Figure 1 presents the topology of the two-level converter containing 6 IGBT and diode pairs. By controlling the gating signals, the switching devices conduct in-turns and the converters generate pulse-width modulated voltage waveform at the AC terminals. The switching devices have non-ideal characteristics during the control cycles with switch-on and switch-off transients as well as conduction voltage drops, which contribute to the switching loss and the conduction loss correspondingly.

The physics model applies differential equations to represent the internal structure of the power electronics devices, which can accurately simulate the transients of the switching process (Kraus and Mattausch, 1998). The behavior model simplifies the parameters which are not necessarily related to the structure and material of the device (Myaing and Dinavahi, 2011). However, it still uses multiple nodes and non-linear components to model a single switching device. Both models are relatively complex using the typical time-step of several to several tens of nanoseconds. Although these models can accurately take the switching and conduction losses into account, they are not applicable to simulate a large system with many converters due to the enormous computation burden.

Two-resistance model uses small resistance for the on-state and large resistance for the off-state, which is one of the most common converter modeling schemes for EMT simulation (Maguire et al., 2018). By adding a voltage source in a series of the resistance representing the threshold voltage and slope resistance during the on-state, the conduction loss can be modeled.

To further reduce the computation effort, the switching function model and the average value model are also commonly used for converter modeling, especially for a large system. These models use controlled sources as the interface, which can avoid the change of conductance matrix and the following complete matrix inversion process (Chiniforoosh et al., 2010). These models can have a one-time step delay in exchanging the voltage and the current values, and it may cause a minor power imbalance between the DC and the AC sides.

Figure 2 shows the topology of the MMC, which is composed of 6 six converter arms. Each arm contains an arm inductor and numerous series-connected sub-modules (SM). The half-bridge (HB) circuit shown in Figure 2 and the full-bridge (FB) circuit are mostly applied, and the sub-module number can be up to several hundred in a typical HVDC transmission system. Due to the relatively complex topology, modeling the MMC using discrete switches are not applicable with thousands of SMs, since the system has too many electrical nodes.

Thevenin equivalence based methods were proposed in (Gnanarathna et al., 2011) to simplify the models of SMs. Series connected Thevenin equivalent interfaces are summed together reducing the electrical nodes significantly. In the real-time simulation, the computation of the Thevenin equivalence process can be further reduced by the equivalent circuit method or the surrogate network method (Maguire et al., 2013; Saad et al., 2015). Like the switching function and average value model for a two-level converter, the above equivalent models for MMC use a controlled source as the interface. Spurious power losses may occur due to some modeling equivalent methods or decoupling method introducing one-step delay. The spurious power losses introduced by the converter model scheme are difficult to calculate and are changing according to the time-step, and the operation state of the circuit.

In literature, device-level modeling schemes were proposed for the switching devices in MMC and other converter topologies (Shen and Dinavahi, 2016; Liang and Dinavahi, 2018; Lin and Dinavahi, 2018). It is noted that such models are not based on iterative solvers such as Spice but are developed for electromagnetic transient programs. The device-level model considers the switching characteristics of all individual switching devices using the values of the firing pulses, the current, and the voltage across each device. The characteristics are obtained from the manufacturer’s datasheet. The device-level model can accurately calculate the instantaneous switching loss and conduction loss. However, it requires high computation effort, which restricts the SM number in MMC applicable to these methods.

The proposed LUT stores the loss ratios for various power ratings, and the instantaneous power is one of the inputs of the LUT. The power loss ratio is also affected by the switching frequency, therefore the switching frequency is another input of the LUT. More variables, such as environment temperature, can also dynamically influence the loss ratio by introducing the third input, while not considered in this work.

The output characteristics and switching loss curves of the switching devices can be obtained from the manufacturer’s datasheet. The curves are fitted by third-order polynomials. Figure 4 shows the IGBT conduction characteristics, IGBT switch-on loss, and IGBT switch-off loss of the Infineon FF 1800R23IE7P IGBT module (Infineon IGBT modules, 2023).

The conduction and switching losses change with the device current, therefore the device current shall be calculated first for certain instantaneous power input. For a two-level converter, the current through each switching device is a portion of the AC side current of each phase. The phase current can be calculated as Eq. 1: i p h = 2 P 3 u p h sin 2 π f n t (1) where i _ph is the phase current; p is the instantaneous power; u _ph is the phase voltage; f _n is the fundamental frequency; t is the simulation time. It is noted that harmonics are neglected in (1) for conduction loss calculation.

For the two-level converter, the phase current either flows through the upper arm or the lower arm. In other words, the conduction current of all devices in the upper and lower arms in the two-level converter is exactly the phase current.

For MMC, the arm current contains both the AC part and the DC part. Without considering the double-frequency circulation current, the bridge arm current for MMC is calculated by the Eqs. 2–4: i d c = P u d c (2) i a r m _ U _ M M C = i d c 3 + i p h 2 (3) i a r m _ L _ M M C = i d c 3 − i p h 2 (4) where i _dc is the DC current; u _dc is the DC voltage; i _{arm_U_MMC} is the upper bridge arm current; i _{arm_L_MMC} is the lower bridge arm current.

The calculation of the device current is based on the assumption that the turn-on time and turn-off time of the IGBT and diode are identical. Taking the upper IGBT and anti-parallel connected diode pair as module V1, and the lower IGBT and anti-parallel connected diode pair as module V2, the current flowing through V1 and V2 can be calculated as Eqs. 5–7: i V 1 = i a r m S j (5) i V 2 = i a r m 1 − S j (6) S j = 1 , V 1  is on − state and  V 2  is off − state 0 , V 2  is on − state and  V 1  is off − state (7) where i _v1 is the V₁ current; i _v2 is the V₂ current; i _arm is the arm current of the converter; S _j is the switching function of the jth submodule.

The conduction losses of both V1 and V2 can be calculated as Eqs. 8–10: P V 1 = 1 T 0 ∫ t − T 0 t 1 2 u C E t i V 1 t + u F t i V 1 t d t (8) P V 2 = 1 T 0 ∫ t − T 0 t 1 2 u C E t i V 2 t + u F t i V 2 t d t (9) P c o n d = P V 1 + P V 2 (10) where T ₀ is the time-period of fundamental frequency; P _V1 is the conduction loss of V₁; P _V2 is the conduction loss of V₂; u _CE is the collector-emitter voltage of IGBT; u _F is the on-state voltage of diode; P _cond is the overall conduction loss of IGBT and diode.

By substituting Eqs. 5, 6, 8, 9 into 10, the conduction loss can be calculated as Eq. 11: P c o n d = 1 T 0 ∫ t − T 0 t 1 2 u C E t i a r m t + u F t i a r m t d t (11)

For switching loss calculation, the switching time instants are assumed evenly distributed within the period of the fundamental frequency. All the current samples during the cycles are used to calculate the switching energy loss. The average switching power loss of all devices in one phase of a two-level converter or a half-bridge SM in MMC can be calculated as Eq. 12: P s w = 1 N ∑ j = 1 N E s w _ o n i j + E s w _ o f f i j + E s w _ r e c i j f c (12) where P _sw is the overall switching loss of IGBT and diode; E _{sw_on} is the switch-on loss of IGBT; E _{sw_off} is the switch-off loss of IGBT; E _{sw_rec} is reverse-recovery loss of diode; f _c is the switching frequency; N is the sample points in a current period. E _{sw_on} and E _{sw_off} can be obtained from Figure 4B. E _{sw_rec} can be obtained from the manufacturer’s datasheet with the same approach of obtaining E _{sw_on} and E _{sw_off} .

Multiplied by the number of repetitive units, the total power loss of the converter can be calculated. The power loss ratio represents the result of dividing power loss by transmission power. The power losses and the power loss ratio of the two-level converter and MMC can be calculated as Eqs. 13–16. P l o s s _ 2 L = 3 P c o n d + P s w (13) P l o s s _ M M C = 6 N M M C P c o n d + P s w (14) R a t i o 2 L = P l o s s _ 2 L P (15) R a t i o M M C = P l o s s _ M M C P (16) where P _{loss_2L} is the power loss of the two-level converter; P _{loss_MMC} is the power loss of MMC; N _MMC is the SM number of one arm; Ratio _2L is the power loss ratio of the two-level converter; Ratio _MMC is the power loss ratio of MMC.

The losses and the corresponding loss ratios are calculated for finite combinations of the instantaneous power and switching frequencies to form the LUT. The balance between the accuracy and efficiency of the proposed method is achieved by using appropriate LUT dimension and linear interpolation.

The power loss ratio of arbitrary combinations can be calculated as Eqs. 17–19: R a t i o P 1 f c = f c 2 − f c f c 2 − f c 1 R a t i o f c 1 , P 1 + f c − f c 1 f c 2 − f c 1 R a t i o f c 2 , P 1 (17) R a t i o P 2 f c = f c 2 − f c f c 2 − f c 1 R a t i o f c 1 , P 2 + f c − f c 1 f c 2 − f c 1 R a t i o f c 2 , P 2 (18) R a t i o f c , P = P 2 − P P 2 − P 1 R a t i o P 1 f c + P − P 1 P 2 − P 1 R a t i o P 2 f c (19) where Ratio is the power loss ratio of the converter; f _c , f _c1 , and f _c2 are the switching frequencies; f _c is between f _c1 and f _c2 , p is between P ₁ and P ₂ .

With the datasheet of Infineon FF 1800R23IE7P, the power loss ratio curves for multiple switching frequencies are generated as shown in Figure 5. The loss ratio first decreases with the increment of the transmission power. Then the power loss ratio increases slightly after reaching the optimal point of the loss ratio. As expected, the switching frequency has a significant impact on the overall loss ratio of the converter.

The transmission power of the converter AC side and DC side is calculated using the instantaneous voltages and currents. The power loss is the transmission power difference between the AC and DC sides.

The reference power loss is calculated by the LUT which is generated in Section Ⅲ. Reference power loss can be obtained by using the interpolation method with inputs of instantaneous power and switching frequency. The AC side instantaneous power is used as the input of the LUT. The switching frequency can be set as either a constant value or a variable by counting the number of switching pulses during the simulation.

To compensate for the power loss during the simulation, a feedback loop with PI control is adopted to track the reference power loss. The PI control can dynamically eliminate the deviation between the reference loss and the measured loss.

Furthermore, the feedforward control is used to improve the response speed of the system, when the reference value or the DC voltage changes rapidly.

A limiter is placed at the output of the control module. The limiter parameter is determined by multiplying the power loss of the rated condition with a coefficient before dividing by the DC voltage. When the power system is in a transient state, the limiter may take effect. The proposed method is mainly aimed for the steady-state condition with little change in power loss. The proposed method is not accurate for drastic transients such as fault conditions, while it has a relatively small effect on the transients due to the limiter.

To track the reference power loss, a variant resistor or a current source can be used as the interface. However, a varaint resistor will change the conductance matrix and require the full solution of the LU-factorization, which is time-consuming. Therefore, a current source is added to adjust the converter power loss according to the reference. The current source can be placed in parallel to the AC side or the DC side. However, the control of the AC side current source requires the phase-locked loop (PLL), which makes it relatively complex. Therefore, the controlled current source is placed at the DC side in this work.