Modeling of MMC for Fast and Accurate Simulation of Electromagnetic Transients : A Review

Abstract: Over the past decade, modular multilevel converters have evolved as the preferred choice for high voltage conversion systems. The design of the converter poses a computational challenge to the classical electromagnetic transient simulation techniques, for which the existing literature proposes numerous simplified and computationally-efficient equivalent representations. Despite the extensive collection of proposed models, the literature lacks a systematic comparison of these representations, which could enable their appropriate selection based on the specific simulation objectives. In light of these requirements, this article presents a comprehensive review of MMC models and discusses their applicability for parallel and real-time simulation. Using PSCAD/EMTDC simulations, the models are objectively compared with regards to computational efficiency and accuracy for various transient conditions. In addition, the paper also presents a brief account of MMC operation and the associated control and modulation system.

Each phase of an MMC-HVDC terminal can consist of several hundreds of identical submodules that switch a module capacitor in and out of the circuit to synthesize stepped AC or constant DC terminal voltages.This operation requires sophisticated control and modulation techniques.Furthermore, accurate modeling and computer simulation of the MMC converter together with the associated power system are essential for its efficient design, validation of the control system and operational planning.Electromagnetic transient solvers are generally utilized for the simulation of such power electronic and electric power integrative systems.However, the design of MMC poses a computational challenge to the classical electromagnetic transient simulations techniques.Independent operation of submodules necessitates their explicit modeling, which leads to hundreds of nodes and semiconductor devices in the equivalent models.This time-varying topology with hundreds of nodes leads to the excessive computational load on solvers based on the classical nodal admittance method.Existing literature presents a number of efficient equivalent models to address the high computational load of MMC simulations [16][17][18][19][20][21][22][23].Using the most recent literature, this paper aims to provide a review of the models that are most suitable for EMT studies.This further enables their hierarchization for different levels of power converter and system studies.
The paper starts with a brief description of topology, operating principles, control and modulation methods for MMC in Section 2. Section 3 presents a literature review of equivalent models, suitable for EMT studies.Simulation results in Section 4 examine the efficacy of the models under transient conditions.Finally, conclusions are drawn in Section 5.

Converter System
The double-star-configured MMC [24][25][26], referred to as the MMC here, is illustrated in Figure 1.Each phase leg comprises two arms that connect DC and AC terminals through N cascaded/series submodules (SMs) and an arm reactor.Submodules are two terminal devices, composed of an IGBT-based converter and a capacitor.These modules can be half-bridge, full-bridge or other hybrid configuration cells [27].Full-bridge submodule MMCs, with their DC fault handling capacity, find specialized application in overhead HVDC transmission systems [28]; while half-bridge submodules (HBSM), with fewer semiconductor devices and subsequently lower losses [29], are the dominant type of submodule for the more widely-used underground and submarine HVDC transmission systems and, therefore, the prime focus of this study.
HBSMs are composed of two IGBTs with freewheeling diodes and a capacitor as illustrated in Figure 2a.With the complementary operation of IGBTs, an HBSM can insert positive (inserted state) or zero (bypassed state) voltage based on the gate signals (T1, T2), irrespective of the direction of the current.Therefore, with this topology, each arm in MMC behaves as a controllable, unipolar voltage source [30].In addition, turning off of both switches results in the blocked state where only freewheeling diodes allow the flow of current, and their conduction is dependent on their voltage and currents.With half-bridge SMs, the converter cannot suppress DC side faults.The blocked state restricts current flow only in one of the two potential directions; therefore, DC breakers or breakers on the AC feed are necessary for such a converter system [30].Table 1 summarizes the operating modes of the half-bridge submodule.

Dynamics
Terminal and internal dynamics of an individual submodule are dictated by the instantaneous and historic values of the arm current/voltage, capacitor voltage and the state of the IGBTs.Nevertheless, for simplicity, the dynamics of all submodules in an arm can be taken as equivalent.As a result, the average dynamics of an arm can be modeled by Equations ( 1)-(3) (refer to the Nomenclature for symbols' interpretation).
where the insertion index, (n (φ±) ), gives the average number of submodules inserted in an arm and v Σ c φ± , i Σ c φ± represent the average unified internal voltage and current dynamics of all submodules.
From Kirchhoff's voltage law, the AC side operation of MMC is defined by [31]: while inserted leg voltage, v leg φ , and circulating current i circ φ , dictate the DC side operation of a leg, i.e., This reveals that with independent control of submodules, each arm of an MMC behaves as a variable voltage source, which allows for independent control on all AC voltages and the DC voltage.The AC side operation of MMC as seen from Equations ( 4)-( 5) is defined by internal electromotive force (emf), e φ (t), and internal impedance, Z eq .Through appropriate selection of insertion indices and the resulting difference of arm voltages, MMC can attain the desired voltage on the AC side together with the desired real and reactive power exchange with the grid, if any.Similarly, the DC voltage as revealed by Equations ( 6)-( 7) is controllable by insertion indices and the resulting sum of arm voltages.The presence of circulating current in phase legs is also revealed by Equations ( 6)- (7).Circulating current in a phase leg is composed of a DC component, which is essential for the transmission of real power.Moreover, voltage imbalance in phase legs results in additional harmonic components in the phase leg.Further analysis of the converter internal dynamics as presented in [32][33][34] reveals the presence of even order harmonics in circulating current, of which the second order is dominant.

Control System
The control system for an MMC is hierarchized as:

Voltage balancing control
The objective of current controllers is to determine the reference voltage waveforms to attain desired voltage and power flow characteristics.This control can be realized by direct or vector methods [35,36] for the grid-connected converter or through droop control [37] in islanded or weak-grid systems.
The circulating current harmonics tend to increase the losses in the system and increase the rating requirements for system components.Thus, mitigation of these harmonics is needed.These harmonics can be regulated by active compensation of voltage imbalance in a phase leg through control methods [38][39][40][41][42][43][44] or can be attenuated by tuned arm filters [45,46].
Voltage balancing control ensures that capacitor voltage is more or less constant across all submodules.This requires instantaneous control to ensure charging of depleting capacitors and discharging of charged capacitors.Superposition of DC and fundamental current in an arm implies that the current through each arm changes its polarity.Therefore, within a one-time cycle, there are instants of charging and discharging currents.Such voltage balancing techniques include circular transposition of carrier waves [47,48] or calculated approaches such as the capacitor voltage rank-based method [3,49] and a combination of averaging and balancing controls [24,[50][51][52][53].

Modulation Strategy
The modulation system for MMCs enables energy balance in the arms of the converter and voltage balancing across individual submodule capacitors and can be classified as follows.

Arm Level Modulation
The objective of this modulation is to generate insertion indices for all arms of MMC, to achieve desired reference values of AC/DC power and or voltage as dictated by the control system.These insertion indices, in turn, determine the number of submodules inserted in an arm at a given instant.In addition, this modulation ensures that the inserted voltage in a phase leg (v leg φ ) and the arm cumulative capacitor voltage (v Σ c φ± ) correspond to the DC link voltage.The modulation can be implemented through the direct calculation of insertion indices or through feedback control as summarized in Table 2.
Ideal open-loop method, assumes v Σ c φ± = v dc and v circ φ = 0. Does not compensate for the harmonics in circulating current or the capacitor voltage ripple.Uncompensated [38,57] Assumes v Σ c φ± = v dc , but determines e re f φ and v circ φ from the control loops.Suppresses harmonics in i circ φ .Compensated [31,[58][59][60] Compensates for the capacitor voltage ripple either using measured or estimated v Σ c φ± (t).Use of measured v Σ c φ± (t) could imply significant communication delays for MMCs with N in order of 100s.

Module Level Modulation
The objective of the module level modulation (MLM) methods is to translate reference values of insertion indices from the arm-level modulation to the gate signals for each submodule.These switching signals for individual submodules not only influence the voltage waveform on the AC and DC sides of the converter as a whole, but also control the charging and discharging of submodule capacitors; therefore, this level of modulation needs to conform with the arm-level modulation and the voltage balancing control.Such methods are summarized in Table 3.
Based on the spatial distribution of carrier waves, MCPWM techniques are classified as [68]: 1.
Level shifted carrier PWM method (a).Here, carrier waveforms have identical peak-to-peak amplitude and frequency and are disposed in such a manner that the adjacent bands fully occupy the range and sub-categorized as phase, phase opposition and alternative phase opposition disposition.

2.
Phase shifted carrier PWM method (b). ( Space-vector domain (SVD) [1,3,74,75] SVD-based modulation techniques are based on the unified control of all phase legs of the converter.In this modulation, converter's discrete output voltages and the reference waveform are represented in the form of stationary and rotating vectors in a complex plane (αβ plane).The desired voltage is synthesized either via nearest vector control, which approximates the reference vector with the closest state vector (vector domain equivalent of NLC [76]), or space vector PWM (SVM), where the converter is switched between nearest state vectors with calculated dwell times and is functionally equivalent to NLM [77].
Selective harmonic elimination (SHE) [78,79] SHE based on the Fourier analysis eliminates particular harmonics from the output waveform by introducing notches in the waveform with calculated switching angles [80].Moreover, this modulation is primarily relevant for a low number of levels.

Modeling
In power electronic systems, switching operation is crucial and the prime source of transients in the system.Therefore, for MMCs, EMT-type solvers are utilized for simulations.These solvers employ the trapezoidal rule of integration to approximate lumped elements as illustrated in Figure 3 [81]; while distributed components are modeled using Bergeron's method [82].With this representation, the system is resolved in the time domain using the nodal admittance method [83].Non-linear characteristics of semiconductor devices and the subsequent time-dependent topology of systems require repetitive re-triangulation and interpolations for exact tracking of switching instances [84][85][86] and tend to make the simulation of power electronics systems computationally intensive [87,88].
Unlike conventional current and voltage source converters, where the series array of semiconductors can be modeled as a single device, MMCs employ submodules with an independent operation.Therefore, MMC simulation expects explicit representation for each semiconductor device for most converter studies.

MMC Classical Modeling
Classical simulation techniques utilize the nodal admittance method on the EMT equivalent representations.Semiconductor devices can be modeled using [23]:

•
Full physics: differential equations or equivalent circuits.
These models preserve the circuit configuration of the MMC and simultaneously solve the entire system using the nodal admittance method.Even with a bi-value resistor representation for semiconductor devices, an arm is mathematically represented by a matrix of dimension 2N + 2 by 2N + 2 in this model.With frequent switching, this solution method requires repetitive re-triangulation and interpolations of the system matrix.Therefore, this modeling method tends to be computationally intensive and does not inherently allow parallel computation, making it unsuitable for real-time simulations.
Full physics representation for MMC (Cigré Type 1 [23]) is not computationally realizable.The works in [19,20,89] present the MMC model using the nonlinear IGBT representation (Cigré Type 2 [23]).Based on bi-value resistor representation for semiconductor devices and EMT equivalent representation for lumped elements, an arm for MMC (Cigré Type 3 [23]) is modeled as Figure 4a and is subsequently refereed as the detailed ideal model (DIM).

MMC Efficient Modeling
Fast and accurate simulations of EMT phenomena require models that capture all of the operating modes of the converter and allow for the natural development of its dynamics.Such efficient yet reasonably accurate models from existing literature include the following.

Isolated Submodule Model
This method as proposed in [18] is identical to the classical modeling approach; however, instead of a unified representation of the entire system, here, the admittance matrix of the system is partitioned into numerous small matrices.That is, each submodule is modeled as a separate subsystem, with its individual system matrix, and the coupling between arms and submodules is reproduced using dependent current and voltage sources.A dependent current source in each submodule mimics its interconnection with the arm, and an arm is replaced by a dependent voltage source, corresponding to the cumulative voltage across all submodules as illustrated in Figure 4b.The work in [90] proposes an enhanced version of this model that renders higher computational efficiency by grouping a number of submodules in each sub-system, while [91] extends this modeling approach to the full-bridge MMC.
This isolated representation for an individual or a group of submodules offers superior computational efficiency of solving numerous small matrices over one large matrix and further enables parallel calculation for each submodule.However, the large number of submodules and frequent switching in the converter imply that even with parallel calculation in this partitioned nodal admittance approach, the resulting model is still not suitable for real-time simulations.Furthermore, the forced decoupling of a coupled system in this model results in an artificial delay in the computation of arm and submodule equivalents.Nevertheless, the continuous nature of arm currents, parallel computation and a small simulation time step mitigate the effect of this computational lag.

Arm Thevenin Equivalent Model
The arm Thevenin equivalent model (EIM) as proposed in [17] utilizes a nested fast and simultaneous solution method to model all submodules in an arm by its Thevenin or Norton equivalent, while internal dynamics of an arm are computed using the linear algebraic relation instead of computationally-intensive matrix operations of the nodal admittance method.Figure 5 illustrates the equivalent representation of N submodules of an arm based on this modeling approach.Here, the Thevenin or Norton equivalent subsystem models inserted and bypassed states of all submodules in the arm, while the blocked state of operation is simulated through additional diodes and switches.The works in [20,89,[92][93][94] further investigate this model.
Thevenin's equivalent for inserted and bypassed states of N submodules. • • Block = 1 when Arm blocked otherwise 0 This modeling method drastically reduces the number of nodes in the system.The resolution of the internal dynamics of an arm no longer utilizes computationally-expensive matrix operations, but is modeled by N linear algebraic relations, determined solely by the gate signals and the last time step internal states of the submodules.As a result, the model is suitable for parallel computation, where dedicated processors based on the IGBT gate signals and historical information update the internal and terminal dynamics of individual arms simultaneously, which are concurrently utilized by the main processor modeling the converter and the rest of the system.This combination of parallel computation and reduced computational load makes this model an ideal candidate for real-time simulations [95,96].However, unlike the nodal analysis method, this representation cannot utilize interpolation for exact tracking of switching instances with a fixed time step solver.As a result, this model, even with a similar IGBT/diode representation, tends to be less accurate as compared to the detailed model representation.Nevertheless, the small time step of simulation implies that the effect is negligible for most of the system dynamics.

Arm Switching Function Model
Unlike EMT models where semiconductor devices are modeled as bi-value resistors or with more detailed representations, the switching function model (SFM) represents the switching operation using binary functions.Such models for MMC are presented in [97][98][99][100].The works in [97][98][99] utilize the state-space formulation, which is computationally intensive and not suitable for fault studies in an HVDC system, while [100] proposes a generalized switching function model that utilizes the binary function to model the individual operation of submodules with high computational efficiency.The references in [101][102][103][104] extend this modeling approach to allow inherent incorporation of the blocked state of the converter.The works in [102,103] present the implementation of these switching function models using generic blocks from simulation software as illustrated in Figure 6.This representation offers easier implementation compared to the hard-coded Thevenin equivalent representation.Similar to the arm Thevenin equivalent model, the decoupled representation for arms allows for parallel computation in this model and makes it suitable for real-time simulations.

Average Model
Average value modeling further simplifies converter representation by ignoring the switching effects in individual submodules and, similar to switching function models, reproduce the AC and DC terminal dynamics of the converter as controlled current and voltage sources [105].Based on the assumption of the identical construction of submodules and instantaneous voltage balance control, Refs.[20,101,106,107] represent all N submodules in an arm by an equivalent module.Despite their idealistic assumptions, these models inherently capture internally-stored energies and are suited for large signal analysis of the MMC.These average models are further classified as:

•
Average arm SFM (AVM): This model incorporates the switching effect by utilizing a discrete set of values for the insertion index using module level modulation, which improves accuracy [22,23].

•
Continuous arm model: This model [31,[108][109][110] does not include module level modulation; therefore, it only captures the fundamental frequency component.This simplification allows for a higher time step for the simulation.
Moreover, further simplified models for MMC have been proposed in the literature [19][20][21][22]31,[107][108][109][110][111][112][113][114][115][116][117][118].The works in [19][20][21][22] extend the classical average modeling technique for 2/3-level converters [119] to MMC and represent its terminal dynamics using controlled sources; [107,[111][112][113][114]120] present enhanced versions of this model that incorporate the blocked state of operation for DC fault simulations and/or consider the effects of distributed submodule capacitor voltage ripple in MMC's terminal dynamics.Cigré [23] categorizes these extensions of conventional voltage source converter average models to MMC as Types 5 and 6.Furthermore, [115,116] reduce MMC to an equivalent buck-boost circuit.These models represent distributed arm capacitors of MMC for all phases by a lumped capacitor and therefore do not capture the internal dynamics of the converter.For large signal transients, the internal dynamics of the converter determine its terminal dynamics.As a result, these simplified models are merely suitable for power-flow and steady-state studies.

Model Validation in Literature
Existing literature provides substantial studies for the validation of the equivalent models using simulations and experimental results.The works in [24,51,121] compare the detailed ideal model against experimental prototypes.These studies conclude reporting a high agreement between the detailed model and experimental prototypes.As a result, DIM serves as the standard for the validation of equivalent representations in various studies [17,90,122,123].The isolated submodule model has been validated in [90,122].The works in [17,20,109,[123][124][125] compare the detailed and EIM models under various transient cases.The continuous variant of EIM has been validated against laboratory prototypes in [110,126].Similarly, the switching function and average models have been validated against detailed models or EIM in the literature [19,20,100,101,106,107].
However, not all of the studies simulate all operating states of the converter, and existing literature lacks an independent collective comparison.Therefore, the following section is dedicated to a comprehensive simulation comparison of the select EIM, SFM and AVM models.

Model Comparison
This section compares the presented equivalent models in terms of accuracy and computational speed under steady-state and transient conditions using the system illustrated in Figure 7.The system mimics a modular multilevel converter system coupling an HVDC link and a strong AC grid.For the validation of the computational efficiency and accuracy of the modeling techniques, the detailed ideal model, the Thevenin equivalent model [17], the switching function model (SFM) [17] and the average value model [101], described in previous sections, are simulated.The reason for simulating just these models is the diversity in their solution methods, while the rest of the presented models can be categorized into one of these four representations and are expected to be simulated with a similar order of accuracy and computational speed.The detailed ideal model, the classical EMT representation with bi-value resistor model for semiconductor devices, serves as the benchmark for the accuracy and computational speed of the three efficient equivalent models.
The models are simulated on a 48-core Windows Server 2012 with PSCAD Version 4.6 using the GFortan 4.2.1 compiler with a time step of 10 µs [49,127].The prime motivation for the selection of PSCAD/EMTDC as the choice of simulation software is its wide use in industry and academia [18,38,49,100,123] for the simulation of the HVDC grid system.However, constrained by the restriction on the maximum number of nodes (200) of the educational license of PSCAD/EMTDC, the maximum number of submodules per arm in these simulations is restricted to 14.
The control scheme for the converter system utilizes vector control to regulate power exchange with the grid.For arm level modulation, uncompensated modulation with circulating current suppression control [38] is employed, together with nearest level control with the capacitor voltage rank-based method for the module level modulation.

Model Accuracy
To investigate the accuracy of the equivalent models, internal and external dynamics of the converter under steady-state and transient conditions are validated against the detailed model.The transients associated with the power converter can be categorized as control system set-point changes, over/under voltage on AC-DC sides and internal faults.Equivalent models investigated here do not allow access to internal components and hence cannot simulate internal faults.To address the remaining transient cases, the paper examines six different scenarios, i.e., real/reactive power reversal at the AC side of the converter, single line/three phase to ground faults at the converter transformer, terminal pole to pole and pole to ground DC faults.The performance of the equivalent models under these extreme transient conditions is representative of the maximum error in equivalent idealized representations, and dynamics under any other scenarios are expected to yield similar, if not better accuracy.
The MMC system being simulated here is restricted to 14 submodules per arm, while for HVDC applications, N is in the order of 100s.Nevertheless, the number of submodules scales with power and voltage ratings and the energy stored per submodule is generally of the same order, and there are no operational differences.As a result, conclusions drawn about the accuracy of the equivalent models are expected to be independent of N.
The simulation results in the following sections present the dynamics before, during and after the transient conditions and the absolute error relative to the detailed ideal model.Furthermore, the overall accuracy of the models is quantified using the normalized mean absolute error (ζ) given by Equation (8).
where, X Model (t) is the value of simulated dynamic for the model under consideration and K is the number of simulation points in the interval under investigation.

Power Flow Reversals
To observe the four-quadrant capability of modular multilevel converters and the accuracy of the equivalent models for power-flow transients, real and reactive power reversal scenarios are simulated, and the results are presented in Figures 8 and 9.The simulation reproduces the dynamics of a converter undergoing a step change in a set point of P, real power exchange with the AC grid, from 0.75 p.u.-−1.0 p.u. at t = 2.0 s, and the step change in the reference point of Q, reactive power exchange with the AC grid, from 0.75 p.u.-−1.0 p.u. at t = 2.5 s.Terminal dynamics, i.e., real power, reactive power and DC current, are presented in Figures 8 and 9a-c, respectively; while internal dynamics, i.e., upper arm current and cumulative capacitor voltage, are illustrated in Figures 8 and 9d,e, respectively.In both scenarios, as seen from Figures 8 and 9a,b, the converter reaches the desired real/reactive power set-point in a short duration without overshoot or oscillations.
These results confirm that EIM, SFM and AVM capture the dynamics of the converter under changing power-flow conditions with high accuracy and normalized mean absolute error of less than 0.5%, 0.7% and 0.7%, respectively, against the DIM.

AC Side Faults
For verification of the presented models under the severe transient condition at the AC side, a single line to ground fault and a three phase to ground fault, both lasting for 200 ms through a negligibly small resistance (0.005 Ω) at the Y terminal of the transformer, are simulated at t = 3.5 s and t = 6.5 s, respectively.Throughout the duration of fault, the system and its control operation are kept unchanged.Terminal dynamics, i.e., AC voltage, current and DC current, are presented in Figures 10 and 11a-c, respectively; while internal dynamics, i.e., upper arm current and cumulative capacitor voltage, are illustrated in Figures 10 and 11d,e, respectively.These results confirm that EIM, SFM and AVM capture the dynamics of the converter under the single-line to ground AC fault condition with high accuracy and normalized mean absolute error of less than 1.5%, 2% and 3.5%, respectively, against the DIM.Errors are significantly high for the three phase faults to ground, but the peak values and envelop of dynamics accurately mimic the detailed model, making equivalent models even suitable for the extreme AC-fault studies.

DC Side Faults
To investigate the modeling of the blocked state of operation and the transients under severe transients on the DC side, a 200-ms pole-pole and pole to ground DC fault are simulated at t = 8.0 s and t = 10 s, respectively.The submodules are blocked as the DC side voltage drops below 80% of its nominal value and unblocked with a delay of 300 ms after the DC-voltage regains its nominal voltage.Terminal dynamics, i.e., AC voltage, current and DC current, are presented in Figures 12 and 13a-c, respectively; while upper arm current and cumulative capacitor voltage are illustrated in Figures 12 and 13d,e, respectively.These results also confirm that EIM, SFM and AVM accurately capture the dynamics of the converter under DC fault conditions with normalized mean absolute error of less than 0.6%, 0.9% and 1.0%, respectively, against the DIM.

Model Computational Load
For comparison of the relative computational efficiency of the presented equivalent models, a 12-s simulation of the system subject to numerous transient conditions was conducted.For this simulation, N was varied between two and 200, and the execution times for the models are presented in Figure 14.These results confirm the computational efficiency of the EIM, SFM and AVM against DIM.Already with 14 submodules per arm, SFM and EIM offer a model with 20-times higher computational speed against the DIM; while AVM has a fixed computational load irrespective of N.
In summary, these simulation results validate the precision of the presented EIM, SFM and AVM in replicating a detailed model with high accuracy.The Thevenin equivalent and switching function models are shown to have high computational speed with little loss in accuracy, while the average model with a constant computational load irrespective of the number of submodule offers very high computational efficiency with a compromise on internal dynamic simulation of the converter.Table 4 summarizes the models and their prospective modeling applications.

Discussion and Conclusions
With a global shift towards renewable energy, MMC with its superior design is becoming the technology of choice for point to point and multi-terminal HVDC systems.This converter topology requires an elaborate control and poses a computational challenge to classical simulation methods, which is the subject of interest in several recent studies.The operation of grid-connected converter requires multi-tier control and modulation schemes, which are classified and discussed in this paper.
The specific objective is to present an overview of MMC models from the literature that are suitable for EMT studies.All of the models discussed reproduce all operating states of the converter and capture its internal and external dynamics under stationary and transient conditions.These models vary in accuracy, as well as in the computational load.HVDC and converter studies involve analyses at the component, system or network levels with varying requirements for computational efficacy and efficiency, making each model a best fit to the simulation application as discussed below: • Detailed models are the next best thing to prototype setups and provide the highest accuracy among all simulation models.These models subject to the accuracy of semiconductor representation, provide precise analyses of switching characteristics and are suitable for studies at the component level such as component optimization, electromagnetic compatibility (EMC) validation, switching loss estimation, etc.

•
For system-level studies, where switching events can be modeled as instantaneous events, isolated submodule, Thevenin equivalent and switching function models offer a computationally-efficient representation.These modeling approaches are suitable for the simulation of the large signal behavior of the converter system and faster submodule level control and modulation schemes.The isolated submodule model allows access to individual submodules and is suited for the simulation of internal faults, while the Thevenin equivalent and switching function models with parallel computation are appropriate for real-time simulations.• Lastly, for the network-level studies, where terminal characteristics are of prime interest, average models with an accurate description of terminal dynamics provide an efficient representation.These models are further appropriate for the design and validation of the converter control and arm modulation schemes.
In short, it is concluded that the arm Thevenin equivalent and the switching function models that individually represent all submodules can precisely replace a detailed model for system-level studies of the converter.These models with parallel computation capabilities are suitable for real-time simulations; whereas the average value model with a constant computational load and accurate terminal characteristics is suited for network-level studies.

Figure 3 .
Figure 3. EMT equivalent representation of lumped elements. i

Figure 4 .
Figure 4. Equivalent representation for an arm.(a) EMT equivalent model; (b) isolated submodule model (ISM)-equivalent representation for an arm. i

Figure 8 .
Figure 8. Real power reversal.(a) Real power injected in the AC-grid P and the model error; (b) reactive power injected in the AC-grid Q and the model error; (c) DC current i dc and the model error; (d) positive arm current i (a+) and the model error; (e) cumulative capacitor voltage v ∑ C(a+) and the model error.DIM, detailed ideal model; EIM, equivalent model; AVM, average arm SFM.

Figure 9 .
Figure 9. Reactive power reversal.(a) Real power injected in the AC-grid P and the model error; (b) reactive power injected in the AC-grid Q and the model error; (c) DC current i dc and the model error; (d) positive arm current i (a+) and the model error; (e) cumulative capacitor voltage v ∑ C(a+) and the model error.

Figure 10 .
Figure 10.Single phase to ground fault.(a) AC voltage v ac a and the model error; (b) AC current i ac a and the model error; (c) DC current i dc and the model error; (d) positive arm current i (a+) and the model error; (e) cumulative capacitor voltage v ∑ C(a+) and the model error.

Figure 11 .
Figure 11.Three phase to ground fault.(a) AC voltage v ac a and the model error; (b) AC current i ac a and the model error; (c) DC current i dc and the model error; (d) positive arm current i (a+) and the model error; (e) cumulative capacitor voltage v ∑ C(a+) and the model error.

Figure 12 .Figure 13 .
Figure 12.Pole-pole DC fault.(a) AC voltage v ac a and the model error; (b) AC current i ac a and the model error; (c) DC current i dc and the model error; (d) positive arm current i (a+) and the model error; (e) cumulative capacitor voltage v ∑ C(a+) and the model error.

Figure 13 .
Figure 13.Pole to ground DC fault.(a) AC voltage v ac a and the model error; (b) AC current i ac a and the model error; (c) DC current i dc and the model error; (d) positive arm current i (a+) and the model error; (e) cumulative capacitor voltage v ∑ C(a+) and the model error.

Figure 14 .
Figure 14.Computational burden of the MMC models.

Table 1 .
Operation of half bridge submodule.

Table 4 .
EMT models for MMC: summary.