Control and Design of Microgrid Components - Power Systems ...

linearity. A value of 30 degrees provides a 4% error in the linear approximation, but a maximum
power angle value of 10 degrees implies that the largest error in the approximation would be
0.5%. A smaller range for the power angle would require too much precision in the
synthesization of the angle at the inverter terminals. The inverter synthesizes the voltage angle
with a certain tolerance depending on the switching frequency. The higher is the operating
frequency the lower is this tolerance. If a large range for the power range is chosen, then the
relative size of the error is small and there are no problems in injecting power. If a small range is
chosen, then the same absolute value of the tolerance becomes relatively large compared to the
range of delta, implying problems in achieving constant power injection. The power would
oscillate, and if the range for the angle is chosen really small, say a fraction of a degree, then it
would be nearly impossible to regulate power due to the fact that the angle would need to be held
with a very high precision.
4.3.1 P versus Q Area of Operation
When designing the inverter, the choice of ratings of the devices plays a key role. The power
electronic block at the minimum has to be rated as the prime mover. If the ratings of the prime
mover exceed the inverter ratings then when the prime mover operates at peak power it would be
impossible to transfer its full power. Each inverter has limits dictated by the ratings on the silicon
devices. These limits appear in the form of maximum voltage that the power electronic device
can safely sustain and the maximum current that it can carry. Based on these considerations, each
designer provides a range of active and reactive powers that the inverter can safely operate at. At
this point there is the first formulation of the answer for the problem of the size of the inductor:
The size of the inductor must be such that it enables delivery of the active and reactive power
that the inverter can provide.
Figure 4.7 shows the plot of the active and reactive power obtained from the formulas included
in Figure 4.5 as a function of the inverter and bus voltage, their angle difference and the
impedance. All quantities are in per unit to maintain generality and the plot is obtained for the
power angle equal to 15 degrees. Smaller inductances allow larger power flows. The inductor
should be chosen small enough to guaranteed deliver of the specified amounts of power: if the
inductor is chosen too large the power may no longer be deliverable. The optimal inductor size
can be seen as the maximum size that the inductor can have, and still being able to satisfy the
delivery requirements.
To develop some intuition on how to solve the problem it is useful to build some maps of the
power that it is possible to deliver given the limits that need to be satisfied.
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