Analysis of high frequency oscillations based on MMC-HVDC system

Modular multilevel converter is widely used in new energy grid-connection and power grid asynchronous interconnection by virtue of its advantages of flexible and easy to expand structure and no commutation failure problem. In recent years, high-frequency oscillations have been observed in several flexible DC transmission projects at home and abroad, while the internal harmonic dynamic characteristics and control system of the MMC system are complex and the mechanism of high-frequency oscillations has not been sufficiently analysed at present. In this paper, the detailed DQ impedance model and the equivalent circuit impedance model on the AC side are firstly derived based on the state-space model of the MMC system, and the impedance sweeping method with superimposed small perturbations is proposed. Meanwhile, the dynamic characteristics of the detailed DQ impedance model are compared with those of the electromagnetic transient model. The amplitude-frequency characteristics of the system impedance under different system delays are further compared, and the mechanism of the high-frequency oscillation of the MMC-HVDC system is revealed.


Introduction
With the continuous development of power system and the construction of energy internet, flexible DC transmission technology based on modular multilevel converter has been rapidly developed by virtue of its superior characteristics in terms of easy scalability, low harmonic content, and controllability [1].However, due to the complexity and uncertainty of power systems, especially in long-distance and largecapacity transmission, HVDC systems are faced with a series of stability problems, among which system instability caused by high-frequency oscillations is one of the important problems.In recent years, highfrequency oscillations have frequently appeared in major flexible DC projects, such as 550 Hz oscillations in the DC side of Xiamen project [2], 1270 Hz in Luxi project, and 700 Hz and 1.8 kHz high-frequency oscillations in Yu-Er-Zhou interconnection project [3].
The main mechanism of high-frequency oscillations triggered by the grid-connected flexible DC transmission system is the interaction of the MMC-controlled link delay with the cable lines of the AC grid, and the resonance point is generally in the frequency interval of negative damping-inductivity.The mechanism of this high-frequency oscillation phenomenon is mostly studied and analysed by eigenvalue analysis and impedance methods.In [4], the effect of the system delay link on high frequency oscillations is considered and a fourth order Pade equivalent is performed for the delay link, and the mutual coupling between the delay link and the AC system is verified to be a key factor in the high frequency oscillations by means of the participation factor and eigenvalues.In [5], a state-space matrix considering the internal harmonic dynamics of the MMC is developed and the effect of the system's phase-locked-loop current-inner-loop voltage-outer-loop loop current suppression link on the high-frequency oscillations is further investigated.In [6], a modular modelling approach is used to model the AC system of the control section of the electrical part of the MMC in chunks and solve the transfer functions of each part separately.In [7], the space equation of state of the MMC system is established and the equivalent impedance of the MMC system is derived using matrix transpose inverse calculations, which more intuitively illustrates the phase angle, amplitude characteristics and stability margins of the MMC system over the full frequency band.
In summary, for the high frequency oscillation problem in the MMC-HVDC system, this paper establishes a detailed impedance model of the MMC system and a cable line impedance model by using small-signal analysis, and carries out impedance scanning and FFT analysis, and compares the results with the theoretical analysis results to validate the accuracy of the model.By comparing the calculated impedance with the measured impedance, the mechanism of high-frequency oscillation of the system is explained, which provides a theoretical basis for suppressing high-frequency oscillation.

Modelling of the MMC-HVDC system
The flexible DC transmission system based on modular multilevel converter described in this paper is divided into two subsystems at the PCC port, i.e., the alternating current grid and the MMC converter station.Meanwhile, the AC grid can be equated to an ideal voltage source in series with the equivalent impedance of the AC side, and the MMC converter station can be equated to an ideal current source in parallel with the MMC conductor.

DQ impedance model for MMC systems
The state space model of the MMC system is shown in equation (2), where x is a state variable and u is an input variable.The modelling process can be referred to [6], and due to space limitation, the detailed expressions of state-space equations are not given here.
( ) By substituting the output variables obtained from the MMC control section into the input variables of the MMC main circuit section, a detailed MMC state space model can be obtained by the method of [8][9][10].The input variables corresponding to the MMC system are ( ,) ,) The small signal model can be obtained by linearising the expansion of equation ( 2) at the stable operating point.For the convenience of subsequent studies, the transformation into transfer function form is continued as shown in equation ( 3): In equation (3), s is a complex variable.From the known input and output relationships, a DQ impedance model for the MMC system can be derived:

Time-delay link equivalent model
Because in the actual engineering, the inverter control system is not in the continuous domain, its control is more through the digital control system DSP, etc., the calculation of the reference value into the modulation of the existence of the computational delay, the modulation of the zero-order keeper characteristics of the wave there will also be a certain delay, as well as on the sampling of the voltage and current will bring about delay, which through the calculation of the equivalent as sT e − .The Pade approximation can be an equivalent alternative to the delay link to the model from the discrete domain into the continuous domain, and then write out the state-space equations.Its corresponding Pade delay function is: ( ) The above pade-equivalent transfer function is transformed into a state space equation, which in turn allows for the introduction of delay-related state variables.Different choices of pade approximation order may lead to different delay characteristics of the model.In this paper, we choose the fifth-order pade equivalent, which can more accurately simulate the delay state of the system.

AC line model
The transmission line is a key part that affects high-frequency oscillations.In this paper, a distributed parameter model is used for theoretical analysis.The AC part of the flexible system is mainly composed of the transmission line and the generator accessing the line as well as the transformer, in which the main part affecting the high-frequency oscillations is the transmission line part, and the state-space model of the series-connected π-type equivalent transmission line is established.The conductance expression of the first section of the π-type equivalent circuit in this model is given by: When each section of π-type circuit is added, it can be calculated by equation ( 8) iteratively: The expression for the final line impedance is: Where s Z is the internal impedance of the power supply, t X is the transformer leakage reactance, g R is the series resistance, g L is the series inductance, g C is the parallel capacitance.

Impedance Scanning Method
To validate the accuracy of the above impedance modelling approach, voltage perturbations can be superimposed on the grid-parallel points and small signal perturbations in each frequency band can be extracted, which in turn can be plotted on the impedance characteristic curve.Regarding the extraction of the signal and the calculation of impedance the specific method is as follows: • Firstly, a three-phase positive-sequence harmonic voltage source is superimposed to the parallel grid point of the MMC system that operates under normal conditions, and the amplitude of this voltage source is 2%-5% the amplitude of the grid voltage source.Set the simulation run model to multiple runs, and set the number of runs and the size of the initial superimposed frequency according to the frequency range and frequency interval of the desired impedance amplitude-frequency curve.• At the end of the simulation, the voltage currents at the tie point at each frequency are extracted and analyzed by FFT.The voltage currents are converted from time domain to frequency domain range, the amplitude and phase angle of the voltage currents at specific frequencies are extracted and the Bode plot of the measured impedance is plotted using the plotting tool.

Comparison of simulation verification
The electromagnetic transient simulation model shown in Figure 1 is built in PSCAD/EMTDC.The object of study in this paper is MMC1, which is controlled by constant active and reactive power control.
MMC system topology structure.

Dynamic response result validation
In order to verify the correctness of the constructed system chapter space, the numerical solution of the dynamic response of the small-signal model in MATLAB is compared with the dynamic characteristics of the electromagnetic transient model built by the PSCAD simulation platform.At the initial moment, the system operates at an active power of 800 MW and a reactive power of 0 Mvar, and at the moment of t=3s, the active power is stepped up to 100 MW, and the comparison of the corresponding active and reactive power time-domain dynamic responses of this system is shown in figure 2 and figure 3.

Simulation and analysis of high frequency oscillation phenomenon
From the above figure, it can be seen that when the active power emitted by the system varies, the detailed electromagnetic transient model and the state-space model simulation results are consistent, which can prove that the state-space model established above considering the harmonic dynamics inside the MMC is accurate.In this section, the validity of impedance modelling and small-signal modelling is verified under two different scenarios with delays of 300 μs and 400 μs, respectively.The bode plots of MMC computed impedance and AC system computed impedance obtained from the impedance model are shown in figure 4 and figure 5 for link delays of 300 μs and 400 μs, respectively.The measured impedance of the MMC obtained from the impedance scan and the measured impedance bode of the AC system are shown in figure 6 and figure 7. The FFT analysis results of the AC system A-phase voltage at 1.5 s are shown in figure 8 and figure 9.That is, the results obtained from impedance scanning and FFT analysis are used to verify the accuracy of the modelling.When the system delay is 300us, it can be seen from the calculated impedance bode plot of the system that the impedance amplitude-frequency curves of the AC system and the MMC are intersected at a frequency of 2200Hz, with a corresponding phase difference of 190.29°, which triggers high-frequency oscillations in the system due to insufficient phase margins.This is because the MMC considering the link delay has a phase of more than 90° in the high-frequency interval, showing a "negative resistanceinductance" characteristic.At this time, the AC system overhead line has a capacitive interval due to the effect of capacitance to ground, resulting in a phase difference of more than 180° between the MMC and AC system at 2200 Hz.Also from figure 6 AC system and MMC measured impedance bode diagram, its high frequency resonance point is 2197Hz, basically consistent with the theoretical modelling results.In order to verify the correctness of the theoretical impedance solution and impedance scanning operation, the system is taken in the delay of 300 μs 1.5s to 2s of the voltage waveform for FFT analysis, from Figure 1 can be seen that the oscillation frequency of the system is 2200Hz, in line with the theoretical analysis results, verifying the correctness of the impedance model constructed in this paper, and likewise illustrates that there is a risk of high-frequency oscillation of the MMC grid-connected.
When the system delay is 400 μs, figure 5 and figure 7 further give the interaction of the MMC impedance with the AC system impedance, and the results obtained from the theoretical analyses and impedance measurements are basically the same, and there are also two high-frequency resonance points at about 1390Hz and 2150Hz, respectively.The FFT analysis of the phase voltage also verifies that the system appears to oscillate at 1389 Hz and 2157 Hz under this operating condition.When the system delay is increased from 300us to 400us, the first negative impedance interval of the MMC impedance will be shifted in the direction of frequency reduction, and the frequency of its first resonance point is also reduced to 1389Hz.It is of concern that a second high-frequency resonance point occurs at a system delay of 400 μs due to the shift to the left of the negative impedance interval.

Analysis of high-frequency oscillation mechanism
The impedance model developed in this paper considers in detail the effects of the phase-locked loop, the voltage outer loop, the current inner loop, and the circulating current suppression loop on its amplitude-frequency characteristics, but the bandwidths of these control links are basically lower than the frequencies of the high-frequency resonance points of the MMC system, and therefore these control links can be neglected in the study of its mechanism.By simplifying the control links, the high-frequency oscillation mechanism can be analysed more intuitively.In the frequency band above 1000 Hz, the equivalent impedance of the MMC system can be described as: The resulting phase expression for the MMC high frequency impedance can be obtained as shown in equation (11): Because the tangent function transforms between -90° and 90°, the simplified MMC impedance phase transforms between 0° and 180°, and exhibits negative damping characteristics when it is greater than 90°.And because of the AC side of the line there is a capacitance to ground, resulting in the intersection frequency of the two, the phase frequency characteristics of the 180 ° difference, which in turn triggers a high-frequency oscillation.Again, it can be concluded that the frequency jumps from 180° to 0° at 1/ fT = , and 90° at . That is, it can be verified that the high-frequency oscillation shifts its negative impedance interval to the left as the delay increases, corresponding to a similar decrease in resonant frequency.aswell as the possible existence of multiple resonance points.At the same time, for a given frequency band, the greater the system delay, the greater the number of times the phase peak occurs in this band.

Conclusion
In this paper, for the high-frequency oscillation phenomenon mainly caused by the system delay, the impedance model of the MMC-HVDC system for analysing the oscillation phenomenon is established, and the conclusions are shown as follows: • The MMC system impedance model and the AC line impedance model proposed in this paper can correspond to the simulation and impedance sweep results, and the results of their impedance calculations are basically error-free, which makes the model universally applicable to the stability analysis of MMC systems.• Different system delays induce different oscillation frequencies.The system delay directly affects the amplitude-frequency characteristics of the high-frequency impedance, and the frequency range of the first negative impedance interval is negatively correlated with the size of the delay.• The high-frequency oscillation mechanism of the MMC-HVDC system is that the converter station presents a negative resistance characteristic in a certain high-frequency range under the influence of delay, which interacts with the capacitive characteristic in the corresponding frequency band of the distributed line and triggers the high-frequency oscillation.• According to the impedance Bode plot, the oscillation risk region can be visualized and the oscillation risk frequency region can be modified by using additional damping and filters, which provides design ideas and theoretical basis for oscillation suppression schemes in practical engineering.
are the dq component of the voltage at the PCC point and , sd sq ii are the dq component of the current at the PCC point.Then the state space equation and output equation of the MMC converter station shown in equation (2) can be established from equation (1).
state matrix input matrix output matrix and feedforward matrix of the MMC system, respectively.