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Three-Phase Photovoltaic Systems: Structures, Topologies, and Control 79

When the forward Euler discretization method is used for both the direct path and feedback path,
an algebraic loop is created. When the backward Euler method is used, the delay on the feedback
path will be doubled; thus, the oscillator goes into instability. When SOGI is implemented in the
continuous domain for a simulation software, such as MATLAB /Simulink, and the discretization
®
is performed by the simulator, the problems mentioned previously can appear. In order to obtain a
stable implementation for SOGI the forward Euler implementation should be used for the direct
path and the backward Euler for the feedback path or vice versa. The difference in the two ways of
implementation consists of a sample delay in H (z):
d

( ω T s − ω T z ) z −1 ω 2 Tz −1
−1
2
s
Hz () = 2 2 −1 −2 ; Hz () = 1 ( 2 2 s − 1 − 2 (4.5)
+ ω T s − )
d
q
1 +( ω T s − ) 2 z + z T 2 z + z
Due to the fact that V is a feedback signal for the orthogonal signal generator (OSG) (as it can
α
be seen in (c) from Figure 4.9), in order to avoid the algebraic loop, a sample delay has to be pres-
ent. Thus, it is preferable to use forward Euler for the discretization of the direct axes and backward
Euler for the feedback axes. The perpendicularity between the alpha and beta components of the
measured grid voltage is not ensured with forward and backward Euler implementation, due to the
different delays on the direct and feedback path (half sampling period delay appears). This becomes
an issue when the PLL has to run with a low sampling rate.
Another commonly used discretization for SOGI is to use the Tustin method for both direct and
feedback path. However, in order to obtain the transfer function, this method requires more calcula-
tions. It has the advantage compared to the forward and backward Euler method, which ensures the
perpendicularity between the alpha and beta components. By substituting the Tustin discretization
from (4.4) into (4.3), the discrete transfer function becomes

2ω T s − 2ω T z −2
Hz () = s
d
2
2
−1
( ω 2 T s + ) +( 2ω 2 T s − ) 8 z +( ω 2 T s + ) 4 z −2
2
4
(4.6)
2
2
2
2
2
2
−2
−1
)
s
Hz ( ) = 2 2 ω T s + ω2 T z + ω T z 2 −2
s
− ) 8 z (
(
q
2
2
−1
2
( ω T s + ) +( ω4 2 T s + ω T s + 4) z
By modifying the transfer function and substituting the parameters of the denominator with “a”
and the parameters of the numerator with “b,” the block diagram from Figure 4.10 can be created.
As H (z) has a term without delay (b0d from Figure 4.10), in order to avoid an algebraic loop in
d
the feedback path of the OSG, a sampling delay has to be inserted (block shown with dashed lines
in Figure 4.10). As it can be observed in both transfer functions ((4.5) and (4.6)), the denomina-
tors are identical; thus, they can be implemented by using only two state variables as shown in
Figure 4.10.
The FLL block from the state point of view consists of one integrator, where forward Euler dis-
cretization is preferred to avoid the algebraic loops. It has to be noted here when FLL is used, the
parameters of the SOGI have to be recalculated at each sampling, which might need extra calcula-
tion power from the digital implementation.
The implementation of the remaining three blocks of the PLL is straightforward: the phase
detector block consists of only mathematical calculations, the LF typically is a proportional inte-
grator (PI) controller where the integrator can be discretized with any of the methods mentioned
earlier, and finally, the VCO can be implemented as a sine and a cosine function while the integra-
tor is using backward or forward Euler.