Small Signal Modeling and Stability Analysis of Modular Multilevel Converter Based on Harmonic State-Space Model

: The stability of Modular multilevel converter (MMC) itself is the premise of analyzing the stability of MMC cascading with other cells, so this paper addressed the small-signal stability of MMC with dc voltage control mode which is the common operation mode of grid-side MMC in high-voltage direct current (HVDC) systems. First, the small signal model of MMC with open-loop control mode and dc voltage control mode while considering the dc voltage controller and current controller are established with the harmonic state-space (HSS) modeling method. Subsequently, eigenvalues and participation factors are analyzed to estimate the stability of MMC system and the inﬂuence of controller parameters, especially the proportional coe ﬃ cients of dual-loop controllers, which can improve the design e ﬃ ciency of control system. Finally, time-domain simulations in MATLAB / Simulink are provided to verify the accuracy of HSS small signal model and the e ﬀ ectiveness of the theoretical stability analysis.


Introduction
Modular multilevel converter (MMC) topology was first introduced in 2003 [1,2] and it has been widely used in high voltage and high power applications, due to its advantages of modularity, scalability, and power controllability. The stability of MMC itself is a significant premise to study the stability of MMC cascading with other cells, e.g., wind farm, PV inverters, weak grid, and so on. While the key is the modeling of MMC itself. However, MMC is characterized by the obvious internal dynamics in sub-module capacitor voltages and arm currents [3,4], which makes its modeling considerably more complicated than the conventional two/three-level voltage source converters (VSCs). Most of the initial researches [5][6][7][8] used the average model, which neglected these internal harmonic characteristics in the modeling process and only obtained the simplified mathematical equations of MMC in fundamental frequency.
Many scholars recently have done deep studies about the modeling of MMC to achieve high accuracy of small signal model of MMC. The state-space model of MMC based on dynamic phasors contains the internal dynamics, which is often used to model a nonlinear time-varying system as a linear time-invariant system [9][10][11][12]. The small signal state-space model of MMC with dual-loop When compared with the traditional state-space equation, the signal x(t) is expressed as an exponentially modulated periodic (EMP) signal to represent harmonic of different orders in the tome domain in Step 3 of Figure 1. On the other hand, because the MMC control system and electrical system are coupled with each other and control loops are coupled with each other, the Toeplitz matrix Γ(A) is applied to express the coupling of state variables with different frequencies in Step 3. Therefore, as compared with the traditional state-space model, the MMC small signal model that is based on harmonic state-space in this paper makes the analysis of dynamic characteristics, steadystate characteristics, and stability of MMC more accurate.
The rest of this paper is organized, as follows. Section 2 introduces the basic topology and operation theory of MMC and it derives the state-space equations in time domain. Section 3 builds the steady-state model and the small signal model of MMC based on the HSS method in open-loop control mode and dc-voltage control mode, respectively. Section 4 uses the eigenvalues and participation factors of the established HSS small signal model to analyze the stability of a dc-voltage controlled MMC system and the influence of controller parameters on the stability of the system. Section 5 provides a case study of dc voltage-mode controlled MMC, and the simulation result verifies the validity of theoretical analyses. Section 6 concludes the work of this paper. Figure 2 shows the basic structure of MMC. The converter consists of three phases, and each phase consists of one upper arm and one lower arm connected in series between two dc terminals. Each arm has N series-connected half-bridge submodules (SM) and one arm inductor L, the equivalent parasitic resistance of which is R. CSM is capacitance of submodule capacitors, whose nominal voltages are all Udc/N.

Basic Theory of MMC
Step 1 Establish model of MMC main circuit and control system                  

                 
Step 4 Analyze of MMC dynamic response , root locus and stability based on small signal state-space equation  Figure 2 shows the basic structure of MMC. The converter consists of three phases, and each phase consists of one upper arm and one lower arm connected in series between two dc terminals. Each arm has N series-connected half-bridge submodules (SM) and one arm inductor L, the equivalent parasitic resistance of which is R. C SM is capacitance of submodule capacitors, whose nominal voltages are all U dc /N.   In outer control loop, the difference of dc voltage reference value and actual dc voltage value goes through a PI controller and generates current reference value in d axis. Inner control loop adopts the basic dq decoupling control strategy, as shown in Figure 3, where idref and iqref are d-axis and q-axis current reference values, respectively. idref is given by the outer dc voltage control loop and iqref is 0. The differences between the current reference values and actual current values go through PI controllers, dq inverse transformation, and nearest level approximation modulation (NLM) and modulation function in abc reference frame are finally generated.   goes through a PI controller and generates current reference value in d axis. Inner control loop adopts the basic dq decoupling control strategy, as shown in Figure 3, where idref and iqref are d-axis and q-axis current reference values, respectively. idref is given by the outer dc voltage control loop and iqref is 0. The differences between the current reference values and actual current values go through PI controllers, dq inverse transformation, and nearest level approximation modulation (NLM) and modulation function in abc reference frame are finally generated. When the MMC operates in steady state, while assuming that capacitance voltages of all submodules are perfectly balanced, the average equivalent circuit of one phase could be constructed, as in Figure 4 [8]. Additionally, Carm is the equivalent capacitance of one arm, where Carm = CSM/N. The switching functions of upper and lower arms are Su and Sl respectively. vs is the ac side phase voltage of the converter, and Vdc is the dc terminal voltage. v∑cu and v∑cl are the sum of the capacitor voltages of the upper and lower arms, respectively. In outer control loop, the difference of dc voltage reference value and actual dc voltage value goes through a PI controller and generates current reference value in d axis. Inner control loop adopts the basic dq decoupling control strategy, as shown in Figure 3, where i dref and i qref are d-axis and q-axis current reference values, respectively. i dref is given by the outer dc voltage control loop and i qref is 0. The differences between the current reference values and actual current values go through PI controllers, dq inverse transformation, and nearest level approximation modulation (NLM) and modulation function in abc reference frame are finally generated.

Basic Theory of MMC
When the MMC operates in steady state, while assuming that capacitance voltages of all submodules are perfectly balanced, the average equivalent circuit of one phase could be constructed, as in Figure 4 [8]. Additionally, C arm is the equivalent capacitance of one arm, where C arm = C SM /N. The switching functions of upper and lower arms are S u and S l respectively. v s is the ac side phase voltage of the converter, and V dc is the dc terminal voltage. v cu and v cl are the sum of the capacitor voltages of the upper and lower arms, respectively.  In Figure 4, the average switching functions of upper arm Su and lower arm Sl are defined as: where m(t) is the modulation function of this phase: m(t) = 2vs(t)/Udc.
While applying Kirchhoff's law to this one-phase circuit of MMC, the equations of the upper and lower arms can be derived as: The dynamic performance of the submodule capacitors can be obtained as: In Figure 4, the average switching functions of upper arm S u and lower arm S l are defined as: where m(t) is the modulation function of this phase: m(t) = 2v s (t)/U dc . While applying Kirchhoff's law to this one-phase circuit of MMC, the equations of the upper and lower arms can be derived as: Energies 2020, 13, 1056

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The dynamic performance of the submodule capacitors can be obtained as: The arm currents of upper and lower arms can be represented as: where i s is ac side phase current of the converter and i c = (i u + i l )/2, which refers to the circulating current flowing through the upper and lower arms of the converter at the same time.
Note that the dc component of i c is I c0 , and then the dc side current I dc can be expressed as: Substituting (7) and (8) into (3)- (6), the state-space equations of one-phase of MMC can be expressed as: . where The above state-space equation of MMC indicates a nonlinear time periodic (NLTP) system. The detailed derivation process of converting this system into a linear time-invariant (LTI) system with the HSS method will be shown in the following section.

HSS-Based Large Signal Model of MMC Power Stage
Any time-varying periodic signal y(t) can be described as the form of Fourier series expansion: where ω 1 = 2π/T 1 and T 1 is the fundamental period of the periodic signal, and h is the harmonic order. Specially, the state variable x(t) is written as an exponentially modulated periodic (EMP) signal to represent the transient evolution of the harmonics: (15) where s = σ + jω is used to modulate the Fourier coefficient for extracting the transient response of harmonic components [18]. Based on the transformations (14) and (15), the NLTP state-space model in (10) of single phase of MMC in time-domain can be expressed as a LTI HSS model in the frequency-domain [19][20][21]: where where X n , U n are, respectively, the nth Fourier coefficient of state variable x(t) and input variable u(t), and n = −h, . . . −1, 0, 1, . . . , h.
where I cn is the nth Fourier coefficient of i c (t), and V cun is the nth Fourier coefficient of v cuo (t).
Other parameters have similar meanings, which are not explained repeatedly here. Γ(A) and N in (16) are expressed as: Γ(A) is a Toeplitz matrix form of A, as shown in (19), which consists of the Fourier series of the matrix A, where A n is the nth Fourier coefficient of (20) is the identity matrix, which has the same order as A.
When n = 0, Energies 2020, 13, 1056 7 of 22 When n 0, (16) is a Toeplitz matrix form of B, being similar to (19), which consists of the Fourier series of the matrix B, where B n is the nth Fourier coefficient of B.
When n = 0, When n 0, Equation (16) represents the steady-state model of single phase of MMC that is based on the HSS method, which is linearized from (10) and is a LTI system. Therefore, the stability of MMC can be analyzed with the classical control theory. This HSS model can not only obtain the information of the 0-hth harmonics, but also reflect the interactions between the harmonics.
Letting the left side of (16) to be zero, the steady-state solution of HSS model of MMC can be calculated by: where the frequency-domain values of state variables can be obtained. Referring to (14), the time-domain values of i co (t), v cuo (t), v clo (t), and i so (t) can be solved.

HSS-Based Small Signal Model of MMC with Open-loop Control
Based on Equations (3)-(6), the small-signal model of MMC around an operation trajectory (characterized by m o (t), v cuo (t), v clo (t), i so (t)) can be derived as: In open-loop control mode, the modulation signal m(t) is directly given, so it could serve as an input variable in the state-space model. While combining Equations (26)-(29), the small signal model of MMC in open-loop control mode, which is also the model of power stage of MMC, can be represented as: Energies 2020, 13, 1056 The elements in (35), such as ∆x n , ∆u n , A sn , and B sn are the nth Fourier coefficients of ∆x(t), ∆u(t), A s (t), B s (t), and n = −h, . . . , −1, 0, 1, . . . , h.

HSS-Based Small Signal Model of MMC with DC-Voltage Control
DC-voltage control is commonly used in an MMC-based HVDC system to maintain the dc bus voltage. Figure 5 shows the control structure, which contains outer dc-voltage control loop, inner current control loop, and phase-locked loop (PLL), where U dc and U dcref are the dc side voltage of MMC and its reference value, respectively. I sd , I sq are the values of MMC ac side current in dq coordinate system. V sd , V sq are the values of MMC ac side voltage in dq coordinate system, and M d , M q are the values of modulation signal in dq coordinate system. ω 1 is the fundamental angular frequency of 314.1 rad/s, and θ is the output phase angle of PLL. H 1 (s), H 2 (s), and H 3 (s) are the PI controllers of dc voltage control, current control, and PLL control, respectively. k id is the decoupling coefficient. The effect of small signal disturbance on PLL can be neglected when the converter is integrated with a strong enough ac grid [24]. Therefore, the small signal model of MMC in this paper does not take the effect of PLL into account.

HSS-Based Small Signal Model of MMC with DC-Voltage Control
DC-voltage control is commonly used in an MMC-based HVDC system to maintain the dc bus voltage. Figure 5 shows the control structure, which contains outer dc-voltage control loop, inner current control loop, and phase-locked loop (PLL), where Udc and Udcref are the dc side voltage of MMC and its reference value, respectively. Isd, Isq are the values of MMC ac side current in dq coordinate system. Vsd, Vsq are the values of MMC ac side voltage in dq coordinate system, and Md, Mq are the values of modulation signal in dq coordinate system. ω1 is the fundamental angular frequency of 314.1 rad/s, and θ is the output phase angle of PLL. H1(s), H2(s), and H3(s) are the PI controllers of dc voltage control, current control, and PLL control, respectively. kid is the decoupling coefficient. The effect of small signal disturbance on PLL can be neglected when the converter is integrated with a strong enough ac grid [24]. Therefore, the small signal model of MMC in this paper does not take the effect of PLL into account. The inner current control loop contains two PI controller. Two additional state variables x i1 , x i2 are introduced to simplify the integral operation: where k p2 and k i2 are proportional coefficients and integral coefficients of the PI controllers in current control loop respectively.
Then the output small signal insertion indexes ∆M d1 , ∆M q1 in dq coordinate system of the inner loop are: The ∆M d1 , ∆M q1 in dq coordinate system need to be transformed into ∆m i (t) in abc coordinate system, where ∆m i (t) is the control signal of phase A that is produced by the inner loop. Firstly, according to the dq transformation matrix, I sd and I sq can be expressed as: where the Park transformation matrix T dq is: From Equation (38), the relationships between I sd , I sq , the amplitude I, and the phase angle ϕ of phase A current are: Combining Equations (14), (36) and (40), Equation (36) in time-domain can be represented in the frequency-domain: According to inverse Park transformation matrix, ∆m i (t) could be calculated as: where the inverse Park transformation matrix T dq −1 is: Energies 2020, 13, 1056 10 of 22 According to the complex form of Fourier expansion, the ±1st coefficients of ∆m i (t) are: Substituting (37) and (41) into (44) the ±1st coefficients of ∆m i (t) are:

Small Signal Model of Outer DC-Voltage Control Loop
As shown in Figure 5, the difference between the actual value of the dc voltage and the reference value would pass through the outer loop PI controller and then the inner loop PI controller, producing small signal insertion index ∆M d2 in d axis. Two additional state variables x v1 , x v2 are introduced to achieve the integral operation of dc components for two integral components: where k p1 and k i1 are the proportional and integral coefficients of the PI controllers in outer DC voltage control loop, respectively. Afterwards, the output control signal ∆M d2 is: According to inverse Park transformation matrix, the control signal ∆m v (t) of phase A in abc coordinate system can be obtained from According to the complex form of Fourier expansion, the ±1st coefficients of ∆m v (t) are: Assume the equivalent impedance on the dc side is Z L . According to Equation (9), ∆U dc is Substituting (47) and (50) into (49), the ±1st coefficients of ∆m v (t) can be represented, as follows: Energies 2020, 13, 1056 11 of 22

Small Signal Model of DC-Voltage Controlled MMC
The models of control system and open-loop system need to be integrated into one state-space equation in order to establish the small signal model of the entire closed-loop system of MMC.
The state variables of the closed-loop system of MMC include four state variables of power stage (∆i c , ∆v cu , ∆v cl , ∆i s ) and four state variables of control system (∆x i1 , ∆x i2 , ∆x v1 , ∆x v2 ): The input variables of the closed-loop system of MMC are the same as the input variables of the power stage of MMC: By combining the equations of control system (36), (45), (46) and (51) with the HSS model of power stage in (34), the HSS model of the entire closed-loop system can be expressed as: where: Additionally, N is the same as (20). The variables ∆x i1 , ∆x i2 , ∆x v1 , ∆x v2 , and ∆U dcref in (55) are all dc components. When these variables are transformed into Fourier series, they only have values at the frequency of 0 Hz, and the components of other frequencies are zero. The control signals ∆m i+1 , ∆m i−1 , ∆m v+1 , and ∆m v−1 in (56) are added to the system matrix A c , and they are only related to the matrix components A 0O , A 0P , A 0N , A −1O , A −1N , A 1O , and A 1P . The other matrix components A 0 , A +1 , A −1 , A +2 , A −2 . . . are only dependent on the main circuit of MMC. Subsequently, the specific expressions of these matrix are presented in the Appendix A.
Similarly, for the input matrix B c , as shown in (57), the controlling signals ∆m v+1 and ∆m v−1 need to be added into the matrix, and they are only related to the matrix components B 0O , B +1O , and B −1O . The specific expressions are also presented in the Appendix A.
In general case, circulating current mainly contains dc and second harmonic components and the capacitor voltages mainly contains dc, fundamental, second and third harmonic components. Therefore, the harmonic order considered in the HSS model is h = 3. Table 1 shows the main parameters of the studied MMC. It is worth mentioning that the number of modules in one arm is 20 to reduce the simulation time. Figure 5 shows the MMC operates in dc voltage control mode. Firstly, the outer loop controller parameters are set as k p1 = 0.87 and k i1 = 10, the inner loop controller parameters are set as k p2 = 0.019 and k i2 = 0.057. It needs to be noted that, since the four state variables of control system ∆x i1 , ∆x i2 , ∆x v1 , and ∆x v2 only have values at the frequency of 0 Hz, they do not have significance at other frequencies (−3ω 1 , −2ω 1 , −1ω 1 , 1ω 1 , 2ω 1 , 3ω 1 ). Thus, there are always 24 eigenvalues at the origin (0, j0), having no effect on the stability analysis of the system. After calculation, it is confirmed that in addition to these 24 eigenvalues, there is another eigenvalue that is located at the origin. Therefore, these 25 eigenvalues will not be discussed in this paper. Additionally, other eigenvalues are plotted as shown in Figure 6. It can be seen that all eigenvalues are on the left half plane of the complex plane, which indicated that the MMC system is stable with these controller parameters. are on the left half plane of the complex plane, which indicated that the MMC system is stable with these controller parameters.  The concept of oscillation mode is introduced, that is, a real root or a pair of conjugate complex roots represent one oscillation mode of the system in order to further analyze the eigenvalues. Additionally, the participation factor analysis is introduced to analyze the interaction between state variables and oscillation modes. The participation factor of the kth state variable in the ith oscillation  The concept of oscillation mode is introduced, that is, a real root or a pair of conjugate complex roots represent one oscillation mode of the system in order to further analyze the eigenvalues. Additionally, the participation factor analysis is introduced to analyze the interaction between state variables and oscillation modes. The participation factor of the kth state variable in the ith oscillation mode of the system can be calculated by [25]:

Stability Analysis Based on the HSS-Based Small Signal Model
where Φ ki is the element in the kth row and ith column of the right eigenvector matrix, and Ψ ik is the element in the ith row and kth column of the left eigenvector matrix. Table 2 lists the oscillation modes of the MMC systems and their main participating state variables, according to the value of the participation factors. ∆i cn , ∆i sn , ∆u cun , ∆u cln , ∆x v1n , ∆x v2n is nth harmonic component of ∆i c (t), ∆i s (t), ∆u cu (t), ∆u cl (t), ∆x v1 (t), ∆x v2 (t), respectively, and n = 0, 1, 2, 3.
Eigenvalue loci when k p1 and k p2 changes are shown in Figure 7a,b, respectively, to analyze the effect of the controller parameters on the stability of MMC. When increasing the parameter k p1 of outer voltage loop with other parameters unchanged, as shown in Figure 7a, oscillation modes 2, 3, 6, 7, 8, 10, 11 remain unchanged, which means that these eigenvalues are not affected by the change of k p1 and most of other oscillation modes move to the right. As k p1 gradually increases, oscillation modes 4, 12, 14, 15 move horizontally to the right whose imaginary parts are nearly unchanged. Additionally, when k p1 is 1.57, oscillation mode 14 firstly moves to the right half plane, which causes the instability of the system. For example, when k p1 = 2.87, the eigenvalues of oscillation mode 14 are (4.196 ± j176.9). The corresponding oscillation period is 2π/176.9 = 0.0355 s.
Based on the analysis of participation factor, the oscillation mode 14 is strongly related to the dc component of the submodule capacitor voltage. It indicates that the instability is due to the change of proportional coefficient k p1 of the outer dc voltage controller, which affects the dc component of submodule capacitor voltage, in turn causing oscillation mode 14 to move to the right half plane and causes instability.
Only the eigenvalues whose imaginary parts remain unchanged could cross into the right half plane, even if k p1 continues to increase. This indicates that with the dc-voltage control mode, the change of controller parameter k p1 will not change the unstable oscillation frequency of the MMC system. submodule capacitor voltage, in turn causing oscillation mode 14 to move to the right half plane and causes instability.
Only the eigenvalues whose imaginary parts remain unchanged could cross into the right half plane, even if kp1 continues to increase. This indicates that with the dc-voltage control mode, the change of controller parameter kp1 will not change the unstable oscillation frequency of the MMC system. When increasing the proportional coefficient kp2 of inner current control loop with other parameters unchanged, as shown in Figure 7b, oscillation modes 2, 3, 6, 7, 8, 10, 11 remain unchanged, which means that these eigenvalues will not be affected by the change of kp2, and most of other oscillation modes move to the right. As kp2 gradually increases, oscillation modes 5, 12, 13, and 15 move horizontally to the right, whose imaginary parts are nearly unchanged. Additionally, when kp2 is 0.042, oscillation mode 12 firstly moves to the right half plane, which causes the instability of the system. For example, when kp2 = 0.16, eigenvalues of oscillation mode 12 are (0.107 ± j756.8). The corresponding oscillation period is 2π/756.8 = 0.0083 s.
The oscillating modes 12 is related to the third frequency component of the ac side current based on the analysis of participation factors. It indicates that the instability is due to the change of proportional coefficient kp2 of inner current controller, which affects the third frequency component When increasing the proportional coefficient k p2 of inner current control loop with other parameters unchanged, as shown in Figure 7b, oscillation modes 2, 3, 6, 7, 8, 10, 11 remain unchanged, which means that these eigenvalues will not be affected by the change of k p2 , and most of other oscillation modes move to the right. As k p2 gradually increases, oscillation modes 5, 12, 13, and 15 move horizontally to the right, whose imaginary parts are nearly unchanged. Additionally, when k p2 is 0.042, oscillation mode 12 firstly moves to the right half plane, which causes the instability of the system. For example, when k p2 = 0.16, eigenvalues of oscillation mode 12 are (0.107 ± j756.8). The corresponding oscillation period is 2π/756.8 = 0.0083 s.
The oscillating modes 12 is related to the third frequency component of the ac side current based on the analysis of participation factors. It indicates that the instability is due to the change of proportional coefficient k p2 of inner current controller, which affects the third frequency component of the ac side current, in turn causing the oscillation mode 12 to move to the right half plane and cause instability.
Even if k p2 continues to increase, only the eigenvalues whose imaginary parts remain unchanged could cross into the right half plane, which indicated that with the dc-voltage control mode, the change of controller parameter k p2 will also not change the unstable oscillation frequency of the MMC system.

Simulation Verifications
The small signal model of MMC that is based on HSS method is built in m. file in MATLAB and the considered order is set to be 3 in the model. Time-domain simulations are carried out in the MTLAB/Simulink and parameters adopted are as shown in Table 1 to validate the theoretical analysis.

Open-Loop Control Mode
When MMC operates in open-loop control mode and the reference of m(t) has a step change from 0.885sin(ω 1 t + ϕ) to 0.87sin(ω 1 t + ϕ) at 3 s, the transient response of i c (t) is presented in Figure 8a and the zoom-in view is shown in Figure 8b, showing good agreement between the small-signal model and the simulation results. Figure 9 shows the harmonic components of ∆i c (t) with open-loop control mode, which indicated that fundamental component's change of m(t) would result in both dc and second harmonic components' changes of i c (t), which proves that the small-signal model that is presented in this paper could reflect the internal harmonic interactions of MMC.

Open-Loop Control Mode
When MMC operates in open-loop control mode and the reference of m(t) has a step change from 0.885sin(ω1t + φ) to 0.87sin(ω1t + φ) at 3 s, the transient response of ic(t) is presented in Figure 8a and the zoom-in view is shown in Figure 8b, showing good agreement between the small-signal model and the simulation results.    and the zoom-in view is shown in Figure 8b, showing good agreement between the small-signal model and the simulation results.

DC-Voltage Control Mode
When MMC operates in dc-voltage control mode and the dc voltage reference value U dcref has a step change from 700 V to 735 V at 3 s, the transient response of i c (t) is presented in Figure 10a, and the transient response of v cu (t) is presented in Figure 10b, all showing good agreement between the small-signal model and simulation results, thus verifying the accuracy of the modeling presented in this paper.

DC-Voltage Control Mode
When MMC operates in dc-voltage control mode and the dc voltage reference value Udcref has a step change from 700 V to 735 V at 3 s, the transient response of ic(t) is presented in Figure 10a, and the transient response of v∑cu(t) is presented in Figure 10b, all showing good agreement between the small-signal model and simulation results, thus verifying the accuracy of the modeling presented in this paper.
Step response comparison of (a) ic(t) with dc voltage control mode. (b) v∑cu(t) with dc voltage control mode.
The outer loop controller parameter kp1 is first changed to 2.87 o verify the predicted oscillation period with different parameters, while other parameters remain unchanged in the simulation model. Figure 11 shows the simulation waveform of ic(t) with kp1 = 2.87. It can be seen that the MMC system is unstable and the measured oscillation period of ic(t) is approximately 0.032 s, which is almost The outer loop controller parameter k p1 is first changed to 2.87 o verify the predicted oscillation period with different parameters, while other parameters remain unchanged in the simulation model. Figure 11 shows the simulation waveform of i c (t) with k p1 = 2.87. It can be seen that the MMC system is unstable and the measured oscillation period of i c (t) is approximately 0.032 s, which is almost consistent with the predicted oscillation period 2π/176.9 = 0.0355 s in Section 4.
Step response comparison of (a) ic(t) with dc voltage control mode. (b) v∑cu(t) with dc voltage control mode.
The outer loop controller parameter kp1 is first changed to 2.87 o verify the predicted oscillation period with different parameters, while other parameters remain unchanged in the simulation model. Figure 11 shows the simulation waveform of ic(t) with kp1 = 2.87. It can be seen that the MMC system is unstable and the measured oscillation period of ic(t) is approximately 0.032 s, which is almost consistent with the predicted oscillation period 2π/176.9 = 0.0355 s in Section 4. Subsequently, the inner loop controller parameter kp2 is changed to 0.162, while other parameters in the simulation model remain unchanged. Figure 12 shows the simulation waveform of ic(t) with kp2 = 0.162. It can be seen that the MMC system is unstable, and the measured oscillation period of ic(t) is about 0.009 s, which is almost consistent with the predicted oscillation period 2π/756.8 = 0.0083 s in Section 4. Subsequently, the inner loop controller parameter k p2 is changed to 0.162, while other parameters in the simulation model remain unchanged. Figure 12 shows the simulation waveform of i c (t) with k p2 = 0.162. It can be seen that the MMC system is unstable, and the measured oscillation period of i c (t) is about 0.009 s, which is almost consistent with the predicted oscillation period 2π/756.8 = 0.0083 s in Section 4.

Conclusions
An accurate and complete small signal model of MMC with dc voltage control mode that is based on the HSS modeling method is derived, which considers the internal and external harmonic interactions, thus achieving higher accuracy. Subsequently, through the analysis of the eigenvalues, participation factors, and the calculation of oscillation frequencies under different controller parameters, the effect of controller parameters on the stability of MMC is studied and the following conclusions can be drawn. Besides, the time-domain simulation built in MATLAB/Simulink verifies the accuracy of the HSS-based MMC model and validity of stability analysis.
(1) When MMC operates with dc-voltage control mode, either too large outer loop proportional coefficient kp1 or inner loop proportional coefficient kp2 might cause the system unstable.

Conclusions
An accurate and complete small signal model of MMC with dc voltage control mode that is based on the HSS modeling method is derived, which considers the internal and external harmonic interactions, thus achieving higher accuracy. Subsequently, through the analysis of the eigenvalues, participation factors, and the calculation of oscillation frequencies under different controller parameters, the effect of controller parameters on the stability of MMC is studied and the following conclusions can be drawn. Besides, the time-domain simulation built in MATLAB/Simulink verifies the accuracy of the HSS-based MMC model and validity of stability analysis.
(1) When MMC operates with dc-voltage control mode, either too large outer loop proportional coefficient k p1 or inner loop proportional coefficient k p2 might cause the system unstable. (2) The too large proportional coefficient k p1 of outer dc voltage control loop will cause an oscillation mode with certain frequency, while the too large proportional coefficient k p2 of inner current control loop will cause another oscillation mode with another certain frequency.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
A 0 is determined by the main circuit of MMC and does not involve the closed-loop control part, which is: A 0O represents the relationship between s∆x 0 and ∆x 0 . According to (46) and (50), ∆U dc , s∆x v1 are related to ∆I c0 , and s∆x v2 is related to both ∆I c0 and ∆x v1 . Thus A 0O could be noted as: A 0N , A 0P represent the relationship between s∆x −1 and ∆x −1 , s∆x +1 and ∆x +1 respectively. According to (45), the first part of ∆m i+1 is related to∆I s+1 and the first part of ∆m i−1 is related to ∆I s−1 . Therefore, A 0N and A 0P are noted as: A +1 is determined by the main circuit of MMC and does not involve the closed-loop control part, which is: A +1N represents the relationship between s∆x 0 and ∆x −1 . According to (41), s∆x i1 and s∆x i2 are both related to ∆I s−1 . Thus A +1N could be noted as: A +1O represents the relationship between s∆x +1 and ∆x 0 . According to (45) and (51), the second part of ∆m i+1 is related to ∆x i1 and the third part of ∆m i+1 is related to ∆x i2 ; the second, third and fourth part of ∆m v+1 are related to ∆I c0 , ∆x v1 , ∆x v2 respectively. The specific expression of A +1O is presented as: Energies 2020, 13, 1056

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A −1 is determined by the main circuit of MMC and does not involve the closed-loop control part, which is: A −1P represents the relationship between s∆x 0 and ∆x +1 . According to (41), s∆x i1 and s∆x i2 are both related to ∆I s+1 . Thus A −1P could be noted as: A +2 is determined by the main circuit of MMC and does not involve the closed-loop control part, which is: Energies 2020, 13, 1056

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A −2 is determined by the main circuit of MMC and does not involve the closed-loop control part, which is:  B +1O represents the relationship between s∆x +1 and ∆U dcref . According to (51), the first part of ∆m v+1 is related to ∆U dcref . Thus B +1O could be noted as: B −1O represents the relationship between s∆x −1 and ∆U dcref . According to (51), the first part of ∆m v−1 is related to ∆U dcref . Thus B −1O could be noted as: