CN106786494B - Direct current micro-grid system with parallel converters and stabilizing method thereof - Google Patents

Abstract

The invention provides a direct current micro-grid system with parallel converters. The system comprises: the constant power load is connected in parallel to the direct current bus; the n direct current micro power supplies are connected in parallel to a direct current bus through respective LC converters; wherein the inductance and capacitance parameters of the LC converter are set such that the difference between the maximum and minimum of all inductance and capacitance products is outside a certain range. The invention analyzes the problem of influence of the parallel connection of LC filters of a plurality of converters on the stability of a system in a direct current micro-grid with a constant power load from the mathematical point of view and provides a system stability criterion containing two and three parallel converters. At the same time, the correctness of the proposed stability criterion was verified by simulation. In fact, it was also demonstrated by quantitative analysis that the proposed stability criterion still applies.

Description

DC Microgrid System with Parallel Converter and Its Stabilization Method

technical field

The invention relates to the technical field of power electronics, in particular, to a DC microgrid with a parallel converter and a stabilization method thereof.

Background technique

In advanced automation systems, when power electronic converters and electric drives, etc. are strictly controlled, they appear as constant power loads at the input and often cause instability problems due to negative impedance. Especially in a DC microgrid with multiple parallel DC-DC converters, the parallel LC filter between the converters and the DC bus may cause oscillation or even collapse of the system.

SUMMARY OF THE INVENTION

In order to solve the above technical problems, the present invention provides a DC micro-grid system with a parallel converter. The system includes:

Constant power load, which is connected in parallel with the DC bus;

n DC micro-power sources, each of which is connected in parallel to the DC bus through its own LC filter;

Wherein, the parameters of the inductance and capacitance of the LC filter are set such that the difference between the maximum value and the minimum value in the products of all inductances and capacitances is outside a certain range.

According to the DC microgrid system with parallel converters of the present invention, when there are two parallel DC-DC converters,

The difference between the maximum and minimum values of all products of inductance and capacitance satisfies the following relationship:

or

It is preferably greater than 0.8.

According to the DC microgrid system with parallel converters of the present invention, when the number of parallel DC-DC converters is 3, the difference between the maximum value and the minimum value in the products of all inductances and capacitances satisfies the following relationship:

where f ₁₁ =M ₁ -M ₂ /M ₀ , f ₁₂ =M ₃ -M ₄ /M ₀ , f ₂₁ =(M ₂ f ₁₁ -M ₀ f ₁₂ )/f ₁₁ , f ₂₂ =(M ₄ f ₁₁ -M ₅ M ₀ )/f ₁₁ , f ₃₁ =(f ₁₂ f ₂₁ -f ₂₂ f ₁₁ )/f ₂₁ , f ₄₁ =(f ₂₂ f ₃₂ -M ₅ f ₂₁ )/f ₃₁ .

Preferably greater than 1.8.

According to another aspect of the present invention, a method for stabilizing a DC microgrid with a parallel converter is also provided, wherein the method comprises the following steps:

The load is connected to the power source through a strictly managed load point converter to form a constant power load, and the dynamic equation of the system is determined and linearized for a specific DC microgrid with n parallel converters;

The stability conditions of the system are obtained by solving the stability conditions of the dynamic equations of the system;

Wherein, the inductance and capacitance parameters of the converter satisfy the following conditions to ensure that the system operates in a stable domain: the difference between the maximum value and the minimum value in the products of all inductances and capacitances is outside a certain range.

According to the method of the present invention for stabilizing a DC microgrid with a parallel converter,

When there are 2 DC-DC converters connected in parallel, the difference between the maximum value and the minimum value in the products of all inductors and capacitors satisfies the following relationship:

or

It is preferably greater than 0.8.

According to the method for a DC microgrid with parallel converters of the present invention, when the number of parallel DC-DC converters is 3, the difference between the maximum value and the minimum value in the products of all inductances and capacitances satisfies the following relationship :

where f ₁₁ =M ₁ -M ₂ /M ₀ , f ₁₂ =M ₃ -M ₄ /M ₀ , f ₂₁ =(M ₂ f ₁₁ -M ₀ f ₁₂ )/f ₁₁ , f ₂₂ =(M ₄ f ₁₁ -M ₅ M ₀ )/f ₁₁ , f ₃₁ =(f ₁₂ f ₂₁ -f ₂₂ f ₁₁ )/f ₂₁ , f ₄₁ =(f ₂₂ f ₃₂ -M ₅ f ₂₁ )/f ₃₁ .

Preferably greater than 1.8.

The benefit of the present invention is that it considers that in a DC microgrid system with multiple microsources, the LC filter connected between the parallel converters and the DC bus may cause system instability. In general, when the products of inductance and capacitance of all parallel LC filters are close to or equal within a certain range, the system will be unstable due to resonance. The invention analyzes the stability problem of the direct current microgrid with n parallel converters from a mathematical point of view. At the same time, the relationship criteria of the LC filter parameters to ensure the stability of the system are also analyzed in the case of two micro-sources in parallel and three micro-sources in parallel. The proposed criteria are simple and have great guiding significance for practical applications.

Other features and advantages of the present invention will be set forth in the description which follows, and in part will be apparent from the description, or may be learned by practice of the invention. The objectives and other advantages of the invention may be realized and attained by the structure particularly pointed out in the description, claims and drawings.

Description of drawings

The accompanying drawings are used to provide a further understanding of the present invention, and constitute a part of the specification, and together with the embodiments of the present invention, are used to explain the present invention, and do not constitute a limitation to the present invention. In the attached image:

FIG. 1 shows a schematic diagram of a power grid system with multiple micro-DC sources in parallel according to an embodiment of the present invention;

FIG. 2 shows a stability domain diagram of a system containing two parallel converters in accordance with one embodiment of the present invention;

Fig. 3 shows the simulation schematic diagram of the system with n parallel converters;

FIG. 4 shows a schematic diagram of the stabilization domain of a system containing two parallel converters according to another embodiment of the present invention;

Figure 5 shows a waveform diagram of the output voltage with two converters;

Figure 6 shows a schematic diagram of the root locus of a system containing two converters; and,

Figure 7 shows a schematic diagram of the root locus of a system with three converters.

Detailed ways

In the past few years, the concept of microgrids has emerged in order to enable the integration of different forms of renewable energy and the electrification of remote areas. Since many sustainable energy sources and loads, such as photovoltaic modules, batteries and LEDs, have their own DC coupling points, DC-DC converters are directly used to connect these micro-sources and loads to form a DC micro-grid. This conversion is more efficient than using AC-DC or DC-AC conversion.

However, when such a DC microgrid directly connected in parallel by a DC-DC converter is connected to a constant power load, it may cause system instability because the constant power load itself exhibits a negative impedance characteristic. In order to solve this problem, many researchers have proposed a large number of technologies and design methods. These methods can be divided into three types from the perspective of the number of converters:

The first type is a single converter with a constant power load, this type is mainly aimed at the study of stability analysis and stabilization methods. One type of prior art mainly adopts the methods of increasing damping and reducing negative impedance to alleviate oscillation and prevent the voltage collapse of the DC bus. There is also a literature that summarizes the use of four analytical methods for large-signal stability analysis, including the Brayton-Moster mixed potential method, the multi-model method, the block-diagonalized quadratic Lyapulov method, and the reverse trajectory tracking method.

The second category is the case where two converters are connected in parallel. The prior art literature introduces a new loop cancellation nonlinear feedback method to eliminate the effects of instability caused by constant power loads. Since increasing the damping is beneficial to the stability of the system, a variety of linear control methods based on virtual resistance are proposed to increase the system damping. But this approach reduces the efficiency of the system or requires adding additional power electronic components to the system.

The third type is the system with n (n≥2) converters in parallel. Based on the Brayton-Moster mixed potential theory, the prior art proposes a stability criterion for a system with a multi-stage LC filter with a constant power load, and this criterion can be applied to an actual constant power load.

In order to make the objectives, technical solutions and advantages of the present invention clearer, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

Figure 1 shows the basic mathematical model of a DC microgrid with a constant power load containing multiple parallel converters. Assuming that the microsources in the model are all Buck converters, the input can be regarded as an ideal voltage source. All loads are constant power loads and all common bus bars have zero resistance. Through analysis, it can be seen that the inductance of the cable has no effect on the stability of the system, but the resistance has an effect on the stability of the system. Therefore, the impedance of the cable can be regarded as a purely resistive impedance. A load point converter and its associated load can be considered a constant power load if the load is connected to the power source through a tightly managed load point converter. Therefore, the load in the mathematical model shown in Figure 1 can be regarded as an ideal constant power load.

From the mathematical model shown in Figure 1, the dynamic equation of the system can be derived as follows:

i＝[i₁ i₂ …i_n]^T，L＝diag{L_j}，C＝diag{C_j}，V＝diag{V_j}。j＝1,2,…n表示第j个微源。L_j及C_j分别表示LC滤波器的电感值和电容值，V_j及u_j分别表示变换器的输入和输出电压。d_j为占空比，

和i_j分别表示电感L_j和第j个变换器的输出电流。where u=[u ₁ u ₂ … u _n ] ^T , d=[d ₁ d ₂ … d _n ] ^T ,

i=[i ₁ i ₂ ... i _n ] ^T , L=diag{ _{Lj}, C=diag{Cj }} , V=diag{ _Vj }. j=1,2,...n represents the jth microsource. L _j and C _j represent the inductance value and capacitance value of the LC filter, respectively, and V _j and u _j represent the input and output voltages of the converter, respectively. d _j is the duty cycle,

and _ij represent the inductor Lj and the output current of the _jth converter, respectively.

For constant power load, there is the following power balance expression:

Among them, P represents the power of the constant power load, u ₀ represents the load voltage, and y _j is the admittance between the jth converter and the constant power load.

It is well known that resonance occurs when the natural frequencies of several oscillators are close to a certain range. This phenomenon also applies to DC microgrid systems with multiple parallel converters. The embodiments of the present invention provide stability analysis and stability criteria of the system, and also perform corresponding stability domain analysis.

1. Stability analysis and stabilization design criteria

Linearizing equation (1) can get

where "^" represents the small signal fluctuation of the system near the stable point. Similarly, by linearizing equation (2), we can get

From this, the partial derivative matrix can be obtained as

表示恒功率负载的等效导纳。令

Y表示系统网络的导纳矩阵，为实对称矩阵。通常情况下电缆线的导纳比负载的大，即

因此有in

Indicates the equivalent admittance of a constant power load. make

Y represents the admittance matrix of the system network, which is a real symmetric matrix. Under normal circumstances, the admittance of the cable is larger than that of the load, that is

Therefore there is

It can be obtained that the matrix Y has at least one negative eigenroot.

The Jacobian matrix of the system is as follows:

make

Therefore, J and J ₁ are in the same spectrum. The characteristic polynomial of J ₁ is

其中K＝C^-1L^-1，Y'＝C^-1/2YC^-1/2。make

Wherein K=C ^-1 L ^-1 , Y'=C ^-1/2 YC ^-1/2 .

为正定矩阵，那么矩阵J₂至少含有一个含有正实部的特征值，即系统不稳定。Lemma 1: If there is a real symmetric indefinite matrix P such that

is a positive definite matrix, then the matrix J ₂ contains at least one eigenvalue with a positive real part, that is, the system is unstable.

为正，并且存在一个列向量x₂使得

为负，那么P为实对称不定矩阵。Lemma 2: If there is a column vector x ₁ such that

is positive, and there exists a column vector x ₂ such that

is negative, then P is a real symmetric indefinite matrix.

Corollary 1: Assumptions

Where b=b ₀ +δ, b ₀ =max{C _j L _j }, and δ is an arbitrary infinitesimal constant, then P is a real symmetric indefinite matrix.

Proof: let

and

x ₂ = [0 0 … 0, 1 0 … 0] ^T ,

then there are

and

Inference 1 is proved.

则有make

then there are

From Corollary 1 we can get

where ΔK=K ^-1 diag{b ₀ -C _j L _j }.

In summary, the following theorem can be drawn:

Theorem 1: For the system described by equation (1) with multiple parallel Buck converters, let

Δ=max{C _j L _j }-min{C _j L _j },

If Δ is small enough within a certain range, the system will be unstable.

是一个正定矩阵。结合引理1，定理1得证。Prove: If Δ is small enough, Y' ² -δI-ΔK will become a positive definite matrix, that is,

is a positive definite matrix. Combined with Lemma 1, Theorem 1 is proved.

2. Stability domain analysis

It can be seen from the above that when the product of the inductance and capacitance of the LC oscillator circuit in multiple parallel converters is close to a certain range Δ, the DC microgrid system with constant power load will be unstable, but it does not give this Exact value of range Δ. To this end, the specific range of Δ will be analyzed below.

For the sake of simplicity, the system with two parallel converters is firstly analyzed from the mathematical point of view, and then the analysis results are extended to the system with three or more parallel converters.

1) Stability domain analysis of a system with two parallel converters

For a DC microgrid system with two parallel converters, we have

For simplicity of calculation, let

K=diag{k ₁ ,k ₂ } (15)

in

Obviously ac<b ² .

From formula (9), we can get

The corresponding characteristic equation is

λ ⁴ +(a+c)λ ³ +(k ₁ +k ₂ +ac-b ² )λ ² +(k ₁ c+k ₂ a)λ+k ₁ k ₂ =0 (17)

The Rouse-Hurwitz criterion table for the system is

The stability condition is

Simplified to get

The stability domain of the system obtained from (20) is shown in Figure 2.

In order to ensure the stability of the system, Δ needs to satisfy the following inequality:

or

From equations (21) and (22), it can be known that Δ is related to k ₁ . That is to say, for different k ₁ , there is a different Δ corresponding to it to ensure the stability of the system.

The gray part in Figure 2 represents the stable region, where the expression of the straight line passing through points A and B is:

The stability domain in the figure is divided into two parts, which are distributed on both sides of the straight line l. The expression of l is:

l:k ₁ =k ₂ (24)

2) Stability domain analysis of a system with three parallel converters

Similarly, when the DC microgrid contains three parallel converters, its system characteristic equation can be expressed as:

λ ⁶ +M ₀ λ ⁵ +M ₁ λ ⁴ +M ₂ λ ³ +M ₃ λ ² +M ₄ λ+M ₅ =0 (25)

The definitions of M _i , i=0, 1, 2, . . . 5 refer to the appendix, and it is obvious that M _i >0.

To ensure stability, the system must meet the following conditions:

where f ₁₁ =M ₁ -M ₂ /M ₀ , f ₁₂ =M ₃ -M ₄ /M ₀ , f ₂₁ =(M ₂ f ₁₁ -M ₀ f ₁₂ )/f ₁₁ , f ₂₂ =(M ₄ f ₁₁ -M ₅ M ₀ )/f ₁₁ , f ₃₁ =(f ₁₂ f ₂₁ -f ₂₂ f ₁₁ )/f ₂₁ , f ₄₁ =(f ₂₂ f ₃₂ -M ₅ f ₂₁ )/f ₃₁ .

Since (26) is a 12th-order inequality equation, the analytical solutions for the range of Δ similar to those given in (22) and (23) cannot be given by calculation, but the numerical solutions are easily obtained. Based on the simulation and numerical analysis, the quantitative analysis of Δ will be given below to obtain the range of Δ.

3) Stability domain analysis of a system with n (n>3) parallel converters

Similarly, when the DC microgrid with constant power load contains n (n>3) parallel converters, conclusions similar to those in 1) and 2) can be drawn and verified by simulation. The range of Δ obtained by simulation and quantitative analysis can provide some reference and guidance for practical application.

The simulation results based on MATLAB/Simulink verify the correctness of the proposed stability criterion. For simplicity, it is assumed that all irrelevant quantities in the parallel converter are equal. Different values of k _j can be realized by taking the same capacitance value C _j and different inductance values L _j . At the same time, an effective method is to simplify the constant power load into a voltage-controlled current source, that is, i=P/v.

为100V。接下来分别对含有两个、三个以及n个并联变换器系统进行仿真分析。Figure 3 shows a schematic diagram of a simulation with n parallel converters. For simplicity, the input voltage of the Buck converter is set as V _in1 =V _in2 =...=V _inn =15V, and the filter capacitor is C ₁ =C ₂ =... =C _n =1F. The power of the constant power load is 1000W, the reference voltage

is 100V. Next, the simulation analysis is carried out for the system with two, three and n parallel converters respectively.

A. Simulation analysis of a system with two parallel converters

For a system with two parallel converters, assuming y ₁ =1S, y ₂ =2S, combining equations (15) and (20), we can obtain:

and its stability domain can be expressed as

Therefore, its stable range can be obtained as shown in Figure 4:

Example 1: When k ₁ =10, k ₂ =2, that is, when L ₁ =0.1H, L ₂ =0.5H, the point (10,2) is in the green area, the system characteristic equation shown in equation (17) Can be simplified to:

The eigenvalue of the solution is

λ _1,2 =-0.1448±2.2945i

λ _3,4 = -0.4931 ± 1.6488i. (29)

Obviously, all the eigenvalues are in the left half-plane of the s-plane, and the system is stable. Figure 5 shows the output voltage waveforms of the two converters.

Example 2: When k ₁ =10,k ₂ =8, and the point (10,8) is outside the green area, the corresponding eigenroot is:

λ _1,2 '=0.4386±2.9604i

λ _3,4 '=-1.0765±2.7881i. (30)

The system is unstable at this time.

In order to illustrate the stability criterion proposed above and the simplicity of the calculation, it is assumed that

k ₁ =10, (31)

k ₂ =0.1～50. (32)

Figure 6 shows the root locus diagram of the system. It can be seen from the figure that when k2 changes from 9.3 to 10.8, there are _two characteristic roots that cross from the left half of the s-plane to the right half and then back to the left half. Therefore, in order to ensure the stability of the system, it is necessary to satisfy

0.7<k ₁ -k ₂ <10 (33)

or

k ₂ -k ₁ >0.8. (34)

B. Simulation analysis of a system with three parallel converters

Similarly, for a system with three parallel converters, suppose y ₁ =1S, y ₂ =2S, y ₃ =3S, and

k ₁ =k ₂ =k=10 (35)

k ₃ =0.1-30. (36)

The corresponding root locus diagram is shown in Figure 7.

It can be seen from Figure 7 that when k ₃ changes from 8.2 to 11.9, namely -1.7≤Δ≤1.8, Δ=k ₃ -k, there are also two characteristic roots that cross from the left to the right and then back to the left, that is to say In this interval, the system is unstable. In order to ensure the stability of the system, the following conditions must be met:

1.7 < kk ₃ < 10 (37)

or

k ₃ -k > 1.8. (38)

C. Simulation of a system with n (n>3) parallel converters

For a system containing n (n>3) parallel converters, a similar rule can be found through quantitative analysis. Since the analysis method is similar to the above method, it will not be repeated here.

Wherein, M _i in formula (23) is defined as follows:

M ₀ =a+c+d

M ₁ =k ₁ +k ₂ +k ₃ +ac+(a+c)db ² -e ² -f ²

M ₂ =(c+d)k ₁ +(a+d)k ₂ +(c+a)k ₃ +acd-2bef-b ² de ² cf ² a

M ₃ =k ₁ k ₂ +(k ₁ +k ₂ )k ₃ +(ad-f ² )k ₁ +(cd-e ² )k ₂ +(ac-b ² )k ₃

M ₄ =dk ₁ k ₂ +ck ₁ k ₃ +ak ₂ k ₃

M ₅ =k ₁ k ₂ k ₃

in

Obviously there are a,b,...,f>0.

It is to be understood that the disclosed embodiments of the invention are not limited to the specific structures, process steps or materials disclosed herein, but extend to equivalents of these features as understood by those of ordinary skill in the relevant art. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only and is not meant to be limiting.

Reference in the specification to "one embodiment" or "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases "one embodiment" or "an embodiment" in various places throughout the specification are not necessarily all referring to the same embodiment.

Although the disclosed embodiments of the present invention are as above, the content described is only an embodiment adopted to facilitate understanding of the present invention, and is not intended to limit the present invention. Any person skilled in the art to which the present invention belongs, without departing from the spirit and scope disclosed by the present invention, can make any modifications and changes in the form and details of the implementation, but the scope of patent protection of the present invention, The scope as defined by the appended claims shall still prevail.

Claims (2)

1. a direct current microgrid system with parallel converter, is characterized in that, described system comprises: Constant power load, which is connected in parallel with the DC bus; n DC micro-power sources, each of which is connected in parallel on the DC bus through its own LC filter; Wherein, the parameters of the inductance and capacitance of the LC filter are set such that the range of the difference between the maximum value and the minimum value in the products of all inductances and capacitances is related to the number n of the DC micro power supply, and is determined by the constant The power of the power load, the reference value of the load voltage, the admittance between the converter and the constant power load, the capacitance value of the LC filter, and the inductance value of the LC filter are determined, among which: When there are 2 DC micro-power supplies in parallel, the difference between the maximum value and the minimum value in the products of all inductors and capacitors in the LC filter satisfies the following relationship:

or

0.7<k ₁ -k ₂ <10 or k ₂ -k ₁ >0.8 When there are 3 DC micro-power supplies in parallel, the difference between the maximum value and the minimum value in the products of all inductors and capacitors in the LC filter satisfies the following relationship:

Wherein, f ₁₁ =M ₁ -M ₂ /M ₀ , f ₁₂ =M ₃ -M ₄ /M ₀ , f ₂₁ =(M ₂ f ₁₁ -M ₀ f ₁₂ )/f ₁₁ , f ₂₂ =(M ₄ f ₁₁ -M ₅ M ₀ )/f ₁₁ , f ₃₁ =(f ₁₂ f ₂₁ -f ₂₂ f ₁₁ )/f ₂₁ , f ₄₁ =(f ₂₂ f ₃₂ -M ₅ f ₂₁ )/f ₃₁ , f ₃₂ =M ₅ =k ₁ k ₂ k ₃ , M ₀ =a+c+d M ₁ =k ₁ +k ₂ +k ₃ +ac+(a+c)db ² -e ² -f ² M ₂ =(c+d)k ₁ +(a+d)k ₂ +(c+a)k ₃ +acd-2bef-b ² de ² cf ² a M ₃ =k ₁ k ₂ +(k ₁ +k ₂ )k ₃ +(ad-f ² )k ₁ +(cd-e ² )k ₂ +(ac-b ² )k ₃ M ₄ =dk ₁ k ₂ +ck ₁ k ₃ +ak ₂ k ₃ M ₅ =k ₁ k ₂ k ₃

k ₁ =k ₂ =k=10 and 1.7<kk ₃ <10 or k ₃ -k>1.8 Among them, P represents the power of the constant power load,

Represents the load voltage reference value, y ₁ , y ₂ and y ₃ represent the admittance between the first, second and third converters to the constant power load, respectively, C ₁ , C ₂ and C ₃ represent the Capacitance values of the 1st, 2nd and 3rd LC filters, L ₁ , L ₂ and L ₃ represent the inductance values of the 1st, 2nd and 3rd LC filters, respectively, k ₁ , k ₂ , k ₃ respectively represent the reciprocal of the product of the inductance and capacitance of the grid-connected lines of the first, second, and third converters. 2. A method for stabilizing a DC microgrid with a parallel converter, wherein the method comprises the following steps: The load is connected to the power source through a strictly managed load point converter to form a constant power load, and the dynamic equation of the system is determined and linearized for a specific DC microgrid with n parallel converters; The stability conditions of the system are obtained by solving the stability conditions of the dynamic equations of the system; Wherein, the inductance and capacitance parameters of the LC filter in the converter satisfy the following conditions to ensure that the system works in the stable domain: the range where the difference between the maximum value and the minimum value in the products of all inductances and capacitors is located is the same as the DC micrometer. The number n of the power supply is related and determined by the power of the constant power load, the reference value of the load voltage, the admittance between the converter and the constant power load, the capacitance value of the LC filter, and the inductance value of the LC filter, where: When there are 2 DC micro-power supplies in parallel, the difference between the maximum value and the minimum value in the products of all inductors and capacitors in the LC filter satisfies the following relationship:

or

0.7<k ₁ -k ₂ <10 or k ₂ -k ₁ >0.8 When there are 3 DC micro-power supplies in parallel, the difference between the maximum value and the minimum value in the products of all inductors and capacitors in the LC filter satisfies the following relationship:

Wherein, f ₁₁ =M ₁ -M ₂ /M ₀ , f ₁₂ =M ₃ -M ₄ /M ₀ , f ₂₁ =(M ₂ f ₁₁ -M ₀ f ₁₂ )/f ₁₁ , f ₂₂ =(M ₄ f ₁₁ -M ₅ M ₀ )/f ₁₁ , f ₃₁ =(f ₁₂ f ₂₁ -f ₂₂ f ₁₁ )/f ₂₁ , f ₄₁ =(f ₂₂ f ₃₂ -M ₅ f ₂₁ )/f ₃₁ , f ₃₂ =M ₅ =k ₁ k ₂ k ₃ , M ₀ =a+c+d M ₁ =k ₁ +k ₂ +k ₃ +ac+(a+c)db ² -e ² -f ² M ₂ =(c+d)k ₁ +(a+d)k ₂ +(c+a)k ₃ +acd-2bef-b ² de ² cf ² a M ₃ =k ₁ k ₂ +(k ₁ +k ₂ )k ₃ +(ad-f ² )k ₁ +(cd-e ² )k ₂ +(ac-b ² )k ₃ M ₄ =dk ₁ k ₂ +ck ₁ k ₃ +ak ₂ k ₃ M ₅ =k ₁ k ₂ k ₃

k ₁ =k ₂ =k=10 and 1.7<kk ₃ <10 or k ₃ -k>1.8 Among them, P represents the power of the constant power load,

Represents the load voltage reference value, y ₁ , y ₂ and y ₃ represent the admittance between the first, second and third converters to the constant power load, respectively, C ₁ , C ₂ and C ₃ represent the Capacitance values of the 1st, 2nd and 3rd LC filters, L ₁ , L ₂ and L ₃ represent the inductance values of the 1st, 2nd and 3rd LC filters, respectively, k ₁ , k ₂ , k ₃ respectively represent the reciprocal of the product of the inductance and capacitance of the grid-connected lines of the first, second, and third converters.

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