Comprehensive analysis system and method for frequency performance evaluation of network construction type converter system

Abstract

The invention discloses a comprehensive analysis system and method for frequency performance evaluation of a network construction type converter system, and belongs to the technical field of virtual synchronous generator control, the system comprises a forward channel and a feedback channel, the forward channel comprises two parts, one part is a transfer function composed of a virtual inertia link and two single integration links; the second transfer function is an integral transfer function formed by a reactive power coupling part, angle-to-reactive power, voltage-to-active power and angle-to-active power; and the feedback channel is active power unit negative feedback. The method comprises the steps of firstly judging whether an active power loop open-loop transfer function is stable; after the stability is determined, the sensitivity can be evaluated by comparing the peak values of the sensitivity functions under different conditions; meanwhile, the robustness is evaluated by judging whether the sum of the sensitivity function peak value and the complementary sensitivity function peak value is smaller than 2 or not. According to the method, the sensitivity, stability and robustness of the system can be comprehensively considered, and the frequency response characteristics of the system can be evaluated from multiple dimensions.

Description

Technical Field

[0001] The present invention relates to the technical field of virtual synchronous generator control, and specifically provides a comprehensive analysis system and method for evaluating the frequency performance of a network-forming converter system. Background Art

[0002] In the control strategy of network-forming converters, virtual synchronous generator (VSG) technology is widely used. By introducing virtual inertia, damping, and synchronous control, it simulates the dynamic characteristics of traditional synchronous generators, thereby achieving grid frequency regulation and stability guarantee. VSG technology has been proven to improve the robustness and stability of the power grid, especially in the context of high penetration of renewable energy, it can effectively alleviate grid frequency fluctuations. However, currently, there are certain technical limitations in the evaluation of the frequency performance of network-forming converter systems, especially in terms of the accuracy of evaluating system robustness and dynamic performance.

[0003] Existing frequency performance evaluation methods mainly rely on sensitivity analysis and stability analysis to quantitatively analyze the static or dynamic characteristics of system responses. Although these methods can evaluate the system's response to disturbances to a certain extent, they often cannot comprehensively and accurately reflect the robustness and stability of the system under complex operating conditions. In particular, in the face of external disturbances, system parameter changes, or internal and external uncertainties, traditional evaluation methods cannot provide sufficient robustness evaluation.

[0004] In addition, most of the existing analysis methods only focus on a single performance index, such as frequency response, steady-state error, etc., lacking the ability to comprehensively evaluate the system under different disturbances and parameter changes. Especially when system parameters (such as damping coefficient, inertia coefficient, etc.) change, the existing methods cannot effectively reflect the changes in system robustness and dynamic stability, resulting in potential instability or performance degradation problems in actual applications.

[0005] Therefore, there is an urgent need for a new evaluation method in the existing technology to comprehensively evaluate the frequency performance of network-forming converter systems under various disturbances and dynamic operating conditions; and this method should be able to effectively reveal the robustness boundaries and stability characteristics of the system. Summary of the Invention

[0006] The technical solution of the present invention aims at the technical problem of the overly single existing technical solution, and provides a solution significantly different from the existing technology. It mainly provides a comprehensive analysis system and method for evaluating the frequency performance of a network-forming converter system, which can comprehensively consider the sensitivity, stability, and robustness of the system, evaluate the frequency response characteristics of the system from multiple dimensions, and reveal the dynamic behavior of the system under different disturbance conditions, providing a scientific basis for the optimization of the system control strategy and parameter tuning.

[0007] The technical solution adopted by the present invention to solve the above technical problems is as follows:

[0008] A comprehensive analysis system for evaluating the frequency performance of a network-forming converter system, including a forward channel and a feedback channel. The forward channel includes two major parts. One is the transfer function composed of a virtual inertia link and two single-integral links; the other is the reactive power coupling part, which is the overall transfer function composed of angle to reactive power, voltage to active power, and angle to active power; the feedback channel is the unit negative feedback of active power.

[0009] Further, the frequency acceleration is obtained through the virtual inertia link; the frequency acceleration obtains the frequency offset through the first single-integral link; the frequency offset is superimposed on the grid angular frequency to form the total angular frequency of the system, and the angular frequency obtains the output phase angle of the virtual synchronous generator through the second single-integral link.

[0010] The present invention also provides a comprehensive analysis method for evaluating the frequency performance of a network-forming converter system. Based on the above comprehensive analysis system for evaluating the frequency performance of a network-forming converter system, the comprehensive analysis method includes the following steps:

[0011] Step 1: Establish a model: the open-loop transfer function G open (s) of the active power loop, the sensitivity function G sent (s), the peak value H s (x n ) of the sensitivity function, and the peak value H t (x n ) of the complementary sensitivity function;

[0012] Step 2: Determine whether G open (s) is stable; when G open (s) is stable, jump to Step 3.1; otherwise, jump to Step 3.2;

[0013] Step 3.1: Under conditions x1 and x2, compare H s (x1) and H s (x2). When H s (x1) is larger, it means that under condition x1, the response speed and sensitivity of the VSG system are faster, the adjustment time is shorter, and the system can reach the stable state faster; otherwise, it means that under condition x2, the response speed and sensitivity of the VSG system are faster, the adjustment time is shorter, and the system can reach the stable state faster; at the same time, compare the sum of the peak value of the sensitivity function and the peak value of the complementary sensitivity function with 2. When the sum of the peak value of the sensitivity function and the peak value of the complementary sensitivity function is less than 2, it means that the system has strong robustness; otherwise, the robustness is relatively weak;

[0014] Step 3.2: It indicates that the VSG system is in an unstable state, and the output variables may exhibit synchronous instability or divergent oscillation.

[0015] To endow the VSG system with a clearer physical meaning and further analyze the robustness of the system, the comprehensive analysis method further includes Step 4: Analyzing based on structured uncertainty modeling, which includes multiplicative uncertainty model and additive uncertainty model. Specifically, it includes:

[0016] Under the multiplicative uncertainty model, the robust stability condition of the system is expressed as:

[0017]

[0018] Under the additive uncertainty model, the robust stability condition of the system is expressed as:

[0019]

[0020] Wherein, and are the sensitivity function and complementary sensitivity function respectively, is a variable in the complex frequency domain, representing the imaginary axis in frequency response analysis, Wm(jω) is the uncertainty weight function of the multiplicative uncertainty model, Wa(jω) is the uncertainty weight function of the additive uncertainty model, and ω represents the angular frequency.

[0021] Judge whether the following inequality is satisfied:

[0022]

[0023] Wherein, G open (s) represents the open-loop transfer function of the active power loop, is a variable in the complex frequency domain, representing the imaginary axis in frequency response analysis; Wm(jω) is the uncertainty weight function of the multiplicative uncertainty model, Wa(jω) is the uncertainty weight function of the additive uncertainty model;

[0024] When the above inequality is satisfied, it indicates that the system can maintain stability and performance in the presence of uncertainties, that is, ensuring the robust performance of the system.

[0025] Furthermore, the robustness of the system is analyzed by plotting the Nyquist curve of the system open-loop transfer function, and the robust stability of the system is determined by checking whether the Nyquist curve encloses the point (-1, 0). When the Nyquist curve encloses (-1, 0) in the left half of the complex plane, the robust stability of the system is relatively weak; when the Nyquist curve passes through (-1, 0) from the right but does not enclose it, the robust stability of the system is relatively strong.

[0026] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0027] (1) By establishing a comprehensive analysis framework, the present invention proposes a new frequency performance evaluation system and method for grid-forming converters. This method can comprehensively consider the sensitivity, stability, and robustness of the system, evaluate the frequency response characteristics of the system from multiple dimensions, reveal the dynamic behavior of the system under different disturbance conditions, and provide theoretical support for improving the robustness and stability of the grid-forming converter system.

[0028] (2) The comprehensive analysis system and method provided by the present invention provide a new physical perspective for the frequency motion trajectory of virtual synchronous generators. By introducing the joint analysis of sensitivity functions and complementary sensitivity functions, the present invention provides a more accurate evaluation method for the stability and robustness of virtual synchronous generators in a variable environment. For example: in the embodiments of the present invention, the effects of damping coefficients and inertia coefficients on the frequency response of virtual synchronous generators are systematically studied, revealing that smaller damping coefficients and larger inertia coefficients may lead to oscillation-related instabilities under certain conditions.

[0029] (3) The present invention also provides a method for analyzing robustness based on structured uncertainty modeling, making the VSG system have a clearer physical meaning and enabling further analysis of the robustness of the system. And the robustness of the system is analyzed by plotting the Nyquist curve of the open-loop transfer function of the system. The Nyquist curve shows the changes of the open-loop gain and phase with frequency, thereby enabling an intuitive evaluation of the stability and performance boundaries of the system under the influence of uncertainties. Therefore, the present invention can comprehensively and accurately reflect the robustness and stability of the system under complex operating conditions in the face of external disturbances, system parameter changes, or internal and external uncertain factors.

[0030] The present invention will be explained and illustrated in detail below in combination with the drawings and specific embodiments. Brief Description of the Drawings

[0031] Figure 1 It is the topological structure of the hardware circuit of the VSG system in the embodiment; in the figure, Inverter represents the converter, EnergyStorage System represents the energy storage system, Transformation(abc-dq) represents the abc-dq transformation, PowerCalculation represents the power calculation; Grid represents the power grid, Generator represents the generator;

[0032] Figure 2 It is the schematic diagram of power coupling of the VSG system;

[0033] Figure 3 It is the frame diagram of the active power closed-loop control in the embodiment;

[0034] Figure 4 It is the analysis flow chart of the VSG response sensitivity and robustness in the embodiment; among them, Establish model means to establish a model; Stabie means stable;

[0035] Figure 5 It is the waveform diagram under different damping coefficients in the embodiment; among them, the abscissa is time (unit: second); the ordinate is frequency (unit: hertz);

[0036] Figure 6 It is the frequency-domain response curve of the sensitivity function under different damping coefficients in the embodiment; among them, the abscissa is frequency (unit: radian / second); the ordinate is amplitude (unit: decibel);

[0037] Figure 7 It is the characteristic root diagram of the dominant pole when the damping coefficient changes in the embodiment; among them, the abscissa is the real axis; the ordinate is the imaginary axis;

[0038] Figure 8 It is the waveform diagram under different inertia support coefficients in the embodiment; among them, the abscissa is time (unit: second); the ordinate is frequency (unit: hertz);

[0039] Figure 9 It is the frequency-domain response curve of the sensitivity function under different inertia support coefficients in the embodiment; among them, the abscissa is frequency (unit: radian / second); the ordinate is amplitude (unit: decibel);

[0040] Figure 10 It is the characteristic root diagram of the dominant pole when the inertia support coefficient changes in the embodiment; among them, the abscissa is the real axis; the ordinate is the imaginary axis;

[0041] Figure 11 It is the robust stability analysis diagram under different working conditions in the embodiment; among them, the abscissa is the real axis; the ordinate is the imaginary axis; Nyquist Diagram means Nyquist curve;

[0042] Figure 12 It is the complementary sensitivity and sensitivity function frequency-domain response curve under different working conditions in the embodiment; among them, the abscissa is frequency (unit: radian / second); the ordinate is amplitude (unit: decibel);

[0043] Figure 13 It is the frequency waveform of the experimental results of the VSG system under different damping coefficients; among them, Disturbance means interference;

[0044] Figure 14 It is the frequency waveform of the experimental results of the VSG system under different inertia support coefficients; among them, Disturbance means interference;

[0045] Figure 15 It is the frequency waveform of the experimental results of the VSG system under different working conditions; where, Disturbance represents interference. Specific implementation manners

[0046] To facilitate the understanding of the present invention, the present invention will be described more comprehensively below with reference to the relevant drawings. Several embodiments of the present invention are given in the drawings, but the present invention can be implemented in different forms and is not limited to the embodiments described in the text. On the contrary, these embodiments are provided to make the disclosure of the present invention more thorough and comprehensive.

[0047] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs. The terms used in the specification of the present invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more of the related listed items.

[0048] Determine the robustness and sensitivity of the frequency motion state of the grid-connected inverter under VSG control. Its control framework includes a forward channel and a feedback channel. The forward channel includes two major parts. One is the transfer function G1(s) composed of a virtual inertia link and two single-integral links, which is used to simulate the inertial response and frequency regulation characteristics of a synchronous generator. Through the virtual inertia link, the system can quickly respond to frequency changes, provide initial inertial support, and reduce the rate of frequency change. The two single-integral links are respectively used to integrate the rate of frequency change to obtain the frequency offset, and to integrate the angular frequency to obtain the output phase angle of the system, so as to achieve dynamic tracking and regulation of the frequency. The other is the reactive power coupling part, including the overall transfer functions from angle to reactive power, from voltage to active power, and from angle to active power. The feedback channel is a unit negative feedback of active power.

[0049] Further, the rate of change of the system output frequency, that is, the frequency acceleration, is obtained through the virtual inertia link , which represents the rate of change of the system frequency with time and is an important index of the system's dynamic response. Subsequently, the acceleration The speed of the system output frequency , that is, the frequency offset, is obtained through the first single-integral link, which reflects the difference between the system frequency and the reference frequency (usually the grid frequency); then, the frequency offset δω and the grid angular frequency ω gThey are superimposed to form the total angular frequency ω of the system. Finally, the angular frequency ω is input into the second single-integration link, and the output phase angle θ of the virtual synchronous generator is obtained through integral operation; the output phase angle θ is the cumulative result of the dynamic change of the system frequency, reflects the phase shift of the virtual synchronous generator relative to the power grid, and is an important parameter for evaluating the synchronization performance and stability of the system.

[0050] Specifically, taking the virtual synchronous generator (VSG system) as an example, as Figure 1 shown in the schematic diagram of the hardware circuit topology of the virtual synchronous generator, the following function model is obtained through the topology and control algorithm:

[0051]

[0052]

[0053] In equations (1)-(2), and respectively represent the dq-axis components of the output current of the VSG system; and respectively represent the dq-axis components of the output voltage of the VSG system; and are respectively the dq-axis components of the grid voltage; , , is the inductance value of the output filter of the VSG system, is the parasitic resistance of the output filter of the VSG system, is the grid-side inductor, is the grid-side resistance; Z is the impedance of the entire system; is the reference angular velocity; is the imaginary unit, and s is the Laplace operator.

[0054] The active and reactive power function expressions output by the VSG system are:

[0055]

[0056] Among them, P and Q are respectively the active power and reactive power injected by the VSG system into the power grid; and are respectively the dq-axis components of the output voltage at the PCC end.

[0057] The dq-axis component expressions of the output voltage of the VSG system are:

[0058]

[0059] In equation (4), E is the output voltage of the VSG system; The phase angle output by the active power closed-loop control framework of the VSG system; , They are the dq axis components of the grid voltage amplitude respectively; is the amplitude of the grid voltage.

[0060] Active power and reactive power are related to the voltage amplitude E of the VSG system output and the output phase angle of the active power closed-loop control framework The functional relationship between them is:

[0061]

[0062] in, is a small signal of active power; is a small signal of reactive power; , are small signals of dq axis components of inverter output current respectively; , They are small signals of the dq axis components of the inverter output voltage respectively.

[0063] The active and reactive power function expressions are obtained by linearizing equations (1) to (5) near the equilibrium operating point, as follows:

[0064]

[0065] The coupling relationship between active power and reactive power is as follows: Figure 2 Specifically, the difference between the active power reference and the actual value is used as input, and after the coupling part, the reactive power is output.

[0066] In formula (6), if Figure 2 , , They are the voltage amplitude E of the VSG system output and the output phase angle of the active power closed-loop control framework. The coefficient of , They are the voltage amplitude E of the VSG system output and the output phase angle of the active power closed-loop control framework. The coefficient of .

[0067] The active power closed-loop control equation of the VSG system is:

[0068]

[0069] The reactive power closed-loop control equation of the VSG system is:

[0070]

[0071] In the above formulas (7)-(8), P refrepresents the reference value of active power, Q ref represents the reference value of reactive power. P and Q respectively represent the active power and reactive power obtained through power calculation. and are respectively the virtual inertia constants of the active and reactive power loops. and are respectively the damping constants of the active and reactive power loops. θ and θ g respectively represent the angular frequencies of the virtual synchronous generator (VSG) and the power grid. E and U g respectively represent the amplitudes of the VSG output voltage and the power grid voltage.

[0072] By combining the above equations (1)-(8), the active power closed-loop control framework as shown in Figure 3 can be obtained. Figure 3 is the control closed-loop for active power.

[0073] Figure 3 The open-loop transfer function expression of the active power loop in

[0074]

[0075] From Figure 2 it can be seen that G1(s) and G2(s) are respectively the transfer functions based on the rotor dynamic equation and the amplitude-phase dynamic equation in formula (1). H 11 (s) and H 22 (s) are respectively the transfer functions from the power angle of the virtual synchronous generator (VSG) to the active power and from the voltage to the reactive power. H 12 (s) and H 21 (s) respectively represent the transfer functions from the VSG power angle to the reactive power and from the VSG output voltage to the active power.

[0076] Figure 2 The components G1(s), G2(s), H 11 (s), H 22 (s) in 12 (s) and H 21 (s) and the power coupling transfer functions H

[0077]

[0078] where represents the impedance transfer function.

[0079] Through Equation (10), an in-depth analysis of the sensitivity of the entire virtual synchronous generator (VSG) system can be carried out. Sensitivity is an important indicator to measure the degree of change in the system response, and it intuitively reflects the damping characteristics of the VSG system under different conditions. Specifically, the magnitude of sensitivity determines the speed at which the system restores balance in the face of disturbances or changes and the stability of the system. When the sensitivity is high, the VSG system can quickly respond to external disturbances and return to a stable state faster. On the contrary, when the sensitivity is low, the system response will be slower, which may lead to an extended oscillation time or system instability.

[0080] To further analyze the sensitivity of the model, it is analyzed through the sensitivity function and the complementary sensitivity function. Among them, the sensitivity function describes the response of the system output to input changes (external disturbances), that is, how the output is disturbed by the input signal; the complementary sensitivity function describes the response of the system output to the reference input and is a supplement to the sensitivity function. These two functions are complementary and together cover all the frequency responses of the system. Specifically as follows:

[0081] Define and as the sensitivity function and the complementary sensitivity function respectively, which can be expressed as:

[0082]

[0083] Define and as the amplitudes of the sensitivity function and the complementary sensitivity function in the VSG system response under the condition x n operating conditions, which can be expressed as:

[0084]

[0085] In the above equation (12) is the angular velocity corresponding to the oscillation frequency point of the VSG output variable under the condition x n conditions, and are the amplitudes of the sensitivity function and the complementary sensitivity function respectively.

[0086] In robust control, the infinity norms of the sensitivity function and the complementary sensitivity function are usually used to evaluate the system performance or robustness. A larger infinity norm indicates that the system is more sensitive to disturbances in certain frequency ranges. Combining Equations (9), (11), and (12), the peak value of the sensitivity function and the peak value of the complementary sensitivity function can be defined as:

[0087]

[0088] The peak value of the sensitivity function is a key indicator for evaluating the robustness of the system, and the peak value of the complementary sensitivity function It reflects the system's sensitivity to high-frequency signals. and The sum of should not be too large, otherwise the stability and performance of the system will be affected, thereby damaging the robustness of the system. The constraints are as follows:

[0089]

[0090] The judgment process of VSG response sensitivity and robustness is as follows: Figure 4 As shown in the figure, the sensitivity of the VSG output response can be measured by H s (x n ) can be quantitatively determined. The robustness of the VSG system can be determined by evaluating whether the sum of the amplitudes of the sensitivity function and the complementary sensitivity function at the oscillation frequency point is less than 2 (i.e., Equation 14).

[0091] The judgment process and criteria are summarized as follows:

[0092] Step 1: Initialization and linearization, building model G open (s), G sent (s), H s (x n ) and H t (x n ).

[0093] Step 2: Determine G open (s) is stable. If G open If (s) is stable, jump to step 3.1; otherwise, jump to step 3.2.

[0094] Step 3.1: Further determine which of the following conditions x1 and x2 If the corresponding H under condition x1 s (x1) is larger, it means that under condition x1, the response speed and sensitivity of the VSG system are faster, the adjustment time is shorter, and the system can reach a stable state faster. Similarly, if the corresponding H under condition x2 is s (x2) is larger, which means that under condition x2, the response speed and sensitivity of the VSG system are faster, the adjustment time is shorter, and the system can reach a stable state faster. At the same time, if the sum of the amplitudes of the sensitivity function and the complementary sensitivity function at the oscillation frequency point is less than 2, it means that the system has strong robustness; otherwise, the robustness is relatively weak.

[0095] Step 3.2: It indicates that the VSG system is in an unstable state, and the output variables may exhibit synchronous instability or divergent oscillations. In this state, the sensitivity of the VSG output response is not considered.

[0096] To verify the accuracy of the proposed model, the damping coefficient of the active power in the VSG control loop is adjusted to different degrees below (including four cases: D p = 1.0, D p = 0.9, D p = 0.8, and D p = 0.7), and the corresponding VSG output frequency waveform diagrams, sensitivity function frequency-domain response result diagrams, and characteristic root diagrams of the dominant poles are obtained through experiments or simulations, as Figures 5 - 7 shown. These diagrams reflect the changes in the system response under different damping coefficient conditions, thus providing an intuitive verification basis for the effectiveness of the model.

[0097] From Figure 5 it can be clearly observed that as the damping coefficient D p of the active power loop gradually decreases, the amplitude of the frequency fluctuation shows a gradually increasing trend. The fundamental reason for this phenomenon is that D p , as the feedback damping loop coefficient, although has a certain influence on the oscillation frequency of the VSG output, its actual change to the oscillation frequency is relatively weak. Therefore, the output oscillation frequency of the VSG basically remains within a stable range. This conclusion is strongly verified in Figure 5 , and under each D p value, the periodic changes of the output frequency are almost the same, showing no significant differences.

[0098] By analyzing the sensitivity function in formula (13), it can be found that it directly reflects the attenuation rate of the VSG output variables. Specifically, the larger the amplitude K sent (x n ) of the sensitivity function, the slower the attenuation process, and the less sensitive the system response to disturbances. As Figure 5 shown, as the D p value increases, the attenuation rate of the VSG output frequency gradually accelerates, and the stability of the system is also significantly improved. This indicates that during this process, the amplitude K sent (x n ) corresponding to the sensitivity function shows a gradually decreasing trend, further proving the close relationship between system stability and sensitivity.

[0099] From Figure 6It can be clearly observed that the oscillating angular frequencies are 35.9 rad / s, 36.0 rad / s, 36.1 rad / s, and 36.2 rad / s respectively, indicating that as D p increases, the oscillating angular frequency gradually increases.

[0100] From Figure 7 it can be clearly observed that as D p increases, the oscillating angular frequency gradually increases, and the dominant pole moves away from the imaginary axis, indicating that the stability of the system gradually increases. This phenomenon is also verified in Figure 6 the following.

[0101] Next, the inertia support coefficient of the active power in the VSG control loop is adjusted to different degrees (including four cases: J p = 0.05, J p = 0.08, J p = 0.1, and J p = 0.15), and the corresponding VSG output frequency waveform diagram, sensitivity function frequency-domain response result diagram, and characteristic root diagram of the dominant pole are obtained through experiments or simulations, as Figures 8 - 10 shown. These diagrams reflect the changes in the system response under different inertia support coefficient conditions, thus providing an intuitive verification basis for the effectiveness of the model.

[0102] From Figure 8 it can be clearly observed that as the active power inertia support coefficient J p gradually increases, the amplitude of the frequency fluctuation shows a gradually decreasing trend. The fundamental reason for this phenomenon is that J p is the inertia damping coefficient, which will greatly affect the oscillation frequency of the system. J p is in the opposite direction to the change of the oscillation frequency. When J p increases, the oscillation period of the frequency waveform increases, and the frequency change rate decreases, thereby reducing the amplitude of the frequency fluctuation.

[0103] From Figure 9 it can be clearly observed that the oscillating angular frequencies are 51.6 rad / s, 40.8 rad / s, 36.1 rad / s, and 29.5 rad / s respectively. When J p increases, the oscillating angular frequency decreases, verifying the phenomenon in Figure 8 the following.

[0104] From Figure 10 it can be clearly observed that as J p increases, the characteristic roots move towards the imaginary axis, indicating that the stability of the system gradually decreases.

[0105] To endow the VSG system with a clearer physical meaning and further analyze the robustness of the system, to ensure the robustness of the system, it is necessary to clearly define and control the possible range of parameter uncertainties. Structured uncertainty modeling and analysis can be considered, which includes multiplicative uncertainty and additive uncertainty.

[0106] Multiplicative uncertainty is more suitable for describing high-frequency model errors or frequency-dependent dynamic uncertainties, which can be defined as:

[0107]

[0108] where \(G_0(s)\) is the nominal transfer function, and \(\Delta(s)\) is the multiplicative uncertainty (usually an unknown but bounded function) used to describe the deviation between the actual system and the nominal model. \(s\) is a variable in the complex frequency domain representing the imaginary axis in frequency response analysis. \(W_m(j\omega)\) is the uncertainty weight function.

[0109] Based on this, the robust stability condition of the system can be expressed as:

[0110]

[0111] Then, the additive uncertainty in structured uncertainty can be used to analyze the nominal performance of the system. Additive uncertainty is more suitable for describing low-frequency model errors, which can be defined as:

[0112]

[0113] where \(G_0(s)\) and \(\Delta(s)\) are the same functions as in formula (15), and \(W_a(j\omega)\) is the uncertainty weight function.

[0114] Under the additive uncertainty model, the robust stability condition of the system can be expressed as:

[0115]

[0116] Based on this, the robustness of the system can be analyzed by plotting the Nyquist curve of the open-loop transfer function of the system. The Nyquist curve shows the variation of the open-loop gain and phase with frequency, thus enabling an intuitive evaluation of the stability and performance boundaries of the system under the influence of uncertainties.

[0117] When considering multiplicative uncertainty and additive uncertainty comprehensively, the magnitudes of the multiplicative uncertainty weight function \(W_m(j\omega)\) and the additive uncertainty weight function \(W_a(j\omega)\) can be set as constants as long as they satisfy the conditions of formulas (14) to (18). To simplify the model analysis, \(|W_a(j\omega)| = |W_m(j\omega)| = 0.5\) can be taken.

[0118] By combining equations (16) and (18), if the following inequality is satisfied, the system can maintain stability and performance in the presence of uncertainties, thus ensuring the robust performance of the system.

[0119]

[0120] The robust stability of the system can be determined by checking whether the Nyquist curve encloses the point (-1, 0). For example, when the Nyquist curve encloses (-1, 0) in the left half of the complex plane, the robust stability of the system is relatively weak. On the contrary, when the Nyquist curve passes through (-1, 0) from the right but does not enclose it, the robust stability of the system is stronger.

[0121] Next, for three different operating conditions, the robust stability analysis diagrams (as Figure 11 shown) and the frequency-domain response curves of the complementary sensitivity and sensitivity functions (as Figure 12 shown) are obtained.

[0122] Operating condition 1: D p = 1.0, J p = 0.1; Operating condition 2: D p = 1.0, J p = 0.05; Operating condition 3: D p = 8.0, J p = 0.1.

[0123] Table I. System parameters

[0124] Grid voltage (line voltage) 380 V (root mean square value) System nominal frequency 50 Hz Operating active power 10 kW L - filter inductance 4.4 mH L - filter resistance 0.4 Ω Grid inductance 6.6 mH Grid resistance 0.2 Ω <![CDATA[Virtual inertia coefficient (J q )]]> 0.5 Modulation signal frequency 50 Hz

[0125] Table I lists the specific parameters of the circuit and control part under the above conditions.

[0126] Figure 11 The Nyquist curves for three different damping coefficients and inertia coefficients are shown. The blue circles represent additive uncertainties, while the red circles represent multiplicative uncertainties. Therefore, the radius of the blue circle is ∣Wa (jω)∣ = 0.5, and the radius corresponding to the red circle is ∣G open (jω)*W m (jω)∣ = ∣G open (jω) / 2∣. The center of the blue circle is located at (-1, 0), while the center of the red circle is the point on the Nyquist curve at the oscillation frequency. According to the physical meaning of equation (19), when the red circle intersects with the blue circle, it indicates that the robustness of the system is weak. On the contrary, when the red circle does not intersect with the blue circle, it indicates that the robustness of the system is strong. It should be noted that when the red circle is contained by the blue circle, this also indicates that the robustness of the system is weak.

[0127] As Figure 11 shown in (a), when D p = 1.0, the Nyquist curve does not enclose the critical point (-1, 0), indicating that the system is in a stable state and has good robust stability. As D p increases to 8.0, as Figure 11 shown in (c), the Nyquist curve still avoids enclosing the critical point (-1, 0), indicating that the system maintains good robust stability. In addition, the blue circle and the red circle no longer intersect, further confirming the enhancement of the system's robustness. Another important observation is that as D p increases, the intersection point of the open-loop Nyquist curve and the X-axis gradually moves away from (-1, 0), thus reflecting the improvement of the system's stability. It should be noted that the red circle is completely contained within the blue circle, which is defined as a specific form of intersection. In this case, the system exhibits relatively weak robustness.

[0128] As Figure 12 shown in (a), the peaks of the Bode plots of the sensitivity function and the complementary sensitivity function are both greater than 1, being 16.1 dB and 15.9 dB respectively. By performing the inverse logarithm operation, it can be easily obtained that the gains are both greater than 1. This does not meet the robust stability condition in formula (16), i.e., H s + H t > 2, resulting in poor robustness of the system. In this case, the system's robustness performance is poor. However, as D p increases, the system stability gradually improves, and the peaks of the Bode plots of the sensitivity function and the complementary sensitivity function also decrease accordingly. As Figure 12 shown in (c), the peaks of the sensitivity function and the complementary sensitivity function decrease to close to 0 dB. By performing the inverse logarithm operation, it can be easily deduced that the gains are both less than 1, i.e., H s + H t < 2, thereby improving the robustness of the system. The good robustness in this case is also well verified by the phenomena presented in Figure 11 (a) and Figure 11 (c).

[0129] It should be noted that Figure 12 the oscillation angular frequencies in (a), Figure 12 (b) and Figure 12 (c) are 36.1 rad / s, 51.9 rad / s and 36.7 rad / s respectively, which correspond to the oscillation angular frequency points in Figure 11 (a), Figure 11 (b) and Figure 11 (c) respectively, further verifying the correctness of the model.

[0130] To further verify the correctness of the model, Figure 13 the experimental results of Cases 1-4 are shown, where, Case 1: D p = 0.7; Case 2: D p = 0.8; Case 3: D p = 0.9; Case 4: D p = 1.0. Cases 1 to 4 correspond to the gradual increase of the damping coefficient D p The experimental setup is based on the Figure 1 topology and control scheme shown.

[0131] The results show that in Cases 1, 2, 3, and 4, the frequency output response of the virtual synchronous generator (VSG) exhibits oscillations, and as D p increases, the oscillation amplitude gradually decreases, while the stable time of the damped oscillation increases. These experimental results are consistent with the simulation results, thus verifying the correctness of the model.

[0132] Similarly, Figure 14 the experimental results of Cases 5-8 are shown, where, Case 5: J p = 0.05; Case 6: J p = 0.08; Case 7: J p = 0.1; Case 8: J p = 0.15. Cases 5 to 8 correspond to the gradual increase of the inertia coefficient J p The experimental setup follows the Figure 1 topology and control scheme shown in.

[0133] The results show that in Cases 5, 6, 7, and 8, the frequency output response of the virtual synchronous generator (VSG) exhibits oscillations, and as J p increases, the system oscillation reaches the stable time gradually extends. These experimental results are consistent with the simulation results, thus verifying the accuracy of the model.

[0134] Figure 15 The experimental results of Cases 9-11 are shown, where, Case 9: D p = 1.0, J p = 0.1; Case 10: D p = 1.0, J p = 0.05; Case 11: D p = 8.0, J p = 0.1.

[0135] Cases 9 and 10 exhibit oscillations when disturbed, indicating poor system robustness. This phenomenon is shown in Figure 12 (a) and Figure 12Verification of (b). In contrast, Case 11 hardly generates oscillations when disturbed, indicating that the system has strong robustness in this case. This conclusion is consistent with Figure 11 (c) and Figure 12 the verification results of (c), further confirming the correctness of the proposed model.

[0136] In summary, by introducing the sensitivity function and the robustness analysis method, the sensitivity and robustness of the output response of the virtual synchronous generator (VSG) can be qualitatively studied, so as to accurately evaluate the stability and robustness of the VSG system under different working conditions. This provides a theoretical basis for further improving the synchronization performance between the VSG and the power grid.

[0137] The present invention is also applicable to the robustness determination of grid-connected inverters under any other control technology.

[0138] The above description of the present invention with reference to the accompanying drawings is exemplary. Obviously, the specific implementation of the present invention is not limited by the above methods. As long as such non-substantial improvements are made by adopting the method concept and technical solution of the present invention, or the concept and technical solution of the present invention are directly applied to other occasions without improvement, they are all within the protection scope of the present invention.

Claims

1. A comprehensive analysis system for evaluating the frequency performance of a network-forming converter system, characterized in that: It includes a forward channel and a feedback channel. The forward channel consists of two major parts. One is the transfer function composed of a virtual inertia link and two single-integral links; the other is the reactive power coupling part, which is the overall transfer function composed of angle to reactive power, voltage to active power, and angle to active power. The feedback channel is the unit negative feedback of active power.

2. The comprehensive analysis system for evaluating the frequency performance of the network-forming converter system according to claim 1, wherein: The frequency acceleration is obtained through the virtual inertia link; the frequency acceleration obtains the frequency offset through the first single-integral link; the frequency offset is superimposed on the grid angular frequency to form the total angular frequency of the system, and the angular frequency obtains the output phase angle of the virtual synchronous generator through the second single-integral link.

3. A comprehensive analysis method for evaluating the frequency performance of a network-forming converter system, based on the comprehensive analysis system for evaluating the frequency performance of the network-forming converter system according to any one of claims 1-2, characterized in that: The comprehensive analysis method includes the following steps: Step 1, establish a model: the open-loop transfer function G open (s), the sensitivity function G sent (s), the peak value H s (x n ) and the peak value H t (x n ); Step 2, determine whether G open (s) is stable; when G open (s) is stable, jump to Step 3.1; otherwise, jump to Step 3.2; Step 3.1: Compare H under condition x1 and condition x2 s (x1) and H s (x2). When H s (x1) is larger, it indicates that under condition x1, the response speed and sensitivity of the VSG system are faster, the adjustment time is shorter, and the system can reach the stable state faster; otherwise, it indicates that under condition x2, the response speed and sensitivity of the VSG system are faster, the adjustment time is shorter, and the system can reach the stable state faster. At the same time, compare the sum of the peak value of the sensitivity function and the peak value of the complementary sensitivity function with 2. When the sum of the peak value of the sensitivity function and the peak value of the complementary sensitivity function is less than 2, it indicates that the system has strong robustness; otherwise, the robustness is relatively weak. Step 3.2: It indicates that the VSG system is in an unstable state, and the output variables may exhibit synchronous instability or divergent oscillation.

4. The comprehensive analysis method for evaluating the frequency performance of the network-forming converter system according to claim 3, wherein: It also includes Step 4: Analyze based on structured uncertainty modeling, which includes multiplicative uncertainty model and additive uncertainty model.

5. The comprehensive analysis method for evaluating the frequency performance of the network-forming converter system according to claim 4, wherein: Under the multiplicative uncertainty model, the robust stability condition of the system is expressed as: Under the additive uncertainty model, the robust stability condition of the system is expressed as: Among them, and are the sensitivity function and complementary sensitivity function respectively, is a variable in the complex frequency domain, representing the imaginary axis in frequency response analysis, Wm(jω) is the uncertainty weight function of the multiplicative uncertainty model, Wa(jω) is the uncertainty weight function of the additive uncertainty model, and ω represents the angular frequency.

6. The comprehensive analysis method for evaluating the frequency performance of the network-forming converter system according to claim 4, wherein: Judge whether the following inequality is satisfied: Among them, G open (s) represents the open-loop transfer function of the active power loop, is a variable in the complex frequency domain, representing the imaginary axis in frequency response analysis; Wm(jω) is the uncertainty weight function of the multiplicative uncertainty model, and Wa(jω) is the uncertainty weight function of the additive uncertainty model; When this inequality is satisfied, it indicates that the system can maintain stability and performance in the presence of uncertainty, that is, to ensure the robust performance of the system.

7. The comprehensive analysis method for evaluating the frequency performance of the network-forming converter system according to claim 6, wherein: Analyze the robustness of the system by plotting the Nyquist curve of the system open-loop transfer function. The robust stability of the system is determined by checking whether the Nyquist curve encloses the point (-1, 0).

8. The comprehensive analysis method for evaluating the frequency performance of a network-forming converter system according to claim 7, characterized in that: When the Nyquist curve encloses (-1, 0) in the left half of the complex plane, the robust stability of the system is relatively weak; when the Nyquist curve passes through (-1, 0) from the right but does not enclose it, the robust stability of the system is relatively strong.

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