MMC-Based HVDC Transmission Systems Models in EMT-Type Program

The modular multilevel converter (MMC) topology utilizes the advantages of both the multilevel structure and pulse width modulation (PWM. Low switching losses and low harmonics (high-quality AC voltage) are the results of low effective switching frequency per device, which significantly minimizes the need for filtering [11].

A Sub-Module is the term for MMC's core element (SM). To acquire the necessary output voltage, the number of SM can be either raised or lowered. Half-bridge (HB) converters, full-bridge (FB) converters, a unidirectional sub-module [12], a clamp double (CD) converter sub-module, three level flying capacitors, three level neutral-point, and five level cross-connected SM [13] are all included in the topology of each switching sub-module.

The number and length of the communication links, the network delay, and the tolerance to faults are all significantly influenced by the control network topology. Ring, tree, and hybrid tree-ring topologies appear to have the most promise among the topologies that can be taken into account.

In order to produce a multilevel voltage waveform at the converter terminal, this converter depends on the cell capacitors. To construct a single valve for DC transmission requirements, hundreds of cells are typically needed. The quality of the AC voltage waveform improves and the harmonic content decreases as the level count rises.

The balancing capacity of this architecture is better than the one with NPC converters because it takes advantage of the redundant combination of module connections for each required AC level [14]. The probability of device and/or system failure is decreased since the MMC performs better than the NPC converter under unbalanced operation and symmetrical/asymmetrical AC faults [15]. The MMC is suited for applications subject to strict grid code requirements since it can bypass many forms of AC faults (wind energy). Since there is no common DC capacitor to transfer ripple between phases, if one phase of the AC system experiences a problem, the other two phases of the converter will continue to function normally, maybe at full per-phase power.

These key features, made of the MMC an attractive choice for VSC HVDC transmission system around the world [16], resulting in a rising interest to develop appropriate modeling technics to meet special purpose analysis (control, transient analysis…) and to reduce the simulation time.

When activated by relatively high-frequency signals, the relatively high number of non-linear semiconductors in MMCs poses some difficulties that necessitate a considerable computational power for re-triangularizing the admittance matrix of the electrical power subsystem [16, 17], making the simulation of the MMC occasionally inconvenient when using a traditional modeling method [18].

In order to adress the afore-mentioned difficulties, the research community propose enhanced reduced order models that increase computational efficiency and maintain the targeted accuracy. A recent tendency is to use average value models (AVMs) that are both simple and capable of providing enough accuracy in dynamic simulations. For MMCs, AVMs and other simplified modeling techniques have been provided by Teeuwsen et al. [7, 19-22].

This section tends to classify the modeling methods of the modular multilevel converter. Then, it makes a detailed descriptions and comparisons among the four models in terms of phenomena covering, accuracy, and application. Furthermore, it aims to help the reader to identify and choose the best-suited model for conducting this analysis.

2.1 Full physics-based model

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Figure 1. Physics based model for IGBT

Due of the high number of cells, modeling MMCs could become very difficult and simulation could take a long time. Depending on the study type and precision needed, several modeling levels can be attained. Models are introduced in this paper in ascending order of complexity. It is anticipated that model complexity reduction will improve computing performance.

Differential equations or an equivalent circuit serve as representations for each semiconductor device. Models of insulated gate bipolar transistors (IGBTs) can be categorized as either behavior models or physical models [23]. A behavior model can properly depict the properties of the device under specific circumstances despite not being based on physical principles. However, when the device parameters (base width, channel length, gate oxide thickness, etc.) are modified, analytical models [24-31] are able to forecast the IGBT behavior. These factors have a significant impact on how well the device works.

Since a MOS transistor and a BJT can be used to approximate an IGBT, an improved analytical model that uses a single-dimension device simulation model for the BJT and an analytical model for the MOS to develop an IGBT equivalent circuit model was proposed by Kao et al. [28]. This model is compared to earlier physics-based models presented by Baliga et al. [24, 25, 32] and creates the IGBT equivalent circuit model shown in Figure 1.

In the converter design phase, the comprehensive physics-based models are helpful for detailed analysis of the behavior of the semiconductor device and for defining the IGBT's size and doping profile. These models are not typically used for power system studies because they require a very small integration time step (in the order of nanoseconds) in the EMTP in order to accurately depict switching losses. This type won't be covered in the parts that follow because of this.

2.2 Detailed model ‎(nonlinear model) ‎

The ideal controllable switch, one series and one anti-parallel diode, and a snubber circuit [33] are used to depict the anti-parallel IGBT/diode pair in Figure 2. A non-linear resistance is used to depict the diode's non-ideal voltage-current characteristic. The nonlinear characteristic can be modified in accordance with the data provided by the manufacturer.

This model can take into account any MMC conduction method and is the most accurate model for EMT-type programs. The advantages include the ability to perform specific studies like blocked states (g1, g2 OFF), submodule details, the converter start-up sequence, and internal converter faults, as well as improved precision in the modeling of the IGBT/diode pair because the non-linear behavior throughout the switching is included. These models can also be used to assess, tune, and validate less complex MMC models like those mentioned in the following.

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Figure 2. Nonlinear IGBT/diode including V-I curve

2.3 Equivalent (or simplified) detailed model

2.3.1 IGBT/Diode representation

This concept is based on the notion that the IGBT device and its anti-parallel diode operate as a bidirectional switch, which is expressed by a two-state resistance, illustrated in Figure 3 by R₁ (small conductive value) and R₀ (big open-circuit value). This method [17] enables the generation of a Norton equivalent for each MMC arm as well as implementing an arm circuit reduction to remove internal electrical nodes. These two controllable resistances are utilized to swap out the two IGTB/diode combinations. Their values are influenced by capacitor voltage, current direction, and gating signals.

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Figure 3. a) IGBT/diode connected in anti-parallel. b) Simplified IGBT/diode model

The IGBT/diode combination may still be thought of as a single two-state resistance, as shown in Figure 4, even if either the diode or the IGBT may be conducting at any one time.

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Figure 4. Single two state resistance model

2.3.2 Submodule capacitor representation

By using the trapezoidal formula of integration and Dommel's expression, the capacitor C_cp can be expressed with its Thevenin or Norton equivalents [34]. For the capacitor, the generic Thevenin and Norton equivalents are determined, respectively.

$\begin{aligned} u_{c p}(t) & =\frac{1}{c_{c p}} \int_{t_1-\Delta t}^{t_1} i_{c p}\left(t_1\right) d t+u_{c p}(t-\Delta t) \\ & =R_{c p} . i_{c p}(t)+u_{H i s t}(t-\Delta t)\end{aligned}$           (1)

The voltage across and current through C_cp, respectively, are represented by u_cp(t) and i_cp(t), where R_{cp =} ∆t/2C_cp. The size of the simulation step is ∆t.

$\begin{aligned} i_{c p}(t) & =\frac{2 C_{c p}}{\Delta t}\left[u_{c p}(t)-u_{c p}(t-\Delta t)\right]-i_{c p}(t-\Delta t) =\frac{u_{c p}(t)}{R_{c p}}+i_{\text {Hist }}(t-\Delta t)\end{aligned}$             (2)

It is clear from (2) that the capacitor can be represented by an equivalent resistance R_cp connected in parallel with the following current source i_Hist(t-∆t):

$i_{H i s t}(t-\Delta t)=-\frac{u_{c p}(t-\Delta t)}{R_{c p}}-i_{c p}(t-\Delta t)$                 (3)

Figures 5 shows the duals of the Thevenin and Norton approximations, which are simply the actual circuit algebraic modifications.

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Figure 5. a) The capacitor and its b) Thévenin and c) Norton EMTP equivalents

The electrical representation of the nth SM half-bridge can be shown in Figure 6 [34] using the Thevenin equivalent in (1) for the SM capacitor.

The nth SM output voltage me be written as:

$\begin{aligned} u_{s m o, n} & =\frac{i_{\text {in }}(t)}{1 / R_{2_{-} n}(t)+1 /\left[R_{1_{-} n}(t)+R_{c p}\right]} +\frac{R_{2_{-} n}(t) \cdot u_{\text {Hist }}(t-\Delta t)}{1 / R_{2_{-} n}(t)+1 /\left[R_{1_{-} n}(t)+R_{c p}\right]} =R_{E_{-} n(t)} \cdot i_{i n}(t)+u_{E_{-} n}(t)\end{aligned}$      (4)

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Figure 6. EMTP Equivalent model of the N^th half bridge submodule

The capacitor current can be expressed as:

$\begin{aligned} u_{s m o, n} & =\frac{i_{i n}(t)}{1 / R_{2_{-} n}(t)+1 /\left[R_{1_{-} n}(t)+R_{c p}\right]} +\frac{u_{s m, n}(t-\Delta t)+R_c \cdot i_{\mathrm{cp}, n}(t-\Delta t)}{R_{2_{-} n}(t)+R_{1_{-} n}(t)+R_{c p}}\end{aligned}$                   (5)

Similarly, the capacitor voltage is calculated as:

$\begin{aligned} u_{s m, n}(t) & =R_{c p} \cdot\left[i_{c p, n}(t)+i_{c p, n}(t-\Delta t)\right] +u_{s m, n}(t-\Delta t)\end{aligned}$                (6)

Thevenin parameters (RE_,n and uE_,n) for the nth submodule are easily obtainable in (4).

2.3.3 Phase arm equivalent

The arm equivalent depicted in Figure 7 is obtained by series connecting the N SMs in the phase arm, where the Thevenin parameters are:

$u_{E_{-,} a r m}(t)=\sum_{n=1}^N u_{E_{-,} \mathrm{n}}(t)$               (7)

$R_{E_{-,} a r m}(t)=\sum_{n=1}^N R_{E_{-,} \mathrm{n}}(t)$              (8)

RE_,n and uE_,n are specified in (4).

Additionally, the model approach described here will need only minor adjustments from the perspective of implementation in order to emulate the full-bridge converter SM [17, 35].

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Figure 7. Thevenin equivalent of the MMC VSC-HVDC phase arm

The reduced detailed model is less computationally intensive than the complete detailed models, although being less accurate [36]. This is important for power system research. The main benefit of this kind is the main system of network equations' significant reduction in the number of electrical nodes. It continues to take into account each SM independently and keeps track of the various capacitor voltages and currents. It is applicable to any quantity of SMs within one arm.

2.4 Averaged models

2.4.1 Averaged model based on the switching function

In this model, the switching function Sn notion of a half-bridge converter is used to average each MMC arm. We make the following assumptions: that all MMC internal variables are perfectly under control, that all SM capacitor voltages are perfectly balanced, and that second harmonic circulating currents in each phase are suppressed. The veracity of this assumption increases as the number of SMs per arm and/or the fluctuation amplitudes of capacitor voltages increase. Each arm can be represented in this scenario by a single module, as seen in Figure 8. State-space modeling, with its intrinsic simplicity of manipulation and capacity for frequency domain analysis, is a favoured method for describing dynamic systems. This method can be used to derive an explicit steady-state expression for the circulating current [37-42].

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Figure 8. Averaged switching model of the MMC phase arm

If the switching frequency is high enough for the averaging method to be accurate and all of the module parameters and modulation commands are balanced at the module level, this model can capture the information of individual capacitors. However, linear conduction losses and circulating currents can be modeled. For control system methods based on internal MMC energy balance, it is also useful to take into consideration the energy transmitted from the ac and dc sides into each arm of the MMC [43, 44].

This method's clear drawback is that because each arm is reduced to an equivalent switching function model, power switches are no longer represented. This is because the information of the separate modules cannot be identified inside an arm because they are all presumed to be equal.

This modeling approach can be used to research harmonics because it uses longer time steps and substantially less computation time than detailed modeling due to the aforementioned simplifications.

2.4.2 Average model based on fundamental frequency

The switching functions are replaced by reference signals in this model, which duplicates the MMC converter's overall dynamics and suppresses the control blocks. The mathematical formulation is the same as in the previous model. Contrary to [33], the MMC can be depicted as a conventional VSC (2- and 3-level topologies) [45]. Therefore, vref are the voltage references produced by the inner controller [46] using a methodology similar to [47-50]. Figure 9(a) displays this model's ac-side.

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Figure 9. Averaged-value model of the MMC converter [48]

The power balancing approach was used to generate the dc-side model in Figure 9(b), which makes no assumptions about energy storage within the MMC converter. The MMC has an inductance in each arm, unlike the traditional VSC model, hence an equivalent inductance should also be supplied on the dc side.

Finally, throughout steady state and dynamic simulations, the two average models that were used in this paper respond satisfactorily. Although the switching function model requires more computation time for the same t, it is more accurate than the typical model based on fundamental frequency. For ac side faults, set point changes, and loss of generation, both the switching model and the AVM can be used to accurately simulate system dynamics.

AVMs can only function properly if the capacitors are big enough to keep the voltage between each MMC submodule roughly constant. The individual capacitor voltages are not estimated, but the large-scale dynamic behavior is adequately represented. An individual dc-side voltage is determined instead.