To compare these algorithms, simulations were performed using Matlab/Simulink. Since the core of the problem deals with the grid-side converter and the performance between the algorithms presented in
Section 3, the generator-side was reduced to a perfect generator. The grid side conversion system was modeled more finely using the parameters reported in
Table 1, which are consistent with the experimental bench described in
Section 5. For different DC-link capacity values ranging from
F to
F, the performance of the different algorithms was compared based on the quality of the DC-link voltage control and of the current injected on the grid.
The DC-link capacity is the key element in the grid side converter: the higher this is, the better the converter will support the power flow. Another possibility to test the different algorithms would have been to set the value of the DC-link capacity and to perform scenarios where the power injected varied over a large range. However, under these conditions, the defects of the other elements in the system would have had a significant influence, in particular the inverter whose losses and voltage drops are related to the flowing currents. This is why we preferred to carry out the experiments with scenarios where the order of magnitude of the powers remained the same and where the performances between algorithms were evaluated for different values of the DC-link capacity.
Two different types of scenarios were performed. In the first scenario, a step of the power supplied by the system on the generator side was considered as well as a step of the reactive power required. In the second scenario, a more realistic configuration was tested where the variability of the power injected by a WECS into the grid was modeled.
4.1. Results with Step Power
In this scenario, no power was injected at the beginning; then, suddenly, 900 W was injected for 2 s; and later, 400 W was injected and, at the same time, a reactive power
of 500 VAR was required for 2 s.
Figure 6 shows the simulation results for the three algorithms described in
Section 3 with a value for the DC-link capacity of
F.
As shown in
Table 1, the DC-link reference voltage was set to
V.
Figure 6a shows that, at the moment where power was injected, the real voltage
presented a jump that was quickly attenuated, which was expected for the designed linear controller in the presence of a perturbation step. After this transient effect, it can be seen that the voltage
was maintained around 400 V with a kind of noise around the reference value. In order to quantify the two phenomena described above, two indicators
and
were introduced. They measured the error on the DC-link voltage using the maximum norm and L2-norm:
where two different periods of time
T were considered: the first one when only active power is injected (
and
) and the second one when both active power is injected and reactive power is required (
and
).
The values of
and
for a given scenario depend on the choice of the algorithm parameters. In order to make a comparative study of the control strategies presented in
Section 3, we sought to adjust the parameters of the linear controller and the nonlinear controllers SMC1 and SMC2 so that the errors
and
were approximately the same for a capacity value of
F.
Concerning the linear control, we define the parameter
related the time response of the closed-loop in
Figure 3;
is also chosen for the time response of the transfer function
. The value of
is required to be equal or greater than
since
is related to the time response of the inner loops for the line currents control (see
Figure 2). Then, the value for the parameters of the PI controller with active damping are computed as following:
where
ms in all experiments.
Concerning the SMC1 controller, the parameter
is related to the inverse of the time constant in the exponential decrease of error
e when the solution remains on the sliding surface
(see Equation (
15) with
). The parameter
must meet the condition
. In order to obtain similar values of
and
as with the linear control when
F, the parameters of the SMC1 controller were adjusted as follows:
where
. In addition,
was set to
because a better control was observed.
Concerning the SMC2 controller, the value of
was computed by Equation (
25) with the maximum relative error on the DC link voltage
(which results in a maximum overshoot of 5 V) and the maximum current produced by the WECS
A (which results in a maximum injected power of
W when
V). In order to obtain similar values of
and
as with the linear control when
F in one hand and to satisfy the required conditions set in Equation (
26) on the other hand, the parameters of the SMC2 controller were adjusted as follows:
where
For each value of the DC-link capacity
C, the scenario of power injection illustrated in
Figure 6 was carried out. The parameters of the different controllers were computed with respect of the value of the DC-link capacity according to Equation (
29) for the linear controller, Equation (
30) for SMC1 and Equation (
31) for SMC2. We postulate that this approach is suitable to compare the performances of the different algorithms for the control of the DC-link voltage. Indeed, the other elements of the conversion chain, the inverter and the choke coil, remain the same and are subjected to currents and voltages of the same order of magnitude in the different simulations and experiments.
Table 2 illustrates a comparison between the
,
and the THD line current indicators obtained by the three algorithms in the case of step power: where only active power is injected (
W and
) and where the active power is injected and the reactive power is required (
W and
VAR).
In the case where only the active power is injected, we can see that and for linear control increased when the value of the capacity decreased, reaching V for and V for when F. In the opposite case, for non-linear controllers (SMC1 and SMC2) the dynamic depended less on the value of the capacity: in the case of SMC1, we observed that there was no great effect when the value of the capacity decreased, with a maximum error V and V when F. This remained almost the same for the case of SMC2, with a maximum error V and V when F.
In addition, we observed that all the strategies had tolerable THD values for injection; however, the linear control proved its efficiency with a THD of around . In contrast, the THD rates for SMC1 and SMC2 remained around and , respectively.
In the case where reactive power was requested ( W and VAR), we observed that the controllers reacted in the same way as in the case where only active power was injected, except that, in this scenario, the errors and were reduced, as the injected power was only 400 W.
We also noticed that the THD rate for the current di not have a significant change for all strategies. However, the simulation results generally showed that the reactive power control was performed independently and had no effect on the control of the voltage and the quality of the injected current.
4.2. Results with Variable Power
In order to simulate the variability of collected wind power, we tested the algorithms with a sinusoidal wind model [
44] where the time evolution of the wind speed
was given by:
where the value of the coefficients
and
are reported in
Table 3.
The mechanical power
supplied by the wind turbine to the electrical generator can be expressed as [
44]:
where
is the power coefficient of the turbine,
A is the area of the turbine, and
is the air density. In our study, we assumed that the WECS extracts the maximum mechanical power; in other words, that
is maintained at its maximum value. The term
is, thus, a constant that has been adjusted in order to model the maximum mechanical power of
kW. In addition, we assumed that the electrical generator is ideal, which implies that
. Thus, the current
that the DC source must provide to the grid-side converter in our simulation and experimental bench was calculated from the value
deduced from the sinusoidal wind model and the DC-link voltage value.
Figure 7 shows the simulation results for the three algorithms with a value for the DC link capacity of
F. As in the first scenario, we observe in
Table 4 that SMC1 and SMC2 had good control on the
voltage since SMC1 and SMC2 generated an error
that changed between
V and
V when the value of the capacity decreased, and
remained relatively low and varied between
V and
V. The linear control generated exponentially increasing errors with the decrease of the capacity with an error
of
V and
of
V when
F.
For the case of THD, we notice that, as in the first scenario, the linear control and SMC2 had good performances in terms of the quality of the current injected into the power grid with a THD that did not exceed a maximum of when F. However, this was not the case for SMC1, which suffered from a relatively higher THD, which was around due to its control nature (the chattering phenomenon).