exhibits more like a weak grid which features a large set of grid impedance values and varieties of background voltage harmonics. The grid impedance can change the loop gain of the grid-connected inverter and thus challenge the control performance or even lead to system instability. The background voltage harmonics will distort the grid current and result in a poor power quality. Therefore, the grid-connected inverter should be designed with strong stability robustness and high harmonic-rejection ability, both of which correlate closely with the inverter output impedance. To shape the inverter output impedance, an impedance shaping method is proposed in this chapter, which introduces a virtual parallel impedance and a virtual series inductor. The virtual parallel impedance is used to maximize the magnitude of output impedance, and meanwhile, a virtual series inductor is used to boost its phase. Thus, the grid-connected inverter exhibits strong rejection ability of grid voltage harmonics and achieves strong stability robustness against the variations of the typical inductive-resistive grid impedance. Experimental results of a 6-kW single-phase grid-connected inverter with LCL filter confirm the effectiveness of the proposed method.

In order to alleviate the effect of the grid voltage on the grid current, Chap. 6 presented the full-feedforward scheme of grid voltages for the single-phase LCL-type grid-connected inverters, and the harmonics of the injected grid current are effectively suppressed. In this chapter, the full-feedforward scheme is extended to the three-phase LCL-type grid-connected inverter. In this chapter, the mathematical models of the three-phase LCL-type grid-connected inverter in both the stationary α–β frame and synchronous d–q frame are derived first. Then, based on the mathematical models, the full-feedforward schemes of the grid voltages for the stationary α–β frame, synchronous d–q frame, and decoupled synchronous d–q frame-controlled three-phase LCL-type grid-connected inverter are proposed. After that, the full-feedforward functions are discussed, and it will be illustrated that the simplification of the full-feedforward function should be taken with caution and simplifying the full-feedforward functions to a proportional feedforward function will give rise to the amplification of the high-frequency injected grid current harmonics. The effect of LCL filter parameter mismatches between the actual and theoretical values is also evaluated. Finally, the effectiveness of the proposed full-feedforward schemes is verified by the experimental results. Meanwhile, the performance of the proposed full-feedforward schemes under unbalanced grid voltage condition is intentionally investigated.

This paper investigates the capacitor-current-feedback active damping for the digitally controlled LCL-type grid-connected inverter. It turns out that proportional feedback of the capacitor current is equivalent to virtual impedance connected in parallel with the filter capacitor due to the computation and pulse width modulation (PWM) delays. The LCL-filter resonance frequency is changed by this virtual impedance. If the actual resonance frequency is higher than one-sixth of the sampling frequency (fs/6), where the virtual impedance contains a negative resistor component, a pair of open-loop unstable poles will be generated. As a result, the LCL-type grid-connected inverter becomes much easier to be unstable if the resonance frequency is moved closer to fs/6 due to the variation of grid impedance. To address this issue, this paper proposes a capacitor-current-feedback active damping with reduced computation delay, which is achieved by shifting the capacitor current sampling instant towards the PWM reference update instant. With this method, the virtual impedance exhibits more like a resistor in a wider frequency range, and the open-loop unstable poles are removed; thus, high robustness against the grid-impedance variation is acquired. Experimental results from a 6-kW prototype confirm the theoretical expectations.

This letter investigates the magnetic integration of the LCL filter in grid-connected inverters. By sharing an ungapped core and arranging the windings properly, the fundamental fluxes generated by the two inductors of an LCL filter cancel out mostly in the common core. Thus, the common core with low flux level can be dramatically reduced. Although the reluctance of the common core can hardly be zero, which implies an inevitable coupling between the integrated inductors, the proposed magnetic integration scheme is still attractive if the cross-section area and magnetic material of the common core are made reasonable. Experimental results from both single-phase and three-phase grid-connected inverters verify the effectiveness of the proposed method.

Due to the significance of extracting the grid voltage information, the grid synchronization system plays an important role in the control of grid-connected power converters, and various grid voltage synchronization schemes have been proposed. This paper adopts the complex-vector-filter method (CVFM) to analyze the grid synchronization systems. With this method, the pairs of scalar signals, for example, the α- and β-axis components in the stationary α-β frame, are combined into one complex vector. As a consequence, the grid synchronization systems can be described with the complex transfer functions, which is very convenient to evaluate the steady-state performance, for example, the fundamental and harmonic sequence decoupling/cancellation, and dynamic performance of these systems. Moreover, the CVFM also provides a more generalized perspective to understand and develop the grid synchronization systems. Therefore, some of the representative systems are reanalyzed with the CVFM in this paper. A generalized second-order complex-vector filter and a third-order complex-vector filter are proposed with the CVFM to achieve better dynamic performance or higher harmonic attenuation. Moreover, a brief comparison of the complex-vector filters analyzed in this paper is presented. The effectiveness of the CVFM and the proposed two complex-vector filters are verified by the simulation and experimental results.

The injected grid current regulator and active damping of the LCL filter are essential to the control of LCL-type grid-connected inverters. Generally speaking, the current regulator guarantees the quality of the injected grid current, and the active damping suppresses the resonance peak caused by the LCL filter and makes it easier to stabilize the whole system. Based on the proportional-integral (PI) and proportional-resonant (PR) compensator together with capacitor-current-feedback active-damping which are widely used for their effectiveness and simple implementations, this paper proposes a simple step-by-step controller design method for the LCL-type grid-connected inverter. By carefully dealing with the interaction between the current regulator and active damping, the complete satisfactory regions of the controller parameters for meeting the system specifications are obtained, and from which the controller parameters can be easily picked out. Based on these satisfactory regions, it is more convenient and explicit to optimize the system performance. Besides, the insight of tuning the controller parameters from these satisfactory regions is also discussed. Simulation and experimental results verify the proposed step-by-step design method.

This paper introduces a novel simplified model based on mathematical functions for modeling the steady-state and the dynamic behavior of Modular Multilevel Converter (MMC). The large number of switches in MMC creates a computational challenge for electromagnetic transient simulation programs, as it is extremely time consuming. The proposed model decreases the order of system's admittance matrix and derives an equivalent function of bridge arm based on capacitor voltage balancing strategy. Simulations are carried out which shows that both the transient and steady-state response of the simplified model well match that of the detailed one. The simplified model is validated to be effective and provides a method for analyzing high level MMC systems.

This paper explores the effect of the computation and pulsewidth modulation (PWM) delays on the capacitor-current-feedback active damping for the LCL-type grid-connected inverter. It turns out that proportional feedback of the capacitor current is equivalent to a virtual impedance connected in parallel with the filter capacitor in digital control. The LCL-filter resonance frequency is changed by this virtual impedance. And if the actual resonance frequency is higher than one-sixth of the sampling frequency (fs/6), where the virtual impedance contains a negative resistor component, a pair of open-loop unstable poles will be generated. As a result, the LCL-type grid-connected inverter becomes much easier to be unstable if the resonance frequency is moved closer to fs/6 due to the variation of the grid impedance. To address this issue, this paper proposes a capacitor-current-feedback active damping with reduced computation delay, which is achieved by shifting the capacitor current sampling instant towards the PWM reference update instant. With this method, the virtual impedance exhibits more like a resistor in a wider frequency range, and the open-loop unstable poles are removed, thus high robustness against the grid-impedance variation is acquired. Experimental results confirm the theoretical expectations.

The injected grid current regulator and active damping of the LCL filter are essential to the control of LCL-type grid-connected inverters. Generally speaking, the current regulator guarantees the quality of the injected grid current, and the active damping reduces the resonance peak caused by the LCL filter and makes it easier to stabilize the whole system. However, the frequency responses of the current regulator and active damping interact with each other, making it difficult to design the proper controller parameters and investigate the effect of the controller parameters. Taking proportional-integral (PI) compensator together with capacitor-current-feedback active-damping as an instance, this paper proposes a step-by-step design method of the current regulator and capacitor current feedback coefficient in both analog control and digital control. By carefully dealing with the interaction between the current regulator and active damping, the satisfactory regions of the controller parameters are obtained based on stability margin and steady-state error checking, making it more convenient and explicit to optimize the performance of the system. The convenient trial-and-error procedures are successfully reduced. Experimental results verify the effectiveness of the proposed design method.

The LCL filter is widely used in grid-connected inverters due to its outstanding performance of attenuating the switching frequency current harmonics. An LCL filter has two individual inductors. Numbers of magnetic cores are required, and large volume has to be reserved for these two inductors. In order to reduce the core volume, magnetic integration of these two inductors is introduced in this paper. Since the attenuating ability of the LCL filter would be weakened by the coupling between the two inductors, decoupled magnetic integration is chosen consequently. Though the reluctance of the common core can hardly be zero, the decoupled magnetic integration scheme is still attractive. A 6-kW prototype is built in the lab, and the experimental results validate the effectiveness of the proposed magnetic integration scheme.

The LCL filter is widely used in grid-connected inverter due to its powerful ability of attenuating the switching-frequency harmonics. However, the frequency response of the LCL filter has a resonance peak, which would amplify the harmonics around the resonant frequency or even cause the inverter to be unstable. Active damping based on the feedback of capacitor current is an effective solution to damp the resonance oscillation. Since the one-timestep delay of the digital signal processor (DSP) can hardly be avoided, the stable margin of the inverter will be weakened. Besides, the optional range of the capacitor-current feedback coefficient will be shrunk. This paper discusses the effect of the one-timestep delay firstly, and proposes a step-by-step design method to choose the parameters of the PI-based current regulator and the capacitor-current feedback coefficient. Based on Jury stability criterion, the selectable 3D region surrounded by the parameters of PI-based regulator and capacitor-current feedback coefficient can be plotted. Further, some specific constraints such as steady-state error and phase margin etc. will decide the suitable values of PI regulator and capacitor-current feedback coefficient. A 6-kW single-phase grid-connected inverter is built to verify the proposed design method.