Harmonic Droop Controller to Improve Voltage Quality - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 335

I 1 ∠ − φ 1 at the fundamental frequency 1
is expressed as the combination of the voltage source V o 1 ∠

V o 1

is shown in Figure 21.1(b). The load of the circuit

0 ◦

and the current source I 1 ∠ φ 1 .

If 1) is close to zero, then what is left on the right-hand side of Figure 21.1(b) is a

current source i h .
The main function of an inverter (or a generator) is to supply the load with the real
power and reactive power at the right voltage level and at the right frequency, which are
regulated by industrial standards and/or law. To be more precise, this should be done at the
fundamental frequency , without harmonics. When multiple inverters are connected in parallel,
they should also share real power and reactive power in proportion to their capacity, again
at the fundamental frequency . Then, what happens with harmonics? Ideally, the harmonics
in the output voltage are expected to be 0, i.e.,

v oh ( h =

), even when there are
harmonic s in the current i . This can be achieved when the voltage drop of the h -th harmonic

v oh = 0( h = 2 , 3 , ···

current √ 2 I h sin( h ω ∗ t + φ h ) on the ou tp ut impedance Z o ( j ω ) is the same as the h -th harmonic

component of the voltage reference √ 2 E h sin( h

ω ∗ t + δ h ), i.e., when

ω ∗ ) ω ∗ )

E h = I h | Z o ( jh | and δ h = φ h + ∠ Z o ( jh . (21.1)

In this chapter, this idea is exploited to design a controller for the inverter so that the harmonics
in the output voltage are considerably reduced, after filling a gap in the theory of power delivery
through an impedance.

21.2 Power Delivered to a Current Source
As discussed in Chapter 19, it is well understood how real power and reactive power are
delivered through an impedance, whether it is inductive, resistive, capacitive or other types,
when the terminal voltage is more or less maintained constant as a voltage source. Moreover,
power sharing schemes, e.g., different droop control strategies (Guerrero et al . 2005, 2007,
2011; Mohamed and El-Saadany 2008a; Yao et al . 2011; Zhong 2012c; Zhong et al . 2011),
have been developed for this case. Here, how real power and reactive power are delivered
when the terminal is connected to a current source with a constant current is studied.
Figure 21.2 illustrates a voltage source v r delivering power to a current source I 0 ◦ through

∠

an impedance Z o ∠ θ . The terminal voltage is

V o =

E

∠ δ −

Z o I

∠ θ

= E cos δ − Z o I cos θ + j ( E sin δ − Z o I sin θ )

and the real power and reactive power delivered to the terminal are, respectively,

Z o I 2 cos P = EI cos δ − θ, (21.2)

Z o I 2 sin Q = EI sin δ − θ. (21.3)

1 Note that the phasors in this chapter may be at different frequencies, which should be clear from the context.