ENTRENAMIENTO VISUAL

The SMC on the input capacitor current has been tested by means of PSIM simulation, based on the scheme of Figure 3.24. The parameters of the DC/ DC converter are the same as in the previous example and are reported in Table 3.1. The parameters of the PI voltage controller, Gv(s), can be selected by

designing the settling time t∈ of the transfer function Tvp(s) in (3.73) according

to Equation (2.22). Given the relationships between the settling time t∈ of the

closed-loop voltage, the damping (ζ), and the natural frequency (ωn) of the

whole PV system, we get ( ) 2 2 2 T s as b s s vp n n = + + ζω + ω (3.74) vP V Hv iP V Gv(s) ivr MPPT Algorithm Algorithm Voltage Controller vre f

T

(s)

G

(s)

Dynamic model of the PV system based on input capacitor current SM control. (a) Double-loop representation. (b) Equivalent single-loop representation.

124 Power Electronics and Control Techniques for Maximum Energy Harvesting

The comparison of Equation (3.74) with Equation (3.73) gives the following relationships for the parameters of the transfer function Gv(s):

2 k C H p in n v = ζω (3.75) 2 k C H i in n v = ω (3.76)

From Table  3.1, the nominal switching period is Tsw 1/fsw0=8.3 .µs Choosing tε/Tsw=20, which means imposing a system settling time 20 times greater than the nominal switching period, provides tε=166µs. The second design consideration is related to the damping of the system, which can be set- tled at ζ = 0.7.

According to the definition of the equivalent time constant, which is τ =ζω1n, the settling time of the closed-loop system can be approximated by tε = τ4 when it is assumed that the settling error step response is smaller than 2%. Finally, if the gain of the voltage sensor is Hv = 0.1, then the PI controller trans-

fer function ensuring the desired t and ε ζ becomes the following:

( ) 24· 24625

G s s

s

v = + (3.77)

In order to verify the stable behavior of the SM on the input capacitor current, the circuit of Figure 3.24 has been tested in the same irradiance condition of the circuit in Figure 3.17 and with the same MPPT parameters, although in this case the P&O algorithm is acting on the voltage reference and not directly on the current reference.

In Figure  3.27 the PV voltage and PV current are shown; an irradiance change from 1000 W/m2 _{to 900 W/m}2 _{occurs at the time instant t = 0.3 s.}

The sudden irradiance variations, having an instantaneous effect on the PV short-circuit current, have been correctly tracked, with the system perma- nently tracking the maximum power point even at the very high rate of irra- diance variation that has been considered. Indeed, comparing the voltage behavior with the simulation proposed in Figure 3.22, now no sensible varia- tion in the PV voltage is visible and the system does not crash. Moreover, it is worth noting that also in this case, a bulk voltage oscillation of 127 V has been considered in the scheme of Figure 3.24; the simulation results put into evidence that the large low-frequency oscillations affecting the bulk volt- age are almost totally rejected at the PV terminals. Moreover, the upper and lower limits of the irradiance variations dG

dt have been calculated by using

Equation (3.68) for the numerical example considered in this section. In the considered case, if _K =0.008 A m /W,⋅ 2

125

MPPT Efficiency: Noise Sources and Methods for Reducing Their Effects

−58.75 W < < m ms dG dt 56.26 W m ms 2 2 (3.78)

In Figure 3.28 an irradiance change from 1000 to 500 W/m2 _{with a slope of}

−50_{m s}W2_µ has been applied at t = 0.3 s; clearly the system maintains an optimal

stable behavior.

Figure 3.29 shows the magnification of the PV voltage in time interval (0.255 s, 0.325 s) in comparison with the bulk voltage. The simulation shows that the PV voltage waveform is free of the 100 Hz oscillations caused by the inverter operation, thus demonstrating that the sliding mode applied to the current of the input capacitor has been able to reject the back prop- agation of the large oscillations of amplitude Δvb affecting the converter

output voltage.

Finally, it is worth noting that in the simulation of Figures 3.27 and 3.28 a nonoptimal TpMPPT parameter has been used; this is because the intent

was to verify the stability of the SMC and not to test the MPPT dynamic per- formances. Anyway, the t specification also allows us to define the optimal ε

MPPT perturbation period Tpof the P&O algorithms, that is, approximately

⋅ ε

1.5 t . Such a value of Tpensures that the PV power has reached its steady state

when the MPPT controller measures it, thus avoiding the MPPT deception as explained in Section 2.4.1 and in [2, 32]. Thus the case shown in Figure 3.28 has been resimulated by using Tp= 250 μs.

Figure 3.30 puts into evidence the proper design of the P&O parameters leading to a three-point behavior of the PV voltage with an MPPT speed that is about one order of magnitude higher than the case shown in Figure 3.28.

vpv = vref iPV 8 7 6 5 4 0.05 0.1 0.15 0.2 0.25 Time [s] Cu rr ent [A] 0.3 0.35 0.4 0.45 0.5 210 220 Voltage [V ] 230 240 250 Figure 3.27

Simulation of the system of Figure 3.24 using the sliding mode and Gvcontrollers. MPPT

126 Power Electronics and Control Techniques for Maximum Energy Harvesting

The same figure shows the waveform of the closed-loop voltage reference

vref and the accurate tracking performed by both Gv(s) and the sliding mode

controller.

The features listed above make the control strategy shown in Figure 3.24 suitable for all PV applications in which the adoption of electrolytic capaci- tors at the DC bus must be avoided or wherever an excellent MPPT perfor- mance is required.

Moreover, the fast MPPT capability makes the proposed architecture suitable for a large range of applications for which the sudden irradiation changes are very common, e.g., in sustainable mobility and for the PV inte- gration on cars, trucks, buses, ships, and so on.

3.4 Analysis of the MPPT Performances