Fast-Slow-Scale Interaction Induced Parallel Resonance and Its Suppression in Voltage Source Converters

. ABSTRACT Multi-timescale interaction of power electronics devices, including voltage source converter (VSC), has made the stability and analysis of high penetrating renewable power systems very complicated. In this paper, the impedance model is used to analyze the multi-timescale characteristics and interaction of the VSC. Firstly, the multi-timescale impedance characteristics of VSC are investigated based on the Bode plots. It is found that the slow-timescale (within the DC-link voltage control scale) and fast-timescale (within the AC current control scale) models are separately consistent with the full-order model perfectly within their low- and high-frequency ranges. In addition, there exists a high impedance peak within the intermediate frequency range (roughly from 10 Hz to 100 Hz). Then, the impedance peak is theoretically estimated and explained by the slow-fast-scale impedance parallel resonance through transfer-function diagram analysis. Moreover, it is found that the impedance peak is more related to some outer controllers, such as the alternative voltage control and active power control. Speciﬁcally, larger proportional coefﬁcients can greatly suppress the resonance peak. Finally, simulations and experiments are conducted to verify the generality of the multi-timescale characteristics and interaction of the VSC. Hence these ﬁndings are not only signiﬁcant to provide a physical insight into the inner key structure of the impedance of VSC, but also expected to be helpful for controller and parameter design of the VSC.


I. INTRODUCTION
With large-scale integration of renewable energies and widespread application of power electronic equipment, the structure of our modern power systems has changed radically. Recently, the emergence of multi-frequency oscillation covering wide frequency bands has seriously threatened the system stability and strongly restricted accommodation of renewable energies around the world [1], [2]. Thereinto, voltage source converter (VSC), usually adopted as a key power electronic device for grid interface of renewable energies, has its intrinsic properties of nonlinear dynamics and multi-timescale interactions of its inner and outer The associate editor coordinating the review of this manuscript and approving it for publication was Feiqi Deng . cascading control. The properties make the system analysis harder, compared to the traditional power systems. Therefore, it becomes urgent to study the interaction induced multi-timescale stability problems and uncover the underlying physical mechanism [3], [4].
Effective mathematical modeling is the foundation for system stability studies. So far, the state-space model and the impedance model are two major methods for the small-signal stability analysis [5]- [8]. In the state-space model, usually the modal analysis on state variables is performed, with all detailed information included, such as controller and system parameters. Modal analysis can be used to assess the stability of the system and analyze the couplings through participation factors and sensitivity analysis [9]- [12]. However, it lacks physical insight into the interactions between different controllers (or timescales) in the VSC. In addition, the impacts of specific controllers on the final calculated results of eigenvalues are not apparent. Hence the modal analysis might be difficult to be applied to study the interactions between different timescales and controllers in the VSC.
On the contrary, by focusing on the terminal characteristics of subsystems in the frequency domain, the impedance model of single components (like VSC, MMC and etc.) can be either theoretically obtained or experimentally measured [6], [7], [13]. Especially, how to reveal the multiple frequency coupling mechanisms in MMC through impedance modeling is a difficulty. It becomes convenient to analyze the interaction between different controllers based on the impedance model. The impedance model has been widely studied in different domains including the dq, sequence, αβ, phasor, etc, whose connections and equivalence have been uncovered in [14]- [16]. Several papers have studied the impact of different parameters and controllers on the amplitude-frequency response of impedance of VSC and analyzed the system stability [17]- [21]. Among them, the impact of the phase-locked loop (PLL) based on the dq impedance was analyzed and it was found that the negative resistance effect induced by the PLL was the key factor for system instability [19]. The impact of time delay was also investigated recently via the dq and sequence impedance models. For instance, it was found that a high time delay of the controller can induce an impedance trough around 1kHz [20], [21].
Generally in all previous papers, only single controls or parameters were concentrated and the mutual influence and interaction between different controllers were overlooked. There are several works on system dynamics analysis within each single timescale. For instance, the slow-scale stability problem of VSC was studied in the range of voltage timescale [22]- [25]. It was reported that the oscillations between dozens and hundreds of Hz can be put into the category of current-timescale, by studying a fast-current-scale model. A model reduction methodology based on a state space model was proposed to assess slow-scale dynamics by calculations of eigenvalues and the participating factor [26]. The fast-scale stability of VSC was studied under different parameters' conditions in the range of alternative current timescale [27]. Slow-scale, fast-scale and coexisting bifurcations were studied in a multi-operating-mode photovoltaic-battery system [28], [29]. Obviously, these papers have studied the dynamics of the VSC within the single timescale and missed the influence of interaction between different timescales or controllers.
The interactions between different VSCs have been analyzed through the amplitude-phase modeling and can be quantified by proposed self-and en-stabilizing coefficients in voltage control timescale [30], [31]. What is more, the interaction between double fed induction generator-based wind turbine and synchronous generator can be quantified through self-and en-stabilizing property based on motion equation in electromechanical timescale [32]. However, the quantitative analysis based on the amplitude-phase model is too complex and the interactions between inner and outer controllers (or different timescales) have not been revealed in the electromagnetic timescale. There was little research on the interactions (or couplings) between different timescales based on the impedance model, to the best of our knowledge.
To fill the above these gaps, we will take a closer look at the details of the full-timescale VSC impedance by using multi-timescale analysis. We uncover a novel parallel resonance within the intermediate frequency region and evaluate its influence factors. These findings are expected to be helpful for further system analysis and controller parameters design for the VSC. Extensive simulations and experiments are conducted to verify the theoretical results.
The rest of the paper will be organized as follows. The impedance models within the fast and slow timescales are established and then compared with state-space models in Section II. In Section III, the impedance characteristics of fast-and slow-timescales are shown in the Bode plot and then theoretically analyzed through the order analysis. Section IV is devoted to the intermediate timescale, by focusing on the phenomenon of interaction-induced impedance parallel resonance and the influence factors of the impedance resonance peak. In Section V, we generalize these findings and results to some other outer controllers. Finally, simulation and experimental results are presented in Section VI, and conclusions and discussions are given in Section VII.

II. IMPEDANCE MODELS OF FAST AND SLOW TIMESCALES
The converters in the power electronics systems share some common features. Fig. 1 schematically shows a two-level three-phase VSC tied to the AC grid and its cascaded controls with certain parameters, and the typical parameters are given in Appendix A. The AC grid is represented by an infinite bus with a line inductance (L g ). The impedance of a single VSC in the full-order form has been well derived and studied in literature. Below we like to start from the concrete form of impedance, dissect it carefully, and obtain the corresponding FIGURE 1. Schematic show for a two-level three-phase VSC tied to AC grid and its cascaded controls. The inner alternative current controller (ACC) and different outer controllers such as the direct voltage control (DVC), alternative voltage control (AVC), and power control (PC) are considered. The PLL for the phase output θ pll is used for grid synchronization and VSC control. VOLUME 9, 2021 impedance models within slow and fast timescales, by using the multi-timescale analysis.
Firstly, let us present the already-existing results of the full-timescale VSC impedance including the alternative current controller (ACC), direct voltage control (DVC), alternative voltage control (AVC), and the PLL for synchronization. According to the detailed derivations of the VSC impedance in Appendix B, all of the transfer functions are in a second-order form. Among them, G L denotes the open-loop output impedance of the converter, G ic is the current control compensator, and G i,pll and G e,pll indicate the relations between the inductor current ( i dq ) and the internal voltage ( e dq ) in the PLL frame and the terminal voltage ( u tdq ) in the synchronous frame, respectively. G iu and G ii represent the relations between the current reference ( i dqref ) in the PLL frame, and the terminal voltage ( u tdq ) and the inductor current ( i dq ) in the synchronous rotation frame, respectively. For more details, see Appendix B [5], [6], [19].
The output impedance of the VSC can be described by where and I denotes a 2 × 2 unit matrix. For a common cascaded control in the VSC, and the bandwidth of the outer voltage controller is usually around five times lower than that of the inner current controller relying on the control design criterion. The power electronics exhibits the intrinsic property of multiple electromagnetic time scales. Essentially based on the knowledge of controller design and understanding of physical equipment, the time scales can be classified into switching (around several kHz), alternative current (around 100 Hz), and dc-link voltage (around 10 Hz) one. The time scale of a phase locking loop (PLL) for synchronization should be considered as between the fast current-control and slow voltage-control scales roughly (10∼100 Hz). As the response speed of switching is very fast, it is usually neglected in the system-level stability analysis.
In this paper we will concentrate on the two electromagnetic time scales mainly: the current-control and voltage-control scales. Though the multi-timescale concepts of the VSC have been proposed in some literature [3], [33], [34], the multi-timescale characteristics and interactions between different controllers (or timescales) have not been compared in detail and investigated from the viewpoint of the frequency domain based on the impedance model.

A. IMPEDANCE MODEL OF SLOW TIMESCALE
Under the DC-link voltage control condition, the dynamic response of the ACC is relatively fast, compared to that of the outer controllers. Thus, the current output of the VSC on the transmission line can be considered equaling the current reference value. In this case, the VSC can be thought as a constant current source and the transmission-line inductance can be regarded as a constant reactance under the working frequency. These assumptions are consistent with [22]- [25]. Therefore, we have Furthermore, by considering (A.1), the slow-timescale impedance including the DVC, AVC, and PLL can be written as where The small-signal stability of this impedance model is consistent with that of the state space models and amplitude-phase models established in [23], [34], [35]. Compared with Z a,full and Z b,full in (2), the dynamic behaviors of the current inner controllers and the line inductance are neglected, and the line currents are equal to the current references. Namely, G ic and G L are neglected.

B. IMPEDANCE MODEL OF FAST TIMESCALE
In contrast, under the current control fast-timescale condition, the action of the outer voltage controllers can be regarded as relatively slow and their outputs can be thought as constants. These assumptions are consistent with [6], [27]. Therefore, Combing it with (A.2), we get e c dq = −G ic i c dq (7) which yields the final impedance within the fast timescale including ACC and PLL: where The small-signal stability of this impedance model is consistent with that of the state space models and amplitude-phase models established in [5], [27], [33]. Compared with Z a,full and Z b,full in (2), now the dynamic behaviors of the voltage outer controllers are neglected, and the line current references are set as constants; both G iu and G ii equal zero.

C. SMALL-SIGNAL STABILITY BASED ON IMPEDANCE MODEL
The systems can be transformed into a simplified circuit composed of the two impedance (or admittance) elements [6], where the transfer function diagram of the impedance-based 90128 VOLUME 9, 2021 where Z L denotes the impedance of the load (here is the line inductance in Fig. 1), and Z dq denotes the impedance of the source (here is the VSC in Fig. 1). Consequently, based on this open-loop information, the so-called generalized Nyquist criterion (GNC) can be used to justify system stability, relying on the multi-variable frequency-domain analysis theory. Accordingly the closed-loop transfer function is Obviously, the system stability can also be judged by its poles.
On the other hand, the system stability can also be determined by a time-domain state space model. The related state space models for the slow-scale, fast-scale, and full-scale are shown in Appendices C, D, and E, respectively. Theoretically, the Smith-McMillan poles of the closed loop transfer function should be equivalent to the eigenvalues calculated from the state matrix due to modern control theory. The calculated results are shown in Tab. 1, where, by comparing the values in the left and right columns, we can see that they fit with each other perfectly.
In addition, by comparing the values on different rows, one can find that the full-scale model does contain the whole information of system while the fast and slow models just contain the information in their frequency range. The inner and outer controllers indeed exhibit the intrinsic property of fast and slow timescales separately. Though the mutual influence of different controllers can be analyzed based on the modal analysis, the multi-timescale characteristics and interactions can be further revealed from the viewpoint of the frequency domain based on the impedance model.

III. CHARACTERISTICS ANALYSIS OF SLOW AND FAST-TIMESCALE IMPEDANCE MODELS A. COMPARISON OF IMPEDANCES WITHIN FAST, SLOW, AND FULL TIMESCALES
As the first step, we plot the above theoretical results of full, slow, and fast timescale impedances in Figs. 3 and 4 for their amplitude and phase responses, respectively. The frequency varies from 1 Hz to 1000 Hz, which we are interested in. The red solid, blue dashed, and green dot-dashed lines denote TABLE 1. Comparison of eigenvalues in state space models and poles of closed-loop transfer functions in impedance models.

FIGURE 3.
Amplitude response of impedances of the VSC within different timescales. The red solid, blue dashed, and green dot-dashed lines denote full-, fast-, and slow-scale behaviors, respectively. In addition, we use red products for the measuring results. Roughly the whole frequency range can be divided by two vertical lines into low (1 ∼ 10 Hz), middle (10 ∼ 100 Hz), and high (100 ∼ 1000 Hz) ones, represented by roman numerals: I, II, and III, respectively.
full-, fast-, and slow-scale behaviors, respectively. In addition, the values of impedance can also be measured through injecting small-signal voltage (or current) disturbances in MATLAB/Simulink. The measuring impedance values are also superimposed in Fig. 3 and 4 with red markers '×', showing a perfect match with the theoretical results.
Based on these comparisons, obviously the slow-timescale impedance matches with the full-timescale impedance very well within the lower frequency range (roughly lower than 10 Hz), whereas the fast-timescale impedance matches with the full-timescale impedance very well within the higher frequency range (roughly above 100 Hz), except that there is a small visible mismatch in Z qd . This mismatch in high frequency range can be narrowed by decreasing the proportional coefficient of AVC (k p,avc ) according to the Bode responses, which is discussed in Appendix F. VOLUME 9, 2021 In the middle region, they both cannot fit with the full-timescale impedance. Therefore, based on these observations, the whole frequency range can be divided into three ranges in each sub-figures of Fig. 3 and Fig. 4, which are emphasized by roman numerals, I, II, and III. The two crossover frequencies have been indicated by vertical dashed lines. Within the middle frequency region II, both curves of the slow and fast timescale impedances substantially deviate from that of the full-timescale impedance. There is not a smooth transition from the slow to fast timescales. An impedance amplitude peak exists within the II region (f ≈ 20 Hz) in the amplitude response plots in Fig. 3, accompanying with a discontinuous phase jumping in the phase response plots in Fig. 4. Recalling the impedance trough phenomenon induced by time delay around 1kHz, which was reported recently in [20], [21], one may reasonably infer that the impedance peak and impedance trough can be thought as a unified phenomenon induced by the changing of dominating scale dynamics when a timescale transition happens. The relevant problems of the impedance peak within the intermediate frequency range will be further studied in detail in Section IV.

B. IMPEDANCE APPROXIMATION OF DOMINANT TERMS WITHIN FAST AND SLOW SCALES
So far, we have seen that the VSC impedance characteristics can be well revealed by the model analysis within the low and high frequency ranges. Below we like to get approximative impedances by studying the dominant terms of the full-order model in the limits of low and high frequencies through order analysis. As in the frequency domain s = jω, in the limit of high frequency, the term with a higher order of s will be dominant. On the contrary, in the limit of low frequency, the term with a lower order of s will be dominant. As Z dq,full in (1) is controlled by Z a,full and Z b,full , we can analyze their orders in terms of s and simply keep the dominant terms as our

FIGURE 5.
Amplitude responses of (a) Z a,slow , Z a,fast , and Z a,full , and (b) Z b,slow , Z b,fast , and Z b,full . Only their dd components are presented. Clearly the two impedances Z dq,slow and Z dq,fast match with Z dq,full within both low and high frequency ranges roughly.
approximation. The analytical result is illustrated in Tab. 2, where Z a,full and Z b,full can be roughly classified into four categories, based on the order s for each item. Directly we have the simplified expressions for the impedances within the lower and higher frequency ranges, respectively, where and where The frequency response of VSC is similar to inductance in high frequency range, which is consistent with our intuition and engineering practice. Comparing these expressions with those of Z dq,slow in (4) and Z dq,fast in (8), one can see that Z dq,slow and Z dq,fast have been simplified comparatively, as only the dominant terms have been kept.
To prove this approximation, Fig. 5 shows the amplitude responses of Z a,slow , Z a,fast , and Z a,full in (a), and Z b,slow , Z b,fast , and Z b,full in (b), respectively. Only their dd components are presented. Apparently the low-frequency and high-frequency terms are separately consistent with the full-expression of Z a,full and Z b,full in the slow-and fasttimescale ranges, indicative of the correctness of the above approximation.

IV. CHARACTERISTICS ANALYSIS OF INTERMEDIATE-TIMESCALE IMPEDANCE
In this section, the phenomenon and its physical mechanism of the impedance peak within the middle frequency range will be concentrated. Furthermore, the influences of different controllers and parameters will be investigated.

A. ESTIMATION OF THE IMPEDANCE PEAK's FREQUENCY
Observing that the impedance peak is located at around 20 Hz in Fig. 3, and meanwhile the amplitude response of Z b,full is smooth in Fig. 5(b) and that of Z a,full in Fig. 5(a) shows a rapid drop within the middle frequency range, we can only calculate the extreme point of the dd element of the Z a,full matrix in Fig. 5(a), which can be expressed in (16), as shown at the bottom of the next page.
Let dZ a,full(dd) (jω)/dω = 0, we have where a = k p,cc k p,dvc , b = (k i,cc k p,dvc + k p,cc k i,dvc ), c = k i,cc k i,dvc , and d = I d0 /C. This yields the peak frequency at 19.73 Hz, which is consistent with the aforementioned numerical observation in Fig. 3. We can also see that this peak frequency is determined by the control parameters of the ACC and DVC, and the DC capacitor, meaning that the peak is contributed by an interplay of both fast-and slow-scale controllers. Further physical explanation and extensive plots under different parameters will show that the impedance peak is a generic phenomenon.

B. PHYSICAL MECHANISM OF THE IMPEDANCE PEAK
Further efforts should be made for uncovering the mechanism of the impedance peak. For this purpose, we examine the detailed transfer function diagram in the full-timescale admittance of the VSC, Y dq,full , in Fig. 6. This is consistent with the form of Z dq,full in (1). Basically its left and right parts represent the contributions of Z a,full and Z −1 b,full , respectively. In view of the top and bottom parts, the top green box and bottom red box separately represent the contributions within the slow and fast scales. In between, G ic G i,pll serves as their common part, representing the impact of the PLL. In addition, the whole right part in Fig. 6 for Y dq,full , Y dq,slow , and Y dq,fast representing admittances within the full, slow, and fast timescales, respectively. Furthermore, based on that G ic G i,pll (serving as a s −2 -order item in Tab. II) is relatively small within the intermediate frequency range, we immediately have: which indicates that within the middle frequency region, the full-scale admittance can be viewed as a sum of the slow-scale and the fast-scale admittance, and hence from the perspective of impedance, the full-scale impedance can be viewed as parallel connection of the slow-scale and fast-scale impedance. Last we supplement that as parallel impedance is considered, it is more convenient to study it in the form of admittance summation, as we have done in the analysis of the transfer function control diagram in Fig. 6.
To check this point, Figs. 7 and 8 show the comparison of amplitude-phase response of the original full-timescale impedance Z dq,full (red solid line) with that of the parallel impedance of the fast-scale and slow-scale impedances, VOLUME 9, 2021 Z dq,fast and Z dq,slow (red dashed line). It can be seen that their difference of amplitude within the middle frequency region is small and their phases are almost the same.
Next we can find that the different characteristics of slowand fast-scale impedance lead to this impedance peak, which can be named after parallel resonance peak. Recall the phenomenon of phase jumping in the phase response in Fig. 4, either from positive to negative in Fig. 4(a) (Z dd ) or from negative to positive in Figs. 4(b)-4(d). This means that around the impedance peak, the electrical characteristics of Z dd , Z dq , Z qd , and Z qq of the fast-and slow-scale impedances are always opposite, i.e., one is inductive (or capacitive) and the other is capacitive (or inductive). Combining this with the parallel impedance rule, we understand that a parallel resonance occurs under the parallel connection of fast and slow-scale impedances. This is similar to the well-known RLC parallel resonance in electrical circuits. Finally, to make this mechanism clearer, we plot the amplitude response of the current-scale ACC impedance (solid curve) and the voltage-scale impedance under the DVC & AVC (dashed curve) in Fig. 9, where the intersection of these two curves is just located at f ≈ 20 Hz (guided by an arrow), indicative of the occurrence of the parallel resonance. The appearance of this parallel resonance within the middle frequency region may play an active role in stability of VSC systems.

C. IMPACT OF CONTROLLERS ON THE PARALLEL RESONANCE PEAK AND THE SYSTEM's STABILITY
The impacts of the parameters of the AVC, DVC, ACC, PLL and current decouple terms on the resonance peak are investigated below. It should be noted that as the figures of all Z dd , Z dq , Z qd , and Z qq are similar, only the amplitude responses of Z dd will be presented. In addition, the dominated eigenvalues  (determined by the sensitivities and participation factors) will be studied and compared under different proportional and integral (PI) parameters, which should be useful for further parameter design.

1) IMPACT OF AVC's PARAMETERS
The amplitude responses of the impedance Z dd under different PI coefficients of the AVC are shown in Fig. 10. It can be seen that the resonance peak decreases dramatically with the increase of k p,avc in Fig. 10(a), while it almost remains unchanged with the increase of k i,avc in Fig. 10    in Fig. 11. It can be seen that the resonance peak decreases with the increase of k p,dvc in Fig. 11(a), while it changes a little with the decrease of k i,dvc in Fig. 11(b). Therefore, it can be gotten that the impact of the DVC on the resonance peak is relatively smaller than that of the AVC. Similarly, the DVC mainly influences the slow-scale eigenvalues, i.e. −6.31 ± j24.41 in Tab. 1, according to the participation factors. The dominated eigenvalues are calculated and concluded in Tab. 4 under different parameters. It can be seen that the real part of eigenvalues keeps away from the imaginary axis with the increase of k p,dvc and decrease of k i,dvc , so a higher k p,dvc and a smaller k i,dvc are beneficial for the system small-signal stability.

3) IMPACT OF ACC's PARAMETERS
The amplitude response of impedance Z dd under different PI coefficients of ACC are shown in Fig. 12. It can be seen that the resonance peak decreases dramatically with the increase of k p,cc in Fig. 12(a), while it almost keeps unchanged with the increase of k i,cc in Fig. 12(b). Hence the resonance peak can be suppressed by increasing k p,cc , corresponding to a larger bandwidth of the ACC.    imaginary axis and the other one keeps away from the imaginary axis with the increasing of k i,cc . For the k p,cc , one pair of eigenvalues continuously keeps away from the imaginary axis with the increasing of k p,cc , and the other one keeps away from the imaginary axis firstly and then approaches it with the increase of k p,cc . Though the changing trends of k i,cc and k p,cc are not the same, generally higher k i,cc and k p,cc are beneficial for the small-signal stability of system, which is consistent with [27].

4) IMPACT OF PLL
The amplitude responses of impedance Z dd with and without PLL are compared and shown in Fig. 13(a). The corresponding results under different bandwidths of the PLL are shown in Fig. 13(b). It can be seen that the influence of PLL is only limited on relatively lower frequency range, i.e., smaller than 15 Hz. Meanwhile, the amplitude response of impedance Z dd almost remains unchanged with the change of the bandwidth of PLL [e.g., 9 Hz, 13 Hz and 30 Hz in Fig. 13(b)]. As a result, VOLUME 9, 2021 it can be concluded that the influence of PLL on the resonance peak is negligible.

5) IMPACT OF THE CURRENT DECOUPLE TERMS
The amplitude responses of the impedance without and with the current decouple terms are compared and shown in Fig. 14. It can be seen that there is only a small mismatch in the high frequency range. Therefore, the current decouple terms have little influence on the characteristics of the impedance. Meanwhile, it can be found that the amplitude responses without and with the current decouple terms are very similar in the intermediate frequency range. Consequently, the impacts of the current decouple terms on the resonance peak can be neglected.
In conclusion, the influence of the current decouple items and the PLL on the resonance peak can be neglected. On the contrary, all the controllers' proportional coefficients have a great impact on the resonance peak. For instance, for the ACC, a larger k p,cc corresponding to a larger bandwidth of the ACC can relieve the resonance peak. Compared with the DVC, the impact of the AVC on the resonance peak is relatively larger. A larger k p,avc corresponding to a larger bandwidth of the AVC can suppress the resonance peak substantially; this point may not be consistent with some design principles and intuition.

V. RESULTS FOR OTHER OUTER CONTROLLERS
In this section, we will present some results for other forms of outer controllers, to further show that the peak phenomenon is generic. Similarly, the impacts of different parameters on the resonance peak will be investigated.

A. PARALLEL IMPEDANCE RESONANCE FOR OTHER OUTER CONTROLLERS
As the first case, we neglect the AVC and consider the DVC only. Under this situation, the q-axis current reference should be set to constant, i.e., i qref = 0 (20) Hence the impedance expression is the same as in (2), and it can be obtained by using the new G iu in Appendix G.
For the second case, we consider the active and reactive power control (PC), and it can be gotten that: where H p = k p,p + k i,p /s and H Q = k p,q + k i,q /s. Similarly, the impedance can be derived by linearizing (21) and updating the forms of G iu and G ii in Appendix H. For these two different cases, we find that all aforementioned findings persist, including three frequency regions for slow, intermediate, and fast timescales with an impedance peak in the intermediate timescale, except that the specific values may change. For example, different from the 1 ∼ 10 Hz, 10 ∼ 100 Hz, and 100 ∼ 1000 Hz in the AVC & DVC for the frequency ranges of the slow, intermediate, and fast timescales, respectively, here they become 1 ∼ 6 Hz, 6 ∼ 100 Hz, and 100 ∼ 1000 Hz in the first case, and 1 ∼ 3 Hz, 3 ∼ 60 Hz, and 60 ∼ 1000 Hz in the second case. In addition, the resonance frequency also shows a system dependence. Different from f ≈ 20 Hz in the AVC & DVC, here it becomes f ≈ 18 Hz in the first case and f ≈ 9 Hz in the second case. For illustration, the results for these two cases are superimposed in Fig. 9, where the parallel resonance corresponding to the intersection of two curves of slow-scale and fast-scale impedances is unchanged. This demonstrates that the mutual interaction between inner and outer controllers with the form of parallel resonance is generic and robust, irrespective of the concrete forms of controllers.

B. IMPACT OF POWER CONTROLLERS' PARAMETERS ON RESONANCE PEAK 1) THE IMPACT OF ACTIVE POWER CONTROLLER
The amplitude responses of the impedance of the VSC under different proportional and integral coefficients of the active power control are also studied. The results of Z dd are shown in Fig. 15(a) and (b), where (k p,p = 0.01, 0.02, 0.3) and (k i,p = 3, 5, 10), respectively. It can be seen that the parallel resonance peak changes dramatically with the increase of k p,p , whereas the resonance frequency only moves right slightly with the increase of k i,p . Clearly the resonance peak can be greatly suppressed by increasing k p,p .

2) THE IMPACT OF REACTIVE POWER CONTROLLER
The amplitude responses of impedance for different PI coefficients of the reactive power control are also studied. Fig. 16(a) and (b) show the results for (k p,q = 0.02, 0.3, 1) and (k i,q = 3, 5, 10), respectively. It can be seen that the resonance peak changes slightly with the change of k p,q . Similarly, k i,q also has a little influence. Comparatively the  impact of the reactive power controller is smaller than that of the active power controller.
To sum up, based on these observations and comparisons, we know that a larger k p,p can relieve the resonance peak substantially while k p,q has a little influence on it. In addition, both the integral coefficients of the power controllers (k i,p and k i,q ) have a limited influence.

VI. SIMULATION AND EXPERIMENTAL VERIFICATION A. SIMULATION RESULTS
In our study, a series of simulations have been conducted to verify the impact of interaction of different timescales on the system small-signal stability. Four examples are illustrated in Fig. 17, which show the system unstable situations under different parameters with different oscillation frequencies.
Meanwhile, the GNC has also been applied to examine the system stability for comparison. Correspondingly, Tab. 3 summarizes the dominating eigenvalues and critical parameters that are calculated from the reduced-order and full-order models by changing the PI parameters. We use a blank cell to represent that the corresponding critical mode does not exist with the change of the corresponding parameter. Cases 1 and 2 illustrate the fast-and slow-scale modes, respectively, while cases 3 and 4 illustrate the intermediate-scale modes which are both influenced by fast and slow timescales.
Case 1 Low-Frequency Mode: We tune the proportional coefficient in the slow-scale DVC. By decreasing k p,dvc to 0.18, the system becomes unstable and begins oscillating at 4 Hz; the time series and frequency spectrum are shown in Figs. 17(a) and 17(b). Based on Tab. 3, one can see that the critical parameters and the dominated modal frequency in the reduced-order and full-order models are very close.  Therefore, for low-frequency oscillations, the slow-scale reduced-order model can indeed catch the dominant system behavior.
Case 2: High-frequency mode. Now we change the proportional coefficient of the ACC. By decreasing k p,cc with a fixed k i,cc = 1250, the system becomes unstable and begins oscillating at 116 Hz. In Fig. 17(c) and (d), the parameters  (k p,cc = 0.16 and k i,cc = 1250) for a critically stability are chosen. Again it can be seen that the critical parameters and the dominated modal frequency in the reduced-order and full-order models are close, meaning that for high-frequency oscillations, the full-scale model can be replaced by the fast-scale one.
Case 3: Intermediate frequency mode one. This mode is motivated by changing the integral coefficient of the ACC. Decreasing k i,cc to 850 with a fixed k p,cc = 0.11, the system becomes unstable and starts oscillating at 89 Hz. The results are shown in Tab. 3 and Fig. 17(e) and (f). It can be seen that now this unstable mode cannot be accurately predicted by the fast-scale model. Meanwhile, it cannot be observed by tuning the corresponding parameters in the slow-scale model.
Case 4: Intermediate frequency mode two. As the fourth case, we change the integral coefficient of the AVC. By increasing k i,avc to 835, the system begins becoming unstable and oscillating at 99 Hz. The corresponding results are shown in Tab. 3 and Fig. 17(g) and (h). Now this unstable mode cannot be found in the two reduced-order models by tuning k i,avc solely any more. Based on these observations and comparisons, one understands that the reduced-order models cannot fully reflect the system dynamics, if intermediate frequency oscillations are considered.
Moreover, the tracking performance of i d versus i dref , and i q versus i qref is tested. As one example, a small-signal disturbance for the DC-link voltage u dc increasing from 1.0 p.u. to 1.1 p.u. at 2 s, and the corresponding time series of i d and i q are shown in Fig. 18. It can be seen that the errors between reference values and measured values are small. Therefore, the tracking performance of the controllers can be thought as good.

B. EXPERIMENTAL RESULTS
The Hardware-in-loop experiments based on the SpaceR are applied for further verification. The SpaceR is a real-time simulation platform, including one part that simulates the AC grid with an equivalent line impedance, and the other one that simulates the VSC, which is shown in Fig. 19. The experimental parameters are the same as those of simulations for cases 1-4. The corresponding time series of i d measured by oscilloscope are shown in Fig. 20. Among them, Fig. 20(a) and (b) show the fast-and slow-scale modes, respectively, whereas Fig. 20(c) and (d) illustrate the intermediate-scale modes. From the time series measured by oscilloscope, it can be seen that the oscillation frequencies of case 1 and 4 are 4Hz and 99Hz, respectively, which is consistent with the above simulations. Correspondingly for cases 2 and 3 [i.e. Fig. 20(b) and (c)], the systems are unstable, and the systems diverge too quickly to determine their oscillation frequencies. To sum up, these experimental results are consistent with the above simulations.

VII. CONCLUSION AND DISCUSSIONS
In conclusion, we have investigated the impedance characteristics and interaction of the VSC based on the multi-timescale decomposition analysis. It has been found that the slow-timescale and fast-timescale impedance models are separately consistent with the full-order impedance model within their low-and high-frequency ranges. Meanwhile, there exists a novel intermediate frequency range (roughly from 10 Hz to 100 Hz) with an impedance peak. We discover that the impedance peak is caused by a parallel resonance between the inner and outer controllers. Furthermore, the influence parameters of the impedance peak are explored fully and it is found that the peak can be suppressed by increasing the proportional coefficients of the AVC (k p,avc ), the active PC (k p,p ) and the ACC (k p,cc ). Simulation and experimental results under different controllers and system parameters verify these theoretical analyses. All these findings are expected to be not only helpful for a physical insight into the multi-timescale interaction in the VSC, but also useful for controller design and system analysis of powerelectronic-dominated power systems.
For discussion, although the impedance model of a single VSC has been well developed before, it remains a great challenge to generalize it to study multi-converter stability and solve wide-frequency-band oscillation problems. Therefore, it is very necessary and important to deep into the inner back-bone structure of impedance, as what we have done in the present work. The findings of multi-scale decomposition and parallel impedance resonance in the paper are significant to better understand full-order impedance in a physical perspective. Furthermore, they might be helpful for system-level analysis of multi-converter systems. In particular, as sub-synchronous and super-synchronous oscillations around the working frequency have been a serious problem in power electrical engineering recently, the parallel impedance resonance within the middle frequency range (from 10Hz to 100Hz) might be highly relevant. In addition, the multi-timescale analyses have been conducted solely on the impedance models within the small signal category in this paper. Some nonlinear multi-timescale characteristics and interactions should be studied further [36], [37]. Moreover, there are some other interesting problems, such as the influences of uncertainties in system parameters on the interaction, and possible application of the impedance model in some advanced controllers including model predicted controller, robust controller, non-linear controller, etc. It is notable that the impedance analysis method can be used for small-signal stability analysis, not for large-signal stability analysis. The impacts of the PI parameters on the small-signal stability of the system are discussed in Section IV, which can be used for further parameter design. where G i,pll and G e,pll indicate the relationships between the inductor current ( i dq ) and internal voltage ( e dq ) in the PLL frame and the terminal voltage ( u tdq ) in the synchronous frame, respectively. G pll = H pll /(s + U s td H pll ) and H pll = k p,pll + k i,pll /s denotes the compensator of the PLL control, where k p,pll and k i,pll are the proportional and integral gains of the PLL.
The d-axis and q-axis current references can be separately obtained from the DVC and AVC, yielding   (1) and (2) in the text. For more details, see Refs. [6], [7], [19].
As for the load, .8) and the load in Fig. 1 is a single inductance, which can be written as Combing the impedance of the source (Z dq ) and the load (Z L ), the small-signal stability of the system can be determined, as shown in section II-C.

C. STATE SPACE MODEL OF SLOW-SCALE MODEL
Referring to [22], [23], [35], the slow-scale (voltagetimescale) state space model can be established by considering the dynamics of the DVC, AVC, capacitor, and PLL. The state variables can be chosen as follows: where x 1 is the state variables of the DVC, x 2 denotes the dynamics of the AVC, u dc denotes the dynamics of the capacitor, and x 3 and ϕ are the state variables of the PLL.
The state matrix is:   L g + L f U t0 h 68 = k p,pll L g k p,cc L g + L f ω 0 + k p,avc k p,cc L g −k i,cc − k p,dvc k p,cc e d0 Cu dc0 L g + L f 2 U t0 h 69 = k p,pll L g −k i,cc L g + L f + k p,avc k p,cc L g −k p,cc ω 0 − k p,dvc k p,cc e q0 Cu dc0 L g + L f 2 U t0 (A.18)

D. STATE SPACE MODEL OF FAST-SCALE MODEL
Referring to [6], [27], [33], the fast-scale (current-timescale) state space model can be established by considering the dynamics of the ACC, line inductance, and PLL. The state variables can be chosen as follows: x = [ ϕ ω e d e q i d i q ] (A. 13) where ϕ and ω are the state variables of the PLL, e d and e q are the state variables of the ACC, and i d and i q denote the dynamics of the line inductance.
The state matrix is (A.14), as shown at the bottom of the previous page, with the detailed information in (A.15):

E. STATE SPACE MODEL OF FULL-SCALE MODEL
Referring to [8], [16], [38], the full-scale state space model can be established by considering the dynamics of the ACC, line inductance, AVC, DVC, and PLL. The state variables can be chosen as follows: x = [ i dref i qref e d e q u dc ω ϕ i d i q ] (A. 16) where i dref and u dc denote the dynamics of the DVC and the capacitor, respectively, i qref is the state variable of the AVC, e d and e q denote the dynamics of the ACC, ϕ and ω are the state variables of the PLL, and i d and i q denote the dynamics of the line inductance.
The state matrix is (A. 17), as shown at the bottom of the 13th page, with the detailed information in (A. 18), as shown at the top of the previous page. Therefore, the eigenvalues of these models in Tab. 1 can be obtained by the modal analysis.

F. EXPLANATION FOR THE MISMATCH IN Z qd
The small mismatch of Z qd in Fig. 3 is discussed here. According to the Bode responses, this mismatch can hardly be narrowed by changing all of the parameters except for the proportional coefficient of the AVC (k p,avc ). The amplitude responses of Z qd with the changing of k p,avc are shown in Fig. 21. It can be seen that the mismatch between the fast-scale and full-scale impedance (Z qd ) in high frequency range decreases with the decrease of k p,avc .