STABILITY CRITERION FOR CONTROL LOOP OF GRID-CONNECTED CONVERTER UNDER WEAK GRID CONDITION

Abstract

The present disclosure provides a method for determining a stability criterion for a control loop of a grid-connected converter under a weak grid condition, and belongs to the field of power control. The method includes performing a sampling, obtaining an expression of a transfer function of each of control loops of a grid-connected converter, further obtaining an expression of a transfer function for indicating stability performance of the system, obtaining an amplitude expression and a phase expression according to the transfer function, and providing a stability criterion for the control loop of the grid-connected converter under a weak grid condition and a stability margin expression of the system. The stability criterion for the control loop is significant and provided by a simple method, can quickly and accurately determine the stability of the grid-connected converter system under the weak grid condition, and is helpful for the parameters design of the controllers.

Claims

1. A method for determining a stability criterion for a control loop of a grid-connected converter under a weak grid condition, wherein the control loop of the grid-connected converter includes a current control loop and a phase-locked loop (PLL), and the determination of the stability criterion for the control loop of the grid-connected converter under the weak grid condition includes the following steps: step 1: performing a sampling to obtain an output current I.sub.g of the grid-connected converter and an output voltage U.sub.g of the grid-connected converter, and providing an expression of a grid impedance link G.sub.1 (s) of the grid-connected converter, an expression of a closed-loop transfer function G.sub.2(s) for the PLL of the grid-connected converter, and an expression of a closed-loop transfer function G.sub.3(s) for the current control loop of the grid-connected converter without considering the influence of the PLL, where the expression of the grid impedance link G.sub.1(s) of the grid-connected converter is as follows: $G_1\left(s\right)=-\frac{I_g}{U_g}\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right),$ where s is a Laplace operator, L.sub.grid is an inductive component of the grid impedance, and R.sub.grid is a resistive component of the grid impedance; the expression of the closed-loop transfer function G.sub.2(s) for the PLL of the grid-connected converter is as follows: $G_2\left(s\right)=\frac{2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2}{s^2+2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2},$ where ξ is a damping ratio of the PLL, and ω.sub.pll is a control bandwidth of the PLL; and the expression G.sub.3(s) of the closed-loop transfer function G.sub.3(s) for the current control loop of the grid-connected converter without considering the influence of the PLL is as follows: $G_3\left(s\right)=\frac{G_{\mathrm{CL}}{G}_{\mathrm{main}}}{1+G_{\mathrm{CL}}{G}_{\mathrm{main}}},$ where G.sub.CL is a transfer function for a proportional-integral (PI) link of the current control loop, and G.sub.main is a transfer function for a filtration link of the grid-connected converter; step 2: providing an expression of an open-loop transfer function G.sub.IL(s) for the current control loop of the grid-connected converter considering the influence of the PLL: $\begin{array}{c}{G}_{\mathrm{IL}}\left(s\right)=G_1\left(s\right)\times G_2\left(s\right)\times G_3\left(s\right)\\ =\left[\frac{I_g}{U_g}\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right)\right]\times \left[\frac{2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2}{s^2+2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2}\right]\times \\ \left[\frac{G_{\mathrm{CL}}{G}_{\mathrm{main}}}{1+G_{\mathrm{CL}}{G}_{\mathrm{main}}}\right]\\ =-\frac{I_g}{U_g}\times \frac{\left(2{\mathrm{\xi \omega }}_{\mathrm{pll}}s+{\omega }_{\mathrm{pll}}^2\right)\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right)}{s^2+2{\mathrm{\xi \omega }}_{\mathrm{pll}}s+{\omega }_{\mathrm{pll}}^2}\times \\ \frac{G_{\mathrm{CL}}{G}_{\mathrm{main}}}{1+G_{\mathrm{CL}}{G}_{\mathrm{main}}}\end{array}$ step 3: providing an expression of a combined transfer function G.sub.pll_grid(s) for a grid impedance and the PLL of the grid-connected converter, and performing identical transformation on the open-loop transfer function G.sub.IL(s) for the current control loop in step 2, where the expression of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter is as follows: $\begin{array}{c}{G}_{\mathrm{pll\_grid}}\left(s\right)=G_1\left(s\right)\times G_2\left(s\right)\\ =-\frac{I_g}{U_g}\times \frac{\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right)\left(2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2\right)}{s^2+2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2};\end{array}$ the identical transformation on the open-loop transfer function G.sub.IL(s) for the current control loop leads to: $\begin{array}{c}{G}_{\mathrm{IL}}\left(s\right)=G_{\mathrm{pll\_grid}}\left(s\right)\times G_3\left(s\right)\\ =G_{\mathrm{pll\_grid}}\left(s\right)/\left(1/G_3\left(s\right)\right)\\ =G_{\mathrm{pll\_grid}}\left(s\right)/(G_{\mathrm{current}}\left(s\right)\end{array},$ where G.sub.current(s)=1/G.sub.3(s), which is labeled as a criterion transfer function for the current control loop of the grid-connected converter without considering the influence of the PLL; and step 4: further providing a stability criterion for the control loop of the grid-connected converter under a weak grid condition according to a transfer function of each link in the foregoing three steps by specifically: step 4.1: obtaining a phase expression and an amplitude expression of each of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter and the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL in the frequency domain, where an amplitude and a phase of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter in the frequency domain are respectively labeled as a combined amplitude A.sub.pll_grid(ω) and a combined phase Ψ.sub.pll_grid(ω), the combined amplitude A.sub.pll_grid(ω) and the combined phase Ψ.sub.pll_grid(ω) being respectively expressed by: $\{\begin{array}{c}{A}_{\mathrm{pll\_gird}}\left(\omega \right)=20\log \left(\frac{I_g}{U_g}\times \frac{\sqrt{{\omega }^2{L}_{\mathrm{grid}}^2+R_{\mathrm{grid}}^2}\times \sqrt{{\left(2{\mathrm{\xi \omega }}_{\mathrm{pll}}\omega \right)}^2+{\omega }_{\mathrm{pll}}^4}}{\sqrt{{\left(2{\mathrm{\xi \omega }}_{\mathrm{pll}}\omega \right)}^2+{\left({\omega }_{\mathrm{pll}}^2-{\omega }^2\right)}^2}}\right)\\ {\psi }_{\mathrm{pll\_gird}}\left(\omega \right)=\arctan \left(\frac{\omega L_{\mathrm{grid}}}{R_{\mathrm{grid}}}\right)+\arctan \left(\frac{2{\mathrm{\xi \omega \omega }}_{\mathrm{pll}}}{{\omega }_{\mathrm{pll}}^2}\right)-\arctan \left(\frac{2{\mathrm{\xi \omega \omega }}_{\mathrm{pll}}}{{\omega }_{\mathrm{pll}}^2-{\omega }^2}\right)\end{array},$ where ω is a rotational angular frequency of the frequency domain; and an amplitude and a phase of the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL in the frequency domain are respectively labeled as a current amplitude A.sub.current(ω) and a current phase Ψ.sub.current(ω), the current amplitude A.sub.current(ω) and the current phase Ψ.sub.current(ω) being respectively expressed by: $\{\begin{array}{c}{A}_{\mathrm{current}}\left(\omega \right)=\frac{.Math.G_{\mathrm{CL}}\left(\omega \right).Math..Math.G_{\mathrm{main}}\left(\omega \right).Math.}{.Math.1+G_{\mathrm{CL}}\left(\omega \right)G_{\mathrm{main}}\left(\omega \right).Math.}\\ {\psi }_{\mathrm{current}}\left(\omega \right)=\angle G_{\mathrm{CL}}\left(\omega \right)+\angle G_{\mathrm{main}}\left(\omega \right)-\angle \left(1+G_{\mathrm{CL}}\left(\omega \right)G_{\mathrm{main}}\left(\omega \right)\right)\end{array},$ where G.sub.CL(ω) is an expression of the transfer function G.sub.CL for the PI link of the current control loop in the frequency domain, |G.sub.CL(ω)| is an amplitude of the transfer function G.sub.CL for the PI link of the current control loop in the frequency domain, G.sub.main(ω) is an expression of the transfer function G.sub.main for the filtration link of the grid-connected converter in the frequency domain, |G.sub.main(ω)|is an amplitude of the transfer function G.sub.main for the filtration link of the grid-connected converter in the frequency domain, |1+G.sub.CL(ω)G.sub.main(ω)| is an amplitude of 1+G.sub.CL(ω)G.sub.main(ω) in the frequency domain, ∠G.sub.CL(Ψ) is a phase of G.sub.CL(ω) in the frequency domain, ∠.sub.main(ω) is a phase of G.sub.main(ω) in the frequency domain, and ∠(1+G.sub.CL(ω)G.sub.main(ω)) is a phase of 1+G.sub.CL(ω)G.sub.main(ω) in the frequency domain; step 4.2: providing the stability criterion for the control loop of the grid-connected converter under the weak grid condition according to the amplitude expression and the phase expression obtained in step 4.1 by determining that the grid-connected converter system is stable when both an amplitude stability condition and a phase stability condition are satisfied; otherwise, determining that the control loop of the grid-connected converter is unstable, where the stability criterion for the control loop of the grid-connected converter under the weak grid condition is determined as follows: the amplitude for the control loop of the grid-connected converter is stable if a phase difference between the combined phase Ψ.sub.pll_grid(ω) and the current phase Ψ.sub.current(ω) is 180°, and A.sub.current(ω.sub.180°)>A.sub.pll_grid(ω.sub.180°); the phase for the control loop of the grid-connected converter is stable if the combined amplitude A.sub.pll_grid(ω) is the same as the current amplitude A.sub.current(ω), and Ω.sub.pll_grid(ω.sub.crossover)−Ψ.sub.current(ω.sub.crossover)−180°>0; and the whole grid-connected converter is stable if the control loop of the grid-connected converter can both satisfy the amplitude stability condition and the phase stability condition, where, ω.sub.180° is a corresponding angular frequency when the phase difference between the combined phase Ψ.sub.pll_grid(ω) and the current phase Ψ.sub.current(ω) is 180°; A.sub.current(ω.sub.180°) is an amplitude of the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL when the angular frequency is ω.sub.180°; A.sub.pll_grid(ω.sub.180°) is an amplitude of the combined transfer function for the grid impedance and the PLL of the grid-connected converter when the angular frequency is ω.sub.⊇°; ω.sub.crossover is a corresponding angular frequency when the combined amplitude A.sub.pll_grid(ω) is the same as the current amplitude A.sub.current(ω); Ψ.sub.current(ω.sub.crossover) is a phase of the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL when the angular frequency is ω.sub.crossover; and Ψ.sub.pll_grid(ω.sub.crossover) is a phase of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter when the angular frequency is ω.sub.crossover; and step 4.3: providing an amplitude margin expression A.sub.M and a phase margin expression P.sub.M of the grid-connected converter system under the weak grid condition according to the phase expression and the amplitude expression of each of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter and the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL in the frequency domain in step 4.1 and in combination with the stability criterion for the control loop in step 4.2: $\{\begin{array}{c}{A}_M=A_{\mathrm{current}}\left({\omega }_{180°}\right)-A_{\mathrm{pll\_grid}}\left({\omega }_{180°}\right)\\ P_M={\psi }_{\mathrm{pll\_grid}}\left({\omega }_{\mathrm{crossover}}\right)-{\psi }_{\mathrm{current}}\left({\omega }_{\mathrm{crossover}}\right)-180°\end{array}.$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0046] FIG. 1 illustrates a topological structure of a grid-connected converter according to the present disclosure;

[0047] FIG. 2 illustrates a Bode diagram of an open-loop transfer function G.sub.IL(s) for a current control loop that takes into account the influence of a PLL when the strength of a power grid is SCR=1.5, the design bandwidth of the current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=64.5rad/s;

[0048] FIG. 3 illustrates a simulation waveform of an output current I.sub.g of a grid-connected converter when the strength of a power grid is SCR=1.5, the design bandwidth of a current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=64.5rad/s;

[0049] FIG. 4 illustrates a Bode diagram of an open-loop transfer function G.sub.IL (s) for a current control loop that takes into account the influence of a PLL when the strength of a power grid is SCR=1.5, the design bandwidth of the current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=320.4rad/s;

[0050] FIG. 5 illustrates a simulation waveform of an output current I.sub.g of a grid-connected converter when the strength of a power grid is SCR=1.5, the design bandwidth of a current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=320.4rad/s;

[0051] FIG. 6 illustrates a Bode diagram of an open-loop transfer function G.sub.IL(s) for a current control loop that takes into account the influence of a PLL when the strength of a power grid is SCR=1.5, the design bandwidth of the current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=452.3rad/s;

[0052] FIG. 7 illustrates a simulation waveform of an output current I.sub.g of a grid-connected converter when the strength of a power grid is SCR=1.5, the design bandwidth of a current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=452.3rad/s;

[0053] FIG. 8 illustrates a Bode diagram of an open-loop transfer function G.sub.IL(s) for a current control loop that takes into account the influence of a PLL when the strength of a power grid is SCR=1.1, the design bandwidth of the current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=32.3rad/s;

[0054] FIG. 9 illustrates a simulation waveform of an output current I.sub.g of a grid-connected converter when the strength of a power grid is SCR=1.1, the design bandwidth of a current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=32.3rad/s;

[0055] FIG. 10 illustrates a Bode diagram of an open-loop transfer function G.sub.IL(s) for a current control loop that takes into account the influence of a PLL when the strength of a power grid is SCR=1.1, the design bandwidth of the current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=193.7rad/s;

[0056] FIG. 11 illustrates a simulation waveform of an output current I.sub.g of a grid-connected converter when the strength of a power grid is SCR=1.1, the design bandwidth of a current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=193.7rad/s;

[0057] FIG. 12 illustrates a Bode diagram of an open-loop transfer function G.sub.IL(s) for a current control loop that takes into account the influence of a PLL when the strength of a power grid is SCR=1.1, the design bandwidth of the current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=387.4rad/s; and

[0058] FIG. 13 illustrates a simulation waveform of an output current I.sub.g of a grid-connected converter when the strength of a power grid is SCR=1.1, the design bandwidth of a current loop is ω.sub.CL=4,750rad/s, and the design bandwidth of the PLL is ω.sub.pll=387.4rad/s.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0059] The present disclosure will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0060] FIG. 1 illustrates a topological structure of a grid-connected converter according to an embodiment of the present disclosure. As shown in FIG. 1, the topological structure includes a direct-current (DC) voltage source U.sub.dc, a DC-side filter capacitor C.sub.dc, a three-phase half-bridge inverter, an L filter, and a three-phase alternating-current (AC) power grid. The DC voltage source U.sub.dc is connected to an input terminal of the inverter through the filter capacitor C.sub.dc. An output terminal of the inverter is connected to the three-phase AC power grid through the L filter. L.sub.grid is a corresponding inductive component of a grid impedance, and is labeled as the inductive component L.sub.grid of the grid impedance. R.sub.grid is a corresponding resistive component of the grid impedance, and is labeled as the resistive component R.sub.grid of the grid impedance.

[0061] In implementation of the present disclosure, the following electrical parameters are provided: The system has a DC voltage source U.sub.dc=750V , a rated capacity of 30 kVA for the grid-connected converter, an effective phase voltage E.sub.a=E.sub.b=E.sub.c=220V for the three-phase power grid, a switching frequency f.sub.sw=10 kHz , sampling time T.sub.s=100 μs, and a filter inductance L=2 mH.

[0062] The present disclosure provides a stability criterion for a control loop of a grid-connected converter under a weak grid condition. The control loop of the grid-connected converter includes a current control loop and a PLL. The determination of the stability criterion for a control loop of a grid-connected converter under a weak grid condition includes the following steps: [0063] step 1: performing a sampling to obtain an output current I.sub.g of the grid-connected converter and an output voltage U.sub.g of the grid-connected converter, and providing an expression of a grid impedance link G.sub.1(s) of the grid-connected converter, an expression of a closed-loop transfer function G.sub.2(s) for the PLL of the grid-connected converter, and an expression of a closed-loop transfer function G.sub.3(s) for the current control loop of the grid-connected converter without considering the influence of the PLL, where [0064] the expression of the grid impedance link G.sub.1(s) of the grid-connected converter is as follows:

[00010]$G_1\left(s\right)=-\frac{I_g}{U_g}\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right),$ [0065] where s is a Laplace operator, L.sub.grid is an inductive component of the grid impedance, and R.sub.grid is a resistive component of the grid impedance; [0066] the expression of the closed-loop transfer function G.sub.2(s) for the PLL of the grid-connected converter is as follows:

[00011]$G_s\left(s\right)=\frac{2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2}{s^2+2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2},$ [0067] where ξ is a damping ratio of the PLL, and ω.sub.pll is a control bandwidth of the PLL; and [0068] the expression G.sub.3(s) of the closed-loop transfer function G.sub.3(s) for the current control loop of the grid-connected converter without considering the influence of the PLL is as follows:

[00012]$G_3\left(s\right)=\frac{G_{\mathrm{CL}}{G}_{\mathrm{main}}}{1+G_{\mathrm{CL}}{G}_{\mathrm{main}}},$ [0069] where G.sub.CL is a transfer function for a proportional-integral (PI) link of the current control loop, and G.sub.main is a transfer function for a filtration link of the grid-connected converter; [0070] step 2: providing an expression of an open-loop transfer function G.sub.IL(s) for the current control loop of the grid-connected converter considering the influence of the PLL:

[00013]$\begin{array}{c}{G}_{\mathrm{IL}}\left(s\right)=G_1\left(s\right)\times G_2\left(s\right)\times G_3\left(s\right)\\ =\left[\frac{I_g}{U_g}\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right)\right]\times \left[\frac{2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2}{s^2+2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2}\right]\times \\ \left[\frac{G_{\mathrm{CL}}{G}_{\mathrm{main}}}{1+G_{\mathrm{CL}}{G}_{\mathrm{main}}}\right]\\ =-\frac{I_g}{U_g}\times \frac{\left(2{\mathrm{\xi \omega }}_{\mathrm{pll}}s+{\omega }_{\mathrm{pll}}^2\right)\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right)}{s^2+2{\mathrm{\xi \omega }}_{\mathrm{pll}}s+{\omega }_{\mathrm{pll}}^2}\times \\ \frac{G_{\mathrm{CL}}{G}_{\mathrm{main}}}{1+G_{\mathrm{CL}}{G}_{\mathrm{main}}}\end{array}$ [0071] step 3: providing an expression of a combined transfer function G.sub.pll_grid(s) for a grid impedance and the PLL of the grid-connected converter, and performing identical transformation on the open-loop transfer function G.sub.IL(s) for the current control loop in step 2, where [0072] the expression of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter is as follows:

[00014]$\begin{array}{c}{G}_{\mathrm{pll\_grid}}\left(s\right)=G_1\left(s\right)\times G_2\left(s\right)\\ =-\frac{I_g}{U_g}\times \frac{\left({\mathrm{sL}}_{\mathrm{grid}}+R_{\mathrm{grid}}\right)\left(2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2\right)}{s^2+2\times \xi \times {\omega }_{\mathrm{pll}}\times s+{\omega }_{\mathrm{pll}}^2};\end{array}$ [0073] the identical transformation on the open-loop transfer function G.sub.IL(s) for the current control loop leads to:

[00015]$\begin{array}{c}{G}_{\mathrm{IL}}\left(s\right)=G_{\mathrm{pll\_grip}}\left(s\right)\times G_3\left(s\right)\\ =G_{\mathrm{pll\_grip}}\left(s\right)/\left(1/G_3\left(s\right)\right)\\ =G_{\mathrm{pll\_grip}}\left(s\right)/(G_{\mathrm{current}}\left(s\right)\end{array},$ [0074] where G.sub.current(s)=1/G.sub.3(s), which is labeled as a criterion transfer function for the current control loop of the grid-connected converter without considering the influence of the PLL; and [0075] step 4: further providing a stability criterion for the control loop of the grid-connected converter under a weak grid condition according to a transfer function of each link in the foregoing three steps by specifically:

[0076] step 4.1: obtaining a phase expression and an amplitude expression of each of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter and the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL in the frequency domain, where [0077] an amplitude and a phase of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter in the frequency domain are respectively labeled as a combined amplitude A.sub.pll_grid(ω) and a combined phase Ψ.sub.pll_grid(ω), the combined amplitude A.sub.pll_grid(ω) and the combined phase Ψ.sub.pll_grid(ω) being respectively expressed by:

[00016]$\{\begin{array}{c}{A}_{\mathrm{pll\_gird}}\left(\omega \right)=20\log \left(\frac{I_g}{U_g}\times \frac{\sqrt{{\omega }^2{L}_{\mathrm{grid}}^2+R_{\mathrm{grid}}^2}\times \sqrt{{\left(2{\mathrm{\xi \omega }}_{\mathrm{pll}}\omega \right)}^2+{\omega }_{\mathrm{pll}}^4}}{\sqrt{{\left(2{\mathrm{\xi \omega }}_{\mathrm{pll}}\omega \right)}^2+{\left({\omega }_{\mathrm{pll}}^2-{\omega }^2\right)}^2}}\right)\\ {\psi }_{\mathrm{pll\_gird}}\left(\omega \right)=\arctan \left(\frac{\omega L_{\mathrm{grid}}}{R_{\mathrm{grid}}}\right)+\arctan \left(\frac{2{\mathrm{\xi \omega \omega }}_{\mathrm{pll}}}{{\omega }_{\mathrm{pll}}^2}\right)-\arctan \left(\frac{2{\mathrm{\xi \omega \omega }}_{\mathrm{pll}}}{{\omega }_{\mathrm{pll}}^2-{\omega }^2}\right)\end{array}.$ [0078] where ω is a rotational angular frequency of the frequency domain; and [0079] an amplitude and a phase of the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL in the frequency domain are respectively labeled as a current amplitude A.sub.current(ω) and a current phase Ψ.sub.current(ω), the current amplitude A.sub.current(ω) and the current phase Ψ.sub.current(ω) being respectively expressed by:

[00017]$\{\begin{array}{c}{A}_{\mathrm{current}}\left(\omega \right)=\frac{.Math.G_{\mathrm{CL}}\left(\omega \right).Math..Math.G_{\mathrm{main}}\left(\omega \right).Math.}{.Math.1+G_{\mathrm{CL}}\left(\omega \right)G_{\mathrm{main}}\left(\omega \right).Math.}\\ {\psi }_{\mathrm{current}}\left(\omega \right)=\angle G_{\mathrm{CL}}\left(\omega \right)+\angle G_{\mathrm{main}}\left(\omega \right)-\angle \left(1+G_{\mathrm{CL}}\left(\omega \right)G_{\mathrm{main}}\left(\omega \right)\right)\end{array},$ [0080] where G.sub.CL(ω) is an expression of the transfer function G.sub.CL for the PI link of the current control loop in the frequency domain, |G.sub.CL(ω)| is an amplitude of the transfer function G.sub.CL for the PI link of the current control loop in the frequency domain, G.sub.main(ω) is an expression of the transfer function G.sub.main for the filtration link of the grid-connected converter in the frequency domain, |G.sub.main(ω)| is an amplitude of the transfer function G.sub.main for the filtration link of the grid-connected converter in the frequency domain, |1+G.sub.CL(ω)G.sub.main(ω)| is an amplitude of in the frequency domain, 1+G.sub.CL(ω)G.sub.main(ω) in the frequency domain, ∠G.sub.CL(ω) is a phase of G.sub.CL(ω) in the frequency domain, φG.sub.main(ω) is a phase of G.sub.main(ω) in the frequency domain, and ∠(1+G.sub.CL(ω)G.sub.main(ω)) is a phase of 1+G.sub.CL(ω)G.sub.main(ω) in the frequency domain; [0081] step 4.2: providing the stability criterion for the control loop of the grid-connected converter under the weak grid condition according to the amplitude expression and the phase expression obtained in step 4.1 by determining that the grid-connected converter system is stable when both an amplitude stability condition and a phase stability condition are satisfied; otherwise, determining that the control loop of the grid-connected converter is unstable, where [0082] the stability criterion for the control loop of the grid-connected converter under the weak grid condition is determined as follows: [0083] the amplitude for the control loop of the grid-connected converter is stable if a phase difference between the combined phase Ψ.sub.pll_grid(ω) and the current phase Ψ.sub.current(ω) is 180°, and A.sub.current(ω.sub.180°)>A.sub.pll_grid(ω.sub.180°); [0084] the phase for the control loop of the grid-connected converter is stable if the combined amplitude A.sub.pll_grid(ω) is the same as the current amplitude A.sub.current(ω), and Ψ.sub.pll_grid(ω.sub.crossover)−Ψ.sub.current(ω.sub.crossover)−180°>0; and [0085] the whole grid-connected converter is stable if the control loop of the grid-connected converter can both satisfy the amplitude stability condition and the phase stability condition, [0086] where, [0087] ω.sub.180° is a corresponding angular frequency when the phase difference between the combined phase Ψ.sub.pll_grid(ω) and the current phase Ψ.sub.current(ω) is 180°; [0088] A.sub.current(ω.sub.180°) is an amplitude of the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL when the angular frequency is ω.sub.180°; [0089] A.sub.pll_grid(ω.sub.180°) is an amplitude of the combined transfer function for the grid impedance and the PLL of the grid-connected converter when the angular frequency is ω.sub.180°; [0090] ω.sub.crossover is a corresponding angular frequency when the combined amplitude A.sub.pll_grid(ω) is the same as the current amplitude A.sub.current(ω);

[0091] Ψ.sub.current(ω.sub.crossover) is a phase of the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL when the angular frequency is ω.sub.crossover; and

[0092] Ψ.sub.pll_grid(ω.sub.crossover) is a phase of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter when the angular frequency is ω.sub.crossover; and [0093] step 4.3: providing an amplitude margin expression A.sub.M and a phase margin expression P.sub.M of the grid-connected converter system under the weak grid condition according to the phase expression and the amplitude expression of each of the combined transfer function G.sub.pll_grid(s) for the grid impedance and the PLL of the grid-connected converter and the criterion transfer function G.sub.current(s) for the current control loop of the grid-connected converter without considering the influence of the PLL in the frequency domain in step 4.1 and in combination with the stability criterion for the control loop in step 4.2:

[00018]$\{\begin{array}{c}{A}_M=A_{\mathrm{current}}\left({\omega }_{180°}\right)-A_{\mathrm{pll\_grid}}\left({\omega }_{180°}\right)\\ P_M={\psi }_{\mathrm{pll\_grid}}\left({\omega }_{\mathrm{crossover}}\right)-{\psi }_{\mathrm{current}}\left({\omega }_{\mathrm{crossover}}\right)-180°\end{array}.$

[0094] In the embodiment, there are two cases of the grid-connected converter system under the weak grid condition:

[0095] First case: The grid-connected converter system has a short-circuit ratio SCR=1.5, and the grid impedance has an inductive component L.sub.grid=10.2 mH and a resistive component R.sub.grid=0.32Ω. When a control bandwidth of the current loop is determined to be ω.sub.CL=4,750rad/s, the control bandwidth of the PLL can be calculated according to the controller design method of the grid-connected converter under the weak grid, and the stability and stability margin of the system can be determined according to the stability criterion for the control loop. Hereinafter, the present disclosure provides descriptions on three cases where the PLL has different control bandwidths: [0096] (1) When the PLL has a control bandwidth ω.sub.pll=64.5rad/s the Bode diagram of the open-loop transfer function G.sub.IL(s) for the current control loop considering the influence of the PLL is as shown in FIG. 2. In the figure, upper and lower portions respectively represent an amplitude-frequency curve and a phase-frequency curve of the Bode diagram. The horizontal coordinate represents angular frequency in units of rad/s. For the amplitude-frequency curve, the vertical coordinate represents amplitude in units of dB. For the phase-frequency curve, the vertical coordinate represents phase in units of deg. As can be seen from FIG. 2, ω.sub.180°=292rad/s, A.sub.current(ω.sub.180°)=0.0165 dB, A.sub.pll_grid(ω.sub.180°)=−14.2 dB, and the amplitude margin A.sub.M=A.sub.current(ω.sub.180°)−A.sub.pll_grid(ω.sub.⊇°)=0.0165 dB−(−14.2)dB=142165 dB>0. Since the amplitude margin is greater than 0, the amplitude satisfies the stability condition of the system. In the full band range, A.sub.current(ω) is always greater than A.sub.pll_grid(ω), such that the phase margin of the system always satisfies the stability condition. The waveform for the output current of the grid-connected converter system is as shown in FIG. 3. According to the waveform for the output current, the system always keeps stable in this condition. In FIG. 3, the horizontal coordinate represents time t in units of second (s), while the vertical coordinate represents the output current I.sub.g of the grid-connected converter in a unit of A. [0097] (2) When the PLL has a control bandwidth ω.sub.pll=320.4rad/s, the Bode diagram of the open-loop transfer function G.sub.IL(s) for the current control loop considering the influence of the PLL is as shown in FIG. 4. In the figure, upper and lower portions respectively represent an amplitude-frequency curve and a phase-frequency curve of the Bode diagram. The horizontal coordinate represents angular frequency in units of rad/s. For the amplitude-frequency curve, the vertical coordinate represents an amplitude in units of dB. For the phase-frequency curve, the vertical coordinate represents phase in units of deg. As can be seen from FIG. 4, ω.sub.180°=999rad/s, A.sub.current(ω.sub.180°)=0.188 dB, A.sub.pll_grid(ω.sub.180°)=−0.223 dB, and the amplitude margin A.sub.M=A.sub.current(ω.sub.180°)−A.sub.pll_grid(ω.sub.180°)=0.188 dB−(−0.233) dB=0.421 dB>0. The amplitude margin is relatively small but still satisfies the stability condition. In the full band range, A.sub.current(ω) is always greater than A.sub.pll_grid(ω), such that the phase margin of the system always satisfies the stability condition. The waveform for the output current of the grid-connected converter system is as shown in FIG. 5. According to the waveform for the output current, the system always keeps stable in this condition. In FIG. 5, the horizontal coordinate represents time t in units of s, while the vertical coordinate represents the output current I.sub.g of the grid-connected converter in units of A. [0098] (3) When the PLL has a control bandwidth co pll =452.3rad/s the Bode diagram of the open-loop transfer function G.sub.IL(s) for the current control loop considering the influence of the PLL is as shown in FIG. 6. In the figure, upper and lower portions respectively represent an amplitude-frequency curve and a phase-frequency curve of the Bode diagram. The horizontal coordinate represents an angular frequency in units of rad/s. For the amplitude-frequency curve, the vertical coordinate represents amplitude in units of dB. For the phase-frequency curve, the vertical coordinate represents phase in units of deg. As can be seen from FIG. 6, ω.sub.180°=1,240rad/s, A.sub.current(ω.sub.180°)=0.29 dB, A.sub.pll_grid(ω.sub.180°)=2.8 dB, and the amplitude margin A.sub.M=A.sub.current(ω.sub.180°)−A.sub.pll_grid(ω.sub.180°)=0.29 dB−(2.8)dB=−2.51 dB<0. The amplitude does not satisfy the stability condition of the system, ω.sub.crossover1=381rad/s, and ω.sub.crossover2=4,290rad/s. The system has the phase margin P.sub.M1=Ψ.sub.pll_grid(ω.sub.crossover1)−Ψ.sub.current(ω.sub.crossover1)−180°=54.43°>0, and P.sub.M2=Ψ.sub.pll_grid(ω.sub.crossover2)−Ψ.sub.current(ω.sub.crossover2)−180°=−38.2°<0, so the phase does not satisfy the stability condition of the system either. ω.sub.crossover1 and ω.sub.crossover2 are respectively first and second angular frequencies when A.sub.pll_grid(ω) and A.sub.current(ω) are the same twice. The waveform for the output current of the grid-connected converter system is as shown in FIG. 7. According to the waveform for the output current, the system is unstable in this condition. In FIG. 7, the horizontal coordinate represents time t in units of s, while the vertical coordinate represents the output current I.sub.g of the grid-connected converter in units of A.

[0099] Second case: The grid-connected converter system has a short-circuit ratio SCR=1.1, and the grid impedance has an inductive component L.sub.grid=13.0 mH and a resistive component R.sub.grid=0.436 Ω. When the current loop has a control bandwidth ω.sub.CL4,750rad/s, the control bandwidth of the PLL can be calculated according to the controller design method of the grid-connected converter under the weak grid, and the stability and stability margin of the system can be determined according to the stability criterion for the control loop. Hereinafter, the present disclosure provides descriptions of three cases where the PLL has different control bandwidths: [0100] (1) When the PLL has a control bandwidth ω.sub.pll=32.3rad/s, the Bode diagram of the open-loop transfer function G.sub.IL(s) for the current control loop considering the influence of the PLL is as shown in FIG. 8. In the figure, upper and lower portions respectively represent an amplitude-frequency curve and a phase-frequency curve of the Bode diagram. The horizontal coordinate represents angular frequency in units of rad/s. For the amplitude-frequency curve, the vertical coordinate represents an amplitude, in a unit of dB. For the phase-frequency curve, the vertical coordinate represents phase in units of deg. As can be seen from FIG. 8, ω.sub.180°=52rad/s, A.sub.current(ω.sub.180°)=0.001 dB, A.sub.pll_grid(ω.sub.180°)=−16.1 dB, and the amplitude margin A.sub.M=A.sub.current(ω.sub.180°)=0.001 dB−(−16.1)dB=16.101 dB>0. The amplitude satisfies the stability condition of the system. In the full band range, A.sub.current(ω) is always greater than A.sub.pll_grid(ω), such that the phase satisfies the stability condition of the system. The waveform for the output current of the grid-connected converter system is as shown in FIG. 9. According to the waveform for the output current, the system always keeps stable in this condition. In FIG. 9, the horizontal coordinate represents time t in units of s, while the vertical coordinate represents the output current I.sub.g of the grid-connected converter in units of A. [0101] (2) When the PLL has a control bandwidth ω.sub.pll=193.7rad/s, the Bode diagram of the open-loop transfer function G.sub.IL(s) for the current control loop considering the influence of the PLL is as shown in FIG. 10. In the figure, upper and lower portions respectively represent an amplitude-frequency curve and a phase-frequency curve of the Bode diagram. The horizontal coordinate represents angular frequency in units of rad/s. For the amplitude-frequency curve, the vertical coordinate represents an amplitude in units of dB. For the phase-frequency curve, the vertical coordinate represents phase in units of deg. As can be seen from FIG. 10, ω.sub.180°=743rad/s, A.sub.current(ω.sub.180°)=0.106 dB, A.sub.pll_grid(ω.sub.180°)=−1.94 dB and the amplitude margin A.sub.M=A.sub.current(ω.sub.180°)−A.sub.pll_grid(ω.sub.180°)=0.106 dB−(−1.94)dB=2.046 dB>0. The amplitude margin is relatively small but still satisfies the stability condition of the system. In the full band range, A.sub.current(ω) is always greater than A.sub.pll_grid(ω), such that the phase satisfies the stability condition of the system. The waveform for the output current of the grid-connected converter system is as shown in FIG. 11. According to the waveform for the output current, the system always keeps stable in this condition. In FIG. 11, the horizontal coordinate represents time t in units of s, while the vertical coordinate represents the output current I.sub.g of the grid-connected converter in units of A. [0102] (3) When the PLL has a control bandwidth ω.sub.pll=387.4rad/s, the Bode diagram of the open-loop transfer function G.sub.IL(s) for the current control loop considering the influence of the PLL is as shown in FIG. 12. In the figure, upper and lower portions respectively represent an amplitude-frequency curve and a phase-frequency curve of the Bode diagram. The horizontal coordinate represents angular frequency in units of rad/s. For the amplitude-frequency curve, the vertical coordinate represents an amplitude in units of dB. For the phase-frequency curve, the vertical coordinate represents phase in units of deg. As can be seen from FIG. 12, ω.sub.180°=1,130rad/s, A.sub.current(ω.sub.180°)=0.24 dB, A.sub.pll_grid(ω.sub.180°)=4.14 dB, and the amplitude margin A.sub.M=A.sub.current(ω.sub.180°)−A.sub.pll_grid(ω.sub.180°)=0.24 dB−4.14 dB=−3.9 dB<0. The amplitude does not satisfy the stability condition of the system, ω.sub.crossover1=275rad/s, and ω.sub.crossover2=5190rad/s. The system has the phase margin P.sub.M1=Ψ.sub.pll_grid(ω.sub.crossover1)−Ψ.sub.current(ω.sub.crossover1)−180°=41.68°>0, and P.sub.M2=Ψ.sub.pll_grid(ω.sub.crossover2)−Ψ.sub.current(ω.sub.crossover2)−180°=−49.8°<0, so the phase does not satisfy the stability condition of the system either. The waveform for the output current of the grid-connected converter system is as shown in FIG. 13. According to the waveform for the output current, the system is unstable in this condition. In FIG. 13, the horizontal coordinate represents time t in units of s, while the vertical coordinate represents the output current I.sub.g of the grid-connected converter in units of A.