1. Introduction
In recent years, as a clean, renewable and relatively proven technology, wind power generation has grown significantly in order to tackle the climate change and replace fossil fuels generators. By the end of 2019, the cumulative installed capacity of wind power worldwide reached 650 GW, of which 60.4 GW was newly added [
1]. With the development of wind power and high voltage direct current transmission system (HVDC), subsynchronous interaction (SSI) has attracted the attention of academia and industry. The SSI is generally classified into the following three types: subsynchronous resonance (SSR), subsynchronous control interaction (SSCI) and subsynchronous torsional interaction (SSTI) [
2]. In 2009, an SSI incident occurred in southern Texas, USA. A doubly fed induction generator (DFIG)-based wind farm was integrated into the grids via a high-series compensation transmission line. This caused a subsynchronous control interaction, resulting in a large number of wind turbine trips [
3,
4]. In 2012, the Guyuan wind farm in China also experienced the interaction between the control of DFIG and series compensation devices, causing the SSI event.
With the increase of grid-connected wind power capacity and the use of long-distance transmission lines, the support from the grids for wind farm connection is weakened [
5]. There is a different type of interaction observed recently between full-converter wind farms and weak AC networks. In 2015, the permanent magnet synchronous generators based wind farm in Xinjiang, China, suffered from a severe oscillation event without series compensation. The oscillation frequency coincided with the torsional vibration frequency of the nearby thermal power unit, which led to the torsional vibration protection action of the thermal power unit resulting in generator shut down [
6]. This type of interaction between full-converter wind farms and weak AC networks is also called subsynchronous oscillation (SSO), which is the topic investigated in this paper.
The dominant elements that affect the subsynchronous interaction characteristics in different scenarios of wind farms connected to the grids [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25]. References [
7,
8,
9] established DFIG-based wind farms interconnected with the grids and analysed the influence of the number of wind turbine generators (WTGs), wind speed, series compensation, line resistance, and outer and inner loop control parameters on subsynchronous interaction. For instance, reference [
8] points out that when the DFIG-based wind farm is connected to series capacitive compensated transmission systems, the system damping decreases with the rise of series compensation or the decrease of total line resistance. Meanwhile, the variations of series compensation also affect the oscillation frequency of subsynchronous interaction. As for SSO in the full-converter wind farm or the VSC connected to AC grid system, the eigenvalue analysis, impedance-based analysis and the complex torque coefficient approach are conducted in [
6,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20] to research the dominant elements that affect the SSO characteristics. AC system strengths, WTGs number, wind speed, converter control parameters, PLL parameters and aggregation characteristics of wind farms are considered in these works. The work in [
6] indicates the SSO will occur with the decrease of the AC system strengths. And control parameters also have great impacts on the SSO characteristics. In addition, SSO caused by the interconnection of direct-drive wind farms via voltage source converter based high voltage direct current (VSC-HVDC) transmission system has been studied in the references [
21,
22,
23,
24,
25,
26,
27,
28]. These elements, including wind farm control parameters, HVDC control parameters, PLL parameters and filter parameters are analysed and the coordinated controller is designed.
Until recently, there were very few papers specifically analysing the impact of the active power output of wind farms on the SSO characteristics. However, the change of active power output during the operation of wind farms will have a more significant impact on system stability. References [
17,
18,
19,
20] established the model of full-converter wind farm integrated into the grids or the VSC connected to AC grid. The SSO characteristics of the system are studied, and the impact of active output is analysed. The works in [
17,
18,
19] pointed out that as the active power output of wind farms increases, the damping of the SSO mode decreases. However, it is revealed in [
20] that increasing the active power output of wind farms will increase the damping of the SSO mode and increase system stability. When the active power output is too small, the system will result in diverging oscillation and loss of system stability.
In the view of the impact of active power output on SSO characteristics, some studies identified that the greater the active power output, the worse the SSO damping characteristics will be [
17,
18,
19]. However, some studies that found that the higher the active output, the better the SSO damping characteristics will be [
20]. Meanwhile, the existing researches are based on a certain set of control parameter without considering the influence of different control parameters. Therefore, it is necessary to study further the relationship between active power output and damping characteristics of SSO mode.
This paper investigates the impact of active power output on SSO characteristics by a small-signal analysis based on analytical models. The correlation between the active power output and the damping of the SSO mode with different control parameters is analysed through dynamic modelling and linear system analysis. First, the critical factor that determines the correlation is identified. Then, based on the eigenvalue analysis results, the strategy to increase the damping of SSO mode and improve system stability is proposed. Case studies and time-domain simulation verify the analysis result.
The rest of the paper is organized as follows:
Section 2 builds the dynamic model of the system with full converter wind farm connected to the AC grids. In
Section 3, both eigenvalue analysis and calculation of participation factors are carried to study the impact of active power output on SSO characteristics. The correlation between the active power output and the damping of the SSO mode is analysed with different control parameters and the critical factors that affect the SSO characteristics are presented. Meanwhile, the strategy to improve the stability of the system is proposed.
Section 4 presents case studies and time-simulation results. Finally, the brief conclusions are given in
Section 5.
2. System Modeling
A full-converter wind model including wind turbine, synchronous generator (SG), machine-side converter (MSC), DC link, grid-side converter (GSC), phase-locked loop (PLL), and converter control system is considered. It is assumed that wind farms usually consist of the same type of wind turbines with similar control parameters and operating conditions. Therefore, a wind farm is represented by an equivalent wind turbine. The schematic diagram of grid-connected wind power system structure is shown in
Figure 1.
Lf1 and
Rf1 represent the filter inductance and filter resistance, respectively.
C1 represents the reactive power compensation parallel capacitor.
R2+j
X2 represents the equivalent impedance of both 25 kV line and 220 kV line.
R3+j
X3 represents the equivalent impedance of the transmission line near the grids.
vpcc denotes the voltage of point of common coupling (PCC).
vgrid denotes the infinite grid voltage.
i1 and
i2 are the grid-side output current and transmission line current, respectively. Since the grid-connected dynamics of full-converter mainly depends on the control features of GSC, this paper ignores the machine-side dynamics. The wind turbine, SG and MSC are simplified as constant power sources [
6].
The following section will establish a dynamic mathematical model of the grid-connected system. There are two
dq reference frames in the dynamic mathematical model, namely the PLL-based
dq frame and the grid-based
dq frame. The PLL-based reference frame aligns its d-axis with the PCC voltage space vector
vpcc through the PLL output phase. Meanwhile, the grid-based reference frame has its d-axis aligned with the grid voltage space vector
vgrid [
10,
17]. Superscripts ‘c’ and ‘g’ represent variables in the PLL-based reference frame and the grid-based reference frame, respectively. Phasor diagram of the component in different reference frames is shown in
Figure 2.
2.1. Modeling of DC-Link
Since the machine-side dynamics are ignored, it is assumed that the active power output of the generator remains constant and is represented by
Pwind. The dynamic mathematical model can be obtained from the DC link active power balance equation as Equation (1).
Pg and Pbase are the GSC power delivered to the grids and base power, respectively. Vdc and Vdc,base are expresses as DC voltage and rated DC voltage, respectively. Superscript ‘pu’ represents per unit variables. Subscripts ‘d’ and ‘q’ respectively notate the d-axis and q-axis components of variables. Hereafter the dc-link dynamic mathematical model is expressed by Equation (2). For convenience, the superscript ‘pu’ is omitted.
2.2. Outer and Inner Control Loop of GSC
The GSC control block diagram is shown in
Figure 3. DC-link voltage control (DVC) and reactive power control are adopted for GSC, which contributes to balancing the power flow through DC link, maintaining DC-link voltage and operating at unit power factor for wind farm. The dynamic mathematical model of the outer and inner loop can be expressed as
where
x1,
x2 and
x3 represent intermediate state variables.
Kpdc and
Kidc notate DVC proportional gain and integral gain, respectively.
Kpi and
Kii are the proportional gain and integral gain of the inner current control loop, respectively. Subscript ‘ref’ denotes the system reference value.
2.3. Phase-Locked Loop Model
PLL uses the three-phase voltage at PCC bus as inputs to obtain the phase of the PCC voltage to achieve synchronization between the wind farm and the grids. The control block diagram of the PLL is illustrated in
Figure 4. The PLL principle has been well documented [
26] and will not be discussed here.
ω0 represents the rated angular frequency of the grids. Δ
ω notates the frequency deviation.
θpll is the voltage phase of the PLL output.
Kppll and
Kipll denote the PLL proportional gain and integral gain, respectively. PLL dynamic mathematical model can be expressed as
2.4. Grid Dynamics
The grid dynamics mainly include shunt capacitor dynamics, filter inductance dynamics, and transmission line equivalent inductance dynamics. The dynamic mathematical model of the grid is established in the grid-based reference frame. The grid dynamic mathematical model can be written as Equation group (7):
Rg and Lg denote the total equivalent resistance and inductance of the grid, including the transformers and the transmission lines. The impedance from the PCC to the grid can be represented as a single impedance Rg + jω0Lg, Rg = RT1 + RT2 + R2 + R3, Lg = LT1 + LT2 + L2 + L3.
5. Conclusions
This paper investigates the influence of active power output on subsynchronous oscillation characteristics in weak grids. Compared to the research in the literature, this is the first of its kind to investigate the distinctive correlations between active power output and the damping of the SSO mode. The reasons for the different correlation between active power output and SSO mode damping have been explained. The findings and contributions of the study include:
The change of active power output in one direction can either improve or reduce SSO mode damping. This work identifies that the correlation between active power variation and damping mainly depends on the proportional gain of the phase-locked loop (PLL).
When the PLL proportional gain is large, the active power output is negatively correlated with the damping of the SSO mode. When the PLL proportional gain is small, the active power output is positively correlated with the damping of the SSO mode. This clarifies the confusions in the understanding of the correlation between active power output and SSO damping.
The PLL integral gain and the DC voltage control proportional gain have little influence on the correlation between the active power output and SSO damping. However, the system stability can be improved by appropriately retuning the PLL integral gain and the DC voltage control proportional gain.
There is a critical range for the PLL proportional gain, in which SSO damping is near consistent irrespective to the change of active power variation. The influence of active power output on the stability can be minimized by selecting proper the PLL proportional gain first when the damping variation is at the critical range. Then adjustment of other parameters will improve the stability. This is valuable for engineering applications in designing PLL parameters.
For full-converter wind power systems, the grid-connected dynamics mainly depend on the control of GSC and are not affected by the wind turbine types. The conclusions of this paper are applicable to full-converter wind farms with induction generators or permanent magnet synchronous generators. DFIG is not covered in the study, and the analysis of the DFIG-based wind farms and the auxiliary control design will be undertaken in future research.