Optimization-Based Suppression Method of Oscillations in Photovoltaic Grid-Connected Systems with Controllable Nonlinear Loads

Abstract

In order to reduce carbon emissions from the power grid, photovoltaic (PV) generation units and controllable nonlinear loads based on power electronic devices are gradually becoming more prevalent in the power system. In a PV grid-connected system featuring controllable nonlinear loads, the interplay among PV grid-connected inverters, the loads, and the grid can potentially lead to voltage oscillations. To tackle this challenge, this paper introduces an optimization-based method for suppressing oscillations, which carefully balances system stability with response performance. Firstly, an impedance model of the system is established by applying the harmonic linearization method, and system stability is analyzed using the “logarithmic frequency stability criterion”. Subsequently, impedance relative sensitivity is used to identify key parameters that affect system stability, and the interaction between key parameters is considered to analyze the stability range for these parameters. On this basis, a parameter optimization method based on the particle swarm optimization algorithm is proposed to balance system stability and response performance. The effectiveness and robustness of this method are verified through a simulation analysis.

1. Introduction

As an important renewable energy source, photovoltaic (PV) generation plays a key role in reducing carbon pollution in the power grid. On the other hand, controllable nonlinear loads based on power electronic interfaces are widely employed in industrial production and domestic living to reduce carbon emissions and enhance energy efficiency on the load side, such as electrolytic hydrogen production, variable speed drives (VSDs), frequency conversion equipment, and electric vehicle (EV) chargers [1]. With the large-scale grid connection of PV and the widespread use of controllable nonlinear loads, the interaction among controllable nonlinear loads, PV, and the Alternating Current (AC) grid may cause system oscillations due to the nonlinear and strong coupling characteristics of power electronic equipment, posing a threat to the safe and stable operation of the grid and its connected equipment [2]. Therefore, it is of vital importance to analyze the oscillation characteristics of the PV grid-connected system with controllable nonlinear loads and to study oscillation-suppression methods.

Currently, the impedance-analysis method is commonly used in studying the stability of renewable energy systems connected to the grid, primarily because of its clear physical meaning and favorable scalability [3]. Based on the harmonic linearization method, references [4,5] established an impedance model of a PV grid-connected system and analyzed the influence of changing the phase-locked loop (PLL) and current loop parameters on the system impedance characteristics. Reference [6] further established a sequence impedance model for PV grid-connected systems considering frequency coupling. Based on the impedance model, the mechanism of frequency coupling in grid-connected inverters and the influence of coupling frequency on system stability were analyzed. Considering the frequency coupling, reference [7] analyzed the impact of PLL on the stability of grid-connected systems under the weak AC grid. Reference [8] analyzed the impact of grid strength in different frequency ranges on the stability of a PV grid-connected system based on the dq-axis impedance model. It was found that under a specific system structure, increasing the impedance of the grid in the sub-synchronous frequency band was beneficial for suppressing oscillations, while increasing the impedance of the grid in the high-frequency range weakened the high-frequency stability of the system.

Although the above references have deeply investigated the interaction mechanism between the PV grid-connected inverter and the grid, the influence of controllable nonlinear loads has been hardly taken into account. At present, research on controllable nonlinear loads mainly focus on the influence of harmonic characteristics, with few literatures analyzing the interaction among controllable nonlinear loads, PV, and the grid [9,10]. Therefore, it is necessary to establish {an} impedance model of the PV grid-connected system with controllable nonlinear loads and analyze the stability of the system.

In terms of oscillation suppression, the parameter optimization method is widely used in practical engineering for oscillation suppression due to its advantages of simplicity, reliability, and ease of implementation. References [4,11,12] analyzed the frequency characteristics of the PLL and optimized its bandwidth by adjusting parameters to reshape the impedance characteristics. Reference [13] analyzed the dominant influencing parameters of the impedance characteristics of grid-connected inverters in different frequency ranges and provided the frequency band distribution ranges of different control loops. References [14,15] identified the dominant parameters that affect the stability of three-phase inverter grid-connected systems using impedance sensitivity and verified the effectiveness of adjusting key parameters to enhance system stability. Although the parameter-optimization method can be directly designed for the frequency band with oscillation risks, it is difficult to improve damping in a wide frequency range as the operating state of the system changes [16]. In addition, the parameter adjustments in the above literature are only for the improvement of system stability and do not consider the impact of parameter changes on response performance. Particle Swarm Optimization (PSO) stands out for its rapid convergence, particularly evident in single-objective optimization problems, making it an attractive approach for addressing various optimization tasks [17]. In this article, a novel parameter-optimization technique based on PSO is proposed for PV grid-connected systems with controllable non-linear loads. This approach balances system stability with response performance, harnessing the robust search prowess of PSO to pinpoint optimal parameter configurations. Compared to traditional parameter-optimization methods, the proposed strategy uniquely incorporates the assessment of response performance into the optimization process, ensuring the more holistic and effective tuning of system parameters.

The rest of this paper is organized as follows. In Section 2, the impedance model of the system is established using the harmonic linearization method, and the stability is analyzed using a logarithmic frequency stability criterion, revealing the mechanism of system oscillation. In Section 3, the impacts of various parameters on system stability are quantitatively scrutinized through the utilization of impedance relative sensitivity, revealing that the proportional coefficient k_p4 of the DC current loop of controllable nonlinear loads and the proportional coefficient k_p2 of the current inner loop of the photovoltaic grid-connected inverter are the key parameters affecting system stability. In Section 4, the stability domain range of the key influencing parameters is analyzed, taking into consideration the interactions among various parameters. Based on this analysis, a parameter-optimization method that balances system stability and response performance is proposed, and the PSO algorithm is used to optimize the key influencing parameters. In Section 5, the correctness of parameter-stability-domain analysis and the effectiveness of the oscillation-suppression method based on parameter optimization are verified through a simulation. Finally, the conclusions are drawn in Section 6.

2. Impedance Modeling of the PV Grid-Connected System with Controllable Nonlinear Loads

The circuit topology of the PV grid-connected system with a controllable nonlinear load studied in this paper is shown in Figure 1, including a PV grid-connected inverter, a controllable nonlinear load, and the grid. Nowadays, controllable rectifiers based on thyristors are widely used for the front-end power conversion of controllable nonlinear loads, such as high-voltage direct current (HVDC) transmission, EV chargers, VSDs, hydrogen storage, and electric arc furnaces, due to their cost-effectiveness and large capacity. Therefore, the controllable nonlinear loads based on front-end thyristor rectification in this article are mainly considered.

In Figure 1, the direct current (DC) side of the PV generation unit adopts a simplified model, which is equivalent to U_dc and connected to the point of common coupling (PCC) through an LCL-filtered inverter. The LCL filter is composed of an inverter-side inductor L₁, a capacitor C_f, a damping resistance R_d, and a grid-side inductor L₂. u_si(i = a, b, c) is the grid voltage, and Z_g represents grid impedance. u_dc and i_dc are the DC voltage and current of the nonlinear load; R₄, L₄, and C₄ are the resistor, inductor, and capacitor on the DC side. Assuming that the influence of the rear-end power conversion on the front-end rectification loop can be ignored, the DC voltage of the rear-end power conversion can be maintained constant, which is equivalent to a constant DC source V_dc0.

According to the impedance equivalent method of the AC system [18], Figure 1 can be equivalent to the small signal model shown in Figure 2, where i_pv, i_nl, Z_pv, and Z_nl are the equivalent current sources and impedances of the inverter and the nonlinear load, respectively. The stability of the system depends on whether Z_g(s)/[Z_pv(s) || Z_nl(s)] satisfies the Nyquist stability criterion if the PV generation unit and the nonlinear load are stable under an ideal voltage source. For the convenience of the later description, the system in which the PV generation unit is connected in parallel with the nonlinear load is referred to as a parallel subsystem. The impedance of the parallel subsystem is as follows:

$$Z_{\mathrm{s}}(s)=Z_{\mathrm{p}\mathrm{v}}(s)\left|\right|Z_{\mathrm{n}\mathrm{l}}(s)$$

(1)

Establishing an impedance model for the PV grid-connected system with controllable nonlinear loads is the basis for stability analysis. There have been many studies on the impedance modeling of LCL-type PV grid-connected inverters, and the PV impedance model used in this article is consistent with the reference [19], which will not be elaborated on here. The following is an introduction to the impedance modeling process for the nonlinear load based on three-phase controllable rectification using the harmonic linearization method.

The circuit topology and control system structure of the nonlinear load based on three-phase controllable rectification are shown in Figure 3. In Figure 3, the PLL is the same as the phase-locked loop in reference [19] with proportional and integral coefficients of k_p3 and k_i3, respectively; G_im(s) is the gain and first-order filtering loop used for the DC current measurement; G_i(s) is a Proportion-Integral (PI) controller for DC current control, with proportional and integral coefficients of k_p4 and k_i4; i_{dc_ref} is the reference value of the DC current; θ_l(0) is the phase lag of the l-th switch; θ_l is the phase angle of the equivalent grid voltage; and α is the trigger angle.

Assume that phase-A voltage u_a(t) at PCC can be expressed as follows:

$$u_{\mathrm{a}}\left(t\right)=V_1\cos \left(2\pi f_{1}t+{\phi }_{\mathrm{V}1}\right)+V_{\mathrm{p}}\cos \left(2\pi f_{\mathrm{p}}t+{\phi }_{\mathrm{Vp}}\right)$$

(2)

where V₁, V_p, f₁, f_p, φ_V1, and φ_Vp refer to the amplitudes, frequencies, and initial phases of the fundamental voltage and positive sequence voltage, respectively. To simplify the modeling process, φ_V1 = 0.

To reduce the harmonics introduced by the front-end rectifier, a 12-pulse rectifier is used for the controllable nonlinear load, of which the specific structure of the rectifier is shown in Figure 4.

Assuming that the DC current i_dc is continuous, the relationships of voltages and currents can be established as follows:

$$\left\{\begin{array}{l}{v}_{\mathrm{d}1}\left(t\right)={\displaystyle \sum _{j=\mathrm{a},\mathrm{b},\mathrm{c}}{s}_{j1}\left(t\right)v_{j1}\left(t\right)}\\ v_{\mathrm{d}2}\left(t\right)={\displaystyle \sum _{j=\mathrm{a},\mathrm{b},\mathrm{c}}{s}_{j2}\left(t\right)v_{j2}\left(t\right)}\\ u_{\mathrm{dc}}\left(t\right)=v_{\mathrm{d}1}\left(t\right)+v_{\mathrm{d}2}\left(t\right)\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{i}_{j1}\left(t\right)=s_{j1}\left(t\right)i_{\mathrm{dc}}\left(t\right), j=\mathrm{a},\mathrm{b},\mathrm{c}\\ i_{j2}\left(t\right)=s_{j2}\left(t\right)i_{\mathrm{dc}}\left(t\right), j=\mathrm{a},\mathrm{b},\mathrm{c}\end{array}\right.$$

(4)

In (3) and (4), v_d1(t) and v_d2(t) are the output DC voltages of the upper and lower bridge arms in the rectification; v_j1(t) and v_j2(t) are the voltages of each phase of the upper and lower bridge arms, j = a, b, c; i_j1 (t) and i_j2(t) are the currents of each phase; and s_j1(t) and s_j2(t) are the switching functions corresponding to each bridge arm.

According to the operating characteristics of thyristors, it can be seen that the conduction of thyristors is determined by the phase-control function, which is generated by α(t) and θ_l(t). Therefore, switching functions are related to α(t) and θ_l(t). Taking phase a1 as an example, s_a1(t) can be expressed as follows:

$$s_{\mathrm{a}1}\left(t\right)=f\left({\theta }_l\left(t\right),\alpha \left(t\right)\right)$$

(5)

Considering the influence of control loops, a double-Fourier series method [20] is used to transform s_a1(t) into the frequency domain, with frequency components of (6 k ± 1)f₁ + n(f_p − f₁). According to the properties of the Fourier series, the lower the frequency is, the larger the component is. Therefore, the steady-state component with n = 0 and the harmonic components with n = ±1 are taken. At this time, the frequency domain model of s_a1(t) is shown in equation (6).

$$s_{\mathrm{a}1}\left(f\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\mathrm{j}\sqrt{3}{\mathrm{e}}^{-\mathrm{j}\left(6k\pm 1\right){\alpha }_0}}{\left(6k\pm 1\right)\mathrm{\pi }}}{\mathrm{e}}^{\mathrm{m}\mathrm{j}{\displaystyle \frac{5\mathrm{\pi }}{6}}}\hfill & f=\left(6k\pm 1\right)f_1\hfill \\ {\displaystyle \frac{\sqrt{3}{\mathrm{e}}^{-\mathrm{j}\left(6k\pm 1\right){\alpha }_0}}{2\mathrm{\pi }}}{\mathrm{e}}^{\mathrm{m}\mathrm{j}{\displaystyle \frac{5\mathrm{\pi }}{6}}}A{e}^{\mathrm{j}{\varphi }_{\mathrm{A}}}\hfill & f=\left(6k\pm 1\right)f_1+\left(f_{\mathrm{p}}-f_1\right)\hfill \\ {\displaystyle \frac{\sqrt{3}{\mathrm{e}}^{-\mathrm{j}\left(6k\pm 1\right){\alpha }_0}}{2\mathrm{\pi }}}{\mathrm{e}}^{\mathrm{m}\mathrm{j}{\displaystyle \frac{5\mathrm{\pi }}{6}}}A{e}^{-\mathrm{j}{\varphi }_{\mathrm{A}}}& f=\left(6k\pm 1\right)f_1-\left(f_{\mathrm{p}}-f_1\right)\hfill \end{array}\right.$$

(6)

The remaining switch functions can be obtained by performing a phase shift at a certain angle and will not be repeated here.

According to the frequency domain convolution theorem, the frequency domain expression of the DC side voltage u_dc(s) can be obtained by combining Equations (3) and (6):

$$u_{\mathrm{dc}}\left(s\right)=\left\{\begin{array}{ll}{\displaystyle \frac{1}{2I_{\mathrm{dc}}}}\left({\displaystyle \frac{S^{\ast }}{12k+1}}+{\displaystyle \frac{-S}{12k-1}}\right){\mathrm{e}}^{-\mathrm{j}12k{\alpha }_0}\hfill & s=12k{f}_1\hfill \\ {\displaystyle \frac{1}{2I_{\mathrm{dc}}}}\left(-QA{e}^{\mathrm{j}{\varphi }_{\mathrm{A}}}-{\displaystyle \frac{S}{12k-1}}{\displaystyle \frac{V_{\mathrm{p}}{\mathrm{e}}^{\mathrm{j}{\phi }_{\mathrm{Vp}}}}{V_1{\mathrm{e}}^{\mathrm{j}{\phi }_{\mathrm{V}1}}}}\right){\mathrm{e}}^{-\mathrm{j}12k{\alpha }_0}& s=12k{f}_1+\left(f_{\mathrm{p}}-f_1\right)\hfill \\ {\displaystyle \frac{1}{2I_{\mathrm{dc}}}}\left(-QA{e}^{-\mathrm{j}{\varphi }_{\mathrm{A}}}+{\displaystyle \frac{S^{\ast }}{12k+1}}{\displaystyle \frac{V_{\mathrm{p}}{\mathrm{e}}^{-\mathrm{j}{\phi }_{\mathrm{Vp}}}}{V_1{\mathrm{e}}^{-\mathrm{j}{\phi }_{\mathrm{V}1}}}}\right){\mathrm{e}}^{-\mathrm{j}12k{\alpha }_0}\hfill & s=12k{f}_1-\left(f_{\mathrm{p}}-f_1\right)\hfill \end{array}\right.$$

(7)

where P and Q are the fundamental active power and reactive power, respectively; S = P + jQ; and the superscript ‘*’ represents conjugation.

The formula for the DC side current i_dc(s) is shown in Equation (8), where Z_dc(s) is the DC side impedance, and the calculation formula is shown in Equation (9).

$$i_{\mathrm{dc}2}\left(s\right)={\displaystyle \frac{u_{\mathrm{dc}2}\left(s\right)}{Z_{\mathrm{dc}}\left(s\right)}}$$

(8)

$$Z_{\mathrm{dc}}\left(s\right)=R_4+s{L}_4+\left(R_4+s{L}_4\right)\parallel {\displaystyle \frac{1}{s{C}_4}}$$

(9)

Combining Equations (4), (6), (8), and (9), the frequency domain expression of the AC side current of the controllable nonlinear load can be derived. The expression for the frequency component f_p of i_{nl_a} is as follows:

$$i_{\mathrm{nl}\_\mathrm{a}}\left[f_{\mathrm{p}}\right]={\displaystyle \frac{36k_{\mathrm{T}}^2}{{\mathrm{\pi }}^2}}{V}_{\mathrm{p}}{\mathrm{e}}^{\mathrm{j}{\phi }_{\mathrm{Vp}}}\times {\displaystyle \sum _{k=-\infty }^{\infty }\left\{\frac{\frac{-12kQ+\mathrm{j}P}{S\left(144k^2-1\right)}{M}_{\mathrm{p}}\left(s-\mathrm{j}{\omega }_1\right)}{Z_{\mathrm{dc}}\left(\mathrm{j}12k{\omega }_1\right)}+\frac{\frac{Q}{S\left(12k-1\right)}{M}_{\mathrm{p}}\left(s-\mathrm{j}{\omega }_1\right)+{\left(12k-1\right)}^{-2}}{Z_{\mathrm{dc}}\left[s+\mathrm{j}\left(12k-1\right){\omega }_1\right]}\right\}}$$

(10)

$$M_{\mathrm{p}}\left(s\right)={\displaystyle \frac{\frac{S}{I_{\mathrm{dc}}}\frac{G_{\mathrm{im}}\left(s\right)G_{\mathrm{i}}\left(s\right)}{Z_{\mathrm{dc}}\left(s\right)}-T_{\mathrm{PLL}}\left(s\right)}{1+\frac{G_{\mathrm{im}}\left(s\right)G_{\mathrm{i}}\left(s\right)}{Z_{\mathrm{dc}}\left(s\right)}\frac{Q}{I_{\mathrm{dc}}}}}$$

(11)

where k_T is the transformer ratio, and k_T = 1 in this article. Similarly, the expression for the frequency component f_n of i_{nl_a} can be derived:

$$i_{\mathrm{nl}\_\mathrm{a}}\left[f_{\mathrm{n}}\right]={\displaystyle \frac{36k_{\mathrm{T}}^2}{{\mathrm{\pi }}^2}}{V}_{\mathrm{n}}{\mathrm{e}}^{\mathrm{j}{\phi }_{\mathrm{Vn}}}\times {\displaystyle \sum _{k=-\infty }^{\infty }\left\{\frac{\frac{12kQ-\mathrm{j}P}{S^{\ast }\left(144k^2-1\right)}{M}_{\mathrm{n}}\left(s+\mathrm{j}{\omega }_1\right)}{Z_{\mathrm{dc}}\left(\mathrm{j}12k{\omega }_1\right)}+\frac{\frac{-Q}{S^{\ast }\left(12k+1\right)}{M}_{\mathrm{n}}\left(s+\mathrm{j}{\omega }_1\right)+{\left(12k+1\right)}^{-2}}{Z_{\mathrm{dc}}\left[s+\mathrm{j}\left(12k+1\right){\omega }_1\right]}\right\}}$$

(12)

$$M_{\mathrm{n}}\left(s\right)={\displaystyle \frac{\frac{S^{\ast }}{I_{\mathrm{dc}}}\frac{G_{\mathrm{im}}\left(s\right)G_{\mathrm{i}}\left(s\right)}{Z_{\mathrm{dc}}\left(s\right)}+T_{\mathrm{PLL}}\left(s\right)}{1+\frac{G_{\mathrm{im}}\left(s\right)G_{\mathrm{i}}\left(s\right)}{Z_{\mathrm{dc}}\left(s\right)}\frac{Q}{I_{\mathrm{dc}}}}}$$

(13)

According to the principle of the sequence impedance calculation, the positive and negative sequence impedances of the controllable nonlinear load can be obtained as shown in Equations (14) and (15) [21].

$$Z_{\mathrm{nl}\_\mathrm{p}}\left(s\right)={\displaystyle \frac{{\mathrm{\pi }}^2}{36k_{\mathrm{T}}^2}}\times {\left\{{\displaystyle \sum _{k=-\infty }^{\infty }\left[\frac{\frac{-12kQ+\mathrm{j}P}{S\left(144k^2-1\right)}{M}_{\mathrm{p}}\left(s-\mathrm{j}{\omega }_1\right)}{Z_{\mathrm{dc}}\left(\mathrm{j}12k{\omega }_1\right)}+\frac{\frac{Q}{S\left(12k-1\right)}{M}_{\mathrm{p}}\left(s-\mathrm{j}{\omega }_1\right)+{\left(12k-1\right)}^{-2}}{Z_{\mathrm{dc}}\left[s+\mathrm{j}\left(12k-1\right){\omega }_1\right]}\right]}\right\}}^{-1}$$

(14)

$$Z_{\mathrm{nl}\_\mathrm{n}}\left(s\right)={\displaystyle \frac{{\mathrm{\pi }}^2}{36k_{\mathrm{T}}^2}}\times {\left\{{\displaystyle \sum _{k=-\infty }^{\infty }\left[\frac{\frac{12kQ-\mathrm{j}P}{S^{\ast }\left(144k^2-1\right)}{M}_{\mathrm{n}}\left(s+\mathrm{j}{\omega }_1\right)}{Z_{\mathrm{dc}}\left(\mathrm{j}12k{\omega }_1\right)}+\frac{\frac{-Q}{S^{\ast }\left(12k+1\right)}{M}_{\mathrm{n}}\left(s+\mathrm{j}{\omega }_1\right)+{\left(12k+1\right)}^{-2}}{Z_{\mathrm{dc}}\left[s+\mathrm{j}\left(12k+1\right){\omega }_1\right]}\right]}\right\}}^{-1}$$

(15)

To verify the correctness of Equations (14) and (15), a simulation model corresponding to Figure 1 is built, for which specific parameters are shown in

Table 1. Parameters of the PV generation unit.

Parameters	Values	Parameters	Values
Udc/V	700	L2/mH	0.2
Effective value of grid’s line voltage/V	380	Cf/uF	6.8
Output active power P of PV/kW	10	Rd/Ω	1.8
Output reactive power Q of PV/kVar	0	Proportional coefficient kp1 of PLL	1.72
f1/Hz	50	Integral coefficient ki1 of PLL	150
Switching frequency fsw/kHz	35	Proportional coefficient kp2 of current loop	10
L1/mH	1.5	Integral coefficient ki2 of current loop	700

and

Table 2. Parameters of the controllable nonlinear load.

Parameters	Values	Parameters	Values
R4/Ω	1	time constant Tim of the Gim(s)	1.2 × 10−3
L4/mH	50	Proportional coefficient kp3 of PLL	7.64 × 10−3
C4/uF	1000	Integral coefficient ki3 of PLL	1.44
Vdc0/V	500	Proportional coefficient kp4 of DC current loop	1.1
gain kim of the Gim(s)	0.01	Integral coefficient ki4 of DC current loop	30

. The impedances are measured using the frequency scanning method, and the results are shown in Figure 5 and Figure 6.

It can be seen from Figure 5 that the impedance models of the controllable nonlinear load are basically consistent with the scanning results. Therefore, it can be considered that the impedance models shown in Equations (14) and (15) are correct. Similarly, frequency scanning is performed on the parallel subsystem, and the results are shown in Figure 6, verifying the correctness of Z_s(s) of the parallel subsystem.

3. Stability Analysis and Key Influencing Factor Analysis of the System

According to the analysis in Section 2, the stability of the PV grid-connected system with controllable nonlinear loads depends on whether Z_g(s)/[Z_pv(s) || Z_nl(s)] satisfies the Nyquist stability criterion. The logarithmic frequency stability criterion extends the Nyquist curve to the Bode diagram. Assuming that the frequency of the intersection of the impedance amplitudes of the grid and the parallel subsystem is f_i., the system is stable if the phase difference is less than 180°. On the contrary, the system is unstable.

Considering the worst situation for system stability, the grid impedance is represented by sL_g, where L_g is the equivalent inductance of the grid. To simulate a weak grid, L_g is taken as 0.003 H. Draw the impedance characteristics of the parallel subsystem and the grid in the Bode diagram, as shown in Figure 7.

From Figure 7, it is obvious that the amplitude–frequency characteristic curves of the parallel subsystem and the grid intersect at 119.04 Hz. At this point, the phase difference between them is 184.02°, which is greater than the stable boundary of 180°. Therefore, there is a risk of oscillation in the PV grid-connected system with controllable nonlinear loads near 119 Hz.

To obtain the key parameters that affect system stability, the concept of impedance relative sensitivity is adopted for analysis [22,23]. Specifically, the formulations detailing the relative sensitivity of both the real and imaginary components of the impedance are presented in Equations (16) and (17), respectively. These equations demonstrate that the relative sensitivity quantifies the extent of variation in the real (or imaginary) part of the impedance when a given parameter undergoes a 1% change. The larger the absolute value of the relative sensitivity for the real part (or imaginary part), the more significant the influence of that component of the system impedance. The positive (or negative) sign of the relative sensitivity for the real part (or imaginary part) indicates whether the corresponding component of the system impedance increases or decreases as the parameter increases or decreases. By comparing and ranking the absolute values of the impedance relative sensitivity for each parameter within the studied frequency band, we can identify the key factors that affect the stability of the system in that frequency range.

Impedance sensitivity analysis was performed for the system parameters in Table 1 and Table 2. The analyzed parameters encompass k_p1 and k_i1 of the PLL in the PV generation unit, k_p2 and k_i2 of the current inner loop, k_p3 and k_i3 of the PLL in the controllable nonlinear load, k_p4 and k_i4 of DC current control, and L₁, L₂, C_f, and R_d of the LCL-type filter. Figure 8 and Figure 9 illustrate the relative sensitivities of the real and imaginary parts of each parameter across the frequency range of 0–150 Hz.

$$H_{\mathrm{Re}}(K_i,s)={\displaystyle \frac{\mathrm{Re}[Z_{\mathrm{total}}(K_i+1\%K_i,s)]-\mathrm{Re}[Z_{\mathrm{total}}(K_i,s)]}{1\%K_i/K_i}}$$

(16)

$$H_{\mathrm{Im}}(K_i,s)={\displaystyle \frac{\mathrm{Im}[Z_{\mathrm{total}}(K_i+1\%K_i,s)]-\mathrm{Im}[Z_{\mathrm{total}}(K_i,s)]}{1\%K_i/K_i}}$$

(17)

where K_i represents each system parameter.

It can be found from Figure 8 that the parameters with higher absolute values of relative sensitivities of the real part are k_p4, k_p2, and k_i2 in the range of 80–150 Hz. Similarly, the parameters with higher absolute values of relative sensitivities of the imaginary part are k_p4 and k_p2, as illustrated in Figure 9. Therefore, it is concluded that the changes in k_p4 and k_p2 in the range of 80–150 Hz have a significant impact on the impedance of the system, and k_p4 and k_p2 are the key parameters affecting system stability.

4. A Parameter-Optimization Method Balancing System Stability and Response Performance

4.1. Stability Domain Analysis of Parameters

From the analysis results in Section 3, it is clear that there is oscillation risk in the PV grid-connected system with controllable nonlinear in the range of 80–150 Hz, and k_p4 and k_p2 are the key parameters that affect system stability. By selecting appropriate values of k_p4 and k_p2, the impedance of the parallel subsystem can be reshaped, thereby improving system stability. However, according to [24,25], there is a certain interaction among control parameters or control loops, which may narrow the selectable ranges of control parameters.

To define the initial optimization range of k_p4 and k_p2, k_p4 and k_p2 are used as independent variables to analyze the variation of the equivalent resistance of the system based on the negative-resistance effect. The result is shown in Figure 10, which adds a reference plane for convenience of observation. Figure 11 is a contour map of the equivalent resistance from Figure 10.

As shown in Figure 10 and Figure 11, the equivalent resistance of the system gradually decreases and even becomes negative as k_p4 and k_p2 increase. In a certain region, the equivalent resistance is less than 0, which indicates that the system may oscillate and become unstable when k_p4 and k_p2 take values within this region. Therefore, the region where the equivalent resistance is less than 0 is defined as an unstable domain, and the remaining region is defined as a stable domain. The range corresponding to the stable domain is the initial optimization range of k_p4 and k_p2.

4.2. Key Parameter Optimization Based on Particle Swarm Optimization Algorithm

By optimizing k_p4 and k_p2, the impedance of the system in the easily oscillating frequency band can be reshaped, thereby improving the stability of the system. Consequently, based on the stability domain analysis of the key parameters in Section 4.1, this subsection proposes a parameter optimization method that balances system stability and response performance.

According to the negative-resistance effect [26], when the equivalent reactance of the system is equal to 0 and the corresponding equivalent resistance is greater than 0, the system is stable. The impedance characteristics of the system based on the negative-resistance effect under initial parameters is shown in Figure 12. It is illustrated in Figure 12 that the imaginary part of the system impedance is equal to 0, and the equivalent resistance of the system is less than 0 at 119 Hz. Therefore, there is an oscillation risk in the PV grid-connected system with a controllable nonlinear load at 119 Hz, which is consistent with the analysis results in Section 3. Moreover, In the range of 80–150 Hz, the equivalent resistance of the system has a minimum value R_min, and its value is less than 0. There are two intersections between the equivalent reactance and the reference line, where the equivalent resistance corresponding to the first intersection is greater than 0 and the second equivalent resistance is less than 0. Negative resistance often appears at the second intersection. If R_min > 0 in the range of 80–150 Hz, the equivalent resistances corresponding to the intersections of the equivalent inductance and the reference line are both positive. Therefore, R_min can be used as a stability optimization indicator.

On the other hand, to quantify the response performance and oscillation-suppression effect of the system, the d-axis current i_d(t) of the PV generation unit and the DC side current i_dc(t) of the controllable nonlinear load are used to define the oscillation energy function ψ, which is shown in Equation (18).

$$\psi ={\displaystyle \underset{t_{\mathrm{s}}}{\overset{t_{\mathrm{s}}+T_{\mathsf{\psi }}}{\int }}\left[\left|i_{\mathrm{d}}\left(t\right)-i_{\mathrm{dref}}\right|+\left|i_{\mathrm{dc}}\left(t\right)-i_{\mathrm{dc}\_\mathrm{ref}}\right|\right]\mathrm{d}t}$$

(18)

where t_s is the starting time for calculation, T_ψ is the calculation time, i_dref(t) is the reference value of i_d(t), and i_{dc_ref} is the reference value of i_dc(t). it is obvious that the smaller the value of ψ is, the better the response performance and oscillation suppression effect are.

Based on the above analysis, a fitness function J is established, of which expression is as follows:

$$J=\min \left[k_1\left(-R_{\min }\right)+k_2\psi \right]$$

(19)

where k₁ and k₂ are the weight coefficients of stability indicator R_min and performance indicator ψ, respectively.

On the basis of the fitness function J, the PSO algorithm is used to optimize k_p4 and k_p2 within the parameter stability domain. The specific optimization process is shown in Figure 13.

5. Simulation Verification

Based on the simulation model built in Section 2, the L_g is set to 0.003 H, and a parameter optimization program is added to verify the correctness of the analyses and the effectiveness of the oscillation-suppression method based on parameter optimization.

5.1. Verification of Stability Analysis

To verify the correctness of the stability analysis in Section 3, L_g is set to 0.003 H, and the remaining parameters are the same as those in Section 2, which will not be repeated here. At this time, the voltages and currents at the PCC of the system are shown in Figure 14. From Figure 14, it can be seen that there is a significant resonance in the voltages and currents at the PCC under the initial parameters. To analyze the resonant frequency of the voltages and currents, the single-phase voltage and current of PCC are collected for FFT, and the results are shown in Figure 15.

According to the FFT results, the resonant components of the voltage and current under initial parameters are mainly concentrated around 120 Hz, with the dominant resonant component being 115 Hz. The current signal also contains a coupling frequency of 15 Hz. The dominant component 115 Hz is close to the oscillation of 119.04 Hz analyzed in Section 3, which verifies the correctness of the stability analysis.

5.2. Verification of Stability Domain Analysis of Parameter

From the analysis in Section 4, it can be known that the range of values for key parameters is reduced when considering the interaction between parameters. To verify the correctness of the stability domain analysis, this subsection selects two sets of data, A and B, for simulation based on the conclusions of Section 4. The values of the parameter groups are shown in

Table 3. The values of the parameter groups.

Region	Name	Value
unstable domain	A	(0.98, 6.6)
stable domain	B	(0.79, 4.7)

, and the location diagram is shown in Figure 16. The simulation results are shown in Figure 17 and Figure 18.

From Figure 17 and Figure 18, it can be seen that the voltages and currents exhibit a significant oscillation and cannot recover stability when group A parameters are used for k_p4 and k_p2. When using group B parameters, the oscillation in voltages and currents is suppressed, and the system returns to stability. Therefore, the simulation results have verified the correctness of the stability domain analysis of the parameters.

5.3. Verification of the Optimization Method

To verify the effectiveness of the proposed parameter-optimization method, this subsection uses the PSO algorithm for parameter optimization. The relevant parameter values of the PSO algorithm are shown in

Table 4. Parameters of PSO algorithm.

Parameter	Value	Parameter	Value
Spatial dimension	2	Population size	100
Inertial weight	0.8	Individual learning factor	0.9
Social learning factor	0.9	Maximum number of iterations	50

. The final optimization results are k_p2 = 2.9578 and k_p4 = 1.3423.

Using the optimized parameters of k_p4 and k_p2, the impedance characteristics of the system are shown by the orange curve in Figure 19. It can be known from Figure 19 that the phase difference at the intersection of the amplitude–frequency characteristics of the parallel subsystem and the grid before optimization is 184.02°, which is greater than the stable boundary 180°, indicating that the system is unstable. After optimizing k_p4 and k_p2, the phase difference is reduced to 169.21°, which is less than the stable boundary 180°, indicating system stability.

After optimizing k_p4 and k_p2, Figure 12 is changed to Figure 20. It can be seen from Figure 20 that in the easily oscillating frequency band, there are two intersections ① and ② between the equivalent inductance of the system and the reference line. The equivalent resistance corresponding to ① is still greater than 0, and the equivalent resistance corresponding to ② increases from the original negative resistance to 0.137. The equivalent resistances corresponding to the easily oscillating frequency band are all greater than 0. According to the negative-resistance effect, the system returns to stability after optimizing k_p4 and k_p2.

Based on the simulation model of the system built earlier, the simulation results after optimizing the parameters of k_p4 and k_p2 are shown in Figure 21, Figure 22, Figure 23 and Figure 24. It can be seen from Figure 21 that there is a significant oscillation in the three-phase voltages and currents of PCC before optimizing the parameters. After using the optimized k_p4 and k_p2 at 1.2 s, the three-phase voltages and currents return to stability. Stable single-phase voltage and current at PCC were collected for Fast Fourier Transform (FFT) analysis, and the results are shown in Figure 22. It is clear in Figure 22 that the fundamental amplitude of the voltage decreased from 235.1 V to 231.3 V, and the harmonic distortion rate decreased from 40.08% to 4.35% after optimization. The fundamental amplitude of the current is still 184.6 A, and the harmonic distortion rate has been reduced from the original 23.62% to 1.51%. In conclusion, after adopting the optimized parameters, the fundamental components of the voltage and current remain basically unchanged, while their harmonic contents are significantly reduced, proving the effectiveness of the proposed parameter-optimization method to suppress system oscillations.

Figure 23 illustrates the d-axis current i_d(t) of the PV generation unit before and after optimizing the parameters. From the graph, it can be seen that in 0.3~1.2 s, the i_d(t) oscillates under the initial parameters. After 1.2 s, by optimizing k_p4 and k_p2, the i_d(t) is maintained at around −21.5 A (the negative sign represents the actual direction of the current being opposite to the reference direction), which meets the corresponding value of the power outer loop control.

Similarly, from Figure 24, it can be seen that the i_dc(t) oscillates under the initial parameters. After optimizing k_p4 and k_p2, the i_dc(t) is maintained at the set value of around 100 A, and the oscillation is suppressed.

Due to the uncertainty of the inductance of the grid in practical engineering, the resonance frequency of the system may shift. To verify the robustness of the parameter optimization results, a simulation analysis is conducted based on changes in the equivalent inductance of the grid. Set the equivalent inductance L_g of the power grid to change from 0.003 H to 0.001 H, and the simulation results are shown in Figure 25, Figure 26, Figure 27 and Figure 28.

From Figure 25, it can be seen that the voltages and currents at the PCC can still remain stable when the equivalent inductance L_g becomes 0.001 H. At this time, the FFT analysis results of the single-phase voltage and current are shown in Figure 26. In Figure 26, the amplitude of the fundamental voltage at PCC is 283.7 V, and the total harmonic distortion (THD) of the voltage is 3.63%; the amplitude of the fundamental current at PCC is 162.2 A, and THD of the current is 1.94%.

The i_d(t) of the PV generation unit and the i_dc(t) of the controllable nonlinear load at L_g = 0.001 H are displayed in Figure 27 and Figure 28, respectively. As shown in the figures, after optimizing k_p4 and k_p2, the PV generation unit and the controllable nonlinear load can operate stably near the set operating condition when L_g = 0.001 H.

In addition to changes in the equivalent inductance L_g, the load level of the controllable nonlinear load may also fluctuate. Figure 29 and Figure 30 show the simulation results when i_{dc_ref} is changing from 100 A to 105 A at 1 s. From Figure 29 and Figure 30, it can be seen that after changing i_{dc_ref} to 105 A, the i_d(t) and the i_dc(t) can both recover stability after short-term fluctuations. Through FFT analysis of the stabilized signals, it can be concluded that the DC component of the i_d(t) is 21.5 A, and the DC component of i_dc(t) is 105 A, both operating near the set value, meeting the requirements.

In conclusion, by analyzing the simulation results of the system under the original parameter, changes in the equivalent inductance L_g and the reference current i_{dc_ref}, the effectiveness and robustness of the parameter optimization results in suppressing system oscillation have been verified.

6. Conclusions

This paper addresses the potential oscillations encountered in grid-connected photovoltaic systems with controllable nonlinear loads and presents an approach based on parameter optimization for suppressing these oscillations. Compared with traditional parameter-optimization methods for suppressing oscillations, this approach considers the impact on response performance during the parameter-optimization process. It aims to improve stability of the system while minimizing the impact on response performance as much as possible, achieving optimal comprehensive performance in terms of both stability and responsiveness under the condition of ensuring no oscillations in the system. The main conclusions of this paper can be summarized as follows:

(1)

Based on the established impedance model of the PV grid-connected system with controllable nonlinear loads, the influence of different parameters on the system impedance characteristics is analyzed using impedance relative sensitivity. This analysis reveals that the key parameters influencing system stability are the proportional coefficient k_p4 of the DC current loop within the controllable nonlinear loads and the proportional coefficient k_p2 of the current inner loop in the photovoltaic grid-connected inverter. By tuning these two parameters, the system impedance can be reshaped, thereby enhancing the overall stability of the system.

(2)

Considering the interactions between the two key parameters, the stability domain of the key parameters is analyzed. A parameter optimization method balancing system stability and response performance is then proposed, which aims to improve stability of the system while minimizing the impact on response performance. The validity of the proposed parameter-optimization algorithm is verified through comparative experiments conducted before and after the system adopts the optimized parameters. Additionally, simulation-based comparative experiments are carried out under varying conditions, specifically changes in the equivalent inductance of the power grid and alterations in load conditions. The experimental results conclusively demonstrate the robustness of the optimized parameters in suppressing system oscillations.