A method for LTP modeling and stability analysis of DSOGI-PLL

Abstract

The present invention relates to the technical field of stability assessment of new energy grid-connected synchronous systems, and discloses a method for LTP modeling and stability analysis of a DSOGI-PLL. The present invention first derives the control differential equation of the DSOGI-PLL in the time domain, and uses the method of small signal linearization to retain the multi-frequency coupling relationship terms and establish an accurate LTP model; further, the established LTP model is represented as an infinite-dimensional LTI model, and the open-loop harmonic transfer function of the three-phase DSOGI-PLL is derived; based on the open-loop harmonic transfer function, the multivariable LTI feedback control system is analyzed using the generalized Nyquist stability criterion, and the boundary of the small signal stability region is obtained by analysis. The present invention accurately describes the periodic time-varying model and small signal stability of the DSOGI-PLL under unbalanced power grid conditions, and ensures the stable operation of the DSOGI-PLL by accurately designing the control parameters.

Description

Technical Field

[0001] The present invention relates to the technical field of stability assessment of a new energy grid-connected synchronous system, and in particular to a method for LTP modeling and stability analysis of a DSOGI-PLL. Background Art

[0002] The three-phase phase-locked loop (DSOGI-PLL) based on the double second order generalized integrator has been applied to the grid-connected part of the actual power system to improve the stability and reliability of the transmission of new energy generators to the power grid. However, there are still unknown possibilities in the modeling and analysis of DSOGI-PLL. The difficulty lies in the fact that DSOGI-PLL is a nonlinear feedback control system. To solve this problem, several linearization techniques have been proposed in recent years to derive the linear model of PLL (Phase-Locked Loop). Most of the existing linearization techniques are based on the linear time-invariant (LTI) system theory. In the modeling process, in order to simplify the modeling of DSOGI-PLL, the time-period dynamic variables of PLL are replaced with nominal values. The DSOGI-PLL applied to the three-phase AC system involves time period quantities in steady state, so it should be linearized into a linear time period (LTP) model. However, the linear model derived under the LTI framework ignores the periodic time-varying nature of DSOGI-PLL. With the increase of nonlinear loads and distributed generation systems in the power grid, the power grid is subjected to more and more frequent shocks, transient faults are increasing, and the three-phase power grid voltage is prone to imbalance and distortion. The stable working state of the phase-locked loop is also facing great challenges. However, the time periodic nature of the DSOGI-PLL has not been modeled in the three-phase AC system, and the stability of the DSOGI-PLL has not been analyzed in detail. In order for the phase-locked loop to maintain stable operation under unbalanced conditions, it is particularly important to accurately model it and obtain the accurate range of the stability domain. Summary of the invention

[0003] In view of the above problems, the purpose of the present invention is to provide a method for LTP modeling and stability analysis of DSOGI-PLL, so as to accurately describe the periodic time-varying model and small signal stability of DSOGI-PLL under unbalanced power grid conditions, and to ensure the stable operation of DSOGI-PLL by accurately designing control parameters. The technical solution is as follows:

[0004] A method for LTP modeling and stability analysis of a DSOGI-PLL comprises the following steps:

[0005] Step 1: According to the three-phase grid voltage signal at the common coupling point in the three-phase AC system, the α-axis and β-axis voltage components in the two-phase stationary coordinate system and their respective orthogonal signals are obtained; combined with the three-phase DSOGI-PLL topology structure, the control differential equation in the time domain is derived, and the multi-frequency coupling relationship terms are retained by using the small signal linearization method to establish an accurate LTP model;

[0006] Step 2: Represent the established LTP model as an infinite-dimensional LTI model and derive the open-loop harmonic transfer function of the three-phase DSOGI-PLL;

[0007] Step 3: Based on the open-loop harmonic transfer function established in step S2, the multivariable LTI feedback control system is analyzed using the generalized Nyquist stability criterion, and the small signal stability region boundary is obtained by analysis.

[0008] Furthermore, the α-axis and β-axis voltage components in the two-phase stationary coordinate system and their respective orthogonal signals obtained in step 1 are specifically:

[0009] The three-phase grid voltage signal v at the common coupling point in the three-phase AC system a 、v b 、v c After Clark transformation, the α and β axis components v of the grid voltage signal in the two-phase stationary coordinate system are obtained. α 、v β , and are processed by a second-order generalized integrator to obtain the α-axis component v of the grid voltage signal α The orthogonal signal and and the β-axis component v β The orthogonal signal and The relevant equation is as follows:

[0010]

[0011] Among them, v x (t) and v β (t) is v x The time domain representation of V x and θ x They are v x The rated amplitude and rated phase of (t) are θ x =∫ωdt, ω is the system angular frequency; To estimate the amplitude, is the estimated phase; the subscript x represents the α-axis or the β-axis.

[0012] Furthermore, in step 1, the control differential equation in the time domain is derived as follows:

[0013] The α and β axis components v of the grid voltage signal α 、v β The orthogonal signals are synthesized to obtain the positive sequence component and The voltage in the two-phase rotating coordinate system is obtained through Park transformation, which is used as the input signal of the positive sequence phase-locked loop. After passing through the phase-locked loop, a phase synchronization signal is generated for power grid synchronization. The relevant equations are as follows:

[0014]

[0015] Among them, k p , k i are the proportional coefficient and integral coefficient of the positive sequence phase-locked loop proportional integral controller, respectively, and k is the SOGI damping coefficient; Estimate the angular frequency for the grid, is the estimated phase of the power grid, ω n is the rated frequency of the power grid; when DSOGI-PLL operates in quasi-static mode, The subscript x represents the α-axis or the β-axis.

[0016] Furthermore, in step 1, establishing an accurate LTP model is specifically as follows:

[0017] The actual variables and estimated variables with small disturbances are defined, and the small signal model of the DSOGI-PLL is obtained by linearizing the small signal of equation (2), and then an LTP model for the DSOGI-PLL is obtained;

[0018] The small signal model of DSOGI-PLL is as follows:

[0019]

[0020] Where Δ and subscript n represent small disturbance and rated value respectively; θ n is the rated phase, ΔV e(x) is the amplitude disturbance, Δθ e(x) is the phase disturbance; is the grid angular frequency disturbance, is the SOGI output phase disturbance under the α axis, is the phase disturbance of SOGI output in β axis; and

[0021] Furthermore, in step S2, the open-loop harmonic transfer function of the three-phase DSOGI-PLL is derived, which is specifically:

[0022] According to the obtained small signal model of DSOGI-PLL, the output signal and After Laplace transformation; redefine ωp =2ω n ,ω p is twice the rated frequency; and according to the Euler formula, the open-loop harmonic transfer function of the three-phase DSOGI-PLL is obtained:

[0023]

[0024] in, is the amplitude disturbance estimate Frequency domain representation of ; is the phase disturbance estimate The frequency domain representation of ; λ is the forward channel gain, and represents the Laplace transform; G PI (s) represents the PI controller transfer function, and

[0025] Furthermore, by m Substituting s and describing the proposed LTP model with a matrix, we obtain the open-loop HTFG(s) as follows, thereby expressing the LTP dynamics of the DSOGI-PLL with the LTI model;

[0026] Y=KG(s)E (5)

[0027] Among them, Y is the output, K is the forward path gain, G(s) is the system open-loop harmonic transfer function, and E is the input error; the specific expression is as follows:

[0028]

[0029]

[0030]

[0031]

[0032] in, M(s)=(0.5G PI (s)+1)I(s),N(s)=0.5G PI (s)I(s); s0, s -1 and +1 They represent the fundamental frequency component, negative sequence frequency component and positive sequence frequency component respectively.

[0033] Furthermore, the step 3 is specifically as follows: determining the stable region of each control parameter of DSOGI-PLL under different grid conditions to ensure that DSOGI-PLL maintains grid synchronization and stability; performing stability analysis on the multiple input multiple output (MIMO) system represented in equation (5) by the generalized Nyquist criterion. For the feedback control system composed of the LTP model, assuming Indicates that it belongs to -jω p / 2≤Im(s)≤jω p The eigenvalue of HTFG(s) in the interval λ / 2 i (s) is the eigenvalue of G(s), Im(s) is the imaginary axis of the Nyquist diagram of the system; N represents the number of poles on the right half of the open-loop harmonic transfer function. The closed-loop LTP system is asymptotically stable if and only if the eigenvalue of the open-loop harmonic transfer function is exactly N times around the point -1 / K (K in Formula (5)) in the counterclockwise direction;

[0034] The beneficial effects of the present invention are:

[0035] 1) The LTP model proposed in the present invention can accurately describe the periodic time-varying properties of the three-phase DSOGI-PLL.

[0036] 2) The present invention adopts the LTP method and proposes an accurate parameter design method.

[0037] 3) The present invention utilizes the proposed LTP model to effectively predict the stable region of DSOGI-PLL. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 A three-phase DSOGI-PLL control structure diagram provided in an embodiment of the present application.

[0039] Figure 2 An LTP model based on DSOGI-PLL is provided in an embodiment of the present application.

[0040] Figure 3 Shown are the system responses of the physical model and the proposed LTP model under amplitude disturbance.

[0041] Figure 4 Shown are the system responses of the physical model and the proposed LTP model under phase perturbation.

[0042] Figure 5 k p =80, k i Nyquist plot of HTFG(s) when =12000.

[0043] Figure 6It is the working waveform of DSOGI-PLL under relevant parameters.

[0044] Figure 7 k = 2, k i Nyquist plot of HTFG(s) when =12000.

[0045] Figure 8 It is the working waveform of DSOGI-PLL under relevant parameters. DETAILED DESCRIPTION

[0046] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0047] A first aspect of the present invention provides a method for LTP modeling and stability analysis of a three-phase phase-locked loop based on a dual second-order generalized integrator, the method comprising the following steps:

[0048] Step S1, combining the three-phase DSOGI-PLL control structure, deriving the control differential equation in the time domain, using the small signal linearization method, retaining the multi-frequency coupling relationship terms, and establishing an accurate LTP model.

[0049] Figure 1 The three-phase DSOGI-PLL control structure diagram is given, including α-axis SOGI, β-axis SOGI and positive sequence phase-locked loop. Among them, the three-phase grid voltage signal v at the common coupling point (PCC) a 、v b 、v c The α and β axis components v in the two-phase stationary coordinate system after Clark transformation α 、v β , respectively, are processed by a second-order generalized integrator (SOGI) to obtain their respective orthogonal signals and The relevant equation is as follows:

[0050]

[0051] Among them, v x (t) and v β (t) is v x The time domain representation of V x and θ x They are v x The rated amplitude and rated phase of (t) are θ x =∫ωdt, ω is the system angular frequency; To estimate the amplitude, is the estimated phase; the subscript x is α or β.

[0052] In step S1, the respective orthogonal signals are synthesized to obtain the positive sequence component, and the voltage in the two-phase rotating coordinate system is obtained through Park transformation, which is used as the input signal of the positive sequence phase-locked loop, and then the phase synchronization signal is generated after passing through the phase-locked loop for grid synchronization. They are v α 、v β The positive sequence component of k p , k i are the proportional and integral coefficients of the positive sequence phase-locked loop proportional integral (PI) controller, k is the SOGI damping coefficient, and the relevant equations are as follows:

[0053]

[0054] Among them, when the DSOGI-PLL operates in quasi-static mode,

[0055] In an optional embodiment, the actual variables and estimated variables with small disturbances are defined as follows:

[0056]

[0057] The small signal model of the DSOGI-PLL is obtained by linearizing the small signal of equation (2):

[0058]

[0059] Where Δ and subscript n represent small disturbance and rated value respectively; θ n is the rated phase, ΔV e(x) is the amplitude disturbance, Δθ e(x) is the phase disturbance; is the grid angular frequency disturbance, is the α-axis SOGI output phase disturbance, is the β-axis SOGI output phase disturbance; and

[0060] The double frequency and cross coupling in the above model reflect the periodic time-varying characteristics of the model. According to the linearized equations, it can be divided into amplitude channel, phase channel and PLL channel, as shown in Figure 2 As shown, an LTP model based on DSOGI-PLL can be obtained.

[0061] Step S2: Based on the harmonic transfer function theory, the established LTP model is expressed as an infinite-dimensional LTI model, and the open-loop harmonic transfer function of the three-phase DSOGI-PLL is derived.

[0062] A single frequency signal as the input of the LTP system will generate multiple frequency components in the output. Therefore, there is coupling between the input and output of the LTP system at different frequencies, and this frequency coupling complicates the transfer function. Next, the open-loop HTF of the LTP model of the proposed DSOGI-PLL is derived using the harmonic transfer function (HTF) theory.

[0063] In an optional embodiment, deriving an open-loop HTF of a three-phase DSOGI-PLL includes:

[0064] The small signal model of the DSOGI-PLL is obtained by synthesis, and the output signal and After Laplace transformation:

[0065]

[0066] in, represents the Laplace transform, M(s)=(0.5G PI (s)+1)I(s),N(s)=0.5G PI (s)I(s).

[0067] In an optional embodiment, define ω p =2ω n , and according to Euler's formula, equation (4) can be expressed as:

[0068]

[0069] According to (5), s m Substituting s, we can further derive (6), and describe the proposed LTP model with a matrix, and obtain the open-loop HTFG(s) as shown in (7), so that the LTP dynamics of DSOGI-PLL can be represented by the LTI model. Multivariable LTI feedback control theory can be applied for stability analysis.

[0070]

[0071]

[0072]

[0073]

[0074] Y=KG(s)E (7)

[0075]

[0076]

[0077]

[0078]

[0079] Among them, s0, s -1 and +1 They represent the fundamental frequency component, negative sequence frequency component and positive sequence frequency component respectively.

[0080] In an optional embodiment, the physical model is analyzed and compared with the simulation verification of the proposed LTP model.

[0081] The DSOGI-PLL control parameters are as follows: K = √2, k p =120, k i =4000,ω n =100*pi / s. The DSOGI-PLL physical model and the proposed LTP model are subjected to amplitude perturbation and phase perturbation respectively, and the waveforms of estimated amplitude, estimated phase difference and estimated frequency are plotted. Figure 3 is the system response of the physical model and the proposed LTP model under amplitude disturbance, Figure 4 The system responses of the physical model and the proposed LTP model under phase disturbance are shown in Figure 2. The proposed LTP model is in good agreement with the simulation results, which verify that the proposed LTP model can accurately describe the dynamic changes of amplitude, phase and frequency under unbalanced grid conditions, as well as the coupling relationship between these variables. The proposed LTP model can accurately describe the periodic time-varying properties of the three-phase DSOGI-PLL under different grid conditions. The proposed LTP model can be used to accurately design system parameters.

[0082] Step S3: Based on the open-loop harmonic transfer function established in step S2, the multivariable LTI feedback control system is analyzed using the generalized Nyquist stability criterion, and the DSOGI-PLL small signal stability region boundary is obtained through analysis.

[0083] Determine the stable region of each control parameter of DSOGI-PLL under different grid conditions to ensure that DSOGI-PLL maintains grid synchronization and stability. The generalized Nyquist criterion is an effective method for stability analysis of the multiple input multiple output (MIMO) system represented in (7). The present invention introduces a Nyquist criterion for calculating the stable region of a closed-loop LTP system. For the feedback control system composed of the proposed LTP model, assume Indicates that it belongs to -jω p / 2≤Im(s)≤jω p / 2 interval. N represents the number of poles on the right half of the open-loop HTF. The closed-loop LTP system is asymptotically stable if and only if the eigenvalue of the open-loop HTF is exactly N times around the -1 / K point (K in Eq. (5) is the forward path gain) in the counterclockwise direction.

[0084] In an optional embodiment, the generalized Nyquist stability criterion is used to obtain the Nyquist diagram under the DSOGI-PLL control parameters. Figure 5 As shown in , when the intersection is located on the left side of the -1 / K point, the image surrounded by the eigenvalues ​​of HTFG(s) surrounds the -1 / K point. According to the generalized Nyquist criterion, the closed-loop LTP system is asymptotically stable. Figure 7 As shown, when the intersection point is located on the right side of the -1 / K point, the image enclosed by the eigenvalues ​​of HTFG(s) does not surround the -1 / K point. According to the generalized Nyquist criterion, the closed-loop LTP system is unstable.

[0085] The control parameters of the embodiment are as follows: p =80, k i = 12000. Draw the Nyquist diagram under this control parameter, as shown in Figure 5 As shown. According to the generalized Nyquist theory, K = λ = kω n / 2, we can get the stable region of gain k, which can be derived as:

[0086] k>1.04 (8)

[0087] like Figure 6 As shown in Figure 2, when the damping coefficient k of the embodiment is set to the critical value of the stable region, the DSOGI-PLL first remains stable and becomes unstable when the control parameter exceeds the stable boundary. These results are consistent with the stable region predicted by the proposed LTP model.

[0088] The control parameters of another embodiment are as follows: k = 2, k i = 12000. Draw the Nyquist diagram under this control parameter, as shown in Figure 7 As shown. According to the generalized Nyquist theory, K = λ = kω n / 2, we can get the gain k p The stable region can be derived as:

[0089] k p >42.3 (9)

[0090] like Figure 8 As shown, the proportionality coefficient k of the embodiment is p Set to the critical value of the stable region, the DSOGI-PLL remains stable at first and becomes unstable when the control parameter exceeds the stable boundary. These results are consistent with the stable region predicted by the proposed LTP model.

[0091] By observing the Nyquist diagram of HTFG(s), the operating trend of the DSOGI-PLL under this control parameter can be intuitively predicted, thereby determining the range of another control parameter when the system is running stably.

Claims

1. A method for LTP modeling and stability analysis of DSOGI-PLL, characterized in that: The steps include: Step 1: According to the three-phase grid voltage signal at the common coupling point in the three-phase AC system, the α-axis and β-axis voltage components in the two-phase stationary coordinate system and their respective orthogonal signals are obtained; combined with the three-phase DSOGI-PLL control structure, the control differential equation in the time domain is derived, and the multi-frequency coupling relationship terms are retained by using the small signal linearization method to establish an accurate LTP model; Step 2: Represent the established LTP model as an infinite-dimensional LTI model and derive the open-loop harmonic transfer function of the three-phase DSOGI-PLL; Step 3: Based on the open-loop harmonic transfer function established in step S2, the multivariable LTI feedback control system is analyzed using the generalized Nyquist stability criterion, and the small signal stability region boundary is obtained by analysis.

2. The LTP modeling and stability analysis method of DSOGI-PLL according to claim 1, characterized in that: The α-axis and β-axis voltage components and their respective orthogonal signals obtained in step 1 in the two-phase stationary coordinate system are specifically: The three-phase grid voltage signal v at the common coupling point in the three-phase AC system a 、v b 、v c After Clark transformation, the α and β axis components v of the grid voltage signal in the two-phase stationary coordinate system are obtained. α 、v β , and are processed by a second-order generalized integrator to obtain the α-axis component v of the grid voltage signal α The orthogonal signal and and the β-axis component v β The orthogonal signal and The relevant equation is as follows: Among them, v x (t) and v β (t) is v x The time domain representation of V x and θ x They are v x The rated amplitude and rated phase of (t) are θ x =∫ωdt, ω is the system angular frequency; To estimate the amplitude, is the estimated phase; the subscript x represents the α-axis or the β-axis.

3. The LTP modeling and stability analysis method of DSOGI-PLL according to claim 2, characterized in that: In step 1, the control differential equation in the time domain is derived as follows: The α and β axis components v of the grid voltage signal α 、v β The orthogonal signals are synthesized to obtain the positive sequence component and The voltage in the two-phase rotating coordinate system is obtained through Park transformation, which is used as the input signal of the positive sequence phase-locked loop. After passing through the phase-locked loop, a phase synchronization signal is generated for power grid synchronization. The relevant equations are as follows: Among them, k p , k i are the proportional coefficient and integral coefficient of the positive sequence phase-locked loop proportional integral controller, respectively, and k is the SOGI damping coefficient; Estimate the angular frequency for the grid, is the estimated phase of the power grid, ω n is the rated frequency of the power grid; when DSOGI-PLL operates in quasi-static mode, The subscript x represents the α-axis or the β-axis.

4. The LTP modeling and stability analysis method of DSOGI-PLL according to claim 3, characterized in that: In step 1, establishing an accurate LTP model is specifically as follows: The actual variables and estimated variables with small disturbances are defined, and the small signal model of the DSOGI-PLL is obtained by linearizing the small signal of equation (2), and then an LTP model for the DSOGI-PLL is obtained; The small signal model of DSOGI-PLL is as follows: Where Δ and subscript n represent small disturbance and rated value respectively; θ n is the rated phase, ΔV e(x) is the amplitude disturbance, Δθ e(x) is the phase disturbance; is the grid angular frequency disturbance, is the SOGI output phase disturbance under the α axis, is the phase disturbance of SOGI output in β axis; and 5. The LTP modeling and stability analysis method of DSOGI-PLL according to claim 4, characterized in that: In step S2, the open-loop harmonic transfer function of the three-phase DSOGI-PLL is derived, which is specifically: According to the obtained small signal model of DSOGI-PLL, the output signal and After Laplace transformation; redefine ω p is twice the rated frequency; and according to the Euler formula, the open-loop harmonic transfer function of the three-phase DSOGI-PLL is obtained: in, is the amplitude disturbance estimate Frequency domain representation of ; is the phase disturbance estimate The frequency domain representation of ; λ is the forward channel gain, and represents the Laplace transform; G PI (s) represents the PI controller transfer function, and 6. The LTP modeling and stability analysis method of DSOGI-PLL according to claim 5, characterized in that: By placing s m Substituting s and describing the proposed LTP model with a matrix, we obtain the open-loop HTFG(s) as follows, thereby expressing the LTP dynamics of the DSOGI-PLL with the LTI model; Y=KG(s)E (5) Among them, Y is the output, K is the forward path gain, G(s) is the system open-loop harmonic transfer function, and E is the input error; the specific expression is as follows: in, M(s)=(0.5G PI (s)+1)I(s),N(s)=0.5G PI (s)I(s); s0, s -1 and +1 They represent the fundamental frequency component, negative sequence frequency component and positive sequence frequency component respectively.

7. The LTP modeling and stability analysis method of DSOGI-PLL according to claim 6, characterized in that: The step 3 is specifically as follows: determining the stable region of each control parameter of DSOGI-PLL under different grid conditions to ensure that DSOGI-PLL maintains grid synchronization and stability; performing stability analysis on the multi-input multi-output system represented by equation (5) by using the generalized Nyquist criterion. For the feedback control system composed of the LTP model, assuming Indicates that it belongs to -jω p / 2≤Im(s)≤jω p The eigenvalue of HTFG(s) in the interval λ / 2 i (s) is the eigenvalue of G(s), Im(s) is the imaginary axis of the system Nyquist diagram; N represents the number of poles on the right half of the open-loop harmonic transfer function. The closed-loop LTP system is asymptotically stable if and only if the eigenvalue of the open-loop harmonic transfer function surrounds the -1 / K point exactly N times in the counterclockwise direction.

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