Recently, high-frequency oscillation of the modular multilevel converter (MMC) based high-voltage direct current (HVDC) projects has attracted great attentions. In order to analyze the small-signal stability, this paper uses the harmonic state-space (HSS) method to establish a detailed frequency domain impedance model of the AC-side of the HVDC transmission system, which considers the internal dynamic characteristics. In addition, the suggested model is also used to assess the system's high-frequency oscillation mechanism, and the effects of the MMC current inner loop control, feedforward voltage links, and control delay on the high-frequency impedance characteristics and the effect of higher harmonic components. Finally, three oscillation suppression schemes are analyzed for the oscillation problems occurring in actual engineering, and a simplified impedance model considering only the high-frequency impedance characteristics is established to compare the suppression effect with the detailed impedance model to prove its reliability.

Modular multilevel converter based high voltage direct current (MMC-HVDC) transmission has been widely used in the fields of new energy generation interconnection, high voltage and large capacity transmission and asynchronous grid interconnection thanks to its advantages of low harmonic content, flexible structure, high controllability, and no phase change failure [

These oscillation situations are excited under certain operating conditions and can be attributed to small disturbance problems. The impedance analysis method can be used to analyze this oscillation stability problem due to its clear physical meaning and explanation of the oscillation mechanism. Building the MMC high-frequency small-signal model under frequency domain response is the key to studying the high-frequency oscillation problem. The main methods for building the MMC small-signal model are the harmonic linearization method [

In this paper, to address the above problems, the high-frequency oscillation problems in the MMC-HVDC system are studied through modelling-simulation-verification-characterization-analysis-suppression strategies of the system.

The small signal model of MMC considering harmonic coupling characteristics in this paper belongs to multi-frequency modeling but is different from the general multi-frequency modeling of DC/DC converters, which is a broader concept than the modeling of harmonic interaction characteristics. Multi-frequency modeling of DC/DC converters is to fully consider the sideband effects caused by the switching frequency coupling, so that the model can accurately describe the dynamic response of the frequency band above one-half of the switching frequency. The purpose of the model proposed in this paper is to consider the interaction of different harmonic components in the internal dynamics of the MMC. The difference between the two is shown in

A MMC-HVDC system’s main structure topology is shown in

If the actual semiconductor switch model is used for each sub-module, the model order will be very high and computationally complex, which is difficult to use in the controller design of MMC. In order to design the MMC’s internal controller, the differences between the individual submodules within each bridge arm can be ignored. The bridge arm averaging model assumes that all sub-modules on the same bridge arm have identical states, thus equating all sub-modules within the bridge arm to a combined DC voltage and current source, greatly simplifying the complexity of the model. The bridge arm averaging model reflects the structural relationship between the bridge arms, ignores the switching dynamics in the MMC, and is a continuous time system model, while preserving the dynamic interactions within the MMC, and is often used as the basic time-domain model in modeling the small signal dynamics of the MMC. In this paper, this model is used to model the MMC based on the system level, ignoring the switching frequency and its sideband harmonics, and approximating that the sum of the switching functions

By averaging the bridge arms, the mathematical model of the MMC is simplified to twelve orders, corresponding to the inductance of the six bridge arms and the capacitance of the averaged submodules of the six bridge arms. An equivalent bridge arm impedance can be expressed as _{0} = _{0} + _{0}. The MMC single-phase averaging equivalent model is shown in _{dc} is the dc voltage; _{g} is the AC-side phase voltage; _{cu} and _{cl} represent the sum of the capacitance voltages of the upper and lower bridge arms, respectively; _{u} and _{l} are the currents flowing through the upper and lower bridge arms, respectively; _{c} is the circulating current; _{g} is the AC-side phase current; _{arm} is the equivalent bridge arm capacitance, and the calculation formula: _{arm} =

The MMC time-domain three-phase state-space expression after perturbation linearization can be obtained from

The variables in the formula are all three-phase (x = a,b,c) periodic time-varying signals, and the superscript s represents the steady-state operating point. The modulation function _{u}, _{l} is also obtained by the MMC control system.

In the MMC transmission system interconnection scenario with the grid, the complete control is shown in

From the control system block diagram, the specific form of the fundamental frequency modulation voltage and the two-fold modulation voltage can be obtained as:

_{inp} and _{ini} denote the current inner loop PI controller scale and integration coefficients; _{cirp} and _{ciri} denote the CCSC PI controller scale and integration coefficients, respectively; _{idq} and _{cdq} denote the current inner loop and CCSC integrators; _{y} denotes the control delay link; _{f} denotes the feedforward voltage coefficient; _{id} denotes the current inner loop cross-decoupling compensation term; _{cd} denotes the CCSC cross-decoupling compensation term.

The linear model of the control section is based on a d-q rotating coordinate system. In order to interface with the MMC main circuit model in the three-phase stationary coordinate system, it is necessary to perform the Park transform as well as the inverse transform to obtain its three-phase output form. The three-phase form of the fundamental frequency voltage and the two-fold frequency voltage is obtained as:

_{1} and _{2} are constant matrices used to implement the coordinate transformation, and the state variables introduced by the control integrator can be expressed as:

The three-phase linear state-space model of the MMC is obtained by associating the control system model with the main circuit model.

According to the theoretical derivation in

Substituting

When the MMC operates in a steady state, the complex variable

The main parameters of the system are shown in

Items | Value |
---|---|

Rated AC-side voltage | 525 kV |

Rated AC-side voltage | 840 kV |

Amount of submodule | 500 |

Capacitance of submodule | 11 mF |

Arm resistor | 0.1 |

Arm inductance | 140 mH |

Ratio of transformer | 525/435 |

Current loop (_{inp}, _{ini}) |
(30, 2500) |

CCSC (_{cirp}, _{ciri}) |
(100, 4000) |

Control delay (_{d}) |
550 us |

Feedforward voltage coefficient (_{f}) |
1 |

In order to verify the correctness of the established impedance model, the equivalent impedance characteristics of the MMC are obtained by the impedance scanning method in the PSCAD electromagnetic transient simulation platform. The MMC electromagnetic transient model with corresponding parameters is connected to the AC power grid mentioned later in the paper.

The principle of impedance measurement is that: first detect the voltage and current on the equipment to be measured when there is no perturbation injection; then, a small perturbation voltage is injected into the system to be measured, and the system to be measured will produce the corresponding current response; finally, the voltage perturbation and current response at the input interface of the power electronic equipment to be measured are detected and subtracted from the voltage and current on the equipment to be measured without perturbation injection. The impedance at the perturbation frequency can be calculated based on the voltage and current difference at the perturbation frequency. The swept frequency result is compared with the calculated result of the method in this paper, and the comparison graph is shown in the

The result shows that the theoretical model and the frequency scanning results are basically consistent, which proves that the model built has high accuracy and can be used to analyze the stability problem.

With the help of the Toplitz matrix, the HSS modeling can be more easily programmed modularly and include the higher harmonic dynamics in the model. Theoretically, HSS modeling includes infinite harmonic dynamics, but due to the presence of inductors, the amplitude of ultra-high harmonics is small, so truncation of finite model order is usually performed. When considering the higher harmonics, the effect of the third and higher harmonics on the impedance characteristics of MMC is shown in

As seen from the figure, the HSS model considering the second harmonic can describe the frequency response of the MMC more accurately, and the HSS model considering the fifth and sixth harmonics remains basically the same. From the impedance characteristics, it can be seen that the effect of higher harmonics on the frequency response is still mainly in the low and middle-frequency band (10–200 Hz). This shows that the HSS method greatly benefits studying oscillations over a wide-frequency range because he can easily consider the higher harmonics. The HSS method is more accurate when analyzing an oscillation problem in the 10 to 200 Hz frequency band due to the problem of insufficient phase margin. When considering the higher harmonics, the impedance characteristics at the intersection of the 60 Hz impedance amplitude-frequency characteristics shown in

However, it is not necessary to consider high harmonic interactions in the analysis of the stability of all frequency bands. For example, in the analysis of low and high-frequency bands, the influence of the higher harmonic components is smaller, and the dynamic characteristics of the high-frequency can be accurately described by considering only the harmonic components within the third order. Therefore, models of different orders can be used when targeting different stability problems.

According to the detailed impedance model, analyzing the law of the influence of each link, it can be obtained that the high-frequency impedance characteristics of the AC-side of the system are mainly influenced by the current inner loop, the feedforward voltage, and the control delay.

Different inner-loop control parameters are designed for AC-side impedance analysis.

When changing the current inner loop scale factor, the amplitude-frequency and phase-frequency characteristics of the frequency band below 1000 Hz change more obviously, and the resonance degree increases when _{inp} increases, but the effect on the characteristics of the high-frequency band above 1000 Hz is small, and the effect on high-frequency oscillation is not significant.

Similar to the analysis of the current inner control, the effect of the feedforward voltage link on the AC-side impedance characteristics of the system is analyzed as shown in

It can be seen from the figure, the impedance amplitude-frequency and phase-frequency characteristics are minimally affected by the feedforward voltage link in the low and medium frequency bands; as the coefficient _{f} rises, the effect on the high-frequency impedance characteristics increases and is more likely to cause stability problems. The MMC impedance characteristics are best when there is no feedforward voltage, but the feedforward voltage link can improve the fault ride-through capability and prevent overcurrent during a system fault, significantly improving the system's transient stability while reducing the system's small disturbance stability.

In order to study the effect of control delay on the impedance characteristics of the MMC, a delay link ^{−sTd} is added to the control system during stable operation, and the delay time Td is gradually increased. The impedance characteristics of the AC-side are shown in

After adding the delay link, it can be seen that the control delay mainly affects the high-frequency impedance characteristics in the system and has little effect on the low and medium frequencies. As the delay time changes, the characteristics change, which may lead to high-frequency oscillations in the system.

For the actual high-frequency oscillation problem, the impedance analysis method is used to analyze its stability. The equivalent sequential impedance model is shown in _{n} and impedance _{n}, and the AC system is equivalent to the DC voltage source _{z} and impedance _{z}.

According to Kirchhoff's law and linear control theory, it can be obtained that the system is stable when _{z} and _{n} satisfy the amplitude-frequency and phase-frequency characteristics. Therefore, the stability problem can be analyzed by the relationship between the impedance forms of the double-ended converter station. From the above stability discrimination principle, in order to further analyze the high-frequency resonance problem in MMC, this paper equates the AC system impedance by connecting two RL and C parallel circuits in series, as shown in

where _{1} = 5 mH, _{1} = 5 _{1} = 0.2 μF; _{2} = 5 mH _{2} = 5 _{2} = 1 μF. Two oscillation frequencies around 700 and 1800 Hz can be simulated with this parameter. The equivalent impedance characteristics of the system are shown in

According to the figure, approaches for suppressing oscillations fall into two primary categories: one is to make the impedance amplitude of the two systems without intersection, and the other is to make the phase difference at the intersection less than 180°. Given that numerous impedance poles exist in the AC line, it is not easy to produce impedance without intersection using the first approach, so we must begin with the second.

The low-pass filter suppression strategy is to set up a filter or quasi-proportional resonance link in the feedforward channel to dampen the high-frequency signal through the attenuation effect of the filter in the high-frequency band. The transfer function of the filter, _{f} (

_{n}, _{n} is the bandwidth of the filter. The smaller the filter bandwidth, the stronger the negative damping removal effect, but too small a bandwidth deteriorates the system's dynamic response. By adjusting the bandwidth of the low-pass filter, it is possible to obtain the optimal impedance characteristic curve of the MMC. In the technical model of this paper, the bandwidth of the low-pass filter is taken equal to 400 Hz. the impedance characteristic diagram of the system after adding the filter suppression is shown in

When the model does not consider the dynamic within MMC, the control link only considers the current inner loop control, feedforward voltage, and control delay link, the simplified impedance model _{jh} is obtained as:

_{0} = 2_{0} = 2_{s} is the delay time, and

It can also be seen from the figure that when the feedforward voltage is introduced to the low-pass filter, the resonant spikes and negative damping characteristics in the high-frequency band of the impedance characteristic curve vanish, while the negative damping band shifts toward the middle frequency. It can be seen that the low-pass filter helps attenuate the high-frequency oscillation characteristics induced by the feedforward voltage and reduces the negative damping band range, but it cannot completely eliminate the negative damping. A project that uses a low-pass filter to suppress the 1700 Hz high-frequency oscillation phenomenon, and after the suppression is completed, a new 800 Hz oscillation phenomenon appears, so it is still necessary to suppress the negative damping of MMC impedance in multiple oscillation risk bands.

Active damping suppression is an additional control strategy used to improve the damping of a specific frequency band. It is typically employed in conjunction with feedforward voltage low-pass filtering. The principle is to superimpose the feedforward voltage transients with the reference current using a first-order high-pass filter link in series with two first-order low-pass filter links to form a third-order dampener. They are then superimposed into the reference voltage. The expression _{damp} is obtained as:

_{H}, _{L1}, and _{L2} are the passbands of the high-pass filter and the two low-pass filters, respectively; _{S} is the gain coefficient. The principle of parameter design is based on the need to know precisely the range of variation of the impedance characteristic of the AC system. The design process is as follows: first, the initial range and step size of the additional damping controller parameters are given, and the upper and lower cutoff frequencies are calculated; then, the parameters whose cutoff frequencies meet the conditions are sorted to obtain the stability domain of the damping controller parameters; finally, the stability domain parameters are introduced into the MMC impedance model to judge whether the oscillation can be suppressed, and the simulation is carried out in the electromagnetic transient model to check its effectiveness. Changing the parameters of the damping controller to adjust the impedance will show the phenomenon of “ebb and flow”, so that the phase difference at the resonance point is less than 180° to avoid the occurrence of high-frequency oscillation. Considering the impedance characteristics, the parameters of the additional damping controller are _{H} = 30 Hz; _{L1} = 300 Hz; _{L2} = 1000 Hz; _{S} = 0.035, and the effect of parameter variation on the impedance characteristics of MMC is shown in

The effect of active damping suppression on the suppression of high-frequency oscillations of the system analyzed under the optimal parameters is shown in

As seen in the figure, the phase angle of the impedance around 0.7 and 1.8 kHz can be coordinated and balanced by using an additional active damping rejection to suppress high-frequency oscillations. However, the high-frequency band of the MMC still has negative damping when the network is capacitive, and the risk of oscillation is not completely eliminated.

The control link optimization enhances the system stability by improving the existing control links. Based on the voltage anticipation low-pass filter, an optimized control structure strategy with a low-pass filter in the proportional current loop link is used to reduce the high-frequency negative damping caused by the control delay. The variable parameters of this strategy include the cutoff frequency of the proportional low-pass filter in the current loop and the cutoff frequency of the voltage anticipation low-pass filter. The impedance characteristics of this parameter are shown in

From the figure, the control link optimization strategy can suppress the oscillations near 1.8 kHz and 700 Hz that occur in the system with proper parameter adjustment but still cannot completely eliminate the negative damping band.

High-frequency oscillation problem of the MMC-HVDC transmission system is becoming increasingly noticeable. This study examines the high-frequency oscillation mechanism and the effects of some factors on the MMC AC-side impedance characteristics. Three oscillation suppression strategies are studied in comparison, and a simplified and accurate high-frequency impedance model of MMC is established for the analysis of high-frequency oscillation problems. The following conclusions are obtained:

1) The effect of higher harmonics on the frequency response is still mainly in the low and middle-frequency band (10–200 Hz), and the dynamic characteristics of the high-frequency can be accurately described by considering only the harmonic components within the third order. The analysis of MMC small signal stability needs to be extended to the wide-frequency range in the future. The HSS model is essential in analyzing the negative damping transfer to the low-frequency band due to the high-frequency suppression strategy and the suppression of oscillation characteristics in the wide-frequency band.

2) High-frequency impedance characteristics can be accurately reflected by the simplified impedance model, which can also be used to analyze high-frequency oscillation issues. The system feedforward voltage link and control delay significantly impact the high-frequency impedance characteristics of the system, which must be controlled by designing a reasonable suppression strategy.

3) Currently, the study of the oscillation instability problem between the converter and the AC system involves a relatively simple AC power grid topology, such as a single long transmission line or an RLC equivalent system for the AC system. However, the actual AC system consists of multiple transmission lines, and the distributed parameter characteristics of the transmission lines cause the impedance of the system to exhibit inductive-volumetric jump characteristics at multiple frequencies, which increases the risk of high-frequency oscillation between the converter and the AC system at various frequencies and presents numerous design challenges for the control system. Further research is needed to characterize the AC system impedance in the frequency domain.

4) The additional filter in the feedforward voltage link can improve the impedance characteristics of the MMC, but it is not sufficient to solve the problem of high-frequency oscillations. Suppressing the extra active damping and optimizing the control system can improve the ability to suppress the high-frequency oscillation to some extent by reasonably adjusting the extra damping controller parameters to reduce the range of the negative damping band, but still cannot completely eliminate the negative damping band of the MMC. These methods also require a known resonant frequency of the transmission line to preset the controller parameters, which is difficult to obtain and estimate in practical applications. Therefore, how to completely eliminate the high-frequency oscillation of the MMC by improving the control still needs further research.

High voltage direct current

Modular multilevel converter

Harmonic state-space

Phase-locked loop

Circulating current suppressing control

Direct current

Alternating current

Insulated gate bipolar transistor

_{g}

Rated AC-side voltage

_{dc}

Rated DC-side voltage

_{cu}

Sum of the capacitance voltages of the upper bridge arms

_{cl}

Sum of the capacitance voltages of the lower bridge arms

_{u}

Currents flowing through the upper bridge arms

_{l}

Currents flowing through the lower bridge arms

_{c}

Circulating current

_{g}

AC-side phase current

Amount of submodule

Capacitance of submodule

_{0}

Arm resistor

_{0}

Arm inductance

_{u}

Upper bridge arms modulation function

_{l}

Lower bridge arms modulation function

_{inp}

Current inner loop PI controller scale coefficients

_{ini}

Current inner loop PI controller integration coefficients

_{cirp}

CCSC PI controller scale coefficients

_{ciri}

CCSC PI controller integration coefficients

_{y}

Control delay link

_{d}

Delay time

_{f}

Feedforward voltage coefficient

We would like to thank all those who have reviewed and contributed to this paper for their valuable assistance.

The authors thankfully acknowledge the support of the project supported by Research on the Oscillation Mechanism and Suppression Strategy of Yu-E MMC-HVDC Equipment and System (2021Yudian Technology 33#).

The authors declare that they have no conflicts of interest to report regarding the present study.