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 Fig. 1:

A MMC-HVDC system’s main structure topology is shown in Fig. 2. The entire converter consists of a three-phase circuit, with each phase consisting of two bridge arms, upper and lower, each containing N half-bridge sub-modules.

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 S of all submodules in each phase bridge arm is equal to the modulated wave s of the bridge arm.

(1)∑k=1NS(k)=Ns

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 Z₀ = R₀ + L₀. The MMC single-phase averaging equivalent model is shown in Fig. 3. In the figure, v_dc is the dc voltage; v_g is the AC-side phase voltage; v_cu and v_cl represent the sum of the capacitance voltages of the upper and lower bridge arms, respectively; i_u and i_l are the currents flowing through the upper and lower bridge arms, respectively; i_c is the circulating current; i_g is the AC-side phase current; C is the sub-module capacitance, C_arm is the equivalent bridge arm capacitance, and the calculation formula: C_arm = C/N.

The MMC time-domain three-phase state-space expression after perturbation linearization can be obtained from Fig. 3 as:

(2){ddtΔicx=−R0L0Δicx−suxs2L0Δvcux−slxs2L0Δvclx−vcuxs2L0Δsux−vclxs2L0ΔslxddtΔvcux=suxsCarm(Δicx+12Δigx)+2icxs+igxs2CarmΔsuxddtΔvclx=slxsCarm(Δicx−12Δigx)+2icxs−igxsCarmΔslxddtΔigx=slxsL0Δvclx−suxsL0Δvcux−R0L0Δigx−vcuxsL0Δsux+vclxsL0Δslx−2L0Δvgx

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 s_u, s_l is also obtained by the MMC control system.

(3){Δsux=−Δvcx∗−Δvsx∗vdcΔs1x=−Δvcx∗+Δvsx∗vdcwhere Δvsx∗ represents the fundamental frequency modulation voltage; Δvcx∗ represents the 2-fold modulation voltage.

In the MMC transmission system interconnection scenario with the grid, the complete control is shown in Fig. 4. In order to reflect the effect of the control delay on system oscillations, an overall delay link is added between the system’s nearest level approximation modulation and the modulating wave generated by the control system.

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:

(4){Δvsdq∗=−GyKinpΔigdq+GyKiniΔxidq+GyKidΔigqd+GyKfΔvgdqddtΔxidq=−Δigdq

(5){Δvcdq∗=−GyKcirpΔicdq+GyKciriΔxcdq+GyKcdΔicqdddtΔxcdq=−Δicdqwhere K_inp and K_ini denote the current inner loop PI controller scale and integration coefficients; K_cirp and K_ciri denote the CCSC PI controller scale and integration coefficients, respectively; x_idq and x_cdq denote the current inner loop and CCSC integrators; G_y denotes the control delay link; K_f denotes the feedforward voltage coefficient; K_id denotes the current inner loop cross-decoupling compensation term; K_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:

(6){Δvsx∗=(n2GyKid−n1GyKinp)Δigx+n1GyKiniΔxix+n1GyKfΔvgxΔvcx∗=(n2GyKcd−n1GyKcirp)Δicx+n1GyKciriΔxcxwhere n₁ and n₂ are constant matrices used to implement the coordinate transformation, and the state variables introduced by the control integrator can be expressed as:

(7){ddtΔxix=−n1⋅Δigx−n2ω1⋅ΔxixddtΔxcx=−2ω1n2⋅Δxcx−n1⋅Δicx

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

(8)x˙(t)=A(t)x(t)+B(t)u(t)

According to the theoretical derivation in Section 2, MMC is a periodic time-varying system with no fixed DC steady-state operating point. In this regard, this section will use the method of harmonic state-space to realize the time domain to frequency domain transformation. The Fourier transform can convert any continuous periodic signal into frequency domain form as shown in Eq. (9).

(9)x(t)=∑n∈Zxnejnω1t

Substituting Eq. (9) into Eq. (8), the HSS based MMC frequency domain model can be obtained after simplification as shown in Eq. (10).

(10)(A−N)X+BU=sXwhere N represents the diagonal matrix, the diagonal elements contain the frequency information; A and B matrices are of the same form and both are Toeplitz matrices; the subscripts represent the Fourier levels. The MMC contains an infinite number of harmonics due to its frequency interaction characteristics. However, in its actual operation, the amplitude of the 3rd and higher harmonics is usually small and almost negligible compared to the first three harmonics. Therefore, only the first three harmonics are considered in this example's model constructed for the MMC. The forms of the matrix A, N and X matrices are shown in Eqs. (11) to (13), respectively.

(11)A=[A0A−1A−2A−3O4×4O4×4O4×4A1A0A−1A−2A−3O4×4O4×4A2A1A0A−1A−2A−3O4×4A3A2A1A0A−1A−2A−3O4×4A3A2A1A0A−1A−2O4×4O4×4A3A2A1A0A−1O4×4O4×4O4×4A3A2A1A0]

(12)N=diag(−j3ω1I,−j2ω1I,−jω1I,O,jω1I,j2ω1I,j3ω1I)

(13)X=[X−3X−2X−1X0X1X2X3]Twhere the elements in the A matrix are also a matrix of the same size as A(t). The number of steps is 126.

When the MMC operates in a steady state, the complex variable s of the HSS model tends to zero, so the inverse of Eq. (13) can establish the relationship between the state variable and the output variable. The output variable contains the external disturbance voltage of the AC-side; the state variable contains the response current of the AC-side, and the AC-side impedance can be calculated by dividing the two.

(14)X=−(A−N)−1B×U

The main parameters of the system are shown in Table 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 Fig. 5.

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 Fig. 6.

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 Fig. 7 can clearly reflect the small-signal instability that may be caused by the improper selection of the system control parameters, while considering only the lower harmonics may lead to wrong conclusions. The LTI model will become highly complicated when analyzing such problems.

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.

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, G_f (s), is obtained as:

(15)Gf(s)=ωn2s2+2ξωns+ωn2where ξ is the damping factor of the second-order low-pass filter, ξ = 0.707; ωn = 2πf_n, f_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 Fig. 14.

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 Z_jh is obtained as:

(16)Zjh=jωL′+R′+(kinp+kinijω)[cos⁡(ωTs)−jsin⁡(ωTs)]1−Kf[cos⁡(ωTs)−jsin⁡(ωTs)]where R₀ = 2R′, L₀ = 2L′, T_s is the delay time, and ω is the perturbation frequency. The real and imaginary parts of the AC-side impedance can be obtained from Eq. (15), which can reflect the amplitude-frequency phase frequency characteristics of the impedance in the high-frequency range. The impedance characteristics before and after adding suppression can also be analyzed by a simplified model, as shown in Fig. 15. The impedance characteristic plots before and after adding suppression in Figs. 14 and 15 show that the simplified model matches well with the detailed model in the frequency band above 200 Hz. It proves that the simplified model can be used to analyze the high-frequency oscillation problem instead.

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 G_damp is obtained as:

(17)Gdamp=kSss+2πfH2πfL1s+2πfL12πfL2s+2πfL2where f_H, f_L1, and f_L2 are the passbands of the high-pass filter and the two low-pass filters, respectively; k_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 f_H = 30 Hz; f_L1 = 300 Hz; f_L2 = 1000 Hz; k_S = 0.035, and the effect of parameter variation on the impedance characteristics of MMC is shown in Fig. 16.

The effect of active damping suppression on the suppression of high-frequency oscillations of the system analyzed under the optimal parameters is shown in Fig. 17.

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 Fig. 18.

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.