A Novel Modulation Function-Based Control of Modular Multilevel Converters for High Voltage Direct Current Transmission Systems

Abstract

In this paper, a novel modulation function-based method including analyses of the modulation index and phase is proposed for operation of modular multilevel converters (MMCs) in high voltage direct current (HVDC) transmission systems. The proposed modulation function-based control technique is developed based on thorough and precise analyses of all MMC voltages and currents in the a-b-c reference frame in which the alternating current (AC)-side voltage is the first target to be obtained. Using the AC-side voltage, the combination of the MMC upper and lower arm voltages is achieved as the main structure of the proposed modulation function. The main contribution of this paper is to obtain two very simple new modulation functions to control MMC performance in different operating conditions. The features of the modulation function-based control technique are as follows: (1) this control technique is very simple and can be easily achieved in a-b-c reference frame without the need of using Park transformation; and (2) in addition, the inherent properties of the MMC model are considered in the proposed control technique. Considering these properties leads to constructing a control technique that is robust against MMC parameters changes and also is a very good tracking method for the components of MMC input currents. These features lead to improving the operation of MMC significantly, which can act as a rectifier in the HVDC structure. The simulation studies are conducted through MATLAB/SIMULINK software, and the results obtained verify the effectiveness of the proposed modulation function-based control technique.

1. Introduction

Owing to remarkable advantages of multilevel modular converters (MMC) in high-voltage and high-power applications including modular structure, dynamic increment of sub-module (SM) numbers, common direct current (DC)-bus and distributed DC capacitors [1,2,3,4], many pulse width modulation (PWM) techniques have been recently proposed to improve control features of these converters [5,6]. However, although there are many existing control methods, designing new control techniques with more simplicity, more efficiency, faster steady state operation and better transient response with respect to the type of application are always required. In [7], an improved PWM method for half-bridge based MMCs is proposed that is able to generate an output voltage with maximally 2N + 1 levels. The method discussed in this article is as great as a carrier-phase-shifted PWM (CPSPWM) method.

A popular PWM technique, which is the most commonly used method in the Cascaded H-bridge converters (CHB) [8,9], is the phase-shifted carrier (PSC). Because of the following features [10], the PSC modulation is also attractive to MMCs [11,12,13,14]: (1) high modularity and scalability of MMC; (2) easily reaching capacitor voltage balancing control; (3) MMC is able to generate an output voltage with a high switching frequency and a low total harmonic distortion (THD); and (4) MMC structure is able to distribute the semiconductor stress and the power of SMs. For instance, in [10], a mathematical analysis of PSC modulation is presented in order to identify the PWM harmonic characteristics of the MMC output voltage and the circulating current. In addition, the influence of carrier displacement angle between the upper and lower arms on those harmonics is evaluated in this paper. The nearest level modulation (NLM) method, which is also known as the round method, is exhaustively discussed in [15,16,17]. This method is suitable for MMCs particularly with a large number of SMs. In comparison with the conventional NLM, a modified NLM method in which the number of output alternating current (AC) voltage levels is as great as the CPSPWM and the improved SM unified PWM (SUPWM) is proposed for MMCs in [18]. Through this method, the number of AC voltage levels increased to 2N + 1, which is almost double; and the height of the step in the step wave is halved, leading to a better quality for the MMC AC output voltage waveform. By adding a zero-sequence to the original modulation signals, a new discontinuous modulation technique is achieved in [19] along with a circulating current control technique the MMC arms are clamped to the upper or lower terminals of the DC-link bus. In order to minimize the switching losses of the MMC, the clamping intervals can be regulated by the use of the output current absolute value. In [19], a significant reduction in the capacitor voltage ripples with low modulation indices is also obtained. A multilevel selective harmonic elimination pulse-width modulation (MSHE-PWM) technique is schemed in [20] to perform a tight control of the low-order harmonics and the lowest switching frequency for the MMC [21]. Moreover, two different modulation patterns for MSHE-PWM as well as a method for selecting the number of SMs in the phase-legs of the converter are proposed in [21]. According to [22,23], the amplitude modulation is widely employed to control MMC-based high voltage direct current (HVDC) transmission systems. The main idea of the method is to first calculate how many SMs should be put into action, and then the capacitors sorting voltage and the final working sequence should be determined by the direction of the arm current. However, in the case of large numbers of SMs, problems related to frequent sorting of capacitor voltage are issued [24].

A novel modulation function accompanied by its index is figured out in this paper by accurately analyzing all MMC voltages and currents in the a-b-c reference frame in order to improve MMC performance in HVDC applications. Working the AC-side voltage out, a combination of the MMC upper and lower’s arm voltages is achieved. The proposed modulation function that completely depends on MMC parameters and also the specifications of MMC input voltages and currents can be derived by using the AC-side voltage. In order to reach an accurate evaluation of MMC operation under different operational conditions, the impact of parameters and input current variations on the proposed modulation function and its index is investigated that ends up improving MMC control. MATLAB/SIMULINK based simulation results show the effectiveness of the proposed modulation function-based control.

2. Modular Multilevel Converter’s Alternating Current-Side Voltages

The structure of a three-phase MMC is illustrated in Figure 1a. The “2N” numbers of SMs are utilized in each arm, whereas their detailed configuration is depicted in Figure 1b. According to the figure, two complementary insulated-gate bipolar transistor (IGBT)-Diode switches are controlled so that each SM may be placed at either connected or bypassed states based on an appropriate switching method and also required controlling aims. In order to suppress the circulating current and also to restrict the fault current during a DC side fault, an inductor is used in either sides of each arm. The series resistor of each arm represents the combination of the arm losses and the inner inductor resistance.

Detailed Calculation of the Alternating Current-Side Voltage

The operation of the used SMs in MMC is highly dependent on the AC-side voltages. As can be inferred from Figure 1a, the AC-side voltages are directly related to the input variables and parameters. Suppose that MMC input voltage and current of phase “a” are:

$$v_{as}=v_m\cos \left(\mathsf{\omega }t\right),i_a=I_{ma}\cos \left(\mathsf{\omega }t+{\mathsf{\alpha }}_a\right)$$

(1)

Based on the proposed MMC structure shown in Figure 1a, the relationship between the input and the AC-side voltages of phase “a” can be written as:

$$v_{as}-v_{at}=L_a\frac{\mathrm{d}{i}_a}{\mathrm{d}t}+R_a{i}_a$$

(2)

By substituting Equation (1) into Equation (2), the AC-side voltage of phase “a” can be achieved as Equation (3):

$$\begin{array}{l}{v}_{at}=\sqrt{\begin{array}{l}{L}_a^2{I}_{ma}^2{\mathsf{\omega }}^2+R_a^2{I}_{ma}^2+v_m^2+\\ 2v_m{L}_a{I}_{ma}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_a\right)-2v_m{R}_a{I}_{ma}\cos \left({\mathsf{\alpha }}_a\right)\end{array}}\times \\ \cos \left(\mathsf{\omega }t+ta{g}^{-1}\left(\frac{\left[v_m+L_a{I}_{ma}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_a\right)-R_a{I}_{ma}\cos \left({\mathsf{\alpha }}_a\right)\right]}{\left[L_a{I}_{ma}\mathsf{\omega }\cos \left({\mathsf{\alpha }}_a\right)+R_a{I}_{ma}\sin \left({\mathsf{\alpha }}_a\right)\right]}\right)-\mathsf{\pi }/2\right)\end{array}$$

(3)

To present more explanations, Equation (3) can be rewritten as Equation (4):

$$\begin{array}{l}{v}_{kt}=\sqrt{\begin{array}{l}{L}_k^2{I}_{mk}^2{\mathsf{\omega }}^2+R_k^2{I}_{mk}^2+v_m^2+\\ 2v_m{L}_k{I}_{mk}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_k\right)-2v_m{R}_k{I}_{mk}\cos \left({\mathsf{\alpha }}_k\right)\end{array}}\times \\ \cos (\mathsf{\omega }t+ta{g}^{-1}\left(\frac{\left[v_m+L_k{I}_{mk}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_k\right)-R_k{I}_{mk}\cos \left({\mathsf{\alpha }}_k\right)\right]}{\left[L_k{I}_{mk}\mathsf{\omega }\cos \left({\mathsf{\alpha }}_k\right)+R_k{I}_{mk}\sin \left({\mathsf{\alpha }}_k\right)\right]}\right)+\frac{2\mathsf{\pi }j}{3}-\frac{\mathsf{\pi }}{2})\end{array}$$

(4)

where $j$ is equal to 0, −1 and 1 for the phases of “a”, “b” and “c”, respectively. The indices of k are the phases sign of “a”, “b” and “c”. Equation (4) shows the general three phase AC-side voltages. As can be realized from Equation (4), the AC-side voltages of MMC can be completely affected by input parameters and variables.

3. Analysis of Proposed Modulation Function

The proposed modulation function is obtained in this section involving AC-side voltages. The voltages placed in entire upper and lower SMs can be aimed to generate signals required for SM switches. Thus, applying Kirchhoff’s voltage law’s (KVL’s) on phase “a” arms of the MMC, the following equations are derived as:

$$v_{at}-v_{dc}/2=L_{au}\frac{\mathrm{d}{i}_{au}}{\mathrm{d}t}+R_{au}{i}_{au}-v_{au}$$

(5)

$$v_{at}+v_{dc}/2=L_{al}\frac{\mathrm{d}{i}_{al}}{\mathrm{d}t}+R_{al}{i}_{al}+v_{al}$$

(6)

By summing up two sides of the equations in Equations (5) and (6) and also assuming $L_{au}=L_{al}=L_{at}$, the following equation is attained:

$$v_{al}-v_{au}=2v_{at}-L_{at}\frac{\mathrm{d}{i}_a}{\mathrm{d}t}-R_{at}{i}_a$$

(7)

By substitution of Equations (1) and (3) into Equation (7), Equation (8) can be achieved as:

$$\begin{array}{l}{v}_{al}-v_{au}=\sqrt{\begin{array}{l}{\left(2L_a+L_{at}\right)}^2{I}_{ma}^2{\mathsf{\omega }}^2+{\left(2R_a+R_{at}\right)}^2{I}_{ma}^2+4v_m^2+\\ 4v_m\left(2L_a+L_{at}\right)I_{ma}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_a\right)-4v_m\left(2R_a+R_{at}\right)I_{ma}\cos \left({\mathsf{\alpha }}_a\right)\end{array}}\\ \cos \left(\mathsf{\omega }t+ta{g}^{-1}\left(\frac{\left[2v_m+\left(2L_a+L_{at}\right)I_{ma}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_a\right)-\left(2R_a+R_{at}\right)I_{ma}\cos \left({\mathsf{\alpha }}_a\right)\right]}{\left[\left(2L_a+L_{at}\right)I_{ma}\mathsf{\omega }\cos \left({\mathsf{\alpha }}_a\right)+\left(2R_a+R_{at}\right)I_{ma}\sin \left({\mathsf{\alpha }}_a\right)\right]}\right)-\frac{\mathsf{\pi }}{2}\right)\end{array}$$

(8)

As been discussed in former section, Equation (8) can be rewritten in a general form as Equation (9):

$$\begin{array}{l}{v}_{kl}-v_{ku}=\sqrt{\begin{array}{l}{\left(2L_k+L_{kt}\right)}^2{I}_{mk}^2{\mathsf{\omega }}^2+{\left(2R_k+R_{kt}\right)}^2{I}_{mk}^2+4v_m^2+\\ 4v_m\left(2L_k+L_{kt}\right)I_{mk}\mathsf{\omega }\sin \left({\alpha }_k\right)-4v_m\left(2R_k+R_{kt}\right)I_{mk}\cos \left({\mathsf{\alpha }}_k\right)\end{array}}\times \\ \cos (\mathsf{\omega }t+\frac{2\mathsf{\pi }j}{3}-\frac{\mathsf{\pi }}{2}+\\ ta{g}^{-1}\left(\frac{\left[2v_m+\left(2L_k+L_{kt}\right)I_{mk}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_k\right)-\left(2R_k+R_{kt}\right)I_{mk}\cos \left({\mathsf{\alpha }}_k\right)\right]}{\left[\left(2L_k+L_{kt}\right)I_{mk}\mathsf{\omega }\cos \left({\mathsf{\alpha }}_k\right)+\left(2R_k+R_{kt}\right)I_{mk}\sin \left({\mathsf{\alpha }}_k\right)\right]}\right))\end{array}$$

(9)

The term “$v_{kl}-v_{ku}$” is used to acquire reference waveforms for shift level pulse width modulation (SLPWM). As evident in Equation (10), the reference signals of the proposed PWM can be changed by input and arm parameters of MMC as well as input voltages and currents characteristics. Considering the reference values of $I_{mk}^{*}$, $v_m^{*}$ and ${\mathsf{\alpha }}_k^{*}$ as input currents and voltages, the proposed modulation index can be written as:

$$\begin{array}{l}{m}_k=\frac{V_{kt}(L_k,R_k,L_{kt},R_{kt},I_{mk}^{*},v_m^{*},{\mathsf{\alpha }}_k^{*})}{v_{dc}}\\ =\frac{1}{v_{dc}}\sqrt{\begin{array}{l}{\left(2L_k+L_{kt}\right)}^2{I}_{mk}^{*2}{\mathsf{\omega }}^2+{\left(2R_k+R_{kt}\right)}^2{I}_{mk}^{*2}+4v_m^{*2}+\\ 4v_m^{*}\left(2L_k+L_{kt}\right)I_{mk}^{*}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_k^{*}\right)-4v_m^{*}\left(2R_k+R_{kt}\right)I_{mk}^{*}\cos \left({\mathsf{\alpha }}_k^{*}\right)\end{array}}\end{array}$$

(10)

Based on Equations (9) and (10), and also assuming the reference values of input variables, the proposed modulation functions can be achieved as:

$$u_{ku}=m_k\left(\begin{array}{l}1-\cos (\mathsf{\omega }t+\frac{2\mathsf{\pi }j}{3}-\frac{\mathsf{\pi }}{2}+\\ ta{g}^{-1}\left(\frac{\left[2v_m+\left(2L_k+L_{kt}\right)I_{mk}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_k\right)-\left(2R_k+R_{kt}\right)I_{mk}\cos \left({\mathsf{\alpha }}_k\right)\right]}{\left[\left(2L_k+L_{kt}\right)I_{mk}\mathsf{\omega }\cos \left({\mathsf{\alpha }}_k\right)+\left(2R_k+R_{kt}\right)I_{mk}\sin \left({\mathsf{\alpha }}_k\right)\right]}\right))\end{array}\right)$$

(11)

$$u_{kl}=m_k\left(\begin{array}{l}1+\cos (\mathsf{\omega }t+\frac{2\mathsf{\pi }j}{3}-\frac{\mathsf{\pi }}{2}+\\ ta{g}^{-1}\left(\frac{\left[2v_m+\left(2L_k+L_{kt}\right)I_{mk}\mathsf{\omega }\sin \left({\mathsf{\alpha }}_k\right)-\left(2R_k+R_{kt}\right)I_{mk}\cos \left({\mathsf{\alpha }}_k\right)\right]}{\left[\left(2L_k+L_{kt}\right)I_{mk}\mathsf{\omega }\cos \left({\mathsf{\alpha }}_k\right)+\left(2R_k+R_{kt}\right)I_{mk}\sin \left({\mathsf{\alpha }}_k\right)\right]}\right))\end{array}\right)$$

(12)

The proposed modulation functions configurations for phase “a” are drawn in Figure 2. With respect to Equations (11) and (12), the proposed index and function are plotted in Figure 3 for $I_{mk}^{*}=50\mathrm{A}$ and ${\mathsf{\alpha }}_k^{*}=0$. As evident in Figure 3, the index modulation is quite close to unity. The effects of MMC parameters and input currents on the proposed modulation functions are comprehensively investigated in the next section. The parameters of $V_{at}$ and ${\mathsf{\theta }}_{at}$ are given in Appendix.

3.1. Parameters Variation Effects on the Proposed Modulation Function

In this section, system parameters are changed to the ones given in

Table 2. Changes in MMC parameters in Condition 2.

Parameter	Value	Unit
Input resistance	1.2	Ohm
Input inductance	25	mH
Arm resistance	1.5	Ohm
Arm inductance	10	mH
AC voltage	6	kV
$I_{mk}$	50	A
${\mathsf{\alpha }}_k$	0	-

in order to evaluate the effects of the parameters variations on the proposed modulation function. The base parameters are as given in

Table 1. Simulated system parameters. AC: alternating current; DC: direct current.

Parameter	Value	Unit
Input resistance	0.6	Ohm
Input inductance	15	mH
Arm resistance	0.5	Ohm
Arm inductance	5	mH
AC voltage	6	kV
DC voltage	12	kV
N	4	-
Input frequency	50	Hz
Carrier frequency	10	kHz
SM capacitance	5	mF
SM voltage	3	kV

. By increasing the system parameters, the proposed modulation indexes are decreased as depicted in Figure 4. The variation trend of the proposed modulation function is also illustrated in Figure 4. According to this figure, in addition to the index changes, the phase angles of both upper and lower modulation functions in three conditions slightly tend to be shifted. For the two obtained modulation functions, a typical shifted-level PWM in intervals of $5\mathrm{ms}\le t\le 10\mathrm{ms}$ is shown in Figure 5. Figure 5 demonstrates that the switching numbers of the second and third levels are decreased, and, instead, the numbers of the lowest level switching is increased. The scenario is inversed for the proposed lower modulation function but not with a similar change in the numbers. As depicted in Figure 6, the presented SLPWM results in a raise in the switching numbers of the second and the third levels and a drop in the switching numbers of the first levels. The sum of the switching generations in both proposed upper and lower modulation functions should lead to a constant value in each level.

3.2. Input Current Variation Effects on the Proposed Modulation Function

The magnitude and phase angle of the input currents impact on the proposed modulation function that is reviewed in this section. The specifications of the input current are changed to I_mk = 100 A and ${\mathsf{\alpha }}_k=-\mathsf{\pi }/6$ at t = 0.2 s. In comparison with parameter variations, the MMC input current variations can make more reduction in the modulation index and phase angle of the proposed modulation functions as illustrated in Figure 7. The effects of the input current changes on the applied SLPWM are shown in Figure 8 and Figure 9. The proposed upper modulation function with its shifted-level triangle waveforms as well as the respective generated signals for two different input currents are drawn in Figure 8. It can be seen that the number of switching signals (SS) in the second level is significantly increased for the MMC operating in the second condition compared with the first one. On the other hand, the first and the second levels of SS are slightly increased for SLPWM applied to the proposed lower modulation function as shown in Figure 9. Considering the interval of $5\mathrm{ms}\le t\le 10\mathrm{ms}$ as a sampling period, the input current changes impact more on the operation of the proposed upper modulation function.

4. Simulation Results

In this section, the control of MMC is executed by the use of proposed modulation function as given in Figure 10. MATLAB/SIMULINK environment in discrete mode is used to perform the overall control structure modelling based on the information given in Table 1 and Table 2. Throughout the evaluation process of MMC operation as a rectifier in HVDC application, the simulation sampling time is selected at the value of one micro second. In addition, initial value of 3 kV is considered for all SM capacitors.

4.1. Parameter Variation Evaluation

The obtained functions in Equations (11) and (12) are considered as carrier waveforms in SLPWM in these simulations. As can be observed, both amplitude and phase angle of the proposed modulation functions can be controlled by varying MMC arm and input parameter changes. In the first section of simulation that is (0, 0.2) seconds, MMC operates in a steady state with parameters given in Table 1. Then, at t = 0.2 s, the MMC parameters are changed to the values given in Table 2. As can be seen in Figure 11, voltages of SMs in phase “a” are kept at their desired values of 3 kV with initial parameters. After parameter variations, the proposed modulation function-based controller is able to acceptably regulate SM voltages, except for a slight deviation from the desired value at t = 0.2 s. Figure 12 shows the DC-link voltage of the MMC. Initially, MMC can reach targeted DC-link voltage after a short transient response. With a very small undershoot, the modulation algorithm continues to attain MMC’s desired DC-link voltage after parameter alterations. Phase “a” current of MMC is illustrated in Figure 13. According to this figure, MMC can generate the assumed current with the amplitude of 50 for both sets of parameters; however, there are negligible transient responses. The active and reactive power sharing of MMC with parameter changes are illustrated in Figure 14. As it can be seen in this figure, the MMC active and reactive powers follow the desired values, even after MMC parameter changes, along with their proportional alterations. The appropriate operation of a designed controller for MMC must lead to minimization of circulating currents. The proposed controller is capable of achieving minimized circulating currents of MMC as depicted in Figure 15. As shown in this figure, the circulating current of phase “a” remains at an acceptable level in both operation states.

4.2. Evaluation of Modular Multilevel Converter Input Current Variation

Changing the input current components of ${\mathsf{\alpha }}_k$ and $I_{mk}$ creates different modulation functions for the proposed modulation function-based controller. Thus, the changes caused by input MMC currents should lead to properly commanding the proposed controller to keep MMC in stable operation. In the primary interval, MMC operates with ${\mathsf{\alpha }}_k=0$, I_mk = 50 A and the parameters given in Table 1. Then, the input MMC currents reach a magnitude of I_mk = 100 A with the phase angle of ${\mathsf{\alpha }}_k=-\mathsf{\pi }/6$ at t = 0.2 s, though keeping the same parameters. The MMC SM voltages of both operation states are demonstrated in Figure 16. As can be understood from Figure 16, the voltages follow the reference value with a slight transient response and also acceptable steady-state error. Moreover, the DC-link voltage of MMC experiences an undershoot after the current variation at t = 0.2 s as depicted in Figure 17. After the transition, the proposed controller shows its dynamic capability in keeping the MMC DC-link voltage with an acceptable deviation from the desired value. Figure 18 contains the MMC input current of phase “a”. Based on this figure, the MMC input current is changed matching the current magnitude to the command, even though with a short period of transient response. Figure 19 shows the active and reactive power of MMC with MMC input current changes. According to this figure, both active and reactive powers of MMC are accurately changed based on the governed MMC input current. The circulating current of MMC is also shown in Figure 20. The curve in this figure implies that minimizing circulating current can be effectively accomplished after variation of the input current.

5. Conclusions

In order to effectively control the operation of MMC in HVDC transmission systems, a novel modulation function with a specified index was proposed in this paper. For this purpose, analysing all MMC voltages and currents in a-b-c reference frames was performed to primarily obtain the AC-side voltage. Then, the combination of the MMC upper and lower arm voltages was achieved by the use of already obtained AC-side voltage. Using this combination led to deriving the proposed modulation function and its modulation index, both depending on MMC parameters, and also the specifications of MMC input voltages and currents. In order to improve the performance of the proposed controller, the impacts of parameters and input current variations on the proposed modulation function and its index were thoroughly investigated in a range of operating points. The main feature of the proposed control technique is its very simple design in a-b-c reference frame, being additionally able to provide a robust performance against MMC parameter changes. MATLAB/SIMULINK allowed verification of the effectiveness of the proposed modulation function-based control technique.