Experimental Validation of Harmonic Impedance Measurement and LTP Nyquist Criterion for Stability Analysis in Power Converter Networks

This paper presents the first experimental validation of the stability analysis based on the online measurement of harmonic impedances exploiting the linear time-periodic (LTP) approach applied to ac networks of power converters. Previous studies have provided the theoretical framework for the method, enabling the stability assessment of an unknown system adopting a black-box approach, relying only on injected perturbations and local measurements. The experimental case study considered in this paper comprises two single-phase converters, one acting as source subsystem and the other as load subsystem. A third converter, the stability measurement unit, is controlled to inject small current perturbations at the point of common coupling (PCC). From the measured small-signal perturbations of PCC voltage, source current, and load current, the harmonic impedances of source and load subsystems are calculated. The LTP Nyquist criterion is then applied to the ratio of the two harmonic impedances in order to assess the stability of the whole system. Theoretical and experimental results from a 5-kW laboratory prototype are provided and confirm the effectiveness of the method. In addition, the measurements do not require sophisticated equipment or control boards and can be easily performed from data sampled by commercial micro-controllers.


I. INTRODUCTION
S TABILITY analysis of interconnected power systems has been an interesting and challenging topic for researchers in the past decades, and it is still nowadays an open field for research [1], [2]. One of the first investigations presented in the literature was provided by Middlebrook [3], who analysed the instability issues due to interactions between DC/DC converters and their input filters. The linearised input and output impedances of the system were first evaluated and the Nyquist Criterion was then applied to assess the stability of the system. This approach was then extended to AC power converters by V. Salis Belkhayat [4], with focus on balanced three-phase AC systems. Exploiting the Park transformation, the system is represented in dq reference frame, hence becoming a MIMO linear system. Then, stability is assessed by applying the Generalised Nyquist Criterion [5].
This, however, is only one of the possible stability-assessment methods [6]. With a focus on balanced three-phase systems, other approaches are: Harmonic Linearisation [7], where a small-signal perturbation is injected into the system in order to measure the small-signal source and load impedances. Stability is then assessed based on the ratio of the two impedances.
Dynamic Phasors, with extension to multi-source and multi-frequency scenarios [8]. DQ-reference frame [9], where it has been demonstrated that for a high power factor system, the only relevant impedance matrices are the dd and qq ones, allowing a simplification of the whole system as two decoupled DC subsystems. Sequence Domain Method [10], which is based on the calculation of the positive and negative sequence impedances and has a relevant connection with the dq-frame approach [11]. Unified Impedance Model [12], where the system impedances and calculations are proposed in the αβ-domain. Finally, the Harmonic State Space, where one of the advantages is that also the switching behaviour can be taken into account in the analysis [13].
In general, some of the aforementioned approaches cannot be directly applied to analyse single-phase systems. In this case, the Dynamic Phasor method can be used considering the real and imaginary parts of the phasor in order to build 2 × 2 impedance matrices and use the Generalised Nyquist Criterion [14]. Also the Harmonic Linearisation method [15] can be applied following a similar approach. In the Apparent Impedance method [16], only the impedance at the measuring point is evaluated, it being effectively a closed-loop transfer function that contains information about stability. Another option is the Harmonic Linearisation defined using signal-flow graphs [17], providing a visual tool in order to understand the various harmonic interactions. Harmonic State Space, which is based on the Linear Time Periodic (LTP) systems theory [18], [19], [20], also provides a straightforward relationship between the various harmonic components of the system.
Focusing on a single-phase scenario, this paper proposes the experimental validation of the Harmonic State Space (HSS) method in combination with the LTP Nyquist Criterion, already presented in detail in [21]. In the previous publication, the ability of the method to assess stability of a black-box single-phase AC system was demonstrated theoretically and in simulation but not experimentally. To fill the gap and prove the practical feasibility of the method, this paper presents its experimental implementation in the same system configuration as discussed in [21]. The case study includes a source and load converter: the source controls the AC voltage of the network and the load controls the current absorbed with unity power factor. The Harmonic Impedances of both subsystems are measured through small-signal current perturbations injected at the PCC by a third converter referred to as the SMU -Stability Measurement Unit. Then the LTP Nyquist Criterion is applied to the ratio of the two Harmonic Impedances in order to assess the stability of the whole system.
The comparison with other existing techniques, the theoretical analysis and an extensive set of simulation results have been already presented in [21]. Hence, this paper is solely intended to discuss the experimental validation of the method. For this reason, the focus will be more on the practical aspects required to apply the technique and replicate the results rather than on the theoretical details, for which the reader is invited to refer to [21]. This paper is organised as follows: Section II describes the AC system under analysis and the SMU; Section III provides JOURNAL OF L A T E X CLASS FILES 3 a practical review of LTP theory and the Harmonic Impedance measurement method; Section IV presents an extensive set of experimental results from the case study; Section V discusses advantages and challenges and Section VI concludes the paper.

II. EXPERIMENTAL SYSTEM: SOURCE, LOAD AND SMU CONVERTERS
The case study for the experimental validation of the stability analysis based on LTP Nyquist Criterion and Harmonic Impedances is reported in Fig. 1, and is the same as used in [21]: a source converter with inner current loop and outer voltage loop that controls the PCC voltage and a load converter that draws a controlled current synchronised with the PCC voltage via a DQ Phase-Locked Loop (PLL). Compared to the system in [21], the small current injection is now implemented by a real converter with closed-loop current control (SMU).
From Fig. 1 it can be seen that control complexity has been deliberately kept to the bare minimum, using only PI controllers and feed-forward terms. The reason is that the focus of the work is on the implementation of the black-box stability analysis method rather than on the performances of the AC system, which is generally unknown to the SMU. In addition, showing that the method is effective even with a simple hardware and control architecture for the SMU demonstrates its ease of implementation, which does not require complex hardware or sophisticated controllers for current injection.
The load and source H-bridge converters are custom designs using IXYS IXF K120N 65X2 power mosfets switching at f source/load pwm = 10 kHz. The load current has been limited to i load rms = 30 A. Three-level modulation is used with double update at 20 kHz, also corresponding to the sampling frequency. All the controllers for source and load converter are implemented in the same control platform, a Texas C6713 DSP with custom FPGA interface, and have been designed in the continuous time domain (as in [21] and in Fig. 1) for a phase margin P M = 60 o and later discretised using the Tustin method. Source The SMU is also connected to a host PC, which through a dedicated Matlab script provides the reference signal for the injections, stores the measured signals with a 20 kHz sampling frequency (one every two samples used by the SMU current control) and finally calculates the LTP Nyquist plot once all the data is collected. The scope of this first implementation is to validate the proposed impedance measurement method and demonstrate that measurements are feasible in practice also having a relatively simple hardware and control software. On the other hand, the SMU has been designed as a fully independent and automated unit, to be as close as possible to a dedicated measurement equipment. The SMU itself, its control and the overall implementation of the harmonic impedance measurement can be greatly improved in terms of measurement speed, data storage requirements, signal to noise ratio and minimum amplitude of injected signal to obtain correct results. All these analyses and optimisations are currently ongoing.
A photo of the experimental rig, excluding the controllers, is shown in Fig. 2. The remaining system parameters, including the controller gains referred to the continuous-time control design, are reported in Table I. Some of the parameters are slightly different from those previously used in [21] due to laboratory constraints.
In order to avoid repetition, the system equations and most of the LTP theoretical analysis will not be reported in this paper and the reader is invited to refer to [21] for a detailed analysis.

III. BASIC PRACTICAL REVIEW OF LTP THEORY
In this Section the basic tools required to perform LTP stability analysis are briefly discussed. For a detailed review, the reader should refer to the original formulation presented by Wereley and Hall [22], [23]. Given the single-phase system in Fig    1, the linearisation around its steady-state leads to the Linear Time Periodic (LTP) model where A(t), B(t), C(t) and D(t) are T -periodic, and T is the period of the steady-state, T = 1/f g = 20ms in the case study.
Exploiting the Exponentially Modulated Periodic (EMP) signal and the Toeplitz transformation, the Harmonic State-Space Model (HSSM) of the system is derived as with N = diag(. . . , N −n , . . . , N −1 , N 0 , N 1 , . . . , N n , . . . ), N n being a diagonal square matrix of the same dimension as A n with diagonal coefficients equal to jnω g . A is the Toeplitz transform of A, defined as : : where M is the truncation order limiting the dimension of A to 2M + 1. The same considerations hold for B, C, D. Stability analysis could now be performed by evaluating the eigenvalues of the A − N matrix. However this requires a knowledge of the system parameters, which is not available when a black-box assessment is required.
From (2), the Harmonic Transfer Function (HTF) of the LTP system is defined through In order to apply the LTP Nyquist Criterion and assess stability with a black-box approach, the Harmonic Impedances must be measured. To do so, a small-signal current perturbation is injected by the SMU at the PCC, as in Fig. 3. The impact of JOURNAL OF L A T E X CLASS FILES 6 Fig. 3. Small-signal current injection at the PCC and harmonic impedances.
an injected perturbation into the system requires careful consideration. In fact, whenĩ x (t) = I x cos(ω inj t) is injected by the SMU, the PCC voltage and the source/load current perturbations can be written in Fourier form with truncation order M : respectively. Thus, the source harmonic impedance at ω inj satisfies with ω inj ∈ (−ω g /2, +ω g /2) [21]. Written in a more compact notation, v o (ω inj ) = Z source (jω inj )i source (ω inj ), and also v o (ω inj ) = Z load (jω inj )i load (ω inj ).
In general, the harmonic impedance will be of the form (7), which is a square matrix of dimension 2M + 1 and where x stands for either source or load. A possible solution to measure these impedances at the frequency ω inj , is to perform a set of 2M + 1 independent small-signal current injections: • n=1: the first injection isĩ x (t) = I x cos((ω inj − M ω g )t).ṽ o (t),ĩ source (t),ĩ load (t) are measured and represented by their Fourier series as in (5) and their complex harmonic coefficients are collected into the column vectors v load (ω inj ) as in (6).
, and the perturbed signals provide the vectors v load (ω inj ) as in (6).
These vectors are then rearranged into matrices as and finally the source and load harmonic impedances at ω inj are calculated as This procedure has to be performed for each ω inj ∈ (−ω g /2, ω g /2) that has been chosen to reconstruct the LTP Nyquist plot experimentally.
Now, from Fig. 3, the LTP Nyquist plot is obtained by evaluating the eigenvalues of the open-loop HTF F(jω inj ), which is equal either to F(jω inj ) = Z source (jω inj ) −1 Z load (jω inj ) or F(jω inj ) = Z load (jω inj ) −1 Z source (jω inj ). The stability of the system is assessed using the following theorem. In this section, the Harmonic Impedance measurement method and the LTP Nyquist Criterion discussed in Section III are applied to the experimental case study discussed in Section II. The proposed method has been developed as a black-box stability assessment, since it relies only on current injections and measured perturbations and does not require any of the system parameters. For the sake of completeness, the results of the black-box stability assessment are also compared with the analytical prediction based on the nominal values of the system parameters and on the LTP model in [21]. In the following, two configurations are analysed: • CASE A: stable mode with the parameters in Table I; • CASE B: bounded unstable mode, with a steady-state high-frequency oscillation in all the waveforms.
The experimental measurement of the Harmonic Impedances has been performed as discussed in Section III. ω inj has been To evaluate the quality of the experimental stability assessment, analytical calculations have been performed using the LTP In the following subsections, the two cases will be evaluated starting from the analytical prediction and comparing it with the stability measured with the black-box approach, estimating the Harmonic Impedances and applying the LTP Nyquist Criterion.

CASE A -Stable System
In this configuration the system works in a stable operating mode, with the parameters reported in Table I  The stability of this operating mode is confirmed by the analytical LTP eigenvalue loci in Fig. 6, calculated by feeding the nominal system parameters in the model derived in [21]. As shown in the figure, all the eigenvalues lie on the left-hand side of the complex plane. It is worth noting that this analytical prediction is not required by the black-box stability assessment method and has been presented here only to compare the experimental results with a theoretical prediction.
In order to apply the LTP Nyquist Criterion, first the Harmonic Impedances, Z source (s) and Z load (s), are experimentally measured using the small-signal current injection method previously described. As examples of waveforms during the injection, available. The solution adopted in this paper to overcome this limitation is to perform a curve-fitting of the single HTFs shown in Fig. 9. In the case study, only the 0-component of F(s), i.e. F 0 (s), shows poles that lie inside the LTP Nyquist Contour plot, and therefore the other components will not be discussed further. However, in general the poles of all the components of F(s) must be checked. The fitting has been based on the package provided by Gustavsen [24], [25] and is reported in Fig. 10, from which it is possible to locate the poles of interest, shown in Fig. 11. It can be seen that F 0 (s) has a pole at the origin and a pair of complex-conjugate poles at s = 53.2 ± j2π50. F 0 (s + jω g ) has the same poles as F 0 (s) but with a shift equal to JOURNAL OF L A T E X CLASS FILES 10 +jω g , meaning that it has poles at s = 53.2, s = 53.2 + 2j2π50 and s = j2π50. Similar considerations apply to F 0 (s − jω g ).
Hence, the poles of F(s) that are inside the LTP Nyquist Contour are s = 0 and s = 53.2, so the system is stable if the LTP Nyquist plot has two counterclockwise encirclements of the critical point (−1, 0). Please note that the phase of F 0 (s) starts from a value close to −180 o , which is due to a pole located in s = −1.22. However, the magnitude of F 0 (s) has an initial slope of −20 dB/dec, confirming the presence of only one pole in the origin.
Finally, the LTP Nyquist plots for Case A are shown in Fig. 12: (a) and (b) show the experimentally measured plot and (c), (d) the one calculated with the analytical model from [21]. As can be seen, the outer encirclement is derived with an infinite enclosure. For each pole at the origin of F(s) (there is a single such pole in the present case study) half a circle must be added between the diverging lines of the LTP Nyquist curve, according to the LTP Nyquist theory. The inner counterclockwise encirclement, shown in the zooms in (b) and (d), provides the second required encirclement, which confirms the stability of the system. The comparison between the zoomed plots in (b) and (d) also shows that the experimentally measured LTP Nyquist is consistent with the analytical one, even though a full match is not to be expected since the analytical one is based on nominal system parameters that may differ from the actual ones in the experimental rig.

CASE B -Bounded Unstable System
In the second case under analysis, the system is in a bounded unstable operating mode, obtained by reducing the phase margin of the load converter current control. The only difference with the parameters reported in Table I   where it can be seen that a steady-state high-frequency oscillation affects all the waveforms of the system. Such instability does not diverge and hence does not cause tripping of the protections. The experimental evaluation of Case B follows the same steps as the one proposed for Case A.
The unstable operating mode is also confirmed by the analytical calculation of the LTP eigenvalue loci, shown in Fig. 14, where some of the eigenvalues lie on the right-hand side of the complex plane.
As in Case A, the first step to determine the LTP Nyquist plot is to measure the source and load Harmonic Impedances.  the unstable nature of the system in this case. The point-to-point measured Harmonic Impedances are reported in Fig. 16. An interesting feature that is worth pointing out is that the difference between the stable and unstable case is solely due to the change in the integral gain of the load current control. However, this change also has the consequence that the source harmonic impedance is different in the two cases, since the harmonic impedances are small-signal models and they change with a change in the steady-state trajectory of the system. approach is the possibility to include in the analysis any number of harmonic components, just by setting the truncation order.
Furthermore, due to the structure of the HTF operator, the interaction between harmonics is automatically taken into account.
This can be achieved also by the Graph Flow based Harmonic Linearisation, or by the Dynamic Phasor method were multiple frequencies are considered. However, in these cases the mathematical derivation becomes a significant constraint and is not always a feasible solution.
One drawback of the proposed Harmonic Impedance measurement method, in its present implementation, is the large number of injections required to obtain accurate LTP Nyquist plots. Nevertheless, at this stage of the research this is not considered a critical limitation since the main goal of the work is to prove the feasibility of the method. Also, in any practical implementation, a stability monitoring system is not likely to run continuously but rather to assess stability at specific discrete time intervals, reducing the impact of a measurement delay. If needed, there are several approaches that can be investigated in order to improve the measurement process, making it faster and computationally more effective. Among all the possible solutions, potential candidates are the injection of a transient to determine the impedance [26], the use of wavelets [27] or impedance measurement techniques based on neural networks [28].

VI. CONCLUSIONS
This paper has presented a set of experimental results on a 5 kW single-phase two-converter AC network to validate the black-box stability analysis method based on Harmonic Impedance measurements and LTP Nyquist Criterion originally proposed in [21]. The stability assessment is based on the measurement of Harmonic Impedances through the injection of small