An ostrowski stability criterion for parallel fractional-order grid-connected inverter suitable for unbalanced grid

Abstract

The application mainly applies to parallel fractional order inverter system under unbalanced grid conditions, and an improved Nyquist stability criterion based on Ostrowski theory is proposed, which can accurately and quickly calculate the stability margin and critical stability point of the fractional order LCL type parallel inverter system under unbalanced grid. First, the positive and negative sequence output admittance matrix of the fractional order LCL type grid-connected inverter under unbalanced grid conditions is established, and the grid impedance is coupled to obtain the return ratio matrix coupled with the positive and negative sequences. Then, the Ostrowski theory is applied to estimate the characteristic roots of the return ratio matrix within a specified frequency bandwidth. Under the condition of meeting the Nyquist stability criterion, the critical resonance point is calculated. The grid-connected inverter critical stability condition judgment technology proposed in the application has the characteristics of low complexity, low conservativeness and high accuracy.

Description

Technical Field

[0001] This invention belongs to the field of power electronics and new energy power generation, specifically relating to an Ostrowski stability criterion for parallel fractional-order grid-connected inverters applicable to unbalanced power grids. Background Technology

[0002] With the development of distributed renewable energy generation systems, multi-unit parallel LCL grid-connected inverters serve as energy conversion interfaces connecting renewable energy generation devices and the power grid (circuit diagram shown). Figure 1 As shown in the figure, it plays an increasingly important role in improving output power and reducing current harmonics.

[0003] First, during the operation of LCL grid-connected inverters with multiple units in parallel, as the number of inverters connected in parallel increases, the output impedance of the inverter and the grid impedance will couple at the common point. Especially when the grid is unbalanced, new resonance points are easily generated, causing output current oscillation and instability, which will impact the load and the grid and affect the stability of the system.

[0004] Secondly, increasing research indicates that real-world inductors and capacitors often exhibit fractional-order characteristics. Using integer-order inductor-capacitor models to model the filter section increases the modeling error between the mathematical model and the actual physical model. Therefore, it is necessary to establish fractional-order filter circuit models to improve modeling accuracy. However, when using traditional integer-order stability criteria to analyze the stability of fractional-order systems, the fractional order leads to a higher order of the overall system model, significantly increasing computational costs. Therefore, this invention establishes a fractional-order mathematical model of multiple inverters in parallel under unbalanced power grid conditions and improves the traditional Nyquist stability criterion using Ostrowski theory, enabling effective determination of the critical stability conditions of fractional-order parallel systems. Summary of the Invention

[0005] Because the Nyquist stability criterion based on integer-order models is not applicable to fractional-order models, this invention proposes an improved Nyquist stability criterion based on Ostrowski theory, which can determine the critical stability condition of a fractional-order parallel inverter grid-connected system under grid imbalance conditions. First, by comparing with integer-order models and actual physical models, the mathematical model of the fractional-order parallel inverter proposed in this invention has higher modeling accuracy and is closer to the actual physical model. Second, under grid imbalance conditions, the improved stability criterion proposed in this invention is applied to estimate the maximum number of parallel inverters in the system, and the effectiveness of the proposed judgment method is verified through the dynamic response process of the point of common current.

[0006] The above-mentioned technical problems of the present invention are mainly solved by the following technical solutions:

[0007] A critical stability condition method for parallel LCL grid-connected inverters based on Ostrowski theory is characterized by: separating the positive and negative sequence voltages under unbalanced grid conditions based on harmonic linearization theory, establishing an expression in the frequency domain for the phase angle error of the phase-locked loop caused by the negative sequence component, and substituting it into the subsequent coordinate transformation stage; converting the inductance and capacitance in the LCL filter into fractional-order inductance and capacitance based on fractional-order calculus theory, and establishing a positive and negative sequence admittance model of the fractional-order LCL grid-connected inverter; establishing the total admittance model of the parallel system based on circuit principles and the fractional-order admittance model of a single unit; and performing stability analysis on the coupling matrix between the total output admittance of the system and the grid impedance based on Ostrowski theory to determine the critical stability condition of the system.

[0008] An analytical method based on positive and negative order separation, Figure 1 Phase a grid side voltage v ga and grid-side current i ga Through positive and negative order separation and Fourier transform, their expressions in the frequency domain are:

[0009]

[0010]

[0011] Where V1, V2, I1, and I2 are the amplitudes of the positive and negative sequence fundamental frequency voltages and currents, respectively; V p V n I p I n , respectively, are the amplitudes of the positive and negative sequence harmonic voltages and currents; f1, f p f n These are the frequencies of the fundamental wave and the positive and negative sequence voltages, respectively. and These represent the phases of the positive and negative sequence fundamental waves and harmonics, respectively.

[0012] Due to the imbalance of the grid voltage, the phase angle obtained by the traditional phase-locked loop (PLL) does not meet the condition of a fundamental frequency of 50Hz. In this invention, the phase angle of the PLL output is defined as θ = θ1 + Δθ. Since the phase angle error Δθ is much smaller than θ1, the approximate relationship cosΔθ≈1 and sinΔθ≈Δθ is satisfied. Thus, the sine and cosine expressions of the output phase angle θ can be obtained as shown in equation (2):

[0013]

[0014] Then, through Fourier transform, the expression for the phase error in the frequency domain can be obtained as shown in equation (4):

[0015]

[0016] in, Substituting equation (3) into equation (2) and performing a Fourier transform, we obtain the sine and cosine expressions of θ in the frequency domain as shown in equations (5) and (6), where T n =[±jH PLL (s)] / [1+V1H PLL (s)].

[0017]

[0018]

[0019] After phase shifting equations (5) and (6), the matrix expression of the Park transform in the frequency domain can be obtained as follows:

[0020]

[0021] After the Park transformation of equation (1) through equation (7), the frequency domain expression of the grid current in the synchronous rotating coordinate system under the condition of grid voltage imbalance can be obtained as follows:

[0022]

[0023]

[0024] The grid current in equations (8) and (9) passes through Figure 2 After the current controller is activated, the inverter-side output voltage e is obtained. d and e q The three-phase voltages after the Park inverse transformation are shown below:

[0025]

[0026] Among them, the current controller is H PI (s)=k pPI +k iPI / s;K dq = (L1+L2)ω0; L1, L2 and C f These represent the inverter-side filter inductor, the grid-side filter inductor, and the filter capacitor, respectively; α1, α2, and β are L1, L2, and C, respectively. f Fractional order.

[0027] Based on the inverse transformation of the matrix in equation (7) and the circuit equations, the relationship between the inverter-side output voltage and the grid voltage and current can be obtained as shown in equation (11):

[0028]

[0029] Where α1, α2 and β are the fractional orders of the inverter-side filter inductor, the grid-side filter inductor and the filter capacitor, respectively.

[0030] Substituting the three-phase output voltage of the inverter side in equation (10) into equation (11), we can obtain the output admittance coupling matrix of the LCL grid-connected inverter under the condition of grid voltage imbalance:

[0031]

[0032] When multiple inverters operate in parallel, such as Figure 3 As shown, the total equivalent output admittance of the parallel system is Where N is the number of inverters connected in parallel in the system.

[0033] According to the equivalent circuit equation, the current at the common point PCC is as shown in equation (13):

[0034]

[0035] According to equation (13), when the grid-side inductance Z g When the value is 0, the system has no poles in the right half-plane, and the parallel system operates stably; as the grid-side inductance increases, the hysteresis matrix... The eigenvalues ​​will gradually surround the point (-1, 0) on the complex plane, which does not satisfy the Nyquist stability criterion and causes the system to become unstable.

[0036] Therefore, this invention proposes to apply Ostrowski theory to estimate the range of the two eigenvalues ​​of the backpropagation matrix L(s), and to estimate the stability margin and critical resonance point of the system by calculating the distance between it and the point (-1, 0).

[0037] Suppose A is an n×n matrix, then the range of the n eigenvalues ​​of matrix A can be represented by a series of disks according to Ostrowski theory.

[0038]

[0039] Among them, a xx Let x be the center of the x-th disk. Let x be the radius of the x-th disk.

[0040] Substituting the back ratio matrix into equation (12), in order to determine the distance relationship between the back ratio matrix features and the point (-1, 0), this invention proposes a calculation method using equation (12), where D p and D nThe stability margin of the system is given by Im and Re, which are the imaginary and real parts of each element in the back ratio matrix, respectively. If the distance from the center of the disk containing the eigenvalue to the point (-1, 0) is greater than the radius of the disk, then the eigenvalue surrounds the point (-1, 0), and the system becomes unstable.

[0041] Attached Figure Description

[0042] Figure 1 This is a circuit topology diagram of the parallel three-phase LCL type grid-connected inverter used in this invention;

[0043] Figure 2 This is a control block diagram of the current controller used in this invention;

[0044] Figure 3 This is a schematic diagram of the equivalent transfer admittance of a parallel inverter;

[0045] Figure 4 The waveform diagram of the common point current instability experiment under the condition of grid voltage imbalance;

[0046] Figure 5 The waveform diagram of the common point current instability experiment under the condition of power grid impedance imbalance; Detailed Implementation

[0047] The technical solutions of the present invention will be clearly and completely described below with reference to preferred embodiments and accompanying drawings. Those skilled in the art will understand that the embodiments described below are only some, not all, embodiments of the present invention, and are used merely to illustrate the invention, and should not be considered as limiting the scope of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0048] This invention proposes an Ostrowski stability criterion for parallel fractional-order grid-connected inverters applicable to unbalanced power grids. Figure 1 The system control block diagram is designed according to the present invention.

[0049] The specific execution steps of this invention are as follows:

[0050] Step S201: Derive the lag matrix equation containing output admittance and grid impedance based on equations (11) and (12).

[0051] Step S202: Set the maximum number of iterations T to 20 and the bandwidth to [1, 1e]. 4 ]Hz, and iterates once every 500Hz.

[0052] Step S203: Calculate D at the current frequency according to equation (15). p With Dn .

[0053] Step S204: If D p With D n If all are greater than zero, proceed to the next iteration; if D at this time... p With D n If the values ​​are less than zero, then the current admittance and grid impedance are the critical impedance values.

[0054] With one Figure 1 Taking a grid-connected inverter as an example, this paper analyzes the critical stability condition judgment capability of an LCL-type grid-connected inverter under grid imbalance conditions. Some parameters are shown in the table below:

[0055] Table 1. Inverter Related Parameters

[0056] When a 10% fifth-harmonic positive-sequence disturbance is added to the grid voltage, the Ostrowski criterion determines that the maximum number of parallel units in the system is 168. Figure 4 In the grid current experimental waveform shown, when the number of parallel converters reaches the critical point of 169, the grid current not only exhibits severe distortion but also waveform divergence, with its amplitude significantly exceeding the reference value of 20A. To achieve overcurrent protection, when the effective value of the grid current exceeds 35A, the grid will disconnect from the inverter, resulting in zero output current and thus achieving protection. Furthermore, the frequency of the grid current is no longer the fundamental frequency of 50Hz; from the partially magnified waveform, it can be observed that the grid current frequency is approximately 2.5kHz, indicating complete system instability.

[0057] Under unbalanced grid impedance conditions, simulation results show that the critical stability point determined by Ostrowski occurs when N=137, i.e., when the number of inverters connected in parallel is 137. The common point current waveform when the number of inverters connected in parallel is 138 is as follows: Figure 5 As shown, the three-phase PCC current exhibits varying degrees of divergence at this point. The current peak will oscillate significantly, clearly exceeding the reference current value of 20A. To prevent damage caused by excessive grid current, the inverter will disconnect from the grid when the effective value of the grid PCC current exceeds 35A. It can be observed from the amplification section that the current frequency at this point is approximately 2.5kHz. The current frequency increases with the increase of the oscillation peak amplitude, indicating severe instability in the system.

[0058] The above embodiments are merely illustrative of the technical concept and features of the present invention, intended to enable those skilled in the art to understand the content of the present invention and implement it accordingly. They should not be construed as limiting the scope of protection of the present invention. The general principles defined in this invention can be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the present invention is not limited to the embodiments shown, but is to be accorded the widest scope consistent with the principles and novel features disclosed in the present invention. All equivalent changes or modifications made according to the spirit and essence of the present invention should be covered within the scope of protection of the present invention.

Claims

1. An Ostrowski stability criterion method for parallel fractional-order grid-connected inverters applicable to unbalanced power grids, characterized in that, The method includes: During the operation of multiple LCL grid-connected inverters in parallel, as the number of inverters connected in parallel increases, the output impedance of the inverter and the grid impedance will couple at the point of common. Especially under unbalanced grid conditions, new resonant points are easily generated, leading to output current oscillations and instability, which impacts the load and the grid, affecting system stability. Secondly, increasing research shows that actual inductor-capacitor filtering devices often exhibit fractional-order characteristics. If an integer-order inductor-capacitor model is used to model the filter part, it will increase the modeling error between the mathematical model and the actual physical model. Therefore, a critical stability condition judgment technique for LCL grid-connected inverters with unbalanced grids based on Ostrowski theory is proposed. The execution steps are as follows: Step 1: Apply harmonic linearization to the unbalanced grid voltage V ga and grid current I ga Positive and negative sequence separation is performed, as shown in Equation 1, where V1, V2, I1, and I2 are the amplitudes of the positive and negative sequence fundamental frequency voltage and current, respectively; V p V n ,I p ,I n , respectively, are the amplitudes of the positive and negative sequence harmonic voltages and currents; f1, f p ,f n These are the frequencies of the fundamental wave and the positive and negative sequence voltages, respectively. and These represent the phases of the positive and negative sequence fundamental waves and harmonics, respectively. Step 2: Due to the unbalanced power grid, the phase angle output of the phase-locked loop will have a certain phase angle error compared with the power grid's fundamental frequency. The sine and cosine expressions cos(θ)[f] and sin(θ)[f] after transforming this phase angle error using Fourier transform are: in T n =[±jH PLL (s)] / [1+V1H PLL (s)],H PLL It is a phase-locked loop PI controller; It is the conjugate of the negative sequence voltage. Step 3: Based on the frequency domain sine and cosine expressions for phase error, the matrix frequency domain expression for the Park transform can be derived as follows: Step 4: The reference and sampled values ​​of the grid current after Park transformation are sent to the current controller; simultaneously, to reduce the resonant spikes of the LCL filter, an active damping method with capacitor current feedback is adopted. After PWM modulation, the expression e of the inverter bridge arm output voltage in the synchronous rotating coordinate system can be obtained. d and e q for in, H PI (s) is the frequency domain expression of the current controller; K dq For coupling terms; i dref and i qref For reference only. Step 5: Using the inverter output voltage in the controller coordinate system obtained from Equation 4 and the equivalent circuit equation of the LCL inverter, the positive and negative sequence output admittance matrix Y of a single LCL inverter can be obtained. o The equivalent output admittance Y of multiple inverters connected in parallel is obtained using the Norton equivalent circuit. eq for: Among them, L1, L2 and C f These are the inverter-side filter inductor, the grid-side filter inductor, and the filter capacitor, respectively; α1, α2, and β are L1, L2, and C, respectively. f fractional order, Step Six: Based on the circuit equations of the parallel equivalent circuit: Among them, i PCC (s) is the frequency domain expression of the current at the common coupling point PCC, Z g (s) represents the grid impedance expression, and N represents the number of inverters connected in parallel. It can be seen that the system stability is determined by the location of the eigenvalues ​​of the hysteresis matrix. It is known that the hysteresis matrix does not have poles in the right half-plane. If the Nyquist curve of the hysteresis matrix encloses the point (-1,0) in the complex plane, the system does not satisfy the Nyquist stability criterion, and the system becomes unstable. Therefore, the stability of this fractional-order LCL parallel inverter system depends on the relationship between the eigenvalues ​​of the hysteresis matrix and the point (-1,0). The hysteresis matrix is ​​shown in Equation 6. Among them, L pp ,L nn and L pn ,L np These are the positive and negative order terms and the positive and negative order coupling terms of the back ratio matrix, respectively; Step 7: Estimate the range of eigenvalues ​​of the backpropagation matrix using Ostrowski theory. The range of eigenvalues ​​S(A) is restricted to a series of disks, as shown in Equation 7, where a xx Let x be the center of the x-th disk. The radius of the x-th disk Step 8: Use Equation 8 to limit the distance between the eigenvalues ​​of the slew rate matrix and the point (-1, 0). The closer the eigenvalues ​​are to the point (-1, 0), the greater the system stability margin D. p and D n The smaller the value of D, the easier it is to become unstable; if D p and D n The values ​​of all values ​​are greater than zero, indicating that the system's characteristic roots do not surround the point (-1,0), and the system is operating stably; if D p and D n If all values ​​are less than zero, the system falls into the instability region, resulting in a deterioration in the quality of the output current waveform, an increase in harmonic content, and a tendency for the waveform to diverge, which can cause impact and damage to the power grid.

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