Sub-synchronous oscillation suppression in MMC-HVDC grid-connected systems under wind farm fluctuation scenarios

Abstract

In order to solve the problem of sub-synchronous oscillation (SSO) of wind power grid-connected system with modular multilevel converter based on high voltage direct current transmission (MMC-HVDC), which is characterized by frequency drift due to the fluctuation of wind power output and the dynamic interaction of MMC. By considering the coupling characteristics of wind power volatility and the internal harmonic dynamics of MMC, we establish a small-signal model of the interconnection system between the equivalent wind farm and MMC-HVDC, study the dominant factors triggering the system oscillation, and use model predictive control (MPC). The results show that the wind farm output, wind farm terminal voltage and terminal current are the dominant factors leading to SSO frequency drift. It is also verified that the MPC-SSDC control not only effectively suppresses SSOs with single frequency drift, but is also effective when multiple SSOs with different frequencies exist simultaneously. It is shown that MPC-SSDC can self-adapt to the multi-oscillation scenario of wind power grid-connected system to provide frequency and damping support for the system.

Keywords

MMC SSO modal analysis MPC damping compensation

1. Introduction

In the context of advances towards a “carbon neutral” economy with a high proportion of renewable energy, offshore wind power will see rapid development [1]. When large wind farms are supplied through HVDC transmission lines, the randomness of wind speed leads to a power system oscillation frequency shift and the interaction between large-scale power electronics and wind farms gives rise to oscillations in the sub-synchronous frequency range [2, 3]. Such multiple SSOs with different frequencies cause system instability [4]. Many offshore wind power projects have exhibited the SSO phenomenon in the commissioning or operation stage [5]. For example, in the Shanghai Nanhui MMC project [6], as the wind field output increases, the system repeatedly oscillates in the frequency band of 20 $\sim$ 30 Hz, and in the Hebei Guyuan flexitransfer project, oscillations in the frequency range of 3 $\sim$ 10 Hz occurred as the power transfer from the power supply side increased [7]. During the commissioning of the Guangdong Nan’ao flexible straight demonstration project [8], the Shucheng converter station and the Qing’ao converter station constituted the two ends of MMC-HVDC operation, and the SSO phenomenon at 30 Hz appeared with increasing power output from the wind field, eventually leading to the system shutdown. In the wind power transmission via DC scenario, the multi-timescale characteristics of the wind power grid-connected system become more and more obvious as large-scale power electronics are put into operation on the network side [9], and the adoption of a reasonable control strategy to suppress oscillations is essential for improving the stability of offshore wind power and grid security [10].

For the SSO that occurs in wind farms via flexible and straight grid connections, at this stage, the suppression measures are mainly based on the optimization of new energy generator set controller parameters and additional damping controller for converter control loop [11], but the control parameter optimization is not applicable to all operating conditions, and the suppression effect is only obvious for specific operating conditions [12]. The existing suppression strategies mainly target a single SSO with a fixed frequency [13, 14, 15], ignoring the coupling between the control loops and making it difficult to accurately track the changes in operating conditions. The active damping control reported in the literature [16], only considers the damping requirements of a certain resonant frequency, and cannot meet the operating conditions in a stochastic environment for the various structures of the system. In addition, the active damping control based on a fixed trap is effective in the range of preset oscillation frequencies [17, 18], but cannot solve the problem that the oscillation frequency shifts with the change of system conditions. The uncontrollability of wind speed and MMC harmonic dynamics make the system unstable with respect to fluctuation with wind field output [19], so that, the SSO modal drift characteristics of the MMC-HVDC grid-connected system are pronounced, and necessitating control strategies for the wind power grid-connected system that are highly adaptive to multi-oscillation scenarios.

To address the problem of SSO frequency drift in the system caused by the wind farm output and state changes, first, according to the wind power grid-connected system of MMC-HVDC, a small signal dynamic model containing wind farm fluctuations is established, and then an MPC-SSDC-based SSO suppression strategy is proposed based on the modal analysis results. The MPC model prediction is used to achieve multivariate wind farm output, wind farm terminal voltage and terminal current, and the SSO component is introduced into the system for real-time feedback with additional damping compensation. Finally, the MMC-HVDC grid-connected system is built in PSCAD to verify the effectiveness and superiority of the suppression strategy.

2. MMC-HVDC-based grid-connected wind power system

In this paper, we study the SSO frequency drift problem for the wind power grid-connected system of MMC-HVDC, and its equivalent system is shown in Fig. 1.

Figure 1.

Grid-connected structure of wind farm via MMC-HVDC.

To accurately reflect the impact of power fluctuation on the system, based on the principle of wind farm simplification, the wind farm adopts the equivalent of a controllable voltage source, converges to the AC bus via 0.69/35 kV machine end transformer, and then feeds into MMC-HVDC via 35/166 kV step-up. MMC-HVDC contains two converter stations, MMC1 and MMC2. MMC1 adopts constant AC voltage control, MMC2 adopts constant MMC1 adopts constant AC voltage control, MMC2 adopts constant DC and constant power control, and finally is integrated into the 166 kV AC system through the converter.

3. Small-signal dynamic model for interconnected systems

Due to the random and uncontrollable nature of wind power generation, the operating points of wind farms change with operating conditions, leading to oscillatory instability during the operation of the wind farm and MMC-HVDC interconnection system. Based on the topology of Fig. 1, a small-signal dynamic model applicable to the wind farm fluctuation scenario is established, as shown in Fig. 2.

Figure 2.

D-PMSG-MMC1 interconnection system.

By incorporating the wind field fluctuation as a new state variable into the system and considering the internal dynamic characteristics of MMC, the model can be used to effectively analyse the SSO frequency drift caused by the unstable wind field output. The small signal model is constructed with the equivalent wind field, AC line and MMC as shown in Eq. (1), and the higher order infinitesimal quantities are neglected.

$\displaystyle\frac{d\Delta x}{dt}=A\Delta x$ (1)

The small-signal dynamic model based on the fluctuation of wind farm proposed in this paper first models the three parts of the equivalent wind field, AC line and MMC as dynamic equations that reflect their internal dynamic characteristics, and then interconnects the dynamic equations into a small-signal model according to the system topology, to realize the overall modelling of the system and lay the foundation for the subsequent efficient analysis of the system stability. $\Delta x$ denotes the state variable of the linearized system, the small-signal dynamic model includes 31 state variables, as described in Appendix A. The actual power output of the corresponding dynamic model of the system after taking into account the fluctuations of the wind farm is:

$\displaystyle\hat{P}=\textit{Pref}+\tilde{P}$ (2)

In the analysis, to accurately respond to the impact of the wind farm output fluctuation on the system, based on the wind farm simplification principle in [20], the wind farm is equated with a controllable voltage source, and the control structure is dominated by the grid-side converter, whose control objective is to maintain DC voltage stability, as shown in Fig. 3.

Figure 3.

Equivalent wind farm control structure.

Therefore, the output of the network-side converter is:

$\displaystyle\left\{{\begin{array}[]{l}L_{s}\frac{d\Delta\hat{I}_{d,q}}{dt}=% \Delta u_{\textit{d,qref}}-\Delta\hat{u}_{d,q}\pm\omega L_{s}\Delta\hat{I}_{q,% d}\\ C_{f}\frac{d\Delta\hat{u}_{d,q}}{dt}=\Delta I_{\textit{d,qref}}-\Delta\hat{I}_% {d,q}\pm\frac{\Delta\hat{u}_{q,d}}{\omega C_{f}}\\ \end{array}}\right.$ (3)

The equivalence wind field small-signal dynamic equation can be derived from the net-side converter output equation, and the detailed equation is shown in Appendix A1.

As shown in Fig. 1, the MMC bridge arm is cascaded by submodules, and the internal harmonic coupling characteristics of the MMC affect on the output characteristics of the AC and DC sides of the MMC. Therefore, the fundamental and diphasic components of the sub-module capacitance voltage, the diphasic components of the AC and DC side currents and their internal circulating currents are selected as the main indicators of the MMC dynamic equations for the construction of the MMC dynamic equations.

The sub-module capacitance voltage in the wind farm output fluctuation state can be expressed as:

$\displaystyle\Delta u_{c}=\Delta u_{mmc\_c}+\Delta u_{ac1}+\Delta u_{ac2}=% \Delta u_{mmc\_c}+\Delta u_{c1}\sin(\omega t+\theta 1)+\Delta u_{c2}\sin(2% \omega t+\theta 2)$ (4)

where $u_{mmc\_c}$ is the DC component of the capacitor voltage, and $u_{c1}$ , $\theta_{1}$ , $u_{c2}$ , $\theta_{2}$ are the amplitude and phase angle of the fundamental and diphasic frequencies, respectively.

The sub-module capacitance voltage linearization is shown in Eq. (5):

$\displaystyle\frac{d\Delta u_{c}}{dt}=\Delta A_{dc}+\Delta A_{1}+\Delta A_{2}$ (5)

where $A_{dc}$ , $A_{1}$ , $A_{2}$ correspond to the fundamental and diphasic components to the DC and AC components of the above equation, respectively.

The dynamic equations corresponding to the dc component, fundamental and doubling frequency component in the dq coordinate system are detailed in Appendices A2–A4.

The DC component of the bridge arm voltage is:

$\displaystyle\Delta u_{\textit{arm\_dc}}=\frac{1}{2}N\Delta u_{\textit{mmc\_c}% }-\frac{1}{4}\textit{NM}\Delta u_{c1}\cos(\alpha-\theta 1)+\frac{\textit{Nu}_{% \textit{cir}}}{2u_{\textit{mmc\_dc}}}\Delta u_{c2}\cos(\varphi-\theta 2)$ (6)

On the AC side, the fundamental frequency voltage component of the bridge arm is:

$\displaystyle\Delta u_{\textit{arm\_ac1}}=\frac{1}{2}\textit{NM}\Delta u_{% \textit{mmc\_c}}\sin(\omega t+\alpha)+\frac{1}{2}N\Delta u_{c1}\sin(\omega t+% \theta 1)-\frac{1}{4}\textit{NM}\Delta u_{c2}\cos(\omega t+\theta 2-\alpha)+% \frac{\textit{Nu}_{\textit{cir}}}{2u_{\textit{mmc\_dc}}}\Delta u_{c1}\cos(% \omega t+\varphi-\theta 1)$ (7)

The two-fold component of the bridge arm voltage is:

$\displaystyle\Delta u_{\textit{arm\_ac2}}=\frac{1}{2}N\Delta u_{c2}\sin(2% \omega t+\theta 2)+\frac{1}{4}\textit{NM}\Delta u_{c1}\cos(2\omega t+\theta 1+% \alpha)+\frac{\textit{Nu}_{\textit{cir}}\Delta u_{\textit{mmc\_c}}}{u_{\textit% {mmc\_dc}}}\sin(2\omega t+\varphi)$ (8)

Then the dynamic equations for the AC, DC sides and the internal doubling frequency components of the MMC in the dq coordinate system can be derived from the corresponding KVL equations, that are given in detail in Appendices A5–A7.

And the small-signal dynamic equation of the AC line is:

$\displaystyle\left\{{\begin{array}[]{l}L_{1}\frac{d\Delta i_{d2,q_{2}}}{dt}=% \Delta u_{d2,q2}-\Delta\hat{u}_{d,q}\mp\omega L_{1}\Delta i_{q2,d2}\\ C_{1}\frac{d\Delta u_{d2,q2}}{dt}=\Delta\hat{i}_{d,q}-\Delta i_{d2,q2}\mp\frac% {\Delta u_{q2,d2}}{\omega C_{1}}\\ \end{array}}\right.$ (9)

where $u_{d2}$ , $i_{d2}$ , $u_{q2}$ , $i_{q2}$ are the dq axis components corresponding to the voltage and current at the MMC end of the AC line, and $L_{1}$ , $C_{1}$ are the inductance and capacitance on the AC line, respectively.

4. SSO mechanism based on small-signal dynamic model

Based on the small-signal dynamic model, the frequency drift SSO phenomenon of the MMC-HVDC grid-connected system is investigated when the wind farm output is unstable and the system state changes. Figure 4 shows the corresponding characteristic root trajectory when the wind farm output rises from 0.2 p.u. to 1.0 p.u.

Figure 4.

The characteristic root locus of the wind farm output from 0.2 p.u. to 1.0 p.u.

As observed from the Fig. 4, the system crosses the imaginary axis into the right half-plane with the increase in the wind field output, resulting in the system instability. The critical outflow force condition is 0.8 p.u., corresponding to an eigenroot of 14.02 $\pm$ 116.05i and an oscillation frequency of approximately 18.47 Hz.

4.1 Eigenvalue analysis

To investigate the oscillation frequency under different operating conditions, the eigenvalues are obtained for the two operating conditions with wind field outputs of 0.8 p.u. and 1.0 p.u. In this paper, only the SSO is studied, and no specific analysis is performed for oscillations in other frequency bands. The parameters of the grid-connected system are given in Appendix B.

Table 1
Eigenvalue analysis in the SSO band

Working conditions	Eigenvalue	Oscillation frequency/Hz	Damping ratio
0.8 p.u.	$\lambda=$ 1.13 $\pm$ 56.36i	8.97	$-$ 0.02
	$\lambda=$ 14.02 $\pm$ 116.05i	18.47	$-$ 0.12
1.0 p.u.	$\lambda=$ 9.50 $\pm$ 64.08i	10.19	$-$ 0.15
	$\lambda=$ 182.62 $\pm$ 136.97i	21.79	$-$ 0.79

There are two sets of oscillation modes with frequencies in the sub-synchronous band for both 0.8 p.u. and 1.0 p.u. conditions, as shown in Table 1. The real part of the eigenvalue reflects the system damping and the imaginary part reflects the oscillation frequency. If the real part is positive, the system damping decreases as the parameter increases, and if the real part is negative, it decreases accordingly; if the imaginary part is positive, the oscillation frequency increases as the parameter increases, and if the imaginary part is negative, it decreases. For the dominant oscillation modes in both operating conditions (bolded in the table), increasing the wind field output, the system damping subsequently decreases and the oscillation frequency increases. It is observed that the wind power grid-connected system based on MMC-HVDC is influenced by wind speed fluctuations to produce SSO with modal frequency drift, and the dominant oscillation modes in both conditions are negatively damped, showing oscillation dispersion, which leads to system instability.

4.2 Participation factor analysis

To investigate the factors influencing the SSO frequency drift generated by the grid-connected system, based on the calculation results, a participation factor analysis was performed for the dominant oscillation modes (denoted as modes 1 and 2) with a positive real part of the eigenvalues and negative damping under different operating conditions.

Figure 5.

The participation factors of the corresponding oscillatory modes at the wind farm output of 1.0 p.u. operating conditions.

As observed from Fig. 5, there are nine state variables in the system that may affect the degree of system oscillation, and the oscillation frequency is shifted with the change in the relevant parameters such as wind field output $\Delta x_{1}$ , wind field end voltage $\Delta u_{d,q}$ and end current $\Delta i_{d,q}$ .

5. SSO suppression policy

5.1 Adaptive trap-based SSO suppression strategy

The adaptive notch filters (ANF) can adaptively adjust the filter parameters as the system conditions change, and are essentially bandwidth filters with the main parameters being the centre frequency and the damping ratio, with the dominant resonant frequency as the centre frequency and the damping ratio determining the bandwidth, as shown in [21], their control structure is shown in Fig. 6.

Figure 6.

ANF control structure.

Where $f_{c}$ is the dominant oscillation frequency obtained by screening. For the oscillations under different working conditions, ANF takes the oscillation mode whose amplitude is greater than its pre-set threshold value captured by the system under different working conditions as the control object, i.e., the dominant oscillation mode of the system, obtains the centre frequency and damping ratio of the ANF corresponding to the mode according to the frequency calculation, adaptively adjusts the ANF parameters, and then embeds and activates the ANF into the system control link to achieve the damping effect of the oscillations at different resonant frequencies to maintain the system stability. However, the ANF needs to determine the dominant oscillation mode of the control with the help of other links, leading to the existence of control delays in the system, and certain errors in the frequency estimation that cause the deviation. Therefore, it is known from the above analysis that the grid-connected system generates SSO with obvious multi-modal frequency drift characteristics due to the change of wind field conditions, and each ANF must be designed individually according to the oscillation modes of different frequencies, so that a single ANF control has only a limited effectiveness for the suppression of oscillations at many different frequencies.

5.2 MPC-SSDC based SSO suppression strategy

According to the above analysis, the oscillation frequency in the MMC-HVDC grid-connected system is not unique and varies with the wind farm output conditions. The traditional single SSO suppression strategy has difficulty adapting to the changing needs of the wind farm dynamic parameters, and cannot effectively solve the multimodal frequency drift SSO problem in wind farm fluctuation scenarios when multiple SSOs with different frequencies exist simultaneously. Therefore, an MPC-SSDC SSO suppression strategy is proposed in this paper.

5.2.1 Multi-objective MPC control structure

The MPC model predicts the future dynamic changes of the system, forms a reference trajectory based on the error between the measured data and the predicted data, corrects the prediction model, refreshes the optimal state in the finite time domain, and applies the optimal control to the controlled object so that it can accurately track the oscillation mode and suppress it effectively, the model architecture is shown in Fig. 7.

Figure 7.

MPC basic control structure diagram.

Considering the influence of wind field output, wind field terminal voltage and terminal current on SSO, discretizing Eq. (3) and using the antecedent difference method to establish the current and voltage prediction model is:

$\displaystyle\left[{\begin{array}[]{l}\hat{i}_{d}(k+1)\\ \hat{i}_{q}(k+1)\\ \hat{u}_{d}(k+1)\\ \hat{u}_{q}(k+1)\\ \end{array}}\right]=\left[{{\begin{array}[]{cccc}0&{\omega T_{s}}&{-T_{s}/L_{s% }}&0\\ {-\omega T_{s}}&0&0&{-T_{s}/L_{s}}\\ {-T_{s}/C_{f}}&0&0&{T_{s}/\omega C_{f}^{2}}\\ 0&{-T_{s}/C_{f}}&{-T_{s}/\omega C_{f}^{2}}&0\\ \end{array}}}\right]\left[{{\begin{array}[]{c}{\hat{i}_{d}(k)}\\ {\hat{i}_{q}(k)}\\ {\hat{u}_{d}(k)}\\ {\hat{u}_{q}(k)}\\ \end{array}}}\right]+\left[{{\begin{array}[]{ccccc}0&0&{T_{s}/L_{s}}&0\\ 0&0&0&{T_{s}/L_{s}}\\ {T_{s}/C_{f}}&0&0&0\\ 0&{T_{s}/C_{f}}&0&0\\ \end{array}}}\right]\left[{{\begin{array}[]{c}{i_{dref}}\\ {i_{qref}}\\ {u_{dref}}\\ {u_{qref}}\\ \end{array}}}\right]$ (10)

Furthermore, the power prediction equation can be obtained as:

$\displaystyle\hat{P}(k+1)=\frac{3}{2}[\hat{u}_{d}(k+1)\hat{i}_{d}(k+1)+\hat{u}% _{q}(k+1)\hat{i}_{q}(k+1)]$ (11)

where $T_{s}$ is the sampling period, $k$ is the current moment, and $k+1$ is the next moment. When the sampling period $T_{s}$ is small enough, it is assumed that the variable reference value at the moment $k+1$ is the same as that at moment $k$ , i.e.:

$\displaystyle\Delta x_{\textit{d(q)ref}}(k+1)\approx\Delta x_{\textit{d(q)ref}% }(k)$ (12)

The tracking control of the wind farm end currents and voltages as well as the wind farm output is implemented according to Eqs (12) and (13) to predict the values of the variables at the next moment. To improve the accuracy of the prediction model and reduce the volatility and randomness of the wind farm output, MPC minimizes the difference between the given value of the prediction window and the predicted value by minimizing the difference between the given value of the prediction window and the predicted value. Assume that the prediction window variable given value is constant and the variable following error satisfies:

$\displaystyle e_{d(q)}(k+1)=\Delta x_{d(q)}(k+1)+e_{d(q)}(k)$ (13)

where the state variables at moment $k$ satisfy:

$\displaystyle e_{d(q)}=|\Delta x_{\textit{d(q)ref}}(k)-\Delta x_{(q)}(k)|$ (14)

Therefore, the absolute value function of the error is:

$\displaystyle G_{1}=\left|{P_{\textit{ref}}(k+1)-\hat{P}(k+1)}\right|+\left|{u% _{\textit{dref}}(k+1)-\hat{u}_{d}(k+1)}\right|+\left|{u_{\textit{qref}}(k+1)-% \hat{u}_{q}(k+1)}\right|+\left|{i_{\textit{dref}}(k+1)-\hat{i}_{d}(k+1)}\right% |+\left|{i_{\textit{qref}}(k+1)-\hat{i}q(k+1)}\right|$ (15)

5.2.2 MPC-SSDC control structure

According to the above analysis, it is known that the randomness of wind speed will cause the SSO of grid-connected system to drift in frequency, and the oscillation often starts from the weak damping or negative damping of the system. To meet the actual damping requirement of the system while suppressing SSO, it is necessary to combine SSDC control on the basis of MPC, and the control structure of MPC-SSDC is shown in Fig. 8.

Figure 8.

MPC-SSDC control structure.

The sub-synchronous component is introduced into the objective function for real-time feedback and additional damping for compensation, and the oscillation component is fed back to the MPC control system in real time according to the adaptive nature of MPC to SSO frequency, avoiding the disadvantages of frequency detection of SSO and the need to control several different frequency SSOs separately, and improving the system damping to meet the damping requirements of the system in the actual fluctuation scenario, this can effectively solve the problem of SSO frequency drift. Specifically, the SSDC selects the current on the AC line of the interconnection system as the input signal of the SSDC, to ensure that the frequency characteristics remain unchanged in the sub-synchronous frequency range in the system, a first-order high-pass filter is used to filter out the DC component, extract the corresponding SSO current component, and then multiply by the sub-synchronous damping coefficient kc output current $i_{\textit{dq\_SSDC}}$ , to avoid adverse effects on the stable operation of the system due to overcontrol. Finally, by limiting the link attached to the wind field control inner-loop current channel to provide positive damping for the system, the amplitude is selected to generally not exceed 0.1 p.u. of the steady-state value of the inner-loop current, the control structure is shown in Fig. 9.

Figure 9.

SSDC control structure.

The sub-synchronous component is introduced into the objective function to perform real-time feedback and additional damping for compensation, then the objective function is:

$\displaystyle G=G_{1}+(\lambda+K_{c})\left[\left|{\hat{i}_{\textit{d\_sub}}(k+% 1)}\right|+\left|{\hat{i}_{\textit{q\_sub}}(k+1)}\right|\right]$ (16)

The specific steps are as follows: first, select the wind field output related control variables $\hat{u}_{d}$ , $\hat{u}_{q}$ , $\hat{i}_{d}$ , $\hat{i}_{q}$ , as MPC model prediction control objects, measure their reference values and measured values at $k$ moments, and then obtain the predicted values of the variables at $k+1$ moments through the model prediction equation to achieve multivariable tracking control; second, attach SSDC control to the equivalence wind field control link, introduce the SSO current component for real-time feedback, and simultaneously perform damping compensation. The size of $\lambda$ affects the variable tracking error, and $k_{c}$ determines the damping compensation, based on the principle that the variable tracking error is minimized and the SSDC current limit cannot exceed 0.1 p.u. of the inner loop current. The initial values of $\lambda$ and kc are both taken as 1. If the tracking error is the same prior to rolling optimization, $\lambda$ and kc are increase; otherwise, they are decrease. Within the current limit, the weight coefficient $\lambda$ and damping coefficient kc are adjusted to refresh the optimal state in the finite time domain until the tracking error tends to 0, so that the system can be controlled optimally.

6. Example analysis

6.1 Oscillation phenomenon under the fluctuation field of wind farm output

To verify the oscillation phenomenon under the wind farm output fluctuation scenario, this paper uses the MMC-HVDC grid-connected system for the example analysis, setting the rated output power of the wind farm to 50 MW, using the wind farm A-phase output voltage Vwind and output power Pwind as the observation objects. To avoid the impact on the simulation results due to system instability at start-up, the simulation time is selected after 3 s. The time domain simulation results are shown in Fig. 10. By changing the wind farm output, the system oscillates to different degrees, and when the output is at 0.8 p.u. and 1.0 p.u. conditions, the system oscillates and becomes unstable, which is consistent with the analysis of the root trajectory in Fig. 4.

Figure 10.

Corresponding output voltage and power of wind farm output from 0.2 p.u. to 1.0 p.u.

6.2 Small-signal dynamic model validation

Based on the small-signal modal analysis, the SSO of the grid-connected system exhibits frequency drift characteristics due to the fluctuation of the wind farm output.

According to the time domain simulation of Fig. 10, the Fast Fourier transform (FFT) analysis is performed for both 0.8 p.u. and 1.0 p.u. operating conditions.

Figure 11.

FFT analysis.

The FFT analysis results show that the dominant oscillation frequencies of the wind farm output voltage and power of the same condition are in the SSO band and meet the coupling, indicating that as the wind farm output increases, the SSO of the MMC-HVDC grid-connected system appears, and the dominant oscillation frequencies are different for different conditions. It is observed that the SSOs of the grid-connected system show the characteristics of frequency drift with the wind farm output. In addition, the FFT analysis and the eigenvalues of wind farm output voltage at 0.8 p.u. and 1.0 p.u. are almost error-free, verifying the accuracy of the proposed small-signal dynamic model.

6.3 SSO suppression policy effectiveness verification

To verify that the control strategy in this paper can dynamically suppress SSO in a wide frequency fluctuation range, the PI, ANF and MPC-SSDC based control strategies are added to the system in Fig. 1 for comparison, and the total simulation time of the system is set to 30 s. The initial operating condition is 0.6 p.u., and the wind farm output condition is dynamically increased to 1.0 p.u. after 10 s of operation, with Vwind and Pwind as the observation objects.

6.3.1 Comparative analysis of suppression strategies at SSO frequency offset

In view of the SSO frequency offset in the wind field fluctuation scenario, the suppression effect of the three control strategies during the whole dynamic simulation period is explored when the grid-connected system in Fig. 1 transitions from 0.6 p.u. to 1.0 p.u.

Figure 12.

Suppression effect at SSO offset.

It is observed from Fig. 12, that the system essentially does not oscillate under the fan output force of 0.6 p.u., and all three control strategies can maintain stable operation. When the fan output rises to 1.0 p.u., the PI control has essentially no suppression effect; ANF control adjusts the trap parameters according to the change of operating conditions, and although there are oscillations at the beginning due to the control delay, the oscillations can be suppressed in the end and the waveform tends to converge and level off; while MPC-SSDC control can accurately track the change of operating conditions and adjust the predictive control in real time according to the SSO oscillation component of the system, to continuously suppress SSO during the whole operation period. The MPC-SSDC control can accurately track the changes in operating conditions and adjust the predictive control in real time according to the SSO oscillation component of the system, enabling continuous suppression of SSO throughout the runtime.

Figure 13.

20 Hz $+$ 10 Hz.

Figure 14.

20 Hz $+$ 10 Hz $+$ 30 Hz.

6.3.2 Comparative analysis of multimodal SSO offset-time suppression

The above analysis, shows that the dominant oscillation frequency of the wind field output at rated operating conditions is 20 Hz, while the SSO with frequency drift in the actual MMC-HVDC project often exists in the multi-modal form, it is also dominated by SSOs with oscillation frequencies of 10 Hz and 30 Hz. To verify that the MPC-SSDC control strategy is effective for SSOs with multi-modal frequency drift characteristics, the SSO voltage components are injected into the equivalent wind farm grid-side converter control system based on the principle that the injected disturbances are consistent with the dominant oscillation components contained in the rated operating conditions in previous simulation context. The simulation results are shown in Fig. 12 based on the injection of 10 Hz SSO in part 1.

It is observed from Fig. 13 that in the first 10 s, the PI control fluctuates slightly, and the ANF and MPC-SSDC control show essentially no fluctuation under the condition of 0.6 p.u. At the rated condition, there are two oscillation modes with the same oscillation component but different frequencies in the system, and the system is unstable under PI control. ANF control is no longer effective; since the external forced disturbance oscillation is different from its own unstable oscillation, the MPC-SSDC control fluctuates under the rated condition, but it is still in a stable range, indicating that the MPC-SSDC control can still effectively suppress the oscillation when there are two SSOs with different frequencies in the system.

Furthermore, SSO with frequencies of 10 Hz and 30 Hz is injected into the system, and the simulation results are shown in Fig. 14.

In Fig. 14, PI control oscillates and loses stability during the dynamic change of working conditions. The oscillation amplitude of ANF control is further increased under 0.6 p.u. condition, while the system under MPC-SSDC control fluctuates, but the amplitude is small, which is almost identical to result in the former case. Under the rated condition, the oscillation amplitude of the system under ANF control and PI control is roughly the same, the oscillation divergence node is advanced, and there is almost no inhibition. Due to the external forced oscillation, the fluctuation amplitude of the system increases under the control of MPC-SSDC, but it can still maintain stable operation. This shows that MPC-SSDC can not only effectively suppress the SSO of single oscillation frequency drift, but also effectively suppress multiple SSOs with different oscillation frequencies in the dynamic change of wind farm output. According to the actual oscillation scenario, MPC-SSDC can accurately track the change in the working conditions, and add effective damping in the sub-synchronous frequency band, this provides frequency and damping support for the system, meets the damping requirements under different working conditions, and maintains the stability of the system.

7. Conclusion

For the SSO problem of frequency drift in MMC-HVDC grid-connected system with increasing wind field output, the following conclusions are drawn in this paper: It is concluded that the wind farm output, wind farm terminal voltage and current are the dominant factors leading to the SSO frequency drift of the grid-connected system. The SSO suppression strategy based on MPC-SSDC takes into account the control effect and response speed, providing frequency and damping support for the system, effectively solving the SSO problem with time-varying frequency and maintaining the stable operation of the system.

The deepening of power electronics in power systems has led to the change of oscillation from a single to a global complex oscillation problem. Based on this research, the future research direction of this topic is to investigate the SSO generation mechanism, oscillation propagation, and to propose better control strategies.

Footnotes

Fund

This work was supported in part by the Shanghai Science and technology program (Project No.: 21ZR1424800).

Appendix A

Among the 31 state variables, the first 11 state variables are the equivalent wind farm and AC line counterparts, and the others are the MMC-HVDC counterparts. $\Delta_{x}=[\Delta x_{1},\Delta x_{2},\Delta x_{3},\Delta i\text{\^{}}_{d},% \Delta i\text{\^{}}_{q},\Delta u\text{\^{}}_{d},$ $\Delta u\text{\^{}}_{q},\Delta i_{d2},\Delta i_{q2},\Delta u_{d2},\Delta u_{q2% },\Delta u_{\textit{mmc\_c1}},\Delta u_{\textit{ac1d1}},\Delta u_{\textit{ac1q% 1}},\Delta u_{\textit{ac2d1}},\Delta u_{\textit{ac2q1}},\Delta i_{\textit{sd1}% },\Delta i_{\textit{sq1}},\Delta i_{\textit{mmc\_dc1}},$ $\Delta i_{\textit{cird1}},\Delta i_{\textit{cirq1}},\Delta u_{\textit{mmc\_c2}% },\Delta u_{\textit{ac1d2}},\Delta u_{\textit{ac1q2}},\Delta u_{\textit{ac2d2}% },\Delta u_{\textit{ac2q2}},\Delta i_{\textit{sd2}},\Delta i_{\textit{sq2}},% \Delta i_{\textit{mmc\_dc2}},\Delta i_{\textit{cird2}},$ $\Delta i_{\textit{cirq2}}]_{31\ast 1}^{T}$ .

The small-signal dynamic equation for the equivalent wind farm is:

(A.1) $\displaystyle\left\{{\begin{array}[]{l}\Delta i_{\textit{dref}}=K_{p1}(\Delta P% _{\textit{ref}}-\Delta\hat{P})+\Delta x_{1}\\ d\Delta x_{1}=\frac{1}{T_{i1}}(\Delta P_{\textit{ref}}-\Delta\hat{P})\\ \Delta u_{\textit{dref}}=K_{p2}(\Delta I_{\textit{dref}}-\Delta\hat{i}_{d})+% \Delta x_{2}+\Delta\hat{u}_{d}-\omega L_{s}\Delta\hat{i}_{q}\\ d\Delta x_{2}=\frac{1}{T_{i2}}(\Delta i_{\textit{dref}}-\Delta\hat{i}_{d})\\ \Delta u_{\textit{qref}}=K_{p3}(-\Delta\hat{i}_{q})+\Delta x_{3}+\Delta\hat{u}% _{q}+\omega L_{s}\Delta\hat{i}_{d}\\ d\Delta x_{3}=\frac{1}{T_{i3}}(-\Delta\hat{i}_{q})\\ \end{array}}\right.$

where $K_{p1}$ , $K_{p2}$ , $K_{p3}$ , $T_{i1}$ , $T_{i2}$ , $T_{i3}$ correspond to the controller parameters, $\omega$ is the reference angular frequency, and the reference value corresponding to each electrical quantity is indicated by the subscript “ref”.

The dynamic equation of the capacitive voltage component of the SM is:

(A.2) $\displaystyle\frac{d\Delta u_{mmc\_c}}{dt}=\frac{1}{6C_{mmc}}\Delta i_{\textit% {mmc\_dc}}+\frac{u_{\textit{mmc\_cq0}}\Delta i_{sq}}{4C_{\textit{mmc}u_{% \textit{mmc\_dc0 }}}}+\frac{u_{\textit{mmc\_cd0}}\Delta i_{sd}}{4C_{\textit{% mmc}u_{\textit{mmc\_dc0}}}}+\frac{u_{\textit{cirq0}}\Delta i_{cirq}}{2C_{mmc}u% _{\textit{mmc\_dc0 }}}+\frac{u_{\textit{cird0}}\Delta i_{cird}}{2C_{mmc}u_{% \textit{mmc\_dc0}}}$

The fundamental frequency component of the SM capacitance voltage is:

(A.3) $\displaystyle\left\{{\begin{array}[]{lll}\frac{d\Delta u_{\textit{ac1d}}}{dt}&% =&-\frac{1}{4C_{\textit{mmc}}}\Delta i_{sd}-\frac{u_{\textit{mmc\_cd0}}\Delta i% _{\textit{mmc\_dc}}}{3C_{\textit{mmcummc\_dc0}}}-\frac{u_{\textit{mmc\_cq0}}% \Delta i_{\textit{cirq}}}{2C_{\textit{mmcummc\_dc0}}}-\frac{u_{\textit{mmc\_c}% }d0\Delta i_{\textit{cird}}}{2C_{\textit{mmcummc\_dc0}}}\\ &&-\frac{u_{\textit{cirq0}}\Delta i_{sq}}{4C_{\textit{mmcummc\_dc0}}}-\frac{u_% {\textit{cird0}}\Delta i_{sd}}{4C_{\textit{mmcummc\_dc0}}}-\omega\Delta u_{% \textit{ac1q}}\\ \frac{d\Delta u_{\textit{ac1}}q}{dt}&=&-\frac{1}{4C_{\textit{mmc}}}\Delta i_{% sq}-\frac{u_{\textit{mmc\_cq0}}\Delta i_{\textit{mmc\_dc}}}{3C_{\textit{% mmcummc\_dc0}}}-\frac{u_{\textit{mmc\_cd0}\Delta i_{\textit{cirq}}}}{2C_{% \textit{mmcummc\_dc0}}}+\frac{u_{\textit{mmc\_c}q0\Delta i_{\textit{cird}}}}{2% C_{\textit{mmcummc\_dc0}}}\\ &&-\frac{u_{\textit{cirq0}}\Delta i_{sd}}{4C_{\textit{mmcummc\_dc0}}+\frac{u_{% \textit{cird0}}\Delta i_{sq}}{4C_{\textit{mmcummc\_dc0}}}}+\omega\Delta u_{% \textit{ac1d}}\\ \end{array}}\right.$

The doubling frequency component of the SM capacitance voltage is:

(A.4) $\displaystyle\left\{\begin{array}[]{l}\frac{d\Delta u_{\textit{ac2d}}}{dt}=% \frac{u_{\textit{cird0}}\Delta i_{\textit{mmc\_dc}}}{3C_{\textit{mmcummc\_dc0}% }}-\frac{u_{\textit{mmc\_cq0}}\Delta i_{sq}}{4C_{\textit{mmcummc\_dc0}}}+\frac% {u_{\textit{mmc\_cd0}\Delta i_{sd}}}{4C_{\textit{mmcummc\_dc0}}}-2\omega\Delta u% _{\textit{ac2q}}+\frac{1}{2C_{mmc}}\Delta i_{\textit{cird}}\\ \frac{d\Delta u_{\textit{ac2q}}}{dt}=\frac{u_{\textit{cirq0}}\Delta i_{\textit% {mmc\_dc}}}{3C_{\textit{mmcummc\_dc0}}}+\frac{u_{\textit{mmc\_cd0}\Delta i_{sq% }}}{4C_{\textit{mmcummc\_dc0}}}+\frac{u_{\textit{mmc\_cq0}\Delta i_{sd}}}{4C_{% \textit{mmcummc\_dc0}}}+2\omega\Delta u_{\textit{ac2d}}+\frac{1}{2C_{\textit{% mmc}}}\Delta i_{\textit{cirq}}\\ \end{array}\right.$

where $i_{sd}$ and $i_{sq}$ are the SM capacitance voltage components in the dq coordinate system; $u_{ac1d}$ , $u_{ac1q}$ , $u_{ac2d}$ , $u_{ac2q}$ are the fundamental and doubling components of the SM capacitance voltage in the dq coordinate system.

The internal two-fold loop current component of the MMC is:

(A.5) $\displaystyle\left\{\begin{array}[]{l}\frac{d\Delta i_{\textit{cird}}}{dt}=% \frac{N_{\textit{ummc\_cd0}\Delta u_{\textit{ac1d}}}}{2L_{\textit{armummc\_dc0% }}}-\frac{N\Delta u_{\textit{mmc\_cucird0}}}{L_{\textit{armummc\_dc0}}}-\frac{% N\Delta u_{\textit{ac2d}}}{2L_{\textit{arm}}}-\frac{N_{\textit{ummc\_cq0}% \Delta u_{\textit{ac1q}}}}{2L_{\textit{armummc\_dc0}}}-2\omega\Delta i_{% \textit{cirq}}\\ \frac{d\Delta i_{\textit{cirq}}}{dt}=\frac{N_{\textit{ummc\_cd0}\Delta u_{% \textit{ac1q}}}}{2L_{\textit{armummc\_dc0}}}-\frac{N\Delta u_{\textit{mmc\_% cucirq0}}}{L_{\textit{armummc\_dc0}}}-\frac{N\Delta u_{\textit{ac2q}}}{2L_{% \textit{arm}}}-\frac{N_{\textit{ummc\_cq0}}\Delta u_{\textit{ac1d}}}{2L_{% \textit{armummc\_dc0}}}+2\omega\Delta i_{\textit{cird}}\\ \end{array}\right.$

where $u_{\textit{cird}},u_{\textit{cirq}}$ , $i_{\textit{cird}}$ , $i_{\textit{cirq}}$ are the corresponding circulation components in the dq coordinate system.

The dynamic equation of the DC side of the MMC is:

(A.6) $\displaystyle\frac{d\Delta i_{\textit{mmc\_dc}}}{dt}=\frac{3N_{\textit{ummc\_% cq0}}\Delta u_{\textit{ac1q}}}{2L_{\textit{armummc\_dc0}}}-\frac{3N_{\textit{% ucirq0}}\Delta u_{\textit{ac2q}}}{2L_{\textit{armummc\_dc0}}}+\frac{3N_{% \textit{ummc\_cd0}}\Delta u_{\textit{ac1d}}}{2L_{\textit{armummc\_dc0}}}-\frac% {3N_{\textit{ucird0}}\Delta u_{\textit{ac2d}}}{2L_{\textit{armummc\_dc0}}}$

The dynamic equation of the AC side of the MMC is:

(A.7) $\displaystyle\left\{{\begin{array}[]{lll}\frac{d\Delta\text{i}_{sd}}{dt}&=&% \frac{N\Delta u_{\textit{ac1d}}}{2L_{eq}}-N\frac{2\Delta u_{\textit{mmc\_cummc% \_cd0}}}{2L_{\textit{eqummc\_dc0}}}-N\frac{\Delta u_{\textit{ac2qummc\_cq}}}{2% L_{\textit{eqummc\_dc0}}}-N\frac{\Delta u_{\textit{ac2dummc\_cd0}}}{2L_{% \textit{eqummc\_dc0}}}\\ &&+N\frac{\Delta u_{\textit{ac1qucirq0}}}{2L_{\textit{eqummc\_dc0}}}+N\frac{% \Delta u_{\textit{ac1ducird0}}}{2L_{\textit{eqummc\_dc0}}}-\omega\Delta i_{sq}% \\ \frac{d\Delta i_{sq}}{dt}&=&\frac{N\Delta u_{\textit{ac1q}}}{2L_{eq}}-N\frac{2% \Delta u_{\textit{mmc\_cummc\_cq0}}}{2L_{\textit{eqummc\_dc0}}}-N\frac{\Delta u% _{\textit{ac2q0ummc\_cd}}}{2L_{\textit{eqummc\_dc0}}}+N\frac{\Delta u_{\textit% {ac2dummc\_cq0}}}{2L_{\textit{eqummc\_dc0}}}\\ &&+N\frac{\Delta u_{\textit{ac1ducirq0}}}{2L_{\textit{eqummc\_dc0}}}-N\frac{% \Delta u_{\textit{ac1qucird0}}}{2L_{\textit{eqummc\_dc0}}}+\omega\Delta i_{sd}% \\ \end{array}}\right.$

where the $L_{eq}=L_{\textit{arm}}/2+L_{t}$ , and the $L_{t}$ is the transformer leakage inductance.

Appendix B

Table 2

Grid-connected system parameters

Equivalent wind farm and AC line	Dual-loop PI for the equivalent wind farm	MMC-HVDC
$f$ /hz 50	$K_{\textit{p1}}$ 0.01	$C_{\textit{mmc1}}$ /mF 1.4
$L_{s}$ / $\mu$ H 18	$T_{i1}$ 0.1	$C_{\textit{mmc2}}$ /uF 1.4
$C_{f}$ / $\mu$ F 100	$K_{\textit{p2}}$ 0.1	N 200
$L_{1}$ /mH 5	$T_{\textit{i2}}$ 1	$L_{\textit{eq}}$ /mH 360
$C_{1}$ /nF 26	$K_{\textit{p3}}$ 0.1	$L_{\textit{arm}}$ /mH 360
	$T_{\textit{i3}}$ 1

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Sub-synchronous oscillation suppression in MMC-HVDC grid-connected systems under wind farm fluctuation scenarios

Abstract

Keywords

1. Introduction

2. MMC-HVDC-based grid-connected wind power system

Table 1 Eigenvalue analysis in the SSO band

5.1 Adaptive trap-based SSO suppression strategy

5.2.1 Multi-objective MPC control structure

6.1 Oscillation phenomenon under the fluctuation field of wind farm output

6.3.1 Comparative analysis of suppression strategies at SSO frequency offset

7. Conclusion

Footnotes

Fund

Appendix A

Appendix B

References

Table 1
Eigenvalue analysis in the SSO band