5.3 P OWER C ONTROL S TRATEGIES UNDER P HASE C URRENT L IMITATION
5.3.2 Reactive Power Control with Limitation of the Reactive Current Component
The same approach as presented for active power control limitation of the active current component can be followed for reactive power control with phase current limitation. The reactive current trajectory resulting from reactive power control with the objective to reduce the reactive power oscillations will in this case have the same orientation as the voltage or Virtual Flux trajectories, while the current trajectory in case
-20 -30 0 -10
20 10 30
0.5 0.6 0.7 0.8 0.9 1 1
1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16
p [pu]
Fault angle, []
Max current vector amplitude [pu]
Fig. 5-5. Profile of maximum current vector amplitude as function of the fault angle and the level of unbalance
5 Virtual Flux-based Power Control Strategies Operating under Current Limitation
114 NTNU 2012
of reactive power control with reduction of active power oscillations will be oriented perpendicularly to the Virtual Flux trajectory. These two cases are therefore investigated separately.
5.3.2.1 Phase current limitation with reduction of reactive power oscillations The starting point for investigating reactive power control with phase current limitation will be the expression for current reference calculation according to (4.45).
Selecting a value of kq in this equation that is between −1 and 0 will correspond to reactive power control with the objective of reducing the second harmonic oscillations in reactive power flow during unbalanced conditions. The current reference vector will then follow the same trajectory as the grid voltage, but 90° phase shifted in time, and will therefore be in phase with the Virtual Flux for the case of reactive power injection to the grid. As explained for the case of active power control, the maximum phase current will then occur in phase a, as long as the fault angle δ is in the range between
−30° and 30°. This situation will be similar to the case illustrated in Fig. 5-4, with the only difference that the reactive current component will be 90° delayed in time. The peak value of the current in phase a can then be derived, as shown in Appendix D.6.1, resulting in (5.10).
The maximum average reactive power flow that can be allowed within the phase current limitation can then be expressed as a function of the control parameter kq and the grid conditions as given by (5.11).
Introducing the maximum transferrable average reactive power back into (4.45), is then resulting in an expression for reactive current reference calculation corresponding to reactive power control with phase current limitation given by (5.12).
These expressions are only directly valid as long as δ is in the range between −30° and 30°. In the same way as explained for the case of active power control with phase current limitation, the derived equations can however easily be made valid for any other orientations of the Virtual Flux trajectory by shifting δ with a multiple of 60°, so that it will always be within the specified range.
By comparing the equations above with the results presented in the previous subsections, it can be noticed that they have the same form as the equations presented for active power control with reduction of reactive power oscillations while operating under phase current limitation. The profile of the maximum allowable current vector amplitude with respect to the fault angle δ and the positive and negative sequence
5.3 Power Control Strategies under Phase Current Limitation
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components of the Virtual Flux will therefore be identical to the result shown in Fig. 5-5. The discussion of when the maximum current vector amplitude corresponding to (5.1) will occur will also be the same.
Considering (5.11) it can however be seen that the maximum average reactive power transfer will be reduced to zero if kq = −1 and the grid is in a “single-phase” fault condition with equal amplitudes of the positive and negative sequence Virtual Flux components. This situation can be considered as a mirrored analogy to the case of active power control with the control objective of eliminating active power oscillations as discussed in section 5.3.1.1. The result of current reference calculation according to (5.12) will in such cases therefore be that maximum allowable current will be injected with a trajectory aligned with the Virtual Flux, causing zero average reactive power flow and only second harmonic active power oscillations.
Considering a case with kq = 0, it can be easily verified that the equations presented above will be valid for the case of balanced three-phase currents. The maximum allowable current vector amplitude will then be equal to the phase current limitation and the maximum average reactive power transfer will be given by the phase current limitation and the amplitude of the remaining positive sequence Virtual Flux component.
5.3.2.2 Phase current limitation with reduction of active power oscillations For values of kq in the range between 0 and 1, the reactive current trajectory will be perpendicular to the voltage and Virtual Flux trajectories. Assuming the fault angle δ to be in the range between 0° and 60°, the maximum current will therefore occur in phase b. The derivation of maximum phase current amplitude, the maximum power transfer within the phase current limitation and the corresponding expression for current reference calculation will then follow the same line as described for the case of active power control with reduction of active power oscillations in section 5.3.1.1. The current trajectory will then be similar to the case illustrated by Fig. 5-2, but the currents in each axis will be phase shifted by 90° in time with respect to the voltages. As shown in Appendix D.6.2, the maximum current occurring in phase b can be derived to be given by (5.13).
The maximum average reactive power that can be transferred within the phase current limitation can then be expressed by (5.14), and the corresponding expression for current reference calculation under phase current limitation is given by (5.15).
5 Virtual Flux-based Power Control Strategies Operating under Current Limitation
116 NTNU 2012
The equations presented above have the same form as the equations presented for active power control with reduction of active power oscillations in section 5.3.1.1 The profile of the maximum allowable current vector amplitude within the phase current limitation as a function of the fault angle δ and the positive and negative sequence Virtual Flux components will therefore be the same as plotted in Fig. 5-3. The expressions can also be made valid for any orientation of the Virtual Flux trajectory during the fault by phase shifting δ by a multiple of 60° so that it will always be within the specified range. However, since the value of kq in this case is limited between 0 and 1, average reactive power can be transferred even during single-phase faults with equal amplitudes of the positive and negative sequence Virtual Flux components.
It can also be noted that the presented equations are valid in case of kq = 0, corresponding to operation with balanced three-phase currents. As for the previously described cases, the objective of balanced three-phase currents will simplify all the equations, and the maximum allowable current vector amplitude will then always be equal to the phase current limitation.
5.4 Simulation of Power Control Strategies Operated with