Study on the Mode and Characteristics of SSOs in Hybrid AC–DC Transmission Systems via Multitype Power Supply

Abstract

The impact of subsynchronous oscillation (SSO) on grid security is becoming increasingly prominent with the rapid development of a large new energy base. However, the SSO modes and characteristics in complex power systems where series-complementary AC systems, DC systems, wind farms, and thermal power plants co-exist simultaneously are still not well understood, and relevant research has yet to be conducted. To address these issues, this study aims to investigate the SSO oscillation modes and the participation of specific influencing factors using eigenvalue and participation factor analysis. Additionally, the influence of system operation mode and control parameters on the SSO characteristics is studied through eigenvalue analysis. The findings of this study suggest that multiple oscillation sources and the co-existence of various oscillation patterns in hybrid AC–DC transmission systems cause SSO problems. The SSOs arise due to inappropriate system operation or parameter selection. As the series compensation increases, the system tends to become unstable. The system stability improves when the wind power output increases or the thermal power output decreases while keeping the output power of other sources constant. On the other hand, the system stability decreases as the DC transmission power gradually decreases. In terms of the control parameters, a higher value of the inner-loop proportionality coefficient of the converter current on the rotor side of the wind turbine results in a more unstable system. However, the rotor-side converter outer loop parameters and the stator-side control loop PI parameters have a negligible effect on the oscillation frequency and damping of the system. Matlab time domain simulations are conducted to verify the accuracy of the theoretical analysis.

1. Introduction

With the introduction of the “carbon peaking and carbon neutrality” objective and efficient exploitation of traditional energy sources [1], China’s power system is gradually transitioning towards a high proportion of renewable energy and an increased reliance on power electronics. To meet this target, various energy companies, including PetroChina, have responded positively to the government’s policy [2] by investing in numerous large-scale wind power bases located in the Gobi desert and desert regions. However, these onshore wind farms are predominantly situated in the north-western region, which is far from the central and eastern load centers. Therefore, the electricity generated must be transmitted over long distances to the recipient grid via extra-high-voltage lines. Due to the volatile nature of wind power, it must be combined with thermal power to improve stability, thereby creating a typical hybrid AC–DC transmission system that consists of multiple power sources. Under certain conditions, the interaction between two or more systems, such as AC systems with series compensation, DC systems, wind farms, and thermal power plants, may result in SSOs [3,4,5,6], which can pose a significant threat to the security of the power system.

More systematic research findings have been procured on the generation mechanism, analysis methods, and suppression measures of SSOs in traditional thermal power units [7,8]. In recent years, with China’s rapid advancement in renewable energy generation, large-scale wind farms have been integrated into the grid, resulting in numerous SSO incidents. Scholars have conducted extensive research on the SSO issues in wind-thermal bundling systems, employing eigenvalue analysis, complex torque coefficient analysis, impedance analysis, and other methods. The literature [9] investigated the variation of the SSO pattern of wind farms at different wind speeds and reactive powers using modal analysis, demonstrating that the grid integration of doubly fed induction generators (DFIGs) would diminish the SSO damping of the system. The literature [10] employed eigenvalue analysis to analyze the stability of a DFIG grid connected to a thermal power unit, revealing that the system stability gradually escalates as the line series compensation progressively decreases. In addition to DFIGs, as the market share of direct-drive permanent magnet synchronous generators (D-PMSGs) gradually expands, the research on related SSOs also increases. Currently, the SSO analysis of D-PMSG grid-connected systems primarily concentrates on two aspects: D-PMSGs connected to weak AC grids and D-PMSGs connected to grids via a Voltage Source Converter-based High Voltage Direct Current (VSC-HVDC). Regarding the stability study of D-PMSG grid-connected systems via VSC-HVDCs, the literature [11,12] utilized impedance analysis to analyze the small disturbance stability of D-PMSG output systems through VSC-HVDCs; the literature [13] employed eigenvalue analysis to investigate the stability of D-PMSG grid-connected output systems via VSC-HVDCs, revealing the presence of an SSO mode involving both D-PMSGs and VSC-HVDCs. The literature [14,15] established a small-signal model of the D-PMSG grid-connected system via VSC-HVDCs and employed eigenvalue analysis to explore wind farm aggregation characteristics and the influence of system parameters on SSO attributes. In addition, system SSOs can also be triggered when wind turbines are connected to weak AC grids in real systems [16,17,18,19,20]. On 1 July 2015, persistent SSO incidents occurred in Hami, Xinjiang when PMSG was connected to the AC grid without series compensation devices, causing the protection devices for thermal power units hundreds of kilometers away to operate [21]. To tackle the SSO issues of wind farms connected to weak AC grids, the literature [22] established a dynamic model of direct-drive wind farms connected to weak AC grids and analyzed the impacts of factors such as the AC system strength, the number of turbines connected to the grid, and turbine control parameters on the SSO characteristics. Additionally, the literature [23] studied the effects of different wind speeds and output powers on the SSO characteristics of PMSGs connected to weak AC grid systems under full operating regions. Moreover, in [24], a conductivity characteristic analysis was employed to investigate the effects of a prefiltering bandwidth, current inner loop, and voltage outer loop on the SSO characteristics of a D-PMSG grid-connected system. In [25], a comparison was made between the effects of a single machine and a multiple machine equivalent model on the SSO characteristics in the scenario of a DFIG grid connected by series compensation.

However, most of the existing papers have discussed the SSOs of a wind-thermal bundling transmission system or AC–DC hybrid system. Limited research has been conducted in the extant literature on SSOs in complex power systems consisting of series-complemented AC systems, DC systems, wind farms, and thermal power plants. As large new energy bases are being rapidly developed, the first multitype power supply and UHV AC–DC hybrid power transmission system has been built, raising the complexity and difficulty of the SSO problem to a new height. It is of great significance to study this problem deeply and solve it effectively. Thus, it is necessary to investigate the modes and characteristics of SSOs in such systems. To this end, this paper employs a combination of eigenvalue analysis and time-domain simulation to analyze the SSO modes in complex power systems containing AC systems with series complements, DC systems, wind farms, and thermal power plants, as well as the influence of system parameters on SSOs. A small-signal model of a hybrid AC–DC transmission system with multiple types of power sources is established, and the eigenvalues of the system are computed to filter out the SSO modes. The relevant state variables involved in the SSO modes are used to determine the subsystems to which the oscillations belong. Moreover, the effects of different influencing factors on the SSO damping characteristics of the system are analyzed using the eigenvalue method and time-domain simulation.

2. The Structure and Model of Hybrid AC–DC Transmission Systems via Multitype Power Supply

This paper establishes a multitype power supply based on the actual grid structure of a region in North China, utilizing a hybrid AC–DC transmission system. The system is composed of four components, including a wind power system, a thermal power system, an AC system with series compensation, and a DC system. The topology of the delivery system is shown in Figure 1.

Considering the complexity of the system under study, a modular block modelling approach was employed to construct the small-signal model of the system. This method involves establishing a state space submodule for each component, with each module establishing a transfer relationship via the input and output signals to derive a small-signal model for the entire system [26]. The modelling process for each module is depicted in Figure 2.

Among them, the thermal power system is composed of several submodules, namely the synchronous generator, shafting, excitation system, machine network interface, and others. On the other hand, the wind power system consists of the asynchronous generator, shaft system, converter, its control strategy, a phase-locked loop, grid interface, and other submodules. Specifically, DFIG is the selected type, and the rotor-side converter (RSC) typically employs a stator voltage directional control method to maintain a stable generator speed and ensure that the stator output reactive power matches a predefined reference value. The primary control objective of the grid side converter (GSC) is to maintain a stable voltage at the machine end of the DFIG. A single machine equivalent model is utilized to represent wind farms and thermal power plants. Meanwhile, the simplified CIGRE standard test model is used to represent HVDCs [26]. In this model, three-phase AC current sources are coupled with DC voltage sources, while capacitor branches are used instead of reactive power compensation and filter branches, as opposed to the thyristor rectifier and inverter bridges. The influence of the AC system on the SSO mode of the entire system is mainly attributed to its series complementarity. Accordingly, the AC system is simplified to an equivalent voltage source and its equivalent impedance, premised on the Davinan equivalence.

The state space equations for each submodule are established separately and shown in Equation (1).

$$\left\{\begin{array}{c}\frac{d}{dt}\Delta X_i=A_i\Delta X_i+B_i\Delta U_i\\ \Delta Y_i=C_i\Delta X_i+D_i\Delta U_i\end{array}\right.$$

(1)

X_i is the state variable of module i, U_i denotes the input variable, and Y_i represents the output variable of module i; and A_i, B_i, C_i, and D_i denote the system matrix, input matrix, output matrix, and direct transfer matrix of module i, respectively. By leveraging the aforementioned system structure and parameters, a modular state space model of the individual components of the studied system was developed. Figure 3 illustrates the interconnections between each module and the corresponding port inputs and outputs.

The state space equation model depicted in Figure 3 was constructed using MATLAB/SIMULINK. Upon completion of the simulation, the small-signal model of the entire transmitting system was obtained, as represented by Equation (2).

$$\frac{d}{dt}\Delta X=A\Delta X+B\Delta U$$

(2)

The system model can be represented by a state space equation of the 40th order. The state variables, denoted by X_i (i = 1–40), can be segregated into four groups based on the submodule they correspond to. These groups are numbered and annotated to facilitate the ease of expression and future analysis. The variables are summarized in

Table 1. The state variables and their corresponding systems.

System Module	Subsystem Module	State Variable Number	State Variable
Thermal power system	Synchronous generator	X1–X5	Id, Iq, If, ID, IQ
	Shafting of thermal power system	X6–X12	ωr, ω2, ω3, ω4, delta21, delta32, delta43
	Excitation system	X13–X14	statvar1_exc, statvar2_exc
	Thermal power system network interface	X15	Deltars
Wind power system	Asynchronous generator	X16–X19	Isd, Isq, Ird, Irq
	Wind power system shafting	X20	ωr
	RSC and its control system	X21–X23	Xi_ω, Xi_Ird, Xi_Irq
	GSC and its control system	X24–X29	Icgd, Icgq, Xi_dc Xi_Qdfig, Xi_Icgd Xi_Icgq
	Converter DC side model	X30	Udc
	Phase-locked loop model	X31	Xi_pll
AC transmission system	/	X32–X35	iLgx, iLgy, ucgx, ucgy
HVDC transmission system	Converter and its control strategy model	X36–X37	var1_CtrlRec_HVDC var2_CtrlRec_HVDC
	DC line model	X38–X40	Ucd, Idr, Idi

, with their respective meanings provided in the

Table A4. The interpretation of the state variables.

State Variable Number	State Variable	Description
X1–X5	Id, Iq, If, ID, IQ	Stator and rotor current of the synchronous generator in dq0 coordinate system
X6–X12	ωr, ω2, ω3, ω4, delta21, delta32, delta43	ωi is the synchronous speed of four mass models of synchronous generator shafting; deltai is the phase difference between the different masses
X13–X14	statvar1_exc, statvar2_exc	Integral variable of excitation system filter
X15	Deltars	Phase angle difference between synchronous rotating coordinate system and rotor rotating coordinate system
X16–X19	Isd, Isq, Ird, Irq	Stator and rotor current of asynchronous generator in dq0 coordinate system
X20	ωr	Rotor angular frequency of wind turbine shafting
X21–X23	Xi_ω, Xi_Ird, Xi_Irq	Integral variable of the rotor-side control loop
X24–X29	Icgd, Icgq, Xi_dc Xi_Qdfig, Xi_Icgd Xi_Icgq	Integral variable of the grid-side control loop
X30	Udc	Voltage of direct current capacity of the DFIG converter
X31	Xi_pll	Integral variable of phase-locked loop
X32–X35	iLgx, iLgy, ucgx, ucgy	The inductive current and capacitance voltage of an AC transmission line
X36–X37	var1_CtrlRec_HVDC var2_CtrlRec_HVDC	Integral variable of the control loop in the HVDC converter
X38–X40	Ucd, Idr, Idi	Capacitance voltage in DC line and current in both the rectifier side and inverter side

in Appendix A.

3. Analysis of the SSO Mode in Hybrid AC–DC Transmission Systems via Multitype Power Supply

The typical operating conditions of the system were selected as follows: a thermal power plant output of 7000 MW, a wind farm output of 7000 MW, a wind speed of 12 m/s, a system series compensation of 29.25%, and a DC line transmission capacity of 10,000 MW.

A linearized model can be obtained by linearizing an n-dimensional dynamic system at its operating point, such as Equation (2), where A is the state matrix. Solving the small-signal model developed in Section 2 for eigenvalues represented by (${\lambda }_{\mathrm{i}}={\sigma }_i+j{\omega }_i$) provides the basis for determining the system stability, with positive and negative eigenvalues being critical. λ_i represents an eigenvalue of the state equation and σ_i is the real part of the eigenvalue, corresponding to a convergence or divergence of the system in the oscillation mode. The negative real part represents the decaying mode, the positive real part represents the diverging mode, and each pair of conjugate eigenvalues corresponds to a mode of oscillation. Additionally, the magnitude of the real part of the eigenvalue reflects how fast the system converges or diverges, while ω_i represents the oscillation angle frequency.

Based on the aforementioned parameter description, the damping ratio is set as Equation (3):

$$\xi =\frac{-\sigma }{\sqrt{{\sigma }^2+{\omega }^2}}$$

(3)

The magnitude of the real part of the eigenvalue indicates the speed of decay (or divergence) of the oscillation, with larger absolute values indicating a faster decay or divergence. If all the real parts of the eigenvalues of the matrix A are less than zero, then the system is stable; otherwise, it is unstable.

Table 2. Subsynchronous oscillation modes and characteristic information.

SSO Mode	Eigenvalue	Modal Frequency/Hz	Damping Ratio
Mode 1 (λ2,3)	−3577.69 ± 268.31i	42.70	0.9980
Mode 2 (λ5,6)	−4.30 ± 423.97i	67.48	0.0101
Mode 3 (λ10,11)	3.67 ± 204.17i	32.46	−0.0183
Mode 4 (λ12,13)	−0.91 ± 187.94i	29.76	0.0050
Mode 5 (λ14,15)	−2.54 ± 163.29i	25.98	0.0156
Mode 6 (λ16,17)	−1.78 ± 97.78i	15.56	0.0182
Mode 7 (λ18,19)	−68.73 ± 76.30i	12.14	0.6693
Mode 8 (λ23,24)	−18.34 ± 24.13i	3.86	0.6051
Mode 9 (λ38,39)	−2.08 ± 359.01i	57.13	0.0058

presents the SSO model obtained by filtering the solution results, and the eigenvalues of the system state matrix are denoted as λ_i (i = 1–40).

The system comprises eight SSO modes and one supersynchronous oscillation mode. All modes, except for mode 3, have negative real parts, indicating that they are stable SSO modes. Mode 3 (λ_10,11) has a positive real part, signifying that it is an unstable SSO mode. The imaginary part of the eigenvalues reveals its oscillation frequency to be 32.46 Hz.

The oscillation modes of the SSO system and the factors influencing them are analyzed in this section using the participation factors. The participation factor of each mode provides a comprehensive measure of the level of interaction between the oscillation mode and the state variables. A higher participation factor indicates a stronger connection between the oscillation mode and the state quantities. The participation factors for each SSO mode are computed and presented in Table 2. The oscillation patterns of these modes are analyzed to identify the factors that influence the SSO.

First, premised on the modal frequencies, it is evident that the torsional modes of the unit are represented by mode 4 (λ_12,13), mode 5 (λ_14,15), and mode 6 (λ_16,17). From the thermal power shaft system parameters, the inherent frequencies of the shaft system are calculated to be 15.5 Hz, 29.93 Hz, and 25.98 Hz. Since the inherent frequency of the shaft system (set to ω_m) is typically not influenced by the mode of operation, its frequency magnitude remains relatively stable. Thus, the imaginary part of the eigenvalues divided by 2π can be compared to determine ω_m. The participation factor analysis is presented in Figure 4.

According to the results of the factor analysis of the participation factors mode 6 (λ_16,17), it can be inferred that the state variables exhibiting high participation rates comprise X₁–X₁₁. The state variables that correspond to X₁–X₅ signify the induction current of the synchronous generator, and the participation factors for these variables are 0.1329, 0.0395, 0.0495, 0.0769, and 0.0363, respectively. The sum of the partial participation factors for these variables is 0.3351. Conversely, the state variables corresponding to X₆–X₁₁ symbolize the shaft system component of the thermal power system, with the participation factors for these variables being 0.0476, 0.0374, 0.1274, 0.1167, 0.063, and 0.1434, respectively. The sum of the participation factors for these variables is 0.5355. Notably, the combined participation factors of the synchronous generator and the axial component reach 0.8706, which is indicative of the dominant role played by this oscillation mode. In addition, the state variables X₃₄ and X₃₅, corresponding to the voltage of the AC line series-complementary capacitor, as well as X₁₆–X₁₉, corresponding to the induction current of the asynchronous generator in DFIGs, also make varying contributions to the oscillation mode. Meanwhile, the participation of the remaining state variables is almost negligible. It can be inferred that the subsynchronous torsional interaction (SSTI) mode is generated by the combined effect of the turbine, the series-complementary capacitor, and the turbine unit shaft system. The subsynchronous frequency component generated by the wind turbine via the series-complementary grid is transmitted along the grid to the thermal power system. When the frequency of this subsynchronous component is in proximity to the complementary shaft system’s intrinsic frequency, it may trigger a severe torsional vibration of the shaft system. Furthermore, mode 4 (λ_12,13) and mode 5 (λ_14,15) can be similarly analyzed and deduced to represent the SSTI.

Second, mode 2 (λ_5,6) and mode 3 (λ_10,11) form a group of electrical resonant modes whose the oscillation frequencies are symmetrically distributed about 50 Hz. The imaginary part of the eigenvalue is associated with an oscillation frequency that can be expressed as 50 ± f_e, where f_e signifies the electrical resonant frequency of the system.

The analysis focuses on the oscillation modes of mode 3 (λ_10,11) (with an oscillation frequency of 32.46 Hz), as they exhibit similar modes to mode 2 (λ_5,6). The participation factor analysis is presented in Figure 5. The participation factor analysis reveals that X₁₆–X₁₉, representing the state variable of the DFIG asynchronous generator induction current, account for the primary factors, with participation factors of 0.1336, 0.1343, 0.1251, and 0.1235, adding up to 0.5165. The synchronous generator induction currents of the thermal system, represented by X₁ and X₄, have a participation factor sum of 0.178. Additionally, X₃₄ and X₃₅, corresponding to the series-complementary capacitor voltages, have a participation factor sum of 0.1303. These three components have a dominant role, with a total participation factor of 0.8248. Moreover, the shaft system and the DC-side parameters of the DFIG converter also contribute to the dominant oscillation mode of the induction generator with DFIG participation, which can be inferred as an IGE.

Third, the participation factor analysis of mode 1 (λ_2,3) and mode 9 (λ_38,39) is presented in Figure 6. The participation factor analysis of mode 1 (λ_2,3) reveals that the turbine stator-side current state variables, X₂₄ and X₂₅, have high participation factors of 0.4826 and 0.4824, respectively, with a total sum of 0.965, indicating their dominant role. In addition, X₃₀ and X₂₇ have low participation factors of 0.0157 and 0.0134, respectively, corresponding to the DFIG stator-side converter reactive power control strategy and the DC-side capacitance. Notably, the absence of the axis system and series complement suggests that the turbine-related state variables are the primary cause of the oscillation, possibly due to the turbine being connected to weak grid conditions, indicating a possible SSO.

The participation factor analysis of mode 9 (λ_38,39) reveals that the d-axis components of the induced currents on the stator and rotor sides of the DFIG, represented by X₁₆ and X₁₈ with participation factors of 0.3034 and 0.2958, respectively, are the primary factors, with a total participation factor sum of 0.5992. Moreover, X₃₉ and X₄₀, representing the state variables for the DC transmission line component, have a participation factor sum of 0.1169. These four factors have a dominant participation factor sum of 0.833. In addition, the DFIG stator-side control parameters and the induced current also contribute. Notably, the shaft system state variables have a negligible participation factor of almost 0, indicating that this oscillation mode is a subsynchronous control interaction (SSCI) mode caused by the interaction of the turbine and the DC converter. The corresponding SSOs may be triggered when the turbine output power is fed out via the DC.

The analysis of the subsynchronous frequency eigenvalues indicates the presence of a SSO problem, in which multiple oscillation forms coexist due to the joint participation of thermal, wind, DC, and series-complement components. The complexity of this problem is reflected in the diversity of oscillation sources, as well as the interaction and complex coupling between multiple components. Additionally, various factors such as the network configuration of the machine, random changes in wind speed, and different control parameters may trigger these oscillations.

4. Analysis of the SSO Characteristics of the System

This section explores the impact of the various operational modes and adjustments to control parameters on the SSO characteristics, based on the participation factors established in Section 3. The principal factors considered comprise system series complementarity, wind and thermal power generation, DC transmission power, grid strength, DFIG control parameters, wind speed, etc.

4.1. The Impact of Various Operational Modes on the SSO Characteristics

(1)

Impact of Series Compensation on the SSO Characteristics

In an actual system, the series compensation escalates and alters as the transmission distance and capacity increase. This section emphasizes the impact of series compensation on the SSO characteristics of the system. With other parameters held constant, the system series compensation varies within a range of 10% to 90%. Utilizing a 10% compensation increment, the corresponding eigenvalues, damping ratio, and oscillation frequency are calculated sequentially with the acquired state space matrix. Figure 7 presents the system eigenvalue computation results for the variation of system series compensation. As observed in Figure 7, the system eigenvalues markedly increase and the overall trend of the system damping ratio diminishes as the degree of series compensation progressively rises. When the system series compensation exceeds 15%, the system damping ratio transitions from positive to negative, causing the system to oscillate. It can be deduced that the system tends toward instability with the increase in system series compensation, and the oscillation frequency significantly declines.

Figure 8 presents the simulation results of the system with various series compensations, which further verifies the aforementioned findings.

Upon introduction of series compensation at 4 s, when the series-complement degree is 50%, the simulated waveform experiences violent oscillations, which subsequently dissipate. This can be attributed to the strong negative damping of the system at the relatively high series compensation degree. Conversely, when the series compensation degree is 30% and 20%, the waveform oscillates, but with a delay in the oscillation time, suggesting an increase in the damping ratio of the system. These results are consistent with the eigenvalue analysis and validate its accuracy. Additionally, the waveform with 50% of the series compensation degree is locally enlarged, and the frequency of oscillation can be approximately deduced from the number of peaks and troughs in a specific time period, which is approximately 27 Hz. This finding aligns with the relationship between the series compensation degree and the frequency shown in Figure 7b, further reinforcing the accuracy of the analysis.

Based on Figure 7a, it can be observed that there is an anomaly in the relationship between the damping ratio of the SSO oscillation mode and the degree of series compensation. Specifically, a sudden decrease in the damping ratio occurs at 70% and 80% of the degree of series compensation, which deviates from the overall trend. There are many SSO modes in the complex power system studied in this paper. The oscillation frequency of the system decreases with the increase in series compensation. When the series compensation is 70%, the oscillation frequency is about 24 Hz, which is complementary to the natural frequency of mode 5 (λ_14,15). Further analysis reveals that the SSTI mode oscillation is excited at 70% and 80% of the degree of series compensation, resulting in the system exhibiting both the SSTI oscillation mode associated with mode 5 (λ_14,15), having a frequency of 25 Hz, and the SSR oscillation mode associated with mode 3 (λ_10,11), having a frequency of 23 Hz. The damping ratios of the two modes are calculated to be −0.0296 and −0.0159, with the SSTI mode being the dominant mode that affects the damping ratio of the SSR counterpart. A fast Fourier transform (FFT) analysis of the system power waveform at 70% series compensation, as depicted in Figure 9, confirms the dominance of the subsynchronous component at 25 Hz, with the subsynchronous component at 23 Hz being slightly less abundant. This is consistent with the results obtained from the eigenvalue calculation.

(2)

The Impact of Wind Power Output on the SSO Characteristics

The impact of wind turbine output power magnitude on the system SSO characteristics, in addition to series compensation, must also be considered. To investigate this impact, the DC transmission power is maintained at a constant level with the thermal power plant output power, while the wind farm output power is adjusted in sequential 0.1 pu output power increments. The wind farm output power is gradually increased from 10% of the rated power to the rated output power. The results of the system eigenvalues are then calculated for the different wind power output levels. Through this analysis, the changes in system damping ratio and frequency are observed and their influence on the system SSO characteristics is studied. Figure 10 illustrates the variation of the damping ratio and frequency with wind turbine output.

As demonstrated in Figure 10, the system eigenvalues decrease gradually with the increasing wind turbine output. As a result, the damping ratio improves and the system stability increases. It is important to note that the frequency remains relatively constant with changes in wind turbine output. Thus, the wind power output does not significantly affect the system oscillation frequency.

(3)

The Impact of Thermal Power Output on the SSO Characteristics

Let the DC transmission power and wind farm output power remain constant, and adjust the output power of each thermal power plant sequentially, still at 0.1 pu output power intervals, so that it progressively increases from 10% of the rated power to the rated output power. The characteristic values of the system parameters at different output powers are computed separately to study their impact on the system SSO characteristics. Figure 11 displays the variation of the damping ratio and frequency concerning the thermal power output. As observed in Figure 11, the system characteristic value gradually increases, the damping ratio progressively decreases, and the system tends toward instability as the thermal power output rises. The frequency first declines and subsequently ascends with the enhancement of thermal power output, yet the overall variation range remains minuscule and negligible. It is evident that an increase in thermal power output does not alter the system oscillation frequency but diminishes the damping ratio of the system, rendering it more susceptible to oscillation.

(4)

Impact of DC Transmission Power on the SSO characteristics

The output power of all thermal power plants and wind farms is maintained constant, while the DC transmission power is sequentially adjusted to incrementally increase from 10% of the rated transmission power to the rated transmission power at intervals of 0.1 pu transmission power. The characteristic values of the system parameters under varying transmission powers are computed to investigate their influence on the SSO characteristics of the system. Figure 12 illustrates the variation of the damping ratio and frequency with respect to DC transmission power.

From the analysis presented in Figure 12, a gradual decrease in the system eigenvalue, coupled with an increase in the damping ratio, is observed as the DC transmission power increases, ultimately leading to system stability. In addition, while the frequency undergoes a slight decrease with an increase in the DC transmission power, the variation range remains negligible. As such, it can be inferred that the DC transmission power does not have a significant impact on the system oscillation frequency. However, an increase in the DC transmission power results in an enhancement of the system damping ratio, thereby contributing to its improved stability. Subsequently, the waveform of wind turbine speed, as depicted in Figure 13 at varying DC transmission power levels, further supports these observations. Specifically, a decrease in the DC transmission power leads to an increase in speed waveform fluctuation, thereby reducing the system stability, consistent with the eigenvalue analysis findings.

(5)

The Impact of Single Turbine Power Output (Wind Speed) on the SSO Characteristics

By increasing the active power output of the single turbine from 0.1 pu to 1.0 pu while maintaining all other conditions constant, the study reveals that the damping ratio of SSOs gradually transitions from negative to positive values, indicative of a shift from an unstable to a stable system. Moreover, the oscillation frequency increases slightly, as illustrated in Figure 14. The findings suggest that higher wind speeds lead to an increase in the stability of the system.

The verification of the conclusion presented earlier was performed through simulation, while keeping all other working conditions constant. The standalone output power at various wind speeds (i.e., 10%, 30%, 70%) were taken as the inputs, and the corresponding active output of the system was observed for each input value, as depicted in Figure 15. As the wind turbine output increased (i.e., wind speed increased), the dissipation of the system decreased, resulting in delayed oscillation instability and a subsequent reduction in oscillation amplitude. Ultimately, the system tended towards stability, which is consistent with the eigenvalue calculation results.

(5)

The Impact of Grid-connected Wind Turbine Ratio on the SSO characteristics

With all other conditions remaining constant, the grid-connected ratio of a DFIG was gradually increased from 10% to 100%. The results, as shown in Figure 16, indicate that when the proportion of grid-connected wind turbines is close to 30% and the damping ratio of oscillation is less than 0, which leads to unstable SSOs. However, it is observed that as the grid-connected wind turbine ratio increases, the damping ratio SSO first decreases and then slightly increases, and the oscillation frequency decreases. Therefore, it is possible that the oscillation damping ratio and the number of wind turbines exhibit a nonlinear relationship. In conclusion, the stability of the system may first be weakened and then enhanced with the increasing proportion of DFIG connected to the grid.

4.2. The Impact of Control Parameters on the SSO Characteristics

(1)

The Impact of Wind Turbine Control Parameters on the SSO Characteristics

Firstly, the impact of the internal loop proportionality coefficient K_{p_Ird} of the converter current on the rotor side of the wind turbine on the SSO is investigated. The coefficient is altered within a range of 0.25 to 5 times the initial value (0.2). Figure 17 displays the curves representing the frequency and damping ratio variations in relation to K_{p_Ird}. As K_{p_Ird} increases, it becomes evident that the SSO frequency first declines and subsequently rises; however, the overall alteration remains below 0.1 Hz, which can be regarded as essentially invariant. Conversely, the damping ratio consistently declines as K_{p_Ird} escalates, causing an increase in the system eigenvalue and a decrease in stability.

Similarly, the investigation proceeds to analyze the impact of the rotor-side converter current inner loop integration coefficient, K_{i_Ird}, on the SSO. K_{i_Ird} is varied within the range of 0.25 to 5 times the initial value (20). As depicted in Figure 18, the damping ratio of this oscillation mode first increases and then decreases as K_{p_Ird} increases, but the overall variation range is approximately 0.001, which can be disregarded in comparison to K_{i_Ird}. Moreover, the frequency experiences a slight decrease with the increase in K_{i_Ird}, but the transformation range is still small enough to be considered unaffected by K_{i_Ird}. Hence, the variation of K_{i_Ird} can be regarded as having a negligible impact on this oscillation mode.

The outer loop parameters of the rotor-side converter and the PI parameters of the stator-side control loop yielded similar results. Specifically, it was observed that the changes in these parameters had a negligible impact on the oscillation frequency and damping of the system.

The verification of the aforementioned findings was accomplished via simulation. Figure 19 displays the simulation results for the system under varying values of K_{p_Ird}. Specifically, when K_{p_Ird} is increased, the system exhibits faster divergence and a larger amplitude of oscillation. This corresponds with the previously obtained eigenvalue calculations, indicating an inclination towards system instability.

(2)

The Impact of HVDC Control Parameters on the SSO Characteristics

In addition to the DC transmission power, the impact of HVDC transmission on the SSO characteristics of the system is primarily determined by two parameters—the rectifier-side control loop proportionality coefficient K_{p_i} and the integration coefficient K_{p_i}. While maintaining the other control system parameters constant, adjustments to K_{i_i} and K_{i_i} were made to observe their effects on the system SSO characteristics. A calculation of the damping ratio revealed minimal variation, with no evident patterns based on the changes in the two aforementioned parameters. These results are illustrated in Figure 20. The simulated waveforms are nearly identical, validating that the effect on the damping ratio and frequency of the system is insignificant, which is consistent with the characteristic value calculation results.

5. Conclusions

In this paper, an investigation was conducted into the characteristics of SSOs in multitype power sources transmitted through an UHV AC–DC hybrid transmission system. The analysis was carried out using eigenvalue analysis and participation factor analysis. The objective was to investigate the effects of diverse system operational modes and control parameters on the damping of SSOs. The research findings are summarized as follows:

(1)

A state-space model of a small-signal and high order 40 was constructed for multitype power sources transmitted through an UHV AC–DC hybrid transmission system, using the modularized chunking modeling approach. This modeling method is suitable for the analysis of SSOs in intricate systems.

(2)

When the operation mode of the system is not appropriate, or the parameters are not selected correctly, multiple types of power sources would send unstable SSO through the UHV AC–DC hybrid transmission system, and lead to a problem of subsynchronous oscillation with the coexistence of multiple oscillation forms involving thermal power, wind power, DC, and a series compensator. In the case of thermal power units, torsional oscillation interactions occur between the grid inductance, series compensation capacitance, DC, and turbine unit shaft systems. On the other hand, for wind farms, control-induced subsynchronous interactions exist between the wind farms and series compensation capacitance, as well as DC transmission.

(3)

The results of the eigenvalue analysis and simulation verification indicate that the SSO characteristics are greatly affected by various factors. Among these factors, the degree of system series compensation has the most significant influence. As the degree of system series compensation increases, the system tends to become unstable, and the oscillation frequency decreases considerably. The system characteristic value gradually decreases as the turbine output increases or the thermal power output decreases while keeping the output power of other sources constant. Furthermore, the damping ratio increases, leading to an improvement in system stability. The system stability decreases gradually as the DC transmission power declines. Conversely, the system stability is enhanced as the wind speed increases. When the proportion of doubly fed turbines connected to the grid increases, the system stability may first weaken and then subsequently be enhanced.

(4)

Concerning the control parameters, the system stability is affected by the internal loop proportionality coefficient K_{p_Ird} of the rotor-side converter current of the wind turbine. Specifically, the larger the value of K_{p_Ird}, the more unstable the system becomes. In contrast, the rotor-side converter outer-loop parameters and the stator-side control loop PI parameters do not have any impact on the system oscillation frequency and damping.

This paper focuses on the modeling of complex power systems and the analysis of SSO characteristics, but does not give specific suppression measures on SSOs. The future research direction is the design of SSO active damping control devices and the protection control after an SSO occurs. The specific system structure, control method, and measured power data will be combined for further analysis, and more effective SSO suppression measures will be proposed.