Page 295 - 2024-Vol20-Issue2

291 | Abdulabbas & Salih

roe - ro = r˜o,and Ce - C = C˜ yield

2 I*o rie 1 - ?2CeLoe + roe 1 - ?2CeLie + Li x? 1 = -rix1 + ? Li x2 - x5 + Vdc ?sd + L˜ i d (iid*) + r˜iiid*-
Sd = Vdc 2 dt

Vg 1 - ?2CeLie ?L˜ iiiq*

(30) Li x? 2 = -? Li x1 - rix2 - + Vdc + L˜ i d iiq* + r˜iiiq*+
2 dt
x6 ?sq

?L˜ iiid*

2 Io* rie?Ceroe + ?Loe + ?Lie 1 - ?2CeLoe + Lo x? 3 = -rox3 + ? Lo x4 + x5 + L˜ o d (iod*) + r˜oiod* - ?L˜ oioq
Sq = Vdc dt

? Ce rie Vg Lo x? 4 = -? Lo x3 - rox4 + x6 + L˜ o d ioq* + ˜roioq* + ?L˜ oiod*
dt
(31)

Cx?5 = x1 - x3 + ?Cx6

Equations (30) and (31) represented the switching function Cx?6 = x2 - x4 - ?Cx5
in steady-state then to find the perturbation state must use the
Lyapunov function. The state variables are clearly defined (32)
as x1 = iid - iid* , x2 = iiq - iiq*,x3 = iod - iod* = iod - I*o,
x4 = ioq - ioq* = ioq , x5 = vcd - vcd*, x6 = vcq - vcq*. A Lyapunov function with energy-like properties can be con-
structed for this system in the following manner.
5) Lyapunov-function based control
The dynamics of a system near its equilibrium point are often V(x) = ?ELi + ?ELo + ?EC = 3 Lix12 + 3 Lix22+
determined using Lyapunov’s direct technique. The equilib- 2 2
rium points of a three-phase grid-tied LCL-filtered qZSI is at (33)
(x1 = 0 ,x2 = 0, x3 = 0, x4 = 0, x5 = 0, x6 = 0). This study 3 3 3 3
aims to establish a control approach that guarantees the over- 2 Lox32 + 2 Lox42 + 2 Cx5 2 + 2 Cx6 2
all asymptotic stability of a grid-tied qZSI at its equilibrium
point. The state variables converge to the equilibrium point as It is evident that the first three features described above are met
the total energy of qZSI is continuously dissipated. The qZSI in Eq. (33). To fulfil the fourth requirement, the function V? (x)
receives its input energy (Ein) from the output voltage (Vdc)
of the qZS network. The energy expended by the components needs to be determined. Now, differentiating the function v(x)
ri, ro, R, and transistors of the qZSI system results in a loss of
a portion of Ein. Nevertheless, the excess energy is conveyed and substituting Eq. (32) yield
to the grid, denoted as Eg.
Capacitors and inductors can store energy instead of releasing V? (x) = 3 Vdc?sd + 3 Vdc?sq - 3ri x12 + x22 - 3ro x32+
it. Thus, the energy contained within the qZSI is allocated 2 x1 2 x2
among the components of the LCL filter, namely the inductor
(Li), capacitor (C), and output filter (Lo). This implies that a x42 + 3L˜ ix1 d (iid*) - ? iiq* + 3L˜ ix2 d iiq* + ?iid* +
portion of Ein is transferred through the interchange of energy dt dt
stored in the components (?ELi, ?ELo, and ?EC) in a bidirec-
tional manner until the overall energy dissipation approaches 3L˜ ox3 d (iod*) - ? ioq* + 3L˜ ox4 d ioq* + ?iod* +
the equilibrium point of the qZSI. dt dt
The control strategy ensuring the overall stability of the grid-
tied qZSI is determined by employing Lyapunov’s direct tech- 3r˜i x1iid* + x2iiq* + 3r˜o x3iod* + x4ioq*
nique. Lyapunov’s direct technique uses a scalar energy-like
function known as the Lyapunov function, represented by V (34)
(x). According to this approach, a system can be ensured to
be asymptotically stable globally if the Lyapunov function Assuming perfect matching between actual and estimation
fulfils several requirements such as V(x) = 0, V(x) > 0 at all parameters the Eq. (34) can be modified by following
x =? 0, V(x) ? 0 as |x| ? 8 ,V? (x) < 0 at all x =? 0. Substi-
tuting the state variables and Eqs. (20)-(25) into Eqs. (14)- V? (x) = 3 Vdc?sd + 3 Vdc?sq - 3ri x12 + x22 - (35)
(19) and assume Lie - Li = L˜ i , Loe - Lo = L˜ o, rie - ri = r˜i, 2 x1 2 x2

3ro x32 + x42

It is apparent that the fourth feature, which is the negative
definiteness of V(x), would always be ensured if ?sd and ?sq
were selected as

?sd = KdVdcx1 (36)