Energy-Based State-Space Representation of Modular Multilevel Converters with a Constant Equilibrium Point in Steady-State Operation

Abstract

The internal currents and voltages of Modular Multilevel Converters (MMCs) contain multiple frequency components in steady state operation and remain time-periodic even when transformed into a synchronously rotating reference frame. This prevents a straightforward state-space representation where a constant equilibrium point is reached and all state variables converge to constant values under steady-state conditions. Such steady-state time-invariant (SSTI) representations are needed for linearization and eigenvaluebased analysis of small-signal stability. This paper presents an energy-based model of an MMC with a modulation strategy where the insertion indices are compensated for the oscillations in the sum arm voltage. The formulation of the model allows for deriving, by the application of Park transformations at three different frequencies, a SSTI representation that accurately captures the internal dynamics of the MMC. This model can be simplified to a reduced order model that maintains accurate reproduction of the external behavior at the ac-and dc-sides while neglecting some of the internal dynamics. The validity and accuracy of these two SSTI MMC models are verified by timedomain simulations and their utilization for eigenvalue-based analysis of MMC dynamics is demonstrated by examples.

Dynamics and Stability of Meshed Multiterminal HVDC Networks

“…The modular multilevel converter is the most promising converter technology for MT-HVDC system in offshore applications. For the purpose of power system studies, its dynamics is described by the continuous model presented in [15]- [17]. Initially, the n submodule dynamic equivalents depicted in Fig.…”

Simplified Modelling of the F2F MMC-Based High Power DC-DC Converter Including the Effect of Circulating Current Dynamics

“…Eq. (2) gives a Steady State Time Periodic (SSTP) model of the MMC, whereas a Steady State Time Invariant (SSTI) model is needed for small signal studies [6]. Dynamic phasor based modelling [8] of the harmonic components is used in order to obtain an SSTI model.…”

“…The second harmonic part is given by (6). (4) to (6) can be extended to the F2F by duplicating all the states except the ac current, because it is common to both sides.…”

“…The building block for the converter, i.e. the MMC, is well developed and models of varying level of detail are already available [6]- [13]. However, because of its symmetry, the F2F lends itself to further simplification.…”

“…This fact was utilized in [14] to develop a simple model of the converter with a single capacitor representing the arm energy dynamics, thus reducing the number of states from 18 to 6. However, as will be shown in this paper, the model is applicable only when using compensated modulation, a special method for calculating the insertion indexes [6]. When using other types of control, such as direct voltage control, there is an uncontrolled interaction between the different harmonic components in the converter, which leads to a poorly damped oscillation at the converter terminals [15].…”