On HPA’ SMPA, the APAF is used to control the staircase waveforms and the techniques used in this application must be fast enough to update the reference levels, otherwise more power is requested from the LPA. The LPA is a critical element on the HPA in terms of losses and can contribute to up to 75% of the HPA losses. If the APAF technique is not fast enough to identify the parameters and adapt the staircase accordingly with the new parameters of the input, more power and consequently losses will be drained from the LPA, making the efficiency of the system lower and creating thermal stresses for the system. This is also true for other applications, such as digitally controlled staircase converters (inverters) that operate with variable output.

#### Detailing of APAF Detection Topologies

For three-phase systems, the OSG is based on Clark’s Transformation (

abc-αβ), which will naturally output the orthogonal (α and β) signals. With these signals and the inverse tangent trigonometric function, it is possible to determine the estimated phase angle, θ, as in (1). Detailed evaluations of three-phase and single-phase APAF (for a single and known frequency) detection are presented in references [

24,

25] and [

2,

3], respectively.

As mentioned before, single-phase amplitude, phase angle and frequency detection also require some alternative way for the OSG. As single-phase systems do not have the additional two phases to apply Clark’s transformation, alternative solutions for the OSG must be used [

19,

22,

26]. The main problem and disadvantage is that they require a fixed and previously known frequency.

Figure 1 shows a conventional Single-Phase SRF-PLL with the OSG at the input of the system. The symbol || or |•| represents the operation with the absolute value. For the purpose of this study, it is defined that the term central frequency, ω

_{cf}, refers to the main or fundamental frequency for those topologies based on fixed or synchronous reference. The central frequency also allows positive and negative variance around it. For example, most topologies are designed for a ω

_{cf} of 50 Hz, but will allow the parameter identification for 47–52 Hz, with an acceptable range of error.

The most common, and simple, OSG is the Second Order Generalized Integrator (SOGI), as presented in

Figure 2 [

19,

27]. The system generates an orthogonal signal,

v_{β}, from an input signal,

v_{α}, with the same amplitude and frequency. It works due to the dual integral blocks, denoted by the ∫ symbol. The central frequency, ω

_{cf}, is used to multiply the intermediate signals, which is the main constraint of the system. The gain

K_{1} has to be determined and it will affect the response of the system in terms of gain, phase, settling time and ringing. Details on determining

K_{1} are presented in reference [

11].

A variation from traditional SOGI-PLL is a system that uses a time delay [

14], or an algorithm-based variable time delay [

25], to create the orthogonal signal. These systems are rather simple and easy to implement, although they depend on a fixed and known central frequency as an SRF system. While reference [

20] evaluates the frequency changing from 47 to 52 Hz, reference [

28] validates the proposed topology with a step from 50 to 55 Hz.

Presented in references [

22,

23] and experimentally validated in references [

4,

22,

23] the Adaptive Notch Filter (ANF) and Amplitude Adaptive Notch filter (AANF) offer a sound strategy for single-phase parameter identification for signals with considerable amplitude variation. In reference [

23], the system is tested for steps on amplitude from 0.5 to 1 pu and 1 to 1.5 pu, and the frequency step is minimal, from 50 to 51 Hz, since the focus of the study is on the adaptive amplitude identification. Both topologies require the input of ω

_{cf} as the initial condition for one of the central integrators, placing this topology within the SRF-PLL category. In reference [

22], the ANF is experimentally validated for a very small amplitude (1 to 1.1 pu) and frequency (60 to 61 Hz) variation, but it is a solid work on how the choice of variables can impact the speed of the detection.

In reference [

24,

25] a valuable study on frequency identification that updates the necessary blocks of the system in order to improve the quality of the signal identification is presented, as an adaptive approach. The systems under study are three-phase, where it is easier to get the orthogonal signals, but the efficiency of the method is validated with highly distorted and unbalanced, with a wide variation on the amplitude of the input signals. The only fragility of the study is that the frequency variation is evaluated only within the range of 50–59 Hz [

24].

Usually the topologies are proposed to identify that central frequency and reject anything else, such as harmonics or noise. In reference [

21], a multiple frequency detection system is proposed, capable of identifying not just the central frequency, but also several frequencies of interest. The proposed technique has a considerable mathematical complexity, which translates to a higher computational burden. Although this is a remarkable contribution, the topology takes a considerable number of cycles, between 10 to 25, depending on some variable definitions, to identify the parameters.

As one of the most recent areas of development, the implementation of Kalman Filters on PLL has received significant attention [

29,

30]. Kalman Filters are used after the orthogonal signal is generated, or simply after the OSG, no matter if the system is single or three-phase. In reference [

29], the study compares the performance of two and three-state prediction models with conventional SRF-PLLs. The work does not cover significant frequency steps, only a step from 50 to 55 Hz and a frequency ramp (+40 Hz/s) from 50 to 53 Hz.

Two topologies under the SRF-PLL category are shown in

Figure 3a and both are recognized for their simplicity and accurate results. The first one is the inverse Park’s PLL [

31,

32], in which the orthogonal signal

v_{β} is generated by the inverse Park’s transformation of the filtered signals from

v_{d} and

v_{q}. The filters are set to the same cut frequency and they determine the APAF dynamic speed and accuracy. In

Figure 3b, the Enhanced-PLL (EPLL) is presented [

33,

34]. This topology reconstructs all components of the input signal, and uses the phase angle to find the orthogonal signal. For both systems the compensator gains,

k_{i} and

k_{p} are designed to adjoin the expected dynamic behavior and disturbance rejection. The main disadvantage of both systems is the constraint of the central frequency, ω

_{cf}.

An alternative for the SP-PLL and traditional SOGI as OSG is presented in

Figure 4. The study presented in reference [

26] can be defined as a frequency-independent topology, considering there is no input of ω

_{cf}, but another constraint is introduced on the system, which is related to a compensation,

C_{1} (2), of the gains of the derivative based (DB or DB

_{TF}(

d/dt)),

k_{d}, and integral based (IB or IB

_{TF}(∫)),

k_{i}, applied to the input signal. The orthogonal signal

v_{β} is given by Equation (3). The function

sign or

signal is used to extract only the signal (positive or negative) of its input. Even though the system presents the above-mentioned limitation and does not present a valid design procedure and analysis of it, there are good advantages on the topology.

If one can ensure that the multiplication of

k_{d} and

k_{i} is unitary, the dependency of this variable is removed. This is the main idea for the proposed system, but most important is the detailed and instructive design procedure, which will be explained in the next section. Still considering reference [

26], there is no coverage on the study related to the actual or desired frequency range, or limits, in which the systems would present satisfactory outputs. In terms of frequency variation, the results for the amplitude detection are evaluated for a steady ω

_{cf} of 50 Hz, and the phase-angle detection for steps between 47–52 Hz.

The literature heavily focuses on the central frequency identification [

35], for grid-connected systems under distorted conditions (mostly with the impact of noise, low and high order harmonics) [

36,

37,

38,

39], but there is a lack of studies in which APAF detection could be performed in significant wide ranges, especially for single-phase systems. With grid-connected converters, the phase-angle and amplitude identification must work with small variations in frequency (0.95 and 1.03 pu) and moderate variations in amplitude (0.45 to 1.2 pu). Now, on the PHIL and HPA applications, the APAF must be able to identify phase angles varying within frequencies from dc to 1000 Hz and amplitudes from 0 to rated voltages (230 Vrms i.e.) [

10,

12]. Not only that, the dynamics related to this utilization are more substantial. Considering this, the APAFs under study and the proposed will be designed for this application, since its limits are more extensive and include those of the grid-connected converters.

To overcome the limitations found in the literature, this study aims to prove that is possible to: (i) propose a multi-purpose solution for amplitude, phase angle and frequency detection that can be used for the connection of converters to grid, act as a signal parameter identification for HPA for Power-Hardware-In-The-Loop applications and active filters, or used as a feed-forward system where frequency reads are necessary; (ii) present an improvement on the OSG topology presented in reference [

26], for a single-phase amplitude, phase angle detection, which will be frequency independent (for a known range) but also self-reliant in term of gains or initial state settings; (iii) benchmark the proposed topology against well know topologies; (iv) create a formal, instructive and effortless design procedure for a selected wide range frequency single-phase input signal; (v) include and evaluate the frequency detection block from the phase angle with traditional derivative action and alternatively employ a modern Kalman filter solution; (vi) prove experimentally the ability to identify the amplitude and phase angle of the system’ signals in a wide range of frequency and amplitude.

Section 2 presents details of the DB and IB blocks modelling and of the proposed design procedure.

Section 3 will present a design example.

Section 4 performs a comparison between the Proposed PLL and some other established topologies.

Section 5 evaluates experimentally the proposed solution for a wide range of amplitude and frequency. Finally,

Section 6 presents the conclusions.