Page 294 - 2024-Vol20-Issue2

290 | Abdulabbas & Salih

be written as: Assume that Lid = Liq = Li, Lod = Loq = Lo , rid = riq = ri,
rod = roq = ro ,and Ccd = Ccq = C in Eqs. (14) - (19). The
ixd 2 ? ixa ? control input is defined as sum of steady-state and perturbed
ixq = T? ixb where x reperesent i or o
3 ixc ? quantities

2 ? vya ? sd = Sd + ?sd (20)

vyd = T? vyb ? where y reperesent g or c
vyq 3 vyc

2 ? Sa ? sq = Sq + ?sq (21)

Sd = T? Sb ? , and Assuming that the inverter, grid currents, and capacitor voltage
Sq 3 Sc track their references in steady-state and all parameters of
inverter and grid inductances and capacitance in two-axis are
T= cos(?t) cos ?t - 2p cos ?t + 2p equal, then steady-state switching functions can be expressing
3 3 Eqs. (14) and (15) as following
- sin(?t) - sin - 2p - sin 2p
?t 3 ?t + 3

is Park transformation 2 Lie d (iid*) + rieiid * + vcd* - ? Lieiiq* (22)
The conversion of a differential equation from a three-axis sys- Sd = vdc dt
tem to a two-axis system can be achieved using the following
equation.

? ixa ? 2 d iiq* + rieiiq* + vcq* + ?Lieiid*
d Sq = vdc Lie dt
2 d ixd ixq (23)
ixq -ixd
3 TLx dt ? ixb ? = Lx dt - ?Lx (12)
ixc
Where Lieand rie are estimate inductance and resistance re-
Multiplying both side of Eq. (11) by Park transformation spectively and iid*,iiq*, vcd*,and vcq* are inverter currents and
and making some algebraic manipulation one can obtain the capacitor voltages, then vcd*,and vcq* can be written from Eqs.
following expression (16) and (17).

21 Sd (13) vcd* = Loe d (iod*) + roeiod* - ? Loe ioq * + vgd (24)
3 TVinv = 2 Vdc Sq dt

Now, the transformation of Eqs. (8) - (10) is given by vcq* d ioq* + roeioq* + ?Loeiod* + vgq
dt
d vdc = Loe (25)
dt 2
Lid (iid) + ridiid = ? Lid iiq + sd - vcd (14) The grid reference current can be converted from a three-phase

d iiq + riqiiq = -? Liq iid + vdc sq - vcq (15) system into two phases, so this current can be expressed as
Liq dt 2
components dq synchronous reference frame oriented with
grid voltage as following iod* = I*o, ioq* = 0 and grid voltage
is given by vgd = Vg , vgq = 0. By substituting the aforemen-
tioned relationships into Eqs. (18)-(19) and (22)-(25) to get

Lod d (iod) + rodiod = ? Lod ioq + vcd - vgd (16)
dt
vc*d = roeI*o + Vg (26)

d ioq + roqioq = -?Loqiod + vcq - vgq (17) vcq* = ?LoeIo* (27)
Loq dt

Cd d (vcd) = iid - iod + ? Cdvcq (18) iid* = Io* 1 - ?2CeLoe (28)
dt

d vcq = iiq - ioq - ?Cqvcd (19) iiq* = ?CeroeI*o + ?CeVg (29)
Cq dt