A SISO model establishing method suitable for multi-machine interaction characteristic analysis

Abstract

The application discloses a SISO model establishment method suitable for multi-machine interaction characteristic analysis, and comprises the following steps: establishing a SISO model of a single grid-following converter connected to a power grid; establishing a SISO model of a single grid-constructing converter connected to the power grid; in step three, the obtained single-machine SISO models are coupled into the same steady-state coordinate system through matrix transformation, and then a dynamic interaction model of the parallel grid-following converter and the grid-constructing converter is established, that is, the SISO model suitable for the multi-machine interaction characteristic analysis is obtained. The model constructed by the application takes the power grid frequency disturbance as an input variable, completely describes the frequency response dynamics of the converter, solves the problem of difficulty in analyzing the multi-input-multi-output system, and finally converts the two parallel converters into two interacting SISO systems, and the synchronization link appears only once in the open-loop transfer function of the system, which is helpful for loop shaping of the phase-locked loop and the power synchronization loop.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of control interaction model construction of hybrid systems, and more particularly, relates to a SISO (Single-Int Single-Output) model establishment method suitable for multi-machine interaction characteristic analysis. BACKGROUND

[0002] With the development of new energy, direct current transmission and energy storage technologies, large-scale power electronic devices are connected to the power grid, which profoundly changes the dynamic characteristics of modern power systems, especially the weakening of grid strength and the reduction of inertia. The load influence, harmonic pollution, reactive power demand and current saturation of power electronic devices will lead to problems such as voltage drop and frequency deviation of the power grid, resulting in weakening of the grid strength. At the same time, the centralized access of large-scale new energy on the power generation side, the intensive direct current transmission and feeding on the power transmission side, the distributed generation, microgrid and direct current distribution network feeding on the power distribution side, and the access of a large number of power electronic loads on the power consumption side all lead to the reduction of system inertia and the increasingly prominent problem of frequency stability.

[0003] To solve the above problems, energy storage systems have been widely used in inertia response, frequency modulation, voltage control and other aspects. Through the improvement of the power outer loop, the grid-connected energy storage system working in the active-reactive control mode can achieve the regulation effect of the power grid. However, the energy storage system synchronized by the phase-locked loop does not have the ability to actively support and cannot independently control the frequency and voltage of the power grid. With the increase of new energy penetration rate, it is insufficient to provide the required system stability of the power grid. On the other hand, under fault conditions, the grid frequency drops with a large initial frequency change rate, and the grid-connected energy storage system shows the defect of insufficient inertia support capacity in response time.

[0004] Under the background of high penetration rate of grid-connected new energy, to provide the required stability of the power system, one feasible way is to configure the grid-connected converter as a grid-forming converter with voltage source characteristics to realize the active support of the power grid. The grid-forming converter has the ability to form a network independently, and the power ring outputs voltage instructions and can be phase-synchronized. Since 2016, many institutions and organizations abroad have published guidance documents, proposing that the converter should have the ability to form a network, and proposing frequency and voltage control, weak grid operation, fault ride-through and other requirements.

[0005] Due to different synchronization modes, the dynamic behaviors of grid-forming and grid-following converters are quite different: the stability margin of grid-following converter based on phase-locked loop (PLL) synchronization gradually decreases with the decrease of short-circuit ratio (SCR) of the power grid, so there is an instability problem under weak grid; however, with the decrease of the strength of the power grid, the interaction between the grid-forming converter based on power synchronization and the AC power grid decreases, and better stability is shown. Therefore, the applicability of devices with different synchronization modes to the strength of the power grid is different, and the synchronization stability problem of the mixed system composed of grid-following and grid-forming devices needs to be further studied. The existing models established for the mixed system are high-order, strongly coupled multi-input multi-output models, which treat the whole system as a "black box", and the internal links are not divided, so it is not conducive to extract single features for theoretical analysis, and the interaction characteristics between the two converters are not clear. SUMMARY

[0006] The application provides a SISO model establishment method suitable for multi-machine interaction characteristic analysis, which takes the grid frequency disturbance as an input variable and completely describes the frequency response dynamics of the converter. The model solves the problem of difficult analysis of MIMO system, and two parallel converters are finally transformed into two interacting SISO systems, and the synchronization link appears only once in the open-loop transfer function of the system, which is helpful for loop shaping of the PLL and power synchronization loop.

[0007] To achieve the above object, the application provides a SISO model establishment method suitable for multi-machine interaction characteristic analysis, comprising the following steps

[0008] Step one, establishing a SISO model of a single grid-following converter connected to a power grid;

[0009] Step two, establishing a SISO model of a single grid-forming converter connected to a power grid;

[0010] Step three, copying the derivation process of the single-machine SISO model obtained in steps one and two, coupling the single-machine SISO models obtained in steps one and two into the same steady-state coordinate system through matrix transformation, and then establishing a dynamic interaction model of parallel grid-following and grid-forming converters, that is, obtaining a SISO model suitable for multi-machine interaction characteristic analysis with grid frequency disturbance as input.

[0011] Further, step one establishes a SISO model of a single grid-following converter connected to a power grid, which specifically comprises:

[0012] Constructing the circuit structure and control block diagram of a single grid-following converter connected to an AC power grid through an LCL filter;

[0013] According to the circuit structure and control block diagram of the single grid-following converter connected to the power grid, PLL dynamic equation, power grid dynamic equation and current controller dynamic equation are written;

[0014] By simultaneously solving the PLL dynamic equation, power grid dynamic equation and current controller dynamic equation, taking the power grid frequency disturbance as input and the frequency measured by the PLL as output, a SISO model of the single grid-following converter connected to the power grid is obtained.

[0015] Further, the writing of the PLL dynamic equation, power grid dynamic equation and current controller dynamic equation according to the control block diagram of the single grid-following converter specifically comprises:

[0016] The output frequency dynamic of the converter is determined by the PLL, and the PLL dynamic equation is expressed as:

[0017]

[0018] Where PI PLL (s)=k ppll +k ipll / s is the transfer function of the phase-locked loop PI controller, and Δθ and Δω are the phase angle and angular frequency disturbance of the rotating coordinate system, respectively; under steady state, the direction of v d is coincident with the d-axis of the system coordinate system, and at this time, the phase angle difference between the PCC point voltage and the power grid voltage is δ; when a small disturbance occurs in the system, v q ≠0, the dynamic process of the PLL will make the system frequency deviate from the steady state value, and the action law is described by the PLL dynamic equation; since the input of the PLL is the voltage at the PCC point, when v gdq is constant, it is related to the output current of the converter, and therefore the frequency dynamic characteristics of the system are coupled with the voltage and current;

[0019] The output port of the converter is connected to the power grid, and the voltage and current satisfy the constraint relationship, and the power grid dynamic equation is:

[0020]

[0021] Where Z g is the power grid impedance matrix, V gdq is the steady-state matrix of the power grid voltage, I dq is the steady-state matrix of the output current of the converter, L g , R g and ω are the power grid inductance, resistance and angular frequency, respectively, Δθ is the phase angle of the rotating coordinate system, and Δθ g is the disturbance of the power grid voltage phase, V gd and V gq are the steady-state values of the power grid voltage, I d and I q are the steady-state values of the output current of the converter;

[0022] The modulation voltage of GFL converter is determined by the current control loop, which is formulated in dq axis of GFL converter as:

[0023]

[0024] Where, PI cl (s)=k pc +k ic / s is the transfer function of current PI controller, the above equation is rearranged, let i dqref =0, eliminate the intermediate variable The current controller dynamic equation is obtained as:

[0025]

[0026] Further, the SISO model of single grid-connected converter connected to the power grid is obtained by simultaneously solving the PLL dynamic equation, the power grid dynamic equation and the current controller dynamic equation, taking the power grid frequency disturbance as input and the frequency measured by the PLL as output:

[0027]

[0028] Where G pll (s)=[0PI PLL (s)].

[0029] Further, step two establishes the SISO model of single grid-connected converter connected to the power grid, specifically including:

[0030] The circuit structure and control block diagram of single grid-connected converter connected to the power grid through LC filter are constructed;

[0031] The power synchronization control equation, the power grid dynamic equation and the admittance model of single grid-connected converter in its own controller coordinate system are written according to the internal control block diagram of single grid-connected converter and its circuit structure connected to the power grid;

[0032] The SISO model of single grid-connected converter connected to the power grid is obtained by simultaneously solving the power synchronization control equation, the power grid dynamic equation and the admittance model of single grid-connected converter in its own controller coordinate system.

[0033] Further, the power synchronization control equation and the voltage and current control loop equation are written according to the internal control block diagram of single grid-connected converter, and the admittance model of single grid-connected converter in its own controller coordinate system is obtained according to the circuit structure of single grid-connected converter connected to the power grid, specifically including:

[0034] The output frequency dynamic of grid-connected converter is determined by PSC, and the power synchronization control equation is represented as:

[0035]

[0036] where, Δp = p ref p is the disturbance of the converter output power, J and D p are the inertia coefficient and the damping coefficient, respectively;

[0037] The AC line is merged into the output impedance of the GFM, and the port constraint is the same as when a single GFL is connected to the grid. The modulation voltage of the converter is determined by the voltage and current control loop, which is formulated as:

[0038]

[0039] where, PI vl (s) and PI cl (s) are the transfer functions of the voltage and current PI controllers, respectively. After rearranging the above equation and setting v dqref = 0, the intermediate variable is eliminated, and the admittance model of the grid-connected converter in its own controller coordinate system is obtained:

[0040]

[0041] Further, by simultaneously solving the power synchronization control equation, the grid dynamic equation, and the admittance model of the grid-connected converter in its own controller coordinate system, the SISO model of a single grid-connected converter connected to the grid is obtained as follows:

[0042]

[0043] where:

[0044]

[0045] Further, the step three of establishing the dynamic interaction model of the parallel grid-connected converter and the grid-connected converter includes:

[0046] Build a general structure of a single GFL connected in parallel with a GFM;

[0047] According to the general structure of a single GFL connected in parallel with a GFM, and using matrix transformation, write the circuit equation of the system in the GFM steady-state coordinate system and the impedance equation of the converter. By simultaneously solving the circuit equation and the impedance equation of the converter, the first characteristic equation set is obtained, which reflects the disturbance of the grid and the power angle of the two converters to the voltage at the PCC point.

[0048] The column writes the second representation equation group that the voltage disturbance and the current disturbance of the two converters respectively enter the synchronization link to influence the power angle of the converter, and the first representation equation and the second representation equation are combined to obtain the SISO model suitable for the multi-machine interaction characteristic with the grid frequency disturbance as the input.

[0049] Further, the first representation equation group that the disturbance of the grid and the power angle of the two converters is transmitted to the PCC point voltage is obtained by combining the circuit equation and the impedance equation of the converter.

[0050] The column writes the second representation equation group that the voltage disturbance and the current disturbance of the two converters respectively enter the synchronization link to influence the power angle of the converter, and the first representation equation and the second representation equation are combined to obtain the SISO model suitable for the multi-machine interaction characteristic with the grid frequency disturbance as the input, specifically including:

[0051] The transformation between the steady-state coordinate systems is performed by using the transformation matrix T, and the circuit equation of the system and the impedance equation of the converter are represented as:

[0052]

[0053] Wherein, the GFL controller coordinate system is referred to as c1 system, the GFM controller coordinate system is referred to as c2 system, Δθ1 and Δθ2 are the disturbance of the phase angle output by the PLL and the PSC respectively, V dq1 and V dq2 represent the steady-state values of the PCC point voltage in the c1 and c2 systems respectively, I dq1 and I dq2 represent the steady-state values of the current of the two converters in their own coordinate systems, and θ 12 is the phase angle of the GFL coordinate system leading the GFM coordinate system in the steady state, and the transformation matrix T is represented as:

[0054]

[0055] In the GFM controller coordinate system, the corresponding equation is written to obtain a dual expression, θ1 is ignored, and the first representation equation group that reflects the mechanism of transmitting the disturbance of the grid and the power angle of the two converters to the PCC point voltage is obtained by combining the circuit equation of the system and the impedance equation of the converter.

[0056]

[0057] In two different controller coordinate systems, the voltage and current disturbances enter the respective synchronization links to generate the power angle of the two converters, form a frequency closed loop, and obtain the second representation equation group that the voltage and current disturbances influence the power angle of the converter.

[0058]

[0059] Solve the first and second representation equation group simultaneously, eliminate And let:

[0060]

[0061] Get the SISO model suitable for multi-machine interaction characteristics with power grid frequency disturbance as input:

[0062]

[0063] The present application has the following beneficial effects:

[0064] 1. The present application takes the power grid frequency disturbance as the input variable, which can completely describe the frequency response dynamics of the converter. The model solves the problem of difficult analysis of MIMO system, converts two parallel converters into two interacting SISO systems, and the synchronization link appears only once in the open-loop transfer function of the system, which is helpful for loop shaping of PLL and power synchronization loop (PSC).

[0065] 2. The model is used to analyze the two-machine interaction mechanism of the parallel GFL and GFM system, which includes two aspects of interaction: the support effect of GFM with different capacity and position on GFL frequency and voltage, and the influence of GFL on the stability of GFM. Further, the key controller parameters under different grid strengths are evaluated, which makes the interaction mechanism of parallel GFM and GFL system clear, and provides reference value for the design of controller parameters. BRIEF DESCRIPTION OF DRAWINGS

[0066] Figure 1 is the circuit structure and control block diagram of GFL converter;

[0067] Figure 2 is the relationship of converter coordinate system under steady state and small disturbance;

[0068] Figure 3 is the SISO model of GFL converter based on frequency dynamics;

[0069] Figure 4 is the circuit structure and control block diagram of GFM converter;

[0070] Figure 5 is the SISO model of GFM converter based on frequency dynamics;

[0071] Figure 6 is the general structure of single GFL parallel GFM grid connection;

[0072] Figure 7 is the multi-machine interaction model based on frequency dynamics of the present application;

[0073] Figure 8 is the comparison and verification result of the model and time domain simulation of the present application;

[0074] Figure 9 is the GFL frequency dynamic model considering GFM access;

[0075] Figure 10 is the PLL output characteristic under different grid strengths, wherein (a) is the Nyquist criterion of the model, and (b) is the Bode plot of the model;

[0076] Figure 11 is the comparison of the system dynamic characteristics before and after approximation;

[0077] Figure 12 is the comparison of the PLL output characteristics before and after GFM access, wherein (a) is the frequency domain characteristic, and (b) is the time domain characteristic;

[0078] Figure 13 is the comparison of the PLL output characteristics after access of different GFM capacities, wherein (a) is the Nyquist criterion of the PLL, and (b) is the Bode plot of the model;

[0079] Figure 14 is the PLL output time domain waveform after access of different GFM capacities;

[0080] Figure 15 is the output characteristic under different electrical distances and PLL bandwidths (SCR = 1.5), wherein (a) is under the condition of different electrical distances, and (b) is under the condition of different PLL bandwidths;

[0081] Figure 16 is the output characteristic of the PLL under different electrical distances (SCR = 5), wherein (a) is the Nyquist criterion of the PLL, and (b) is the Bode plot of the PLL system;

[0082] Figure 17 is the comparison of the PSC output characteristics before and after GFL access, wherein (a) is the frequency domain characteristic, and (b) is the time domain characteristic;

[0083] Figure 18 is the output characteristic diagram of the PSC under different grid strengths, wherein (a) is the Nyquist criterion diagram thereof, and (b) is the Bode plot. DETAILED DESCRIPTION

[0084] In order to make the objectives, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and do not limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0085] The embodiment of the present application proposes a SISO model establishment method suitable for multi-machine interaction characteristic analysis, which takes the power grid frequency disturbance as an input variable and completely describes the frequency response dynamics of the converter. The model solves the problem of difficult analysis of MIMO systems, and finally converts two parallel converters into two interacting SISO systems, and the synchronization link only appears once in the open-loop transfer function of the system, which is helpful for loop shaping of the PLL and the power synchronization loop (PSC). Using the model, the interaction mechanism analysis is carried out on two different types of converters, GFL and GFM, which specifically includes two aspects of interaction: the support effect of GFM access with different capacity and position on the frequency and voltage of GFL, and the influence of GFL access on the stability of GFM. Further, the key controller parameters under different grid strengths are evaluated, and the simulation results prove the effectiveness of the proposed modeling method and analysis.

[0086] The method specifically comprises the following steps:

[0087] Step one: establish a SISO model of a single GFL (grid-following) converter connected to the power grid. Figure 1 For the dq coordinate system, the circuit structure and control block diagram of the three-phase power converter connected to the AC power grid through the LCL filter are as follows: let e dq =[e d e q ] T be the converter bridge arm output potential vector, the PCC point voltage v dq =[v d v q ] T , the output current i dq =[i d i q ] T , the parameters of the LCL filter are L f1 ,L f2 ,C f , the power grid is simulated by an ideal voltage source in series with an inductance L g , and the grid voltage vector is represented as v gdq =[v gd v gq ] TThe converter uses an SRF-PLL for grid synchronization, with the reference value for the current control loop derived from the upstream active and reactive power control loops. Existing research indicates that the main factor affecting the stability of the GFL converter under weak grid conditions is the coupling between the PLL and the current control loop. Changes in the PCC voltage propagate through the PLL, severely impacting the converter current, and the resulting self-synchronization loop significantly reduces the synchronization stability of the GFL converter. To simplify the model, this study ignores the power control loop, i.e., constant current command control; simultaneously, since the high-frequency current component flowing through the filter capacitor has little impact on synchronization stability, the capacitor branch can be disconnected, and the inverter-side filter inductor L... f1 With the grid-side filter inductor L f2 After merging, use L f express.

[0088] The output frequency of the converter is dynamically determined by the PLL, as follows:

[0089]

[0090] Where PI PLL (s)=k ppll +k ipll / s represents the transfer function of the phase-locked loop PI controller, and Δθ and Δω represent the phase angle and angular frequency disturbance in the rotating coordinate system, respectively. The superscript 'c' indicates the physical quantity in the controller coordinate system, and '~' indicates the disturbance. For simplicity, the '~' symbol will be omitted hereafter, and lowercase letters will always represent disturbances.

[0091] Figure 2 This is the phasor diagram of a power converter. In steady state, v d The direction coincides with the d-axis of the system coordinate system. Figure 2 d s (axis), at this time the phase angle difference between the voltage at point PCC and the grid voltage is δ. When a small disturbance occurs in the system, causing v q When the voltage is not equal to 0, the dynamic process of the PLL will cause the system frequency to deviate from the steady-state value. Its operating law is described by equation (1). Since the input of the PLL is the voltage at point PCC, when v gdq When constant, it is related to the output current of the converter, so the frequency dynamic characteristics of the system are coupled with the voltage and current.

[0092] The converter's output port is connected to the power grid, and the voltage and current satisfy the following constraints:

[0093]

[0094] Z g Let Δθ be the power grid impedance matrix. g V represents the disturbance in the phase of the grid voltage. gd and Vgq is the steady state value of the grid voltage, I d and I q is the steady state value of the converter output current.

[0095] The modulation voltage of the converter (i.e. the reference input for generating the PWM signal) is determined by the current control loop, which can be formulated in the dq-axis of the converter as:

[0096]

[0097] where PI cl (s) = k pc + k ic / s is the transfer function of the current PI controller, rearranging equation (3) and setting i dqref = 0, the intermediate variable is eliminated, the admittance model of the converter in its own controller coordinate system is obtained:

[0098]

[0099] Equations (1), (2) and (4) describe the PLL dynamics, grid dynamics and current controller dynamics respectively. By combining equations (1), (2), (4) with the grid frequency disturbance as input and the PLL measured frequency as output, a SISO model integrating the whole system dynamics is obtained:

[0100]

[0101] where G pll (s) = [0 PI PLL (s)].

[0102] Figure 3 The above transfer function model is described. Equation (5) describes how the converter output frequency tracks the grid frequency under the action of the PLL and contributes to the synchronization process. Compared to the traditional modeling method focusing on the output voltage and current of the converter, this model investigates the output frequency characteristics of the converter. Since the components of the two-axis system do not need to be considered, this modeling method avoids the complex analysis of MIMO models. The instability of the converter synchronization link PLL and PSC is due to the instability of internal states such as axis rotor angle, phase-locked angle, etc., which in turn excites the instability of the output voltage / current phase, which belongs to the category of synchronization stability. This model fully captures the grid synchronization dynamics of the converter, so it is very suitable for the study of synchronization stability.

[0103] Although the control objectives and synchronization methods are different, the method proposed in the last section is still applicable to the modeling of GFM type converters.

[0104] Step two: Establish the SISO model of single GFM (network type) converter accessing the grid. Figure 4 is the circuit structure and control block diagram of single GFM accessing the grid through LC filter, L f and L c are filter inductance and line inductance respectively. The converter uses power synchronous control (PSC) to realize synchronization with the grid, and alternating current voltage control (AVC) generates the d-axis reference value of alternating current voltage, which is sent to the voltage and current double closed loop controller in the rear stage to generate the modulation voltage, and the voltage controller controls the voltage v Cdq on the capacitor. In contrast to GFL, GFM shows better stability under weak grid, but under strong grid, due to the coupling between PSC loop and voltage loop, the power angle disturbance of the grid is fed forward through the voltage and current loop, which reduces the synchronization stability of the system. Since the reactive power regulation part has little effect on the synchronization stability of GFM, the AVC loop is omitted in the present application, a constant alternating current voltage control is adopted, and the capacitor branch is still disconnected, and the model is established in the controller coordinate system of GFM.

[0105] The output frequency dynamic of the converter is determined by PSC, which is expressed as:

[0106]

[0107] where Δp=p ref -p is the disturbance of the output power of the converter, J and D p are inertia coefficient and damping coefficient respectively. Similar to GFL, when the power of the converter appears disturbance, the dynamic process of PSC will make the system frequency deviate from the steady state value, and the action law is described by formula (6).

[0108] The alternating current line is combined into the output impedance of GFM, and the port constraint is still formula (2), the modulation voltage of the converter is determined by the voltage and current control loop, which is formalized as:

[0109]

[0110] where PI vl (s) and PI cl (s) are the transfer functions of voltage and current PI controllers respectively, formula is arranged, v dqref =0 is set, and intermediate variables are eliminated, to obtain the admittance model of the converter under its own controller coordinate system:

[0111]

[0112] The SISO model integrating the dynamics of the whole system is obtained by combining formulas (2), (6) and (8):

[0113]

[0114] wherein:

[0115]

[0116] Figure 5 is the SISO model of GFM converter based on frequency dynamics. Equation (9) has the same structure as equation (5). The current controller of GFL corresponds to the voltage controller of GFM, and the PLL loop of GFL corresponds to the PSC loop of GFM. Both equations describe the response of a single converter to grid frequency, and separate the grid dynamics from the dynamics of the synchronization link, which helps to shape the loop.

[0117] Step three: Establish the dynamic interaction model of parallel GFL and GFM converters. Now consider the case of a GFL and a GFM connected in parallel to the grid. Figure 6 is the generalized grid structure of a single GFL connected in parallel with a GFM. Zc is the line impedance between the GFM outlet and the PCC. The superscripts c1 and c2 represent the GFL controller coordinate system and the GFM controller coordinate system, respectively. The subscript 1 represents the physical quantities of the GFL, and the subscript 2 represents the physical quantities of the GFM.

[0118] Due to the existence of line impedance, the steady-state coordinate systems of the two converters do not coincide. Here, the transformation matrix T is used to transform between the steady-state coordinate systems. The circuit equation of the system and the impedance equation of the converter are represented as:

[0119]

[0120] where Δθ1 and Δθ2 are the disturbances of the PLL and PSC output phase angles, respectively. V dq1 and V dq2 represent the steady-state values of the PCC point voltage in the c1 and c2 systems, respectively. The grid voltage is the same. I dq1 and I dq2 represent the steady-state values of the currents of the two converters in their own coordinate systems. Let θ 12 be the phase angle of the GFL coordinate system leading the GFM coordinate system in the steady state. The transformation matrix T is represented as:

[0121]

[0122] The description of equation (11) is done in the c2 coordinate system. Similarly, the corresponding equation can be written in the c1 coordinate system to obtain a dual expression. Note that θ 12 is generally small, so it is ignored. The first characteristic equation set is obtained by combining each equation in equation (11):

[0123]

[0124] The equation (13) describes how the disturbance of the grid and the power angle of the two converters is transferred to the voltage of the PCC point. In the two different controller coordinate systems, the disturbance of the voltage and current enters the respective synchronous loop, and generates the power angle of the two converters, forming a frequency closed loop, and obtaining the second characterization equation of the voltage and current disturbance affecting the power angle of the converter:

[0125]

[0126] The equations (13) and (14) are combined, and and

[0127]

[0128] The SISO model suitable for multi-machine interaction characteristics is obtained with the grid frequency disturbance as input:

[0129]

[0130] Figure 7 The above equation is a transfer function model with the grid frequency disturbance as input. In the model, the two converters take the phase Δθ1, Δθ2 of the respective coordinate system as output, and show different dynamic responses to the frequency disturbance of the grid, and there is a coupling relationship between them. The model well describes the synchronization process. Since the voltage and current of the converter are only intermediate variables, from the input and output ports, the model can be described by two SISO transfer functions, so as to avoid the difficulty of MIMO system analysis.

[0131] Figure 8 For the comparison results of the frequency domain model and the time domain simulation, a step disturbance of 1 rad / s is applied to the grid frequency in Simulink, and the step response of the above model is consistent with the result of the electromagnetic transient simulation, verifying the correctness of the modeling.

[0132] In equation (15), the steady-state value of the low-frequency current is small in order of magnitude and can be ignored, and it is approximately considered that V dq = V gdq , combined with equation (4) and equation (8), without considering the line impedance, Y 11 , Y 12 and Y 21 have the following approximate relationship:

[0133]

[0134] Where S represents the capacity of the converter.

[0135] Figure 9is the GFL converter SISO model considering GFM access. Based on the above analysis, the dynamic interaction between GFM and GFL under different grid strength, electrical distance and capacity is analyzed. In equation (16), Δθ 1- Δθ g , Δθ 2- Δθ g is the main element, Δθ g is eliminated to obtain:

[0136]

[0137] Further, we get:

[0138]

[0139] Let the transfer function in the above figure be G mf (s). At this point, the GFM branch can be converted into the frequency loop of the GFL, and the GFL converter SISO model considering GFM access is obtained. The red part represents the influence of GFM on the frequency dynamics of GFL. The open-loop transfer function is shown in equation (20):

[0140]

[0141] Compared with GFL alone, the interaction between GFM and GFL is reflected in two aspects. One part is due to the access of physical impedance Y IV2 (s), which is related to the circuit parameters of GFM and the parameters of voltage and current controllers, and affects the value of the first term Y 11 in the above equation. The parallel access of GFM is represented as the addition of the admittance of the two. The other part is due to the difference in the synchronization link. There is no such term for converters with the same synchronization mode in parallel. The inertia and damping characteristics of GFM are reflected in this term.

[0142] Now we will consider the influence of grid strength, GFM capacity and electrical distance on the support characteristics of GFM by equation (20).

[0143] 1) Under different grid strengths

[0144] When the system short-circuit ratio SCR changes between 1.4 and 4.4, the frequency of the PLL output is considered under the following conditions: GFL alone access, and GFL parallel access of 20% capacity GFM. Figure 10 (a) describes the Nyquist diagram of the parallel system, which is used to determine the stability of the system. When the grid strength increases, the Nyquist curve gradually approaches the critical point (-1, 0j), and the system begins to lose stability. Figure 10(b) The frequency support effect of GFM to GFL under different grid strength is given. After GFM is connected, the characteristics of the interconnected system in low frequency band are changed significantly, and the instability problem under weak grid is improved obviously. But with the further increase of grid strength, the stability margin of the system is reduced, until SCR=4.4, the phase-frequency characteristics of the system cross the-180° phase line at about 6Hz and enter instability. The oscillation in this frequency band is caused by GFM under strong grid. Therefore, when the grid is strong, the damping coefficient of GFM should be increased properly to improve the stability of the interconnected system.

[0145] It is noted that the bandwidth of PSC is much smaller than that of PLL, and if the line impedance is not considered, there is σ(Y IV2 ) >> σ(Y IV1 ) (σ represents singular value of matrix) psc Therefore, the amplitude of G 22 Y 21 in equation (20) is much smaller than that of G pll Y 12 , and it is neglected to obtain:

[0146]

[0147] Further, in low frequency band, the amplitude of G pll Y 12 >>1, and the second term in the denominator is neglected; in high frequency band, the amplitude of G psc Y 11 decays rapidly, and the first term in the numerator is neglected. Therefore, the final open-loop transfer function can be approximated as:

[0148]

[0149] where ω1 and ω2 are certain segment point frequencies, which are about 10-20Hz in this example.

[0150] Figure 11 are the Bode plots of the approximate open-loop transfer functions before and after, under weak grid, equation (22) can describe the frequency characteristics of the system more accurately. From this we can see that after GFM is connected, the effect of PLL in low frequency band will be ignored, and only in high frequency band. The low frequency characteristics of the system will be determined by the PSC loop. Comparing the low frequency characteristics of GFM connected alone in equation (23), which is the dominant part in low frequency band in equation (22).

[0151] G GFM (s) = G psc (E-Z g Y IV2 ) -1 V gdq ≈ G psc Y 11(23)

[0152] Figure 12 In case (a), the GFM is connected to the PCC point of the GFL, and the GFL output frequency is shown to vary under weak grid conditions. The system short circuit ratio (SCR) is about 1.5 in this case. Before the GFM is connected, the phase margin PM of the Bode plot is about 0, Figure 12 In case (b), large oscillations appear in the time domain waveform, while after the GFM is connected, the system characteristics in the low frequency band are significantly changed. Due to the bandwidth limitation of the PSC loop, the frequency response speed of the GFL is slowed down, and the phase in the low frequency band starts to decrease from -90 deg, thus leaving a phase margin of about 60 deg around the crossover frequency. The overshoot in the time domain waveform (d) disappears, and the oscillation is significantly suppressed.

[0153] 2) Different GFM capacities

[0154] Under certain grid strength, the GFM needs to support the GFL, and its capacity has a lower limit. For example, Figure 12 As shown in case (a), when SCR = 1.4, as the GFM capacity decreases, the Nyquist curve gradually approaches the (-1, 0j) system stability margin. Figure 12 Case (b) describes the changes in the GFL frequency response Bode plot with the GFM capacity. The GFM has a significant impact on the low frequency band, and the connection of a GFM with larger capacity reduces the system crossover frequency and slows down the response, while the stability margin is also improved. At this time, the system oscillation mode is also below 10 Hz, indicating that the oscillation is caused by the GFM loop. Because the grid strength under the GFM angle is gradually increasing as the capacity decreases. Figure 14 The frequency waveform of the PLL output after the connection of GFM with different capacities is given.

[0155] As can be seen, even the connection of a GFM with only 0.2 p.u. capacity can significantly improve system stability, because its voltage source control method makes it have a large small signal admittance in the low frequency band (see equation (8)). At the same time, we find that a GFM with too small capacity will affect the entire system due to its own loop instability, at which time the bandwidth of the PSC loop needs to be reduced to increase the stability margin.

[0156] 3) Different electrical distances

[0157] The electrical distance is represented by the size of the line impedance of the GFM connected to the PCC point. To investigate its impact on the stability of the parallel system, two cases of SCR = 1.5 and SCR = 5 are taken. Figure 15(a) Bode plot of the PLL output frequency when the line impedance is changed for SCR = 1.5. The results show that when the electrical distance between GFM and GFL is too far, the existence of line impedance weakens the voltage source characteristic of GFM, which is not enough to support the stable operation of GFL. The change of line impedance is equivalent to changing the small signal admittance of GFM port, which is described in equation (24), and its influence is mainly concentrated in the middle and high frequency band. Figure 15 (b) The output frequency characteristics of the PLL with different bandwidths when the line impedance is fixed at 0.3 p.u. Reducing the bandwidth of the PLL can improve the stability margin of GFL, and the frequency of oscillation also decreases. At the same time, since GFM is closer to the voltage source characteristic in the low frequency band, when the electrical distance between GFM and GFL is far, the bandwidth of the PLL should be appropriately reduced to improve stability.

[0158] Figure 16 The influence of line impedance on system characteristics under strong grid (SCR = 5) is presented. The existence of GFM connecting line can enhance its own stability, and too small line impedance may cause the instability of GFM loop, as shown in Figure 16 (a). From Figure 16 (b), it can be seen that the larger the line impedance, the higher the stability margin of the system, and the slower the system response speed. In order to improve the stability of GFM under strong grid, virtual resistance (VR) control is usually introduced to simulate the effect of line resistance.

[0159] Under weak grid, the access of GFM can significantly enhance the synchronization stability of GFL, and under strong grid, it is also necessary to consider how the access of GFL affects the stability of GFM. The original model is dual processed, and the influence of GFL on GFM is converted to the GFM branch, and the open-loop transfer function with GFM frequency as the output variable is obtained:

[0160]

[0161] After similar approximation, we get the output frequency characteristics of GFM in the low frequency band:

[0162]

[0163] where the small amount of G psc Y 21 +G pll Y 22 -1 in the denominator and the small amount of G psc Y 11 in the numerator are ignored.

[0164] It can be seen from the comparison between formula (25) and formula (22) that both of them have almost the same characteristics in the low frequency band, which further explains the frequency support effect of the GFM access on the whole system.

[0165] Figure 17 The change of the GFM output frequency before and after the GFL access under a strong power grid is described, and the system short circuit ratio (SCR) at this time is about 5. Figure 17 (a) It is shown that the access of the GFL almost does not affect the response speed of the GFM (fast response in parallel with slow response, which is slow as a whole), but reduces the phase of the GFM in the low frequency band, and reduces the stability margin. Figure 17 (b) In the time domain, it is embodied that the access of the GFL enhances the oscillation of the GFM under a strong grid. Therefore, the access of the GFL actually reduces the stability of the GFM.

[0166] Figure 18 The output frequency characteristics of the PSC under different grid strengths are investigated. Since the frequency characteristics of the system do not change much before and after the GFL access, the same as a single GFM, the instability phenomenon occurs under a strong grid. Figure 10 With Figure 18 The critical stable grid strength in the formula coincides with the grid strength when the SCR is 4.4, and the oscillation frequency of about 6 Hz occurs.

[0167] In summary, the application provides a SISO model suitable for multi-machine interaction characteristic analysis. The model converts two parallel converters into two interacting SISO systems for the problem of MIMO system analysis, and the synchronization link only appears once in the open loop transfer function of the system, which is helpful for the loop shaping of the PLL and the power synchronization loop (PSC). By using the model, the interaction mechanism analysis of two different types of converters, GFL and GFM, is carried out, which specifically includes two aspects of interaction: the frequency and voltage support effect of the GFM access of different capacities and positions on the GFL, and the influence of the GFL access on the stability of the GFM. Further, the key controller parameters under different grid strengths are evaluated.

[0168] Compared with the traditional model, the application can reveal the internal mechanism better, and solves the problem of difficult analysis of the MIMO system. By using the model, the frequency support characteristics of the grid energy storage on the GFL under different grid strengths, different GFM capacities and different electrical distances, and the influence of the GFL access on the stability of the GFM are analyzed, and the following conclusions are obtained:

[0169] 1. When the GFM and the GFL are close in electrical distance, the instability problem of the GFL under a weak grid is significantly improved. However, when the grid is too strong or the GFM capacity is too small, the GFM itself will lead to system instability, and at this time, the damping coefficient of the GFM needs to be increased to improve the stability.

[0170] 2. When the electrical distance is too far, GFM cannot effectively support the frequency and voltage of GFL, GFL generates oscillation under weak power grid, at this time, the bandwidth of GFM voltage loop needs to be increased, the voltage source characteristics of GFM needs to be enhanced, or the bandwidth of PLL needs to be appropriately reduced, the coupling between PLL and current loop needs to be weakened.

[0171] 3. The access of GFL will enhance the coupling of GFM internal loop, destroy the stability, therefore, when designing the parameters of GFM, a certain stability margin needs to be reserved.

[0172] Although the preferred embodiments of the application have been described, those skilled in the art will be able to make additional changes and modifications to these embodiments once they have the basic inventive concept. Therefore, the appended claims are intended to be interpreted as including all the preferred embodiments and all the changes and modifications falling within the scope of the application.

[0173] Obviously, those skilled in the art can make various modifications and variations to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application fall within the scope of the claims of the present application and their equivalent technologies, the present application also intends to include these modifications and variations.

Claims

1. A method for establishing a SISO model suitable for multi-machine interaction characteristic analysis, characterized in that, The method comprises the following steps: Step one, establishing a SISO model of a single grid-following converter connected to a power grid; Step two, establishing a SISO model of a single grid-forming converter connected to a power grid; Step three, coupling the SISO models of the single grid-following converter and the single grid-forming converter obtained in steps one and two into the same steady-state coordinate system by matrix transformation, and then establishing a dynamic interaction model of the parallel grid-following converter and the grid-forming converter, to obtain a SISO model suitable for multi-machine interaction characteristic analysis with the power grid frequency disturbance as the input; Step one of establishing a SISO model of a single grid-following converter connected to a power grid comprises the following steps: constructing a circuit structure and a control block diagram of the single grid-following converter connected to an AC power grid through an LCL filter; writing PLL dynamic equations, power grid dynamic equations and current controller dynamic equations according to the circuit structure and the control block diagram of the single grid-following converter connected to the power grid; obtaining a SISO model of the single grid-following converter connected to the power grid by simultaneously solving the PLL dynamic equations, the power grid dynamic equations and the current controller dynamic equations, with the power grid frequency disturbance as the input and the frequency measured by the PLL as the output; the writing of the PLL dynamic equations, the power grid dynamic equations and the current controller dynamic equations according to the control block diagram of the single grid-following converter comprises the following steps: the output frequency dynamic of the converter is determined by the PLL, and the PLL dynamic equation is represented as: ； Where PI PLL (s)=k ppll + k ipll / s is the transfer function of the phase-locked loop PI controller, Δθ, Δ These represent the phase angle and angular frequency perturbation of the rotating coordinate system, respectively; in steady state, v d The direction coincides with the d-axis of the system coordinate system, and at this time, the phase angle difference between the voltage at point PCC and the grid voltage is... When a small disturbance occurs in the system, causing v q When the voltage is not equal to 0, the dynamic process of the PLL will cause the system frequency to deviate from the steady-state value. Its effect is described by the PLL dynamic equation. Since the input of the PLL is the voltage at the PCC point, at v... gdq When constant, it is related to the output current of the converter, therefore the frequency dynamic characteristics of the system are coupled with the voltage and current; the output port of the converter is connected to the power grid, and the voltage and current satisfy a constraint relationship, and the power grid dynamic equation is: ； where Z g is the grid impedance matrix, V gdq is the grid voltage steady-state matrix, I dq is the converter output current steady-state matrix, L g , R g , are the grid inductance, resistance and angular frequency, respectively, Δθ is the phase angle of the rotating coordinate system, Δθ g is the disturbance of the grid voltage phase, V gd and V gq are the steady-state values of the grid voltage, I d and I q are the steady-state values of the converter output current; the modulation voltage of the GFL converter is determined by the current control loop, and the current control loop is formulated in the dq axis of the GFL converter as: ； where, PI cl (s)=k pc + k ic / s is the transfer function of the current PI controller, rearranging the above equation, let i dqref = 0, eliminate the intermediate variable e , the current controller dynamic equation is: 。 2. The SISO model establishing method suitable for multi-machine interaction characteristic analysis according to claim 1, characterized in that: the SISO model of the single grid-following converter connected to the power grid obtained by simultaneously solving the PLL dynamic equations, the power grid dynamic equations and the current controller dynamic equations with the power grid frequency disturbance as the input and the frequency measured by the PLL as the output is: ； where G pll (s) = [0 PI PLL (s)].

3. The SISO model establishing method for multi-machine interaction characteristic analysis according to claim 1, characterized in that, Step two of establishing a SISO model of a single grid-forming converter connected to a power grid comprises the following steps: constructing a circuit structure and a control block diagram of the single grid-forming converter connected to the power grid through an LC filter; writing power synchronization control equations, power grid dynamic equations and an admittance model of the single grid-forming converter in the controller coordinate system according to the control block diagram inside the single grid-forming converter and the circuit structure of the single grid-forming converter connected to the power grid; obtaining a SISO model of the single grid-forming converter connected to the power grid by simultaneously solving the power synchronization control equations, the power grid dynamic equations and the admittance model of the single grid-forming converter in the controller coordinate system.

4. The SISO model establishing method for multi-machine interaction characteristic analysis according to claim 3, characterized in that, the writing of the power synchronization control equations and the voltage and current control loop equations according to the control block diagram inside the single grid-forming converter, and the obtaining of the admittance model of the grid-forming converter in the controller coordinate system according to the circuit structure of the single grid-forming converter connected to the power grid comprise the following steps: the output frequency dynamic of the grid-forming converter is determined by the PSC, and the power synchronization control equation is represented as: ； where Δp = p ref - p is the perturbation of the inverter output power, J and D p are the inertia and damping coefficients, respectively; the AC line is merged into the output impedance of the GFM, the port constraint is the same as when the single GFL is connected to the grid, the modulation voltage of the converter is determined by the voltage and current control loop, and the voltage and current control loop equation is formulated as: ； where, PI vl (s), PI cl (s) are the transfer functions of the voltage, current PI controllers, respectively. The above equation is rearranged by letting v dqref = 0 and eliminating the intermediate variable, resulting in the admittance model of the grid-forming converter in the controller coordinate system: 。 5. The SISO model establishing method for multi-machine interaction characteristic analysis according to claim 4, characterized in that, By simultaneously solving the power synchronization control equation, the grid dynamic equation and the admittance model of the grid-forming converter in its own controller coordinate system, a SISO model of a single grid-forming converter connected to the grid is obtained as follows: ； Wherein: 。 6. The SISO model establishing method for multi-machine interaction characteristic analysis according to claim 1, wherein, The step three of establishing the dynamic interaction model of the parallel grid-following converter and the grid-forming converter, specifically includes: Build a single GFL parallel GFM grid-connected general structure; According to the single GFL parallel GFM grid-connected general structure and using matrix transformation, write the circuit equation and the impedance equation of the converter in the GFM steady-state coordinate system, and by simultaneously solving the circuit equation and the impedance equation of the converter, obtain the first characteristic equation set reflecting the disturbance quantity of the grid and the power angle of the two converters being transmitted to the PCC point voltage; Write the second characteristic equation set reflecting the voltage disturbance quantity and the current disturbance quantity of the two converters entering the synchronization link to affect the power angle of the converter, and by simultaneously solving the first characteristic equation and the second characteristic equation, obtain the SISO model suitable for the multi-machine interaction characteristics with the grid frequency disturbance as the input.

7. The method for SISO model development for multi-machine interaction characteristic analysis according to claim 6, wherein, According to the single GFL parallel GFM grid-connected general structure and using matrix transformation, write the circuit equation and the impedance equation of the converter in the GFM steady-state coordinate system, and by simultaneously solving the circuit equation and the impedance equation of the converter, obtain the first characteristic equation set reflecting the disturbance quantity of the grid and the power angle of the two converters being transmitted to the PCC point voltage; Write the second characteristic equation set reflecting the voltage disturbance quantity and the current disturbance quantity of the two converters entering the synchronization link to affect the power angle of the converter, and by simultaneously solving the first characteristic equation and the second characteristic equation, obtain the SISO model suitable for the multi-machine interaction characteristics with the grid frequency disturbance as the input, specifically including: Use the transformation matrix T to perform the steady-state coordinate system transformation, and the circuit equation and the impedance equation of the converter are expressed as: ； Where, GFL controller coordinate system is referred to as c1 system, GFM controller coordinate system is referred to as c2 system, and Δθ1 and Δθ2 are disturbance of PLL and PSC output phase angle respectively, V dq1 and V dq2 represent steady-state values of PCC point voltage in c1, c2 system respectively, I dq1 and I dq2 represent steady-state values of current of two converters in their own coordinate system, let be the phase angle of GFL coordinate system leading GFM coordinate system in steady state, and the transformation matrix T is represented as: ； In the GFM controller coordinate system, write the corresponding equation to obtain a dual expression, ignore θ1, and by simultaneously solving the circuit equation and the impedance equation of the system, obtain the first characteristic equation set reflecting the action mechanism of the disturbance quantity of the grid and the power angle of the two converters being transmitted to the PCC point voltage: ； In the two different controller coordinate systems, the voltage and current disturbance quantities also enter the respective synchronization links to generate the power angle of the two converters, form a frequency closed loop, and obtain the second characteristic equation set reflecting the influence of the voltage and current disturbance on the power angle of the converter: ； Solving the first set of characterization equations and the second set of characterization equations simultaneously, eliminating , and letting: ； Obtain the SISO model suitable for the multi-machine interaction characteristics with the grid frequency disturbance as the input: 。

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