Damping Characteristic Analysis of LCL Inverter with Embedded Energy Storage

Abstract

This paper investigates the system architecture and circuit topology of grid-connected inverters with embedded energy storage (EES), encompassing their modulation strategies and control methodologies. A mathematical model for an EES grid-connected inverter is derived based on capacitor current feedback control, from which the expression for the inverter’s output impedance is obtained. Building on this foundation, this study analyzes the influence of control parameters—such as the proportional coefficient, resonant coefficient, and switching frequency—on the inverter’s output impedance. Subsequently, the stability of single and multiple inverter grid-connected systems under various operating conditions is assessed using impedance analysis and the Nyquist criterion. Finally, the validity of the stability analysis based on the established mathematical model is verified through simulations conducted on the Matlab/Simulink platform, where models for both a single inverter and a two-inverter grid-connected system are constructed.

1. Introduction

With the rapid expansion of renewable energy sources, the pressure from the fossil fuel crisis has been reduced. However, the widespread adoption of renewable energy has transformed the nature of traditional power systems, resulting in a dual trend: increased renewable energy penetration and a higher proportion of power electronic devices in the system [1,2,3]. Within modern power systems, the interplay between power electronic devices and the grid has made system stability more complex. A large presence of power electronic devices in the grid can often lead to broadband oscillations, which may hinder the stable operation of the grid-connected system [4]. For example, in 2009, a sub-synchronous oscillation occurred in a renewable energy grid system in Texas, USA, leading to the disconnection of many wind turbines [5]. Between 2012 and 2016, sub-synchronous oscillations repeatedly affected wind farms in Guyuan, Hebei Province, China, with oscillation frequencies between 4 and 10 Hz, posing a threat to the grid’s stable and safe operation [6,7]. In 2015, a sub-synchronous oscillation with a frequency of 34 Hz took place in the Hami region of Xinjiang, China, due to the integration of permanent magnet direct-drive turbines in a weak grid. This issue resulted in high-frequency oscillations caused by the connection of multiple renewable energy power plants to the grid through long-distance transmission lines [8,9,10].

Wide-range-frequency oscillations fundamentally differ from the instabilities observed in traditional power systems, making it challenging to apply the stability analysis methods used in traditional power systems to study the stability issues of systems characterized by the high penetration of renewables and a high proportion of power electronic devices. Consequently, the stability of such dual-high systems has become a current research hotspot both domestically and internationally. Among the components of these systems, grid-connected inverters, as crucial interface devices for new energy grid-connected power generation, have received widespread attention [11]. Common research methods for analyzing the stability of inverter systems include eigenvalue analysis and impedance analysis. The former involves establishing a state-space model based on the grid-connected system, linearizing the system’s state equations using the small signal method, and then determining the system’s stability by calculating the eigenvalues and eigenvectors of the state space. However, any changes in the system structure or control parameters necessitate re-establishing the system’s state equations, complicating the modeling process. With the integration of new energy sources, grid impedance varies widely, further complicating stability analysis and reducing the method’s generality [12,13]. Currently, the commonly used method is impedance analysis [14]. This approach divides the grid-connected inverter and the grid into two independent systems, modeling their impedances separately. Changes in the parameters of one system do not affect the model of the other, allowing for stability analysis based on impedance stability criteria. When grid impedance changes, there is no requirement to rebuild the model of the grid-connected inverter. This method is widely applicable, and many studies have employed it to perform detailed analyses of the stability of voltage source inverter–grid coupled systems, resulting in several valuable conclusions.

As a crucial interface device between new energy systems and the power grid, grid-connected inverters require appropriate filters to suppress switching harmonics generated by switching devices during operation, ensuring that the output grid-connected current meets harmonic requirements. The main types of filters are L-type and LCL-type. Under the same filtering effect, LCL-type filters offer lower costs and smaller sizes, making them commonly used [15,16]. However, LCL-type inverters exhibit resonance peaks, where the phase undergoes a −180° shift at the resonant frequency [17], making the system prone to resonance and instability. Therefore, damping of these resonance peaks is necessary [18]. Current damping methods are categorized into passive damping, which involves series or parallel resistors in the circuit [19], and active damping, which introduces state variable feedback control [20,21]. The literature [22,23] analyzes passive damping methods for LCL, LLCL, and LLCCL filters, comparing the suppression effects of the Rd damping method and the Rd-Cd hybrid resistor–capacitor passive damping method. The former can dampen resonance but affects the gain characteristics in the high-frequency range, reducing high-frequency harmonic attenuation. The latter, with the addition of capacitance, can dampen resonance without affecting high-frequency harmonic attenuation. Passive damping methods are simple and effective but introduce losses. In weak grids, when grid impedance varies widely, these methods cannot adjust damping levels in a timely manner, limiting their suppression effectiveness. Active damping methods, on the other hand, do not introduce additional losses and offer more flexible damping implementation. Currently, there are multiple active damping methods, mainly divided into two approaches: one that does not require additional sensors, achieving damping through grid-connected current feedback or filter parameter estimation; and another that requires additional sensors, feeding back electrical quantities such as voltage or other currents to achieve virtual damping control. The literature [24] examines the core principles of active damping methods, demonstrating the feasibility of control techniques, including inverter-side and grid-side current feedback. The literature [25] proposes an active damping control method based on a band-pass filter, which does not require additional sensors and analyzes the damping effects of introducing the band-pass filter at different locations. The literature [26] proposes an improved active damping control method based on traditional grid-connected current feedback active damping, expanding the frequency range of positive damping. The literature [27] analyzes the damping characteristics and parameter selection based on capacitor current feedback under PI control. The literature [28] proposes an improved active damping method using differential feedback of the filter capacitor voltage, achieving resonance peak suppression without affecting filter performance. In summary, the most widely used method is active damping based on proportional feedback of capacitor current, which is equivalent to the passive damping method of paralleling a resistor across the capacitor. This method can provide virtual resistance for damping, and proportional feedback is easy to implement without introducing additional losses. The above comparisons of different methods are also presented in the following

Table 1. A comparison of different methods.

Methods		Description
Resonance Peak Damping Methods	Passive Damping	Achieved by series or parallel resistors in the circuit Simple and effective, but introduces losses
	Active Damping	No additional losses introduced, more flexible damping implementation
Analysis of Passive Damping Methods	Rd Damping Method	Dampens resonance but affects gain characteristics in the high-frequency range, reducing high-frequency harmonic attenuation
	Rd-Cd Hybrid Resistor–Capacitor Passive Damping Method	Dampens resonance without affecting high-frequency harmonic attenuation by adding capacitance
Analysis of Active Damping Methods	Methods Without Additional Sensors	Achieved through grid-connected current feedback or filter parameter estimation
	Methods Requiring Additional Sensors	Feedback voltage or other currents for virtual damping control
Most Widely Used Method	Active Damping Based on Capacitor Current Proportional Feedback	Equivalent to passive damping by paralleling a resistor across the capacitor Provides virtual resistance for damping, easy to implement without introducing additional losses

.

In recent years, the modular multilevel converter with embedded energy storage has emerged as a significant development direction in storage transmission. In this configuration, battery energy storage units are decentralized and integrated into the submodules within the converter arms, offering advantages such as multi-dimensional coordination and rapid response. However, to date, there has been a lack of relevant literature analyzing its impedance characteristics. To address these concerns, this paper studies this type of converter’s characteristics and the contributions are as follows:

(1)

The impedance model of BESS-MMC has been established, where the control and grid system parts are considered, which is realized through a Quasi-Proportional–Resonant (QPR) controller.

(2)

The key factors influencing the inverter’s impedance and its stability characteristics are investigated, and the recommendations for enhancing stability are also provided.

The organization of this paper is as follows: Section 2 introduces the operating principle of an LCL inverter with embedded energy storage, Section 3 analyzes the stability mechanism, Section 4 conducts simulation verifications, and Section 5 concludes the paper.

2. Operating Principle of LCL Inverter with Embedded Energy Storage

2.1. System Structure and Control Strategy

In analyzing the characteristics of Modular Multilevel Converters (MMCs), common assumptions are made to simplify the models and highlight the key phenomena. These include assuming ideal switching devices (neglecting losses and delays), a constant DC-side voltage, balanced capacitor voltages across sub-modules, negligible circulating currents, linear system behavior in steady-state or small signal analyses, negligible parasitic parameters, and steady-state operating conditions. These assumptions provide a foundational framework, though their application may require adjustments based on specific system needs and operational contexts.

The topological structure of a photovoltaic–wind complementary power generation system equipped with BESS-MMC is shown in Figure 1. It mainly consists of photovoltaic panels, wind turbines, energy storage batteries, grid-connected inverters, transformers, and other components. The relevant new energy sources and the BESS-MMC system adopt a grid-side inductor current feedback control strategy and are connected to the grid through an LCL-type grid-connected inverter.

Figure 2 illustrates the circuit and control diagram of a single-phase grid-connected inverter for embedded energy storage when digital control is employed. S₁–S₄ represent the switching devices. L₁ and L₂ represent the inductors on the inverter and grid sides, respectively. L_g is the inductive grid impedance, and C is the filter capacitor. U_dc is the DC-side voltage, u_g is the grid voltage, u_inv is the output voltage of the inverter, u_pcc is the voltage at the point of common coupling (PCC) between the inverter and the grid, i₁ is the inductor current on the inverter side, i_C is the filter capacitor current, and i₂ is the grid-side current. H_i1 and H_i2 are the feedback coefficients for the capacitor current and the grid current sampling, respectively, and G_i(s) is the current regulator.

Firstly, the phase of the voltage at the point of common coupling is obtained by the Phase-Locked Loop (PLL) to ensure that the grid-connected current is in phase with the voltage. I_ref is the amplitude of the reference current, and the current regulator is used to make the output grid-connected current track the reference current i_2ref. In addition, to mitigate the resonance of the LCL filter, the Capacitor Current Feedback Active Damping (CCF-AD) control method is utilized. The feedback is applied to the output of the current regulator to obtain the modulation wave v_M, which is then used to generate pulse signals through the modulation process to drive the switching devices.

The proper selection of a current regulator ensures that the output grid-connected current complies with the grid connection standards. Currently, commonly used current regulators in engineering include the Proportional–Integral (PI) controller and the Proportional–Resonant (PR) controller, and the Quasi-Proportional–Resonant (QPR) controller. Due to its excellent performance and simple control structure, the PI controller is widely used in practical engineering applications. However, it can only achieve zero steady-state error tracking for DC quantities, so it is often used in the dq coordinate system. The ideal PR controller has infinite gain at the fundamental frequency, but it is difficult to adapt to grid frequency fluctuations in practical engineering applications. The QPR controller, on the other hand, has better robustness than the PR controller and can adapt to grid frequency fluctuations, achieving nearly zero steady-state error tracking for AC quantities. It is often used in the αβ coordinate system. The Bode plots of these two controllers are shown in Figure 3. To conduct a quantitative analysis, we added the frequency drift tolerance comparison based on 0 dB-cross bandwidth, namely, the bandwidth covered 0 dB. It can be seen from Figure 3 that the frequency drift tolerance of QPR control is 43 rad/s, while PR control is only 7 rad/s, thus, the frequency drift tolerance of QPR control is about 6 times of PR control.

In summary, this paper selects the Quasi-Proportional–Resonant (QPR) controller, whose expression is given by

$$G_i(s)=K_{\mathrm{p}}+{\displaystyle \frac{2K_{\mathrm{r}}{\omega }_{i}s}{s^2+2{\omega }_{i}s+{\omega }_0^2}}$$

(1)

where K_p and K_r represent the proportional coefficient and the resonant coefficient, respectively; w_i is the bandwidth of the resonant term; and w₀ is the fundamental angular frequency.

2.2. Modulation Strategy

The main circuit of the inverter consists of a DC power supply, an inverter bridge, and a filter. In this paper, the inverter adopts unipolar double frequency sinusoidal pulse width modulation (SPWM). The inverter bridge is composed of IGBT switches and antiparallel diodes. The schematic diagram of unipolar double frequency modulation is shown in Figure 4, where v_M represents the modulation wave, ±U_tri represent the triangular carrier waves, S₁ and S₂ are switches on the same leg, and S₃ and S₄ are switches on the other leg. The pulse signals for the left leg are obtained by comparing v_M with U_tri, and the pulse signals for the right leg are obtained by comparing v_M with −U_tri. The specific process is as follows: When v_M > U_tri, S₁ is turned on and S₂ is turned off; when v_M < U_tri, S₁ is turned off and S₂ is turned on. Similarly, when v_M > −U_tri, S₃ is turned off and S₄ is turned on; when v_M < −U_tri, S₃ is turned on and S₄ is turned off. From the schematic diagram, it can be seen that during the positive half-cycle of the modulation wave, the voltage v_ab between the legs has two levels: +U_dc and 0. Similarly, during the negative half-cycle of the modulation wave, the voltage v_ab between the legs also has two levels: −U_dc and 0. The frequency of v_ab is twice that of the triangular carrier wave. According to the literature [27], the Fourier series expression for the inverter output voltage v_ab under unipolar double frequency modulation is given by

$$\begin{array}{cc}{v}_{ab}(t)& =M{U}_{dc}\sin {\omega }_{0}t\hfill \\ & +{\displaystyle \frac{4U_{dc}}{\pi }}{\displaystyle \sum _{m=2,4,\cdots }^{\infty }{\displaystyle \sum _{n=\pm 1,\pm 3,\cdots }^{\pm \infty }\frac{J_n(mM\pi /2)}{m}}}\cos {\displaystyle \frac{m\pi }{2}}\sin (m{\omega }_{\mathrm{sw}}t+n{\omega }_{0}t)\hfill \end{array}$$

(2)

where M denotes the modulation index, w₀ represents the fundamental angular frequency, ω_sw is the angular frequency of the triangular carrier wave, m and n are the multiples of the frequencies of the triangular carrier wave and the modulation wave, respectively, and J_n(x) is the Bessel function, expressed as

$$J_n(x)={\displaystyle \sum _{k=0}^{\infty }\frac{{(-1)}^k}{k!\left(k+n\right)!}}{\left({\displaystyle \frac{x}{2}}\right)}^{2k+n}$$

(3)

2.3. Mathematical Model

Based on the circuit and control structure of the embedded energy storage LCL grid-connected inverter shown in the figure, the mathematical model of the single-phase LCL grid-connected inverter is derived, as illustrated in Figure 5.

Where Z_L1, Z_C, and Z_L2 represent the impedances corresponding to the inverter-side inductance, filter capacitor, and grid-side inductance, respectively. K_PWM is the transfer function from the modulation wave to the inverter output voltage u_inv. G_d(s) accounts for the one-sample calculation delay and the 0.5-sample modulation delay introduced by digital control, with its expression given as

$$G_d(s)=e^{-1.5s{T}_s}$$

(4)

Through block diagram transformation, Figure 5 can be further simplified into Figure 6. The expressions for G_X1(s) and G_X2(s) are presented in Equations (5) and (6), respectively.

$$G_{\mathrm{X}1}(s)={\displaystyle \frac{K_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)G_i(s)}{s^2{L}_{1}C+sC{H}_{i1}{K}_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)+1}}$$

(5)

$$G_{\mathrm{X}2}(s)={\displaystyle \frac{s^2{L}_{1}C+sC{H}_{i1}{K}_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)+1}{s^3{L}_1{L}_{2}C+s^2{L}_{2}C{H}_{i1}{K}_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)+s(L_1+L_2)}}$$

(6)

As shown in Figure 6, the system’s loop gain can be expressed as:

$$\begin{array}{cc}T(s)& =G_{\mathrm{X}1}(s)G_{\mathrm{X}2}(s)H_{i2}\hfill \\ & ={\displaystyle \frac{H_{i2}{K}_{\mathrm{PWM}}{G}_i(s)}{s^3{L}_1{L}_{2}C+s^2{L}_{2}C{H}_{i1}{K}_{\mathrm{PWM}}+s(L_1+L_2)}}\hfill \\ & ={\displaystyle \frac{H_{i2}{K}_{\mathrm{PWM}}(K_{\mathrm{p}}+2K_{\mathrm{r}}{\omega }_{i}s/s^2+2{\omega }_{i}s+{\omega }_0^2)}{s^3{L}_1{L}_{2}C+s^2{L}_{2}C{H}_{i1}{K}_{\mathrm{PWM}}+s(L_1+L_2)}}\hfill \end{array}$$

(7)

Therefore, the grid-connected current i₂(s) can be expressed as:

$$\begin{array}{cc}{i}_2(s)& ={\displaystyle \frac{1}{H_{i2}}}{\displaystyle \frac{T(s)}{1+T(s)}}{i}_{2\mathrm{ref}}(s)-{\displaystyle \frac{G_{\mathrm{X}2}(s)}{1+T(s)}}{u}_{pcc}(s)\hfill \\ & =i_s(s)-{\displaystyle \frac{u_{pcc}(s)}{Z_{\mathrm{o}}(s)}}\hfill \end{array}$$

(8)

The first part, i_s(s), in Equation (8) represents the equivalent ideal current source of the grid-connected inverter, while the second part, Z_o(s), denotes the output impedance of the inverter, which can be expressed as:

$$Z_{\mathrm{o}}(s)={\displaystyle \frac{1+T(s)}{G_{\mathrm{X}2}(s)}}$$

(9)

2.4. Impedance Characteristic Analysis

By substituting Equations (6) and (7) into Equation (9), the output impedance expression of the inverter using the CCF-AD control method is derived as

$$\begin{array}{l}{Z}_{\mathrm{o}}(s)\\ ={\displaystyle \frac{s^3{L}_1{L}_{2}C+s^2{L}_{2}C{H}_{i1}{K}_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)+s(L_1+L_2)+K_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)G_i(s)H_{i2}}{s^2{L}_{1}C+sC{H}_{i1}{K}_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)+1}}\\ ={\displaystyle \frac{s^3{L}_1{L}_{2}C+s^2{L}_{2}C{H}_{i1}{K}_{\mathrm{PWM}}{e}^{-1.5s{T}_s}+s(L_1+L_2)+K_{\mathrm{PWM}}{e}^{-1.5s{T}_s}{G}_i(s)H_{i2}}{s^2{L}_{1}C+sC{H}_{i1}{K}_{\mathrm{PWM}}{e}^{-1.5s{T}_s}+1}}\end{array}$$

(10)

From Equation (10), it can be observed that the output impedance of the inverter is related to the circuit parameters, control parameters, sampling coefficients, capacitor current feedback coefficients, and switching frequency. It can also be noticed that the time delay appears in the equation, where the delays may reduce the system’s stability margin and shrink the stable operating region of the system.

Below, we will analyze the influence of each parameter on the output impedance. The parameters of the circuit are provided in

Table 2. Key parameters of the inverter.

Parameters	Value
Grid voltage (ug/V)	110
DC-side voltage (Udc/V)	300
Triangular carrier amplitude (Utri/V)	1
Inverter-side inductance (L1/mH)	2
Filter capacitor (C/μF)	10
Grid-side inductance (L2/mH)	400
Grid-connected current sampling coefficient (Hi2)	0.15
Capacitor current feedback coefficient (Hi1)	0.002

.

(1)

Influence of Control Parameters

The output impedance of the inverter is influenced by the control parameters, including K_p and K_r. This section analyzes the impact of varying these control parameters on the output impedance.

Figure 7 and Figure 8 present the Bode plots of the inverter’s output impedance for different values of the control parameters. When the resonant coefficient K_r is fixed and the proportional coefficient K_p is increased, the passive region of the output impedance gradually decreases. As K_p continues to increase to a critical value, the system resonates and loses stability. Similarly, when the proportional parameter K_p is fixed and the resonant coefficient K_r is increased, the passive region of the output impedance initially increases. However, as K_r continues to increase beyond a certain value, the passive region of the output impedance begins to decrease. When K_r reaches a sufficiently high value, the system resonates and loses stability.

(2)

Influence of Switching Frequency

The delay resulting from digital control can influence the inverter’s output impedance. The magnitude of this delay is correlated with the switching frequency (f_sw). Below is an analysis of how the switching frequency impacts the output impedance characteristics. Through block diagram transformation, it can be observed that the effect of the CCF-AD control strategy is equivalent to the passive damping method that involves paralleling a resistor across the terminals of the filter capacitor, as illustrated in Figure 9.

The delay introduced by digital control causes the equivalent resistance to become a frequency-dependent impedance, Z_eqc, which can be expressed as:

$$Z_{\mathrm{eqc}}(s)={\displaystyle \frac{Z_{L1}(s)Z_{\mathrm{C}}(s)}{H{{}_{i1}K}_{\mathrm{PWM}}{G}_{\mathrm{d}}(s)}}={\displaystyle \frac{L_1}{CH{{}_{i1}K}_{\mathrm{PWM}}{e}^{-1.5s{T}_s}}}$$

(11)

Based on the expressions for the equivalent resistance and equivalent reactance, their frequency characteristic curves are plotted. As shown in Figure 10, the equivalent resistance is no longer constant but varies with changes in the switching frequency and sampling frequency. It becomes negative damping at the frequency of f_s/6. The equivalent reactance exhibits capacitive behavior at the frequency of f_s/3. When the grid impedance varies over a wide range, the system’s resonant frequency shifts.

As the frequency approaches f_s/6, the system becomes unstable. To reduce the effects of digital control delay, optimizing the delay is essential to improve system stability.

In summary, to ensure the stability of the inverter itself and its interaction stability under a strong grid, the control parameters for the inverter adopting capacitor current feedback control are selected as follows: K_p = 0.25, K_r = 10, and f_sw = 5 kHz.

3. Stability Mechanism Analysis

3.1. Stability Analysis of a Single Inverter

The fluctuations in grid impedance under weak grid circumstances can influence the stability of the system’s interaction. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 illustrate the Nyquist curves of the equivalent loop gain of the system under different grid impedances. As the grid impedance changes from L_g = 100 mH to L_g = 700 mH, the Nyquist curve of the equivalent loop gain of the grid-connected system shifts from being far away from the (−1, j0) point to asymptotically encircling and then completely encircling the (−1, j0) point. As grid impedance increases, the grid becomes weaker, and the system’s stability deteriorates. When L_g = 400 mH, the system approaches critical stability.

When the grid impedance L_g is 700 mH, the loop gain Z_g(s)/Z_inv(s) of the system does not satisfy the Nyquist criterion. Figure 12 illustrates the magnitude–frequency characteristic curve, where the zero-crossing point—indicating the interaction between the inverter and grid impedances—occurs at the crossover frequency of 1650 Hz. At this frequency, the phase exceeds 180°, violating the Nyquist criterion and rendering the system unstable. In other words, the resonant frequency is 1650 Hz.

3.2. Stability Analysis of Multiple Inverters

When several inverters are connected to the grid at the same time, each can be represented at the Point of Common Coupling (PCC) by a Norton equivalent circuit, which consists of a current source i_sn(s) in parallel with an impedance Z_invn(s). The corresponding equivalent circuit is shown in Figure 13.

Z_all(s) represents the total parallel impedance at the port. According to the superposition theorem, the voltage expression at the Point of Common Coupling (PCC) can be derived as follows:

$$\begin{array}{cc}{u}_{\mathrm{pcc}}(s)& ={\displaystyle \frac{Z_{\mathrm{all}}(s)Z_{\mathrm{g}}(s)}{Z_{\mathrm{all}}(s)+Z_{\mathrm{g}}(s)}}{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{n}}{i}_{\mathrm{sm}}}(s)+{\displaystyle \frac{Z_{\mathrm{all}}(s)}{Z_{\mathrm{all}}(s)+Z_{\mathrm{g}}(s)}}{u}_{\mathrm{g}}(s)\hfill \\ & ={\displaystyle \frac{1}{1+\frac{Z_{\mathrm{g}}(s)}{Z_{\mathrm{all}}(s)}}}\left({\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{n}}{i}_{\mathrm{sm}}}(s)Z_{\mathrm{g}}(s)+u_{\mathrm{g}}(s)\right)\hfill \end{array}$$

(12)

The mathematical model is derived from the equivalent circuit diagram of multiple inverters connected to the grid, as shown in Figure 14.

The expressions for the grid-connected current i_gm(s) of each inverter and the total grid-connected current i_g(s) at the Point of Common Coupling (PCC) are given by:

$$i_{\mathrm{gm}}(s)=i_{\mathrm{sm}}(s)-{\displaystyle \frac{u_{\mathrm{pcc}}(s)}{Z_{\mathrm{invm}}(s)}}\begin{array}{c}\hspace{1em}\left(m=1,2,\cdots ,n\right)\end{array}$$

(13)

$$i_g(s)={\displaystyle \frac{u_{\mathrm{pcc}}(s)-u_g(s)}{Z_g(s)}}$$

(14)

Similar to the analysis of a single inverter connected to the grid as mentioned above, the stability conditions for multiple inverters connected to the grid are as follows:

• Under a strong grid condition, each inverter ensures its own stability.
• The equivalent loop gain Z_g(s)/Z_all(s) for multiple inverters connected to the grid meets the Nyquist criterion. The magnitude–frequency characteristic curve of the equivalent parallel impedance Z_all(s) at the port does not intersect with that of the grid impedance. If they do intersect, the phase at the crossover frequency f_c must be positive, meaning that the phase margin is satisfied.

$$\mathrm{PM}=180°-(\angle Z_g(2\mathsf{\pi }{f}_{\mathrm{c}})-\angle Z_{\mathrm{all}}(2\mathsf{\pi }{f}_{\mathrm{c}}\left)\right)>0$$

(15)

A verification analysis is conducted based on two grid-connected inverters.

Figure 15 demonstrates the functioning of two inverters when linked to the grid, employing the traditional CCF-AD control strategy. Inverter 1 is a photovoltaic (PV) grid-connected inverter, while inverter 2 is an embedded energy storage grid-connected inverter. The circuit and control parameters of the inverters are listed in

Table 3. Key parameters of the two grid-connected inverters.

Parameters	Value
Grid voltage (ug/V)	110
DC-side voltage (Udc/V)	300
Rated capacity (S/VA)	4000
Inverter-side inductance (L1/mH)	2
Filter capacitor (C/μF)	10
Grid-side inductance (L2/mH)	400
Gi(s) proportionality coefficient (Kp)	0.25
Gi(s) Resonance coefficient (Kr)	10
Switching frequency (fsw/kHz)	5
Sampling frequency (fs/kHz)	10
Grid-connected current sampling coefficient (Hi2)	0.15
Capacitor current feedback coefficient (Hi1)	0.002

.

Figure 16 presents the Nyquist plots of the system’s equivalent loop gain for various grid impedances. As the grid impedance varies from L_g = 50 mH to L_g = 350 mH, the Nyquist curve of the equivalent loop gain of the grid-connected system shifts from being far away from the point (−1,j0) to gradually encircling and then completely encircling the point (−1,j0). As grid impedance increases, the grid weakens, and the system’s stability deteriorates. When L_g = 200 mH, the system approaches critical stability.

When the grid impedance L_g is 350 mH, the loop gain Z_g(s)/Z_all(s) of the multi-inverter system does not satisfy the Nyquist criterion. As shown in Figure 17, which depicts the magnitude–frequency characteristic curve, the point where the curve crosses zero—indicating the point where the inverter output impedance intersects with the grid impedance—occurs at a crossover frequency of 1640 Hz. At this frequency, the phase exceeds 180°, violating the Nyquist criterion and rendering the system unstable. This indicates that the resonant frequency is 1640 Hz.

4. Simulation Verifications

This section presents the construction of a simulation model for a single-phase embedded energy storage grid-connected inverter in MATLAB/Simulink 2023a, aimed at validating the inverter parameters and stability analysis using the CCF-AD control.

4.1. Verification of the Impact of Control Parameters

As the proportional coefficient K_p increases, the passivity region of the output impedance gradually decreases. However, when K_p reaches a certain value, the system resonates on its own and loses stability. Figure 18 presents four operating conditions with different proportional parameters, K_p. When K_p increases from 0.1 to 0.2, the current and voltage waveforms remain stable. As K_p continues to increase to 0.4, the current waveform resonates, and the system becomes unstable. When K_p further increases to 0.8, the degree of resonance intensifies. The simulation results demonstrate that increasing K_p leads to a decline in system stability, which aligns with the theoretical findings. Similarly, when the proportional coefficient K_p is fixed and the resonance coefficient K_r increases, the passivity region of the output impedance gradually expands.

However, a notable phenomenon occurs when the resonance parameter K_r attains a specific boundary value. At this juncture, the passivity region of the output impedance begins to diminish, and the system, in an unexpected turn of events, starts to resonate autonomously. This self-induced resonance causes the system to lose its inherent stability, leading to potential operational disruptions. Figure 19 vividly illustrates four distinct operating conditions, each characterized by a different value of the resonance parameter K_r. As K_r progressively increases from an initial value of 10 up to 350, a clear transformation in the output current waveform can be observed. Initially stable, the waveform gradually becomes divergent, a clear indication of the system’s resonance and subsequent instability. The simulation results provide compelling evidence that as the resonance coefficient K_r continues to escalate, the overall stability of the system correspondingly diminishes. This empirical finding is in perfect harmony with the theoretical analysis, thereby reinforcing the validity of the theoretical framework and providing valuable insights into the system’s behavior under varying resonance conditions.

4.2. Verification of the Stability Analysis in Single Inverter Case

Figure 20 shows the simulation waveforms of current and voltage for a single-inverter grid-connected system at different grid impedances: L_g = 100 mH, L_g = 400 mH, and L_g = 700 mH. In the first two cases, as the grid impedance rises from 100 mH to 400 mH, the system remains stable. The grid current is in sync with the voltage at the Point of Common Coupling (PCC), and the Total Harmonic Distortion (THD) of the current is under 5%, satisfying the grid connection requirements. However, when the grid impedance reaches L_g = 700 mH, a resonance between the current and voltage occurs, leading to system instability. This observation aligns with the stability analysis discussed earlier.

Figure 21 shows the FFT analysis of the grid-connected current when the grid impedance is 700 mH. The analysis indicates that the dominant harmonic frequency of the current is 1650 Hz, which matches the findings from the equivalent impedance Bode plot in Figure 12. These simulation results confirm the accuracy of the established model.

4.3. Verification of the Stability Analysis in Multiple Inverters Case

When wind–solar hybrid inverters 1 and 2 with embedded storages are put into operation, three operating conditions are set. For the first operating condition, with a grid impedance of L_g = 50 mH, the voltage and current at the Point of Common Coupling (PCC) of the grid-connected system are observed, as illustrated in Figure 22a. The analysis reveals that when the grid impedance is L_g = 50 mH, the Total Harmonic Distortion (THD) is 0.32%, and the current meets the grid connection standards. When the two inverters operate simultaneously in grid-connected mode, the system remains stable.

Maintaining the inverter parameters unchanged: when the grid impedance is increased to L_g = 200 mH, the voltage and current at the Point of Common Coupling (PCC) of the grid-connected system are observed, as shown in Figure 22b. The analysis indicates that with a grid impedance of L_g = 200 mH, the Total Harmonic Distortion (THD) is 0.74%, and the current meets the grid connection standards. When the two inverters operate simultaneously in the grid-connected mode, the system remains stable.

When the grid impedance is further increased to L_g = 350 mH, the voltage and current at the Point of Common Coupling (PCC) of the grid-connected system are observed, as depicted in Figure 22c. The analysis reveals that when the grid impedance is L_g = 350 mH, resonance occurs in both the current and voltage, leading to an unstable system, which is consistent with the theoretical analysis of Figure 23.

5. Conclusions

This study investigates the stability of grid-connected inverter systems incorporating embedded energy storage (EES). The research findings reveal the following:

(1)

The effect of the control parameters on a grid-connected inverter is demonstrated by a reduction in system stability as the proportional gain K_p increases. When K_p reaches a certain threshold, the system becomes unstable. Similarly, as the resonant coefficient K_r increases to a specific value, the system experiences resonance. To increase the stability of the inverter in practical applications, the values of K_p and K_r should be limited and, once the resonance occurs, decreasing the two parameters can be effective to suppress the oscillations.

(2)

The stability of the coupled system is affected by both the magnitude of grid impedance and the number of grid-connected inverters. As grid impedance increases, transitioning the system from a strong grid to a weak grid, system stability diminishes. Furthermore, even with a constant grid impedance, an increase in the number of grid-connected inverters also degrades system stability. Thus, the operation modes of the practical system should be paid attention to for the inverters connected to the grid.

However, because this study mainly focuses on voltage-sourced converters, an inverter based on current-sourced converters is not analyzed, but will be investigated in future.