T ABLE 52, D EFINITION OF THE VIRTUAL VECTOR SEQUENCE MODULATION SCHEME 1 Predetermined sequences of 4 states shall be applied This corresponds with half a

SVM or CB PWM period. Two types of sequences are possible:

a. Virtual vector sequence type 1 based on regular size triangles and incorporating only modified vectors or virtual vectors type 1. No DM distortion is generated in this case.

b. Virtual vector sequence type 2 based on large size triangles and incorporating also virtual vectors type 2. DM distortion is generated in this case.

2. The choice of sequence is based on small triangles (according to virtual vector type 1 spacing)

3. The actual sequences to be applied are based on triangles of original or double size.

4. Some small triangles (e.g. outside the star region) refer to the same sequence. 5. Multiple sequences may be stored per triangle. The sequence to be applied is

chosen based on operating point (load angle) or a chosen optimization criteria. Namely, there are several options in the low modulation depth range, where all phases feature partner states.

6. Redundant states in phases not related to the partner states are chosen online (modification of the predetermined sequences) according to the state selection scheme also used in the standard modulators (depending on FC voltage deviation and phase current)

α β 100% 96.5% 81% 65% 50% Hexagon Regular Vectors Modified Vectors Virtual Vectors 1 Virtual Vectors 2

6.1.4.2

Availability of virtual vector sequences

Four different regions have been identified:

1. Region at low modulation depth where virtual vector sequences type 1 are available for all of the three phases (yellow in Figure 94). This region performs well for all load angles.

2. Region at intermediate modulation depth where virtual vector sequences type 1 are available only for one phase (blue in Figure 94). This region extends the effective operating range, while keeping a non distorted DM output. This region performs best in reactive power operation.

3. Region at high modulation depth where virtual vector sequences type 2 are available for one phase (red in Figure 94). This region generates higher switching losses and increases DM distortion, but it is effective up to the over-modulation region. It performs best for reactive power but also improves the NP control capacity for active power operation.

4. Region at high modulation depth where no virtual vector sequences are available at all. In these regions, any standard modulation and NP control scheme can be applied.

Figure 94, Availability of virtual vector sequences

The red area outside of the star shaped area defined by the green dots has virtual vector sequences type 2; however, there will be no perfect balance of the flying capacitors but improved NP controllability. A scheme without that extension can easily be implemented by a different sequence selection scheme. The whole area of the hexagon is covered well with virtual vector sequences, which indicates that the concept is powerful for all physically possible modulation depths including the non sinusoidal over modulation region.

6.1.5

Performance impact of modified and virtual vectors

6.1.5.1

Minimum and maximum NP currents in function of θθθθ

Minimum and maximum NP currents can be calculated for balanced and undistorted operating conditions by simply applying a matrix of operating points to the controller. Limits for pure reactive power operation are given in Figure 95. The impact on minimum and maximum NP

α β 100% 96.5% 81% 65% 50% Hexagon Regular Vectors Modified Vectors Virtual Vectors 1 Virtual Vectors 2

current is obviously very large (compare with last column of TABLE 23). A full set of graphs for different load angles and modulation schemes is given in appendix 9.5.3.

Figure 95, (a) Maximum and (b) minimum NP currents in function of modulation depth and voltage angle θθθθ in pure reactive power operation

6.1.5.2

Average minimum and maximum NP currents

The average minimum and maximum NP currents over the fundamental period are a measure for the NP control capacity. For low modulation depth, virtual vector sequences are available for several phases. As a result, the NP current control capacity can be increased to 50% (of the peak phase current) for all load angles at low modulation depths down to zero. For high modulation depth, virtual vector sequences are available for one phase only and the effectiveness is depending on the load angle. Nevertheless, there is a significant impact on the NP current control capacity in both reactive and active power operation, as can be seen from Figure 96. Notably in the over modulation region, the modified and virtual vector scheme still provides effective NP control capacity. Note that for zero load angle, the unconstrained DC CM injection performs the same as the real time NP current scheme; the green curve is not visible in the graph as it coincides with the blue curve. Note that the red lines in Figure 96 (b) for minium and maximum average NP currents over a fundamental period correspond with an averaging of the functions in Figure 95 over θ.

Figure 96, minimum and maximum average NP currents in function of load angle, modulation type and modulation depth. Red: modified and virtual vector scheme, Blue: real time NP current scheme,

Green: unconstrained DC CM injection scheme. (a) ϕϕϕϕ = 0 (cos(ϕϕϕϕ) = 1), (b) ϕϕϕϕ = 1.56 (cos(ϕϕϕϕ) = 0.01)

θ θ

m m

(b) (a)

A full set of graphs for different load angles ϕ is given in appendix 9.5.4.

6.1.6

Implementation of a virtual vector modulator and experimental

verification

The virtual vector modulator has been implemented based on lookup tables for the first segment. All other vectors are transformed into that first segment to calculate states and application times. The calculations are done based on a non perpendicular coordinate system, very similar to the algorithm presented in [67]. A seamless integration of the different sequences is crucial, as there may be multiple changes in type of sequence within a fundamental period (see Figure 94). Also, the modulator needs to be compatible with standard modulators so that a smooth transition between the different operating modes is guaranteed. These features have been confirmed both by simulation and experiment.

Figure 97, Virtual vector modulation at low modulation depth (m=0.2) and high modulation depth (m=0.9) in pure reactive power operation with zoom on transition period

Figure 97 shows a step in modulation depth from m=0.2 to m=0.9. This step includes a transition from virtual vector sequences type 1 to virtual vector sequences type 2, which seems to go very smooth. For low modulation depth, only virtual vector sequences type 1 are applied. They generate systematic 2 level steps in the phase voltages, but they are not visible in the phase to phase voltages (apart from glitches generated by commutation). Accordingly, there is no distortion in the current compared to any standard modulation scheme. The NP current can be kept zero in this operating mode and there is no low frequency ripple in the NP voltage. For high modulation depth, virtual vector sequences type 2 are applied. They also generate 2 level steps in the phase voltages, but these are not completely cancelled out and are partly visible also in the phase to phase voltages as expected from theory and simulation. For high modulation depth, the NP current cannot be cancelled out completely and a third harmonic appears.

-300 -200 -100 0 100 200 300 -0.03 -0.02 -0.01 0 0.01 Time V o lt a g e -36 -24 -12 0 12 24 36 Cu rr e n t U_phase1 U_phase2 U_21 U_NP I_phase2

Figure 98, NP current ripple for real time NP current scheme (a) and virtual vector scheme (b) at medium modulation depth (m=0.75)

Figure 98 shows the impact of the modulation scheme on the minimum NP voltage ripple. The virtual vector scheme can significantly reduce the NP voltage ripple at the cost of increased losses and increased output voltage distortion.

(a) (b)

6.2

Online calculated optimal sequences by the use of optimal control

schemes

When considering NP and FC control, all required quantities can be controlled separately, as has been done with previously proposed control schemes: Output voltage (for torque, flux, power or voltage control) can be set in open loop by the appropriate αβ vector (and feed forward control taking into account actual DC voltages); flying capacitors can be controlled by the appropriate share of redundant states; the NP can be controlled with the application of a CM and suitable choice of redundant states.

Alternatively, all parameters can be controlled at the same time, applying an optimal control scheme. This chapter introduces an approach to optimized SVM by means of optimal control for the generation of sequences of converter states to be applied. This is related to the work by Geyer/Papafotiou/Morari [51, 52] or J. Rodriguez et al. [54, 57], who use MPC (model predictive control) to implement modulators integrating control of internal and external quantities. Those published concepts act directly on the individual switching states, resulting in a “next best vector” type of modulation (see 4.4.3) and variable frequency operation. In contrast, the method proposed in this chapter uses model based prediction and optimal control but applies those concepts on a number of generic sequences rather than next optimal vectors. As a result, the operation of the converter is much closer to classic CB PWM or SVM with constant switching frequency with the notable difference that multiple quantities are controlled at the same time.

Due to the large number of available states in ML converters, any sequence based on SVM can be generated in a large number of different ways. To start with, the length of a SVM sequence is not inherently given. A minimum of three discrete vectors is required to approximate any continuous space vector in αβ0. However, more vectors are possible. It has been shown that CSPD PWM results in optimized harmonic performance for ML converters [35]. Such a modulator results in 7 states over one full period of the carrier. If there are two samples per carrier period, there are still 4 vectors to be applied per sampling period. Apart from the degree of freedom in choosing a certain type of sequence, there is also the degree of freedom in choosing redundant states within a given sequence frame. For the choice of redundant states there are many constraints and optimization criteria, which can be applied.

TABLE 53,CONSTRAINTS AND OPTIMIZATION CRITERIA FOR OPTIMAL SEQUENCE SVM