Figure 1 shows the structure of a typical photovoltaic system. The MPPT control of PV panels is achieved by a DC/DC converter, which can also maintain the voltage stability of the DC bus. The PV inverter (DC/AC converter) can achieve the active/reactive power control.
2.1. Perturbation and Observation (P&O) MPPT Algorithm
Conventionally, MPPT is embedded in a converter to determine the duty cycle that maximizes the PV power yield [
22]. A perturbation and observation method (P&O) is used to find the maximum power point [
23] by altering the array terminal voltage and then comparing the PV output power with its previous value. If the power increases while voltage increases, the PV array is operating in the correct direction; otherwise, the operational point should be adjusted to its the opposite direction [
24,
25]. The main advantage of P&O lies in its simplicity. This method shows its effectiveness, provided that solar irradiation does not change very quickly. As shown in
Figure 2, there are four operational points, A, B, C and D. From point A to point B, the PV power increases while the voltage of point B is higher than that of point A. Therefore, the next perturbation voltage keeps increasing, or vice versa. If the operation starts from point C to D, PV power decreases while the voltage of point C is higher than the voltage of point D. The next operation should be changed to the opposite direction. Therefore, the next perturbation reduces the voltage so as to redirect the trajectory towards the maximum power point. Accordingly, four scenarios are summarized in
Table 1.
A boost DC/DC converter is adopted to adjust the terminal voltage by regulating the duty ratio
D. It will therefore incur the change of the equivalent power output which in turn realizes the maximum power output of a PV array. A flowchart of the P&O method is shown in
Figure 3.
2.2. A Boost DC-DC Converter and the Equivalent Circuit
In this study, a boost converter is adopted to operate the voltage via changing
D. A boost converter is expected to connect with the output of PV arrays to produce the equivalent output voltage equal to the voltage
at the maximum power point, with equivalent resistance equal to
with the current
over the output circuit, as shown in
Figure 4. Selecting a proper DC-DC converter with reasonable circuit parameters is essential.
To design a boost converter for a PV system, key elements in the selection include the inductance of an inductor , the capacitance of an input capacitor , the input resistance , capacitance of an input capacitor , and the load resistance . The purpose of this boost converter is to realize the equivalent circuit resistance to Therefore, the maximum output power under the current condition can be produced via this boost converter from a PV array.
The maximum power point resistance is calculated based on the maximum power point voltage and the maximum power point current by using a simple calculation in Equation (1). can be calculated by Equation (2). The duty ratio can be derived from Equation (3). Assuming there is no power loss in circuit, an energy balance equation can be established (Equation (4)).
Equation (5) presents the relationship between resistance and voltage. The output voltage is calculated from Equation (5) to Equation (6).
The inductance can be estimated by many methods [
26,
27]. A general calculation is given by Equation (7).
where
and
refer to the switching frequency and the inductor current ripple factor.
Input capacitor can be calculated according to Equation (8).
Output capacitor
can be calculated according to Equation (9) [
28].
where the current ripple factor
and the voltage ripple factors
,
are refined within 5%.
2.3. Division of the Overmodulation Area
Figure 5 presents the main circuit topology of the three-phase grid-tie inverter, where
is the DC-link voltage,
,
, and
are the inverter output currents,
L is the filter inductance, and
R is the filter inductance equivalent series resistance, respectively.
In the inverter, the on-state of the upper arm switches and the off-state of the lower arm switches are defined as “1”; otherwise, they are “0”. Therefore, the three bridge arms of the inverter have eight switch states, corresponding to eight basic voltage space vectors:
(000),
(100),
(110),
(010),
(011),
(001),
(101) and
(111). The SVPWM modulation voltage vector and the sector distribution of the PV inverter are demonstrated in
Figure 6. The amplitude of
~
is
, and the phase angles of
~
differ by 60°.
According to the volt-second balance principle:
where
is the given output voltage vector,
is the action time for
,
is the action time for
, and
is the switching period.
The output space rotating vector with a constant rotating speed and a constant amplitude is achieved by the vector addition of adjacent vectors. The output range is in the inscribed circle of the regular hexagon constituted by the vectors
, which are also known as the linear modulation areas. The maximum amplitude of the output phase voltage is given by:
If the PV inverter is controlled by the six-step wave mode outside of the linear modulation area, the amplitude of the phase voltage can be obtained [
29].
The region from outside the linear modulation area to the six-step maximum output voltage area is called the overmodulation area. The modulation coefficient
m is defined as:
There are three different regions as per the modulation coefficient. In the linear modulation area,
; in the overmodulation area I,
; in the overmodulation area II,
.
Figure 7 shows the trajectory of the synthesized voltage vector in the overmodulation areas. The simplified formulas for overmodulation areas I and II are given by [
30,
31,
32,
33].
Take sector I for example. In overmodulation area I, the rotational speed of the output voltage vector remains constant and the amplitude is limited by the hexagon. The vertex of the trajectory follows the thick solid line of ABCD. In overmodulation area II, the rotation speed of the output voltage vector changes and the amplitude is limited. When , the output voltage vector is . When , the trajectory of the output voltage vector is the BC solid line. When , the output voltage vector is . When , the output voltage vector traces at the vertex of the regular hexagon, and the modulation coefficient is the maximum (m = 1). The remaining sectors are the same.
2.4. Full Modulation Region Voltage Vector
The key to controlling the voltage vector in the full modulation area is to determine the action time of the voltage vector according to the modulation coefficient m.
Take sector I for example again:
is synthesized by two basic voltage space vectors
(100) and
(110) and it is known that
,
. According to the sine theorem:
The action time and can be further calculated. When , the zero vector (000) or (111) is used to fill the remaining time .
(1) SVPWM linear modulation area ()
It can be obtained from Equation (16):
(2) Overmodulation area I ()
(i) When or , , and are calculated in the same way as Equation (17).
(ii) When ,
(3) Overmodulation area II ()
(ii) When
,
(iii) When
,