By formulating the upper and lower inner AC voltage loop laws in Figure 2.1a, the following differential equations can be obtained on a per-phase level,
−uS 2 + ud
| {z }
−uU
+
uku+ Riku+ Ldiku dt
+ uk = 0 (2.25)
−uS
2 + ud
| {z }
uL
+
ukl + Rikl + Ldikl dt
− uk= 0 (2.26)
where uk= RTiT k+ LTdidtT k + uT k+ ucm.
As only the fundamental harmonic of the line current needs to be controlled, the common-mode voltage component can be neglected in the disturbance vector. The continuous-time state space representation can be formulated by considering the states xk= (iT k ikcirc)T, the manipulated variables1 uk = (uku ukl)T as well as the disturbances wk= (uT k uU L)T.
1The symbol of the k-th phase manipulated variable vector uk should not be confused with the fundamental harmonic component of the k-th phase converter voltage uk.
2.4. Current Control Considerations for an MMC
Defining furthermore Leq= L+2LT and Req= R+2RT, the following relation is obtained
˙xk= Axk+ Buk+ Dwk (2.27)
The fact that A is a diagonal matrix shows that a decoupled current control of these two magnitudes is at hand. This statement is quite important and implies that in fact the input power can be controlled independently from the output power [49]. In the studied case, the circulating current has been modeled to bear a DC component associated with the output power. In the case where the common terminals are connected to a single-phase grid, such a property also holds with an appropriate and equivalent definition of ikcirc.
Figure 2.4 shows the block diagrams for the proposed line and circulating current control loop implementations, again on a per phase leg basis. The block Gpe(s)represents the first-order transfer function which models the application-specific delays Tpeline and Tpecirc, caused by the modulator, measurement and computational system as well as sampling time [106].
Figure 2.4: Current control loop block diagrams for: (a) line current and (b) circulating current.
The two respective system transfer functions GlineS (s) and GcircS (s) are expressed as
GlineS (s) = 1
Req+ sLeq (2.28)
GcircS (s) = 1
2 (R + sL) (2.29)
which shows that they are typically of first-order. This is similar to [56] but considering the branch resistances as well.
2.4.1 Line Current Control
The line current control can be performed as for any grid-connected voltage source converter in a stationary reference frame (SRF) or rotating reference frame (RRF), and according to the transfer function GlineS (s) given by (2.28). Two different methods have been chosen for the cases of the three- and two-phase systems and are described in the following.
Three-phase MMC
A proportional-resonant (PR) controller has been chosen for the three-phase MMC case, which is expressed by (2.30) in a generalized manner [107].
GlineP R(s) = Kpline+ X
h=2n+1
2Kihlineωcs
s2+ 2ωcs + (hω)2 (2.30)
The transfer function of (2.30) comprises a proportional plus several resonant terms.
These can be tuned to control odd harmonic frequencies (hω), which are the most prominent in a typical current spectrum [107,108]. This is especially beneficial in cases of harmonically polluted networks, where a Modular Multilevel Converter might also be required to provide active filtering services. The cutoff frequency ωc is utilized in order to avoid the practical stability, real-time implementation, as well as parameter (such as grid frequency) variation issues associated with a theoretically infinite gain at the chosen AC frequencies [107,108]. An additional advantage of the PR controller is the fact that grid voltage disturbance feed-forwarding is not necessary, in contrast to the case of a rotating reference frame current controller where such an action is usually preferred. This is illustrated in Figure 2.4a. In such a case, the three-phase system will be controlled in a SRF, therefore two resonant controllers need to be implemented.
Two-phase MMC
A PR control would be suitable for the two-phase Modular Multilevel Converter as well.
In this thesis report, however, an alternative solution is presented for the case where the RRF implementation is preferred. In order to imitate the behavior of a three-phase system and achieve a straightforward active/reactive power regulation through a current vector proportional-integral (PI)-based control scheme, a fictive axis emulator is utilized [109].
The general idea behind the concept is to utilize the β-component of the phase voltage uT β as well as the vector control output uβab, in order to generate the imaginary component
2.4. Current Control Considerations for an MMC of the line current iT β by means of the system transfer function itself. The block diagram of the algorithm is depicted in Figure 2.5. However, a modification needs to take place, in order to account for the branch inductances and resistances as well. Thus, the quantities LF AE and RF AE for the two-phase structure are given by (2.31).
LF AE = Leq , RF AE= Req (2.31)
It is noted that the imaginary component of the grid phase voltage uT β can be generated through the use of a second-order generalized integrator (SOGI) [110]. Finally, the block named ucmd refers all the associated delays linked to modulation, measurements and computational time, which need to be modeled in order to keep the symmetry between the two axes.
sLF AE1 RF AE
ucmd
−
− iT β
uT β ab
uβ
Figure 2.5: The Fictive Axis Emulation (FAE) concept for line current vector control of a two-phase Modular Multilevel Converter.
2.4.2 Circulating Current Control
For the circulating current control, a proportional term Kpcirc has been implemented as the transfer function of GcircP (s). Figure 2.4b shows that in this case, a feed-forward disturbance rejection signal of the DC-link voltage uU L is also used. Since an abc frame is kept, three such controllers need to be implemented in this case.
By using a proportional term for the circulating current control, a steady state error will appear between the reference and the measured values, especially in the case where a sinusoidal value has to be tracked. For the elimination of this error, two more terms could be added to the respective controller transfer function. Therefore, it would consist of a proportional term, an integral action for controlling the DC part of the circulating current as well as a resonant term for the imposed second-order harmonic component.
This is shown in (2.32).
GcircP IR(s) = Kpcirc+Kicirc
s + 2Ki2circωcs
s2+ 2ωcs + (2ω)2 (2.32)
In such a case, the rejection of the disturbance uU L in the sense of feed-forwarding would not be necessary.
2.4.3 Simulation Results
The dynamic performance of the closed-loop system has been tested with the parameters taken from Table 2.1, both for the cases of the three- and two-phase Modular Multilevel Converter. The time-domain implementation of the current controllers with the associated signal processing blocks are depicted in Figure 2.6.
T∗
Figure 2.6: Time-domain MMC current control implementation with estimation of capacitor voltage ripples: (a) three-phase and (b) two-phase converter cases.
The control of line and circulating currents of the three-phase MMC is shown in Figure 2.7a, where for reasons of graphical clarity only the first phase is depicted. At t = 120 ms and while the line current of phase a is at its peak (≈ 24 A), a power flow reversal is executed to an active current of ˆiT a = 22 A. This is immediately followed by an injection of reactive power at t = 160 ms, which corresponds to a current component with a peak
2.5. Experimental Tests