Page 296 - 2024-Vol20-Issue2

292 | Abdulabbas & Salih

?sq = KqVdcx2 (37) vcd + X5 +- ?sd + sd
vcd* - Kcd +
-
iiiidd* + X1 +
-
Kd * Vdc -
+
Where Kd < 0 and Kq < 0 are constant. 15 + Sd
While Eqs. (36)-(37) can ensure global stability, if there is I?*? ??2CeLoe * rie - +
a parameter mismatch, it is advisable to choose Kd and Kq + + 2
to be as large as feasible in order to have dominance over 30 ??????
the last term in Eq. (34), ensuring that the criterion for V? (x) I?*? rie +- ?sq
to be less than zero is guaranteed. The level of resonance
damping is unsatisfactory, in order to address the issue of ??2CeLie * roe dq
resonant damping, the authors in [19] suggested incorporating
the capacitor voltage feedback into the perturbed switching q PWM
function in the following manner.
abc

roe

??2CeLie

?sd = KdVdcx1 - Kcdx5 (38) Vg X6 + sq
vcq + Kcq +
vcq* -
X2
iiq +
iiq* - Kq * Vdc

?sd = KqVdcx2 - Kcqx6 (39) rie?Ceroe +
+
Where Kcd > 0 and Kcq > 0 are constant. Now, substituting ?Loe + 2 Sq
equations (38)-(39) into (35) and assume Kd = Kq and Kcd = ??3CeLie * Loe + +
Kcq gives ?Lie + + ??????

V? (x) = 3 Vdc2 Kd x12 + x22 - 3 VdcKcd (x1x5 + x2x6) - Vg ?Cerie
2 2

3ri x12 + x22 - 3ro x32 + x42 < 0 Fig. 8. Control using a Lyapunov-based function.

(40)

Then, considering that r1 and r2 contribute to an extra passive It is important to mention that the control laws in Eqs. (42)-
damping effect, the stability of the Lyapunov function would (43) can accomplish the intended control objectives if the dc
be at its worst if r1 and r2 were not there. link voltage of the qZS network is regulated to the correct
level as shown in Fig. 8 (which should be higher than the peak
Kcd = Kcq > VdcKd x12 + x22 (41) voltage at the inverter output) and the reference function (I*o)
(x1x5 + x2x6) is accurately produced.

Therefore, the total switching function, which includes both 6) Control of qZSI network by shoot-through method
steady-state switching function and perturbed switching func- The regulator of shoot-through duty ratio (DST) is basically
tion, may be expressed from Eqs. (30)-(31) and Eqs. (38)-(39) in performance wanted operation of the qZSI network, to
yield realize this, the inductance current and capacitance-voltage
can be controlled in a circuit. Proportional-Resonant (PR)
2 Io* rie 1 - ?2CeLoe + roe 1 - ?2CeLie + control can be utilized for this purpose. The fact that a PR
sd = Vdc controller demonstrates excellent performance in accurately
following a reference sinusoidal signal is widely recognized.
Vg 1 - ?2CeLie + KdVdcx1 - Kcdx5 The transfer function of an ideal (PR) controller is shown in
equation below.
(42)

2 GPR(s) = Kp + 2 Krs (44)
sq = Vdc s2 + ?2
Io* rie?Ceroe + ?Loe + ?Lie 1 - ?2CeLoe +

?CerieVg + KqVdcX2 - KcqX6 The variables Kp and Kr represent the proportional and reso-
nant gains, respectively. The symbol ? represents the resonant
(43) frequency. If the frequency of the reference grid current is cho-