I. INTRODUCTION
An interfacing converter plays an important role in the integration of renewable energy into modern electric grids [1]–[3]. It functions as the bridge to connect the source, load, and grid. The modular multilevel converter (MMC) is the most suitable converter for HVDC transmission [4], [5], medium- or high-voltage motor drive applications [6], [7], and electric railway supplies because of its recognized features in
terms of modularity, reliability, and efficiency, [8]. A switching process of MMC can affect the response of AC/DC link under some disturbances, and a circulating current that flows within its three-phase units is generated. Given that the circulating current control increases the converter, power loss results in a complex and non-linear system phenomenon or even instability issues [9]–[11].
New modulation methods have been proposed to suppress the circulating current and to eliminate harmonics by avoiding its influence on the AC terminal. PWM methods, including carrier-phase-shifted PWM [12], submodule (SM) unified PWM [13], and selective harmonic elimination method [14], share some common features, such as high switching frequency, large switching losses, and complex implementation process. Unlike the aforementioned methods, the nearest level modulation (NLM) method [15] does not need a complex calculation of the trigger angles. Therefore, NLM is the simplest and the most practical modulation method for MMC.
The upper and lower arm voltage references of MMC consist of constant DC voltage and fundamental frequency component [16]. Thus, the restriction of arm voltage reference is determined by the value of the modulation index. At the same time, the circulating current control employs additional component into the arm voltage reference, which influences the characteristics of DC-side impedance and maximum modulation index of MMC. Hence, overmodulation can be avoided, and DC voltage utilization is reduced. Preceding research focused on circulating current suppression and on reducing DC voltage ripple distortion. However, the effects of the DC impedance model and its frequency characteristics are still not fully explained to the approaches of predicting any possible resonance problems, which are important in exploring the impedance of the converter on the DC side. The DC voltage ripple transfer is also a challenge from the fault converter to other converters.
This study aims to derive the maximum modulation index and DC-side equivalent impedance of MMC considered in the HVDC system. A detailed model of MMC is first presented, and an analysis of the modulation index to DC-side impedance frequency characteristic is provided. When the fault occurs in a terminal of the two-terminal MMC-HVDC system, the effect of this terminal on the other terminal is studied with DC-side impedance model. Thus, the DC voltage ripple propagation between the two terminals in HVDC system under unbalanced conditions is predicted. Finally, the simulation and experimental results verify the validity of the derived model and DC voltage ripple prediction results.
II. BASIC CONFIGURATION AND CONTROL METHOD FOR MMC
A. Basic Configuration
The currents, voltage, and insertion indices must be defined to perform the following calculation. Fig. 1 shows the configuration of MMC [17]. A three-phase MMC consists of three-phase legs with upper and lower arms. Each arm has a series-connected N SMs and inductor L0 and arm equivalent resistance R0. A single SM has two switching states (i.e., on and off) and one capacitor (i.e., Csub), and the corresponding output voltage of SM is vcp or vcn in the upper or lower arm. The subscript “p” denotes the upper arm, whereas “n” denotes the lower arm. The equivalent circuit of the single MMC under steady state is depicted in Fig. 1, where vj (j = a, b, c) is the converter output voltage of phase j. The reference for ij is the corresponding line current. Thus, the inserted voltage and capacitor currents are proportional to the insertion indices in each other arm. Similarly, the voltage vp (vn) can be obtained by multiplying the insertion index with the capacitor voltage, which is shown as
where μp and μn are the inserted SM numbers to the total SM numbers in each arm, and they can be expressed as in Equ. (2):
where Vdc is the DC link voltage, and N is the assumption total number of SMs.
Fig. 1.Circuit configuration of MMC.
According to the single-phase unit circuit in Fig. 1, the AC-side voltage can be described with the following based on Kirchhoff’s law:
B. Control Method
The active and reactive power are in control because the outer control loop provides the references for the inner current control loop [18]. The inner current control loop aims to regulate the d-q axis currents, and its output is the electromotive force vdq. The disturbance inputs can be cancelled with feedforward compensation usdq [19]. The modulation and input voltage balancing control is presented with Equ. (4), and the diagram of inverter control system is shown in Fig. 2. Finally, the main control parameters are selected as follows: kp = 8.05 × 10−4, ki = 0.115, ω = 60 Hz. The corresponding closed loop root locus is illustrated in Fig. 3. The proportional control parameter ki (kp) increases when kp (ki) is constant. The closed loop dominant poles P1 and P2 also move toward the imaginary axis. Hence, the stability of the control system can be guaranteed.
Fig. 2.Control diagram of MMC.
Fig. 3.Closed-loop root locus of the control diagram.
The circulating current flows among the three phases of MMC. Thus, it increases the RMS current, thereby increasing the power loss on the devices in SMs. According to [20], an extra compensating component (ncir) is added to the insertion indices np and nn. Hence, insertion indices can be expressed as in Equ. (5).
III. MAXIMUM MODULATION INDEX CALCULATION FOR MMC
The difference of charge and discharge of the phase unit SM capacitor of MMC leads to an imbalance in SM capacitor voltage, thereby reducing service life and causing circulating current power loss. These negative effects can be avoided by introducing a new insertion index μcir into the arm voltage reference to offset the capacitor voltage of the SM.
According to Equ. (3), the AC-side voltage can be expressed as
On the basis of Equ. (7), the DC-side voltage equivalent model can be written as
In MMC, the fundamental component and second-order frequency make obtaining the unbalanced voltage and the insertion indices difficult. Given that R0 is small, the following can be obtained:
SM capacitance C is generally large enough to render the deviation of each capacitor voltage ripple, which is very small. Meanwhile, the AC-side current, DC-side SM capacitor voltage, and circulating current can be controlled. Therefore, the SM capacitor voltage ripple can be neglected in a steady-state system with the following assumptions:
where 0.5(vcp + vcn) is the steady-state values for the average voltage of the upper and lower arms.
The insertion index ncir can be calculated as follows by substituting Equ. (10) into Equ. (9):
According to the equivalent circuit of MMC at the DC and AC sides in [16], the DC-side active power and AC-side active power can be expressed as in Equ. (12):
At the same time, Pdc = Pac can be obtained by neglecting the power loss of the converter with the power balance at the DC and AC terminals, which keep the balance of energy exchange between the phase leg and DC link. The average value of the circulating current idc represents the current that must flow through the phase leg to keep a certain power flow on the AC side [21]. idc can be obtained by solving Equ. (12).
where M is the modulation index, which is defined as 2vj/Vdc. φ represents the power angle between AC phase voltage and current.
The SM capacitor voltage of the upper and lower arm can be expressed as
The sum of the SM capacitor voltages and the differences of upper and lower arms are shown in Equs. (15) and (16) by combining Equs. (5) and (11)–(14).
ncir can be obtained as in Equ. (17) by inserting Equs. (15) and (16) into Equ. (11):
The insertion index for the upper and lower arms can be rewritten as Equ. (18) by combining Equs. (6) and (17), which is shown as
where n'p + n'n = 1 , and insertion index should satisfy
From these results, the maximum and minimum of the upper and lower arm insertion indices are obtained when cos(φ) equals its minimum value:
where ε is the maximum relative SM capacitor voltage ripple. The relationship is shown as follows by letting the capacitor have a relative voltage deviation (i.e., ε around its reference value vc,):
The capacitor energy per SM is presented with Equ. (23) [22]
where Ps is the AC-side output power of the converter, and φ is the power factor angle. According to [23], the maximum reactive power is generally defined as half of the maximum active power. Therefore, the power factor at the full apparent power is 0.866, and the maximum capacitor voltage ripple ε can be expressed as in Equ. (24):
The constraint of the modulation index can be obtained as follows by combining Equs. (19) to (21):
The preceding discussion indicates that the insertion index is related to the operating conditions M, Ij, φ, and Vdc and system parameters C and N. The maximum modulation index is smaller than 1. It is also related to SM capacitor voltage ripple.
IV. DC-SIDE IMPEDANCE FREQUENCY CHARACTERISTICS AND VOLTAGE RIPPLE TRANSFER PREDICTION ANALYSIS
A. Impedance Frequency Characteristics Analysis
According to Equs. (5) and (14), expressing the SM voltages as follows is reasonable:
The sum of the three-phase arm current is the direct current:
Then, the sum and the difference of the upper and lower arms of the SM capacitor voltage of the phase unit can be obtained
According to Equs. (9) and (26) to (29),
The balanced converter-side current is proved achievable in the system performance with both controllers in a two-terminal MMC-HVDC, whereas the second-order DC-side voltage ripple cannot be eliminated by the constant AC active power control scheme. To deal with this issue, the compensating component is inserted in this study instead of the traditional circulating current control and suppressing the second-order DC-side current under unbalanced conditions. The expression of DC-side equivalent impedance is related only to R0, L0, C, and ncir through the small-signal perturbation frequency domain analysis of Equ. (30) [24], [25]. The DC equivalent impedance model of MMC in the frequency domain can be determined as follows:
The DC-side impedance frequency domain model without compensating component is a series of R0, L0, and C. It can be defined as
The DC voltage ripple can be predictive by analyzing the DC-side impedance frequency characteristics and through system parameters R0, L0, C, and ncir. Thus, the following equations can be derived based on Equs. (31) and (32):
where fdc and fdc_cir represent the DC-side equivalent impedance frequency with and without a compensating component, respectively. MMC DC-side impedance is changed between capacitive and inductive following the different values of R0, L0, C, and ncir. The MMC performs as a capacitive impedance when f < fdc (fdc_cir). By contrast, the MMC performs as inductive impedance.
B. DC Voltage Ripple Transfer Prediction
According to the preceding derivation of DC-side equivalent impedance for MMC, the equivalent circuit of HVDC system based on MMC can be obtained. An inverter with distorted DC voltage is applied instead of back-to-back HVDC system to avoid the low-frequency harmonics induced by overmodulation during unbalanced system assuming that the DC voltage of the rectifier is constant. The equivalent impedance model of DC side is shown in Fig. 4, where Zs is the internal impedance of the source, Zline is the impedance of the power cable, which is represented by π model, and Zdc is the equivalent impedance of the DC side of the inverter.
Fig. 4.Equivalent inner impedance in the inverter side.
The induced ripple voltage is shown as vripple when faults occur in MMC-HVDC system. On the basis of the model of the equivalent impedance, the effect of the first side of the fault to the other side can be predicted. The effects of the length of the cable and its equivalent inductance and capacitance are discussed in the simulation. Fig. 5(a) illustrates that vmmc_inv/vripple is reduced with the increasing length of the cable. The effect of internal inductance, capacitance, and compensating component ncir on vmmc_inv/vripple is further analyzed. The capability of fault transfer is reduced by increasing the equivalent inductance and capacitance while keeping the distance the same. A linear relationship can be determined with increasing capacitance.
Fig. 5.Effect of system parameters on vmmc_inv/vripple.
V. SIMULATION RESULTS AND EXPERIMENTAL VERIFICATIONS
A. Simulation Results
In Section IV, the equivalent of DC impedance is derived. The parameters related to DC-side equivalent impedance are obtained. The simulation model of a three-phase MMC-HVDC is also established in MATLAB/Simulink. The main circuit parameters of the simulation system are provided in Table I.
TABLE IPARAMETERS OF MMC-HVDC
The steady-state values vc+, vc-, icirj, and ncirj are derived in Equs. (15), (16), (18), (20), and (21) to verify the calculated maximum modulation index, and the simulation results are shown in Fig. 6. The MMC must be operated in the condition with a power factor of , which is the worst case. For this case, the modulation signal starts to hit the limit of M = 0.92. Fig. 7 shows the upper (lower) arm current with different modulation indices. The upper (lower) current is in normal operation when M = 0.90, and the modulation signal is close to 1. However, this signal has some notches and also exceeds the rated current value when M = 0.95. Meanwhile, the modulation signal exceeds 1. This phenomenon is called overmodulation. The simulation results verify the theoretical analysis on the maximum modulation index limitation.
Fig. 6.Steady-state simulation results of MMC-HVDC.
Fig. 7.Upper (lower) arm current under different modulation indices of MMC.
As mentioned, ncir is related to the frequency characteristics in Equ. (31). The DC terminal output voltage and current can be monitored with frequency sweeping between 5 and 990 Hz, which can be adjusted according to the studied bandwidth. The DC impedance can also be derived by applying Fourier decomposition. The frequency domain analysis result of DC-side impedance is shown in Fig. 8. The equivalent circuit and simulation results without and with circulating current control closely match the theoretical analysis from Equs. (31) and (32), and ncir has a small effect on the DC-side impedance frequency characteristics.
Fig. 8.DC impedance frequency characteristic.
The simulation results in fault condition are shown in Fig. 9 when the oscillation in the DC voltage occurs in the rectifier side of the system and is transferred to the inverter side. Fig. 9(a) illustrates that when the fault voltage is 4000 or 2000 V and ncir = 0, transferring through the power cable with the same parameters and high oscillation in the inverter stage can be determined at the ending terminal with high oscillation in the starting terminal. Fig. 9(b) illustrates that a similar upper or lower arm current can be detected in the inverter side under the same case.
Fig. 9.Effect of system parameters for DC voltage ripple transfer and upper or lower arm current.
The DC voltage ripple with a minimal value is shown in Fig. 9(c) when the voltage ripple value is fixed and the compensating component of modulation index is added. However, the upper or lower arm current is reduced in Fig. 9(d). Hence, the effect of injecting compensated terms into the modulation index is higher on the bridge current than on voltage fluctuation.
B. Experiment Verifications
To verify the preceding analysis, a 1 kW downscaled three-phase MMC prototype was built. The prototype consists of 2 SMs per arm, and the rated DC voltage is 100 V. The SM capacitance is 2700 μF, and the arm inductance is 0.5 mH. The carrier frequency is 1 kHz. The load resistance is 5 Ω. The DC-side inductance is 0.5 mH, and AC-side inductance is 1.5 mH. A single full-bridge converter is used to generate voltage ripple into the DC side of the rectifier in the two-terminal MMC-HVDC system.
Fig. 10 illustrates the experimental results for the worst case when the power factor is and M is 0.92, which is the critical value. The circulating current has some small notches when the modulation index reached the maximum value. This phenomenon occurs because of overmodulation, and a theoretical analysis is discussed in Section III.
Fig. 10.Experimental results with cosφ = and M = 0.92.
Fig. 11 shows the experimental results of the upper and lower arm currents and the inverter side voltage ripple for the MMC-HVDC system when the rectifier-side voltage ripple vripple is 5 V and the circulating current compensating component ncir is 0. Given that the voltage ripple is transferred from the DC side of the rectifier, the second-order harmonics appears at the DC side of the inverter for MMC-HVDC.
Fig. 11.DC voltage ripple transfer component and arm current with ncir = 0 and vripple = 5 V.
Results of the voltage ripple in the DC side of the inverter and upper and lower arm currents when vripple is increased to 10 V are shown in Fig. 12. Unlike Fig. 11, Fig. 12 illustrates that the arm current and the inverter-side DC voltage ripple increase obviously with a large voltage ripple and without a compensating component (ncir = 0).
Fig. 12.DC voltage ripple transfer component and arm current with ncir = 0, vripple = 10 V.
Fig. 13 shows that the voltage ripple is the same as that in Fig. 12, which is kept as 10 V. The compensating component ncir = 0.04 is introduced in the modulation index. Unlike that in Fig. 12, the voltage ripple in Fig. 13 does not change significantly. However, the circulating current in Fig. 13 becomes considerably smaller than that in Fig. 12. Therefore, a high-voltage ripple leads to an increased voltage ripple that transfers from one side to the other, and the compensating component in the modulation index leads to decreased arm currents.
Fig. 13.DC voltage ripple transfer component and arm current with ncir = 0.04, vripple = 10 V.
VI. CONCLUSION
A DC-side equivalent impedance model in frequency domain that describes the analytical expression in MMC-HVDC systems simply and accurately is proposed. On the basis of this model, the influences of DC cable and inverter parameters on DC voltage ripple distribution are analyzed in detail under unbalanced conditions according to an analysis of simulation results and experimental verifications. The following can be concluded:
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