A Control Strategy of LCL-Type Grid-Connected Inverters for Improving the Stability and Harmonic Suppression Capability

Abstract

The conventional inverter-side current single-loop feedback control scheme is weak in suppressing the grid-side current harmonics, posing a challenge for an inverter to inject high-quality current under distorted grid voltage. With capacitor current compensation added, the control scheme achieves controllability of the grid-side current harmonics so that it can effectively suppress some specific harmonic components. However, due to the stability requirements, only a few low-order harmonic resonance controllers can be applied, which limits the mitigation of high-order harmonics. To tackle this problem, the grid-side current feedback control with inductor–capacitor–inductor (LCL) resonance damping is proposed in this paper. In this case, a higher LCL resonance frequency can be set compared to the inverter-side current single-loop feedback control scheme. Thereby, more resonance controllers can be applied to suppress high-order grid-side current harmonics. The active damping method of capacitor current proportional feedback plus capacitor voltage proportional feedback is adopted because of its high robustness to grid impedance variations. Furthermore, this paper reveals that the applied active damping method has a limitation in that it only considers a single inverter under inductive grid impedance, which cannot eliminate the risk of resonance caused by the interaction of multiple inverters and the grid. To address this issue, a phase lead compensator (PLC) is proposed, eliminating the resonance risk by removing the non-passive region of the inverter output admittance. To retain the advantage of the inverter-side current single-loop feedback control scheme, i.e., only a few measuring devices are required, a digital differentiator is used to calculate the capacitor current from the capacitor voltage. The difference between the measured inverter-side current and the calculated capacitor current is taken to approximate the grid-side current for the feedback control. The control performance is comparable to using the grid-side current for feedback. Simulation and experimental results demonstrate that the proposed control scheme endows the inverter with good stability and current quality without extra measurement devices.

1. Introduction

In a renewable energy generation system, grid-connected pulse-width modulation (PWM) inverters play a significant role in connecting renewable energy to the utility grid. Compared with the L filter, the LCL filter has an excellent high-frequency harmonic attenuation performance with a smaller volume, making it a common solution for filtering harmonics produced by the PWM inverter [1]. However, two problems have to be faced with an LCL filter. One is the LCL resonance, which may challenge the stability of the inverter control system [2]. Assume the total delay time is 1.5T_s for the digital calculation and PWM process. T_s represents the sampling period [3]. For the inverter-side current single-loop feedback control, the stable region of LCL resonant frequency f_r is (0, f_s/6), and the unstable region is (f_s/6, f_s/2). f_s is the sampling frequency. f_s/6 is the critical frequency that divides the stable and unstable regions. For the grid-side current single-loop feedback control, the stability region is divided exactly the opposite way to the former. When the LCL resonant frequency is in the corresponding stable region and the current regulator parameters are reasonably designed, the inverter-side or the grid-side current single-loop feedback control can enable an inverter to operate stably [4]. With the grid impedance changes, the actual resonant frequency will vary widely and may enter the unstable region, making the system unstable [5]. Therefore, many stability enhancement methods such as active damping based on filters [6,7], state variable feedback [8,9,10,11,12], delay time control [1,4] and some resonance analysis studies [13,14] have been proposed in recent years. Another problem is that compared with the L filter, the total inductance employed by the LCL filter is smaller. As a result, the grid voltage harmonics are more likely to distort the current injected into the grid by the inverter [15]. The multi-resonant regulators in the stationary frame [16,17], proportional–integral (PI) regulators [18] in the multi-synchronous reference frames or grid voltage feedforward [19,20] are the common choice to solve this problem. For most inverter products, a group of current sensors is set to measure the inverter-side current, providing overcurrent protection for the inverter [21]. Compared with the grid-side current feedback control, the inverter-side current feedback control is cost-saving since it does not require the grid-side current sensors [15]. However, the grid-side current is not directly controlled, which may leave it distorted. This issue can be tackled by adding harmonic information to the inverter-side reference current [15,22], but no LCL resonance damping applied will result in a strong interaction between the frequency characteristics of the LCL resonance and those around the resonance controller. To mitigate its effect on stability, the resonance frequencies of the resonance controllers should be much lower than that of the LCL filter [23]. The inverter-side current single-loop feedback control requires an LCL resonant frequency below f_s/6, which further limits the application of high-order resonant controllers. The grid-side current single-loop feedback control permits setting the LCL resonant frequency in (f_s/6, f_s/2). Accordingly, the application range of resonant controllers is extended, but as the grid impedance increases, the LCL resonant frequency may enter the instability region of (0, f_s/6), causing system instability [5]. To improve the robust stability against grid impedance changes, this paper suggests the grid-side current feedback control with the damping method as proportional feedback of capacitor current plus proportional feedback of capacitor voltage, for which the maximum allowable LCL resonant frequency is close to the Nyquist frequency [24], allowing for more high-order harmonic resonance controllers to be incorporated, further improving the quality of the injected current. Additionally, the damping method has high robustness to the variation of the grid equivalent inductance or the LCL filter parameters. However, it is only studied for a single inverter under inductive grid impedance. Further analysis is needed for its applicability in more complex situations.

With the development of renewable energy, more and more inverters are connected to the grid, making the system increasingly complex [25]. The interaction among multiple inverters, reactive power compensation devices, loads and transmission lines may cause the system to resonate at various frequencies. Enabling each inverter to meet internal stability (considering the inverter itself) and external stability (considering the overall system) can eliminate the risk of resonance [26]. For the adopted method, internal stability is well satisfied [24]. External stability is determined by the output admittance of the inverter, as it reflects the external characteristics of the inverter. External stability can be assessed according to the frequency-domain passivity theory [27]. A system is stable if each component is passive [28]. The grid admittance is generally passive. Therefore, the system stability can be guaranteed if the output admittance of each inverter in the system is passive. The inverter output admittance Y_o is passive if it satisfies the following two conditions: (1) Y_o has no right-half-plane pole, (2) the real part of Y_o is non-negative in the controllable frequency range, i.e., [0, f_s/2] [29]. Condition 1 also implies that an inverter can operate stably if the influence of the other components in the system is negligible, which corresponds to internal stability. Condition 2 corresponds to external stability [29]. Reference [28] proposed a passive enhancement method for inverters with single-loop feedback control, which cannot guarantee the output admittance passive throughout the controllable frequency range. Reference [26] proposed a passive enhancement method for inverters with grid-side current feedback control and capacitor current feedback active damping, which requires a proper selection of LCL filter parameters to prevent the LCL resonant frequency from entering the unstable region. In this paper, the adopted active damping method [24] can almost eliminate the unstable region of the LCL resonant frequency, making it easier to design the LCL filter parameters than in [26].

This paper reveals that for the method proposed in [24], the real part of Y_o is negative above a certain frequency, implying the existence of a non-passive admittance region. In this case, LC resonance may be generated between the inverter output admittance and the equivalent grid admittance [28], indicating system instability. To eliminate the non-passive region of Y_o, this paper proposes to insert a phase lead compensator (PLC) cascaded in series with the current regulator. In this way, the inverter satisfies internal and external stability over the entire controllable frequency range.

The proposed scheme improves the grid-side current quality with enhanced system stability, which requires the grid-side current for feedback control and the capacitor current for active damping. It is preferable to avoid investments in the corresponding measurement devices, which leads to the development of the sensorless methods [30,31,32]. This paper calculates the capacitor current from the capacitor voltage measured for the phase-locked loop (PLL). The estimated capacitor current is subtracted from the measured inverter-side current, by which the approximate grid-side current can be obtained. Through the above method, the applied measurement devices are identical to that of the inverter-side current single-loop feedback control scheme, retaining the latter’s advantage that only a few measurement devices are required.

The rest of this paper is organized as follows. Section 2 revisits the inverter-side current single-loop feedback control with capacitor current compensation, unveiling that the restriction on LCL resonant frequency results in limited improvement in harmonic suppression. Section 3 reveals the presence of a non-passive region with the adopted damping method, which can be eliminated by the proposed PLC, and illustrates that special care should be taken when using the calculated capacitor current for active damping. Section 4 investigates the effectiveness of the PLC in improving external stability through a case study. Section 5 demonstrates the excellent harmonic suppression capability of the proposed method. Section 6 verifies the theoretical analysis through simulations and experiments. Section 7 concludes this paper.

2. Problem Formulation for Current Single-Loop Feedback Control

2.1. System Description

Figure 1a shows the single-phase equivalent circuit of a three-phase balanced wind power generation system. A permanent magnet synchronous generator (PMSG) is connected to the grid through a back-to-back dual-PWM converter. Replacing the DC side with other renewable energy sources does not affect the analytical conclusions. The grid-side inverter is connected to the grid through an LCL filter composed of the inverter-side inductor L₁, the filter capacitor C_f and the grid-side inductor L₂. L_g is the grid inductor. i_L1, i_c, i_L2, v_c, v_g and V_in are the inverter-side current, the capacitor current, the grid-side current, the measured capacitor voltage, the grid voltage and the DC-link voltage, respectively.

The current control is implemented in the αβ reference frame. The current regulator consists of a proportional controller, several quasi-resonant controllers for reference tracking and selective harmonic suppression [33], expressed as

$$G_i(s)=K_p+{\displaystyle \sum _{h=1}^M{K}_{ih}\frac{s\cdot \cos ({\theta }_h)-{\omega }_h\cdot \sin ({\theta }_h)}{s^2+{\omega }_h^2}}$$

(1)

where K_p is the proportional gain, ω_h is the angular frequency at hω₀, ω₀ is the fundamental angular frequency, h is the harmonic order, K_ih is the resonant gain at ω_h, M is the maximum order of the quasi-resonant controller and θ_h (h = 1, 2, …, M) is the phase lead for mitigating the h th-order resonant controller’s effect on the phase margin, which helps to extend the harmonic suppression range of the current regulator from below the open-loop crossover frequency to approach the LCL resonant frequency [23].

2.2. Analysis of the Harmonic Suppression Capability under the Single-Loop Feedback Control

The current injected into the grid should comply with relevant standards, such as [34], which specify the allowable harmonic current injected into the grid. For the conventional inverter-side current single-loop feedback control, the inverter-side current is taken as the controlled object. Since the inverter-side reference current contains only fundamental components, the resonant controllers can only effectively suppress the harmonic components at the inverter-side rather than that at the grid-side, which may make the grid-side current fail to meet relevant requirements.

2.2.1. The Improved Inverter-Side Current Single-Loop Feedback Control

Adding the capacitor current harmonic information to the inverter-side reference current can directly suppress the grid-side current harmonics [15,22]. Figure 1b1 extracts the capacitor current harmonics from the capacitor voltage using multiple differentiators based on second-order generalized integrators [22]. Figure 1b2 calculates the capacitor current from the capacitor voltage using non-ideal generalized integrator derivatives. The calculated capacitor current is then added to the harmonic resonance controllers’ branch, from which the harmonic information of the capacitor current can be obtained [15]. Incorporating the harmonic information into the inverter-side current reference value results in the inverter-side current containing the 6k ± 1th order harmonics of the capacitor current, k = 1, 2, …, M, which indicates that the harmonic currents will flow through the L₁–C_f current loop instead of the L₂–C_f current loop, making the grid-side current free of them. However, the improvement on the grid-side current is limited. First, the methods in [15,22] are equivalent to the grid-side current single-loop feedback control for only specific resonant frequencies, while for other frequencies, they are still equal to the inverter-side current single-loop feedback control. In addition, only a handful of resonance controllers can be used due to the limitation of the relatively low LCL resonant frequency. The usage of resonance controllers is further restricted when the LCL resonant frequency decreases as the grid inductance increases. Taking the method in [15] as the analysis object, the LCL parameters in this section are set as follows: L₁ = 860 μH, L₂ = 90 μH and C_f = 230 μF. Only C_f differs from the corresponding value in

Table 1. Parameters of the main circuit and the controller.

Parameters	Value	Parameters	Value
Grid voltage Vg/V	220	Resonant frequency fr_pro/Hz	2788
Grid fundamental frequency f0/Hz	50	Current sensor gains Hi, Hi2	0.15
Inverter DC-link voltage Vin/V	360	Feeder inductance Lc/μH	196
Amplitude of the triangular carrier Vtri/V	4.58	Feeder capacitance Cc/μF	0.24
Sampling frequency fs/kHz	20	Load equivalent capacitance Cload/μF	0.6
LCL—Inverter-side inductor L1/μH	860	Proportional gain Kp	0.405
LCL—Grid-side inductor L2/μH	90	hth resonator gain Kih	32
LCL—Capacitor Cf /μF	5	Capacitor current feedback coefficient Hi1	−0.06
Resonant frequency fr0_inv/Hz	1163	Capacitor voltage feedback coefficient K	−1600
Resonant frequency fr_inv/Hz	411	Lead compensator factor αlead	1.42
Resonant frequency fr0_pro/Hz	7885	Lead compensator factor τlead	4 × 10−5

. The parameters are set in this way to make the frequency characteristic close to that of [15] and to facilitate the comparison with the proposed scheme. The initial resonant frequency of an LCL filter is expressed as

$$f_{r0}=\frac{1}{2\pi }\sqrt{\frac{L_1+L_2}{L_1\cdot L_2\cdot C_f}}.$$

(2)

When the grid inductance L_g = 0, taking the above LCL parameters into (2), the initial resonant frequency is obtained as f_{r0_inv} ≈ 1163 Hz. When L_g increases to 10% per unit, i.e., 2.6 mH [24], replacing L₂ in (2) with L_T = L₂ + L_g, the actual resonant frequency is obtained as f_{r_inv} ≈ 411 Hz.

Figure 2 displays a series of equivalent control block diagrams of Figure 1b2, in which Figure 2a is successively simplified to Figure 2b–d. To make the transformation process concise, the grid voltage is first omitted.

In Figure 2a, H_i represents the sensor gain for both currents at the grid side and at the inverter side, A represents the harmonic resonance controller, B represents the proportional controller plus the fundamental resonance controller, C represents the transfer function from the current regulator output u to the inverter output voltage u_inv, composed of the control delay, i.e., 1.5T_s and the inverter gain K_pwm, D represents the transfer function from inverter output voltage u_inv to the inverter-side current i_L1, E represents the transfer function from the inverter-side current i_L1 to the grid-side current i_L2. A, B, C, D and E are expressed in order as

$$A={\displaystyle \sum _{h=2}^M{K}_{ih}\frac{s\cdot \cos ({\theta }_h)-{\omega }_h\cdot \sin ({\theta }_h)}{s^2+{\omega }_h^2}},$$

(3)

$$B=K_p+K_{i1}\frac{s\cdot \cos ({\theta }_1)-{\omega }_1\cdot \sin ({\theta }_1)}{s^2+{\omega }_1^2},$$

(4)

$$C=e^{-1.5s{T}_s}\cdot K_{PWM},$$

(5)

$$D=\frac{1}{L_1{L}_T{C}_f}\cdot \frac{L_T{C}_f{s}^2+1}{s(s^2+{\omega }_{res}^2)},$$

(6)

$$E=\frac{1}{L_T{C}_f{s}^2+1}.$$

(7)

2.2.2. Closed-Loop Response Analysis

The block diagram in Figure 2d is redrawn in Figure 3a1 for easy visibility. From Figure 3a1, the grid-side current i_L2 is derived as

$$i_{L2}(s)=G_{cl}(s)i_{L1}^{*}(s)$$

(8)

where G_cl (s) denotes the closed-loop transfer function of the inverter-side current reference value i_L2^* to i_L2, expressed as

$$G_{cl}(s)=\frac{H_i(A+B)CDE}{1+H_i(DA+B)CE}.$$

(9)

The Bode plot of G_cl (s) is given in Figure 3b1, in which the amplitudes at the characteristic harmonic frequencies are all close to 0 dB, indicating that the original method cannot suppress the characteristic harmonics induced by the reference current. When disconnecting the path from the grid-side current reference i_L1^*(s) through controller A to the controller output u, the corresponding control block diagram is redrawn in Figure 3a2. The grid-side current i_L2 is expressed as

$$i_{L2}(s)=G_{cl}^{\prime }(s)i_{L1}^{*}(s)$$

(10)

where G_cl^′(s) is the corresponding closed-loop transfer function from i_L1^* to i_L2, expressed as

$$G_{cl}^{\prime }(s)=\frac{H_{i}BCDE}{1+H_i(DA+B)CE}.$$

(11)

Figure 3b2 shows the Bode plot of G_cl^′(s), indicating that the characteristic harmonics brought by i_L1^* can be effectively suppressed. Figure 4a shows the Bode plots of G_cl^′(s) when different numbers of harmonic resonance controllers are used. The amplitude–frequency response of G_cl^′(s) has a spike around the LCL resonance frequency, which will increase as more resonance controllers are applied. During the inverter startup or when the reference current undergoes a step change, the harmonic components around the spike frequency will appear in the reference current and be amplified in the grid-side current.

2.2.3. Open-Loop Response Analysis

The open-loop transfer function can be obtained from Figure 3a2 as

$$G_{ol}(s)=CD(AE+B)H_i.$$

(12)

When L_g = 0, the 5th, 7th, 11th and 13th harmonic resonance controllers are inserted. Figure 4b shows the Bode plot of G_ol (s). The phase margin PM₁ is 55.6°, and the gain margin GM₁ is 11.4 dB. Both of them satisfy the stability margin requirements in classical control theory: PM ϵ (30°, 60°), GM ≥ 3–6 dB. Figure 5a shows the Bode plot of G_ol (s) after the 17th order resonance controller is inserted. One positive crossing occurs near the 17th harmonic frequency, while no negative crossing occurs. Moreover, the open-loop transfer function G_ol (s) has no right-half-plane pole. According to the Nyquist stability criterion, the closed-loop transfer function G_cl^′(s) has a pair of right-half-plane poles, which indicates that the system is unstable. To leave a sufficient gain margin, harmonic resonance controllers’ usage is further limited. Figure 5b shows the corresponding open-loop Bode plot with only the 5th order resonant controller inserted when L_g increases to 10% per unit, i.e., 2.6 mH [24]. One negative crossing occurs near the 5th harmonic frequency, while no positive crossing occurs. Moreover, the open-loop transfer function has no right-half-plane pole. Similar to the previous analysis, the system is unstable, which indicates that the stability of the resonance controllers will be affected by the LCL resonance if it is not properly damped. For this reason, resonant controllers are usually set below the open-loop crossover frequency [2,35].

2.2.4. Harmonic Suppression Analysis

Figure 6a shows the control block diagram for analyzing the effect of the grid voltage v_g. F is the open-loop transfer function from v_g to i_L2, denoted as

$$F=\frac{L_T{C}_f{s}^2+1}{L_1{L}_T{C}_f{s}^3+(L_1+L_T)s}.$$

(13)

The closed-loop transfer function from v_g to i_L2 can be derived from Figure 6a as

$$G_{vcl}(s)=-\frac{F}{1+CD(AE+B)H_i}.$$

(14)

Figure 6b shows the Bode plot of G_vcl (s), which illustrates that the grid-side current can be free from the 1st, 5th, 7th, 11th and 13th order components of the grid voltage.

2.2.5. Simulation Verification

To verify the above analysis, a set of simulations are performed with MATLAB/Simulink.

Simulation 1: Set L_g = 0. The inverter-side current reference value i_L1^* jumps from 30 A to 40 A at 0.225 s and from 40 A back to 30 A after 0.08 s. The voltage source contains 6k ± 1 order harmonic voltage with an amplitude of 30 V, where k = 1, 2. Harmonic resonance controllers of the corresponding orders are inserted. Figure 7a,b show the grid-side current waveform and its fast Fourier transform (FFT) analysis result, respectively. In Figure 7a, during the half fundamental period after i_L1^* jumps, the grid-current is seriously distorted, which is caused by the harmonics around the spike frequency. Due to the space limitation, the FFT analysis result is not presented. After half a fundamental cycle, the grid-side current distortion basically disappears. An FFT analysis is performed on the periodic waveform marked red in Figure 7a, and the results are shown in Figure 7b. The total harmonic distortion (THD) of i_L2 is 1.64%, which proves that the resonance controllers can effectively suppress the 6k ± 1 order harmonics, where k = 1, 2.

Simulation 2: Set L_g = 0, the inverter-side reference current i_L1^* = 30sinω₀t A. The voltage source contains harmonic components of the 5th, 7th, 11th, 13th and 17th orders with an amplitude of 30 V. Harmonic resonance controllers with the corresponding orders are used. Figure 8a,b show the grid-side current waveform and its FFT analysis result, respectively. In Figure 8a, the grid-side current oscillates when the inverter starts. An FFT analysis is performed on the periodic waveform marked red in Figure 8a, with the results shown in Figure 8b. The main disturbance components are the harmonics around 850 Hz, indicating that the 17th order resonance controller destroys the system stability. The THD of i_L2 is 117.97%.

Simulation 3: Set L_g = 2.6 mH, the inverter-side reference current i_L1^* = 30sinω₀t A. The voltage source contains the 5th order harmonic with an amplitude of 30 V. The 5th order resonance controller is used. Figure 9a,b show the grid-side current waveform and its FFT analysis result, respectively. In Figure 9a, the grid-side current is severely distorted. An FFT analysis is performed on the periodic waveform marked red in Figure 9a, with the results shown in Figure 9b. The main disturbance components are the harmonics at and around 250 Hz, indicating that the 5th resonance controller not only fails to suppress the 5th harmonic but also causes harmonic amplification, making the system unstable. The THD of i_L2 is 107.14%.

The above analysis and simulation results indicate that the improved single-loop feedback control of the inverter-side current [15,22] allows only a few low-order resonant controllers to be used, which provides limited improvement on the grid-side current quality.

2.2.6. The Equivalent Grid-Side Current Single-Loop Feedback Control

Figure 1b3 shows the equivalent grid-side current single-loop feedback control, for which the stability region of the LCL resonant frequency is (f_s/6, f_s/2) [4]. Compared with the inverter-side current single-loop feedback control in Figure 1b1,b2, a higher LCL resonant frequency can be set. Accordingly, more high-order harmonic resonance controllers can be incorporated. However, when the grid inductance increases, the LCL resonant frequency may enter the instability region. The system instability can be avoided by setting the possible minimum LCL resonant frequency greater than f_s/6, but a large filter inductor is required to ensure the switching ripple attenuation performance [24].

3. The Proposed Method for Passivity Enhancement

3.1. The Proposed Control Strategy

Figure 1b4 presents the control strategy proposed in this paper. A digital differentiator is applied to calculate the capacitor current. Next, the calculated capacitor current is subtracted from the detected inverter-side current, by which the equivalent grid-side current is obtained for feedback control. The calculated capacitor current and the detected capacitor voltage are used for active damping, which eliminates the LCL resonant frequency instability region. In this way, a higher LCL resonant frequency can be set compared with [15,22] without violating the system stability. More high-order harmonic resonant controllers can then be inserted, leading to a sinusoidal grid-side current. The PLC marked in Figure 1b4 is employed to enhance the system stability. Detailed analyses are presented in the following.

Figure 10 shows the block diagram of Figure 1b4. G_i (s) is the current regulator. G_lead (s) represents the PLC. G_d (s) represents the time delay in the digital control, i.e., G_d (s) = e^−1.5Tss, where T_s is the sampling period. K_pwm is the inverter bridge gain, i.e., K_pwm = V_in/V_tri, where V_in is the inverter DC-link voltage, and V_tri is the triangular carrier amplitude. H_i1 and K are the proportional coefficients for active damping. H_i2 is the sensor gain of the grid-side current i_L2. L₁, C_f and L₂ compose the LCL filter. v_PCC represents the voltage at the point of common coupling (PCC). The parameters of the primary circuit and the controller are listed in Table 1. For comparison purposes, some parameters are the same as those in [24].

3.2. Modeling of the Inverter Output Admittance

Figure 11a shows the simplified form of Figure 10, which is obtained through a series of equivalent transformations, where

$$G_{x1}(s)=\frac{G_i(s)G_d(s)K_{PWM}}{s^2{L}_1{C}_f+s{C}_f(H_{i1}+\frac{K}{s})G_d(s)K_{PWM}+1},$$

(15)

$$G_{x2}(s)=\frac{s^2{L}_1{C}_f+s{C}_f(H_{i1}+\frac{K}{s})G_d(s)K_{PWM}+1}{s^3{L}_1{L}_2{C}_f+s^2{L}_2{C}_f(H_{i1}+\frac{K}{s})G_d(s)K_{PWM}+s(L_1+L_2)}.$$

(16)

The open-loop transfer function is derived from Figure 11a as

$$T(s)=G_{x1}(s)G_{x2}(s)H_{i2}=\frac{H_{i2}{G}_i(s)G_d(s)K_{PWM}}{s^3{L}_1{L}_2{C}_f+s^2{L}_2{C}_f(H_{i1}+\frac{K}{s})K_{PWM}{G}_d(s)+s(L_1+L_2)}.$$

(17)

To perform the passive analysis of the inverter output admittance, the grid-connected inverter is represented by the Norton equivalent circuit [19], as shown in Figure 11b. The grid-side current is derived as

$$i_{L2}(s)=G_{cl}(s)i_{L2}^{*}(s)-v_{PCC}(s)\cdot Y_o(s)$$

(18)

where i_L2^*(s) is the grid-side reference current, G_cl (s) is the closed-loop transfer function, Y_o (s) is the inverter output admittance. G_cl (s) and Y_o (s) are denoted as

$$G_{cl}(s)=\frac{T(s)}{1+T(s)},$$

(19)

$$Y_o(s)=\frac{G_{x2}(s)}{1+T(s)}.$$

(20)

3.3. Passive Analysis of the Inverter Output Admittance

Recalling the passivity theory [27], the inverter output admittance Y_o is passive if it satisfies (21), where ω_s is the sampling angular frequency.

$$\mathrm{Re}\left\{Y_o(j\omega )\right\}\ge 0, 0\le \omega \le {\omega }_s/2$$

(21)

Substituting s = jω into (20) and taking its real part results in

$$\mathrm{Re}\left\{Y_o(j\omega )\right\}=N\cdot (H_{i2}{K}_{p}K{C}_f+(H_{i2}{K}_p-H_{i2}{K}_p{L}_1{C}_f{\omega }^2+H_{i1}{L}_1{C}_f{\omega }^2)\cdot K_{PWM}\cos (1.5\omega T_s)-K{L}_1{C}_f\omega K_{PWM}\sin (1.5\omega T_s)).$$

(22)

In (22), N is positive for 0 ≤ ω ≤ ω_s/2. Therefore, it is irrelevant to the passivity of Y_o. The expression is not listed due to its complexity. Figure 12 shows the frequency characteristic curve of Re {Y_o (jω)}.

In Figure 12, Re {Y_o (jω)} is negative above 9472 Hz, indicating that the frequency range from 9472 Hz to 10,000 Hz (the Nyquist frequency f_s/2) is the non-passive region of Y_o (jω).

3.4. The Proposed Passivity Enhancement Method

To eliminate the non-passive region, i.e., (9472 Hz, 10,000 Hz), this paper proposes to use a PLC [26] that corrects the phase of Y_o in this region to within ±90°.

The transfer function of the PLC is given as

$$G_{lead}(s)=\frac{1+{\alpha }_{lead}{\tau }_{lead}s}{1+{\tau }_{lead}s}$$

(23)

where α_lead and τ_lead are the parameters, written as

$${\alpha }_{lead}=\frac{1+\sin {\phi }_{m\_lead}}{1-\sin {\phi }_{m\_lead}},$$

(24)

$${\tau }_{lead}=\frac{1}{\sqrt{{\alpha }_{lead}}\cdot {\omega }_{lead}}.$$

(25)

where φ_{m_lead} is the phase lead provided at ω_lead. The maximum phase lag to compensate is about 6.9°, located at the Nyquist frequency. Accordingly, set φ_{m_lead} = 30° and ω_lead = ω_s/2 initially. α_lead = 1.42 and τ_lead = 4 × 10⁻⁵ can be calculated from (24) and (25), respectively, listed in Table 1. Figure 13 shows the phase diagrams of the output admittance Y_o (Y_o^′) before and after using the PLC.

Before using the PLC, the phase angle of Y_o satisfies ∠Y_o (f) < −90°, ∀ f > 9472 Hz. With the PLC applied, the output admittance is renamed Y_o^′. The phase angle of Y_o^′ satisfies ∠Y_o^′(f) ϵ (−90°, 90°), i.e., Re (Y_o^’(f)) > 0, ∀ f ϵ (0, f_s/2), which means the passivity is guaranteed for any Y_o^′ within f ϵ (0, f_s/2). To summarize the proposed method, a flowchart is depicted in Figure 14.

3.5. Impact of the PLC on Other Critical Characteristics

This section explores what will happen to the performance of the original system when the PLC is added. When L_g = 0, it is verified that the original system has no open-loop right-half-plane pole [24]. The PLC has no open-loop right-half-plane pole either. Thus, no open-loop right-half-plane pole exists in the upgraded system. For the grid-side current feedback control with active damping [24], Figure 15 shows the open-loop Bode plots before and after using the PLC under L_g = 0.

In Figure 15, within the open-loop bandwidth, the characteristic curves of the original system and the upgraded system coincide essentially, while in the high-frequency range, both the amplitude–frequency and phase–frequency characteristic curves are raised by the PLC. At the Nyquist frequency, the amplitude increases the most, which is 4.8 dB. In fact, the slope beyond the LCL resonance frequency is close to −60 dB/dec, indicating that the high-frequency harmonic attenuation ability is slightly degraded but still meets the requirements. The phase margin PM₂ is 56°, an increase of 10° from PM₁ of the original system. The gain margin GM₂ is 9.43 dB, a decrease of 0.21 dB from GM₁ of the original system. PM₂ and GM₂ satisfy the stability margin requirements of no less than 45° and 3 dB [24], respectively, indicating that internal stability is preserved. Furthermore, a larger φ_{m_lead} results in a more sufficient phase margin for Y_o^′, but will further reduce the attenuation of the high-frequency harmonics. For this reason, this paper selects φ_{m_lead} = 30° to achieve a balance between the stability and the harmonic attenuation.

The s-domain transfer functions from the inverter output voltage v_inv (s) to the grid-side current i_L2 (s), to the capacitor current i_c (s) and to the capacitor voltage v_c (s) can be derived from Figure 10 as follows:

$${\mathrm{G}}_{iL2}(s)=\frac{i_{L2}(s)}{v_{inv}(s)}=\frac{1}{s{L}_1}\frac{{\omega }_{L_{T}C}^2}{s^2+{\omega }_{res}^2},$$

(26)

$${\mathrm{G}}_{ic}(s)=\frac{i_c(s)}{v_{inv}(s)}=\frac{1}{s{L}_1}\frac{s^2}{s^2+{\omega }_{res}^2},$$

(27)

$${\mathrm{G}}_{vc}(s)=\frac{v_c(s)}{v_{inv}(s)}=\frac{1}{L_1{C}_f}\frac{1}{s^2+{\omega }_{res}^2},$$

(28)

where ${\omega }_{L_{T}C}=\sqrt{\frac{1}{L_T\cdot C_f}}$, ${\omega }_{res}=\sqrt{\frac{L_1+L_T}{L_1\cdot L_T\cdot C_f}}$, $L_T=L_2+L_g$.

Discretizing (26)–(28) using the zero-order holder method yields

$$G_{iL2}(z)=\frac{i_{L2}(z)}{v_{inv}(z)}=\frac{T_s}{(L_1+L_T)(z-1)}-\frac{\sin ({\omega }_{res}{T}_s)}{{\omega }_{res}(L_1+L_T)}\frac{z-1}{z^2-2\cos ({\omega }_{res}{T}_s)z+1},$$

(29)

$$G_{ic}(z)=\frac{i_c(z)}{v_{inv}(z)}=\frac{\sin ({\omega }_{res}{T}_s)}{{\omega }_{res}{L}_1}\frac{z-1}{z^2-2\cos ({\omega }_{res}{T}_s)z+1},$$

(30)

$$G_{vc}(z)=\frac{v_c(z)}{v_{inv}(z)}=\frac{L_T}{L_1+L_{\mathrm{T}}}\frac{(1-\cos ({\omega }_{res}{T}_s))z+1-\cos ({\omega }_{res}{T}_s)}{z^2-2\cos ({\omega }_{res}{T}_s)z+1},$$

(31)

The current regulator (1) is discretized with reference to [23]. A forward Euler integrator-based discretization method is used. The denominator with the sixth-order Taylor series is for enhancing the resonant pole accuracy. We obtain

$$G_i(z)=K_p+{\displaystyle \sum _{h=1}^M{K}_{ih}{T}_s\frac{\cos ({\theta }_h)-z^{-1}\cos ({\theta }_h-{\omega }_h{T}_s)}{1-2z^{-1}\cos ({\omega }_h{T}_s)+z^{-2}}}.$$

(32)

Discretizing the PLC (23) using the bilinear transform with prewarping at f_s/2, we obtain

$$G_{lead}(z)=\frac{a_{1}z+a_2}{b_{1}z+b_2}$$

(33)

where a₁ = 1.257, a₂ = −0.4782, b₁ = 1, b₂ = −0.2215.

Figure 16 shows the discrete time-domain control model corresponding to Figure 10. The grid voltage as an interference term does not affect the analysis results of this section, so it is ignored. To simplify the analysis process, the harmonic resonance controller in (32) is also neglected.

In Figure 16, the control model comprises three feedback loops. The corresponding open-loop transfer functions are denoted as G_inner (z), G_middle (z) and G_outer (z), with the expressions as

$$G_{inner}(z)=\frac{z^{-1}{K}_{PWM}}{1+z^{-1}{K}_{PWM}{H}_{i1}{G}_{ic}(z)},$$

(34)

$$G_{middle}(z)=\frac{G_{inner}(z)}{1+G_{inner}(z)G_{vc}(z)CK},$$

(35)

$$G_{outer}(z)=G_i(z)\cdot G_{lead}(z)\cdot G_{middle}(z)\cdot G_{iL2}(z)\cdot H_{i2}.$$

(36)

According to (36), Figure 17 plots the closed-loop pole diagram with the grid inductance variation, where the closed-loop pole diagram given in [24] is also plotted for comparison. The pair of closed-loop poles introduced by the resonant part of the current regulator is not given since they almost do not move. In Figure 17, even if the grid inductance varies widely, all the closed-loop poles are in the unit circle with a sufficient distance from it, demonstrating that the inverter still has good stability–robustness after applying the PLC.

3.6. Effect of the Calculated Capacitor Current on the Control Performance

In practical applications, the capacitance current can be calculated from the capacitance voltage utilizing a digital differentiator in (37), instead of installing a set of measurement devices, avoiding additional investment.

$$G_{dd}(z)=\frac{2}{T_s}\cdot \frac{z^2+0.5z-0.5}{z^2+0.25z-0.25}\cdot \frac{z-1}{z+1}$$

(37)

Figure 17 shows the Bode plots of the digital differentiator in (37) and the pure derivative for comparison. In [24], the grid inductance L_g may change from 0 to 2.6 mH. The corresponding LCL resonant frequency will vary over a wide range, i.e., [2788 Hz, 7885 Hz], as marked in Figure 18.

From Figure 18, the digital differentiator G_dd (z) closely matches the pure derivative in a wide frequency range. Furthermore, in the low-frequency range, the reactance of C_f is much larger than that of L₁ and L₂, resulting in a negligible capacitance current, so the calculation error of the capacitance current is negligible. Therefore, for this frequency range, the difference between the measured inverter-side current and the calculated capacitor current can be used to achieve the equivalent grid-side current feedback control [36]. In contrast, in the high-frequency range, the phase–frequency curve drops rapidly, which may cause the digital differentiator to lose its derivative characteristic so that the calculated capacitance current will lose its usefulness in active damping.

i_c (z) is replaced with the calculated capacitor current i_c^′(z) for feedback, whose feedback path is depicted by a dashed line in Figure 16. Figure 19 shows the corresponding closed-loop pole map according to (35), where G_ic (z) in (33) is substituted by C_f •G_dd (z) •G_vc (z). In Figure 19, a pair of closed-loop conjugate poles is outside the unit circle when L_g = 0, indicating an unstable system. As L_g increases, the closed-loop conjugate poles gradually move into the unit circle, which means the system becomes stable. From Figure 18 and Figure 19, the system stability is influenced by the damping effect of i_c^′, determined by the proximity of i_c^′ to i_c. For the LCL filter parameters in Table 1, the corresponding LCL resonant frequency is above the highest frequency that i_c^′ can provide effective damping. Consequently, to use the calculated capacitor current for active damping, these LCL filter parameters need to be modified. A higher LCL resonant frequency requires smaller LCL elements, for which above f_s/6 is cost-effective. However, the ripple attenuation capacity decreases as the LCL resonant frequency increases. For this reason, a compromise should be made when designing an LCL filter. A further study on this is out of the scope of this paper.

4. A Case Study on Stability Evaluation

4.1. System Description

Figure 20 displays a typical per-phase topology diagram of multiple renewable energy inverters connected to the grid through a power cable. The power cable is represented by the Π-type equivalent circuit, assuming that each section has the same parameters. C₁ represents the capacitive load connected to point A, L_g represents the grid inductance and C_CPF represents the power factor correction capacitor. View from the PCC point, when C_CPF is installed, the grid is inductive or capacitive, and when it is not installed, the grid is inductive. For both cases, the grid impedance is generally passive. The above parameters are listed in Table 1.

Figure 21 depicts the admittance-based equivalent circuit of Figure 20. Four inverters are considered in this case. The inverter at point A is taken as the analysis object, which is represented by the Norton equivalent circuit. The other inverters are represented by the output admittance Y_o. For the inverter connected to point A, the equivalent load admittance Y_Load is the series-parallel combination of the grid admittance, the admittance of each section of the power cable, the output admittance of other inverters, and the admittance of the capacitive load connected to point A.

4.2. Stability Evaluation of the Proposed Method

Considering the equivalent load admittance Y_Load’s effect, the closed-loop response is expressed as

$$i_{L2}(s)=\frac{1}{1+Y_o(s)/Y_{Load}(s)}\cdot G_{cl}(s)\cdot i_{L2}^{*}(s).$$

(38)

Internal stability can be satisfied if the closed-loop transfer function G_cl (s) is stable. For an inverter with internal stability, its external stability depends on the minimum feedback loop formed by Y_o (s) and Y_Load (s) [28]. Figure 22 shows the frequency response of the inverter output admittance Y_o (Y_o^′) and the equivalent load admittance Y_Load (Y_Load^′) before and after using the PLC.

In Figure 22a, before using the phase lead compensator, the intersection point of the amplitude–frequency response of Y_Load and Y_o is located in the non-passive region of Y_o. At the intersection frequency, the phase angle difference between Y_Load and Y_o is above 180°, which means the phase margin of the minimum feedback loop is negative, indicating that an LC resonance may be triggered. With the PLC, Y_o^′ is passive all over the controllable frequency range, i.e., (0, f_s/2). Since all components of Y_Load^′ are passive, Y_Load^′ is also passive [28]. As illustrated in Figure 22b, the phase angle difference between Y_o^′ and Y_Load^′ is less than 180° all over (0, f_s/2), which means the phase margin of the minimum feedback loop is positive, indicating that the LC resonance risk has been eliminated. It can be concluded from above that internal and external stability can be satisfied for inverters with the proposed control strategy.

5. Analysis of the Harmonic Suppression Capability of the Proposed Method

When the grid inductance L_g = 0, taking the LCL parameters from Table 1 into (2), the initial resonant frequency can be obtained as f_{r0_pro} ≈ 7885 Hz. When L_g increases to 10% per unit, i.e., 2.6 mH [24], replacing L₂ in (2) with L_T = L₂ + L_g, the actual resonant frequency is obtained as f_{r_pro} ≈ 2788 Hz. Both f_{r0_pro} and f_{r_pro} are much higher than f_{r0_inv} and f_{r_inv} in Section 2.2. Therefore, the LCL resonant frequency characteristics are less restrictive for the use of resonant controllers. The following will verify that more high-order resonant controllers can be inserted to improve the grid-side current quality.

According to the analysis in Section 2.2, to reduce the current distortion during inverter start-up or step change of the reference current, Figure 23 is substituted for Figure 10, where A denotes the harmonic resonance controllers, B denotes the proportional controller plus the fundamental resonance controller, as given in (3) and (4), respectively.

Performing a series of equivalent transformations on Figure 23 obtains the equivalent block diagrams given in Figure A1 of Appendix A, from which the following transfer functions can be derived.

The open-loop transfer function from i_L2^* to i_L2 is expressed as

$$G_{ol\_pro}(s)=\frac{G_{lead}(s)G_d(s)K_{PWM}{H}_{i2}B}{L_1{L}_T{C}_f{s}^3+L_T{C}_f(s{H}_{i1}+K)K_{PWM}{G}_d(s)s+s(L_1+L_T)+G_{lead}(s)G_d(s)K_{PWM}{H}_{i2}A}.$$

(39)

The closed-loop transfer function from i_L2^* to i_L2 is expressed as

$$G_{cl\_pro}(s)=\frac{G_{lead}(s)G_d(s)K_{PWM}{H}_{i2}B}{L_1{L}_T{C}_f{s}^3+L_T{C}_f{G}_d{K}_{PWM}(H_{i1}s+K)s+(L_1+L_T)s+G_{lead}(s)G_d(s)K_{PWM}{H}_{i2}(A+B)}.$$

(40)

The closed-loop transfer function from v_g to i_L2 is expressed as

$$G_{vcl\_pro}=\frac{L_1{C}_f{s}^2+G_d(s)K_{PWM}{C}_f{H}_{i1}s+G_d(s)K_{PWM}{C}_{f}K+1}{L_1{L}_T{C}_f{s}^3+G_d(s)K_{PWM}{L}_T{C}_f{H}_{i1}{s}^2+(L_1+L_T+G_d(s)K_{PWM}{L}_T{C}_{f}K)s+(A+B)H_{i2}{G}_{lead}(s)G_d(s)K_{PWM}}.$$

(41)

The Bode plot of G_{ol_pro} (s) is shown in Figure 24a when L_g = 0. To make it easy to observe, the highest order of the harmonic resonance controller applied is the 67th order. The phase margin PM₁ is 47.5°. The gain margin GM₁ is 9.44 dB. They both satisfy the stability margin requirements of the classical control theory, i.e., PM ϵ [30°, 60°], GM ≥ 3–6 dB [10].

The Bode plot of G_{ol_pro} (s) is shown in Figure 24b when L_g = 2.6 mH. For ease of observation, the highest order of the harmonic resonance controller applied is the 31st order. The phase margin PM₁ and the gain margin GM₁ of G_{ol_pro} (s) are 58° and 5.66 dB, respectively. They both satisfy the stability margin requirements of the classical control theory.

It can be drawn from above that by suppressing the LCL resonance, the restriction of it on the stability of resonant controllers is mitigated. A higher LCL resonant frequency than that of [15] further extends the harmonic rejection range.

The Bode plots of G_{cl_pro} (s) are shown in Figure 25a,b when L_g = 0 and L_g = 2.6 mH, respectively. Note that unlike the amplitude–frequency response of G_cl^′(s) in Figure 4a, G_{cl_pro} (s) has no spike near the LCL resonant frequency, which will not cause the harmonic current amplification when a step change of the grid-side current reference value occurs.

The Bode plots of G_{vcl_pro} (s) are shown in Figure 26a,b when L_g = 0 and L_g = 2.6 mH, respectively, which illustrates that the resonant frequency components of the grid-side current, including the fundamental component and harmonic components, can be effectively attenuated.

To sum up, as for the improved inverter-side current single-loop feedback control schemes [15,22] in Section 2.2, when L_g = 0, the highest order of the harmonic resonance controller available is the 13th order. When L_g = 2.6 mH, it is hard to use a resonance controller to suppress any characteristic harmonic. In this paper, the available order of the harmonic resonance controller is much higher than that of [15] and [24] in both cases, which means that the current quality at the grid side can be significantly improved.

6. Simulation and Experimental Verification

For validating the analysis in the above sections, simulations and experiments are performed with the parameters listed in Table 1. The simulations are conducted on Matlab/Simulink. The experimental setup is shown in Figure 27. The modulation strategy is space vector modulation with a 10 kHz switching frequency for IGBT.

6.1. Case 1: Verification of the Analysis in Section 3.6

The active damping is performed with the capacitance current measurement value i_c before t₁ and the calculated value i_c^′ after t₁. The simulated waveform of the grid-side current and the FFT analysis of the part marked red are shown in Figure 28a1,a2. The corresponding experimental waveform is shown in Figure 28a3. When i_c^′ is used instead of i_c for control, the LCL resonance damping fails, and the protection device shuts off the inverter from the grid, preventing the inverter from being damaged by overcurrent. The results confirm the correctness of the analysis results in Section 3.6.

6.2. Case 2: Verification of the Stability-Enhancing Effect of the Proposed Method

The PLC is applied before t₁ and exited at t₁. The simulated waveform of the grid-side current and its FFT analysis result are shown in Figure 28b1,b2. The corresponding experimental waveform is shown in Figure 28b3. Figure 28b1 illustrates that the inverter operates stably with the proposed PLC. When the PLC is exited, the grid-side current starts to oscillate and diverge, indicating the system is unstable [37]. Figure 28b2 shows the corresponding FFT analysis result. From Figure 28b2, the disturbance components are close to the crossover frequency in Figure 22a, which proves that the oscillation of the grid-side current is caused by the resonance between the inverter output admittance and the equivalent load admittance, validating the analysis in Section 4.2. Figure 28b3 shows that the resonated current trips the overcurrent protection after the enable signal is triggered. The results verify the stability-enhancing effect of the proposed method.

6.3. Case 3: Verification of the Harmonic Suppression Capability of the Proposed Method

The performance of the proposed control method is tested under severely distorted grid voltage. The grid-side reference current i_L2^* experiences a step change from 30 A to 40 A at t₁ and jumps back to 30 A after 0.08 s.

For L_g = 0, the grid voltage contains harmonic components of 6k ± 1th with an amplitude of 30 V, k = 1, 2, …, 11. Correspondingly, the harmonic resonance controllers of 6k ± 1th are applied, k = 1, 2, …, 11. In Figure 29a1, during the half cycle after i_L2^* jumps, the waveform of i_L2 is not distorted. In Figure 29a2, the THD of i_L2 is 3.35%, much lower than the 5% required by the grid standard [34]. The experimental result in Figure 29a3 is consistent with the simulation results.

For L_g = 2.6 mH, the grid voltage contains harmonic components of 6k ± 1th with an amplitude of 30 V, k = 1, 2, …, 6. Correspondingly, the harmonic resonance controllers of 6k ± 1th are applied, k = 1, 2, …, 6. In Figure 29b1, during the half cycle after i_L2^* jumps, the waveform of i_L2 is close to a sine wave. In Figure 29b2, the THD of i_L2 is 2.67%, which is in compliance with the grid requirements. The experimental result in Figure 29b3 is consistent with the simulation results.

The results verify that the proposed control strategy achieves excellent grid-side current harmonic suppression.

7. Conclusions

The emphasis of this paper lies in enhancing the stability and harmonic suppression capability of LCL-type grid-connected inverters. The main contributions can be summarized as follows:

(1) The improved inverter-side current single-loop feedback control achieves controllability of the grid-side current harmonics, but the harmonic rejection range is narrow due to the stability requirement. The grid-side current feedback control with active damping permits higher-order resonance controllers to be used, further mitigating the grid-side current harmonic distortion.

(2) The PLC can elevate the phase of the inverter output admittance, making the real part of the inverter output admittance positive in the controllable frequency range, which means that the inverter output admittance is passive.

(3) When multiple inverters are connected to a non-ideal grid, making each inverter adopt the proposed control method, the potential resonance risk in the system can be eliminated.

(4) The calculated capacitor current can achieve the equivalent grid-side current feedback control and active damping, enabling the inverter to perform well without additional measuring devices.

Simulation and experimental results have confirmed the effectiveness of the proposed method.