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Faced with the problem of sub-synchronous oscillation (SSO) caused by the interaction between permanent magnetic synchronous generator (PMSG)-based wind farms and weak AC grids, we construct a transient energy function model that follows the structure of a PMSG. The transient energy composition of a PMSG is analyzed, and the dissipation energy expression that can intuitively reflect the development trend of SSO is derived, reflecting the damping level of a grid-connected wind power system. Furthermore, in response to the problem of mutual coupling between the control links of the PMSG during the SSO process, which hinders oscillation characteristic assessment, a method based on oscillation energy is proposed to analyze the oscillation characteristics. Considering the dynamic changes in the output of the phase-locked loop during sub-synchronous oscillation, the transient energy dominated by various control links is derived, and the effects of the phase-locked loop, current inner loop, and voltage outer loop on transient energy and oscillation characteristics are analyzed. The simulation verifies the effectiveness of the analysis of the transient energy model.
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With the rapid development of new energy represented by wind power, permanent magnet synchronous generators (PMSGs) have been widely used because of their high efficiency and low failure rate. A PMSG employs the power electronic converter as the interface for the power grid, which makes its power generation dynamic characteristics disparate from a traditional synchronous generator. When the converter interacts with the power grid, it can induce resonance or oscillation issues that potentially trigger unit tripping, cause equipment damage, and compromise the safe and stable operation of the power grid (
Several techniques have been currently employed for analyzing SSO in wind power grid-connected systems, including eigenvalue analysis, complex torque coefficient analysis, impedance analysis, and time-domain simulation, and henceforth, certain results are obtained. Some references use the eigenvalue analysis to study sub-synchronous oscillation, which investigates the system’s dynamic response by solving the eigenvalue of a linearized model for small disturbances (
SSO in power systems occurs when there is an energy exchange between the oscillating units and the system at a frequency that is lower than the synchronous frequency, which is manifested as power oscillation. Recently, low-frequency oscillations and SSO based on the transient energy method have been studied (
We have analyzed the SSO of a PMSG by employing the transient energy flow method, and the contributions of different time scale control links to the SSO process have been examined. On the grounds of the relevant mathematical model, the transient energy flow calculation formula applicable to a PMSG is derived in
The machine-side converter of a PMSG is decoupled from the grid, and the time scale of wind speed change is far greater than the time scale of converter control. The influence of the grid-side converter (GSC) of a PMSG should be emphatically considered while analyzing SSO in grid-connected wind power systems (
Configuration of the PMSG’s GSC connected to the grid.
Spatial position and phase relationship of the coordinate system.
The GSC of a PMSG includes a DC capacitor and an AC inductor, which are two energy storage components to realize energy exchange with the grid. To maintain the stability of the state variable of the DC capacitor and AC inductor, DC voltage control and AC current control are designed for the GSC (
The GSC utilizes the phase-locked loop (PLL) to sense the phase of point of interconnection voltage
Structure of the PLL.
Owing to the different energy storage capacities of the DC capacitor and AC inductor, the bandwidths of control loops designed according to their state variable are also different. When disturbances occur in the power grid, both the DC capacitor voltage and AC inductance current of the converter change and drive each controller to act. Since the bandwidths of both the DC voltage and AC current controllers differ, the speed and sequence of their response to grid disturbances are also different; i.e., the PMSG control system shows the characteristics of sequential action.
Owing to the sequential action of the controller, the response process of the PMSG to different time scale disturbances is different; i.e., the PMSG exhibits multiple time scale dynamic response characteristics to the power grid. According to the characteristics of energy storage elements and the response speed of their corresponding controllers,
Dynamic time scale division of inverter control.
The essence of oscillation is the accumulation and consumption of disturbance energy between devices. The energy function construction is based on the pure mathematical definition, but the potential energy and dissipated energy contained in the energy function have the same expression as the energy in the real physical process. In the work of
We expand Eq.
Equation
In Eq.
The transient energy of the PMSG in the dq-axis coordinate system is given as
According to
By measuring the power, voltage, and current information of the PMSG, the transient energy and its variation trend of the PMSG can be calculated.
The transient energy of the electrical device obtained through integration cannot be distinguished between potential and kinetic energy. The transient energy of a component consists of two parts. One is a conservative term that is independent of the path, corresponding to the periodic variables, which reflects the periodic change of the transient energy of the electrical device, i.e., kinetic and potential energy over time. The other part is a non-conservative term related to the path, corresponding to the non-periodic variables, which reflects the transient energy consumed by the electrical device, which corresponds to damping. This part of energy is defined as the dissipated energy.
The integration-based approach for obtaining the transient energy of an electrical device cannot differentiate between potential energy and kinetic energy, which is not conducive to analyzing the characteristics of the electrical device. Dissipated energy corresponds to damping in a physical sense. When SSO occurs, the change in dissipated energy determines the damping level of SSO, which is the main basis for analyzing the SSO. In actual calculations, each variable can be represented by its deviation from the steady-state value.
Assuming that there is a symmetrical sub-synchronous sinusoidal disturbance current in the three-phase current output by the GSC of the PMSG, the phase current and voltage can be described as
To obtain the PMSG’s transient energy, we first convert the voltage and current values of the PMSG from the static abc coordinate system to the rotating dq-axis coordinate system. Considering the PLL response, the voltage and current converted to the rotating dq-axis coordinate system are as follows:
The dissipated energy is the integral of the non-periodic component in the transient energy expression. We calculate the derivatives
Substituting Eqs
Therefore, the dissipated energy at the GSC of the PMSG is given as
Equation
During the process of sub-synchronous oscillation, the interdependence of each control link of the GSC complicates the analysis of oscillation characteristics. Here, the dynamic effects of the phase-locked loop, current inner loop, and DC voltage outer loop control are considered, and the influence of different time scale control links on transient energy and oscillation characteristics is studied.
According to Eq.
While SSO occurs, the PLL phase angle is not equal to the steady-state voltage phase angle at the measuring point. This induces a misalignment between the dq-axis coordinate system in the control system and that of the power grid, and the included angle is the variation of the phase-locked angle
The transformation relationship between the voltage or current in the dq-axis coordinate system of the control system and the voltage or current in the dq-axis coordinate system of the power grid is
Simultaneous Eqs
Since
To calculate
Equivalent control system diagram of the GSC of the PMSG.
In the dqs-axis coordinate system of the control system, which is obtained from the current inner loop control equation,
Simultaneously, Eqs
Since
Then,
According to Eq.
In engineering, the control bandwidth of the current loop is usually about twice that of the PLL and 10 times that of the voltage outer loop control (
Furthermore, calculating the differential of the relevant terms coupled with the PLL control, it is considered that
It is assumed that the current loop and PLL control have achieved steady state when calculating the differential of the voltage outer loop control output, which yields the expression
From the simultaneous Eqs
Here,
The other items are the transient energy items that are directly affected by the control link. The second item of the expression is the transient energy dominated by the current loop control. The third item is the transient energy controlled by the PLL, and the fourth item is the transient energy controlled by the DC voltage outer loop.
The third item is the transient energy dominated by the PLL control, whose positive and negative values mainly depend on the PLL link, though the magnitude of the transient energy is still affected by the current loop. The GSC of the PMSG uses a decoupling control technique for the active and reactive power, which causes the d-axis current to govern the output of active power and the q-axis current to govern the output of reactive power:
By combining Eq.
Therefore, the influence of the controls of the PLL, current loop, and DC voltage outer loop of the GSC on the transient energy characteristics of the PMSG is obtained. In the case of the transient energy
In PSCAD/EMTDC, a simulation example system is built to demonstrate the integration of a direct drive wind farm with a weak AC power grid. The wind farm’s overall generating capacity is 900 MW (1.5 MW*600), and the terminal voltage of the PMSG is 0.69 kV. Furthermore, the power grid is connected after the voltage rise (0.69 kV/220 kV). We set different operating conditions to measure the transient energy and dissipated energy of the PMSG and check the reliability of the PMSG’s transient energy model. Parameters of the GSC of the PMSG are shown in
Parameters of the GSC of the PMSG.
Symbol | Quantity | Values |
---|---|---|
|
Rated output power of the wind turbine | 1.5 MW |
|
Rated voltage of the wind turbine | 0.69 kV |
|
Rated frequency | 50 Hz |
|
DC voltage control reference value | 1.15 kV |
|
DC capacitor | 0.09 F |
( |
DC voltage outer loop control parameters of the PMSG | (10, 1,000) |
( |
Phase-locked loop control parameters | (2, 50) |
( |
AC current loop control parameters | (0.2, 50) |
We set different oscillation types, (1) Sub-synchronous divergence oscillation occurs in the grid-connected wind power system
At an operational condition of 4 m/s, the AC grid’s equivalent impedance changes, where the grid strength is reduced from SCR = 3.4 to 1.5 at t = 5s, thus causing divergent SSO in the example system. The transient and dissipated energy of the PMSG are measured.
The upper half of (2) Sub-synchronous convergence oscillation occurs in the grid-connected wind power system
Power waveform, transient energy, and dissipated energy waveform of the PMSG under the divergent SSO in the system.
At an operational condition of 7 m/s, the AC grid’s equivalent impedance changes, where the grid strength is reduced from SCR = 3.4 to 1.5 at t = 5s, thus causing convergent SSO in the example system. The transient and dissipated energy of the PMSG are measured.
The upper half of
Power waveform, transient energy, and dissipated energy waveform of the PMSG for the sub-synchronous convergence oscillation in the system.
As mentioned above, the dissipated energy of the PMSG can be employed to gauge the damping level of the grid-connected wind power system besides understanding the sub-synchronous oscillation characteristics.
The direct drive wind farm connected to the weak power grid faces SSO risk. Since there are differences in the operating conditions of each sub-wind farm, their influence on the SSO characteristics can also vary. Here, it is assumed that the operating conditions of the PMSGs in each sub-wind farm are identical.
We divide the 600 PMSGs in the example system into two sub-wind farms, each with 300 PMSGs. The operating condition of sub-wind farm 1 is 4 m/s, and the operating condition of sub-wind farm 2 is 7 m/s. The transient energy and dissipated energy of the PMSGs of two sub-wind farms are measured.
According to
Power waveform of sub-wind farm 1 and sub-wind farm 2.
According to
Transient energy curve of sub-wind farm 1 and sub-wind farm 2.
Therefore, the corresponding manifestation mode of SSO is power oscillation, which primarily depends on the energy exchange between the wind farm and the power grid.
The GSC parameters of the PMSG are provided in
Transient energy of each control link of the PMSG.
According to (1) The impact of the proportional coefficient of the current inner loop on transient energy
We change the proportion coefficient (
Power waveform of the PMSG and transient energy of each link of the PMSG when
According to (2) The influence of the integral coefficient of the current inner loop on transient energy
We change the integral coefficient (
Power waveform of the PMSG and transient energy of each link of the PMSG when
Comparing
Thus, the participation of the PMSG in the SSO is primarily associated with the control of the GSC. The transient energy dominated by each control link of the GSC is derived with the support of simulation results.
In the process of SSO, the transient energy contributed by the PLL is positive, which increases the transient energy of the PMSG and easily leads to oscillation divergence. The transient energy contributed by the current loop is negative, which decreases the transient energy of the PMSG and is beneficial to the oscillation convergence.
Both the order of magnitude of the transient energy contributed by the voltage outer loop and its effect on SSO are negligible.
For the GSC’s current loop parameters of the PMSG, increasing the proportional coefficient and decreasing the integral coefficient can reduce transient energy, which is beneficial for suppressing oscillations.
Starting from the physical essence, this article describes the dynamic development process of SSO by employing the transient energy and, based on that, constructs a method for analyzing SSO. Focusing on the coupling effects between control links at different time scales, the transient energy dominated by various control links is derived, and the effects of the PLL, current inner loop, and voltage outer loop on transient energy and oscillation characteristics are analyzed. The results of the simulation demonstrate that the transient energy analysis can provide valuable insights into the SSO characteristics of the grid-connected wind power system, and the impact of each PMSG under different operating conditions on the sub-synchronous oscillation characteristics of the system can be understood. In the control link of the GSC, the transient energy contributed by the PLL and current loop plays a dominant role in the transient energy characteristics, determining the characteristics of SSO.
A striking number of PMSGs are typically present in a direct drive wind farm, and there is mutual influence between PMSGs during the process of SSO. The transient energy characteristic analysis of the PMSG by considering the coupling relationship is the focus of our future work.
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
GY: writing–review and editing. YW: writing–original draft and writing–review and editing. CY: writing–review and editing. LY: writing–review and editing.
The authors declare financial support was received for the research, authorship, and/or publication of this article. This work was supported by the National Natural Science Foundation of China (No. U1866601).
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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