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I. INTRODUCTION
The interest in energy generation through renewable energy sources (such as solar energy and wind energy) has increased in recent years. Moreover, the benefits obtained by this form of generating energy are widely accepted and friendly to the environment [1-2]. The multilevel inverters are electronic devices that switch DC energy into AC energy and have been used to link the AC grid and renewable energy sources with the classic topologies (NPC, flying capacitor, and cascaded H-bridge) [3].
In addition to using multilevel inverters for conditioning the energy, it is necessary to have a control process to find the phase angle of the AC grid to synchronize the energies [4]. One way to achieve this is by using Phase-Locked Loop Algorithms (PLL) [5]. It consists of a control loop that generates an AC signal that is compared with a reference signal, obtaining information such as frequency, amplitude, and phase [6]. The first synchronization schemes were based on an open-loop estimation of the phase angle by detecting zero crosses [7]. Nevertheless, the presence of phenomena related to energy quality causes limitations for controllers of this type. To improve these negative aspects, phase tracking was implemented, and good results were obtained, and a harmonic oscillator controlled by voltage for compensating harmonic destabilization was added [8]. One of the most used methods is the fixed reference frame due to its positive performance [9-10]. However, its performance is adequate only with balanced systems. Therefore, other control schemes such as PI controllers and resonant controllers have been used for compensating the unbalances and harmonics on the grid.
The literature presents different PLL schemes for synchronizing multilevel inverters with the AC grid. For example, Rodriguez-Trujillo et al. [11] present the design, modeling, and implementation of a one-phase inverter connected with the mains by a PLL that generates a quadrature component with the voltage signal obtained from a delay. Rashed et al. [12] used a repetitive resonant controller for a grid-tied inverter with very distorted energy, where the proposed control obtains the angle, frequency, and amplitude of the grid fundamental. Moreover, Yue and Tolbert [13], a CHB of 11 levels synchronizes energy generated by photovoltaic solar panels with the AC grid through a PLL with delay to generate a signal in quadrature. As a feature, it is necessary that the amplitude and phase are obtained quickly and precisely, independent of the technique used, even if the input signal is distorted.
In general, there is an increase in the use of synchronization schemes, specifically PLLs for grid-tied multilevel inverters with renewable energy sources. This paper presents a grid-tied multilevel inverter with a PLL based on an adaptive observer. The PLL creates a reference signal that serves as a modulated signal, and then it is compared with triangular carriers signals of high frequency, which generates the pulse width modulation (PWM) for turning on and off the power semiconductor devices composing the inverter, thus, obtaining a robust control under uncertainties of the grid such as voltage sag, noise, and distorted signal.
II. METHODOLOGY
A. Multilevel Inverter and PLL Development
The topology selected is the one-phase five-level CHB, consisting of two H bridges, each one with four power semiconductor devices and a DC voltage source.
According to the literature, there are several techniques for obtaining a PWM for multilevel inverters. The PWM used in this paper is the high-frequency phase disposition PWM (PD-PWM) [14]. This PWM requires two carriers for each bridge which is compared in amplitude with a sinusoidal signal (modulated signal), as shown in Figure 1. In addition, Table 1 presents the output voltage obtained by turning on and off the power semiconductors of the inverter.
Table 1 Output voltage for the switches turning on and off.
| 𝑆11 | 𝑆12 | 𝑆13 | 𝑆14 | 𝑆21 | 𝑆22 | 𝑆23 | 𝑆24 | 𝜐𝐶𝐷 |
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 𝜐𝐶𝐷 |
| 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 2𝜐𝐶𝐷 |
| 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | −𝜐𝐶𝐷 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | −2𝜐𝐶𝐷 |
The connection plan is depicted in Figure 2, where the signal obtained from the PLL serves as a modulated signal in the PD-PWM.
For the PLL design, the input is considered a harmonic oscillator, and a state estimator with damping is built as a copy of the input, as shown in Equations (1) and (2). The damping is added to rebuild the phase angle. Subsequently, a clean version of the input is obtained, and the fundamental frequency is estimated.
Where
represents the error signal given by the difference between the input and the estimated,
is an auxiliary signal in quadrature regarding the estimate of the input time derivative,
is the estimate of the phase angle of the input signal,
is the estimate for the input signal, and λ is a positive design parameter for introducing the damping. An adaptive law is proposed for rebuilding θ, following a Lyapunov approach as a storage energy function in Equation (3), taking the error signals over the trajectories and introducing γ as an adaptive gain.
Where,
. The adaptive law
is proposed for making the derivative regarding the time, negative semi-definite as
[15].
B. Adjustment Parameters of the PLL
A first approximation for adjusting the parameters λ and γ is considering that the system is operating in-balance conditions, that is, 𝜓 = 𝜐𝑆, and assuming
, where ω0 is the angular frequency of the input signal and
is the estimate of the angular frequency. This yields the following system:
Next, a change of variables is proposed with,
, and
, obtaining the Equations (6), (7), and (8).
Then, the gains adjustment is based on the desired bandwidth for a second-order system. For design purposes, it is considered a damping ratio of
. The bandwidth (BW) must be less than 2𝜔0, where 𝜔0 is the nominal value of the AC grid angular frequency. It is common in practice to select the bandwidth between 1/2 and 1/10 of that frequency. Under this consideration, the parameters can be adjusted with the Equations (9) and (10).
III. RESULTS
Numerical results were obtained using Simulink of MATLAB, for the PLL based on the adaptive observer shown in Figure 3. Figure 4 shows the PLL capability to respond under abrupt change of signal. The change is from a 1V square signal to 1Vpp sinusoidal signal, both at 60Hz of frequency. The two signals (input and output) are spliced at the same graphic.
Then, Figure 5 shows a zoom in the PLL response where the input is a square signal of 1.6Vpp at 60Hz. The PLL recovery time is around 0.00004 seconds.
Next, Figure 6 presents the phase portrait for the input signal versus the output signal, where the input is a sinusoidal signal. This evidences the similarity between signals.
The PLL creates a reference signal obtained from the AC grid, and this signal serves as a modulated signal in the phase disposition PWM. This scheme is simulated to obtain the response shown in Figure 7.
The prototype shown in Figure 8 was built for the experimental test having a 15kW inverter with the PLL and the modulation implemented through operational amplifiers [16]. The red rectangle shows the IGBTs, and this represents the power stage. The yellow square exhibits the high-frequency phase disposition PWM. The blue square is the PLL, and finally, the green rectangle is the signal conditioning (optocouplers, drivers, dead time).
To corroborate the numerical results obtained for the PLL in Simulink, the PLL implemented through operational amplifiers was probed under changes of signal, signal with noise, and the phase portrait. Figure 9 depicts the PLL response under signal change from 1V square signal to 1Vp sinusoidal signal at 60Hz. The input signal comes from a Rigol signal generator. The peaks over the signal are produced by noise in the oscilloscope tips.
Then, Figure 10 shows the PLL phase portrait for a sinusoidal input signal.
Finally, the PLL is connected to the AC grid and its output to the phase disposition PWM. Figure 11 depicts the result for the voltage synchronization between the multilevel inverter output and the AC grid.
IV. DISCUSSIONS AND CONCLUSIONS
This article presented a grid-tied multilevel inverter with a PLL based on an adaptive observer. The design process considered the AC input signal as a harmonic oscillator, and a state estimator was proposed as a copy of the model, which incorporated an adaptive law to reconstruct the angular frequency. Regardless of whether the signal is square, triangular, sinusoidal, etc., as long as the signal is periodical, the PLL can reproduce it and obtain a clean version of it.
The PLL does not need to detect zero crosses or delays; it has a fast response, and the model of the algorithm is based directly on the AC grid without a fixed reference frame for obtaining the angle.
The algorithm is computationally efficient and can be implemented through basic elements such as operational amplifiers, resistors, and capacitors, reducing its difficulty in executing it in a system. A conventional technique is used (PD-PWM) for achieving the synchronization to the AC grid. Numerical results and experimental results were consistent.
Future research could finish the prototype and connect it to renewable energy sources such as photovoltaic solar cells.
