1. Introduction
To fulfill the constantly increasing worldwide energy demand, renewable energy sources such as photovoltaic (PV), wind, geothermal, and biomass, are being explored. Currently, PV energy is becoming one of the most widely used renewable energy sources, due to its indubitable known advantages and the decreasing installation costs.
In order to improve the PV conversion efficiency, maximum power point tracking (MPPT) algorithms are widely adopted in both grid-connected and stand-alone PV systems. Among the different methods known in the literature, the most popular and effective are perturb and observe (P&O) [
1] and incremental conductance [
2]. Typical problems for these methods are the identification of a suitable perturbation step size, and the limited maximum power point (MPP) tracking dynamic capability during sudden variations in solar irradiance. In this class of P&O algorithms, either fixed or variable step size are adopted to improve the settling time in transient conditions and the MPP resolution in the steady-state operating points [
3].
Similar to P&O techniques, the extremum seeking (ES) algorithm employs “ad hoc” perturba-tions [
4]. A multi-variable extremum seeking algorithm based on a single control loop for cascaded dc/dc photovoltaic micro-converters has been proposed in [
5]. A Newton-based ES algorithm has been adopted to improve the dynamic performances of gradient-based ES in [
6]. All aforementioned algorithms implemented the ES control by injecting an external perturbation signal into the duty cycle. On the contrary, ripple correlation control (RCC) algorithms exploit the inherent PV voltage and current oscillations to track the MPP. In particular, for single-phase PV systems the 2
nd harmonics are exploited [
4,
7,
8,
9,
10,
11,
12,
13]. The RCC algorithm has good dynamic performance comparing with the P&O algorithms and it does not require additional perturbations in tracking the MPP. A comprehensive analysis and comparison between RCC and ES methods has been carried out in reference [
9] in case MPPT algorithms for PV systems.
The basic RCC-MPPT method has been investigated in references [
9,
12,
14] where two low-pass filters and two high-pass filters are used for ripple extraction and implementation. In reference [
10], a modified RCC method has been proposed by using the moving average concept instead of high/low-pass filtering to improve the dynamic response, without the need to tune the time constant of the low/high pass filters. Moreover, the scheme has been simplified by using only the sign of the product of power and voltage ripple to drive the PV operating point toward the MPP. In reference [
13], a hybrid RCC-MPPT algorithm has been proposed to improve stability during sudden solar irradiance transients. In case of three-phase two-level (2L) inverters, the RCC algorithm cannot be applied due to the inexistence of inherent (natural) low-order harmonic oscillations on the PV side of the inverter. In this case, the ES algorithm can be successfully applied by means of additional perturbations, as reported in reference [
15].
In recent years, multilevel inverters have become more attractive for single- and three-phase systems thanks to their advantages over the conventional inverters [
7,
15,
16,
17,
18,
19]. They offer improved output waveforms, lower total harmonic distortion (THD) and smaller grid filter size [
20,
21,
22]. The most common multilevel converter configurations, presented in literature, are the cascaded H-bridge (CHB), neutral-point-clamped (NPC) and flying capacitor (FC). These types of converters are also adopted in PV applications due to the aforementioned advantages. Multilevel flying capacitor inverters have some specific distinct advantages over the other two topologies such as: no need for many isolated dc sources when compared to CHB inverters, no need for clamping diodes in contrast to NPC inverters, but still preserving the ability to self-balance the capacitor voltages.
With reference to the FC inverter, several analyses have been presented in the literature. A mathematical model for the dynamic behavior of a simple FC inverter using a sampled-data modeling approach has been derived in reference [
23], with useful information on the power circuit characteristics and its natural balancing property. In reference [
24], a modified pulse with modulation (PWM) strategy is introduced to improve the balancing rate of capacitor voltages, mainly for small output voltages, by optimizing the use of redundancy of switching states. A new PWM scheme has been proposed and analyzed in reference [
25], which results in better balancing properties than the normal phase-shifted (PS) PWM. In particular, a five-level configuration has been considered and a modified PS-PWM scheme has been studied, solving the slow-balancing problems of the normal PS-PWM method for odd-level of FC converters.
In general, adopting a multilevel inverter introduces different low-order voltage and current harmonics on the dc-link side compared to the case of a single-phase H-bridge inverter where only the 2
nd order harmonics are present. The application of the RCC algorithm in case of a single-phase inverter with level doubling network (LDN) has been introduced and examined in reference [
10]. Due to the multiple harmonics, the basic implementation of the RCC-MPPT scheme becomes deficient, leading to a misestimating of the voltage derivative of the power (
dPpv/
dVpv). For this reason, a modified RCC scheme extracting the amplitude of a definite harmonic form PV voltage and current (dc-link) waveforms has been proposed. In order to maximize the resolution, the proposed solution makes reference to the highest amplitude harmonic, leading to a more effective estimation of
dPpv/
dVpv.
Though numerous RCC-MPPT algorithms for single-phase grid-connected PV systems have been developed, no analysis of RCC-MPPT in case of three-phase multilevel inverters has been reported. This paper presents an RCC-MPPT algorithm for a grid-connected three-phase three-level FC inverter (
Figure 1) with reference to centered level-shifted (LS) carrier-based PWM modulation. Moreover, a complete analysis of PV voltage and PV current harmonic components is accomplished, being the foundation of the ripple correlation control. In this case, dc-link voltage and current harmonics are introduced by the instantaneous power oscillations between the three flying capacitors and the dc bus, consisting mainly of 3
rd harmonic components. These voltage and current harmonics are exploited as embedded perturbations to determine the MPP of the PV array. The considered grid-connected three-phase PV generation system is presented in the block diagram of
Figure 1.
2. Modulation Principle for the Three-Level FC Inverter
With reference to linear and balanced sinusoidal modulation, the modulating signals (
uA,
uB, and
uC) correspond to the inverter output voltages (
vA,
vB, and
vC) normalized by
Vpv and averaged over the switching period (
Tsw = 1/
fsw):
where ϑ = ωt is the phase angle, ω is the fundamental (angular) frequency,
m is the modulation index,
are the reference (normalized) output voltages (
i = A, B or C), and
Cm is the common-mode signal to maximize the linear modulation range.
The voltages across the flying capacitors
CA,
CB and
CC are spontaneously regulated to the half of the DC-link voltage if a proper modulation technique with self-balancing capability is adopted [
19]. Introducing for the upper switch of each phase
i (
i = A, B, C) the switching function
(averaging is denoted in the following by overline) results in:
With reference to phase A, considering Equations (1) and (2), it becomes:
The switching functions for the other phases B and C are readily obtained by exploiting the modulation symmetry among the three phases, according to Equation (1).
In order to improve the so-called sinusoidal PWM (SPWM), in which there is no common-mode signal injection,
Cm = 0, one of the most popular ways to maximize the modulation index is the so called “centered” PWM (CPWM), consisting of the injection of a common-mode signal able to center the modulating signals:
In case of sinusoidal reference voltages, Equation (1),
Cm can be rewritten as:
Figure 2 shows an example of the considered carrier-based CPWM strategy for three different modulation indices.
4. Proposed RCC-MPPT Algorithm
Figure 6 presents the block diagram of the adopted control scheme, including MPPT and dc-link (PV) voltage regulation. In particular, the proposed RCC algorithm estimates the voltage derivative of the power,
dPpv/
dVpv, used to drive the working point to the maximum power point. Integrating
dPpv/
dVpv, the reference dc-link voltage
vpv* is simply obtained and a proportional-integral (PI) voltage regulator can be used to determine the reference grid current amplitude
Iac*. In order to inject a sinusoidal current into the grid with unity power factor, a traditional
dq current controller has been used.
An example of PV voltage and current ripple in case of a three-phase FC inverter configuration is given in the following.
Figure 7 shows the results considering the PV conversion scheme for a given operating point. As expected, the inverter voltage has nine levels for a higher modulation index. Also, it is noticeably clear that PV voltage and current contain a huge 3
rd harmonic component (150 Hz) and are in phase opposition.
In case of three-phase PV systems, with the inverter being directly connected to the PV array, the FC inherent 3
rd (and odd multiple) order harmonic appears in PV current and PV voltage (as shown in
Figure 7). As is known, ripple correlation control algorithm is able to exploit the amplitude of these oscillations to provide information about the operating point of the PV array. Considering the working point
Q, the voltage derivative of the PV power
dPpv/
dVpv can be s written as:
where
dIpv/
dVpv is defining the voltage derivative of the current as:
A conventional RCC-MPPT algorithm is implemented by multiplying the voltage oscillation by the current oscillation, and integrating over the harmonic period [
10]. As an alternative implementation, the estimation of
dIpv/
dVpv (and then
dPpv/
dVpv) can be carried out by considering a specific harmonic order, which is the 3
rd in the case of the FC inverter. Equation (23) can be then rewritten as:
By following the approach of the conventional RCC-MPPT implemented in [
10], the estimation of the voltage derivative of the current
dIpv/
dVpv can be calculated, considering the third harmonic component as:
It should be noted that
dIpv/
dVpv derived by Equation (25) is calculated by integrating the product of the PV voltage and current over the fundamental period
T, as defined by Fourier series (FS) to calculate the
3rd harmonic. The block diagram of
Figure 8 shows a possible implementation of Equation (25), with the estimation of
dPpv/
dVpv considering only the 3
rd harmonic component.
5. Simulation Results
In order to validate the proposed analysis, the three-phase grid-connected PV generation system shown in
Figure 1 was simulated in Matlab-Simulink. The circuit parameters are summarized in
Table 1. A string of series/parallel connected PV modules has been adopted as PV source, according to the data given in
Table 2 in standard test conditions (STC). In particular, the considered PV modules are SP-305 type (96 cells, monocrystalline).
The first simulation test is carried out in order to verify the input/output steady-state waveforms of the PV generation system with considered three-phase three-level FC inverter.
In particular,
Figure 9 shows grid voltage and current (top traces), PV voltage (medium trace) and PV current (bottom trace). It can be noticed that the grid current is almost sinusoidal, with unity power factor. As expected, the PV voltage and PV current have 3
rd harmonic component oscillations (150 Hz), in phase opposition. The proposed RCC-MPPT algorithm can well track the maximum power point at sun irradiance
E = 1000 W/m
2 (in this case
Vmpp = 875 V and
Impp = 123 A).
The following simulation tests are carried out to verify the dynamic performance of the proposed RCC-MPPT algorithm in case of fast solar irradiance transients. In particular, two scenarios have been considered:
The cell temperature has been considered constant (25 °C) during simulations for both scenarios.
The simulation results depicted in
Figure 10 and
Figure 11 show the behavior of the considered system during the first scenario. A linear ramp increase of solar irradiance from 500 W/m
2 to 1000 W/m
2 is applied in a transient period of 200 ms, between 1.3 s and 1.5 s. The results clearly indicate that as soon as the transient occurs, the estimation of
dPpv/
dVpv is oscillating. However, it is almost acceptable during the steady-state (
Figure 10b), which means that the PV current and the PV voltage perfectly follow the MPP under this kind of smooth irradiance change (
Figure 10c,d).
Figure 10e presents the waveform of the injected grid current. The injected grid current is perfectly sinusoidal and well controlled thanks to the dynamic performance of the
dq current controller.
Figure 11 shows the
Ppv(
Vpv) and
Ipv(
Vpv) diagrams corresponding to the transient depicted in
Figure 10. The operating point moves from the first MPP, before the solar irradiance transient, to the new MPP, in the steady-state condition after the transient.
Simulation results for ramp decrease in solar irradiance are presented in
Figure 12 and
Figure 13. In this case, the initial value of solar irradiance is 1000 W/m
2, decreasing linearly to 500 W/m
2 in a period of 200 ms. As can be seen, the estimation of
dPpv/
dVpv is correct during the steady state (
Figure 12b). At the end of the transient, the estimation of
dPpv/
dVpv becomes correct again, and the operating point is correctly driven toward the MPP. The current injected into the grid is perfectly sinusoidal and follows the sun irradiance profile (
Figure 12e). The path of the operating points is displayed on the
Ppv(
Vpv) and
Ipv(
Vpv) diagrams and presented in
Figure 13.
6. Conclusions
In this paper, an original RCC-MPPT algorithm suitable for three-phase three-level flying capacitor photovoltaic systems has been proposed and analyzed in detail. The three-phase FC inverter introduces voltage and current harmonics on the input (i.e., PV) side. In particular, a significant 3rd order harmonic is clearly noticeable, whereas higher harmonic components have much lower amplitudes. Both PV voltage and current harmonic amplitudes are analytically calculated in the whole modulation range in case of centered PWM, offering the possibility of a precise and effective design of the DC-link capacitor in order to obtain the desired ripple amplitude.
Because of the presence of different harmonics, a ripple correlation control scheme extracting the amplitude of a specific harmonic from PV voltage and current waveforms has been proposed in order to track the maximum power point of the PV arrays. Specifically, the estimation of dIpv/dVpv (and then dPpv/dVpv) is carried out by considering the 3rd order harmonic of PV voltage and current oscillations.
Numerical tests have been performed to prove the effectiveness of the whole PV generation scheme, including three-phase three-level FC inverter and the proposed RCC-MPPT algorithm. Tests have been carried out considering both steady-state conditions and fast solar irradiance transients in order to verify the dynamic performance of the proposed RCC algorithm, resulting in an effective and original MPPT method in the case of three-phase PV systems.