Frequency Scanning of Weakly Damped Single-Phase VSCs With Chirp Error Control

The chirp sweep is a go-to wide-band impedance measurement technique when speed and simplicity are of main concern. In addition, the time-frequency trait of chirp scans provide unique benefits for systems exhibiting frequency couplings: a phenomenon often encountered in single-phase voltage source converters (VSCs). Unfortunately, time domain interpretation of chirp responses are inaccurate if the VSC-based system exhibits weak damping, as the transient response induced by the chirp cannot be safely neglected. We quantify this error for linear(ized) time-invariant and time-periodic systems, with impedance representations of scalar transfer functions and harmonic transfer functions, respectively. An observer termed the multiple chirp reference frame filter is proposed which enables real-time estimation of the harmonic transfer functions and their corresponding chirp errors. By controlling the relative errors to negligible values, the chirp exhibits low rate around resonances and high rate elsewhere. An experimental admittance sweep of a single-phase STATCOM operated by dispatchable virtual oscillator control, consolidates the proposed approach as a simple frequency scan technique for weakly damped and frequency coupled systems.

or symmetrical components allows for eliminating time from the dynamic model. However, time usually enters into the equations periodically, and thus, compact models can be formed by changing to harmonic bases. The linear time-periodic (LTP) theories of the harmonic state space (HSS) and harmonic transfer function (HTF) [1], dynamic phasors [2], extended harmonic domain [3], and general DQ frames [4] all recognize this in slightly different ways [5], [6]. Regardless of choice, they all provide frameworks for what the power electronics community refers to as frequency couplings which have been reported extensively in single-phase [7], [8], [9], three-phase unbalanced systems [2], [10], [11], and even dc systems [12]. From a practical viewpoint, the coupling phenomenon stems from a combination of nonlinear interactions (e.g., ac/dc modulation, power calculation), asymmetric control systems, and harmonics in steady state. Single-phase VSCs have it all: the dc-side voltage contains double-line oscillations, and orthogonal system generators (OSGs) are only orthogonal at the fundamental frequency.
Naturally, there are implications for single-phase impedance (or admittance) scans. One exceptionally useful assumption of frequency response measurements is that a system responds to a sinusoidal excitation at the excitation frequency only, in steady state. Wideband, periodic stimuli builds on this assumption, and provides high resolution frequency scans in a relatively short time. As frequency couplings and their importance in power electronics stability assessments started to draw attention from the early 2000's [13], [14] and continue to do so, this assumption was forfeited by necessity. The majority of impedance-based stability analyses which incorporate coupling effects have since relied on single-tone sweeps for model verification [15], [16], [17] and even black-box input-output identification [18]. Some examples from other fields adopt wideband injections by either performing multiple independent injections [19] or by carefully designing the input signal to avoid spectral overlap [20], [21]. Either way introduces additional complexity to the design and postprocessing procedures. The chirp is one wideband stimulus with long tradition for linear time invariant (LTI) systems as a fast, simple sweep with low crest factor (peak to rms ratio) [22]. While the chirp is usually described from its frequency domain characteristics, it also has the time domain trait of locally resembling a single constant-frequency sinusoid. A frequency scan building on this observation was proposed in [23], though it is limited to LTI transfer functions. Furthermore, it assumes that the chirp response locally approaches a single-tone response, which generally does not hold true for weakly damped systems. This constitutes the first knowledge gap this article aims to bridge: quantification of the local discrepancy between the timedomain response of a chirp, and that of a single-tone response. Then, we show that it is possible to limit this discrepancy-this chirp error-through real time observation and control. The result is a time-domain black-box frequency scan which captures an HTF with a single chirp sweep: as fast as possible without inducing a large chirp error. This chirp scan relies only on simple operations (integration, reference frame rotation, logic, and arithmetic) and can therefore be easily realized in most simulation and experimental environments.
The circuit that will be scanned is inspired by recent reports on the increasingly popular grid-forming virtual oscillator-based controllers, e.g., dispatchable virtual oscillator control (dVOC). While this scheme exhibits almost global asymptotic stability under certain transmission line circumstances [24], [25] and can be directly extended to single-phase VSCs through an OSG [26], small-stability issues which require periodic models have been investigated in [27]. Furthermore, on a similar note, as shown for single-phase VSCs in [28], a reduced dc-side capacitance is desirable to reduce cost and volume as high-lifetime film capacitors take larger market shares. An unintended consequence is stronger frequency couplings and weaker stability margins. The device under test for the frequency scans in this article consists of exactly that: a VSC in STATCOM mode with low dc capacitance as in [29] and [30], with power and current loops replaced with dVOC. An analytic model with HSS serves as comparison to the experimentally measured impedance. The HSS model relies on a harmonic balance approach to find the periodic steady state (PSS) [31], [32], which enables parametric stability studies. A few such parametric studies, combined with participation factors, are used to highlight the poor stability margins that may occur in spite of reasonable tuning of control parameters.
The rest of this article is organized as follows. In Section II, the issue of a time-domain interpretation of the chirp response is laid out and tackled through derivation of the chirp error. In Section III, the error expression is leveraged-in combination with an observer-to design a chirp rate controller. Before experimental verification is shown for a dVOC operated VSC in Section V, the grid-connected VSC is presented in Section IV along with a brief parametric stability study. Potential improvements and applications of the contributions in this article are discussed in Section VI. Finally, Section VII concludes this article.

II. TIME-DOMAIN INTERPRETATION OF THE CHIRP RESPONSE
A chirp stimulus u(t) can be generally defined as with phase function ϕ, instantaneous frequency ω = dϕ dt and a so-called chirp rate k = 1 2π dω dt . In the trivial case of k = 0, the stimulus u(t) corresponds to a single tone. In the nontrivial case, the instantaneous chirp variables can take any form, provided ω is continuously differentiable. The time-domain response y(t) (2) Interpretation of the chirp response in time-domain is particularly useful in presence of frequency couplings. We adopt the Harmonic Transfer Function to represent the frequency coupled transfer functions that arise (see Appendix A). An HTF chirp response would, similar to that of the scalar H, ideally look like with α z = 2, α p = 5, ω z = 2π35, and ω p = 2π45 features a sharp resonance and antiresonance. A sweep with a complex exponential chirp between 30 and 60 Hz with k = 10 Hz/s is shown in Fig 1, along with the ideal transfer function. The figure also includes the true (simulated) chirp error y − H(jω)e jϕ . The amplitude response |y| is grossly inaccurate around the resonance, and may even be wrongly interpreted as multiple, and close, resonances and antiresonances. Elsewhere, the response is quite accurate. Yet, the appeal of frequency scanning is its black box trait, thus we cannot assume to know the pole locations in advance. How then can we ensure that (2) and (3) are valid? We begin the answer by quantifying the time-domain discrepancy between a chirp and single-tone response: the chirp error.

A. Chirp Error
Given a proper single-input, single-output (SISO) transfer function H(s) describing a linear (or linearized) system Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. and its time-domain differential equation To facilitate analysis of the difference between a chirp and a single-tone, let ϕ(t) be decomposed as so that we can look at u(t) around t = 0, for any values of single-tone phase ϕ s and frequency ω s . Δϕ(t) conveys the phase difference between a chirp and single tone. In other words, if Δϕ(t) = 0, the input corresponds to a single tone. If nonzero, it is a chirp, which now can be written as As Δϕ(t) can take on any form, an analytic approach to the particular solution is not feasible. This limitation is bypassed by an approximation of e jΔϕ(t) with a second-order Maclaurin series with k = 1 2π dω dt | t=0 as the local chirp rate. The input is now separated into two terms: 1) u s (t) is a single tone and 2) u (t) is the difference between the chirp and single-tone. The principle of superposition applies so that the particular solution becomes in which the first term is the ideal single-tone response y s (t), while the second term u (t) induces the chirp error (t). As u (t) is a product of a complex exponential and a second-order polynomial, we use the approach of undetermined coefficients and assume a solution of (t) on the same form with P (t) = [At 2 + Bt + C]. After some amount of algebralaid out in Appendix B-the coefficients are determined as Here, denotes derivative with respect to jω. Insertion of (12) into (11) gives (13) As mentioned, this procedure holds for any single-tone frequency ω s and phase ϕ s . The additional information provided by t around the point of linearization, is redundant. ω s and ϕ s can be replaced with the chirp frequency and phase from (1), though we drop the time dependency to emphasise that ω and ϕ has been treated as independent variables in the derivation The full particular solution becomes (15) but what is still missing is a way to deduce the total response. In absence of an analytic expression, additional insight comes from looking at inputs that would induce the second term in (15), given complete relaxation of the system modes. An intuitive guess of the chirp error including the transient response would be where * represents time-domain convolution and h is the system impulse response, yet (16) is not entirely correct. The zeros in H(s) do not cause any delay in the chirp response, and a more accurate expression is where h d is the impulse response pertaining to the denominator only.
The error expressions are plotted in Fig. 1 for the introductory example, and visualizes how 1 inaccurately predicts a large error around the antiresonance. However, the rest of the article will consider only 1 , as its input may be computed in real time through simple means. Either way, the assumption of (2) is only valid if the chirp error relative to H, defined as can be neglected.
Simplifications of (18) allow further insights. Assume that a transfer function H is dominated by a pole p w = σ w + jω w around its frequency ω w , with residue r w When (19) is plugged directly into (18), the residue is cancelled out and the error around the pole ε w becomes which attains its maximum ε w,max at ω = ω w Clearly, the damping σ w and chirp rate k determines the worst case error. A well-damped system will allow a high k without any error, while a negligible error for weakly damped system is only achievable through low rate.

III. VARIABLE CHIRP RATE THROUGH ERROR CONTROL
While the derivation process of the error can come across as quite tedious, the resulting expression in (18) is compact and easily interpreted. It concisely quantifies the dependency of ε on k, indicating that the latter can be used to manipulate the former, loosely speaking, increase the rate as the error is low and vice versa. Such a scheme must perform several functions, all of them which must be implemented in real time.  1) Observe the HTFs H n (jω) from y(t).
3) Manipulate k to control ε. An overview of the chirp scan procedure is given in Fig. 2. The three parts are modular to some extent, as there are multiple ways to realize them. The following two subsections are devoted to these parts.

A. Multiple Chirp Reference Frame Filter
In the following, we design a filter to separate y into chirps with known instantaneous frequency. The approach starts by transforming the error signal into reference frames aligned with the input chirp instantaneous phase and its frequency shifted chirp phases, i.e., φ n . The error signal is integrated in each reference frame, before it is rotated back to its original frame. As such, it acts as an multiple second-order generalized integrator (MSOGI) OSG does on a harmonic signal, but in reference frames attuned to the chirp frequencies. Henceforth, we refer to it as the multiple chirp reference frame (MCRF) filter for convenience.
One CRF is shown in Fig. 3, and the MCRF filter consists of settingŷ = ŷ n , with at least as many CRF filters attuned to shifted chirp frequencies as there are couplings in the system. If operating points exist, such as fundamental or harmonic currents/voltages, additional reference frame transformation filters can and must be included for the relevant frequencies.
The MCRF filter provides a measurement of d 2 dt 2 |H n | through a filtered derivative, from which the chirp error amplitude can be computed However, to avoid adverse feedback effects of dividing by the rate k, it is substituted with the lower gain limit k low .

B. Chirp Error Control
The objective of the controller is to sweep as fast as possible (i.e., high rate k) while limiting the chirp error to an acceptably low value. That means, at times, that the rate should be allowed to be low, down to a few Hz/s. A simple scheme, as in Fig. 4, controls the rate proportionally to the chirp error, when the error is less than a user specified maximum |ε max | (e.g., 0.2). The controller outputs a maximum rate k max when the error is zero-which it usually approaches for frequencies far away from any pole or zero-and reduces the rate as the error increases. A saturation block limits the lower chirp rate to avoid standstill mode, which may happen due to imperfect error measurements or around very weak poles. As a side note, integral action would cause windup and is not actually needed; the objective is to limit the chirp error below a user-specified limit, not to regulate it to a specific value. A high k max may induce instability around resonances. This can be addressed with a gain scheduling loop, as shown in red in Fig. 4: on the same principle as the proportional controller but on a slower time scale, adjust the effective gain when |ε| < |ε| * . The gain scheduling loop should be fast enough to react to incoming poles but slow enough that it does not interact with the faster dynamics of the proportional controller and observer.

C. Tuning
The control objectives of a fast and accurate sweep can be set with k max high and |ε| * low, respectively. Note that ambitious values of k max and |ε| * will pose challenges for the subsequent regulatory tuning. Furthermore, accuracy depends on the tracking and disturbance rejection of the combined observer-controller chirp sweep. Nevertheless, the functionality of the MCRF filter and error controller imposes three strict rules as follows. Table II shows suitable ranges we have found heuristically. Unfortunately, the MCRF filter is a bottleneck toward an improved tradeoff between speed and accuracy. For reasons discussed in Section VI, the MCRF sometimes generates disturbances internally at integer multiples of ω 0 /2. The remedial actions in this article has been to 1) choose relatively high time constants and acceptable error 2) reduce the speed through k max in the expected dysfunctional frequency range. We envision the MCRF filter issues to be addressed in future research so that |ε| max |, |ε| * , and all T s can take on lower values and the gain be kept high.  [26], [34]. The voltage control law φ is given in Appendix C.

IV. CASE STUDY: A DVOC-OPERATED VSC
A single-phase, grid-connected VSC operated with the increasingly popular dVOC serves as case study, as shown in Fig. 5. While believed to possess good stability properties in balanced three-phase systems [25], the time-scale separation argument does not hold as frequency couplings emerge. We devote this section to show not only that the case exhibits frequency couplings and weak damping, but also why. The indication should be clear: any single-phase converter-based system, even if tuned after time-scale separation principles, may end up as frequency coupled and weakly damped.
DVOC can be used for single-phase VSCs, given that they use an OSG to generate the current space vector i αβ . The asymmetric behavior of the OSG combined with a nonlinear voltage control loop, gives rise to frequency couplings. Still, the induced periodicity in the linearized state space can arguably be neglected under normal operation. That assumption does not hold if one includes nonlinear and periodic dynamics in other parts of the circuit. The dc voltage is a typical example: small capacitance leads to significant double-line oscillations in steady state. The single-phase VSC considered in this case study has no dc-side energy source, that is, it operates as a STATCOM. The average of the squared dc voltage is controlled through the active power reference of the dVOC. To avoid the detrimental effects on stability of compensated modulation [15] (for single-phase VSCs) yet retain its power quality benefits, this article adopts a notch filter-based compensated modulation scheme. All variables in Fig. 5 are in per unit. The dVOC structure is slightly modified from its typical form, so that the synchronization gain η and inverse voltage control gain μ can be directly interpreted as the ω − P and V − Q droop slopes, respectively, in per unit.
To show why this system is weakly damped, we conduct a few parametric stability studies through a combination of numeric root locus and participation factors. Participation factors provide an additional information layer to root locus plots: which states contribute to the eigenvalues as one parameter is swept. This combination is particularly useful when several states participate in one mode across various harmonics. Participation factors also provide an elegant way to sort out the redundant information carried by the eigenvalue copies; HSS eigenvalues are folded about the imaginary line corresponding to each eigenvalue center. It is not easy to automate the selection of the centermost eigenvalue due to the spurious eigenvalues caused by truncation. Nor should one limit oneself to the fundamental strip as it does not necessarily contain all unique modes; power electronics systems are often mildly periodic which allows low HSS model order, though exhibit eigenvalues in a wide frequency range. An easy and robust approach is to select the modes which have their highest participation in a center (zeroth) harmonic state. If these states are defined in suitable reference frames, a particular locus will resemble its LTI system counterpart. The dVOC states are transformed to the dq-frame of the grid for this reason, which is a simple procedure laid out in Appendix C.
A first study is conducted without interference of the dc dynamics, by using a larger capacitor such that c dc = 500 p.u. The electromagnetic equivalent time constant of the transmission line is altered by sweeping the virtual resistance r vir from 0.01 to 0.1, and the resulting root locus is depicted in Fig. 6(a). In the smaller plots, participation of the state variables i a , v d , v q , and u dc are included as opacity and offer qualitative insight into each mode. i a is the dominating participating state in the electromagnetic mode which is, as should be expected, better damped as resistance increases. v q and v d participation indicate that synchronization and voltage regulation are decoupled. From u dc participation, it is clear that the dc voltage is slow and does not interact with the faster states. It should be noted that several fast modes have been excluded for clarity and brevity, and that the SOGI states naturally participate in all ac-related modes.
The same sweep is shown in Fig. 6(b) but now the capacitance is, as in Table I, as will be experimentally verified in Section V. The dc voltage time constant now overlaps with the fast ac states, as indicated by u dc participation. The virtual resistance can only provide limited additional damping of the oscillatory, weak mode, which is marginally stable for r vir ≈ 0.03. Several states of different harmonics participate, and the mode is sensitive to parameter changes. For example, a sweep of μ would reveal instability for μ > 0.06, which consequently limits the possible Q-V droop slopes. To summarize, the analytic modeling shows  Table I.   TABLE I CONVERTER PARAMETERS: BASE CASE that the dVOC-operated VSC with low dc capacitance exhibits weak poles and nonnegligible periodic behavior.

V. EXPERIMENTAL RESULTS
An all-hardware single-phase VSC with parameters, as in Table I, serves to experimentally verify the frequency sweep. The control, protective functions, and sweep are all implemented in embedded C-code on a Myway DSP, as shown in Fig. 7. A voltage amplifier serves as grid, and is fed a reference signal  Table II, with N denoting the truncation order and N pss the number of harmonics in the PSS. For reasons discussed in Section VI, only even couplings and odd ac-side steady-state harmonics are measured. For the same reason, the maximum gain is reduced to 30 Hz/s when the frequency reaches 350 Hz.
A few relevant variables from the MCRF filter and the error controller is shown in Fig. 8. The sweep starts at 1 kHz and Fig. 9. Bode plot of the elements in the middle HTF admittance column. The opacity of the phases for the chirp is set to the ratio of each individual HTF amplitude wrt. the maximum HTF amplitude at each frequency, as the phases are very noisy for low amplitude HTFs.
is very fast in the high-frequency range as the transfer function curvature is very low in this inductive-dominated range. The maximum gain is reduced at 350 Hz, below that the rate varies between ≈5-25 Hz/s in inverse proportion to the error (and thus, in inverse proportion to the transfer function curvature).
The admittance shown in Fig. 9 is computed directly from the MCRF filter, after division of the voltage perturbation magnitude of 0.003 p.u. The match with single-tone scanning and analytic model is reasonable, yet a few discrepancies are evident. In particular, the noise level and frequency overlap issues required-for this case-a large time constant T. That means the measured HTFs in the MCRF filter will be slightly delayed, which is evident around the fundamental frequency where the transfer functions exhibit high curvature due to poles and zeros. This is a real-time measurement issue for the MCRF filter which can be addressed a posteriori through the use of the piecewise polynomial chirp decomposition (PPCD) as in [33]. The PPCD computes the admittance from the scope measurements of current and chirp phase, the latter is output from DSP to oscilloscope due to storage limits in the digital scope. Another practical aspect is that the injection source has a limited bandwidth (≈ 2 kHz), which leads to a slight mismatch at higher frequencies.
The time-domain waveforms of the u g , u dc , i a are all shown in Fig. 10, along with the chirp phase. The PPCD admittance is shown in Fig. 11, in which the admittance between 15 and 180 Hz has been replaced with a PPCD computed admittance. While the PPCD does not suffer from real-time measurement delay, it is limited to postprocessing of the chirp response. However, the PPCD does benefit from a rate-controlled chirp sweep, the length and order of the splines can be uniformly designed as the transfer function curvature (with respect to time) is limited by the error controller.

VI. DISCUSSION
The limitations of the proposed approach can be traced to the frequency response of LTP systems at integer multiples of ω 0 /2. The injected voltage is a real sinusoid which through the Euler's rule is a sum of two complex exponentials rotating in opposite directions, that is, they contain positive and negative frequencies. When the absolute value of two CRF filter frequencies overlap, there is no information in the output signal that allows differentiation between the two HTFs. As an example, when the chirp input frequency reaches 75 Hz, the −3 coupling reaches −75 Hz. At these frequencies, the MCRF filter may malfunction, particularly when poles or zeros are close. The issue is reduced, but not removed, by considering only even couplings. The same issue happens due to the operating point filters, which may be addressed by high time constants if the operating point can be assumed to be constant.
Otherwise, the chirp sweep with error control is quite robust and simple to implement. Only one chirp sweep must be performed, and only ordinary control engineering operations are used, namely, reference frame rotation, discrete integration, logic, and arithmetic. No postprocessing is required as the HTF is computed in real time. A summary of the qualitative arguments advocated in this article is shown in Table III.  TABLE III  QUALITATIVE ASSESSMENT OF BLACK-BOX IMPEDANCE SCAN APPROACHES FOR SINGLE-PHASE VSCS

VII. CONCLUSION
This article has presented a black-box chirp frequency sweep for frequency coupled impedances and admittances, represented by HTF. Through time-domain interpretation of the chirp response, a single chirp input may characterize the entire HTF. Quantification of the error of such a time-domain interpretation is provided, and leveraged to design a chirp error controller which varies the rate in inverse proportion to the error. The control is enabled through a proposed real-time observer termed the MCRF filter. The frequency scan was experimentally demonstrated for a single-phase dVOC operated VSC with low dc-side capacitance. Parametric stability studies on a HSS model complement the main contribution by showing that the dVOC operated VSC may exhibit weak margins and strong couplings.

APPENDIX A FREQUENCY COUPLINGS WITH HSS AND HTF
Most VSC-based systems are inherently nonlinear but can be linearized and modeled in the harmonic state space as The bold font in (23) implies a Fourier series representation of the state-space vectors and matrices, the latter in block Toeplitz form. N is a diagonal matrix containing integer multiples of the fundamental frequency. The periodic steady state can be found by invoking the principle of harmonic balance and iterating until the left-hand side of (23) equals zero, as in [31]. The eigenvalues and eigenvectors of A − N convey the modal properties of the LTP system, with the same interpretation and use as its LTI counterpart [1].
As for LTI systems, it is straightforward to go from HSS to HTF y(s) = C (sI − (A − N)) −1 B + D H(s) u(s) (25) for which the HTF H(s) is a matrix, but its structural properties allow a column form which is more suitable for a single frequency input [35]. The HTF admittance is generated by specifying the voltage as input and current as output, and for impedance, vice versa.

APPENDIX B UNDETERMINED COEFFICIENTS
The general Leibniz rule gives the nth derivative of (t) as (n) (t) = 2 k=0 n k (jω s ) n−k P (k) e j(ω s t+ϕ s ) (27) and equivalent for u(t). Insert both in (6) and collect the power terms on each side t 2 : N n=0 a n (jω s ) n A = jkπ and solve for A, then B and finally C.
Here, denotes derivative with respect to jω, evaluated at jω s .

APPENDIX C VSC HSS MODEL
The states, inputs and outputs of the state-space model are chosen as Some algebraic equations are included to ease readability (32) Differential equations: The dVOC dynamics captured by v α and v β are replaced by the dq-frame equivalents v d and v q through a simple 50-Hz coordinate rotation, written in complex coordinates