A low-frequency stability reduced-order modeling analysis method for weakly interconnected DC systems

Abstract

The application provides a weakly interconnected DC system low-frequency stability order reduction modeling analysis method, including the following steps: ignoring the current inner loop control dynamics and the electromagnetic transient process of the filter circuit, considering the influence of the DC voltage control link, the reactive power control link and the phase-locked loop dynamics under the weak network characteristics, establishing a multi-branch transfer function model of the DC voltage control unit; through simplification and order reduction processing on the multi-branch equivalent transfer function model, an equivalent order reduction circuit model of the DC voltage control unit connected to the weak network is established; under the DC voltage control time scale, an equivalent circuit model of the constant power control unit is established; combining the DC voltage control unit, the constant power control unit and the DC network characteristics, a full-system order reduction model of the weakly interconnected DC system is constructed; a second-order characteristic equation of the weakly interconnected DC system is obtained, and when the weakly interconnected DC system operates in an under-damped working condition, the system damping coefficient and the oscillation frequency are solved.

Description

Technical Field

[0001] This invention belongs to the field of DC system stability modeling and analysis, specifically involving a method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems, which is particularly suitable for dynamic stability analysis of DC systems under weak network characteristics. Background Technology

[0002] With the continuous development of high-proportion renewable energy integration and high-proportion power electronic equipment, DC systems have become a key research and development direction in the power sector due to their advantages such as flexible power control and efficient integration of renewable energy and energy storage devices. However, in long-distance, high-capacity power transmission scenarios, DC systems are often connected through weak AC grids, which results in a strong coupling interaction between the converter and the AC system. In a weakly interconnected environment, the short-circuit capacity of the AC system is relatively low, and the equivalent impedance increases, making it prone to low-frequency oscillations and other small disturbance stability problems, directly threatening the safe and stable operation of the DC transmission system.

[0003] Currently, the commonly used methods for addressing stability issues in such systems, both domestically and internationally, are the "impedance method" and the "state-space method," respectively. The literature [J. Lyu, J. Yin and L. Gao, "Analysis and Control of Multi-Mode Oscillations in Multi-Infeed MMC-HVDC Systems," IEEE Transactions on PowerDelivery, vol. 40, no. 2, pp. 1203-1213, April 2025.] studies multi-mode oscillation phenomena in multi-input MMC-HVDC systems based on the whole-system impedance analysis method, and introduces parameter sensitivity analysis to quantitatively analyze the influence of different parameters on the oscillation modes. The literature [H. Liu and J. Sun, "Voltage Stability and Control of Offshore Wind Farms With AC Collection and HVDC Transmission," IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 2, no. 4, pp. 1181-1189, Dec. 2014.] established an impedance model for a direct-drive wind farm connected to a flexible DC grid and used the Nyquist criterion to conduct system stability analysis. The literature [D. Lee and G. Jang, "Sub-Synchronous Oscillation Constrained Communication-Free Grid Frequency Control of Offshore Wind Farm Linked HVDC System," IEEE Transactions on Power Systems, vol. 39, no. 2, pp. 4397-4408, March 2024.] established a global small-signal state-space model for an offshore wind farm connected to a VSC-HVDC grid and used eigenvalue analysis and participation factor method to reveal the subsynchronous oscillation characteristics and key influencing factors of the system.The literature [Wang Y., Wang Z., Lei M., Qiu F. Analysis and Control of DC Voltage Dynamics Based on a Practical Reduced-Order Model of Droop-Controlled VSC-MTDC System in DC Voltage Control Timescale[J]. IEEE Transactions on PowerDelivery, 2024, 39(2): 1031-1039] constructs a reduced-order impedance model for the droop-controlled VSC-MTDC system under the DC voltage control timescale, neglecting the influence of high-frequency dynamics, and reveals the influence of control parameters on system stability. However, traditional impedance analysis and state-space analysis methods can only rely on root locus and Nyquist criterion to complete the system stability judgment, and can only complete the qualitative assessment of stability. It is difficult to support the optimization of control parameters and the improvement of stability margin in engineering scenarios, and cannot provide direct and practical guidance for the stable control of actual systems. Summary of the Invention

[0004] To address the above problems, this invention proposes a method for reduced-order modeling and analysis of low-frequency stability in weakly interconnected DC systems. The specific technical solution is as follows: A method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems includes the following steps: S1, neglecting the dynamics of the current inner loop control and the electromagnetic transient process of the filter circuit, and taking into account the influence of the DC voltage control link, reactive power control link, and phase-locked loop dynamics under weak network characteristics, a multi-branch transfer function model of the DC voltage control unit is established; S2, by simplifying and reducing the order of the multi-branch equivalent transfer function model, an equivalent reduced-order circuit model for the DC voltage control unit connected to the weak network is established. S3. Under the DC voltage control time scale, establish the equivalent circuit model of the constant power control unit; combine the characteristics of the DC voltage control unit, the constant power control unit and the DC network to construct the full-system reduced-order model of the weakly interconnected DC system. S4, the second-order characteristic equation of the weakly interconnected DC system is obtained. When the weakly interconnected DC system is operating under underdamped conditions, the system damping coefficient and oscillation frequency are obtained by solving. S5. Based on the equivalent circuit model, and combined with the damping coefficient and oscillation frequency dynamic evaluation index, the impact of weak grid access on the time-scale stability characteristics of DC voltage control of DC system is analyzed.

[0005] Furthermore, the method of S1 is as follows: Active power Δ P and phase angle increment Δ θ The increment Δ of the d-axis component of the outer loop current output of the active control type is used respectively. i d The increment Δ of the q-axis component of the outer loop current output in reactive power control type i q and the PLL output phase angle Δ θ pll The specific form is as follows: (1) in, m 1, m 2, m 3, n 1 and n The specific expression of 2 is as follows: (2) In the formula, v g0 and v t0 These represent the steady-state values ​​of the amplitudes of the infinite grid voltage and the grid connection point voltage, respectively. X g Indicates AC line inductance L g The reactance value, θ This represents the phase of the grid connection point voltage, and in steady state... θ 0= θ pll0 ; Based on the specific control strategy of the reactive power control loop, the input variable reactive power Δ Q and voltage amplitude increment Δ v t It can be represented in the following general form: (3) In the formula, Δ R This represents the general input variables for the reactive power control loop; When constant AC voltage control is used, Δ R , k 1. k 2 and k The specific form of 3 is as follows: (4) When constant reactive power control is used, Δ R , k 1. k 2 and k The specific form of 3 is as follows: (5) In summary, the DC voltage control unit can be obtained d axial AC current component Δ i d With active power Δ P The basic transfer function model between them, where... H pll (s) and H R (s) are the equivalent transfer functions for the PLL loop and the reactive power control loop, respectively, and their specific forms are as follows: (6) In the formula, G pll ( s ) represents the transfer function of a phase-locked loop (PLL) element. G vac ( s ) represents the control function of the AC voltage control loop; The basic transfer function model is broken down and reorganized to obtain its multi-branch transfer function form, which contains a total of 4 branches: the first branch T 1(s) corresponds to the inherent electrical circuit connection of the system, the second branch. T 2(s) corresponds to the dynamic influence of PLL control, the third branch T 3(s) corresponds to the dynamic influence of reactive power control components, the fourth branch. T 4(s) represents the dynamic interaction between the PLL and reactive power control components; the specific form is as follows: (7) When considering the influence of the DC voltage control loop, the transfer function of the DC voltage control loop is... G vdc ( s )and d axial AC current component Δ i d Multi-branch transfer function between active power ΔP T 1( s ), T 2( s ), T 3( s )and T 4( s Further simplification yields the equivalent transfer function of the DC voltage control unit. G vdceq ( s Multi-branch form: (8) In the formula, G i (s) indicatesG vdceq (s) i The equivalent transfer function of the branch, v dc0 This represents the steady-state value of the VSC output voltage; The DC voltage control unit for weak network access is decomposed into four branches with clear physical meanings: the DC voltage control link directly affects... G 1(s); Dynamic Influence of DC Voltage Control and PLL Interaction G 2(s); Interactive dynamic influence of DC voltage control and reactive power control components G 3(s); The combined effect of DC voltage control, reactive power control components, and PLL. G 4(s), its specific form is as follows: (9).

[0006] Furthermore, in S2, the equivalent output impedance of the equivalent reduced-order circuit model... Z s for: (10) In the formula, R s and L s Equivalent output impedance Z s The equivalent resistance and equivalent inductance, C s For VSC output capacitor, P 0 represents the steady-state value of active power.

[0007] Furthermore, in S3, the total parallel impedance of the reduced-order model of the entire system... Z sum Represented as: (11) In the formula, Z p This is the equivalent output impedance of the constant power control unit. C p Indicates port capacitance. R p This represents the equivalent resistance of the constant power control unit.

[0008] Furthermore, in S4, the second-order characteristic equation of the weakly interconnected DC system is obtained from equation (11), and its specific form is as follows: (12) When the weakly interconnected DC system operates in an underdamped condition, i.e., 0 < When <1, the system damping coefficient can be obtained by solving. and oscillation frequency ω d .

[0009] Furthermore, in S4, the system damping coefficient and oscillation frequency ω d The parsing expression is: (13).

[0010] Advantages of the present invention: (1) For the DC voltage control unit connected to the weak network, a multi-branch equivalent RLC circuit model considering DC voltage control, reactive power control and the dynamic influence of phase-locked loop is established to clarify the path of multiple control loops and their interaction affecting system stability.

[0011] (2) Combining the characteristics of constant power control unit and DC network, construct an overall RLC reduced-order model of weak network access DC system, and reveal the dynamic stability mechanism of DC voltage control time scale from the perspective of equivalent circuit.

[0012] (3) Using the proposed method, the impact of different reactive power control links on system stability was compared and analyzed intuitively. By using the equivalent resistance and equivalent inductance of each path, the influence of grid strength, DC transmission power and control parameters was quantitatively assessed, and the key branches that dominate the dynamic stability of the system were identified accordingly. Attached Figure Description

[0013] Figure 1 This is a topology diagram of a weakly interconnected DC system with a reduced-order modeling and analysis method for low-frequency stability of a weakly interconnected DC system, as described in this invention.

[0014] Figure 2 This is a schematic diagram of the VSC topology and its control strategy for a method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems, as described in this invention.

[0015] Figure 3 This invention relates to a method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems. d axial AC current component Δ i d To active power Δ P Schematic diagram of the evolution process of the inter-transfer function model (wherein, Figure 3 -a indicates the basic transfer function model. Figure 3 -b indicates a multi-branch transfer function model.

[0016] Figure 4 This is a schematic diagram illustrating the evolution of the multi-branch transfer function model of the DC voltage control unit in the low-frequency stability reduction modeling and analysis method for weakly interconnected DC systems involved in this invention (wherein, Figure 4 -a indicates the basic transfer function model. Figure 4 -b indicates a multi-branch transfer function model.

[0017] Figure 5 This is a schematic diagram of the equivalent circuit model of the DC voltage control unit for weak network access in the low-frequency stability reduction modeling and analysis method of a weak interconnected DC system involved in this invention.

[0018] Figure 6 This is a schematic diagram of the equivalent circuit model of the constant power control unit in the low-frequency stability reduction modeling and analysis method of a weakly interconnected DC system involved in this invention.

[0019] Figure 7 This is a schematic diagram of the equivalent circuit model of a weak network DC system, which is the basis for the low-frequency stability reduction modeling and analysis method of a weak interconnected DC system involved in this invention.

[0020] Figure 8 This is a schematic diagram of the topology and reduced-order model of a BTB DC system connected to a weak interconnection network, which is part of the low-frequency stability reduction modeling and analysis method for weak interconnection DC systems involved in this invention. Figure 8 -a indicates a detailed model. Figure 8 -b indicates a reduced-order equivalent circuit model.

[0021] Figure 9 The AC inductor in the AC voltage control mode of the example described in this invention L s Schematic diagram of the impact (where, Figure 9 -a indicates equivalent conductance. Figure 9 -b indicates equivalent susceptance.

[0022] Figure 10 The AC inductor in the reactive power control mode of the example described in this invention L s Schematic diagram of the impact (where, Figure 10 -a indicates equivalent conductance. Figure 10 -b indicates equivalent susceptance.

[0023] Figure 11 This is a schematic diagram comparing the damping coefficient and oscillation frequency under different control modes in the examples involved in this invention.

[0024] Figure 12 AC inductors in different control modes in the examples involved in this invention L s Schematic diagram of simulation results when changes occur (where, Figure 12 -a indicates AC inductance in AC voltage control mode. L s Simulation results during changes Figure 12-b indicates AC inductance in reactive power control mode. L s Simulation results during changes).

[0025] Figure 13 The AC voltage control proportional coefficient in the AC voltage control mode of the example involved in this invention. k puac Schematic diagram of the impact (where, Figure 13 -a indicates equivalent conductance. Figure 13 -b indicates equivalent susceptance, Figure 13 -c indicates the damping coefficient. Figure 13 -d indicates the oscillation frequency.

[0026] Figure 14 The AC voltage control proportional coefficient in the AC voltage control mode of the example involved in this invention. k puac A schematic diagram of the simulation results during the change.

[0027] Figure 15 The integral coefficient of AC voltage control in the AC voltage control mode of the example involved in this invention. k iuac Schematic diagram of the impact (where, Figure 15 -a indicates equivalent conductance. Figure 15 -b indicates equivalent susceptance, Figure 15 -c indicates the damping coefficient. Figure 15 -d indicates the oscillation frequency.

[0028] Figure 16 The integral coefficient of AC voltage control in the AC voltage control mode of the example involved in this invention. k iuac A schematic diagram of the simulation results during the change.

[0029] Figure 17 The reactive power control proportional coefficient in the reactive power control mode of the examples involved in this invention. k pQ Schematic diagram of the impact (where, Figure 17 -a indicates equivalent conductance. Figure 17 -b indicates equivalent susceptance, Figure 17 -c indicates the damping coefficient. Figure 17 -d indicates the oscillation frequency.

[0030] Figure 18 The reactive power control proportional coefficient in the reactive power control mode of the examples involved in this invention. k pQ A schematic diagram of the simulation results during the change.

[0031] Figure 19The reactive power control integral coefficient in the reactive power control mode of the example involved in this invention. k iQ Schematic diagram of the impact (where, Figure 19 -a indicates equivalent conductance. Figure 19 -b indicates equivalent susceptance, Figure 19 -c indicates the damping coefficient. Figure 19 -d indicates the oscillation frequency.

[0032] Figure 20 The reactive power control integral coefficient in the reactive power control mode of the example involved in this invention. k iQ A schematic diagram of the simulation results during the change. Detailed Implementation

[0033] The present invention will now be described in conjunction with the accompanying drawings and embodiments: (1) System topology and control strategy The weak network access DC system topology considered in this invention is as follows: Figure 1 As shown, based on differences in operating control characteristics, various power electronic devices connected to the DC grid can be divided into two main categories: DC voltage control modules and constant power control modules. Among them, photovoltaic, wind power, and other new energy sources, power electronic equipment, and DC loads, all possessing constant power operation characteristics, can be considered constant power control units. The DC voltage control unit is a voltage source converter (VSC) connected to a weak grid, primarily used to maintain the stability of the DC side voltage of the system.

[0034] The VSC topology and its control strategy considered in this invention are as follows: Figure 2 As shown, v dc and i odc These are the VSC output DC voltage and DC current, respectively. v g ∠0 represents an infinite grid voltage. v t ∠ θ This indicates the voltage at the point of common coupling (PCC). L f Indicates the filter inductance; P and Q These represent the active and reactive power outputs on the AC side, respectively, with the positive direction being the flow towards the AC power grid. L g Indicates the inductance of an AC line. C s and i inThese represent the output capacitor of VSC and the DC current injected into VSC, respectively.

[0035] The VSC control system includes DC voltage control, reactive power control, phase-locked loop (PLL) control, and current inner loop control. v dcref and G vdc ( s These represent the DC voltage reference value and the controller in the DC voltage control loop, respectively; the reactive power control loop includes two control methods: AC voltage control and constant reactive power control. v tref and G vac ( s These represent the AC voltage reference value and the controller in the AC voltage control loop, respectively. Q ref and G Q ( s ) represent the reactive power reference value and transfer function of the constant reactive power control loop, respectively. G pll ( s This indicates a PLL control loop, typically using a PI controller. θ pll This indicates the phase-locked angle obtained from the PLL. The inner current loop control ensures... d, q Two-axis actual current i d , i q Precisely and quickly follow the current reference value output by the outer loop. i dref , i qref .

[0036] like Figure 2 As shown, this invention defines two synchronous rotating coordinate systems: the power grid side... DQ Coordinate system and control side dq Coordinate system. Wherein, DQ The coordinate system is based on the grid voltage vector. v g Let ∠0 be the orientation reference, and let D The axis coincides with the grid voltage vector. Q Axis hysteresis D The axis is 90°, forming a synchronous rotating coordinate system for the power grid. dq The coordinate system uses the phase angle output by the PLL. θ pll As the orientation benchmark, let d The axis coincides with the voltage vector output by the PLL. qAxis hysteresis d The axis is 90°, forming a control synchronous rotating coordinate system. Under weak power grid conditions, due to the dynamic influence of the PLL, a phase deviation Δ exists between the two coordinate systems. θ = θ pll The two coordinate systems are mapped to each other through the orthogonal rotation transformation of equation (1).

[0037] (1) (2) Reduced-order modeling of DC voltage control unit connected to weak network For the DC voltage control unit, linearization is performed at the system's steady-state operating point, and the dynamics of the inner current loop control are ignored. The dynamics of the outer loop control are as follows: (2) Where Δ represents the small signal increment of the variable, Δ i d and Δ i q They represent dq under reference frame d Axis current components and q Axis current component. Δ R ref and Δ R These represent the reference and actual values ​​of the general variables for the reactive power control loop, respectively. When using constant AC voltage control, G R ( s )= G vac ( s ), Δ R ref =Δ v tref Δ R =Δ v t When using constant reactive power control, G R ( s )= G Q ( s ), Δ R ref =Δ Q ref Δ R =Δ Q Δ v dc Δ is a small disturbance to the DC voltage output of VSC. v t For small disturbances in the grid connection point voltage, Δ Q It represents a small disturbance in reactive power.

[0038] After linearization, the phase-locked loop (PLL) control can be expressed as: (3) Where, Δ θ pll This represents the phase difference of the phase-locked angle obtained by the PLL. Δ θ This indicates the voltage at the point of common coupling (PCC). v t The phase angle difference relative to the infinite grid voltage; the subscript "0" indicates the steady-state value of each variable; After completing the small-signal linearization derivation using the coordinate transformation rules given in equation (1), the small-signal disturbance equations corresponding to the AC current of the DC voltage control unit and the voltage at the point of common coupling can be obtained. The specific expressions are as follows: (4) in ， Δ i D and Δ i Q They represent D Axis current components and Q Axial current component; Δ v D and Δ v Q They represent D Axis voltage components and Q Axis voltage components; X g Indicates line inductance L g The reactance value, θ pll This represents the phase-locked angle obtained from the PLL; based on DQ The current and voltage components in the synchronous rotating coordinate system can be used to express the active power ΔP and reactive power ΔQ delivered by the VSC to the grid as follows: (5) Increment of voltage amplitude at grid connection point Δ v t and phase angle increment Δ θ satisfy: (6) Combining equations (1)-(6), the active power Δ P and phase angle increment Δ θ The output Δ of the outer loop of the active control class is used respectively. i d The output Δ of the outer loop of reactive power control iq and the PLL output phase angle Δ θ pll The specific form is as follows: (7) in, m 1, m 2, m 3, n 1 and n The specific expression of 2 is as follows: (8) in, v g and v t These represent the amplitudes of the infinite grid voltage and the grid connection point voltage, respectively. θ This represents the phase of the grid connection point voltage, and in steady state... θ 0= θ pll0 ; Similarly, depending on the specific control strategy of the reactive power control loop, its input variable (reactive power Δ) can be determined. Q and voltage amplitude increment Δ v t It can be represented in the following general form: (9) In the formula, Δ R This represents the general input variable for the reactive power control loop.

[0039] When constant AC voltage control is used, Δ R , k 1. k 2 and k The specific form of 3 is as follows: (10) When constant reactive power control is used, Δ R , k 1. k 2 and k The specific form of 3 is as follows: (11) Based on power balance, and after small-signal linearization, the dynamic disturbance equations for the DC-side output voltage and current of the DC voltage control unit can be derived. Here, we only... Figure 2 The definitions of output voltage and current in the equation, after linearization and adding Δ, yield the following result: (12) In summary, the DC voltage control unit can be obtained d axial AC current component Δi d With active power Δ P The basic transfer function model between them, such as Figure 3 As shown in -a. In the transfer function, ΔP is the output of the basic transfer function, Δ i d This is the input in the basic transfer function. H pll (s) and H R (s) are the equivalent transfer functions for the PLL loop and the reactive power control loop, respectively, and their specific forms are as follows: (13) To more intuitively reveal the impact of different control loop dynamics and interactions on system stability, further research is needed. Figure 3 The basic transfer function model shown in -a can be broken down and rearranged to obtain, as follows: Figure 3 The multi-branch transfer function form shown by -b contains a total of 4 branch transfer functions: the first branch T 1(s) corresponds to the inherent electrical circuit connection of the system, the second branch. T 2(s) corresponds to the dynamic influence of PLL control, the third branch T 3(s) corresponds to the dynamic influence of reactive power control components, the fourth branch. T 4(s) represents the dynamic interaction between the PLL and reactive power control components. The specific form is as follows: (14) When considering the influence of the DC voltage control loop, the complete transfer function model of the DC voltage control unit connected to the weak network is as follows: Figure 4 -a is shown.

[0040] Combining equations (2) and (12), the transfer function of the DC voltage control unit is... G vdceq ( s )and d axial AC current component Δ i d Multi-branch transfer function between active power ΔP T 1( s ), T 2( s ), T 3( s )and T 4( s Further simplification yields the equivalent transfer function of the DC voltage control unit. G vdceq ( s The multi-branch form of ) is shown in equation (15): (15) in, G i (s) indicates G vdceq (s) i The equivalent transfer function of the branch; at this time, such as Figure 4 As shown in (b), the DC voltage control unit for weak network access can be decomposed into four branches with clear physical meaning: the DC voltage control link directly affects G 1(s); Dynamic Influence of DC Voltage Control and PLL Interaction G 2(s); Interactive dynamic influence of DC voltage control and reactive power control components G 3(s); The combined effect of DC voltage control, reactive power control components, and PLL. G 4(s), in the following specific form: (16) although Figure 4 The multi-branch transfer function model of the DC voltage control unit shown in -b can, to some extent, characterize the effects of DC voltage control, phase-locked loop, and reactive power control, and their interactions, on the low-frequency dynamic stability of the weakly interconnected DC system. However, this model still lacks intuitive physical representation capabilities, and the intrinsic physical mechanisms by which various control dynamics affect the system's stability characteristics are not clearly explained. Therefore, in [the following section will be discussed]... Figure 4 Based on the multi-branch transfer function model of the DC voltage control unit obtained by -b, the reduced-order equivalent circuit model of the DC system connected to the weak network is derived, and the dynamic stability of the system is analyzed clearly and intuitively.

[0041] Combining equations (12) and (15), the dynamic DC voltage of the DC voltage control unit can be expressed as follows: (17) In the formula, v ref This indicates the equivalent DC voltage source of the DC voltage control unit. Z s This represents the equivalent output impedance of the DC voltage control unit; its specific form is as follows: (18) To investigate the impact of weak grid access on the dynamic stability characteristics of the weakly interconnected DC system, a targeted reduction and simplification of the original high-order model will be carried out. While fully preserving the frequency response characteristics of the dominant low-frequency modes surrounding the weakly interconnected DC system, the complex frequency parameters will be... s =j ω 0 (of which) ωSubstituting 0 (the angular frequency of the dominant oscillation mode of the system) into the transfer function G i (s), according to the following formula, complete the first-order equivalent simplification of the transfer function.

[0042] (19) In the formula, k pueqi and k iueqi They are respectively G i The equivalent proportional coefficient and integral coefficient of (s).

[0043] Combining equations (15) and (19), G vdceq The equivalent PI element of (s) can be expressed in the following form: (20) Combination Figure 4 -b, Equivalent output impedance of DC voltage control unit Z s This can be further expressed as: (twenty one) From equation (21), it can be seen that the equivalent output impedance Z s It is composed of an equivalent RLC parallel circuit, and the specific correspondence is as follows: (twenty two) in, R Ti , L Ti These are the resistance and inductance parameters for the corresponding branch.

[0044] Combining equations (21) and (22), and based on Norton's theorem, we can obtain the equivalent reduced-order circuit model of the DC voltage control unit, as follows: Figure 5 As shown. The DC voltage control unit's equivalent current source current disturbance is shown. I dcref The following relationship must be satisfied: (twenty three) (3) Reduced-order modeling of constant power control unit Using a similar method, the constant power control unit can be represented by the RC parallel circuit shown in Figure 6, used for stability analysis under DC voltage control time scale. The specific expression is as follows.

[0045] (twenty four) In the formula, I pref This refers to the small-signal current disturbance of the equivalent current source of the constant power control unit. v p and i p Corresponding sequentially to the small-signal disturbances of the DC voltage and port current at the constant power control unit ports, this invention uniformly stipulates that the outflowing DC bus is positive. P cpl0 and v p0 These are the steady-state power of the constant power control unit and the steady-state value of the port voltage, respectively. C p Indicates port capacitance. R p This represents the equivalent resistance of the constant power control unit. .

[0046] (4) Reduced-order modeling of the entire DC system connected to the weak network Based on the equivalent circuit models of the DC voltage control unit and constant power control unit obtained from equations (21)-(24), and combined with the characteristics of the DC network, the equivalent reduced-order circuit model of the DC system connected to the weak network can be obtained, such as... Figure 7 As shown. Among them, Z linei ( i =1, 2, ..., n () indicates the line impedance. Z pi ( i =1, 2, ..., n The ) represents the equivalent output impedance of the constant power control unit. The total parallel impedance of this model is... Z sum Equivalent output impedance of DC voltage control unit Z s Equivalent output impedance of constant power control unit Z p The parallel connection can be represented as: (25) From equation (25), the specific form of the second-order characteristic equation of the system can be further obtained as follows: (26) When the weakly interconnected DC system operates in an underdamped condition (0 < When <1), the damping coefficient of the system can be obtained by solving. and oscillation frequency ω d The parsing expression: (27) Based on the proposed reduced-order equivalent circuit model, combined with the damping coefficient and oscillation frequency ω d By using core dynamic evaluation indicators, the impact of weak grid access on the time-scale stability characteristics of DC voltage control in DC systems can be analyzed.

[0047] Analysis and Verification Not generally, with Figure 8 Taking the back-to-back (BTB) DC system connected to a weak network as an example (shown in section -a), analysis and verification are performed. Two control modes, AC voltage control and reactive power control, are used in the reactive power control stage for comparison. The equivalent parallel circuit model of the back-to-back system connected to the weak network is as follows: Figure 8 As shown in -b. The detailed model of the energy storage system was simulated and verified in PSCAD / EMTDC. The basic system parameters are shown in Table 1.

[0048] Table 1 System Main Circuit and Control Parameters

[0049] First, we analyze the impact of AC inductance on system stability under two different control modes in the reactive power control loop. Under AC voltage control mode, the AC inductance... L s The effect of parameters on the stability characteristics of the system is as follows: Figure 9 As shown in the figure. A comparison of damping coefficient and oscillation frequency under different control modes is shown in the figure below. Figure 11 As shown. Combined with Figure 9 The variation law of the parameters shown can be obtained as follows: Accompanying AC inductance L s The equivalent conductance of the branch controlled solely by DC voltage increases continuously, 1 / R T1 The conductance of the branch with DC voltage control and phase-locked loop coupling effect 1 / R T2 All showed attenuation, while the conductance of the multi-stage coupling branch of DC voltage control, AC voltage control, and phase-locked loop was 1 / R T4 This increases, ultimately resulting in an overall parallel conductance of 1 / R s The susceptance decreases (see Figure 9-a). Simultaneously, the equivalent susceptance of the multi-link coupled branch 1 / L T4 Increase, overall parallel susceptance 1 / L s The inductance is increased synchronously (see Figure 9-b). Comparing the results with Figure 11, it can be seen that increasing the AC inductance... L sThis means that the power grid strength continues to weaken, the system damping decreases (see Figure 11-a), and the system oscillation frequency increases (see Figure 11-b).

[0050] For reactive power control conditions, AC inductance L s The effect of parameters on the system's stability characteristics is shown in Figure 10. Combining the parameter variation patterns in Figure 10, we can derive the following: [The text abruptly ends here, likely due to an incomplete sentence or missing information.] L s The equivalent conductance of the dominant branch controlled solely by DC voltage increases continuously, 1 / R T1 The conductance of the branch with DC voltage control and phase-locked loop coupling effect 1 / R T2 The conductance of the branch where DC voltage control and reactive power control are coupled is 1 / R T3 All showed attenuation, while the conductance of the multi-stage coupled branch was 1 / R T4 There has been an increase; the overall parallel conductance of the DC voltage control unit is 1 / R s It has declined (see) Figure 10 -a). Total parallel susceptance of DC voltage control unit 1 / L s Decrease (see Figure 10-b). Comparing the results with Figure 11, it can be seen that increasing the AC inductance... L s That is, the grid strength continues to weaken, the system damping performance deteriorates (see Figure 11-a), and the system oscillation frequency decreases (see Figure 11-a). Figure 11 -b). It also indicates that compared to the AC voltage control mode, the power-free control mode has poor dynamic stability under weak grid conditions.

[0051] AC inductor under two different control modes L s The time-domain simulation results of the impact on system stability are as follows: Figure 12 As shown. By Figure 12 As can be seen, the DC voltage dynamics of the proposed reduced-order model are almost identical to those of the detailed model, verifying the effectiveness of the proposed method. Furthermore, when... L s When the AC inductance Ls = 0.3H, the DC voltage fluctuation in reactive power control mode is 1.07V, which is larger than that in AC voltage control mode, with a longer convergence time and poorer stability, consistent with the theoretical analysis above. To further verify the effectiveness of the established model, taking reactive power control mode as an example, when the AC inductance Ls = 0.3H, the DC voltage exhibits a low-frequency oscillation response after being subjected to power disturbance, with an oscillation angular frequency of approximately 1.25 rad / s and a complete oscillation period of 5.04 s. This simulation dynamic result is consistent with... Figure 11 The oscillation frequency of 1.30 rad / s obtained by the -b analysis is highly consistent, which fully verifies the effectiveness of the proposed modeling and analysis method.

[0052] Secondly, the influence of the AC voltage control coefficient on system stability is analyzed when the reactive power control link adopts AC voltage control mode. The effect of the AC voltage control proportional coefficient on the system stability characteristics is shown in Figure 13, and the corresponding time-domain dynamic simulation verification results are shown below. Figure 14 Combining the parameter variation patterns in Figure 13, we can conclude that as the proportional parameter of the AC voltage control continues to increase, the equivalent conductance 1 / of the multi-link coupled branch of the DC voltage control, AC voltage control, and phase-locked loop increases. R T4 The equivalent conductance of the remaining branches remained largely unchanged; influenced primarily by the parameters of this branch, the overall parallel conductance of the DC voltage control unit was 1 / R s As the system damping increases (see Figure 13-a), the system damping also increases (see Figure 13-a). Figure 13 -c); DC voltage control unit total parallel susceptance 1 / L s Almost unchanged (see) Figure 13 -b), the system oscillation frequency decreases (see...). Figure 13 -d). By Figure 14 It can be seen that increasing the AC voltage control proportional coefficient reduces the DC voltage fluctuation amplitude and enhances the system damping, consistent with the theoretical analysis. Next, we investigate the effect of the AC voltage control integral coefficient on the system stability characteristics, and the corresponding results are shown in Figure 15. According to the parameter variation patterns in Figure 15, as the AC voltage control integral coefficient continuously increases, the equivalent conductance 1 / ... R T4 The current has increased somewhat; due to the dominant effect of this branch, the overall parallel conductance of the DC voltage control unit is 1 / R s As the voltage increases synchronously (see Figure 15-a), the system damping increases accordingly (see Figure 15-c); simultaneously, the equivalent susceptance of the multi-link coupled branches 1 / L T4 A slight increase, resulting in an overall parallel susceptance of 1 / L s Slightly increasing (see Figure 15-b), the system oscillation frequency increases (see...). Figure 15 -d). By Figure 16 It can be seen that increasing the integral coefficient of AC voltage control reduces the amplitude of DC voltage fluctuation and enhances system damping, which is consistent with theoretical analysis.

[0053] Finally, the impact of the reactive power control coefficient on system stability is analyzed when reactive power control mode is adopted in the reactive power control stage. The effect of the reactive power control proportional coefficient on system stability characteristics is as follows: Figure 17 As shown. The corresponding time-domain simulation verification results are shown in [the image / document]. Figure 18 Combining the parameter variation patterns in Figure 17, we can conclude that as the reactive power control proportional parameter continues to increase, the equivalent conductance 1 / of the multi-link coupled branch consisting of DC voltage control, reactive power control, and phase-locked loop increases. R T4 The equivalent conductance of the remaining branches gradually increases, while the equivalent conductance of the remaining branches remains relatively stable. Due to the dominant influence of this branch, the overall parallel conductance of the DC voltage control unit is 1 / R s A slight increase (see Figure 17-a), the overall parallel susceptance 1 / L s The values ​​remain essentially unchanged (see Figure 17-b). Combining this with the time-domain simulation results in Figure 18, it can be seen that increasing the reactive power control proportional coefficient slightly reduces DC voltage fluctuations and slightly improves the system damping level. Simultaneously, it can be observed that the dynamic curves of the DC bus voltage before and after the change in the reactive power control proportional coefficient show minimal difference and almost identical trends, indicating that when reactive power control is used in reactive power control components, the proportional coefficient has minimal impact on system stability, consistent with theoretical analysis. The effect of the reactive power control integral coefficient on system stability characteristics is as follows: Figure 19 As shown. The corresponding time-domain simulation verification results are shown in [the image / document]. Figure 20 Based on the parameter variation patterns in Figure 19, we can conclude that as the integral coefficient of reactive power control increases, the equivalent conductance 1 / ... of the multi-link coupled branch consisting of DC voltage control, reactive power control, and phase-locked loop... R T4 Gradually increase the conductance of the branch where DC voltage control and reactive power control are coupled (1 / R T3 Reduce, overall parallel conductance 1 / R s Slight increase (see) Figure 19 -a) The system damping is slightly enhanced; the total parallel susceptance of the DC voltage control unit is 1 / L s Basically unchanged (see Figure 19 -b). Combining the time-domain simulation results in Figure 20, it can be seen that increasing the reactive power control integral coefficient slightly reduces DC voltage fluctuations and slightly enhances system damping. Simultaneously, it can be observed that the dynamic curves of the DC bus voltage before and after the change in the reactive power control integral coefficient show minimal difference and almost identical trends, indicating that when reactive power control is used in reactive power control components, its integral coefficient has minimal impact on system stability, consistent with theoretical analysis. This verifies the feasibility of implementing this patent.

Claims

1. A method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems, comprising the following steps: S1, neglecting the dynamics of the current inner loop control and the electromagnetic transient process of the filter circuit, and taking into account the influence of the DC voltage control link, reactive power control link, and phase-locked loop dynamics under weak network characteristics, a multi-branch transfer function model of the DC voltage control unit is established; S2, by simplifying and reducing the order of the multi-branch equivalent transfer function model, an equivalent reduced-order circuit model for the DC voltage control unit connected to the weak network is established. S3. Under the DC voltage control time scale, establish the equivalent circuit model of the constant power control unit; combine the characteristics of the DC voltage control unit, the constant power control unit and the DC network to construct the full-system reduced-order model of the weakly interconnected DC system. S4, the second-order characteristic equation of the weakly interconnected DC system is obtained. When the weakly interconnected DC system is operating under underdamped conditions, the system damping coefficient and oscillation frequency are obtained by solving. S5. Based on the equivalent circuit model, and combined with the damping coefficient and oscillation frequency dynamic evaluation index, the impact of weak grid access on the time-scale stability characteristics of DC voltage control of DC system is analyzed.

2. The method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems according to claim 1, characterized in that, The method for S1 is as follows: Active power Δ P and phase angle increment Δ θ The increment Δ of the d-axis component of the outer loop current output of the active control type is used respectively. i d The increment Δ of the q-axis component of the outer loop current output in reactive power control type i q and the PLL output phase angle Δ θ pll The specific form is as follows: (1) in, m 1, m 2, m 3, n 1 and n The specific expression of 2 is as follows: (2) In the formula, v g0 and v t0 These represent the steady-state values ​​of the amplitudes of the infinite grid voltage and the grid connection point voltage, respectively. X g Indicates AC line inductance L g The reactance value, θ This represents the phase of the grid connection point voltage, and in steady state, θ 0= θ pll0 ; Based on the specific control strategy of the reactive power control loop, the input variable reactive power Δ Q and voltage amplitude increment Δ v t It can be represented in the following general form: (3) In the formula, Δ R This represents the general input variables for the reactive power control loop; When constant AC voltage control is used, Δ R , k 1. k 2 and k The specific form of 3 is as follows: (4) When constant reactive power control is used, Δ R , k 1. k 2 and k The specific form of 3 is as follows: (5) In summary, the DC voltage control unit can be obtained d axial AC current component Δ i d With active power Δ P The basic transfer function model between them, where... H pll (s) and H R (s) are the equivalent transfer functions for the PLL loop and the reactive power control loop, respectively, and their specific forms are as follows: (6) In the formula, G pll ( s ) represents the transfer function of a phase-locked loop (PLL) element. G vac ( s ) represents the control function of the AC voltage control loop; The basic transfer function model is broken down and reorganized to obtain its multi-branch transfer function form, which contains a total of 4 branches: the first branch T 1(s) corresponds to the inherent electrical circuit connection of the system, the second branch. T 2(s) corresponds to the dynamic influence of PLL control, the third branch T 3(s) corresponds to the dynamic influence of reactive power control components, the fourth branch. T 4(s) represents the dynamic interaction between the PLL and reactive power control components; the specific form is as follows: (7) When considering the influence of the DC voltage control loop, the transfer function of the DC voltage control loop is... G vdc ( s )and d axial AC current component Δ i d Multi-branch transfer function between active power ΔP T 1( s ), T 2( s ), T 3( s )and T 4( s Further simplification yields the equivalent transfer function of the DC voltage control unit. G vdceq ( s Multi-branch form: (8) In the formula, G i (s) indicates G vdceq (s) i The equivalent transfer function of the branch, v dc0 This represents the steady-state value of the VSC output voltage; The DC voltage control unit for weak network access is decomposed into four branches with clear physical meanings: the DC voltage control link directly affects... G 1(s); Dynamic Influence of DC Voltage Control and PLL Interaction G 2(s); Interactive dynamic influence of DC voltage control and reactive power control components G 3(s); The combined effect of DC voltage control, reactive power control components, and PLL. G 4(s), its specific form is as follows: (9)。 3. The method for reduced-order modeling and analysis of low-frequency stability of weakly interconnected DC systems according to claim 1, characterized in that, In S2, the equivalent output impedance of the equivalent reduced-order circuit model Z s for: (10) In the formula, R s and L s Equivalent output impedance Z s The equivalent resistance and equivalent inductance, C s For VSC output capacitor, P 0 represents the steady-state value of active power.

4. The method for reduced-order modeling and analysis of low-frequency stability of weakly interconnected DC systems according to claim 1, characterized in that, In S3, the total parallel impedance of the reduced-order model of the entire system Z sum Represented as: (11) In the formula, Z p This is the equivalent output impedance of the constant power control unit. C p Indicates port capacitance. R p This represents the equivalent resistance of the constant power control unit.

5. The method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems according to claim 1, characterized in that, In S4, the second-order characteristic equation of the weakly interconnected DC system is further obtained from equation (11), and its specific form is as follows: (12) When the weakly interconnected DC system operates in an underdamped condition, i.e., 0 < When <1, the system damping coefficient can be obtained by solving. and oscillation frequency ω d .

6. The method for reducing the order of low-frequency stability modeling and analysis of weakly interconnected DC systems according to claim 5, characterized in that, In S4, the system damping coefficient and oscillation frequency ω d The parsing expression is: (13)。

Images