In Figure 2.1a a structural representation of a three-phase Modular Multilevel Converter is illustrated. As expected in a practical application, a large number of series-connected submodules results in a high apparent multilevel PWM frequency. The latter leads to an accurate average approximation of the branch quantities, motivating the modeling of the converter utilizing a unique equivalent submodule per branch. The mid-point of the common link is grounded for facilitation of the analysis. When the MMC is connected to a DC line, LS and RSrepresent the filter or line inductance and uS is a DC voltage source.
In the case of a transformerless single/three-phase direct AC/AC converter they denote the grid choke that handles short-circuit currents [12], and uS is the single-phase AC
grid voltage. On the three-phase side, LT and RT represent the transformer or machine leakage inductance and cable resistance, whereas the branch inductances and resistances are symbolized with L and R accordingly. Finally, uT k is the grid or motor back emf, respectively.
It is noted that throughout this chapter, the following subscripts/superscripts are utilized:
k∈ {a, b, c} referring to the studied phase and j ∈ {u, l} to the upper/lower branches of the same phase leg.
Upper Loop
Figure 2.1: (a) Multi-phase Modular Multilevel Converter, (b) Detailed representation of a converter branch with one equivalent submodule.
2.2. Modeling Principles of an MMC
2.2.1 Derivation of Total Submodule Voltage Ripples
In order to obtain the analytical expression for the submodule capacitor voltages of the Modular Multilevel Converter, the fundamental capacitor current equation is considered, taking as an example the upper branch of a converter phase leg. This relates the current ikcu that flows through the equivalent capacitance Cbranch with the sum of all the submodule voltages Σukusm in one converter branch.
CbranchdΣukusm
dt = ikcu⇔ Csm
N dΣukusm= mkuikudt⇔ Csm
N dΣuuksmΣukusm= ukuikudt (2.1) In the above equation (2.1), N denotes the number of submodules in one converter branch, each one of which having a capacitance of Csm. The quantity mku represents the normalized modulation reference for the upper branch and uku is the voltage applied by the series connection of the branch submodules, which are PWM-controlled. By integrating (2.1), the following expression is derived,
Csm
where the appearing integration constant ΣE0sm refers to the DC value of the total submodule stored energy. It is therefore clear that the product ukuiku corresponds to the power contribution of the submodules to the branch, and its integral forms the total stored capacitive energy. This equation is the basis for the voltage estimation of the capacitor voltage ripples.
Taking into consideration Figure 2.1a, the following relation (2.3) is formed for the capacitive instantaneous power,
where the phase voltage uk comprises the fundamental frequency (uV k) plus a common-mode (ucm) components. The terms pbranch, pR and pL correspond to the total branch power, the losses in the branch resistance as well as the reactive power consumed by the inductor, respectively. For the lower branch, the power and energy equations become:
pklc = uklikl =
= [−uL+ (uV k+ ucm)] ikl
| {z }
pklbranch
− ikl2
R
| {z }
pklR
− Ldikl dt ikl
| {z }
pklL
(2.4)
The integration finally gives the respective capacitive energy variation in upper and lower branches. Therefore, it becomes evident that (2.2) can be used for the accurate estimation of the submodule voltage ripples. It is noted that these equations are valid for any given DC/AC or AC/AC Modular Multilevel Converter configuration. The difference lies in the desirable shapes of the voltage uS, current iS (DC or AC respectively) as well as the intentional imposition of the so-called circulating currents within the converter phase-legs, which will cause different frequency components in the branch currents.
In the next section, two specific examples are chosen in order to apply this theory, namely the three-phase and two-phase DC/AC MMC cases. As a remark and for the simplification of the analysis, the voltage drop on RS and LS is disregarded in the modeling procedure. This implies that the voltage uU L is considered to be known. The latter forms a fair assumption, since this magnitude can be either actively adjusted (in the case of a current-controlled single-phase AC grid), or directly measured (in typical DC/AC systems).
Capacitor Voltage Natural Charge Level Mechanism
The integration constant ΣE0sm of (2.2) is of significant importance and is not to be confused with the inner current control of an MMC, since their action mechanisms are different. As depicted in Figure 2.1b, the equivalent submodule should provide a specific voltage uku. Once the voltage reference uk∗u is defined as a result of the two current loops described in the following sections, the choice of mku can be considered furthermore as a degree of freedom for the capacitor charge level, since its input/output power equilibrium is dictated by the physics of the system. Indeed, the following relation holds,
mku= uku∗
Σukusm0+ Σ˜ukusm (2.5)
where Σ˜ukusm denotes the sum of the branch capacitor voltage ripples.
The normalization constant Σukusm0 can be freely chosen to charge/discharge the capacitor voltages as intended. This is better illustrated in the example of Figure 2.2. By increasing the normalization constant at t = 40 ms, the modulation reference mku obviously decreases.
The instantaneous difference of the voltage sum (uku+ ukl)from the DC link voltage uU L is capable of inducing a specific amount of current is/min the branch (m being the number of parallel-connected phase-legs), which in turn charges the capacitors at a higher level.
When the new equilibrium is reached, the quantities mku and Σukusm0 are changed, in order to maintain unaltered voltages uU O and uk, respectively. It should be noted that this
2.3. Accurate Capacitor Voltage Ripple Estimation for MMC